Uniformity & Asymmetry
Measurements

Horn orientation

Horn design is often a cut-and-try operation, where you modify the horn in the hope of improving its performance and then make measurements to see how the performance has changed. When measuring amplitudes, it is important that the horn always retain the same orientation for each measurement. Otherwise, you will not be able to properly judge the effects of the modifications. To maintain proper orientation, you must mark the horn at some spot such that the mark will not be erased during the subsequent modification. You must then maintain the same orientation between yourself and the marked spot for all amplitude measurements.

Consider the example of a 100 mm spool horn (HRD 060) whose axial resonant frequency is 19700 Hz. The face amplitudes are shown in figure í1Ê, for which the asymmetry is 0.643. Note the black triangle on the one of the eight dividing lines, which is the orientation mark. The location of this orientation mark was arbitrary, but once chosen its location was never changed. You can see that the lowest amplitude (10.0 microns) is at a 9 o'clock position with respect to the orientation mark. The highest amplitude (28.0 microns) is exactly opposite, at a 3 o'clock position with respect to the orientation mark.

Now, since the horn face was already properly machined, the horn was tuned from the stud surface to increase its axial resonant frequency. This means that the converter orientation would change when it was next screwed to the horn. The question of interest was this: would the asymmetry pattern change when the converter was reoriented or would the pattern remain constant regardless of the converter orientation?

Figure í3Ê«» (left) shows the amplitude pattern after the horn was tuned from the back to 19958 Hz. You can see that the asymmetry decreased to 0.317. However, the pattern remained the same, with the lowest amplitude (15.1 microns) at the 9 o'clock position and the highest amplitude (22.1 microns) at the 3 o'clock position. Thus, since the asymmetry pattern did not change when the converter was reoriented, the asymmetry seems to be associated with the horn rather than the converter.

To verify this conclusion, a different converter was used to make the amplitude measurements without any alterations to the horn. The results are shown in figure í3Ê. Again, the asymmetry pattern is preserved. Thus, we have strong evidence that the asymmetry is due to the horn itself and not to the influence of the converter. Note that if we had not been consistent in our orientation of the horn (by using the orientation mark), then we could not have drawn these conclusions.

What should you use to make the orientation mark? I usually use some type of indelible marker, which does not cause any stress concentration in the horn. Be careful, however, because the marks can sometimes be erased by machining fluids. If a horn identification number has been stamped on the horn, then you can use this as a reference.

For cylindrical horns that are symmetric about the stud axis, I usually divide the face into eight equal pie-shaped segments and then take measurements on each of the eight lines. We will look at other types of horns later.

Epirirical data versus FEA

For amplitude instruments such as the A 455 or Fotonic sensor, the amplitude cannot be measured exactly at the edge of of a surface (e.g., the horn face). This is because of the finite probe diameter and possible edge effects. (See the chapter on "Amplitude".) Thus, these instruments will not give the true amplitude at the edge of the horn face, especially for horns where the amplitude changes rapidly at the edge. In this case, specifying that the amplitude was measured at the "edge" of the face is not sufficient; the distance from the probe centerline to the edge of the face should be specified when amplitude data is presented. This is especially important when comparing empirical data to FEA results.

For the A 455, the probe boot can be modified for edge amplitude measurements. (See figure ífÊ«» and the chapter on Amplitude.) Then the amplitude will always be measured at a fixed distance from the edge. (The boot that I have used has an offset of 3.8 mm from the edge.)

Note:

FEA "EDGE" AMPLITUDES ARE TAKEN EXACTLY AT THE EDGE.

Face uniformity for solid, unslotted cylindrical horns

In this section we will look at how the uniformity of solid, unslotted cylindrical horns is affected by the face diameter, the material, and the frequency.

The solid, unslotted cylindrical horn is the simplest shape for making uniformity measurements and for analyzing the resultant data. Hence, we will start with this horn and then proceed to more complex horns in later sections.

Measurement method

For solid, unslotted cylindrical horns the maximum average amplitude always occurs at the center of the horn face. The minimum averge amplitude always occurs at the edge of the horn face. This assumes that the horn has not been specially modified to increase its face uniformity, as with a spool horn.

