Test methods

Overview

For organization purposes, I have divided the discussion of material properties into two chapters: elastic properties and inelastic properties. For elastic properties, I have included those that are needed for computer design of resonators, including the density. The only inelastic material property is Q (quality factor), which is related to the loss of the material. This chapter will discuss elastic properties; the following chapter will discuss inelastic properties.

Elastic material properties are very important for computer design of resonators. If the material properties are not correct, then the computer predictions will also err. Material directionality and variations of properties with heat treatment are also important in the manufacture of resonators. These can cause variations in resonator tuned lengths. We will first define the properties of interest and then discuss four methods of determining these properties: static tests, resonant tests, high-frequency (NDT) tests, and finite element analysis. (The resonant and NDT tests are together called dynamic tests.) Temperature effects and the effect of grain direction on large horns are covered at the end of the chapter.

Introduction

In this section we will look at interrelations among material properties. We will also look at causes of variations in material properties which make measurement more difficult. In designing resonators by computer, the elastic material properties must be accurately known so that the computer- predicted results (frequencies, amplitudes, stresses) are correct. Since handbook values for the material properties are usually not sufficiently accurate, the properties must be determined experimentally.

Assumptions

The following discussion is relevant to common resonator materials (e.g., aluminum, steel, titanium). It does not necessarily apply to nonresonator materials such as rubber and composites.

Principal material properties

The following is a brief description of the principal material properties. The associated units that we will be using are given in brackets [ ]. (These properties are covered in more detail in the chapters on "Fundamental Relations" and "Wave Motion".)

Elastic material properties

Elastic material properties are those which can be determined by deforming the material:

1. Young's modulus, E. (Also called modulus of elasticity.) Young's modulus describes the resistance of a material to being stretched, with respect to its supporting area and length. [MPa]

2. Shear modulus, G. The shear modulus describes the resistance of a material to being twisted, with respect to its supporting area and length. [MPa]

3. Poisson's ratio, nu. Poisson's ratio describes the transverse dimensional change as the material is stretched in the axial direction. [No units]

Density

1. Density, p. The density is the mass of material occupying a given volume. [kg/m3]

Wave properties

Wave-speed properties depend on both the elastic material properties and the density. For our purposes the most important wave speeds are:

1. Thin-wire wave speed, C&a+45V&dDO&a-45V&d@. The thin-wire wave speed is the speed at which a pressure wave travels along the length of a thin wire. [m/sec]

2. Dilatational wave speed, C&a+45V&dDd&a-45V&d@. The dilatational wave speed is the speed at which a pressure wave travels through a medium whose lateral dimensions (i.e., perpendicular to the wave propagation direction) are essentially infinite in comparison to the wavelength. [m/sec]

3. Shear wave speed, C&a+45V&dDS&a-45V&d@. The shear wave speed is the speed at which a shear wave travels through a medium. The dimensions of the medium have no effect on the shear wave speed. [m/sec]

The above properties are not necessarily constant under all conditions for a specified material. For instance, most of the above properties are significantly affected by temperature. However, unless stated otherwise, we will assume that the material property values have been evaluated under "normal" conditions.

Isotropic materials

Suppose we take a small sphere of material and tested a particular property for all possible orientations of the sphere. If the test results for each orientation give exactly the same value, then that particular material property is said to be isotropic.
If all of the significant material properties are isotropic, then the material itself is isotropic. No material is completely isotropic. However, many aluminums and steels are reasonably isotropic.
When a material is isotropic, some simple relations exist among the above material properties:
E
1) G = ÄÄÄÄÄÄÄÄÄÄ
2 (1 + nu)
Ú E ¿1/2
2) CO = ³ ÄÄÄ ³
À p Ù
Ú (1 - nu) ¿1/2 3) C&a+45Vd&a-45V = CO ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³
À (1 + nu) (1 - 2 nu) Ù
Ú G ¿1/2
4) CS = ³ ÄÄÄ ³
À p Ù
For equations Æ2µ, Æ3µ, and Æ4µ, see the chapter on "Wave Motion".
If we substitute equations Æ1µ and Æ2µ into equation Æ4µ, then the shear wave speed can be expressed in terms of CO and nu:
Ú 1 ¿1/2
5) C&a+45VS&a-45V = CO ³ ÄÄÄÄÄÄÄÄÄÄ ³
À 2 (1 + nu) Ù
Since the properties are interrelated by the above equations, the measurement of some property values will allow the remaining property values to be calculated. In fact, for an isotropic material, only two properties (other than the density) are needed to determine all of the others.
raw stock types
We will also need some definitions that describe the material configuration of raw stock -- i.e., the unmachined stock as it comes from the manufacturer:
1. Bar stock. Bar stock has one relatively long dimension compared to the transverse dimensions. For example, typical bar stock dimensions might be 4m x 150mm x 60mm. Normally, bar stock could have either round or rectangular cross section. However, we will use the term "bar" to refer exclusively to rectangular stock. Bar horns are often made from bar stock. 2. Rod stock. Rod stock has a cylindrical shape, with a small diameter compared to its length. For example, typical rod stock dimensions might be 4m x í38.1. Unslotted and small-diameter slotted cylindrical horns are usually made from rod stock. 3. Plate stock. Plate stock is rectangular, with large length and width compared to the thickness. For example, typical plate stock dimensions might be 2m x 1m x 75mm. Bar horns, rectangular horns, and large slotted cylindrical horns can all be made from plate stock.
4. Sheet stock. Sheet stock is the same shape as plate stock, except the thickness is much smaller. For example, at TIMET the distinction between plate and sheet occurs at a thickness of 4.8 mm. (18, p. 7) Sheet stock is generally used only for flexural resonators (e.g., polar mount diaphragms). (See the chapter on "Flexure-Plate Design".) Because different processing techniques may be used to produce the different raw stock types, the resulting materials may have somewhat different material properties. Thus, for example, 7075-T651 aluminum may have slightly different properties depending on whether it came from bar, rod, or plate stock. These differences may have some effect on the final performance of the resonator.
ANISOTROPIC AND ORTHOTROPIC materials If the grains of a polycrystalline material are small compared to the sample and if the grains are randomly oriented, then the material properties will be approximately isotropic. When a material is processed (such as by casting, working, or heat- treating), then the crystals assume certain preferred orientations due to the differing ease of slip along the various atomic planes. (See Fine, p. 59, and Varley, pp. 44 - 45.) In this case, some material properties will depend on the direction in which the material is tested. An anisotropic material is one for which every direction has a (possibly) unique set of elastic properties. In this case, 21 elastic constants (in addition to the density) are required to completely define the material. (Sokolnikoff, p. 61) Fortunately, resonator materials are not anisotropic. An orthotropic material is one which has three mutually perpendicular planes of elastic symmetry. If a material is orthotropic, then nine elastic constants (in addition to the density) are required to describe its behavior. (Sokolnikoff, pp. 63 - 64) Because of the way they are processed, bar and plate stock are generally orthotropic to some degree. A transversely isotropic material has at least one plane where the material properties are independent of the test direction. If a material is transversely isotropic, then five elastic constants (in addition to the density) are required to describe its behavior. Rod stock is generally transversely isotropic in the plane perpendicular to the rod axis.

When a material is not isotropic, you must specify the direction in which the material property is measured. In relation to the stock as it comes from the mill (e.g., the uncut raw stock), there are three possible material directions: 1. Longitudinal. This is the direction that has the longest grains. For bar or rod material, this will be along longest dimension. For a plate, the longitudinal direction is normally along the longest dimension, although this will depend on the processing.
2. Long-transverse. The long transverse direction is along the largest dimension that is transverse to the longitudinal direction. Sometimes this is simply called transverse, rather than long-transverse.
3. Short transverse. The short transverse direction is along the shortest dimension that is transverse to the longitudinal direction.
.f.Figure Õ1¸®Ÿelastic\matl_dir.mat¯®typical machining direction for bar and plate stock;¯ shows the three material directions for bar and plate stock. For example, if a piece of titanium bar raw stock has dimensions of 4m x 150mm x 60mm, then the longitudinal direction is along the 4m dimension, the long-transverse direction is along the 150mm dimension, and the short-transverse dimension is along the 60mm dimension. .f.Figure Õ2¸®Ÿelastic\bowen1.plt¯®grain directionality in Ti-6Al-4V bar;¯®photocopy from bowen¯ (Bowen, 21, p. 1273) shows the grain direction for a section of titanium bar. For bar and plate material (orthotropic), the elastic constants should be specified in all three of the above directions. For rod (transversely isotropic) material, only two material directions are specified: longitudinal and transverse. .f.Figure Õ3¸®Ÿelastic\al_grain.plt¯®grain directionality in rolled aluminum;¯ shows the grain direction for a section of aluminum rod.
If a material's directionality is not known, then it can be etched with an acid to bring out the grain structure. Aluminum is etched with a solution called Keller's etch (1% HF, 1.5% HCl, 2.5% HNO3, balance water). Titanium is etched with a solution called Krolls etch (1% HF, 12% HNO3, balance water). (TIMET, p. 8)
Relation to direction of vibration This material directional nomenclature does not imply a specific relation to the direction of resonator vibration. Bar horns. Most of Branson's bar horns are machined with the stud axis along the long-transverse material direction, with the width of the horn in the longitudinal material direction and the horn thickness in the short-transverse material direction. (See figure Õ1¸®Ÿelastic\matl_dir.mat¯.) This practice originated when most bar horns were machined from bar stock, which allowed very wide horns to be machined from the bar. For shaped (i.e., E-dimensioned) bar horns, this also allowed easiest machining of the nodal radius for wide horn blanks, which were later cut into narrower individual horns. This machining direction continues for titanium bar horns, even though they are now made of plate (rather than bar) material. Aluminum bar horns are still normally made from bar stock. (Note: For increased fatigue resistance, bar horns are sometimes machined with the stud axis along the longitudinal material direction and the horn width along the long transverse material direction. This aligns the direction of vibration with the grain direction. See the chapter on "Material Life Testing".) Cylindrical horns. Branson's small diameter cylindrical aluminum horns are normally machined from rod stock with the stud axis along the longitudinal material direction. For larger diameter aluminum horns, the horns are machined from plate stock with the stud axis along the short-transverse material direction. Branson's cylindrical titanium horns are normally machined from rod stock with the stud axis along the longitudinal material direction. (Very large diameter titanium horns, which would require plate stock, are rarely made because the required slots are not easily machined.)
Problems with equations
Fortunately, many materials are only mildly orthotropic, so that, as a first approximation, they are assumed to be isotropic. When a material is orthotropic, each elastic property (E, G, and nu) must be specified in each of the three directions. Equations Æ1µ, Æ3µ, and Æ5µ no longer give the correct relations among the material properties. Equation Æ2µ remains correct for a specified direction since the thin-wire wave speed is unaffected by material properties in the transverse directions. (See H. J. McSkimin, p. 325.) Equation Æ4µ also remains correct, since Poisson's ratio is not involved. Aluminum and steel. Most resonator materials are not truly isotropic, but aluminum and steel are reasonably close. Therefore, we will tentatively assume that equations Æ1µ..Æ5µ are valid for aluminum and steel.
Titanium. Depending on its type and method of processing, titanium can be quite orthotropic. .f.Figure Õ5¸®Ÿelastic\ti_heats.plt¯®effect of heat-to-heat variations on young's modulus for Ti-6Al-4V sheet;¯ (TIMET, p. 14) shows the difference in longitudinal and transverse modulus for Ti-6Al-4V sheet from eight different heats. At room temperature, the average transverse modulus is about 6% higher than the average longitudinal modulus. For one section of annealed Ti-6Al-4V bar (235 mm wide x 57 mm thick), Bowen (21, p. 1272) reports the following Young's moduli from static tests:
% difference from
Test direction Young's modulus Longitudinal Longitudinal 113.8 0.0
Long-transverse 129.0 13.4
Short-transverse 113.8 0.0

Differences were also found in the yield, tensile, and fatigue strengths.
TIMET (18, p. 15) reports that Poisson's ratio for Ti-6Al-4V depends on the material texture and test directions. For ten measurements with two-element rosette strain gages, the mean Poisson's ratio was 0.342, with a range from 0.288 to 0.391. (The type of titanium raw stock is not known.) Thus, equations Æ1µ, Æ3µ, and Æ5µ must be used with considerable care for titanium. Because of its orthotropic behavior, titanium resonators will have different tuned lengths, depending on how they are machined relative to the material direction. (See the section "Effect of Grain Direction in Horns with Large Lateral Dimensions". Also see the sections on measurement of titanium's material properties.)
HOMOGENEITY
Let's start with a large piece of raw material stock, from which test specimens are then machined at various locations. Each specimen is then tested in the same material direction (e.g., in the longitudinal material direction). If the tested material property is the same (within experimental error) for all of the samples, then the material is homogeneous with respect to that material property. Otherwise, the material in nonhomogeneous with respect to that material property. The material homogeneity does not depend on whether it is isotropic. Thus, a material could be anisotropic, yet still be homogeneous. Conversely, a material could be isotropic, yet be nonhomogeneous.
Localized stock size effect
During their manufacture, metals must be raised to high temperatures. The rate of subsequent cooling affects the material's properties. If the cooling process is gradual (e.g., controlled oven cooling), then the temperature will remain relatively uniform throughout the stock and the microstructure will be fairly uniform. However, if the cooling process is very rapid (e.g., quenching), then the outside will cool much more rapidly and will have a finer grain structure than the interior. The difference in properties from the outside to the inside will depend on the material's thermal conductivity and the stock size. Materials with low conductivity and large size will cool less uniformly and will have greater property variation across the cross-section. Thus, homogeneity depends on the size of the raw stock and the manner in which it is processed. .f.Figure Õ6¸®Ÿelastic\ti_size2.plt¯®effect of raw stock size on the cross-sectional tensile properties of sta ti-10v-2fe-3al;¯ (Boyer, p. 445) shows the variation in tensile and yield strengths across the section for Ti-10V-2Fe-3Al which has been water quenched. Note that the strength variation decreases as the section size decreases from 127 mm to 76 mm. Tiktinksy (pp. 1013 - 1014) found little variation in tensile or yield strengths across the section for annealed Ti-6Al-4V die forging and forged billets up to 6" thick, in either the longitudinal or transverse directions. Bowen (15, p. 2101) reports similar results for annealed Ti-6Al-4V rod (í 114.3). (See the chapter on "Material Life Testing" for data.) However, Young's modulus increased by 5.7% from the surface to the center of the rod. (See .f.figure Õ7¸®Ÿelastic\ti_e_dia.plt¯®variation in young's modulus across a 114.3 mm diameter rod of annealed ti- 6al-4v;¯.) This may have been due to microstructure and texture variations across the rod due to temperature gradients in the bar during working, with possible overheating at the bar center. (p. 2102, 2107.) This variation is modulus across the section may not be typical, but it does indicate that such variations can exist and may therefore affect the wave speed. Not only may the material be nonhomogeneous across the section, but it may be nonhomogeneous along the length. The section "Electrostatic Resonant Tests" shows possible variations in the longitudinal and long-transverse modulus and in the wave speed between the center and ends of a Ti-6Al-4V bar. The homogeneity also depends on the care in processing. For example, cast materials are sometimes nonhomogeneous due to voids.
General stock size effect
In the above discussion, we noted that larger stock sizes were more likely to have property variations across their cross- sections. In addition, there may also be a general change in properties as the stock size increases. For instance, the tensile and yield strengths of Ti-6Al-4V decrease as the cross- section increases. (See
.f.figure Õ8¸®Ÿelastic\ti_size1.plt¯®effect of raw stock section size on tensile properties of sta Ti-6Al-4V;¯, taken from TIMET data, p. 12.) Bowen (15) and others have also found that titanium's fatigue strength varies with the material stock size. (See the chapter on "Material Life Testing".) heat-to-heat and Bar-to-bar variations TIMET literature (p. 14) notes that the tensile modulus depends on the heat treatment. Figure Õ5¸®Ÿelastic\ti_heats.plt¯®timet, p. 14¯ shows this effect for Ti-6Al-4V sheet from eight heats from one producer. The spread between lowest and highest modulus values is about 10% for either the longitudinal or transverse directions.
Thus, at least for Ti-6Al-4V, normal variations between heats can significantly affect the modulus. This, in turn, affects the wave speed and the tuned length. Thus, resonators of nominally identical dimensions (e.g., titanium boosters) may have significantly different tuned lengths. This is supported by manufacturing experience. It is not known if such variability can be more closely controlled or if all titaniums exhibit the same degree of variability from heat to heat. (Note: this was supposedly a particular problem with Ti-6Al-4V titanium bar and rod stock, which is one reason that Branson changed to Ti-7Al-4Mo. It is not known if this reason for changing is valid.)
It is also possible that different titanium bars within the same heat may have different moduli. No direct substantiating evidence could be located. However, since property variations along a single bar are possible (see the section "Electrostatic Resonant Tests"), it seems likely that property variations among bars may also occur.
MATERIAL PROPERTIES: DENSITY
In this section we will give measurements of density for Ti-7Al-4Mo, various aluminums, and D2 tool steel. The density is the easiest property to determine. It only requires measurement of the mass and volume of a sample:
mass
6) p = ÄÄÄÄÄÄ
volume
Table 2 shows density measurements for various materials. A quick glance at the data indicates that the data scatter is quite small. Calculation of the 95% confidence intervals of the mean confirms this impression. For instance, for the titanium bar we can be 95% confident that the true density lies within (ñ4 kg/m3) of the calculated average density (4464 kg/m3). We also see that the density of a particular material (e.g., aluminum) seems to be relatively independent of either the alloy, stock type, or vendor.
Based on the data from table 2, table 1 gives the average densities for usual acoustic materials:TABLE 2
&a+180H
DENSITIES FOR ACOUSTIC MATERIALS &a+180H
&a+180H(0U(s0p12h10vs3b3TDensity, p &a+180H(0U(s0p12h10vs3b3T HRD # Material Alloy Stock Vendor &a+180H(0U(s0p12h10vs3b3T(kg/m3) Date
401 Aluminum 2024 Rod Kaiser 2787 11/18/80 &a+180H
402 Aluminum 2024 Bar ALCOA 2762 4/2/81
403 Aluminum 2024 Bar ALCOA 2795 4/2/81
404 Aluminum 2024 Bar ALCOA 2764 4/2/81
405 Aluminum 2024 Bar ALCOA &dD2776&d@ 4/2/81 &a+180H
2774
&a+180H
(24)
&a+180H
406 Aluminum 7075T651 Rod, QQA225 ALCOA 2819 11/18/80 &a+180H
422 Aluminum 7075 Rod, QQA225 MM 2808 1/27/81
407 Aluminum 7075 Bar, QQA225 MM 2807 3/6/81
408 Aluminum 7075 Bar, QQA225 MM &dD2806&d@ 3/6/81 &a+180H
2807
&a+180H
(5)
&a+180H
409 Aluminum Aero Rod MM 2872 10/14/81 &a+180H
410 Aluminum Aero Bar MM 2815 11/12/81
(s3BTABLE (s0B2(s3B (CONTINUED)(s0B &a+180H
&a+180H(0U(s0p12h10vs3b3TDensity, p &a+180H(0U(s0p12h10vs3b3T HRD # Material Alloy Stock Vendor &a+180H(0U(s0p12h10vs3b3T(kg/m3) Date
441 Titanium 7-4 Rod TIMET 4467 7/21/80
442 Titanium 7-4 Rod TIMET &dD4465&d@ 7/21/80 &a+180H
4466
&a+180H
(12)
&a+180H
397 Titanium 7-4 Bar TIMET 4463 2/17/81
398 Titanium 7-4 Bar TIMET 4455 2/17/81
399 Titanium 7-4 Bar TIMET 4467 3/17/81
400 Titanium 7-4 Bar TIMET 4471 3/17/81
412 Titanium 7-4 Bar TIMET 4459 3/17/81
413 Titanium 7-4 Bar TIMET 4470 3/17/81
414 Titanium 7-4 Bar TIMET 4477 3/17/81
415 Titanium 7-4 Bar TIMET 4459 3/17/81
416 Titanium 7-4 Bar TIMET 4467 3/17/81
417 Titanium 7-4 Bar TIMET 4456 3/17/81
418 Titanium 7-4 Bar TIMET 4465 3/17/81
419 Titanium 7-4 Bar TIMET 4474 3/17/81
420 Titanium 7-4 Bar TIMET 4458 3/17/81
421 Titanium 7-4 Bar TIMET &dD4460&d@ 3/17/81 &a+180H
4464
&a+180H
(4)
&a+180H
411 Steel D2 Rod, R&a+45Vc&a-45V 54
-- 7671 3/13/81 &a+180H
Notes:
1. The test samples from each of the following sets was machined from the same piece of raw stock:
(HRD-412 --> HRD-416)
(HRD-417 --> HRD-421)
(HRD-402, HRD-403)
(HRD-404, HRD-405)
(HRD-407, HRD-408)
2. Numbers in each column directly below a solid line are averages. Numbers in parentheses are the associated 95% confidence interval of the mean.
Static Tests
In this section we will look at tests where the material properties are determined by slow deformation of the material. If a material is slowly stretched or twisted, its material properties can be determined from the applied forces and resulting deformations. The two basic tests are the tensile test (to determine Young's modulus E and Poisson's ratio nu) and the torsion test (to determine the shear modulus G). Tensile tests: METHOD
In a tensile test, a specially shaped part (.f.figure Õ10¸®Ÿelastic\tensile1.ela, photo¯®ti tensile test specimen with attached strain gage;) is stretched. To determine Young's modulus, the force and dimensional change along the specimen axis are measured. If the transverse dimensional changes are also measured, then Poisson's ratio can be calculated.
The equation that relates Young's modulus to uniaxial stress and strain is —
åi
7) Ei = --
îi
&a+180H
&a+180H
where E = Young's modulus
å = tensile stress
î = tensile strain
i = specified material direction
(e.g., longitudinal, long-
transverse, etc.) (See chapter on "Fundamental Relations" for discussion of tensile stress and tensile strain.)
Equation Æ7µ can be rearranged to give: 8) åi = Ei îi
We can recognize this as the equation of a straight line relation between åi and îi, where Ei is just the slope of the line for the material tested in the i direction. Thus, if we take measurements of strain at several stress levels, then these data pairs can then be plotted on a graph of stress versus strain, for which Young's modulus is just the slope of the resulting curve. (This is best determined by linear regression curve fitting. See the chapter on "Statistics".)
If we also measure strains transverse to the stress direction, we can determine Poisson's ratio for that direction, which is the ratio of the strain in a transverse direction to the strain in the direction of applied stress:
îj
9) nui,j = - --
îi
&a+180H
&a+180H
where nu = Poisson's ratio
î = strain
i = direction of applied stress
j = specified transverse direction As with the equation for Young's modulus, the above equation can be rearranged to the appearance of a straight line: 10) îi = nui,j îj
Thus, nui,j is just the slope of the straight line relating îi to îj. As with Young's modulus, we only need to take several measurements of the two strains and then plot the resulting data pairs. Poisson's ratio will then be the slope of the resulting curve, again best determined by linear regression curve fitting. For materials that are isotropic, only a single test is needed to determine the value of Young's modulus and Poisson's ratio, since these values are the same for every direction in the material. Thus, for isotropic materials, the subscripts that denote the directions are not needed.
For orthotropic materials such as titanium, tensile tests in three perpendicular directions are needed to completely define all values of Young's modulus and Poisson's ratio. Also, for orthotropic materials nui,j <> nuj,i, but these are related by the equation:
nui,j nuj,i
11) ÄÄÄÄÄ = ÄÄÄÄÄ
Ei Ej
Ti-7Al-4Mo TITANIUM BAR tensile tests Lucius Pitkin testing lab has tested a single sample of Branson's TIMET Ti-7Al-4Mo titanium bar in the long-transverse and longitudinal material direction. (For normal Branson bar horns, the stud is aligned in the long-transverse material direction.) Long-transverse tensile tests
Table 3 shows the stress-strain data for the long-transverse tensile test.
TABLE 3 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
STRESS-STRAIN DATA FOR (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONG-TRANSVERSE TENSILE TEST OF TIMET Ti-7Al-4Mo (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Long-transverse Long-transverse (10U(s3t12vpsb10H&a+108H³&a+4752H³
Stress, å&a+45V&dDLT&a-45V&d@ (MPa) Strain, î&a+45V&dDLT&a-45V&d@ (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
&a+1152H69.0&a+1728H575
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H137.9&a+1656H1082
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H206.8&a+1656H1638
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H275.0&a+1656H2285
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H334.8&a+1656H2724
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H413.7&a+1656H3347
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H482.6&a+1656H3884
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H551.6&a+1656H4444
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