To measure the face amplitudes, the face was divided into eight approximately equal pie-shaped elements. Eight elements is convenient because the face can quickly be divided into equal quarters, and then these quarters can be divided into half. Also, eight elements provides a sufficient number to determine an average amplitude. (Note that the elements need only be approximately equal, since the amplitudes around the edge will be averaged anyway.) One of the eight lines on the face was specially marked so that it could be used for orientation.

All amplitude measurements were made with an A 450 using a hand-held probe. Ideally, the amplitude measurements around the periphery of the horn face should have been made exactly at the edge. However, because the probe has a finite diameter [mm] and because of probe edge effects (see the chapter on "Amplitude"), the "edge" amplitude measurements were actually taken slightly inboard of the edge of the horn. For most of the edge measurements, the center of the amplitude probe was located 3.8 mm from the edge of the horn, for which a special fixture was used.

Table å1Ï gives uniformity measurements for 18 cylindrical horns made of various materials.

Pochhammer equation for axial amplitude

Zemanek[1A] (1) gives a theoretical equation for the axial amplitude distribution (equation 18 lower, p. 271). This equation is based on the Pochhammer solution (sometimes called the Pochhammer-Chree solution or the Pochhammer-Love solution) for wave propogation in an infinitely long cylinder. For a cylinder of radius R, the equation compares the amplitude at a radial distance r from the cylinder axis to the amplitude that exists along the axis of the cylinder. (See Appendix A. Also see the chapter on "Wave Motion".)

Figure í4Ê plots the uniformity data from table å1Ï. Also shown are two uniformity curves calculated from the Pochhammer equation. The Pochhammer equation requires the correct material constants, which in this case were taken for Aerospace aluminum (thin-wire wave speed Co = 5081 m/sec; Poisson's ratio_ nu = 0.319), since most of the measured data is for this material. Note: these constants had been previously determined by fitting the Mori equation for Aerospace aluminum. (See the chapter on "Elastic Material Properties".)

The lower curve shows the uniformity that should theoretically be measured if the amplitude probe could be positioned exactly at the edge of the horn face. The FEM computer analysis data point, for which the uniformity can be determined exactly at the edge of the horn face, falls very close to this curve.

The upper curve of figure í4Ê shows the Pochhammer equation evaluated at r = R   3.8 mm (i.e., at 3.8 mm from the edge of the horn face), which is where the actual amplitude measurements were made. Since this curve corresponds to the physical amplitude measurements (see above), it gives much better agreement with the empirical data then does the lower curve. Note that the error between the lower and upper curves becomes increasingly severe for larger horn diameters. This indicates that large horns suffer from very rapid amplitude decrease near the edge of the face. (Also see below.)

Since the Pochhammer equation also allows the amplitude distribution across the horn face to be determined, this has been plotted in figure í5Ê for Aerospace horns of various diameters. For this figure, R is the radius of the horn and r is the radius at which the amplitude is to be determined. For example, suppose you have a 100 mm diameter horn (R = 50 mm). If you want to determine the relative amplitude that occurs at a radius of r = 35 mm, then r/R = 35/50 = 0.7. The 100 mm curve shows that the relative amplitude is 0.81    i.e., at 35 mm from the center of a 100 mm diameter horn, the amplitude is 0.81 of the amplitude at the center of the horn face.

Best horn size

These curves indicate that, for a given part size, you should use the horn with the smallest possible diameter. For example, suppose that you have a part whose diameter is 100 mm and that must be welded across the entire surface. Then the outermost weld would occur at a radius of 50 mm. If you use a 100 mm horn for the weld, the welding will occur over a surface whose uniformity is 0.65. (See the 100 mm curve with r/R = 50/50 = 1.0.) What would happen if you used a 125 mm diameter horn to do the same weld? Then r/R = 50/62.5 = 0.8, for which the uniformity of the 125 mm horn is only 0.56. Thus, by using the 100 mm horn instead of the 125 mm horn, the weld uniformity has been increased from 0.56 to 0.65.

Best booster size

Increase back dia to be equal to nodal dia decreases back uniformity across joint area? If so, by how much? However, also increases booster gain, requiring less undercut of 2.5:1 boosters. Also, less machining on back of booster.

Effect of material constants

For a horn of given diameter, the uniformity depends on the material constants C_o (thin-wire wave speed) and nu (Poisson's ratio). Thus, the previously plotted Pochhammer curves are strictly valid only for Aerospace aluminum. However, titanium and steel would give nearly identical curves, since their wave speed and Poisson's ratio are very similar to Aerospace aluminum.

Since the uniformity is material dependent, it might be possible to find a material with significantly different material constants that would give improved uniformity. For instance, reducing Poisson's ration would improve the uniformity. If Poisson's ratio could be reduced to zero (for which the horn would have no lateral bulge while vibrating), then Pochhammer's equation indicates that the uniformity would be 0.1, regardless of the diameter. Alternately, we could try to find a material with a higher wave speed, which would also improve the uniformity.

2090 Aluminum. Aluminum-lithium alloys (2090) have a thin-wire wave speed of about 5400 m/sec (computed from modulus and density values given by matweb.com), which is about 6% higher than Aerospace aluminum. The uniformity curve from the Pochhammer equation is shown in figure í7Ê (for which Poisson's ratio is 0.34 from matweb.com). Compared to Aerospace aluminum, the 2090 alloy offers substantial uniformity improvement, especially at the larger diameters.

Beryllium. Another interesting metal is beryllium, whose thin-wire wave speed is about 12600 m/sec (approximately 2.5 times higher than Aerospace aluminum or titanium). (See 8, p. 4; also see Case, table on page 59; also see Machine Design, Metals Reference Issue, Feb 14, 1974 and contact R. E. Strock Jr. or R. C. Fullerton-Batten, Beryllium Metal Sales, Kawecki Berylco Industries Inc., Reading, PA.) Its Poisson ratio is only 0.6 - 0.8 for low strains, although for some grades it increases with increasing strain to about 0.3 near the yield point. (See Floyd, pp. 332 - 333.) The resulting Pochhammer uniformity is shown in figure í7Ê.

Of course, such alternate material would also have to have other acceptable characteristics, such as good fatigue resistance, low loss, good machinability, etc. Aluminum-lithium and beryllium are discussed further in the chapter on "Fatigue Testing". Beryllium: high cost $300+/pound, depending on shape and size. Also, beryllium toxic airborne dust and beryllium has poor impact resistance. Also, sensitive to surface damage from machining. (See 9, pp. 197 - 228.) However, good fatigue resistance. Mean structural damping coefficient = 21*10⁴ for 0.020" SR200 sheed in the frequency range between 20 and 100 Hz. See Aerospace Structural Metals Handbook, vol 5, code 5101, p. 2. Some beryllium alloys (Lockalloy) may overcome some beryllium limitations. See Aerospace Structural Metals Handbook, vol 5, code 5102, pp. 1 - 9.

Transverse (radial) amplitude at the face

Because of Poisson coupling, horns have transverse motion (motion that is perpendicular to the stud axis). Transverse motion has been measured at the edge of the horn face for several unshaped solid cylindrical horns. This motion can be important because it may cause marking of the plastic part. Table å1Ï shows this information, where the transverse motion has been expressed as a percent of the axial horn amplitude at the center of the horn face:

⁸) Relative transverse face amplitude

Ö Transverse ampl at edge of horn face Ì
= °                                      °
Ŵ Horn face centerline amplitude ì

(Note: if no data has been taken, then this is indicated by "   " in the table, irrespective of whether the horn is shaped or unshaped.)

The data and fitted curve are plotted in figure í9Ê. The curve-fit equation is —

⁹) Relative transverse face amplitude


Ö D ̲ Ö D ̳
= b₂ °     ° + b₃ °     °
Ŵ L_o ì Ŵ L_o ì


where D = horn face diameter
L_o = thin-wire half-wavelength
b₁, b₂ = curve fit constants

for which the fit constants are:

¹⁰) b₁ = -0.0101

¹¹) b₂ = 0.0532

and the standard error of estimate is —

¹²) Y_SE = 0.405

The data and plot (for L_o = 126.6 mm at 20 kHz) show that the transverse face amplitude does not exceed 5% of the horn centerline amplitude so long as the horn diameter is smaller then 125 mm. Thus, for unshaped solid cylindrical horns, marking of the part should not be a problem. (In a later section we will see that this transverse face motion is a problem in designing large diameter spool horns.)