(Note: The original Lucius Pitkin report (12/3/81) gave the stress in English units with one significant digit -- e.g., 50000 lbf/in2. These have been converted to metric units in the above table, where MPa = Mega Pascal = 106 Pa.) The above data is plotted in
.f.figure Õ11¸®Ÿelastic\lp_ti2a.plt¯®tensile test, 7-4 titanium bar: long-transverse material direction;¯. The linear regression equation shows that the slope of the line (i.e, Young's modulus in the long-transverse direction ELT) is: 12) ELT = 124.0 GPa
&a+180H
&a+180H
where GPa = Giga Pascal &a+180H
= 109 Pa During the above tensile test, strain measurements were also taken in the short-transverse material direction for each stress load. Table 4 shows the corresponding long-transverse and short- transverse strain measurements: &a+180H
TABLE 4 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
STRAIN DATA FOR (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONG-TRANSVERSE TENSILE TEST OF TIMET Ti-7Al-4Mo (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Long-transverse Short-transverse (10U(s3t12vpsb10H&a+108H³&a+4752H³
Strain, î&a+45V&dDLT&a-45V&d@ Strain, î&a+45V&dDST&a-45V&d@ (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H575&a+1800H242
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H1082&a+1800H428
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H1638&a+1800H620
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H2285&a+1800H775
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H2724&a+1800H910
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H3347&a+1728H1090
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H3884&a+1728H1265
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1152H4444&a+1728H1441
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

.f.Figure Õ12¸®Ÿelastic\lp_ti2b.plt¯®tensile test, 7-4 titanium bar: long-transverse material direction;¯ shows a plot of this data. The slope of the line is the ratio of the strain in the short-transverse direction to the strain in the long-transverse direction due to a stress in the long-transverse direction. This is Poisson's ratio nuLT,ST, where the subscript "LT" indicates the long-transverse material direction and "ST" indicates the short-transverse material direction. Determining the slope by linear regression, we have:
îST
13) nuLT,ST = ÄÄÄ
îLT
&a+180H
&a+180H
= 0.303
Ideally, strain measurements should also have been made in the longitudinal direction. This would have permitted the calculation of the Poisson's ratio nuLT,L. However, this was not done.
Longitudinal tensile tests
Lucius Pitkin also tested TIMET 7-4 titanium bar in the longitudinal direction. The data required to calculate Young's modulus in the longitudinal direction is given in table 5 and .f.figure Õ13¸®Ÿelastic\lp_ti1a.plt¯®tensile test, 7-4 titanium bar: longitudinal material direction;¯. (Again, I emphasize that the term "longitudinal", when referring to a material direction, means parallel to the grain direction. In this context, it has no relation to any specific direction of possible resonator vibration.)
TABLE 5 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
STRESS-STRAIN DATA (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONGITUDINAL TENSILE TEST OF TIMET Ti-7Al-4Mo (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Longitudinal
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Stress, å&a+45V&dDL&a-45V&d@ (MPa) Strain, î (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1368H69.0&a+1656H650
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H137.9&a+1584H1240
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H206.8&a+1584H1870
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H275.8&a+1584H2410
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H334.8&a+1584H2895
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H413.7&a+1584H3585
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H482.6&a+1584H4170
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H551.6&a+1584H4762
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

We see from figure Õ13¸®Ÿelastic\lp_ti1a.plt¯ that the slope of the line (i.e., Young's modulus in the longitudinal direction EL) is —
14) EL = 117.8 GPa
Comparing equation Æ12µ to equation Æ14µ, we see that Young's modulus in the long-transverse material direction ELT is about 5.3% higher than Young's modulus in the longitudinal material direction EL. Thus, TIMET's Ti-7Al-4Mo is anisotropic with respect to Young's modulus. (For the effects on tuning, see the section "Effect of Grain Direction in Horns with Large Lateral Dimensions".)
Table 6 and .f.figure Õ14¸®Ÿelastic\lp_ti1b.plt¯®tensile test, 7- 4 titanium bar: longitudinal material direction;¯ give the data needed to calculate Poisson's ratio nuL,ST. The slope of the line in figure Õ14¸®Ÿelastic\lp_ti1b.plt¯ is the ratio of strain in the short-transverse material direction to the strain in the longitudinal material direction due to a stress in the longitudinal material direction -- i.e., nuL,ST. By linear regression this value is —
îST
15) nuL,ST = ÄÄÄ
îL
&a+180H
&a+180H
= 0.334
In comparison to nuLT,ST (equation Æ13µ), nuL,ST is about 9.3% higher.
TABLE 6 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
STRAIN DATA FOR (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONGITUDINAL TENSILE TEST OF TIMET Ti-7Al-4Mo (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Longitudinal Short Transverse (10U(s3t12vpsb10H&a+108H³&a+4752H³
Strain, î&a+45V&dDL&a-45V&d@ Strain, î&a+45V&dDST&a-45V&d@ (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1296H650&a+1800H216
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H1240&a+1800H414
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H1870&a+1800H603
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H2410&a+1800H806
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H2895&a+1800H985
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H3585&a+1728H1185
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H4170&a+1728H1396
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H4762&a+1728H1584
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

Thin-wire calculations
Now that we know both the density and Young's modulus for 7-4 titanium bar, we can calculate the thin-wire wave speed CO from equation Æ2µ. Since Young's modulus depends on the material direction, CO will also depend on the material direction. In the long-transverse material direction (the usual direction of the stud axis for Branson bar horns), CO for 7-4 titanium bar is —
Ú ELT ¿1/2
16) CO,LT = ³ ÄÄÄ ³
À p Ù
&a+180H
&a+180H
Ú 124.0*109 Pa ¿1/2
= ³ ÄÄÄÄÄÄÄÄÄÄÄÄ ³
À 4464 kg/m3 Ù
&a+180H
&a+180H
= 5270 m/sec
The thin-wire half-wavelength âO is related to CO by (see the chapter on "Wave Motion"):
CO
17) âO = ÄÄÄ
2 f
Thus, at 19950 Hz the long-transverse thin-wire half-wavelength is —
CO,LT
18) âO,LT = ÄÄÄÄÄ
2 f

5270 m/sec
= ÄÄÄÄÄÄÄÄÄÄÄÄ
2 * 19950 Hz
&a+180H
&a+180H
= 0.1321 m
&a+180H
= 132.1 mm
In the longitudinal material direction, CO for titanium 7-4 bar is —
Ú ELT ¿1/2
19) CO,L = ³ ÄÄÄ ³
À p Ù
&a+180H
&a+180H
Ú 117.8*109 Pa ¿1/2
= ³ ÄÄÄÄÄÄÄÄÄÄÄÄ ³
À 4464 kg/m3 Ù
&a+180H
&a+180H
= 5137 m/sec
At 19950 Hz, the corresponding longitudinal thin-wire half- wavelength is —
CO,L
20) âO,L = ÄÄÄÄ
2 f
&a+180H
&a+180H
5137 m/sec
= ÄÄÄÄÄÄÄÄÄÄÄÄ
2 * 19950 Hz
&a+180H
&a+180H
= 0.1287 m
&a+180H
= 128.7 mm
Thus, for 7-4 titanium bar, we see that the thin-wire wave speed in the long-transverse material direction CO,LT is about 2.5% higher than in the longitudinal material direction CO,L. Correspondingly, the thin-wire half-wavelength in the long- transverse material direction (âO,LT) is about 2.5% higher than in the longitudinal material direction (âO,L). Shear tests: METHOD
In shear tests, a cylindrical sample (.f.figure Õ14a¸®Ÿelastic\torsion1.ela, photo¯®ti torsion test specimen with attached strain gage;) is twisted while the twisting torque and angle of twist are measured, from which the shear stress and shear strain can be calculated. The shear modulus (modulus of rigidity) G which relates the shear stress and shear strain is given by —
ç
21) G = ÄÄÄ
where G = modulus of rigidity
ç = shear stress
= shear strain (See the chapter on "Fundamental Relations" for discussion of shear stress and shear strain.) Rearranging equation Æ21µ into the form of a straight line, where G is the slope:
22) ç = G
For a material that is anisotropic, such as titanium, the above equations no longer apply, since two perpendicular shear moduli are actually involved. (See Kuenzi, p. 39. Kuenzi describes a method for determining the shear modulus of orthotropic materials, pp 37 - 40.) None-the-less, we will use equations Æ23µ and Æ24µ below to give an "effective" shear modulus Gi in the i test direction:
çi
23) Gi = --
i
24) çi = Gi i
Ti-7Al-4Mo TITANIUM BAR shear tests The Lucius Pitkin lab provided the following data on a sample of TIMET Ti-7Al-4Mo bar material. Long-transverse shear tests
For tests of TIMET Ti-7Al-4Mo bar in the long-transverse material direction, the data is given in table 7, which is plotted in .f.figure Õ15¸®Ÿelastic\lp_ti4.plt¯®shear test, 7-4 titanium bar: long-transverse material direction;¯. &a+180H
TABLE 7 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONG-TRANSVERSE SHEAR TEST OF TIMET Ti-7Al-4Mo (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Shear Shear (10U(s3t12vpsb10H&a+108H³&a+4752H³
&dDStress, ç&d@&a+45V&dDLT&a-45V&d@&dD (MPa)&d@ &dDStrain, &d@&a+45V&dDLT&a-45V&d@ (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H146.2&a+1656H2350
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H289.9&a+1656H4650
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H433.0&a+1656H6940
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H574.7&a+1656H9200
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H720.5&a+1584H11500
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

The linear regression equation of figure Õ15¸®Ÿelastic\lp_ti4.plt¯ shows the slope of the curve (GLT) as:
25) GLT = 62.5 GPa
Longitudinal shear tests
For TIMET Ti-7Al-4Mo bar in the longitudinal material direction, the shear data is given in table 8 and plotted in .f.figure Õ16¸®Ÿelastic\lp_ti3.plt¯®shear test, 7-4 titanium bar: longItudinal material direction;¯. &a+180H
&a+180H
&a+108H(10UTABLE 8 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONGITUDINAL SHEAR TEST OF TIMET Ti-7Al-4Mo (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Shear Shear (10U(s3t12vpsb10H&a+108H³&a+4752H³
&dDStress,&d@ ç&a+45V&dDL&a-45V&d@ (M&dDPa)&d@ &dDStrain, &d@&a+45V&dDL&a-45V&d@ (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H154.8&a+1728H2650
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H306.1&a+1728H5250
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H457.8&a+1728H7800
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H615.7&a+1656H10500
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1224H770.2&a+1656H13150
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

From the linear regression equation of figure Õ16¸®Ÿelastic\lp_ti3.plt¯, we see that the slope of the curve (i.e., GL) is —
26) GL = 58.7 GPa
In comparison to the longitudinal modulus of rigidity (GL), we see that the long-transverse modulus of rigidity (GLT) is 6.7% higher. This again confirms that titanium is anisotropic. Shear wave speed
While the shear wave speed is of less interest than the thin-wire wave speed, we will calculate its value for comparison to results obtained by NDT testing. For the long-transverse direction:
Ú GLT ¿1/2
27) CS,LT = ³ ÄÄÄ ³
À p Ù
&a+180H
&a+180H
Ú 62.5*109 Pa ¿1/2
= ³ ÄÄÄÄÄÄÄÄÄÄÄ ³
À 4464 kg/m3 Ù
&a+180H
&a+180H
= 3742 m/sec
In the longitudinal direction:

Ú GLT ¿1/2
28) CS,L = ³ ÄÄÄ ³
À p Ù
&a+180H
&a+180H
Ú 58.7*109 Pa ¿1/2
= ³ ÄÄÄÄÄÄÄÄÄÄÄ ³
À 4464 kg/m3 Ù
&a+180H
&a+180H
= 3626 m/sec
Thus, as with the thin-wire wave speed, the shear wave speed is higher (3.1%) in the long-transverse material direction than in the longitudinal direction.
Note: For each direction of shear wave propagation, two shear wave speeds can be specified, corresponding to two perpendicular directions of particle motion. (See McSkimin, pp. 325 - 326, Thurston, p. 90, and Cady, p. 106.) Thus, a shear wave moving in the long-transverse direction can have two shear wave speeds; similarly in the longitudinal direction. It is not apparent which of these wave speeds correspond to the above calculations. It is possible that the above calculations represent average shear wave speeds in the specified directions.

CONVERTER DRIVEN RESONANT Tests In this section we will look at how the thin-wire wave speed CO and Young's modulus E can be determined by testing a full-wave horn. We will then apply this technique to specific materials. ThEORY
From the chapter on Wave Motion", the relation between the thin- wire wave speed CO and the thin-wire half-wavelength âO is: 29) CO = (2 âO) f
&a+180H
&a+180H
where f = frequency corresponding to the
tuned thin-wire length âO Thus, if we can determine the thin-wire half-wavelength âO, then we could also determine the thin-wire wave speed CO and, from equation Æ2µ, Young's modulus. METHOD: HALF-WAVE hornS?
If we tune an unshaped half-wave horn to a specified frequency, we might consider using the resulting length in equation Æ29µ. Unfortunately, this approach has three problems: 1. Stud effect. To attach the horn rigidly to the converter, the horn must have a stud in one end. If the horn is titanium or aluminum, then the steel stud will be heavier than the horn material that it displaces. This will make the horn tuned length shorter than the a "pure" horn without any stud. &a+180H
Of course, we might make the stud from the same material as the horn. However, the portion of the stud that extends into the converter would then have the wrong mass, since converters are tuned to resonate with a steel stud.
(Note: McMahon (p. 86) has used silicone grease to couple the horn to the converter, which is similar to the method used in ultrasonic nondestructive testing (NDT). This requires a converter that is correctly tuned without a stud.) 2. Converter effect. When a converter is attached to a horn, the resulting resonant frequency is the frequency of the entire stack. If the resonant frequency of the converter alone is different from that of the stack, then the length of the horn will be affected. For instance, we may have a converter whose resonant frequency is 19800. If we tune the converter and half- wave horn to 19950, then the half-wave horn must be somewhat short to make up for the low frequency of the converter. Thus, the length of the half-wave horn would not be correct for use in equation Æ29µ.
&a+180H
Well, why not tune the converter to exactly to 19950 and eliminate this problem? Unfortunately, this is difficult to do precisely because the converter, by itself, will run high in frequency due to the tapped stud hole in its front driver. (See the chapter on "Converters".)
3. Finite diameter. Even neglecting the above problems, we know that our half-wave horn will have some finite diameter. Due to

the effect of Poisson's coupling, this finite diameter makes the tuned horn somewhat shorter than if it had truly been a "thin- wire". This effect is called dispersion. (See the chapter on "Wave Motion".) .f.Figure Õ17¸®Ÿwave\wlendia1.plt¯®aerospace aluminum half-wavelength at 19950 Hz;¯ shows this effect for Aerospace aluminum. Of course, we could make the half-wave horn as thin as possible, but then the effect of the stud would be even greater, since the stud would occupy a greater percentage of the total horn volume.
(Note: In a later section, we will look at a resonance test in which the horn is driven electrostatically, which essentially eliminates problems 1 and 2 above. However, the electrostatic method requires special equipment.) METHOD: FULL-WAVE horns
Suppose we adopt the following procedure. Let's start with a converter and a half-wave horn that have been tuned to a specified frequency, say 19950 Hz. For the present discussion, the shape of this original half-wave horn is not important. (See the top of .f.figure Õ18¸®Ÿelastic\add_hw.dwg¯®addition of half-wave section to tuned horn;¯.) Now let's suppose that we glue an additional half-wave section onto the original horn and that the resulting frequency is again exactly 19950. (See the bottom of figure Õ18¸®Ÿelastic\add_hw.dwg¯.) We must then conclude that the added half-wave section has a resonant frequency of exactly 19950, since otherwise the frequency of the entire stack would have changed when this section was added. By using this approach, we have avoided the first two problems mentioned above, since there is no stud in the glued half-wave section and because we precisely know the stack frequency before gluing the additional half-wave section. Note that we have not specified a shape for the added half-wave horn. However, our calculations will be greatly simplified if this added half-wave horn is a solid unshaped cylindrical section -- i.e., similar to the Titanium Standard but without a stud.
To differentiate between the original half-wave horn (any shape, with stud at converter end) and the added half-wave section (unshaped without stud), I will refer to the added section as the front half-wave horn (for lack of a better term). Now we have resolved the first two problems mentioned above. Note, however, that this front half-wave horn is not a thin-wire horn (the third problem), since it has a finite diameter. While the above procedure will theoretically give the desired results, gluing horns together is not very practical. However, we can simply reverse the above process to achieve the desired effect. Let's start with a one-piece full-wave horn of the required material. (The front half-wavelength of this horn should be unshaped.) We'll tune this horn to a specified frequency near the nominal converter frequency (e.g., 19950 could be chosen for 20 kHz, but any reasonable frequency is acceptable). We then machine away the front half-wave section

and retune the remaining back half-wave section to the original frequency. If we subtract the length of this remaining half-wave horn from the tuned length of the original full-wave horn, the difference is just the half-wavelength of the front section at the tuning frequency.
CORRECTING FOR FINITE HORN DIAMETER By starting with a full-wave horn, we have avoided the first two problems mentioned at the beginning of this section. However, we still need to address the remaining problem of the finite diameter of the resonator. Since we no longer have to have a stud in the front half-wave section, we could theoretically make the front half-wave section very thin. However, this could cause flexing which might affect tuning. Since we cannot make a true thin-wire resonator, we cannot measure âO directly and therefore cannot directly use equation Æ29µ to find the thin-wire wave speed CO. However, if we have a full-wave resonator whose front half-wave section is slender (diameter d) but not actually thin, we can still use the above procedure to determine the tuned length â of the front section. Then, to get the thin-wire half-wavelength âO, we must multiply â by a correction factor to account for the finite diameter d. Thus:
30) âO = â * (Correction factor) A possible correction factor can be determined from either the Rayleigh equation or the Mori equation. Both of these equations are approximate. The Rayleigh equation is simpler and is often mentioned, but is less accurate than the Mori equation. (See the chapter on "Wave Motion" for the derivation and discussion of these equations.)
Rayleigh's equation
For unshaped cylindrical horns, one form of Rayleigh's equation can be written in terms of d/â as:
Ú Ú ¿2 ¿1/2 31) âO = â ³ 1 + (1/8) ³ ã nu (d/â) ³ ³
À À Ù Ù &a+180H
&a+180H
where âO = thin-wire half-wavelength
at the specified frequency
â = length of front half-wave
section of diameter d at
the specified frequency
nu = Poisson's ratio Since the quantity within the square-root brackets is always greater than 1.0, the correction factor is always greater than 1.0, and âO will always be larger than â. Using equation 31, the predicted value of âO is always less than the actual thin-wire half-wavelength. .f.Figure Õ19¸®Ÿelastic\ray_err1.plt¯®rayleigh error: prediction of thin-wire half-wavelength;¯ shows the approximate error. (The graph is valid for nu ÷ 0.3.) Mori equation

Mori has developed a more correction equation that is substantially more accurate at large diameters:
Ú 2 B5 ¿1/2
³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 32) âO = â ³ Ú ¿1/2 ³
³ B6 - ³ B62 - (4 B5) ³ ³
À À Ù Ù &a+180H
&a+180H
where B6 = 1 + B1 B3 B4
B5 = B2 B3 B4 &a+180H
Ú d ¿2
B4 = ³ ÄÄÄ ³
À â Ù &a+180H
Ú ã ¿2
B3 = ³ ÄÄÄ ³
À 2 à Ù &a+180H
B2 = 1 - 3 nu2 - 2 nu3 &a+180H
B1 = 1 - nu2 &a+180H
à ÷ 1.84 + 0.68 nu This equation is cumbersome but is easily implemented on a computer or programmable calculator. Using equation 32, the predicted value of âO is always greater than the actual thin-wire half-wavelength. (See Derks, pp. 43 - 44.)
Mori versus Rayleigh
Figure Õ17¸®Ÿwave\wlendia1.plt¯ shows how the Rayleigh and Mori equations differ as the horn diameter increases. (This figure is discussed in detail in the section "Converter Driven Resonance Tests" --> "Aluminum".) If the front half-wave horn section is slender, then Mori's equation gives only slightly better accuracy than Rayleigh's equation. However, Mori's equation is substantially more accurate for larger horn diameters. Since the Mori equation gives better results for larger diameter horns, it will be consistently used to make the correction required for equation 30. Once âO has been calculated, we can use equation Æ29µ to find CO.
Accuracy considerations
In the above discussion, the basic process consisted of five steps:
1. Tune the full-wave horn to a specified frequency near the converter nominal frequency (e.g., 19950 Hz). 2. Cut off the front half-wave section and tune the remaining section with the stud to the same frequency. 3. Subtract the resonator length in step 2 from the length in step 1 to give the front half-wavelength â at diameter d.

4. Apply a correction to convert â to the thin-wire half- wavelength âO.
5. Substitute âO and f into equation Æ29µ to give the thin-wire wave speed.
Linear regression. Steps 1 and 2 require that the horn be tuned exactly to a specified frequency. In practice, however, this is not the best approach. First, it will generally be difficult to obtain exactly the same frequency for both the full-wave and half-wave horns. Second, this method ignores valuable tuning data that would help to reduce random tuning error. A better approach is to determine a linear regression equation of resonator length versus frequency for all tuning cuts in the vicinity of the desired tuned frequency. For instance, if the resonator nominal frequency is 20  kHz, then a linear regression equation could be fit to all tuning cuts between 19 kHz and 21 kHz. (For a 40 kHz resonator, the linear regression equation would be for tuning cuts between 38 kHz and 42  kHz). We would then have one linear regression equation for the full-wave horn of step 1 and another linear regression equation for the back half-wave horn of step 2. The difference between these two linear regression equations at the specified tuning frequency (e.g., 19950 Hz) would give the true half-wavelength â at that frequency. By this method, all of the relevant tuning data has been used. (This process is illustrated later.) Effect of temperature. We must also be concerned about the horn temperature rise that occurs during machining in steps 1 and 2. This will be most noticeable for titanium and, if not corrected, will cause the frequency to read too low. To prevent this, the resonator should be cooled for several minutes in a bucket of room-temperature water before measuring the frequency. (See the section "Temperature Dependence" below.) Effect of Poisson's ratio. In step 4, we apply either the Rayleigh or Mori correction equations to determine âO. However, note that if d and â are experimentally measured, then the correction equations actually involves two unknowns: âO and nu. If an incorrect value of nu is used, then âO will also be in error. However, the effect of using an incorrect nu decreases as (d/â) decreases -- i.e., as the resonator becomes increasingly slender. Thus, as (d/â) approaches zero, large errors in nu have little effect on the calculated value of âO. Conversely, as (d/â) becomes large, small errors in nu have a significant effect on âO. Thus, the front half-wave section should be as slender as possible to minimize Poisson errors. If an experimental value for nu is not available, then a handbook value can be used. (For acoustic materials, nu usually ranges from about 0.29 to 0.34.) Best resonator shapes
The above method will work with any full-wave horn whose front half-wavelength is cylindrical and unshaped. As noted above, the front half-wave section should preferably be slender. Although the rear resonator half-wave section (the section with the stud) can be any shape, two shapes have been most commonly used:

1. Unshaped horn. The rear half-wave section is machined to the same diameter as the front half-wave section. This gives a full- wave unshaped horn of constant diameter. Such a horn is very easy to machine.
2. S-N horn. The rear half-wave section is machined for very high gain. (.f.Figure Õ20¸®Ÿsn\sn_dims2.20¯®20 kHz s-n horn;¯ shows one possible design.) After tuning the resulting full-wave horn, it can then be used for fatigue testing. (See the chapter on "Material Life Testing" for further information.) When the front half-wave section fails from fatigue, it can be machined away to determine the thin-wire wave speed as described in the above sections. Because the S-N horn has high gain, it can also be used for Q measurements. (See the chapter on "Inelastic Material Properties" for Q measurements.) Thus, while an S-N horn is more expensive to machine, it can provide more data than the unshaped full-wave horn.
Half-wave tests
As discussed above, the stud and the converter both have an effect on the resonant length of a half-wave horn. Therefore, the full-wave resonance tests should be used whenever possible. Occasionally, however, data may be available from a half-wave horn whose shape starts out as an unshaped cylindrical horn. For instance, spool horns may begin as unshaped cylindrical horns before the final spool shape is machined. In such a case, the data may still be useful, especially if corrections for the stud can be applied or the magnitude of the error can be estimated. However, the results from half-wave horns should be considered less reliable that data from full-wave horns. The effects of the converter and stud will be minimized under the following circumstances:
1. The half-wave horn is very massive. A large diameter horn of dense material will be less affected by the converter and stud than a small diameter horn of lower density. Thus, for an equivalent diameter, a titanium horn will be less affected than aluminum. As mentioned above, however, the accuracy of the Rayleigh or Mori equations will suffer if an incorrect value of Poisson's ratio is used when calculating âO for large horns. 2. âO and CO are calculated at the converter design frequency -- 19950 Hz for 20 kHz converters or 39900 Hz for 40 kHz converters. We will consistently use this approach for both full-wave and half-wave resonators.
3. The stud is small. A í9.5 stud will have less effect than a í12.5 stud. For certain experimental horns, a special step stud with only a í4.7 thread in the horn has been used. 4. The effect of the stud can be calculated. The effect of the stud may be determined by analyzing the horn with a computer. As an approximation, the tuned length of the horn can be increased by the amount by which mass of the stud exceeds the mass of the horn material that it displaces, according the following equation:
&a+180H

&a+180H

Ú dS ¿2 Ú Stud density ¿ 33) ëLh = LS ³ ÄÄÄÄ ³ ³ ÄÄÄÄÄÄÄÄÄÄÄÄ - 1 ³
À dh Ù À Horn density Ù &a+180H
&a+180H
where: ëLh = increased horn length
LS = length of stud within
the horn
dS = stud diameter
dh = horn diameter &a+180H
&a+180H
You can see that when the stud and horn have the same density (e.g., a steel stud in a steel horn), the stud has no effect according to the above equation. Also, as the horn diameter is increased or the stud volume is decreased, the stud effect diminishes. For a í12.7 x 19.0 deep steel stud in an aluminum horn, FEA computer analysis shows that the corrected horn half- wavelength using equation 33 errs by less than 0.2% from the true half-wavelength if no stud was present. The accuracy will be better for smaller studs or titanium horns. The use of half-wave resonators to estimate the wave speed is illustrated for converter driven tests of aluminum (below). Ti-7Al-4Mo bar stock
Converter driven resonance tests have been used to find the material properties to TIMET Ti-7Al-4Mo bar stock in both the long-transverse and longitudinal directions. We will illustrate the method in considerable detail for the long-transverse direction and then give a cursory look at the longitudinal direction.
Long-transverse direction
In titanium bar stock of which resonators are made, the long- transverse direction has a maximum stock length of about 150 mm. Since this is too short for a full-wave horn at 20 kHz, we have machined our long-transverse test samples to resonate at 40 kHz. The first full-wave test sample (HRD-397, 2/17/81) has a diameter of í19.0 over its entire length. For each tuning cut on the face of this horn, the length and frequency were recorded (table 9). The data for this horn is plotted at the top of .f.figure Õ21¸®Ÿelastic\hrd_397.plt¯®wave speed measurement: Ti-7Al-4Mo bar, long-transverse;¯.) &a+180H
&a+180H

&a+108H(10UTABLE 9 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
TUNING DATA FOR FULL-WAVE HORN, (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
TIMET Ti-7Al-4Mo BAR STOCK, (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONG-TRANSVERSE MATERIAL DIRECTION (HRD-397) (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Length (mm) Frequency (Hz) (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1008H137.34&a+1728H38604
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1008H133.48&a+1728H39511
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1008H132.48&a+1728H39772
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1008H131.55&a+1728H39999
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1008H130.53&a+1728H40238
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

The linear regression equation for this data is: 34) Length (full-wave) = (-4.155*10-3 mm/Hz) f + 297.71 mm &a+180H
&a+180H
where Length ÍÍ> mm
f ÍÍ> frequency (Hz) for which the correlation coefficient is: 35) r (full-wave) = - 0.99988 Equation 34 is plotted at the top of figure Õ21¸®Ÿelastic\hrd_397.plt¯. Having tuned the full-wave horn, we need only cut the horn length approximately in half and retune the remaining section (i.e., the section with the stud). The tuning data for the back half-wave section is given in table 10.
&a+180H
&a+180H

&a+108H(10UTABLE 10 (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
TUNING DATA FOR BACK HALF-WAVE HORN, (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
TIMET Ti-7Al-4Mo BAR STOCK, (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
LONG-TRANSVERSE MATERIAL DIRECTION (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³
Length (mm) Frequency (Hz) (10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+180H
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1080H68.00&a+1728H38936
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1080H66.09&a+1728H39701
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1080H65.33&a+1728H40000
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+1080H64.64&a+1728H40269
(10U(s3t12vpsb10H&a+108H³&a+4752H³ &a+108H(10U

The linear regression equation for this data is: 36) Length (back half-wave)
&a+180H
= (- 2.518*10-3 mm/Hz) f + 166.04 mm &a+180H
&a+180H
where Length ÍÍ> mm
f ÍÍ> frequency (Hz) for which the correlation coefficient is: 37) r (back half-wave) = - 0.99998 The bottom of figure Õ21¸®elastic\hrd_397.plt¯ shows the above data and linear regression line. The length â of the front half-wave resonator at any specified frequency f can be determined from the graph as the difference between the top and bottom lines. Alternately, we can subtract equation Æ36µ from equation Æ34µ to give â as a function of f: 38) â (Front half-wavelength) &a+180H
= Length (Full-wave) - Length (Half-wave) &a+180H
&a+180H
Ú ¿
= ³ (-4.155*10-3) f + 297.71 ³
À Ù &a+180H
Ú ¿
- ³ (-2.518*10-3) f + 166.04 ³
À Ù Note that by plotting the linear regression lines or using the above equation, we can get â for any frequency near 40 kHz. Thus, we do not have to be especially careful about tuning the resonator to exactly a predetermined frequency. Also, using the

linear regression lines tends to smooth out any random tuning errors that may have occurred. Let us choose a frequency near 40 kHz -- say, the nominal tuning frequency of 39900 Hz. For this frequency, we can find the length â from equation Æ38µ (or from figure Õ21¸®Ÿelastic\hrd_397.plt¯) as: 39) â (Front half-wavelength @ 39900 Hz) &a+180H
= Length (Full-wave) - Length (Half-wave) &a+180H
= 131.93 - 65.57
&a+180H
= 66.35 mm
In looking at equation Æ39µ, we see that the back half-wave section with the stud (65.57 mm) is 0.78 mm shorter than the front half-wavelength â (66.35 mm). Since these sections are otherwise identical, we can attribute the shorter length of the back half-wave section to the stud effect, as discussed above. This has been verified by computer analysis. Wave speed calculations. Using Mori's equation (Æ32µ) we can calculate the thin-wire half-wavelength. Since we do not know the exact value of Poisson's ratio, we will use 0.30, which is the value determined from the static tests of titanium in the long-transverse direction. Mori's equation then gives: 40) âO = (66.35 mm) * 1.0058
&a+180H
= 66.73 mm
where we have used:
&a+180H
nu = 0.30
d = 19.0 mm
â = 66.35 mm Note that â has been determined at a specified frequency (39900 Hz in this case) and, as a result, the âO calculated above is valid only at this frequency. Thus, âO cannot be specified independent of the frequency.
(Note: if we had used Rayleigh's equation for the calculation, the correction factor would have been 1.0045, for which âO would have been 66.65 mm. This is 0.12% lower than the Mori value.) Using equation 29, the resulting thin-wire wave speed is: 41) CO = (2 âO) f
&a+180H
= (2 * 66.73 mm) * 39900 Hz &a+180H
= 5325000 mm/sec
&a+180H
= 5325 m/sec
The above results were determined at a frequency of 39900 Hz. What result would we expect if we had chosen a different frequency, say f = 39500 Hz. Using the same approach as above, except at 39500, we find that â is:

42) â (Front half-wavelength @ 39500 Hz) &a+180H
= Length (Full-wave) - Length (Half-wave) &a+180H
= 133.55 - 66.58
&a+180H
= 66.97 mm
Again using the Mori equation, the thin-wire half-wavelength will be:
43) âO = (66.97 mm) * 1.0058
&a+180H
= 67.36 mm
Thus, the thin-wire half-wavelength at 39500 Hz is longer that at 39900 Hz. Of course, we should have expected this from experience, since we know that lower frequencies give longer horns. What about the wave-speed? Using the same method as above (but at a frequency of 39500 Hz), we find that CO is: 44) CO = (2 * 67.36 mm) * 39500 Hz &a+180H
= 5321000 mm/sec
&a+180H
= 5321 m/sec
Comparing the wave speed calculated in equation Æ41µ at 39900 Hz to that of equation Æ44µ at 39500 Hz shows that there is no significant difference in wave speed as the frequency changes. Thus, for an unshaped cylindrical resonator, the following conclusions are valid:
1. The half-wavelength â depends on the material, the frequency, and the diameter.
2. The thin-wire half-wavelength âO depends on the material and the frequency. Since âO depends on the frequency, it cannot be considered a material property. 3. The thin-wire wave speed CO depends only on the material. These conclusions are generally accepted and will be further substantiated below.
Young's modulus calculation. We can solve equation 2 to give Young's modulus in terms of the density and wave speed: 45) E = p CO2
Substituting the resonant test value of CO for Ti-7Al-4Mo bar in the long-transverse direction: 46) ELT = p (CO,LT)2
&a+180H
= (4463 kg/m3) (5319 m/sec)2 &a+180H
= 126.3 GPa
(Note: See HRD-397 of table 2 for density.) Additional tests. To check how the above material properties vary among samples, additional horns have been tested in the long-transverse direction. The results are summarized in table 11. (Note: HRD-397 from above is repeated in this table

for comparison. Also, the Mori equation has been used in order to be consistent with later calculations.) As can be seen, the CO data scatter is small. Using the Student-t distribution, we can be 95% confident that the true thin-wire wave speed lies between 5304 and 5364 m/sec. Recalling that the static tests gave a thin-wire wave speed of 5270 m/sec in the long-transverse material direction, we see that this falls outside the above 95% confidence interval. This difference may arise either because of random sampling error, because the materials are somehow different, because the two test methods produce inherently different results (see the section "Comparison among Test Methods"), or because of experimental error. (Note: The subsequent tables give the 95% confidence interval of the mean for both the thin-wire wave speed CO and Young's modulus E. This interval is determined from the Student-t distribution. If CO is t-distributed (which we do not know for sure), then E cannot be t-distributed, since it is a nonlinear transformation of CO -- i.e., E is proportional to CO2. However, for lack of any better information, we will assume that all data are t-distributed.)

TABLE 11
&a+180H
TIMET Ti-7Al-4Mo BAR STOCK:
&a+180H
LONG-TRANSVERSE MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, 40 kHz FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Horn Tuned &a+180H(0U(s0p12h10vs3b3TC&a+45VO,LT&a-45V E&a+45VLT&a-45V &a+180H(0U(s0p12h10vs3b3T HRD # Date Shape Length (mm) â&a+45VO&a-45V (mm) &a+180H(0U(s0p12h10vs3b3T(m/sec) (GPa) &a+180H
397 2/17/81 A 66.35 66.74 5326 126.6
398 2/17/81 A 66.30 66.69 5322 126.4
290 10/22/84 B 66.26 66.33 5293 125.1
277 9/25/85 B 67.16 67.24 5366 128.5
289 9/25/85 B 67.17 67.25 5366 128.5
329 9/25/85 B 66.70 66.78 &dD532&d@&dD9&d@ &dD126.8&d@ &a+180H
5334 127.0
&a+180H
(30) (1.4)

Notes:
1. Shapes: 40 kHz (nominal) full-wave horns
A --> Unshaped, í19.05 nominal diameter.
B --> S-N horn, 8.64 nominal front diameter.
(See .f.figure Õ22¸®Ÿsn\sn_dims1.40a¯®40 kHz s-n horn;¯.)
2. Tuned length is the actual length of the front half-wave section at the tuning frequencies given below. Minimum tuning correlation (r) = -0.998
3. âO is the thin-wire half-wavelength calculated from the Mori equation at the tuning frequencies given below. 4. Tuning frequencies:
HRD-397 --> 39900 Hz
HRD-398 --> 39900 Hz
HRD-290 --> 39927 Hz
HRD-277 --> 39913 Hz
HRD-289 --> 40019 Hz
HRD-329 --> 39900 Hz
5. Poisson's ratio = 0.30 (assumed for Mori equation). Density = 4464 kg/m3 (to calculate E).

6. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.

Longitudinal direction
In the same manner as the resonance tests in the long-transverse material direction, two samples (HRD-399 and HRD-400) from different pieces of stock were tested in the longitudinal direction. .f.Figure Õ23¸®Ÿelastic\hrd_399.plt¯®wave-speed measurement: 7-4 titanium bar, parallel to grain;¯ and .f.Õ24¸®Ÿelastic\hrd_400.plt¯®wave-speed measurement: 7-4 titanium bar, parallel to grain;¯ show the graphs of the test data.
As shown from the summary data in table 12, the two test samples showed significant differences in material properties. The percent difference between the thin-wire wave speeds is 3.5%, while the percent difference between the Young's modulus is 6.6%. While a 3.5% difference in wave speed may not seem large, such difference results in a 3.5% difference in half-wavelength of an unshaped bar, which is approximately 4 mm at 20 kHz. Possible causes for these differences might be: 1. Testing errors. Although testing errors are possible, they seem somewhat unlikely, since both samples were tested on the same day with the same equipment. Also, the graphs of the tuning data (figure Õ23¸®elastic\¯ and 24elastic\) show very little data scatter, with all regression lines having a correlation of better than 0.998.
&a+180H
If these tuning tests could be repeated on identical samples, statistical analysis indicates that the resulting wave speed for HRD-399 would not vary by more than ñ5 m/sec for every 95 out of 100 such tests -- i.e., the 95% confidence interval would be (4984 ñ 5) m/sec. Similarly, for HRD-400 the 95% confidence interval is (5149 ñ 11) m/sec. Thus, it seems unlikely that the difference between the two wave speeds (165 m/sec) could be attributed to testing error.
2. Normal material variability. As mentioned in the section "Introduction" --> "Heat-to-Heat and Bar-to-Bar Variations", the titanium modulus can vary significantly from heat-to-heat. This modulus variation would also cause a variation in wave speed. 3. Defective material. One of the two samples may have been defective in some manner. If you compare sample HRD-400 with the results of static tests in the longitudinal direction, you will find only a 0.04% difference in thin-wire wave velocity and a 0.3% difference in Young's modulus. Thus, HRD-400 gives good agreement with the static tests. On the other hand, HRD-399 gives relatively poor agreement with the static tests, and so may be defective. However, I consider this to be somewhat unlikely.

TABLE 12
&a+180H
TIMET Ti-7Al-4Mo BAR STOCK:
&a+180H
LONGITUDINAL MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Horn Tuned &a+180H(0U(s0p12h10vs3b3TC&a+45VO,L&a-45V E&a+45VL&a-45V &a+180H(0U(s0p12h10vs3b3T HRD # Date Shape Length (mm) â&a+45VO&a-45V (mm) &a+180H(0U(s0p12h10vs3b3T(m/sec) (GPa) &a+180H
399 3/17/81 A 61.95 62.46 4984 110.9
400 3/17/81 A 64.04 &dD64.53&d@ &dD5149&d@ &dD118.4&d@ &a+180H
63.50 5066 114.6
&a+180H
(13.14) (1051) (47.6) &a+180H

Notes:
1. Shapes: 40 kHz (nominal) full-wave horns
A --> Unshaped, í19.05 nominal diameter. 2. Tuned length is the actual length of the front half-wave section at 39900 Hz. Minimum tuning correlation (r) = -0.998. 3. âO is the thin-wire half-wavelength calculated from the Mori equation at 39900 Hz.
4. Poisson's ratio = 0.33 (assumed for Mori equation). Density = 4464 kg/m3 (to calculate E). 5. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.

Ti-7Al-4Mo TITANIUM ROD STOCK
With titanium rod stock, horns are always made parallel to the length of the rod -- i.e., in the longitudinal material direction. Since this direction is long enough to accommodate a full-wave 20  kHz horn, all of the testing has been done at 20 kHz.
Full-wave tests
The results of the full-wave testing are given in table 13, where the thin-wire half-wavelength âO and wave speed CO have been calculated using the Mori equation (32). Because Poisson's ratio is not known for rod material, I have assumed a value of 0.33, as was determined from the static tests of Ti-7Al-4Mo bar stock in the longitudinal material direction. The resulting average CO is 5037 m/sec.
Half-wave tests
The above data has been supplemented with two tests on large diameter half-wave resonators. The data is given in table 14. Computer analysis shows that the effect of the studs on the tuned horn length is less than 0.08%. the Mori equation (Æ32µ) has been used to calculate the thin-wire half-wavelength âO. Again, a Poisson's ratio of 0.33 has been assumed. The resulting average thin-wire wave speed is 5135 m/sec. Stock size effect?
In reviewing the results of both the full-wave and half-wave tests, we see that the thin-wire wave speed does not seem very consistent. It ranges from 4992 m/sec to 5177 m/sec, with a standard deviation of 70 m/sec. By contrast, CO for Aerospace aluminum rod (to be given later) has a range between 5071 m/sec and 5131 m/sec, with a standard deviation of only 15 m/sec. One possible explanation for the variation in titanium data is that Poisson's ratio should be some other value than 0.33. By reducing Poisson's ratio, the calculated âO and CO will become smaller. This effect will be especially strong for the larger diameter horns, so that the CO values for the large and small diameter horns will tend to converge. A nonlinear curve-fitting of the Mori equation gave a best-fit Poisson's ratio of 0.23. However, this value is much too low to be considered credible. Another possibility is that the titanium material properties depend on the size of the raw stock from which the horn was machined. .f.Figure Õ25¸®Ÿelastic\tico_dia.plt¯®7-4 titanium rod: wave speed dependence on raw stock size;¯ shows the Mori calculated thin-wire wave speed from tables 13 and 14 as a function of the raw stock diameter. If the trend indicated by figure Õ25¸®Ÿelastic\tico_dia.plt¯ is true, then a horn machined from a large diameter stock will have a longer tuned length than an identical horn from smaller stock. This would be most unusual. (Note: in plotting this data, a constant Poisson's value of 0.33 was assumed. It may be possible that both CO and Poisson's ratio vary with the stock size.) It is also possible that Mori's equation is not adequate for orthotropic materials such as titanium.