Transverse (radial) amplitude at the node

In addition to the face motion, we may also be interested in transverse motion at the node. The nodal transverse motion is mainly important in designing boosters, where the transverse motion causes loss and energy transfer to the structure that supports the booster.

Pochhammer equation

The Pochhammer equation given by Zemanek[1A] (1) (equation 18 upper, p. 271) predicts the radial amplitude in an infinitely long cylinder. (See Appendix [_].) This equation has been plotted in figure í10Ê for Aerospace aluminum. This is approximately the transverse motion that would occur at the node of an unshaped cylindrical horn.

Poisson approximation

The Pochhammer equation can be approximated by a much simpler equation involving Poisson's ratio, which I will call the Poisson equation. [Possibly put much of this derivation in chapter on "Fundamental Relations" or in an appendix.] For an isotropic material, Poisson's ratio nu is defined as:

Ö __j Ì
¹³) nu = - °    °
Ŵ __i ì


where nu = Poisson's ratio
_ = Strain
i = Direction of applied stress
j = Specified transverse direction

For an unshaped cylindrical horn of reasonable diameter, the stress occurs almost entirely in the axial direction. Thus, "i" will refer to the axial z direction and "j" will refer to the radial r direction:

Ö __r Ì
¹⁴) nu = - °    °
Ŵ __z ì

The general equation of strain is —

Ö ds_x Ì
¹⁵) __x = °     °
Ŵ dx ì


where __x = Strain in the x direction
s_x = Amplitude in the x direction
d = Differentiation

Multiplying equation ó19Â through by -__Z and then substituting equation ó20Â (where the approxpriate subscript is substutited for x):

Ö ds_r Ì Ö ds_z Ì
¹⁶) °     ° = -nu °     °
Ŵ dr ì Ŵ dz ì


where s_r = Amplitude in the radial (r)
direction
s_z = Amplitude in the axial (z)
direction

For an unshaped cylindrical horn, the amplitude distribution along the center of the horn is —

Ö Ð Ì
¹⁷) s_z = S_z cos°     Z °
Ŵ L ì


where s_z = Axial amplitude distribution
S_z = Peak amplitude at the center
of the horn face
L = Half-wavelength of horn at
specified frequency
Z = Axial distance from horn face

In this equation, the argument of the cos function is a value in radians. Differentiating s_z with respect to Z and evaluating the result at Z = Ð/2 (the node) gives:

Ö ds_z Ì Ö Ð Ì
¹⁸) __z = °     ° = S_z °     °
Ŵ dx ì Ŵ L ì

Substituting this into equation ó21Â and integrating across the radius R at the node gives the radial amplitude S_r:

¶R Ö Ð Ì
¹⁹) S_r = ° -nu S_z °     ° dr
¾0 Ŵ L ì


Ö Ð Ì
= -nu S_z °     ° R
Ŵ L ì

Thus, the ratio of radial amplitude at the node to axial amplitude at the face (i.e., the relative radial amplitude RRA) is —

Ö S_r Ì
²⁰) RRA = °    °
Ŵ S_z ì


Ö Ð R Ì
= -nu °     °
Ŵ L ì

The only problem with using this equation is that L depends on the diameter of the horn. However, this can easily be calculated using the Mori equation. (See the chapter on "Wave Motion".) Equation ó25Â is plotted in figure í10Ê and labeled as "Poisson". (Note that the minus sign has been dropped for convenience in plotting.) The Poisson equation agrees with the Pochhammer equation up to about 40 mm diameter. Above this, the error is no greater than about 5%.

Note that the above equation could be used to empirically determine Poisson's ratio, simply by measuring the radial and axial amplitudes and by measuring the tuned length. The horn would have to be driven at high amplitudes to get an appreciable radial amplitude.

Empirical data

Figure í10Ê shows that both the Pochhammer and Poisson equations agree well with the measurements on a ø125 Aerospace horn (HRD 156).

Anomalies

Horn HRD 054 (the third data point on the fitted curve, face diameter = 57.4 mm, EDP = 318 002 020), seems to have abnormally high uniformity as compared to the fitted curve. This could be because the fitted curve is too low at this point or because of error in measuring the uniformity. It may also be that the large back diameter of this horn (102 mm) somehow improves the face uniformity. Include figure of horn dimensions.