Evidently, more testing is required to pin down the material properties of titanium rod.

TABLE 13
&a+180H
TIMET Ti-7Al-4Mo ROD STOCK:
&a+180H
LONGITUDINAL MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Horn Raw Stock Tuned &a+180H(0U(s0p12h10vs3b3TC&a+45VO,L&a-45V E&a+45VL&a-45V &a+180H(0U(s0p12h10vs3b3T HRD # Date Shape Size (mm) Length (mm) â&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3T(mm) (m/sec) (GPa) &a+180H
470 7/21/80 A 38.1 124.10 125.11 4992 111.2
427 11/18/80 A 38.1 125.09 126.09 5031 113.0
299 8/2/84 B 70.0 127.35 &dD127.55&d@ &dD5089&d@ &dD115.6&d@ &a+180H
126.25 5037 113.3
&a+180H
(3.05) (114) (5.1)
&a+180H
&a+180H

Notes:
1. Shapes: 20 kHz (nominal) full-wave horns
A --> Unshaped, í38.1 nominal diameter.
B --> S-N horn, í17.2 nominal front diameter,
machined from í70 diameter stock.
(See figure Õ20¸®Ÿsn\sn_dims2.20¯.) 2. Tuned length is the actual tuned length of the front half-wave section at 19950 Hz. Minimum tuning correlation (r) = -0.995 3. âO is the Mori thin-wire half-wavelength at 19950 Hz. 4. Poisson's ratio = 0.33 (assumed for Mori equation). Density = 4464 kg/m3 (to calculate E). 5. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.

TABLE 14
&a+180H
TIMET Ti-7Al-4Mo ROD STOCK
&a+180H
LONGITUDINAL MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, HALF-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Horn Raw Stock Tuned &a+180H(0U(s0p12h10vs3b3TC&a+45VO,L&a-45V E&a+45VL&a-45V &a+180H(0U(s0p12h10vs3b3T HRD # Date Dia (mm) Size (mm) Length (mm) â&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3T(mm) (m/sec) (GPa) &a+180H
285 5/31/84 76.2 76.2 123.12 127.56 5093 115.8
286 5/31/84 101.6 101.6 120.97 &dD129.75&d@ &dD5177&d@ &dD119.6&d@ &a+180H
128.66 5135 117.7
&a+180H
(13.92) (530) (24.2) &a+180H
&a+180H

Notes:
1. Tuned length is the actual horn tuned length at 19950 Hz with a í9.5 x 14.3 deep stud. Minimum tuning correlation (r) = - 0.9996
2. âO is the Mori thin-wire half-wavelength at 19950 Hz. 3. Poisson's ratio = 0.33 (assumed for Mori equation). Density = 4464 kg/m3 (to calculate E). 4. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.
5. Minimum tuning correlation (r) = -0.9996

ALUMINUM
Full-wave tests
Converter driven resonant tests with full-wave horns have been performed on several different aluminums. The results are summarized in tables 15 - 19.
The tests of ALCOA 2024-T3 (table 16) bar indicate very little difference between material properties in the long-transverse and longitudinal material directions. This might be expected, since aluminum is generally considered to be isotropic. The Martin Marietta 7075 QQA225 aluminum (table 18) shows little difference in material properties between rod and bar stock. The Aerospace aluminum (table 19) shows greater difference. However, because only two samples have been tested, it is not known if the Aerospace difference is statistically significant or whether it is simply due to normal data scatter. Half-wave tests: Mori and Rayleigh equations Several half-wave unshaped cylindrical horns have been tuned prior to their conversion to spool horns. Table 20 gives a summary of this information for Aerospace rod material. Using the Mori equation (Æ32µ), we can estimate both the thin-wire wave speed and Poisson's ratio.
Column 4 of table 20 shows the actual tuned length of the horn with a í12.7 x 19.0 deep stud. The stud makes the horn tune shorter than if no stud was present. Unlike the titanium half- wave horns discussed earlier, the stud effect in these aluminum horns cannot be neglected. This is because the stud is denser in comparison to the aluminum than is the titanium. In addition, these aluminum horns have larger studs (í12.7 x 19.0 deep) than the titanium horns (í9.5 x 14.3 deep) discussed previously. The effect of the stud was estimated by equation 33. The corrected horn length is given in column 5 of table 20. A comparison of columns 4 and 5 shows that the stud effect is most pronounced at the smaller horn diameters, where the stud is a greater percentage of the total horn mass. The data from column 5 has been plotted in figure Õ17¸®Ÿwave\wlendia1.plt¯. (Some of the data points cannot be distinguished because they lie so close to adjacent data points.) Horn HRD-409 (a full-wave horn of the same material from table 19) has also been included in the plot.
Mori equation. The thin-wire half-wavelength âO and Poisson's ratio nu have been estimated by fitting the Mori equation (Æ32µ) to the data using a nonlinear least-squares curve fitting routine. The best fit occurs when nu is 0.319 and âO is 127.34 mm at 19950 Hz. The resulting thin-wire wave speed is 5081 m/sec, which is only 0.04% higher than the results obtained from testing the full-wave horn HRD-409. The Poisson's ratio also appears reasonable (perhaps a bit low), based on handbook values for similar materials. (Note: HRD-436 was excluded from the curve-fit data, since this point seems to be in error. If this point is included, the thin-wire wave speed increases to 5084 m/sec and Poisson's ratio increases to 0.320.)

A plot of the resulting Mori equation is shown in figure Õ17¸®Ÿwave\wlendia1.plt¯. The fit of the curve to the data appears good. However, because the Mori equation is only approximate, the fitted values of CO and nu are also approximate. Derks (pp. 43-44) shows that when Mori's equation, using empirical ("true") values for CO and nu, is plotted against the empirical data, the Mori curve falls below empirical data. For the aluminum data which Derks cites (nu = 0.344, CO = 5150 m/sec, p. 38), the tuned length predicted by Mori is about 3.5% too low when the horn diameter equals the thin-wire half-wavelength (i.e., a horn diameter of approximately í125 at 20 kHz). .f.Figure Õ26¸®Ÿwave\wlendia2.plt¯®effect of poisson's ratio on half-wavelength of aerospace al at 19950 Hz;¯ shows a plot of the Mori equation with CO = 5081 m/sec, and nu of 0.319 and 0.34. Increasing nu above 0.319 lowers the Mori curve below the empirical data, which suggests that the true value of nu should be greater than 0.319. In the section on "Finite Element Analysis", it is shown that nu = 0.333 gives the best agreement between empirical and FEA predicted amplitudes for certain cylindrical horns.
Dispersion. Figure Õ17¸®Ÿwave\wlendia1.plt¯ shows how the horn half-wavelength decreases as the horn diameter increases. For an unshaped horn, the half-wavelength â is related to the wave speed C by —
47) C = (2 â) f
Hence, for a given frequency, if â decreases with diameter, then C must also decrease proportionately with diameter. Thus, the shape of the curve in figure Õ17¸®Ÿwave\wlendia1.plt¯ shows both the change in horn length and the change in wave speed as the horn diameter increases.
Stock size effect?
For the Aerospace aluminum data given in table 20, the raw stock size ranged from about í75 to í125. Looking at the thin-wire wave speed (column 7), we see that CO is essentially constant, regardless of the raw stock size. This contrasts with the titanium rod stock, for which there was a very strong positive correlation between raw stock size the thin-wire wave speed.

TABLE 15
&a+180H
KAISER 2024 QQA225 ALUMINUM MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Stock Material Horn C&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3TE G &a+180H(0U(s0p12h10vs3b3T HRD # Date Type Direction Shape (m/sec) &a+180H(0U(s0p12h10vs3b3T(GPa) (GPa)
401 11/18/80 Rod L A 5158 74.15 27.86
&a+180H
&a+180H

Notes:
1. Material direction:
L --> Longitudinal
2. Shapes: Full-wave horns
A --> 20 kHz unshaped, í38.1 nominal diameter. 3. Density = 2787 kg/m3
Poisson's ratio = 0.33
(assumed value; required in order to calculate G). 4. Minimum tuning correlation (r) = -0.99992

TABLE 16
&a+180H
ALCOA 2024-T3 ALUMINUM MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Stock Material Horn C&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3TE G &a+180H(0U(s0p12h10vs3b3T HRD # Date Type Direction Shape (m/sec) &a+180H(0U(s0p12h10vs3b3T(GPa) (GPa) &a+180H
402 4/2/81 Bar LT A 5163 73.63 27.68
404 4/2/81 Bar LT A &dD5169&d@ &dD73.85&d@ &dD27.&d@&dD76&d@ &a+180H
5166 73.74 27.72
&a+180H
(38) (1.40) (0.51)
&a+180H
403 4/2/81 Bar L A 5180 75.00 28.19
405 4/2/81 Bar L A &dD5176&d@ &dD74.&d@&dD38&d@ &dD27.96&d@ &a+180H
5178 74.69 28.08
&a+180H
(25) (3.94) (1.46)
&a+180H

&a+180H

Notes:
1. Material direction:
LT --> Long-transverse
L --> Longitudinal
2. HRD-402 and HRD-403 are from the same bar. HRD-404 and HRD-405 are from the same bar. 3. Shapes: Full-wave horns
A --> 40 kHz unshaped, í19.05 nominal diameter. 4. Density:
HRD-402 --> 2762 kg/m3
HRD-404 --> 2764 kg/m3
HRD-403 --> 2795 kg/m3
HRD-405 --> 2776 kg/m3
5. Poisson's ratio = 0.33
(assumed value; required in order to calculate G). 6. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean. 7. Minimum tuning correlation (r) = -0.9994

TABLE 17
&a+180H
ALCOA 7075 QQA225 ALUMINUM MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Stock Material Horn C&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3TE G &a+180H(0U(s0p12h10vs3b3T HRD # Date Type Direction Shape &a+180H(0U(s0p12h10vs3b3T(m/sec) (GPa) (GPa) &a+180H
406 11/18/80 Rod L A 5056 72.06 27.09
431 12/30/80 Rod L B &dD5132&d@ &dD74.2&d@5 &dD27.91&d@ &a+180H
5094 73.16 27.50
&a+180H
(483) (13.91) (5.21)
&a+180H

&a+180H

Notes:
1. Material direction:
L --> Longitudinal
2. Shapes: Full-wave horns
A --> 20 kHz unshaped, í38.1 nominal diameter.
B --> 20 kHz S-N, í17.25 nominal diameter.
(See figure Õ20¸®Ÿsn\sn_dims2.20¯.) 3. Density = 2819 kg/m3
Poisson's ratio = 0.33
(assumed value; required in order to calculate G). 4. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.
5. Minimum tuning correlation (r) = -0.9997

TABLE 18
&a+180H
MARTIN MARIETTA 7075 QQA225 ALUMINUM MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Stock Material Horn C&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3TE G &a+180H(0U(s0p12h10vs3b3T HRD # Date Type Direction Shape (m/sec) &a+180H(0U(s0p12h10vs3b3T(GPa) (GPa) &a+180H
422 1/27/81 Rod L A 5061 71.90 27.03
432 1/29/81 Rod L B 5052 71.64 26.93
433 1/29/81 Rod L B 5046 71.47 26.87
434 1/29/81 Rod L B &dD5052&d@ &dD71.64&d@ &dD26.93&d@ &a+180H
5053 71.66 26.94
&a+180H
(10) (0.28) (1.05)
&a+180H
407 3/6/81 Bar LT C 5063 71.96 27.05
408 3/6/81 Bar LT C &dD5064&d@ &dD71.98&d@ &dD27.06&d@ &a+180H
5064 71.97 27.06
&a+180H
(9) (0.13) (0.06)
&a+180H

&a+180H

Notes:
1. Material direction:
L --> Longitudinal
2. Shapes: Full-wave horns
A --> 20 kHz unshaped, í38.1 nominal diameter.
B --> 20 kHz S-N, í17.25 nominal diameter.
(See figure Õ20¸®Ÿsn\sn_dims2.20¯.)
C --> 40 kHz unshaped, í19.05 nominal diameter. 3. HRD-407 and HRD-408 are from the same bar. 4. Density = 2807 kg/m3
Poisson's ratio = 0.33
(assumed value; required in order to calculate G). 5. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.
6. Minimum tuning correlation (r) = -0.9993

TABLE 19
&a+180H
AEROSPACE ALUMINUM MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Stock Material Horn C&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3TE G &a+180H(0U(s0p12h10vs3b3T HRD # Date Type Direction Shape (m/sec) &a+180H(0U(s0p12h10vs3b3T(GPa) (GPa) &a+180H
409 10/15/81 Rod L A 5079 74.09 28.06
&a+180H
410 11/12/81 Bar L B 5131 74.11 28.07
&a+180H

&a+180H

Notes:
1. Material direction:
L --> Longitudinal
2. Shapes: Full-wave horns
A --> 20 kHz S-N, í17.25 nominal diameter.
(See figure Õ20¸®Ÿsn\sn_dims2.20¯.)
B --> 20 kHz S-N, í17.25 nominal diameter,
machined from 152 x 50 bar stock
(See .f.figure Õ28¸®Ÿsn\sn_dims3.20¯®20 kHz s-n horn machined from 152 x 50 bar stock;¯.) 3. Density:
HRD-409 --> 2872 kg/m3
HRD-410 --> 2815 kg/m3
4. Poisson's ratio = 0.32 (from curve-fit of data in table 20 to Mori equation; required in order to calculate G). 5. Minimum tuning correlation (r) = -0.9998

&a+180H&a+2052HTABLE 20
&a+180H&a+2340H
&a+180H&a+1512H(10U(s3t12vpsb10HAEROSPACE ALUMINUM ROD: &a+180H&a+2340H
&a+180H&a+1188H(10U(s3t12vpsb10HLONGITUDINAL MATERIAL PROPERTIES &a+180H&a+2340H
&a+180H&a+756H(10U(s3t12vpsb10H(CONVERTER RESONANCE TESTS, HALF-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Horn Tuned Tuned &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V E G &a+180H(0U(s0p12h10vs3b3T HRD # Date Dia (mm) Length&a-45VA&a+45V (mm) Length&a-45VB&a+45V (mm) â&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3T(mm) (m/sec) (GPa) (GPa) &a+180H
438 10/10/85 50.0 124.22 125.74 127.38 5082 73.43 27.81 &a+180H
437 10/10/85 70.2 123.09 123.90 127.36 5082 73.43 27.81 &a+180H
061 7/9/82 75.0 122.65 123.37 127.39 5083 73.45 27.82
065 7/16/82 75.0 122.43 123.15 127.18 5075 73.22 27.73
082 9/2/82 75.0 122.48 123.20 127.23 5077 73.28 27.76 &a+180H
079 8/25/82 90.0 120.36 120.89 127.16 5073 73.17 27.72
436 10/8/85 89.3 122.05 122.56 128.60 5131 74.85 28.35 &a+180H
060 7/94/82 100.0 119.08 119.54 127.71 5096 73.83 27.97
064 7/14/82 100.0 118.96 119.42 127.60 5091 73.69 27.91
080 ÄÄÄÄÄÄÄ 100.0 118.55 119.01 127.23 5077 73.28 27.76 &a+180H
081 8/25/82 110.0 116.18 116.58 127.28 5079 73.34 27.78 &a+180H
059 7/9/82 125.0 111.18 111.53 127.37 5082 73.43 27.81
073 8/26/82 125.0 111.15 111.50 127.35 5081 73.40 27.80
156 9/17/85 125.0 110.83 111.18 &dD127.10&d@ &dD5071&d@ &dD73.1&d@&dD1&d@ &dD27.69&d@ &a+180H
&a+180H
127.42 5084 73.49 27.84 &a+180H

(0.22) (9) (0.25) (0.10) &a+180H

TABLE 20(s3B (CONTINUED)(s0B Notes:
1. Tuned lengthA is the actual horn tuned length at 19950 Hz with a í12.7 x 19.0 deep stud. The minimum tuning correlation (horn tuned length versus frequency) was -0.9989 for all horns. 2. Tuned lengthB is the horn half-wavelength at 19950 Hz after correcting for the effect of the stud. 3. âO was calculated using the Mori equation (Æ32µ). 4. HRD-438 was machined from HRD-437. 5. Density = 2843 kg/m3 (to calculate E). 6. Poisson's ratio = 0.32 (from curve-fit of Mori equation to data in table 20; required in order to calculate G). 7. Numbers in each column directly below the solid line are averages. The number in parentheses are the 95% confidence interval of the mean.

D-2 TOOL STEEL
A converter driven resonant test have been performed on a full- wave D-2 S-N horn. The results are summarized in table 21. Note that this horn has been heat-treated to Rc 54 according to Branson's specifications. As compared to the non-heat-treated state, the heat-treatment always reduces the thin-wire wave speed somewhat (on the order of 1%).

TABLE 21
&a+180H
D-2 TOOL STEEL ROD STOCK, Rc 54: &a+180H
LONGITUDINAL MATERIAL PROPERTIES &a+180H
(CONVERTER RESONANCE TESTS, FULL-WAVE HORNS) &a+180H(0U(s0p12h10vs3b3T Horn Tuned &a+180H(0U(s0p12h10vs3b3TC&a+45VO,L&a-45V E&a+45VL&a-45V G &a+180H(0U(s0p12h10vs3b3T HRD # Date Shape Length (mm) â&a+45VO&a-45V (mm) &a+180H(0U(s0p12h10vs3b3T(m/sec) (GPa) (GPa) &a+180H
411 3/31/81 A 130.51 130.70 5210 208.2 80.08 &a+180H

&a+180H

Notes:
1. Shapes: 20 kHz (nominal) full-wave horns
A --> S-N horn, í19.0 nominal front diameter 2. Tuned length is the actual horn tuned length at 19931 Hz. 3. âO is the thin-wire half-wavelength calculated from the Mori equation at 19931 Hz.
4. Poisson's ratio = 0.30 (assumed for Mori equation). Density = 7671 kg/m3 (to calculate E).

&a+180H
REPEATABILITY
For comparison to other property measurement methods, we need to determine the experimental repeatability of the converter driven resonance tests. By experimental repeatability, I mean the ability to achieve consistent test results when the significant test conditions are reproduced as closely as possible. I am deliberately excluding variability associated with the material itself, since this condition is not directly under our control. For full-wave converter driven resonance tests, we would like to know how the data would vary if the test could be repeated many times on the same horn. Of course, we cannot repeatedly test a single horn because it is essentially destroyed when the front half-wave section is machined away. However, the linear regression data for a single horn can be analyzed statistically to provide some conclusions about the experimental repeatability. For each full-wave test, what we would like to know is the repeatability of the difference between the full-wave and half- wave linear regression lines, from which the thin-wire wave speed can be calculated. For all of the full-wave tests of titanium, aerospace aluminum, and D-2 tool steel, the average 95% confidence interval for repeatability of CO is ñ9 m/sec. Thus, on average, if a single horn could be tested 100 times, then 95 of these tests for that particular horn would give a wave speed within ñ9 m/sec of the value that was actually measured by a single test. Since the wave speed is typically about 5200 m/sec, the above experimental variation is only ñ0.17% of this value, which is quite good. We will see later that the high-frequency NDT tests were not as experimentally repeatable as the converter- driven full-wave tests. (Admittedly, however, the NDT tests were rather quickly done.)

ELECTROSTATIC RESONANT Tests
In this section we will discuss the method and results of electrostatic resonant tests, where a half-wave section is driven at its resonant frequency by electrostatic means. Method
In our discussion of converted driven resonant tests, we noted that precise results were not possible with half-wave resonators because of the effects of the converter and stud. To eliminate these problems, a full-wave test horn was used. Another way to eliminate these problems is to eliminate the converter and stud altogether. This can be done by driving the test sample with a noncontact means, either electrostatically or electromagnetically, depending on the material. (See Fine, pp. 49 - 50; McSkimin, pp. 304 - 306. Also see Hill for some experimental data on pure titanium and vanadium-titanium alloys.) The method described below uses an electrostatic drive source. .f.Figure Õ29¸®Ÿelastic\electro.dwg¯®electrostatic resonance equipment;¯ shows the equipment used to electrostatically drive a half-wave resonator. A cylindrical test sample with flat ends is lightly clamped at its midpoint. At a small gap from one end of the sample, a flat disk is connected to a variable frequency voltage source, which excites the sample electrostatically. At the other end of the sample, an identical (but passive) disk is used to monitor the amplitude of the sample according to the change in capacitance across the gap. (Note: Other equipment, such as the Fotonic sensor, could be substituted to monitor the sample amplitude.)
As the frequency of the driving voltage is adjusted, the amplitude of the sample will reach a maximum at resonance. Thus, by monitoring the amplitude, the resonant frequency can be determined. The lowest possible resonant frequency is the half- wave resonance -- i.e., the sample is just one half wavelength long with a node at the midpoint. Now we have determined the resonant frequency for a half-wave sample of length â and diameter d. To determine the thin-wire half-wavelength âO, the Mori equation (Æ32µ) is used to correct the length â to account for the finite rod diameter d. Knowing the resonant frequency and the thin-wire half-wavelength, the thin-wire wave speed can be calculated from equation Æ29µ. The test results given in tables 22 and 23 are from tests performed at TIMET (Jim Hall, Supervisor of Metallurgy) on 2/9/77, which I was present to verify. Samples were approximately í12.7 x 76.2 mm long. Rod stock samples were machined in the longitudinal direction. Bar stock samples were machined in both the long-transverse and longitudinal material directions, with samples taken from both the middle and ends of the bar to check for material nonhomogeneity. 7-4 Titanium Rod Stock
Table 22 shows the results for two samples machined from í63.5 rod stock. To determine the repeatability of the measurements, the tests on sample B were completely repeated four times. As

can be seen from the very small 95% confidence interval, the tests showed very little data scatter. Assuming that this repeatability is representative of all of the electrostatic tests, then the "actual" thin-wire wave speed of a given sample should fall within ñ1 m/sec of the experimentally measured value, at the 95% confidence level. (The "actual" thin-wire wave speed assumes that there are no biasing errors that will make the electrostatic measurements either consistently too high or too low.) Note that the above confidence interval is applied to a particular sample. If sample-to-sample variations in CO are larger than ñ1 m/sec, these differences should then be attributed to inherent differences in the material properties of the samples.
Averaging sample A with the average value of sample B gives the following:
48) CO = 5250 m/sec
49) EL = 123.0 GPa
This thin-wire wave speed is 4.2% higher than the average of the converter full-wave resonance tests for titanium rod (table 13). Young's modulus is 8.6% higher than the average of the converter full-wave resonance tests.
7-4 Titanium Bar Stock
Table 23 shows the test results for titanium bar-stock tests in the long-transverse and longitudinal material directions. As indicated by the 95% confidence interval of the mean, both of these material directions show considerable data scatter. Because of the high testing repeatability shown in the electrostatic tests of 7-4 rod, it must be assumed that the 7-4 bar data variability is due to actual variations (nonhomogeneity) in the material properties, rather than testing errors. The material properties do not show any consistent pattern from the ends to the center of the forging. Compared to the converter driven resonance tests (Tables 11 and 12), the electrostatic wave speed is 2.5% lower in the long- transverse material direction and 4.4% higher in the longitudinal direction. Similarly, Young's modulus is 4.9% lower in the long- transverse direction and 9.2% higher in the longitudinal direction.
(Note: NDT measurements were later made on samples D, F, and H.) Interpreting the data
Neither the rod nor the bar stock tests are consistent between the electrostatic and the converter-driven resonance tests. If both the bar and rod results had been either consistently higher or lower than the converter tests, we might suspect that the electrostatic tests were biased. However, this is not the case. Also, the simplicity of the tests would tend to rule out biasing, assuming that the test instruments themselves (frequency meters, calipers) are not biased.
Most likely, the differences between the electrostatic and converter-driven tests can be attributed to heat-to-heat variations in the material. (See the section "Introduction" -->

"Heat-to-Heat and Bar-to-Bar Variations".) It is also possible that the method of manufacturing the titanium changed slightly between the time of the electrostatic tests (1977) and the converter driven resonance tests (1980 or later). EQUIPMENT PROBLEMS
In order to test additional material samples, the Branson R&D lab tried to duplicate the equipment setup used at TIMET. This could not be done, although the reason is unknown.

TABLE 22
&a+180H
TIMET Ti-7Al-4Mo ROD MATERIAL PROPERTIES &a+180H
(ELECTROSTATIC RESONANCE TESTS) &a+180H(0U(s0p12h10vs3b3T Material Length Freq &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V E &a+180H(0U(s0p12h10vs3b3T Part Direction Location (m) (Hz) &a+180H(0U(s0p12h10vs3b3T(m/sec) (GPa)
A L --- 76.15 34378 5245 122.8 &a+180H
B L --- 76.15 34431 5254 123.2
B L --- 76.15 34426 5253 123.2
B L --- 76.15 34434 5254 123.2
B L --- 76.15 &dD34426&d@ &dD5253&d@ &dD123.2&d@ &a+180H
34429 5254 123.2 &a+180H
(6) (1) (0.0)
&a+180H
Notes:
1. All samples came from the same rod. 2. Material direction:
L --> Longitudinal
3. CO (thin wire wave speed) calculated from the Mori equation (Æ32µ).
4. Density = 4464 kg/m3 (Average from other tests.) 5. Numbers in each column directly below the solid line are averages. Numbers in parentheses are the associated 95% confidence interval of the mean.

TABLE 23
&a+180H
TIMET Ti-7Al-4Mo BAR MATERIAL PROPERTIES &a+180H
(ELECTROSTATIC RESONANCE TESTS) &a+180H(0U(s0p12h10vs3b3T Material Length Freq &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V E &a+180H(0U(s0p12h10vs3b3T Part Direction Location (m) (Hz) &a+180H(0U(s0p12h10vs3b3T(m/sec) (GPa)
C L Middle 76.96 33747 5204 120.9
D L End 77.47 34773 5398 130.1
E L End 77.37 34096 &dD5286&d@ &dD124.7&d@ &a+180H
5290 125.2
&a+180H
(246) (11.4)
&a+180H
F LT Middle 77.39 33356 5172 119.4
G LT End 77.34 34010 5270 124.0
H LT End 76.37 33293 &dD5161&d@ &dD118.&d@&dD9&d@ &a+180H
5201 120.8
&a+180H
(149) (7.0)
&a+180H
Notes:
1. All samples came from the same bar. 2. Material direction:
LT --> Long-transverse
3. CO (thin wire wave speed) calculated from the Mori equation (Æ32µ).

HIGH FREQUENCY (NDT) Tests
In this section we will look at tests of material properties using high frequency NDT (nondestructive testing) equipment. Since both the dilitational and shear wave speeds are measured, Poisson's ratio can be calculated if the material is assumed isotropic.
Theory
In the introductory section on material properties, one of the material properties that we listed was the dilitational wave speed C&a+45Vd&a-45V. For a dilitational wave, the lateral dimensions of the propagation medium must be very large in comparison to the wavelength.
In the frequency range of power ultrasonics (20 to 50 kHz), the lateral dimensions of the resonator are less than half of the wavelength. Thus, it is not practical to generate a dilitational wave at these frequencies. However, since the wavelength is related to the frequency (see equation Æ29µ), we can achieve the required short wavelength by increasing the frequency. If the frequency is increased to from 20 kHz to 10 megahertz (10&a-45V6&a+45V Hz = 10 MHz), the wavelength decreases by a factor of 500. Thus, at 10 MHz we can easily make reasonably sized samples whose lateral dimensions are large compared to the 10 MHz wavelength, for which dilitational waves can then be generated and measured. Waves in the megahertz frequency region can be generated using ultrasonic nondestructive test (NDT) equipment. Why would we be interested in determining C&a+45Vd&a-45V if such a wave can only be generated in the megahertz frequency range? Well, let's take another look at equations Æ3µ and Æ5µ (rewritten below):
Ú (1 - nu) ¿1/2 3) C&a+45Vd&a-45V = CO ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³
À (1 + nu) (1 - 2 nu) Ù
Ú 1 ¿1/2
5) C&a+45VS&a-45V = CO ³ ÄÄÄÄÄÄÄÄÄÄ ³
À 2 (1 + nu) Ù

We see that there are two equations in four unknowns:
C&a+45Vd&a-45V = dilitational wave speed
C&a+45VS&a-45V = shear wave speed
CO = thin-wire wave speed
nu = Poisson's ratio Since the NDT technique allows us to determine both C&a+45Vd&a-45V and C&a+45VS&a-45V, we can then solve the equations for CO and nu, which are needed for horn calculations. You will recall that we could not determine nu from our earlier resonance tests, except by curve-fitting the Mori equation, for which the result was still approximate. Thus, NDT has a potential advantage over the resonance tests. To make our calculations more convenient, let us define á (beta) as the square of the ratio of C&a+45VS&a-45V to C&a+45Vd&a-45V:
Ú C&a+45VS&a-45V ¿&a-45V2&a+45V 50) á = ³ -- ³
À C&a+45Vd&a-45V Ù
á can be calculated from the results of NDT tests. Note that á is the square of the ratio of CS and Cd. For acoustic materials, beta will typically be between 0.2 and 0.3. Solving equations Æ3µ and Æ5µ for nu and expressing the result in terms of á gives:
Ú 1 - 2 á ¿
51) nu = ³ ÄÄÄÄÄÄÄÄÄ ³
À 2 (1 - á) Ù
Substituting equation Æ51µ into equation Æ3µ gives CO in terms of á and Cd.
Ú á (3 - 4á) ¿1/2 52) CO = Cd ³ ÄÄÄÄÄÄÄÄÄÄ ³
À (1 - á) Ù
Method
To determine the wave speed with NDT, a transducer is coupled to one end of the sample with a thin layer of grease. The transducer generates a high-frequency pulse of very short duration into the sample. The pulse type (either longitudinal or shear) will be determined by the transducer design. The time required for the pulse to travel to the far end of the sample and return to the transducer is measured. By measuring the length of the sample, the wave speed can be determined by —
Ú 2 * Sample_length ¿
53) C = ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³
À Transmission_time Ù
If the pulse was a longitudinal type, then C will represent the dilitational wave sped Cd. If the pulse was a shear type, then C will represent the shear wave speed CS. (Note: The above is a general description of the NDT technique. Many variations provide added sophistication. See McSkimin, pp. 312 - 317.) Orthotropic materials
For an isotropic material such as aluminum, Cd and CS should be constant (excluding experimental errors), regardless of the direction in which the material is tested. For an orthotropic material like titanium, the situation is much more complex. For

each direction in which the material is tested, there will be one value of Cd and two possible values of CS, corresponding to the two mutually perpendicular directions of particle motion in a shear wave. (See D. A. Berlincourt, p. 193.) Thus, in orthotropic materials, the direction of shear wave particle motion is important. For materials that are markedly orthotropic, the above equations may give results with considerable error. (See Richards, pp. 91 - 92.) Aluminum test results
When we earlier ran the converter resonance material tests using full wave horns, the unshaped front half-wave section had been cut off. Those front wave sections were not wasted, but were saved for use in the high frequency NDT tests. The results of the NDT aluminum tests are given in table 24. In comparing the CO values calculated from the NDT tests with the those from the full-wave converter resonance tests (see Tables 16 and 18), the NDT tests have consistently lower CO values, coupled with considerably higher 95% confidence interval. Also, material A shows much greater difference in CO between the long-transverse (LT) and longitudinal (L) material directions with the NDT tests than with the resonance tests. Such a difference in CO is unexpected, since aluminum is usually considered to be relatively isotropic, for which CO should be the same in all directions. The calculated Poisson's ratios are slightly higher than the values given in most handbooks (typically 0.33 to 0.34). Titanium bar stock results
High frequency NDT tests have been performed on TIMET 7-4 titanium bar stock. (See table 25.) The samples include not only the cylindrical front sections from converter-driven full- wave tests but also rectangular sections machined from the fronts of two bar horns. (Samples HRD-412 ... HRD-416 were all machined from the first bar horn. Samples HRD-417 ... HRD-421 were all machined from the second bar horn.)
Although the dilitational and shear wave velocities have been measured, the interpretation of the results is unclear. This is because titanium is orthotropic, for which equations Æ3µ and Æ5µ (or Æ51µ and Æ52µ) no longer apply. However, the data is still useful, since it gives some indication of the repeatability of testing a single sample several times. (This is discussed separately below.) We can also see if the Cd measurements make sense as compared to the CO values calculated earlier for the same samples using the converter-driven resonance method. Dilitational wave speed, C&a+45V&dDd&a-45V&d@ From the previous converter-driven resonance tests for titanium bar stock, we found that the thin-wire wave speed CO was higher in the long-transverse material direction than in the longitudinal direction. (See tables 11 and 12.) Does this dependence on material direction repeat for the dilitational wave speed Cd?
The bottom of the fourth column of tables 25 and 26 gives the average results for Cd in the long-transverse and longitudinal

material directions. These averages were determined by considering the means of the measurements for each of the samples (i.e., the values below the solid lines). For example, in the longitudinal material direction, the average dilitational wave speed Cd,L was calculated from the values 6134, 6050, 6110, 6149, 6177, and 6198.
If we compare the average of all of the dilitational wave speed measurements, we see that the long-transverse material direction has a higher average (Cd,LT = 6249 m/sec) than the average in the longitudinal material direction (Cd,L = 6136 m/sec). Thus, the trend of wave speed dependence on the material direction agrees with the thin-wire wave-speed results (CO) of both the static tests and converter resonance tests. Shear wave speed
At the time of the shear wave measurements, no consideration was given to the direction of particle motion in the shear wave. This will have introduced unknown variability into the results and, thus, little can be concluded from these measurements. Miscellaneous
One measurement was made for a sample oriented 45Ø to both the longitudinal and long-transverse material directions. This sample (HRD-416) gave a dilitational wave speed of Cd = 6177 m/sec and a shear wave speed of CS = 3345 m/sec. (Note: this sample was machined from the front of the same horn as samples HRD-412 ... HRD-415.)
D-2 tool steel results
The results of D-2 tool steel tests are given in table 27. The thin-wire wave speed gives excellent agreement with the results previously obtained for full-wave resonance tests (table 21). The calculated Poisson's ratio is close to the nominal values given in most handbooks (typically 0.29 to 0.30). Repeatability of NDT C&a+45V&dDD&a-45V&d@ measurements From the multiple testing of titanium bar samples, we can estimate the repeatability of the Cd measurements on the individual samples, as indicated by the 95% confidence interval of the data. Considering those samples for which multiple Cd tests were run (HRD-397, 399, 400, 414, and 421), the average 95% confidence interval is \( \pm \)42 m/sec, which is 0.68% of the average dilitational wave speed for these samples. (Remember that the 95% confidence interval is the interval into which the wave speed measurement should fall in 95 out of 100 tests, on average. For example, if we were to test HRD-397 100 times under "identical" conditions, we should expect 95 of the wave speed measurements to fall within (6224 \( \pm \)42) m/sec, while 5 of the measurements would be outside of this range.) Because of the small number of repeat tests of each sample, the average confidence interval calculated above is subject to some error. None-the-less, the 0.68% value is still almost 4 times as large as for the converter full-wave resonance tests (0.18%). Validity of NDT C&a+45V&dDD&a-45V&d@ measurements

The dilitational wave speed was defined as the speed of a pressure wave whose wavelength is very small in comparison to the lateral dimensions of the propagation medium. The dilitational wave speed of these samples has been around 6300 m/sec, so that a frequency of 10 MHz would give a wavelength of: 54) Wavelength = Cd / f
&a+180H
&a+180H
6300 m/sec
= ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
10*106 cycle/sec &a+180H
&a+180H
= 6.3 * 10-5 m &a+180H
= 0.063 mm
The lateral dimensions of these samples was about 13 mm, which is about 200 times larger than the wavelength. Thus, the sample dimensions seem adequate.

TABLE 24
&a+180H
VARIOUS ALUMINUMS:
&a+180H
HIGH FREQUENCY NDT TESTS
&a+180H(0U(s0p12h10vs3b3T Material C&a+45Vd&a-45V C&a+45VS&a-45V &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3THRD # Date Material Direction (m/sec) (m/sec) á &a+180H(0U(s0p12h10vs3b3Tnu (m/sec) % Difference
402 4/2/81 A LT 6335 3134 0.245 0.338 5124 -0.76%
404 4/2/81 A LT &dD6373&d@ &dD3155&d@ &dD0.245&d@ &dD0.338&d@ &dD5156&d@ -&dD0.25%&d@
6354 3144 0.245 0.338 5140 -0.50%
(242) (135) (0.0) (0.0) (23)
&a+180H
403 4/2/81 A L 6370 3081 0.234 0.347 5065 -2.24%
405 4/2/81 A L &dD6363&d@ &dD3053&d@ &dD0.230&d@ &dD0.350&d@ &dD5020&d@ -&dD3.06%&d@
6366 3067 0.232 0.348 5042 -2.65%
(45) (180) (0.026) (0.018) (287)
&a+180H
407 3/6/81 B LT 6248 3034 0.237 0.344 4999 -0.89%
408 3/6/81 B LT &dD6&d@&dD251&d@ &dD3063&d@ &dD0.240&d@ &dD0.342&d@ &dD5018&d@ -&dD1.27%&d@
6250 3048 0.489 0.343 5008 -1.08%
(18) (189) (0.019) (0.009) (117)
&a+180H
Notes:
1. Date refers to the date when the original converter full-wave resonance tests were run. (See tables 16 and 18.) All NDT tests were run on 6/18/81.
2. Material:
A --> ALCOA 2024-T3 Bar
B --> Martin Marietta 7075 QQA225 bar 3. Material direction:
LT --> Long-transverse
L --> Longitudinal
4. HRD-402 and HRD-403 are from the same bar. HRD-404 and HRD-405 are from the same bar. HRD-407 and HRD-408 are from the same bar.

5. % Difference is the percent difference between the CO value in this table and the corresponding CO value from full-wave resonance tests in tables 16 and 18.

TABLE 24 (CONTINUED)
6. Numbers in each column directly below a solid line are averages. Numbers in parentheses are the associated 95% confidence interval of the mean.

TABLE 25
&a+180H
TIMET Ti-7Al-4Mo BAR
&a+180H
LONG-TRANSVERSE MATERIAL DIRECTION: &a+180H
HIGH FREQUENCY NDT TESTS
C&a+45Vd&a-45V,LT C&a+45VS&a-45V
&dDHRD #&d@ &dDDate&d@ &dDTest #&d@ &dD(m/sec)&d@ &dD(m/sec)&d@
397 2/18/81 Cd-1 6226
Cd-2 6248
Cd-3 6210
Cd-4 &dD6210&d@
6224
(57) &a+180H
CS-1 3282
CS-2 &dD3269&d@
3276
(116) &a+180H
398 2/18/81 Cd-1 6261 &a+180H
CS-1 3213
CS-2 3165
CS-3 3165
CS-4 &dD3165&d@
3177
(76) &a+180H
412 6/17/81 -- 6246 3238 &a+180H
413 6/17/81 -- 6251 3228 &a+180H
417 6/17/81 -- 6253 3205 &a+180H
418 6/17/81 -- 6251 3205 &a+180H
419 6/17/81 -- 6259 3345 &a+180H
ÍÍÍÍ ÍÍÍÍ
6249 3239
(11) (52)

TABLE 25 (CONTINUED)
Notes:
1. HRD-397 and HRD-398 are the remaining front sections from the associated full-wave converter resonance tests at 40 kHz. Dimensions are approximately í19.0 x 55 mm long. The dates refer to when the original resonance tests were run. (See table 11.) 2. HRD-412 and HRD-413 were both machined from the front of the same bar horn on 6/17/81. Dimensions are approximately 12.7 mm square x 40 mm long.
3. HRD-417, HRD-418, and HRD-413 were all machined from the front of the same bar horn on 6/17/81. Dimensions are approximately 14.3 mm square x 27 mm long.
4. A Cd-X test number indicates the Xth test for dilitational wave speed for the associated sample -- e.g., Cd-2 for HRD-397 indicates the second dilitational test on HRD-397. Similarly for CS-X. A "--" in the "Test #" column indicates that only a single test was run.
5. All NDT tests were run on 6/18/81. 6. Numbers in each column directly below a single solid line are averages for the sample. Numbers in parentheses are the associated 95% confidence interval of the data. 7. Numbers at the bottom of each column directly below the double (ÍÍÍÍ) line are averages for all of the samples. Numbers in parentheses are the associated 95% confidence interval of the mean.

TABLE 26
&a+180H
TIMET Ti-7Al-4Mo BAR
&a+180H
LONGITUDINAL MATERIAL DIRECTION: &a+180H
HIGH FREQUENCY NDT TESTS
C&a+45Vd&a-45V,L C&a+45VS&a-45V
&dDHRD #&d@ &dDDate&d@ &dDTest #&d@ &dD(m/sec)&d@ &dD(m/sec)&d@
399 3/17/81 Cd-1 6134
Cd-2 &dD6134&d@
6134
(0) &a+180H
CS-1 3142
CS-2 3157
CS-3 &dD3152&d@
3150
(33) &a+180H
400 3/17/81 Cd-1 6050
Cd-2 &dD6050&d@
6050
(0) &a+180H
CS-1 3056
CS-2 &dD3053&d@
3054
(27) &a+180H
414 6/17/81 Cd-1 6116
Cd-2 &dD6104&d@
6110
(108) &a+180H
CS-1 3180
CS-2 &dD3139&d@
3160
(368)

TABLE 26(s3B (CONTINUED)(s0B
C&a+45Vd&a-45V,L C&a+45VS&a-45V
&dDHRD #&d@ &dDDate&d@ &dDTest #&d@ &dD(m/sec)&d@ &dD(m/sec)&d@
415 6/17/81 Cd-1 6149 &a+180H
CS-1 3198
CS-2 3167
CS-3 &dD3160&d@
3175
(87) &a+180H
420 6/17/81 Cd-1 6177 &a+180H
CS-1 3190
CS-2 &dD3195&d@
3192
(45) &a+180H
421 6/17/81 Cd-1 6200
Cd-2 &dD6195&d@
6198
(45) &a+180H
CS-1 3254
CS-2 3279
CS-3 3249
CS-4 &dD3244&d@
3256
(49) &a+180H
ÍÍÍÍ ÍÍÍÍ
6136 3165
(55) (69)

TABLE 26(s3B (CONTINUED)(s0B Notes:
1. HRD-399 and HRD-400 are the remaining front sections from the associated full-wave converter resonance tests at 40 kHz. (See table 12.) Dimensions are approximately í19.0 x 55 mm long. The dates refer to when the original resonance tests were run. 2. HRD-414 and HRD-415 were both machined from the front of the same bar horn on 6/17/81. Dimensions are approximately 12.7 mm square x 31 mm long.
3. HRD-420 and HRD-421 were both machined from the front of the same bar horn on 6/17/81. Dimensions are approximately 14.3 mm square x 39 mm long.
4. A Cd-X test number indicates the Xth test for dilitational wave speed for the associated sample -- e.g.,Cd-2 for HRD-399 indicates the second dilitational test on HRD-399. Similarly for CS-X.
5. All NDT tests were run on 6/18/81. 6. Numbers in each column directly below a single solid lines are averages for the sample. Numbers in parentheses are the associated 95% confidence interval of the data. 7. Numbers at the bottom of each column directly below the double line (ÍÍÍÍ) are averages for all of the samples. Numbers in parentheses are the associated 95% confidence interval of the mean.