ø125 Aero Al spool anomalies:

HRD Date D₂/D₁ Conv U

073 4 8/26/82 1.000 CU90215D 0.44
073-7 7/22/83 0.854 CU90215D 0.709
073-8 7/22/83 0.854 CU80754D 0.715

059 2 6/17/82 1.000 CU90215D 0.443
059 8 7/20/82 0.839 CU90215D 1.016

Both ø125 spools have same stud size and depth. FEA gives good agreement with HRD 059 8. Include figure of amplitude trends as D₂/D₁ is reduced.

Effect of horn stud and joint

In this section we will look at how the horn stud and the horn joint affect the face and stud-surface uniformity. These effects are especially important when using FEA to determine the horn uniformity and stress. (See the chapter on "Stress".)

Introduction

Consider an unshaped horn. Assume that the front half of the horn is identical to the back half of the horn    i.e., the horn is symmetric about its midplane. Also assume that the horn vibrates in its axial mode without any attached resonators (i.e., no attached booster or converter). Then, the amplitudes of the face will be exactly the same as the amplitudes of the stud surface.

Introducing a stud will upset the dimensional symmetry of the horn. Since the stud is a relatively small portion of the total horn, we might think that the stud would have little effect. This is not true. The stud can significantly affect horn amplitudes and, as a result, the horn gain and stress.

Adding a resonator (booster or converter) to the horn can also affect the horn amplitudes. When two resonators are joined together, they will not affect each other if:

1. Both had the same frequency before being joined.
2. Both had the same amplitude distributions over their mated surfaces before being joined.

For axial resonators whose amplitudes are essentially axial, these conditions are approximately true. Thus, for "normal" horns at axial resonance, the joint has little effect on horn amplitudes. However, for horns that have significant transverse motion, the joint can significantly affect horn amplitudes. The joint can also affect on the resonant frequencies of nonaxial modes. (See the chapter on "Frequency Separation".)

The following sections give examples that show the effect of the stud and/or joint.

Solid, unshaped cylindrical aluminum horns

For solid, unshaped cylindrical aluminum horns, we will first determine the effect of a stud on face and stud-surface uniformity. We will then be able to determine the effect of the joint on face and stud-surface uniformity. Except as noted, the results of this section were determined by FEA. Figure ífÊ shows a typical FEA model.

Effect of the stud

For FEA models of the type shown in figure ífÊ, figure ífÊ shows how the face uniformity of solid, unshaped cylindrical aluminum horns changes as the horn diameter increases. A ½ 20 x 19.0 deep stud improves the face uniformity slightly compared to a horn without a stud. The largest improvement occurs for the largest diameter horns (about 8% improvement for a ø125 horn). Note that the Pochhammer equation agrees more closely with the horns that have a stud. The reason is not known.

Figure ífÊ shows that a ½ 20 x 19.0 deep stud reduces the stud-surface uniformity compared to a horn without a stud. The reduction is significant for most horn diameters and is largest for the largest diameter horns (about 19% uniformity reduction for a ø125 horn).

With a ½ 20 x 19.0 deep stud, the gain decreases as the horn diameter increases. For a ø125 horn, the gain is only 0.903. (See the chapter on "Gain".)

Effect of the joint

To determine the effect of the joint, the cylindrical horn is modeled with an attached Aluminum Standard. The Aluminum Standard simulates either a booster or converter. Figure ífÊ shows a typical FEA model with a ½ 20 x 38.1 stud centered at the joint. A corresponding FEA model of the horn without a joint is shown in figure ífÊ.

Figure ífÊ shows that adding the Aluminum Standard horn to an unshaped cylindrical aluminum horn has negligible effect on the face uniformity. The stud-surface uniformity (.f.figure ífÊ) improves slightly when the Aluminum Standard horn is added. The improvement is about 7% for a ø125 horn.

The joint has a small effect on horn gain. With a ½ 20 x 38.1 stud at the joint, the gain of a ø125 horn increases from 0.903 to 0.937 (3.8%) when the Aluminum Standard is added.