TABLE 27
&a+180H
D-2 TOOL STEEL ROD STOCK, Rc 54: &a+180H
HIGH FREQUENCY NDT TESTS
&a+180H(0U(s0p12h10vs3b3T Material C&a+45Vd&a-45V C&a+45VS&a-45V &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3THRD # Date Material Direction (m/sec) (m/sec) á &a+180H(0U(s0p12h10vs3b3Tnu (m/sec) % Difference
411 3/31/81 Rod L 5972 3245 0.295 0.291 5207 -0.06% &a+180H
&a+180H
Notes:
1. Date refers to the date when the original converter full-wave resonance tests were run. (See table 21.) The NDT test was run on 6/18/81.
2. Material direction:
L --> Longitudinal
3. % Difference is the percent difference between the CO value in this table and the corresponding CO value from full-wave resonance tests in table 21.

FINITE ELEMENT ANALYSIS
In this section we will look at how finite element analysis (FEA) can be used to determine Poisson's ratio when the other material properties are already known.
Converter-driven resonance tests and electrostatic tests give valid results for Young's modulus if the samples are thin in comparison to the wavelength, so that Poisson's ratio has little effect on the calculations. Unfortunately, these tests are not well suited to determining Poisson's ratio. However, if Young's modulus has been determined by other methods, then FEA may permit an estimate of Poisson's ratio, provided that a proper horn is chosen for FEA modeling.
procedure
We need to find some acoustic parameter (amplitudes, frequencies, losses, etc.) that is sensitive to Poisson's ratio. Using this parameter, we then compare empirically measured values of this parameter against the values predicted by FEA using various values of Poisson's ratio. The best agreement between empirical and FEA indicates the best estimate of Poisson's ratio. One parameter that is sensitive to Poisson's ratio is the horn amplitude, especially for larger horns. (See the chapter on "Amplitude".) Other possible parameters are the horn tuned length or the resonant frequency. Of these parameters, the amplitude is the most sensitive, although this depends on the horn design. (See below.)
Thus, the procedure is —
1. Choose the material whose Poisson's ratio you wish to determine. The material must be reasonably isotropic. Otherwise, at least three values of Poisson's ratio will have to be simultaneously determined, which will not be easy. 2. Choose a horn design whose amplitude distributions show significant variation with Poisson's ratio. This choice is mainly by trial and error. However, large, unslotted horns (either cylindrical or bar) are good candidates. 3. Machine and tune the horn.
4. Measure amplitudes at various locations on the horn. Use particular care to insure that these measurements are accurate. If the horn has significant amplitude asymmetry, then it should not be used because the amplitudes may not compare well with those predicted by FEA, regardless of Poisson's ratio. 5. Model and analyze the tuned horn by FEA for an estimated value of Poisson's ratio. Use the previously determined values of Young's modulus and density.
6. Make sure that the FEA results have converged by running mesh convergence studies.
7. Once an appropriate mesh has been found, choose various values of Poisson's ratio until the FEA amplitudes agree most closely empirical amplitudes. This is the estimated value for Poisson's ratio.
This procedure will be illustrated for Aerospace aluminum using unslotted cylindrical horns.

í125 spool horn
.f.Figure 30 shows a í125 spool horn (HRD-059-10, 7/20/82) made of Aerospace aluminum. Because the horn is cylindrical, I assume that it was made from rod stock, although this has not been documented. (It might possibly have been made from plate stock.) Using an A-200A, the horn had an axial resonance at 19943 Hz and axisymmetric nonaxial resonances at 22427 Hz and 23495 Hz. (Note: the horn may have had additional axisymmetric resonances above 24 kHz, but these were not checked. The horn also had several asymmetric modes which have not been listed.) Figure Õ30¸®Ÿhorn\059_10.hrn¯ shows the locations where the horn amplitudes were measured using an A-450 amplitude meter. The axial and radial edge amplitudes were measured with the A-450 probe centered exactly 3.8 mm from the horn's edge, using a special probe boot (figure 31). Such accurate probe positioning is necessary where amplitudes change rapidly over the horn's surface. This will be seen, for example, when we discuss figures 32 and 33. Multiple amplitude measurements have been made around the face, back surfaces, and sides to assure that the horn does not have significant asymmetry and to obtain a better estimate of the "true" amplitude at these locations. To facilitate comparison to subsequent FEA results, a "relative amplitude" parameter will be used:
Amplitude at specified location 55)  = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
_&a+2448H&l1C
&l8C
_&a+2448H&l7C
&l8C
Face centerline amplitude where the double underline indicates relative amplitude. Where multiple amplitude measurements have been made, the "amplitude at specified location" is the average of these multiple measurements. For example, eight axial amplitudes were measured at the edge of the horn face (20.4, 20.3, 20.8, 20.9, 20.4, 20.4, 20.9, and 20.9 microns), for an average of 20.625 microns. Since the face centerline amplitude was 20.2 microns, the relative face axial amplitude at the horn edge is —
20.625
56)  = ÄÄÄÄÄÄ
_ &l1C
&l8C
_ &l7C
&l8C
20.2
&a+180H
&a+180H
= 1.021
These relative amplitudes are labeled as "empirical data" in subsequent graphs. (See figure Õ33¸®Ÿ125pl3.plt¯ for the above empirical data point.)
.f.Figure Õ34¸®Ÿfea\h059c28a.fea¯®fea model of í125 aero al spool horn with 400 series converter;¯ shows a FEA axisymmetric model of this horn joined to a 400 series converter. The shown horn mesh gives converged FEA results. (See later discussion.) Figure Õ32¸®Ÿfea\z125spl3.plt¯ shows the horn face axial FEA amplitude distribution from the horn centerline to the edge of the horn for nu between 0.32 and 0.35. Note the significant change in face amplitude with nu. Figure Õ33¸®Ÿfea\z125spl4.plt¯

shows FEA amplitude distribution on the stud surface of the í125 spool horn.
.f.Figures Õ35¸®Ÿfea\z125spl5.plt¯®EFFECT OF POISSON'S RATIO ON RELATIVE FACE RADIAL AMPLITUDE OF 125 MM AERO AL SPOOL HORN;¯ and .f.Õ36¸®Ÿfea\z125spl6.plt¯®EFFECT OF POISSON'S RATIO ON RELATIVE BACK RADIAL AMPLITUDE OF 125 MM AERO AL SPOOL HORN;¯ show the face and back radial amplitudes. Note that the face has 60% higher radial amplitude than the stud surface. Correspondingly, the face axial amplitudes show a much greater effect of Poisson's ratio than do the axial amplitudes of the stud surface. (Compare figures Õ33¸®Ÿfea\z125spl3.plt¯ and Õ34¸®Ÿfea\z125spl3.plt¯.) .f.Figure Õ37¸®Ÿfea\f125spl1.plt¯®EFFECT OF POISSON'S RATIO ON RESONANT FREQUENCIES OF 125 MM AERO AL SPOOL HORN;¯ shows the effect of nu on resonant frequencies. The effect is small for the axial mode -- the frequency decreases by 89 Hz as nu increases from 0.32 to 0.34. The nonaxial modes show a slightly greater frequency change. Thus, resonant frequency would not have been a good parameter for estimating nu. í110 spool horn
.f.Figure Õ38¸®Ÿhor\081_7.HRN¯®110 MM AERO AL SPOOL HORN;¯ shows a í110 spool horn (HRD-081-7, 7/21/83) made of Aerospace aluminum. .f.Figure Õ39¸ shows the FEA model. The horn mesh is essentially the same as for the í125 spool horn above. .f.Figure Õ40¸®Ÿfea\z110spl1.plt¯®EFFECT OF POISSON'S RATIO ON RELATIVE FACE AXIAL AMPLITUDE OF 110 MM AERO AL SPOOL HORN;¯ shows the face amplitude distribution. analysis
To determine the best value of nu, we can compare the measured amplitudes against those predicted by FEA. .f.Figure Õ41¸ shows the three locations where the axial amplitudes were measured on the í125 and í110 spool horns. For each location and for a specified nu, we can compute the FEA amplitude error as:
Ú FEA - MEASURED ¿ &a+2304H_ _ &l1C &l8C
&a+2304H_ _ &l7C &l8C
57) FEA amplitude error (%)= ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ * 100
À MEASURED Ù &a+2520H_ &l1C
&l8C
&a+2520H_ &l7C
&l8C
Then, averaging the errors at the three locations for a specified nu gives the overall amplitude error for the specified nu. Table 28 shows the results:

amplitudes -- either value of nu gives good agreement with the empirical data. Thus, we cannot use this horn design to determine the correct value of nu.

comparison among test methods
In this section we will compare the static test with the three dynamic tests (converter-driven resonance tests, electrostatic tests, and NDT tests), and compare the three dynamic tests among themselves.
static VERSUS DYNAMIC tests
Static tests give somewhat lower modulus because of temperature, strain rate, and mode-of-testing differences. Temperature considerations
In static tests, the strain rate is so slow that any generated heat is dissipated to the surroundings, so that the specimen remains at a constant temperature (isothermal condition). In contrast, the strain rate of dynamic tests is so rapid that the heat cannot dissipate, so that the temperature of the specimen increases (adiabatic condition). For cubic materials such as aluminum and steel, Fine (p. 60) gives the difference between Young's modulus for isothermal versus adiabatic tests as:
1 1 T à2
58) ÄÄÄ - ÄÄÄ = ÄÄÄÄ
EI EA p Cp
&a+180H
&a+180H
where EI = Young's modulus (isothermal)
EA = Young's modulus (adiabatic)
T = Absolute temperature
à = Coefficient of thermal expansion
p = Density
Cp = Specific heat at constant
pressure Assuming that the experiment is otherwise precisely controlled, then the temperature effects of dynamic testing will give slightly higher values for E than the static tests. For example, Young's modulus from dynamic aluminum tests should be about 0.6% higher than the static tests. (Also see Cady, p. 63; Richards, p. 94.)
Strain-rate effect
When a load is applied slowly as in static tests, the material tends to creep (slowly change length without any additional change in load). Thus, static measurements give a "relaxed" elastic modulus. With dynamic loading, however, the material does not have sufficient time to creep between the tensile and compressive cycles. Thus, dynamic measurements give an "unrelaxed" elastic modulus, which is higher than the static measurements. (See Richards, p. 94.) Tensile/compressive testing
Richards (p. 92) reports results of Bauschinger, which show that Young's modulus for steel is greater in compression than in tension. Similarly, RMI information (17, p. 5) shows that Ti-6Al-4V STA has a compressive modulus about 4% higher than the tensile modulus. For 2024-T852, the compressive modulus is about 2.7% higher than the tensile modulus in the longitudinal, long-

transverse, and short-transverse directions (22, vol. 3, code 3203, p. 13).
Dynamic tests, for which the specimen is stressed in both tension and compression, should give an average of the tensile and compressive moduli. Therefore, the dynamic modulus should be higher than the modulus of a static tensile test. Thus, all of the above factors should generally give a dynamic modulus that is larger than the static modulus. comparison among dynamic tests If we assume from the above discussion that dynamic tests are preferable to static tests, then we still have at least three choices: converter-driven full-wave resonator tests, electrostatic tests, and NDT tests. The converter-driven full- wave resonator tests have several advantages over the other two: 1. The test equipment is already available: it is same equipment that is used for tuning horns. 2. If S-N horns are used to determine the elastic material properties, then they can also provide data about the material fatigue and loss properties.
3. This method comes closest to testing the materials as they will be used in actual practice. 4. Using linear regression for frequency versus length moderates the effect of any erroneous data reading. The converter-driven full-wave resonator tests do have some disadvantages compared to the electrostatic and NDT tests: 1. Specimens (typically S-N shaped) are more complex. 2. Measurements (including tuning) are more time consuming. 3. Because of specimen length restrictions, measurements cannot be taken in some directions. For instance, full-wave horns cannot be made from bar stock in the short-transverse direction, whose dimension is typically less than 75 mm. 4. The test provides no information about Poisson's ratio, except as can be inferred from the Mori equation for specimens with large diameters. If FEA is available, then it can be used to estimate Poisson's ratio. Otherwise, you might possibly use the handbook values for Poisson's ratio. An incorrect estimate of Poisson's ratio will give small frequency errors, but possibly significant amplitude errors. (See the section on "Finite Element Analysis".)
For materials which are orthotropic, such as titanium, I would recommend static tests to determine Poisson's ratio. If the static tests also measure Young's modulus, then this can provide some corroboration of the converter full-wave resonance values. In my opinion, converter-driven full-wave resonator testing is the best method. It can be supplemented by finite element analysis to determine Poisson's ratio.

"BEST" ELASTIC MATERIAL PROPERTY VALUES Table 29 gives the current best estimates of the material properties for normal resonator materials, based on the converter-driven resonator tests. Remember that the table values are based on limited data. Especially for the titanium, further testing is needed to verify the elastic properties. Table 29 also lists values for the coefficient of thermal expansion (à). These have not been measured at Branson (see the listed references). They have been included for later use in the section "Temperature Dependence". Piezoelectric ceramic data has been included for use in the chapter on "Converters". The values have been taken from references of ceramic manufacturers. The listed ceramic values have given good results for FEA converter design.

TABLE 29
&a+180H
"BEST" MATERIAL PROPERTY VALUES &a+180H
AT ROOM TEMPERATURE

&a+180H(0U(s0p12h10vs3b3T Stock Material Density Poisson's &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V &a+180H(0U(s0p12h10vs3b3TMaterial Condition Type Direction à (1/ØC) (kg/m3(0U(s0p12h10vs3b3T) Ratio &a+180H(0U(s0p12h10vs3b3TE (GPa) G (GPa) (m/sec)

Aluminum 2024 Rod & Bar ÄÄÄ 22.7*10-6 2777 0.34 74.2 22.7 5170 &a+180H
Aluminum Aerospace Rod L 23.2*10-6 2840 0.333 73.5 27.4 5084 Aluminum Aerospace Bar L 23.2*10-6 2840 0.333 74.1 27.6 5131 &a+180H
Titanium 7-4 Rod L 9.5*10-6 4460 0.33 112.9 ---- 5037 Titanium 7-4 Bar L 9.5*10-6 4460 0.33 117.6 58.7 5138 Titanium 7-4 Bar LT 9.5*10-6 4460 0.30 126.6 62.5 5326 &a+180H
Steel D-2, Rc 54 Rod L 11.7*10-6 7670 0.29 208.2 80.1 5210 Steel 302 SS 17.3*10-6 &a+180H
Ceramic PZT-8 ÄÄÄ 33 ÄÄÄ 7600 0.30 118 45 3950 &a+180H
Notes:
1. Nomenclature:
L ÍÍ Longitudinal material direction
LT ÍÍ Long-transverse material direction
à ÍÍ Coefficient of thermal expansion
E ÍÍ Young's modulus (Modulus of elasticity)
G ÍÍ Shear modulus
CO ÍÍ Thin-wire wave speed 2. à is assumed to be independent of the material direction. The table values were taken from the following references: &a+180H
Aluminum: Byars, p. 371
(For temperature dependence of 2024, see 22, vol. 3,
code 3203, p. 14, figure 2.0142.) Titanium: Beer, p. 585
(For temperature dependence of Ti-6Al-4V, see 22,
vol. 4, code 3707, p. 9, figure 2.0141; for

temperature dependence of Ti-6Al-4V, see 22, vol. 4,
code 3708, p. 9, figure 2.014.) Steel: Beer, p. 585
3. E and CO are determined from converter-driven resonance tests. 4. The values for shear modulus and Poisson's ratio are the least reliable of this data.
5. For 7-4 titanium rod stock, E and CO may depend on the raw stock diameter. The values in the table are for a diameter of í50.
6. Piezoelectric ceramic data: The ceramic data is taken from handbooks. The density and Young's modulus (open circuit) are taken from Vernitron (pp. 28 - 29) at low electric field strength. Poisson's ratio is taken from Ferroxcube (pp. 18 - 19) for PZE41, probably at low field strength. The shear modulus and wave speed are calculated from these values. Ceramic properties are quite variable. (See the chapter on "Converters".)

TEMPERATURE DEPENDENCE
In this section we will look at the corrections that must be made to the material properties when the material is tested at a nonstandard temperature. (All of the above assumes that the tests were conducted at room temperature -- approximately 24ØC.) THEORY
We are primarily interested in the effect of a temperature change on the wave speed CO, since CO is fundamental to determining the frequency or tuned length. Since CO is given by equation 2:
Ú E ¿1/2
59) CO = ³ ÄÄÄ ³
À p Ù
we must determine the change in E and p as a function of temperature.
The change in Young's modulus due to a change in temperature is —
ëE
60) ET2 - ET1 = ÄÄÄ (T2 - T1)
ëT
&a+180H
&a+180H
where ET2 = Young's modulus at
temperature T2
ET1 = Young's modulus at
temperature T1
(ëE/ëT) = a constant which gives the
rate of change in the
modulus for each degree
change in temperature The above equation assumes that the modulus changes linearly with temperature, which is usually reasonable up to a certain maximum temperature determined by the material. Equation Æ60µ can be solved for ET2:
Ú Ú 1 ëE ¿ ¿ 61) ET2 = ET1 ³ 1 + ³ ÄÄÄ -- ³ (T2 - T1) ³
À À ET1 ëT Ù Ù Since (ëE/ëT) is a negative number, the modulus will decrease as the temperature increases. This makes sense, since materials are easier to stretch as they become hotter. Thus, increasing the temperature will decrease the modulus, which tends to decrease the wave speed CO (equation 59). Now that we have found Young's modulus at temperature T2, we need to find the density at this temperature:
pT1
62) pT2 = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
Ú ¿3
³ 1 + à (T2 - T1) ³
À Ù &a+180H
&a+180H
where pT2 = density at temperature T2
pT1 = density at temperature T1

à = coefficient of thermal
expansion The coefficient of thermal expansion à is a constant that depends on the particular material. à usually increases somewhat with temperature, but this will be ignored for the present discussion. (For the effect of temperature on à of 2024 aluminum, see 22, vol. 3, code 3203, p. 14, figure 2.0142. For the effect of temperature on à of Ti-6Al-4V see 22, vol. 4, code 3707, p. 9, figure 2.0141. For the effect of temperature on à of Ti-7Al-4Mo see 22, vol. 4, code 3708, p. 9, figure 2.014.) Since à is a positive number, the density will decrease as the temperature increases. This occurs because the volume expands as the temperature increases. Thus, increasing the temperature decreases the density, which tends to increase the wave speed CO (equation 59).
The net effect of temperature on the thin-wire wave speed CO can be found by substituting equations Æ61µ and Æ62µ into equation Æ59µ:

Ú ET2 ¿1/2
63) CO,T2 = ³ ÄÄÄ ³
À pT2 Ù
&a+180H
&a+180H
Ú ET1 ¿1/2 ÚÚ Ú 1 ëE ¿ ¿¿1/2
= ³ ÄÄÄ ³ ³³ 1 + ³ ÄÄÄ -- ³ (T2 - T1) ³³
À pT1 Ù ÀÀ À ET1 ëT Ù ÙÙ &a+180H
Ú ¿3/2
* ³ 1 + à (T2 - T1) ³
À Ù &a+180H
&a+180H
ÚÚ Ú 1 ëE ¿ ¿¿1/2
= CO,T1 ³³ 1 + ³ ÄÄÄ -- ³ (T&a+45V2&a-45V - T&a+45V1&a-45V) ³³
ÀÀ À ET1 ëT Ù ÙÙ &a+180H
Ú ¿3/2
* ³ 1 + à (T2 - T1) ³
À Ù The effect of modulus on wave speed is determined by the factor
Ú 1 ëE ¿
64) ³ ÄÄÄ -- ³
À ET1 ëT Ù
whose value is typically on the order of 0.001/ØC. The effect of density on wave speed is determined by the coefficient of thermal expansion à, whose value is typically on the order of 0.00001/ØC. Hence:
THE EFFECT OF TEMPERATURE ON WAVE SPEED (OR TUNED LENGTH, OR FREQUENCY) DERIVES PRIMARILY FROM THE CHANGE IN MODULUS. THE TEMPERATURE-RELATED CHANGE IN DENSITY (OR HORN LENGTH) HAS RELATIVELY LITTLE EFFECT ON WAVE SPEED (OR TUNED LENGTH, OR FREQUENCY).
Thus, ignoring the effect of density change, equation Æ63µ becomes:
Ú Ú 1 ëE ¿ ¿1/2 65) CO,T2 = CO,T1 ³ 1 + ³ ÄÄÄ -- ³ (T2 - T1) ³
À À ET1 ëT Ù Ù Now, if we know the material values CO and E at temperature T1 and if we also know the rate of change of modulus with temperature (ëE/ëT), then we can calculate the wave speed CO at any temperature T2. The temperature T1 will typically be taken as room temperature.
The factor given in equation Æ64µ will typically be less than 0.001. Thus, if the temperature difference T2 - T1 is limited to several hundred degrees, equation Æ65µ can be approximated as a linear equation:
ÚÚ 1 Ú 1 ëE ¿ ¿¿ 66) CO,T2 = CO,T1 ³³ 1 + ÄÄÄ ³ ÄÄÄ -- ³ (T2 - T1) ³³
ÀÀ 2 À ET1 ëT Ù ÙÙ

The slope of the line that relates CO to T is —
ëCO 1 Ú CO,T1 ëE ¿
67) ÄÄÄ = ÄÄÄ ³ ÄÄÄÄÄ -- ³
ëT 2 À ET1 ëT Ù
An exactly analogous equation to Æ66µ can be written for the effect of temperature on the shear wave speed CS, based on the change in shear modulus G with temperature:
ÚÚ 1 Ú 1 ëG ¿ ¿¿ 68) CS,T2 = CS,T1 ³³ 1 + ÄÄÄ ³ ÄÄÄ -- ³ (T2 - T1) ³³
ÀÀ 2 À GT1 ëT Ù ÙÙ Table 30 summarizes the effect of temperature on the properties to acoustic materials.
TITANIUM
Using the values in table 30 in equations 61, 62, and 63, .f.figure Õ46¸®Ÿelastic\temp_ti.plt¯®effect of temperature on properties of ti 7al-4mo;¯ shows the relative change in E, p, and CO with temperature. Figure Õ46¸®Ÿelastic\temp_ti.plt¯ extends only to 125 ØC, because normal acoustic operations are unlikely to exceed this temperature. However, the curves continue to be linear up to 650 ØC.
(Note: Because titanium is orthotropic, the curve of modulus versus temperature is affected by the direction in which the modulus is measured. The curve of figure Õ46¸®Ÿelastic\temp_ti.plt¯ uses an average titanium modulus value. The resulting error is small and can be neglected in comparison to the effect of the converter and booster on the stack frequency, which we have not considered.) As discussed above and as shown by figure Õ46¸®Ÿelastic\temp_ti.plt¯, the relative change in E is much greater than the relative change in density (about 20:1). Thus, the wave speed is mainly affected by the change in E. Note that the effect of temperature on wave speed CO is almost linear, as discussed in reference to equation 66. Also note that the rate of change of wave speed with temperature is just 1/2 of the rate of change of modulus with temperature. This is because of the square root in equation Æ65µ or the 1/2 factor in equation Æ66µ as compared to the modulus equation (Æ61µ). For Ti-7Al-4Mo (22, vol. 4, code 3708, p. 23), the change in the shear modulus with temperature is —
ëG
69) ÄÄÄ = -0.027 GPa/ØC
ëT
Example
Titanium bar (long-transverse) has a wave speed of 5326 m/sec at 25 ØC. What the wave speed will be if the temperature increases to 60 ØC?
From table 30, the change in wave speed with temperature is - 1.3 m/sec/ØC (or -0.025%/ØC). Thus, for a temperature increase of 35 ØC, the change in wave speed is -45 m/sec (or 0.875%). Thus, the resulting wave speed is approximately 5280 m/sec. Tuning rate

Table 30 shows that the wave speed changes at -0.025%/ØC. The resonator frequency (which is proportional to the wave speed, equation Æ29µ) will change at an equal rate. At 20 kHz, this would represent a frequency change of -5.4 Hz/ØC. This assumes that the resonator is at uniform temperature, and it ignores any outside influences. For instance, if the resonator is a horn, then the frequency change of -5.4 Hz/ØC ignores the effect of booster and converter, which are made of different materials and may be running at different (probably lower) temperatures. Thus, the actual effect of a temperature increase on the &dDstack&d@ frequency will probably be less than -5.4 Hz/ØC. Example
A horn similar to that shown in .f.figure Õ47¸®Ÿhorn\p&g2.bar¯®114 x 48 x 67 Eared Ti Bar Horn;¯ ran at a temperature at 60 ØC when driven by a 500 series silver booster in air (no load). (The ear width is probably 6.3.) The converter and booster ran at lower temperatures. Using the
-5.4 Hz/ØC value, the &dDhorn&d@ frequency should drop by about 190 Hz compared to room temperature (25 ØC). The measured frequency drop of the &dDstack&d@ (horn, booster, and converter) was 140 Hz. Thus, as expected, the stack frequency drop was less than the predicted horn frequency drop. STEEL
Garofalo tested a number of ferritic and stainless steels to determine the effect of temperature on the properties. The results are given in table 30. Using the values in table 30 in equations 61, 62, and 63,
.f.figure Õ48¸®Ÿelastic\temp_st.plt¯®effect of temperature on properties of ferritic steel;¯ shows the relative change in E, p, and CO with temperature.
Garfalo (p. 22) found the following effects of temperature on shear modulus:
ëG
70) ÄÄÄ = -0.027 GPa/ØC
ëT
aluminum
For titanium and steel, Young's modulus decreases at a constant rate near room temperature -- i.e., the slope of the modulus- temperature graph is constant. For aluminum, however, Young's modulus drops off more rapidly as the temperature increases -- i.e., the curve is nonlinear, especially at higher temperatures. This is shown in .f.figure Õ49¸®Ÿelastic\temp_a75.plt¯®effect of temperature on properties of 7075 aluminum;¯ for 7075 aluminum and .f.figure Õ50¸®Ÿelastic\temp_a24.plt¯®effect of temperature on properties of 2024 aluminum;¯ for 2024 aluminum. (Note: the modulus curves for 7075 and 2024 are cubic regression fits of the data (respectively) from 22, vol. 3, figure 3.061, code 3207, p. 30 and figure 3.0623, code 3202, p. 28. In a private conversation, the ALCOA technical people indicated that aluminum's modulus is not affected by the temper.)

For 7075, there is no single value that gives the rate of change of E or CO with temperature. However, table 30 gives the rate of change of these properties at room temperature (25 ØC). For 2024, the modulus and wave speed are nearly linear in the range of 0 ØC to 125 ØC, and are much less affected by temperature than any of the other discussed materials. The density change with temperature is reasonably linear up to 400 ØC for 2024-T4. (See 22, vol. 3, figure 2.0211, code 3203, p. 15.)

TABLE 30
&a+180H
EFFECT OF TEMPERATURE ON MATERIAL PROPERTIES

&a+180H(0U(s0p12h10vs3b3T Modulus E Density p Poisson's Wave Speed &a+180H(0U(s0p12h10vs3b3TC&a+45VO&a-45V Maximum &a+180H(0U(s0p12h10vs3b3TMaterial Condition GPa/ØC %/ØC (%/ØC) Ratio (nu) (m/sec)/ØC &a+180H(0U(s0p12h10vs3b3T%/ØC Temp (ØC) Aluminum 7075 -0.052&a-45V1&a+45V -0.071&a-45V1&a+45V -0.0070&a-45V2&a+45V Unknown -1.8 -0.035 25&a-45V1&a+45V Aluminum 2024 -0.017&a-45V8&a+45V -0.023&a-45V8&a+45V -0.0070&a-45V2&a+45V Unknown -0.36 -0.007 125&a-45V8&a+45V Titanium 6-4 or 7-4 -0.060&a-45V3&a+45V -0.050&a-45V3&a+45V -0.0028&a-45V4&a+45V No effect&a-45V5&a+45V -1.3 -0.025 650&a-45V3&a+45V Steel Ferritic -0.074&a-45V6&a+45V -0.036&a-45V6&a+45V -0.0035&a-45V4&a+45V No effect&a-45V7&a+45V -0.93 -0.018 375&a-45V6&a+45V Notes:
1. Maximum Temp is the highest known temperature for which &dDall&d@ the listed values for the specified material are valid. This temperature may be conservative for individual properties. 2. Superscript numbers denote references: &a+180H
1 ÍÍ> 22, vol. 3, figure 3.061, code 3207, p. 30 2 ÍÍ> Byars, p. 371
3 ÍÍ> Ti-6Al-4V: 22, vol. 4, code 3707, p. 26
Ti-7Al-4Mo: 22, vol. 4, code 3708, p. 22 4 ÍÍ> Beer, p. 585 (Also see 22, code 3798, p. 9) 5 ÍÍ> 22, vol. 4, figure 3.061, code 3707, p. 25 6 ÍÍ> Garfalo, pp. 18 - 20
7 ÍÍ> Garfalo, p. 21
8 ÍÍ> 22, vol. 3, figure 3.0623, code 3203, p. 28

effect of grain direction in horns with large lateral &a+180H
dimensions
In this section we will look at how the grain direction (relative to the vibration direction) affects the performance of horns whose lateral dimensions (transverse to the stud axis) are not small.
If a material is isotropic (the properties are the same in all directions), then the performance of the finished horn should not be affected by the direction in which the horn was machined from the raw stock. However, if a material is anisotropic, then the performance will be affected by the direction of vibration relative to the grain direction of the raw stock. The degree of this effect will depend on the degree of anisotropy and may depend on the lateral dimensions of the horn. For this discussion, the following nomenclature will be used: 1. LT-vibration -- the long-transverse material direction is parallel to the stud axis. Then the longitudinal material direction (the grain direction) will run parallel to the horn width. This is the usual manner for machining Branson horns. 2. L-vibration -- the longitudinal material direction is parallel to the stud axis. Then the long-transverse material direction will run parallel to the horn width. For all cases considered below, the short-transverse material direction is parallel to the horn thickness. unshaped titanium bar horns
For Ti-7Al-4Mo bar, table 29 shows that the longitudinal wave speed is 3.7% lower than the long-transverse wave speed. This should affect the tuned length of practical titanium bar horns, depending on the direction of horn vibration relative to the material direction.
Now, although we expect differences in tuned length for LT-vibration versus L-vibration, these differences may not be exactly the 3.7% given for the thin-wire. This is because practical horns have substantial dimensions transverse to the direction of vibration (unlike the thin-wire). Then the material properties perpendicular to the vibration direction may affect the final tuned length. Also, practical horns are often shaped, which may also affect length differences between vibration in the longitudinal versus long-transverse material directions. .f.Figures Õ51¸®Ÿhorn\grain423.hrn¯®20 kHz Ti-7Al-4Mo horn: LT- vibration;¯ and .f.Õ52¸®Ÿhorn\grain424.hrn¯®20 kHz Ti-7Al-4Mo horn: L-vibration;¯), show two unshaped 7-4 titanium bar horns without slots. These horns have the same nominal width and thickness dimensions. However, horn HRD-423 vibrates parallel to the long-transverse material direction (per normal Branson practice), while HRD-424 vibrates parallel to the longitudinal material direction. Table 31 shows the data measured for each of these horns. To assure that the data was repeatable, the testing was repeated once for each horn, between which the horn-booster joint was disassembled.

Table 31 shows the original measured values. It also shows adjusted values, which have been corrected to a common frequency and amplitude so that comparisons between horns can be made on a common basis. These adjustments are discussed below. Tuned length
The horn lengths have been corrected to a common frequency of 19950  Hz, assuming an inverse relation between horn length with frequency:
Ú f ¿ 71) Adjusted_length = Measured_length ³ ÄÄÄÄÄ ³
À 19950 Ù &a+180H
&a+180H
where f = frequency corresponding to
Measured_length The adjusted length of HRD-424 (L-vibration) is 2.6% less than HRD-423 (LT-vibration). This is less than the 3.7% of the thin- wire measurements.
Loss
Since we are interested in the horn alone, the horn loss at the measured amplitude can be determined by subtracting the converter loss (approximately 10 watts for a 400 series converter) and booster loss (approximately 17 watts for a 2.5:1 500 series titanium booster) from the measured stack loss. (See the chapter on "Loss" for estimates of converter and booster loss.) The resulting value is then adjusted to a nominal horn amplitude of 50 microns, assuming that the power varies with the square of the amplitude:
72) Adjusted_horn_loss
Ú  ¿2
= (Measured_stack_loss - 27 watts) ³ ÄÄÄÄÄÄÄÄÄÄ ³
À 50 microns Ù &a+180H
&a+180H
where  = horn amplitude (microns)
corresponding to
Measured_stack_loss The adjusted horn loss of HRD-424 (L-vibration) is about 21% higher than for HRD-423 (LT-vibration). (For further discussion of this loss data, see the chapter on "Loss".) Uniformity
The uniformity of both horns is substantially the same, regardless of the vibration direction. This conclusion might not be valid for substantially wider horns, where possible uniformity differences could be more easily measured. shaped titanium bar horns
.f.Figures Õ53¸®Ÿhorn\grain425.hrn¯®76 x 11 x 51 Ti-7Al-4Mo bar horn: LT-vibration;¯ and
.f.Õ54¸®Ÿhorn\grain426.hrn¯®76 x 11 x 51 Ti-7Al-4Mo bar horn: L-vibration;¯), show two shaped 7-4 titanium bar horns without slots. These horns have the same nominal dimensions. However,

horn HRD-425 vibrates parallel to the long-transverse material direction (per normal Branson practice), while HRD-426 vibrates parallel to the longitudinal material direction. Table 32 shows the data measured for each of these horns. To assure that the data was repeatable, the testing was repeated once for each horn, between which the horn-booster joint was disassembled. As with the unshaped horns, table 32 shows both the measured values and adjusted values for the horn lengths and loss. Tuned length
The adjusted length of HRD-426 (L-vibration) is 2.1% less than HRD-425 (LT-vibration). This is less than the 3.7% from thin- wire measurements.
Uniformity
No uniformity data was taken for these horns. Loss
The net horn loss has been corrected to a common amplitude of 125 microns:
73) Adjusted_horn_loss
Ú  ¿2
= (Measured_stack_loss - 27 watts) ³ ÄÄÄÄÄÄÄÄÄÄÄ ³
À 125 microns Ù &a+180H
&a+180H
where  = horn amplitude (microns)
corresponding to
Measured_stack_loss Then the net loss of HRD-426 (L-vibration) is about 47% higher than for HRD-425 (LT-vibration). (For further discussion of this loss data, see the chapter on "Loss".) unshaped aluminum bar horns
.f.Figure Õ55¸®Ÿhorn\eh_grain.bar¯®effect of grain orientation on resonant frequencies of 2024-t4 aluminum bar horn;¯ shows impedance plots for a 2-slotted aluminum bar horn (.f.figure Õ56¸®Ÿhorn\200x25a.hrn¯®2024 aluminum horn, 20 kHz;¯) that was machined to a fixed length with grain parallel or transverse to the stud axis. (The type of aluminum and the manufacturer are unknown. The best guess is that the aluminum was probably 2024.) For L-vibration, the axial resonant frequency was approximately 125 Hz (0.6%) lower than for LT-vibration. Thus, for this aluminum the wave speed is nearly identical, regardless of the grain direction. This is consistent with the usual assumption that aluminum is nearly isotropic. The frequency separation between the axial and the adjacent nonaxial resonances is relatively unaffected by the grain orientation. No loss or uniformity data are available for this horn.
conclusions
The thin-wire half-wavelength for 7-4 titanium in the longitudinal material direction is 3.7% lower than in the long- transverse material directions. Working horns with significant

lateral dimensions (above) show somewhat lesser length differences for L-vibration and LT-vibration. The loss of titanium with L-vibration is substantially higher than for horns with LT-vibration. The reason is unknown. (See the chapter on "Loss" for further discussion.) (Note: TIMET (p. 16) notes that the loss is directionally dependent for Ti-6Al-4V.)
Based on the limited titanium data, it appears that the material direction does not affect the horn uniformity. Additional tests on wider horns would be needed to confirm this conclusion. For aluminum horns, the material direction should have only slight effect on tuned frequency and negligible effect on frequency separation.

TABLE 31
&a+180H
DIRECTIONAL PROPERTIES OF
&a+180H
UNSHAPED TIMET Ti-7Al-4Mo BAR HORNS

&a+180H(0U(s0p12h10vs3b3T Measured Adjusted &a+180H(0U(s0p12h10vs3b3T Stack Horn &a+180H(0U(s0p12h10vs3b3TMeasured Adjusted &a+180H(0U(s0p12h10vs3b3T HRD # Date Dir Test # f (Hz) U (%) A` (æ) Loss (W) Loss (W) &a+180H(0U(s0p12h10vs3b3TLength Length &a+180H
423 3/23/81 LT 1 19821 83.9 50.3 60 1/2 33.1 131.27 130.4
423 3/23/81 LT 2 ÄÄÄÄÄ ÄÄÄÄ 50.3 62 34.6 131.27 130.4 &a+180H
424 3/17/81 L 1 19756 84.8 51.6 71 41.3 128.32 127.1
424 3/17/81 L 2 ÄÄÄÄÄ ÄÄÄÄ 51.6 70 40.4 128.32 127.1 &a+180H
Notes:
1. Tests were run with 400 series converter CU02113D and 500 series 2.5:1 titanium booster at 930 RF volts, RF dip. 2. The horn-booster joint was disassembled between the first and second tests. The joint was torqued to 34 N-m (300 lbf-in) with MolyKote grease.
3. Abbreviations:
Dir --> Direction of vibration, referenced to the
material direction.
L --> Vibration in Longitudinal material
direction (parallel to grain).
LT --> Vibration in Long-transverse material
direction.
&a+180H
f --> Frequency
U --> Uniformity across horn width.  --> Output amplitude at the center of the horn face. 4. The "Measured Stack Loss" is the power of the entire stack at the measured horn amplitude. The "Adjusted Horn Loss" is given by equation 72.
5. The "Adjusted Length" adjusts the "Measured Length" to a nominal frequency of 19950 Hz. See equation 71.

TABLE 32
&a+180H
DIRECTIONAL PROPERTIES OF
&a+180H
SHAPED TIMET Ti-7Al-4Mo BAR HORNS

&a+180H(0U(s0p12h10vs3b3T Measured Adjusted &a+180H(0U(s0p12h10vs3b3T Stack Horn &a+180H(0U(s0p12h10vs3b3TMeasured Adjusted &a+180H(0U(s0p12h10vs3b3T HRD # Date Dir Test # f (Hz) U (%) A` (æ) Loss (W) Loss (W) &a+180H(0U(s0p12h10vs3b3TLength Length &a+180H
425 8/11/81 LT 1 19970 ÄÄÄÄ 130.8 78 1/2 47.0 139.22 139.4
425 8/11/81 LT 2 ÄÄÄÄÄ ÄÄÄÄ 129.0 76 46.0 139.22 139.4 &a+180H
426 3/17/81 L 1 19917 ÄÄÄÄ 123.7 93 1/2 67.9 136.68 136.5
426 3/17/81 L 2 ÄÄÄÄÄ ÄÄÄÄ 123.7 94 68.4 136.68 136.5 &a+180H
Notes:
1. Tests were run with 400 series converter CU02113D and 500 series 2.5:1 titanium booster at 930 RF volts, RF dip. 2. The horn-booster joint was disassembled between the first and second tests. The joint was torqued to 34 N-m (300 lbf-in) with MolyKote grease.
3. Abbreviations:
Dir --> Direction of vibration, referenced to the
material direction.
L --> Vibration in Longitudinal material
direction (parallel to grain).
LT --> Vibration in Long-transverse material
direction.
&a+180H
f --> Frequency
U --> Uniformity across horn width.  --> Output amplitude at the center of the horn face. 4. The "Measured Stack Loss" is the power of the entire stack at the measured horn amplitude. The "Adjusted Horn Loss" is given by equation 73.
5. The "Adjusted Length" adjusts the "Measured Length" to a nominal frequency of 19950 Hz. See equation 71.

REFERENCES AND NOTES
1. M. E. Fine, "Dynamic Methods of Determining the Elastic Constants and Their Temperature Variation in Metals", Symposium on Determination of Elastic Constants, Special Technical Publication No. 129, American Society for Testing and Materials, Philadelphia, 1952, pp. 43 - 67.

2. D. A. Berlincourt, D. R. Curran, and H. Jaffe, "Piezoelectric and Piezomagnetic Materials and Their Function in Transducers", Physical Acoustics, Principles and Methods, volume I, part A, Warren P. Mason ed., Academic Press, New York, 1964, pp. 169 - 270.
3. H. J. McSkimin, "Ultrasonic Methods for Measuring the Mechanical Properties of Liquids and Solids", Physical Acoustics, Principles and Methods, volume I, part A, Warren P. Mason ed., Academic Press, New York, 1964, pp. 271 - 334.

4. Erwin Meyer and Ernst-Georg, Physical and Applied Acoustics, Academic Press, New York, 1972, pp. 25 - 28. Describes a novel method of determining the shear and compressional wave speeds of metals using underwater spark discharge and Schlieren photography.

5. G. W. McMahon, "Experimental Study of the Vibrations of Solid, Isotropic, Elastic Cylinders", The Journal of the Acoustical Society of America, vol. 36, number 1, January 1964, pp. 85 - 91. McMahon gives results for Onoe's method, by which Poisson's ratio may be calculated from the ratios of resonant frequencies of particular vibration modes (p. 87).

6. Morio Onoe, "Contour Vibrations of Isotropic Circular Plates", The Journal of the Acoustical Society of America, vol. 28, number 6, November 1956, pp. 1158 - 1162.

7. Lord Rayleigh (John William Strutt), The Theory of Sound, vol 1, Dover Publications, New York, 1945, pp. 251 - 252.

8. Pierre Louis Leonard Marie Derks, The Design of Ultrasonic Resonators with Wide Output Cross-Sections, Nederlandse Philips Bedrijfen B.V., 5600 MD Eindhoven, The Netherlands, 1984. A good discussion of Mori's equation. pp. 41 - 45.

9. E. Mori, K. Itoh, A. Imanuza, "Analysis of a Short Column Vibrator by Apparent Elasticity Method and its Application", Ultrasonics International, 1977, Conference Proceedings, IPC Science and Technology Press, Surrey, England, 1977, pp. 262 - 266.
10. F. Garofalo, P. R. Malenock, G. V. Smith, "The Influence of Temperature on the Elastic Constants of Some Commercial Steels", Symposium on Determination of Elastic Constants, Special Technical Publication No. 129, American Society for Testing and Materials, Philadelphia, 1952, pp. 10 - 27.

11. E. W. Kuenzi, "Methods for Determining the Elastic Constants of Nonmetallic Materials", Symposium on Determination of Elastic Constants, Special Technical Publication No. 129, American Society for Testing and Materials, Philadelphia, 1952, pp. 10 - 27.
12. R. N. Thurston, "Wave Propagation in Fluids and Normal Solids", Physical Acoustics, Principles and Methods, volume I, part A, Warren P. Mason ed., Academic Press, New York, 1964, pp. 271 - 334.
13. Walter Guyton Cady, Piezoelectricity, Dover Publications, New York, 1964.
14. C. Kleesattel, "Uniform Stress Contours for Disk and Ring Resonators Vibrating in Axially Symmetric Radial and Torsional Modes", Acustica, Vol. 20, 1968, pp. 2 - 13. 15. A. W. Bowen and C. A. Stubbington, "The Effect of à + á Working on the Fatigue and Tensile Properties of Ti-6Al-4V Bars", Titanium Science and Technology, R. I. Jaffee and H. M. Burte (eds.), Plenum Press, New York, 1973, pp. 2097 - 2108. 16. John T. Richards, "An Evaluation of Several Static and Dynamic Methods for Determining the Elastic Moduli", Symposium on Determination of Elastic Constants, Special Technical Publication No. 129, American Society for Testing and Materials, Philadelphia, 1952, pp. 71 - 98. 17. "Facts about Titanium: RMI Ti-6Al-4V", RMI Company, Niles, OH 44446.
18. "Properties and Processing Ti-6Al-4V", TIMET, Pittsburgh, PA 15230, 1986.
19. R. R. Boyer and H. W. Rosenberg, "Ti-10V-2Fe-3Al Properties", unknown source, pp. 441 - 456.