Thus, with or without the joint or stud, the face uniformity of unshaped cylindrical horns shows little change. The joint has a small effect on stud-surface uniformity, while the stud has a relatively large effect on stud-surface uniformity. These conclusions apply to a ø12.7 stud that extends 19.0 into the horn and/or booster. Other stud diameters are considered in the next section.

ø125 unshaped aluminum horn

Since the effect of stud and joint are largest for the larger diameter horns, we will look more closely at a ø125 unshaped Aerospace aluminum horn. Several horns of this size have been measured empirically, which will permit comparison to FEA results.

Effect of the stud diameter. Figure ífÊ shows a FEA model of a ø125 aluminum horn that was used to determine the effect of stud diameter on horn amplitudes. The stud diameter was adjusted as required, while the rest of the mesh remained unchanged. All studs were 19.0 deep into the horn.

Figures ífÊ and .f.ífÊ show the amplitude distributions for the face and stud surface. For figure ífÊ, the amplitudes have been normalized with respect to the amplitude at the center of the face. For figure ífÊ, the amplitudes have been normalized with respect to the amplitude at the center of the stud surface. .f.Figure ífÊ shows the face and stud-surface uniformities. These figures show that the stud mainly affects the stud surface; it's effect on the face is much smaller. As expected, the largest diameter studs have the largest effect._

The empirical data of figure ífÊ generally agrees with the FEA data, despite the fact that the FEA model did not have a joint. For HRD 156 4, the amplitudes were measured at four locations at each of three diameters on the face and stud surface. (See figure ífÊ.) For HRD 059 2 and HRD 073 4, the amplitudes were measured at eight locations at a single diameter on the face and stud surface.

Effect of the joint. Figures ífÊ and ífÊ show the FEA model of the ø125 horn, without and with an attached Aluminum Standard.

For this ø125 horn, the effect of the joint is relatively small. Figures ífÊ and ífÊ shows the relative face and stud-surface amplitudes, with and without a joint. The joint decreases the horn's face uniformity from 0.372 to 0.368 (a 1.1% decrease) and increases the stud-surface uniformity from 0.278 to 0.298 (a 7.2% increase). For the stud surface, the amplitude difference grows most quickly over the region of the joint (i.e., up to a radius of 19.0 mm). Thereafter, up to a radius of 62.5 mm, the amplitudes actually converge.

ø125 unshaped titanium horn

The effect of the stud has also been analyzed for a ø125 unshaped Ti 7Al 4Mo horn (no joint). Figures ífÊ and .f.ífÊ show the effect of a ½ 20 x 12.7 deep stud on the face and stud-surface amplitudes. The effect on the face is negligible and the effect on the stud surface is small (2%).

A comparison of figure ífÊ with figure ífÊ and figure ífÊ with figure ífÊ shows that the amplitudes of the titanium horn are much less affected by the stud than is the aluminum horn. Also, the gain is reduced less when the stud is added to the titanium horn (0.967 for a ø125 titanium horn versus 0.903 for a ø125 aluminum horn). These results are reasonable, since the stud is inserted less deeply in the titanium horn than in the aluminum horn (12.7 mm versus 19.0 mm, per normal usage of ø12.7 studs) and since the density of the steel stud is closer to that of titanium than aluminum. Thus, the stud appears less foreign to the titanium horn than to the aluminum horn.

ø176 unshaped aluminum horn

Figure ífÊ shows a ø175 x 19.0 long aluminum horn vibrating at axial resonance (19956 Hz). (Note: this mode can also be construed as a radial mode because of the high radial amplitude. See the chapter on Modeshapes.) Because the horn is symmetric, the axial mode is also symmetric.

Now we join an Aluminum Standard without stud to the horn. (The axial frequency of the Aluminum Standard is 19953 Hz, 3 Hz lower than the ø176 horn.) The frequency of the resulting stack is 20239. The modeshape of the ø176 horn becomes contorted (figure ífÊ«axial" modeshape for ø176 al horn on al standard horn, no joint stud;».) The resulting mode may, in a sense, still be considered axial, since it is derived from the union of two axial resonators and because it is descended from horns of smaller diameter whose modes are clearly axial. See the chapter on Modeshapes for further discussion.