20. "Fatigue Characteristics of Titanium Alloy Forgings for Rotary Wing Vehicles", M. Tiktinsky, The Science, Technology and Application of Titanium, R. I. Jaffee and N. E. Promisel (eds.), Pergamon Press, New York, 1970, pp. 1013 - 1021. 21. A. W. Bowen, "The Effect of Testing Direction on the Fatigue and Tensile Properties of a Ti-6Al-4V Bar", Titanium Science and Technology, R. I. Jaffee and H. M. Burte (eds.), Plenum Press, New York, 1973, pp. 1271 - 1281. 22. Aerospace Structural Metals Handbook, 1986 edition. 23. Ferdinand P. Beer and E. Russell Johnston, Jr., Mechanics of Materials, McGraw-Hill Book Co., New York, 1981. 24. I. S. Sokolnikoff, Mathematical Theory of Elasticity, Robert E. Krieger Publishing Co., Malabar, Florida, 1987. 25. Piezoelectric Technology Data for Designers, Vernitron Piezoelectric Division, Bedford, Ohio, 44146. 26. Piezoelectric Ceramics, Application Book, J. van Randeraat, R. E. Setterington (ed), Ferroxcube Corp., Saugerties, NY, 1974. 27. Edward F. Byars and Robert D. Snyder, Engineering Mechanics of Deformable Bodies, International Textbook Company, Scranton, PA, 1968.

changes in this version
7/8/87:
1.  (which printer will not print) in equations replaced by ë. 2. Style sheet changed from My.sty to New.sty. 3. Equations reformatted with brackets spanning 3 lines. 4. Most '*' operators removed from equations. 5. Outline now available.
L --> Longitudinal
2. Density = 4464 kg/m3 (Average from other tests.) 3. Numbers in each column directly below the solid line are averages. Numbers in parentheses are the associated 95% confidence interval of the mean. 9/8/89
1. Co replaced by CO. âo replaced by âO. (using search and replace, combined with macro)

obsolete material
9/11/89 Taken from the section "Aluminum" --> "Half-wave Tests". Rayleigh equation. The Rayleigh equation (Æ31µ) can be solved for â as a function of âO and nu:
Ú Ú ¿2 ¿1/2 74) â = âO ³ 1 - (1/8) ³ ã nu (d/âO) ³ ³
À À Ù Ù Using the above curve-fit constants for âO and nu, this equation has been plotted in .f.figure Õ17¸®Ÿwave\wlendia1.plt¯ for comparison to the to the data from table 32 and to the Mori equation. The Rayleigh equation shows good agreement up to about í30, for which the horn can still be considered slender. Above í30, the Rayleigh equation shows increasing error. .f.Figure Õ19¸®Ÿelastic\ray_err1.plt¯®rayleigh error: prediction of thin-wire half-wavelength;¯ shows the approximate error when the Rayleigh equation is used to predict the tuned half- wavelength â, assuming that the values of âO (i.e., CO) and nu are correctly known.
&a+180H
9/13/89 Taken from the section "Finite Element Analysis". ®.c.:notes
Mesh refinement shows that the above results have essentially converged for the chosen mesh. .f.Figures Õ29o¸®Ÿfea\m125spl1.plt¯®EFFECT OF AXIAL MESH DENSITY ON RELATIVE AMPLITUDE OF 125 MM UNSHAPED AERO AL SPOOL HORN;¯ and .f.Õ29p¸®Ÿfea\m125spl2.plt¯®EFFECT OF AXIAL MESH DENSITY ON RESONANT FREQUENCIES OF 125 MM UNSHAPED AERO AL SPOOL HORN;¯ show the amplitude and frequency convergence for the í125 spool. The smallest of these meshes (approximately 1.7 mm high x 6.3  mm wide) was used for the above FEA models. All horn elements are 4-noded quadrilaterals, including the stud. The presence (or absence) of the converter does not substantially affect the horn amplitudes. All horn models used a Young's modulus of 73.5 GPa and a density of 2777 kg/m3 for the Aerospace aluminum.¯
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&a+900H
&a+900H(10U(s3t12vpsb10HFootnotes for equations. DO NOT ERASE. &a+900H

&a+900H(10U(s3t12vpsb10HFigure 1 &a+900H (10U(s3t12vpsb10HTypical Machining Direction for Bar and Plate Stock6 &a+900H(10U(s3t12vpsb10HFigure 2 &a+900H (10U(s3t12vpsb10HGrain Directionality in Ti-6Al-4V Bar&a+1008H7 &a+900H(10U(s3t12vpsb10HFigure 3 &a+900H (10U(s3t12vpsb10HGrain Directionality in Rolled Aluminum 7 &a+900H(10U(s3t12vpsb10HFigure 5 &a+900H (10U(s3t12vpsb10HEffect of Heat-to-Heat Variations on Young's Modulus &a+900H (10U(s3t12vpsb10Hfor Ti-6Al-4V Sheet&a+2304H8 &a+900H(10U(s3t12vpsb10HFigure 6 &a+900H (10U(s3t12vpsb10HEffect of Raw Stock Size on the Cross-Sectional &a+900H (10U(s3t12vpsb10HTensile Properties of STA Ti-10V-2Fe-3Al 9 &a+900H(10U(s3t12vpsb10HFigure 7 &a+900H (10U(s3t12vpsb10HVariation in Young's Modulus across a 114.3 mm &a+900H (10U(s3t12vpsb10HDiameter Rod of Annealed Ti-6Al-4V&a+1152H10 &a+900H(10U(s3t12vpsb10HFigure 8 &a+900H (10U(s3t12vpsb10HEffect of Raw Stock Section Size on Tensile &a+900H (10U(s3t12vpsb10HProperties of STA Ti-6Al-4V&a+1656H10 &a+900H(10U(s3t12vpsb10HFigure 10 &a+900H (10U(s3t12vpsb10HTi Tensile Test Specimen with Attached Strain Gage16 &a+900H(10U(s3t12vpsb10HFigure 11 &a+900H (10U(s3t12vpsb10HTensile Test, 7-4 Titanium Bar: Long-Transverse &a+900H (10U(s3t12vpsb10HMaterial Direction&a+2304H18 &a+900H(10U(s3t12vpsb10HFigure 12 &a+900H (10U(s3t12vpsb10HTensile Test, 7-4 Titanium Bar: Long-Transverse &a+900H (10U(s3t12vpsb10HMaterial Direction&a+2304H19 &a+900H(10U(s3t12vpsb10HFigure 13 &a+900H (10U(s3t12vpsb10HTensile Test, 7-4 Titanium Bar: Longitudinal &a+900H (10U(s3t12vpsb10HMaterial Direction&a+2304H20 &a+900H(10U(s3t12vpsb10HFigure 14 &a+900H (10U(s3t12vpsb10HTensile Test, 7-4 Titanium Bar: Longitudinal &a+900H (10U(s3t12vpsb10HMaterial Direction&a+2304H21 &a+900H(10U(s3t12vpsb10HFigure 14a &a+900H (10U(s3t12vpsb10HTi Torsion Test Specimen with Attached Strain Gage23 &a+900H(10U(s3t12vpsb10HFigure 15 &a+900H (10U(s3t12vpsb10HShear Test, 7-4 Titanium Bar: Long-Transverse &a+900H (10U(s3t12vpsb10HMaterial Direction&a+2304H24 &a+900H(10U(s3t12vpsb10HFigure 16 &a+900H (10U(s3t12vpsb10HShear Test, 7-4 Titanium Bar: LongItudinal Material &a+900H (10U(s3t12vpsb10HDirection&a+2952H25 &a+900H(10U(s3t12vpsb10HFigure 17 &a+900H (10U(s3t12vpsb10HAerospace Aluminum Half-Wavelength at 19950 Hz 29 &a+900H(10U(s3t12vpsb10HFigure 18 &a+900H (10U(s3t12vpsb10HAddition of Half-Wave Section to Tuned Horn 29 &a+900H(10U(s3t12vpsb10HFigure 19 &a+900H (10U(s3t12vpsb10HRayleigh Error: Prediction of Thin-Wire Half- &a+900H (10U(s3t12vpsb10HWavelength&a+2880H31 &a+900H(10U(s3t12vpsb10HFigure 20 &a+900H (10U(s3t12vpsb10H20  kHz S-N horn&a+2520H34 &a+900H(10U(s3t12vpsb10HFigure 21 &a+900H (10U(s3t12vpsb10HWave Speed Measurement: 7-4 Titanium Bar, Long- &a+900H (10U(s3t12vpsb10HTransverse&a+2880H36 &a+900H(10U(s3t12vpsb10HFigure 22 &a+900H (10U(s3t12vpsb10H40  kHz S-N horn&a+2520H43

&a+900H(10U(s3t12vpsb10HFigure 23 &a+900H (10U(s3t12vpsb10HWave-Speed Measurement: 7-4 Titanium Bar, Parallel &a+900H (10U(s3t12vpsb10Hto Grain&a+3024H44 &a+900H(10U(s3t12vpsb10HFigure 24 &a+900H (10U(s3t12vpsb10HWave-Speed Measurement: 7-4 Titanium Bar, Parallel &a+900H (10U(s3t12vpsb10Hto Grain&a+3024H44 &a+900H(10U(s3t12vpsb10HFigure 25 &a+900H (10U(s3t12vpsb10H7-4 Titanium Rod: Wave Speed Dependence on Raw &a+900H (10U(s3t12vpsb10HStock Size&a+2880H47 &a+900H(10U(s3t12vpsb10HFigure 26 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Half-Wavelength of &a+900H (10U(s3t12vpsb10HAerospace Al at 19950 Hz&a+1872H51 &a+900H(10U(s3t12vpsb10HFigure 28 &a+900H (10U(s3t12vpsb10H20  kHz s-n Horn Machined from 152 x 50 Bar Stock 57 &a+900H(10U(s3t12vpsb10HFigure 29 &a+900H (10U(s3t12vpsb10HElectrostatic Resonance Equipment&a+1224H63 &a+900H(10U(s3t12vpsb10HFigure 30 &a+900H (10U(s3t12vpsb10H125  mm Aero Al Spool Horn&a+1800H83 &a+900H(10U(s3t12vpsb10HFigure 31 &a+900H (10U(s3t12vpsb10HAmplitude Probe with Special boot&a+1224H83 &a+900H(10U(s3t12vpsb10HFigure 32 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Relative Face Axial &a+900H (10U(s3t12vpsb10HAmplitude of 125 mm Aero Al Spool Horn 83 &a+900H(10U(s3t12vpsb10HFigure 33 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Relative Back Axial &a+900H (10U(s3t12vpsb10HAmplitude of 125 mm Aero Al Spool Horn 83 &a+900H(10U(s3t12vpsb10HFigure 34 &a+900H (10U(s3t12vpsb10HFea Model of í125 Aero Al Spool Horn with 400 Series &a+900H (10U(s3t12vpsb10HConverter&a+2952H84 &a+900H(10U(s3t12vpsb10HFigure 35 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Relative Face Radial &a+900H (10U(s3t12vpsb10HAmplitude of 125 mm Aero Al Spool Horn 84 &a+900H(10U(s3t12vpsb10HFigure 36 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Relative Back Radial &a+900H (10U(s3t12vpsb10HAmplitude of 125 mm Aero Al Spool Horn 84 &a+900H(10U(s3t12vpsb10HFigure 37 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Resonant Frequencies of &a+900H (10U(s3t12vpsb10H125 mm Aero Al Spool Horn&a+1800H84 &a+900H(10U(s3t12vpsb10HFigure 38 &a+900H (10U(s3t12vpsb10H110  mm Aero Al Spool Horn&a+1800H85 &a+900H(10U(s3t12vpsb10HFigure 39 &a+900H (10U(s3t12vpsb10H&a+3600H85 &a+900H(10U(s3t12vpsb10HFigure 40 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Relative Face Axial &a+900H (10U(s3t12vpsb10HAmplitude of 110 mm Aero Al Spool Horn 85 &a+900H(10U(s3t12vpsb10HFigure 41 &a+900H (10U(s3t12vpsb10H&a+3600H85 &a+900H(10U(s3t12vpsb10HFigure 42 &a+900H (10U(s3t12vpsb10HEffect of Poisson's Ratio on Relative Amplitude &a+900H (10U(s3t12vpsb10HError of í125 and í110 Aero Al Spool Horns 85 &a+900H(10U(s3t12vpsb10HFigure 43 &a+900H (10U(s3t12vpsb10H125 MM AERO AL SPOOL HORN&a+1800H86

&a+900H(10U(s3t12vpsb10HFigure 44 &a+900H (10U(s3t12vpsb10HEFFECT OF POISSON'S RATIO ON RELATIVE FACE AXIAL &a+900H (10U(s3t12vpsb10HAMPLITUDE OF 125 MM UNSHAPED AERO AL SPOOL HORN 86 &a+900H(10U(s3t12vpsb10HFigure 45 &a+900H (10U(s3t12vpsb10HEFFECT OF POISSON'S RATIO ON RELATIVE BACK AXIAL &a+900H (10U(s3t12vpsb10HAMPLITUDE OF 125 MM UNSHAPED AERO AL SPOOL HORN 86 &a+900H(10U(s3t12vpsb10HFigure 46 &a+900H (10U(s3t12vpsb10Heffect of temperature on properties of ti 7al-4mo 96 &a+900H(10U(s3t12vpsb10HFigure 47&a+3528H97 &a+900H(10U(s3t12vpsb10HFigure 48 &a+900H (10U(s3t12vpsb10Heffect of temperature on properties of ferritic &a+900H (10U(s3t12vpsb10Hsteel&a+3240H98 &a+900H(10U(s3t12vpsb10HFigure 49 &a+900H (10U(s3t12vpsb10Heffect of temperature on properties of 7075 &a+900H (10U(s3t12vpsb10Haluminum&a+3024H98 &a+900H(10U(s3t12vpsb10HFigure 50 &a+900H (10U(s3t12vpsb10Heffect of temperature on properties of 2024 &a+900H (10U(s3t12vpsb10Haluminum&a+3024H98 &a+900H(10U(s3t12vpsb10HFigure 51 &a+900H (10U(s3t12vpsb10H20  kHz Ti-7Al-4Mo horn: LT-vibration 101 &a+900H(10U(s3t12vpsb10HFigure 52 &a+900H (10U(s3t12vpsb10H20  kHz Ti-7Al-4Mo horn: L-vibration&a+936H101 &a+900H(10U(s3t12vpsb10HFigure 53 &a+900H (10U(s3t12vpsb10H76 x 11 x 51 Ti-7Al-4Mo bar horn: LT-vibration 102 &a+900H(10U(s3t12vpsb10HFigure 54 &a+900H (10U(s3t12vpsb10H76 x 11 x 51 Ti-7Al-4Mo bar horn: L-vibration 102 &a+900H(10U(s3t12vpsb10HFigure 55 &a+900H (10U(s3t12vpsb10Heffect of grain orientation on resonant frequencies &a+900H (10U(s3t12vpsb10Hof 2024-t4 aluminum bar horn&a+1512H103 &a+900H(10U(s3t12vpsb10HFigure 56 &a+900H (10U(s3t12vpsb10H2024 aluminum horn, 20 kHz&a+1656H103

&a+900H&dDable 1&d@Average Densities for Acoustic Materials 13 &a+900H&dDTable 2&d@
&a+900H (10U(s3t12vpsb10HDensities for Acoustic Materials&a+1296H14 &a+900H&dDTable 3&d@
&a+900H (10U(s3t12vpsb10HStress-Strain Data for Lont-Transverse Tensile Test &a+900H (10U(s3t12vpsb10Hof TIMET Ti-7Al-4Mo&a+2232H18 &a+900H&dDTable 4&d@
&a+900H (10U(s3t12vpsb10HStrain Data for Long-Transverse Tensile Test of &a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo&a+2448H19 &a+900H&dDTable 5&d@
&a+900H (10U(s3t12vpsb10HStress-Strain Data: Longitudinal Tensile Test of &a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo&a+2448H20 &a+900H&dDTable 6&d@
&a+900H (10U(s3t12vpsb10HStrain Data for Longitudinal Tensile Test of TIMET &a+900H (10U(s3t12vpsb10HTi-7Al-4Mo&a+2880H21 &a+900H&dDTable 7&d@
&a+900H (10U(s3t12vpsb10HLong-Transverse Shear Test of TIMET Ti-7Al-4Mo 25 &a+900H&dDTable 8&d@
&a+900H (10U(s3t12vpsb10HLongitudinal Shear Test of TIMET Ti-7Al-4Mo 25 &a+900H&dDTable 9&d@
&a+900H (10U(s3t12vpsb10HTuning Data for Full-Wave Horn, TIMET Ti-7Al-4Mo Bar &a+900H (10U(s3t12vpsb10HStock, Long-Transverse Material Direction (HRD-397) &a+900H (10U(s3t12vpsb10H37 &a+900H&dDTable 10&d@
&a+900H (10U(s3t12vpsb10HTuning Data for Back Half-Wave Horn, TIMET &a+900H (10U(s3t12vpsb10HTi-7Al-4Mo Bar Stock, Long-Transverse Material &a+900H (10U(s3t12vpsb10HDirection&a+2952H38 &a+900H&dDTable 11&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Bar Stock, Long-Transverse Material &a+900H (10U(s3t12vpsb10HProperties (Converter Resonance Tests, 40 kHz Full- &a+900H (10U(s3t12vpsb10HWave Horns)&a+2808H43 &a+900H&dDTable 12&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Bar Stock Longitudinal Material &a+900H (10U(s3t12vpsb10HProperties (Converter Resonance Tests, Full-Wave &a+900H (10U(s3t12vpsb10HHorns)&a+3168H45 &a+900H &dDTable 13&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Rod Stock Longitudinal Material &a+900H (10U(s3t12vpsb10HProperties (Converter Resonance Tests, Full-Wave &a+900H (10U(s3t12vpsb10HHorns)&a+2880H48 &a+900H&dDTable 14&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Rod Stock Longitudinal Material &a+900H (10U(s3t12vpsb10HProperties (Converter Resonance Tests, Half-Wave &a+900H (10U(s3t12vpsb10HHorns)&a+3168H49 &a+900H&dDTable 15&d@
&a+900H (10U(s3t12vpsb10HKaiser 2024 QQA225 Aluminum Material Properties &a+900H (10U(s3t12vpsb10H(Converter Resonance Tests, Full-Wave Horns) 53 &a+900H&dDTable 16&d@
&a+900H (10U(s3t12vpsb10HALCOA 2024-T3 Aluminum Material Properties &a+900H (10U(s3t12vpsb10H(Converter Resonance Tests, Full-Wave Horns) 54 &a+900H&dDTable 17&d@
&a+900H (10U(s3t12vpsb10HALCOA 7075 QQA225 Aluminum Material Properties &a+900H (10U(s3t12vpsb10H(Converter Resonance Tests, Full-Wave Horns) 55 &a+900H&dDTable 18&d@
&a+900H (10U(s3t12vpsb10HMartin Marietta 7075 QQA225 Aluminum Material

&a+900H (10U(s3t12vpsb10HProperties (Converter Resonance Tests, Full-Wave &a+900H (10U(s3t12vpsb10HHorns)&a+3168H56 &a+900H&dDTable 19&d@
&a+900H (10U(s3t12vpsb10HAerospace Aluminum Material Properties &a+900H (10U(s3t12vpsb10H(Converter Resonance Tests, Full-Wave Horns) 57 &a+900H&dDTable 20&d@
&a+900H (10U(s3t12vpsb10HAerospace Aluminum Rod Material Properties &a+900H (10U(s3t12vpsb10H(Converter Resonance Tests, Half-Wave Horns) 58 &a+900H &dDTable 21&d@
&a+900H (10U(s3t12vpsb10HD-2 Tool Steel Rod Stock, Rc 54: Longitudinal &a+900H (10U(s3t12vpsb10HMaterial Properties (Converter Resonance Tests, &a+900H (10U(s3t12vpsb10HFull-Wave Horns)&a+2160H61 &a+900H&dDTable 22&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Rod Material Properties &a+900H (10U(s3t12vpsb10H(Electrostatic Resonance Tests)&a+1368H66 &a+900H&dDTable 23&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Bar Material Properties &a+900H (10U(s3t12vpsb10H(Electrostatic Resonance Tests)&a+1368H67 &a+900H &dDTable 24&d@
&a+900H (10U(s3t12vpsb10HVarious Aluminums: High Frequency NDT Tests 74 &a+900H &dDTable 25&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Bar, Long-Transverse Material &a+900H (10U(s3t12vpsb10HDirection: High Frequency NDT Tests 76 &a+900H &dDTable 26&d@
&a+900H (10U(s3t12vpsb10HTIMET Ti-7Al-4Mo Bar, Longitudinal Material &a+900H (10U(s3t12vpsb10HDirection: High Frequency NDT Tests 78 &a+900H &dDTable 27&d@
&a+900H (10U(s3t12vpsb10HD-2 Tool Steel Rod Stock, Rc 54: High Frequency &a+900H (10U(s3t12vpsb10HNDT Tests&a+2664H81 &a+900H&dDTable 28&d@
&a+900H (10U(s3t12vpsb10HFEA Amplitude Error Dependence on Nu&a+1008H85 &a+900H&dDTable 29&d@
&a+900H (10U(s3t12vpsb10H"Best" Material Property Values at Room &a+900H (10U(s3t12vpsb10HTemperature91 &a+900H&dDTable 30&d@
&a+900H (10U(s3t12vpsb10HEffect of Temperature on Material Properties 99 &a+900H&dDTable 31&d@
&a+900H (10U(s3t12vpsb10HDirectional Properties of Unshaped TIMET Ti-7Al-4Mo &a+900H (10U(s3t12vpsb10HBar Horns&a+2880H105 &a+900H&dDTable 32&d@
&a+900H (10U(s3t12vpsb10HDirectional Properties of Shaped TIMET Ti-7Al-4Mo &a+900H (10U(s3t12vpsb10HBar Horns&a+2880H106 &a+900H&l1H

 

 

References

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