The contorted modeshape results from a mismatch of joint amplitudes. By itself, the ø175 horn has high radial motion at a radius of 19 mm, which would be at the edge of the joint if a booster was attached. (See figure ífÊ.) The Aluminum Standard has very low radial motion at its free end. The mismatch in joint amplitudes when the Aluminum Standard is joined to the ø176 horn causes a high bending moment on one side of the ø176 horn. This asymmetric loading causes the contorted modeshape. (Note the high distortion at the joint end of the Aluminum Standard in figure ífÊ.)

100 X 100 unshaped aluminum horn

Figure ífÊ shows the dimensions for a 100 x 100 unshaped aluminum horn (HRD 456 3). (The material is aluminum. The type is probably 7075, although this is not certain.) .f.Figure ífÊ shows the face and stud-surface amplitudes of this horn when driven by converter CU01296D. As compared to the face corner amplitudes, the stud-surface corner amplitudes are 18% lower. As compared to the face mid-edge amplitudes, the stud-surface mid-edge amplitudes are 4% lower.

Move?Because of the asymmetry of the mid-edge amplitudes of the face and stud surface, this horn was retested on converter CU02207D. The amplitudes are shown in figure ífÊ. As compared to the face corner amplitudes, the stud-surface corner amplitudes are 21% lower. As compared to the face mid-edge amplitudes, the stud-surface mid-edge amplitudes are 3% lower.

Because the relative face and back amplitudes are essentially the same for both converters and because the orientation with respect to the reference mark does not change when the converter is changed, we may assume that the amplitudes are inherent to the horn. The reason for the asymmetry of the face and stud-surface mid-edge amplitudes is discussed in the section on "Amplitude Asymmetry".

Since this horn is symmetric about its midplane except for the horn joint and stud, the difference in amplitudes between equivalent locations on the face and stud surface must be due to the effect of the joint and stud.

127 X 127 double slotted rectangular horn

We will look at two 127 x 127 horns double slotted aluminum horns. The first (HRD 002) is symmetric about its midplane. The second (HRD 001) is not. For both horns, studs of different lengths were used to see the effect on horn amplitudes.

HRD-002

Figure ífÊ shows a 127 x 127 horn made of 2024 QQA225 plate. Except for the 19.0 mm deep stud hole and the stud, the horn is essentially symmetric about its midplane.

(Note: Just prior to the horn configuration shown in figure ífÊ, the rear web was 22.0 mm long. The horn was then tuned from the stud surface, reducing the depth of the stud hole to 16.6 mm. It is not known if the stud hole was then retapped to a depth of 19.0. If not, then the stud would protrude 21.4 mm instead of 19.0 mm.)

Figure ífÊ shows the face and stud-surface amplitudes when a standard ½ 20 x 38.1 stud (per A STD 1580 1) is bottomed in the stud hole. (All amplitudes are normalized to 100 microns with respect to the amplitude at the center of the horn face.) In the corners, the face amplitude is about 2% higher than the stud-surface amplitude. At the mid-edges, the face amplitude is about 9% lower than the stud-surface amplitude. Since the horn dimensions are symmetric except for the stud and converter-horn joint, the amplitude differences between the face and stud surface must be due to the effect of the stud and/or joint.

To determine the effect of the stud, the standard stud was replaced by a ½ 20 x 19.1 stud with only 6.3 mm extending into the stud hole (i.e., unbottomed stud). (See figure ífÊ. The stud protrusion is approximate.) With the shorter stud, the face and back amplitudes changed to those in .f.figure ífÊ.

At the corners, the face amplitude is 0.5% lower than the stud-surface amplitude. At the mid-edges, the face amplitude is 1% higher than the stud-surface amplitude. Thus, when a shorter stud that is inserted less deeply into the horn, the face and back amplitudes become almost equal. This indicates that the stud exerts significantly more influence over the horn's amplitude than does the joint.

Comparing the amplitudes of figures ífÊ and ífÊ, we see that reducing the stud length increased all amplitudes in relation to the face center amplitude. Also, the highest amplitude on the stud surface moved from the mid-edge to the corner.

With a shorter stud, the overall face uniformity improved from 0.855 (central) to 1.106 (central). This suggests that there may be a compromise stud length for which the face uniformity would be more nearly equal to 1.0. The face edge uniformity improved from 0.932 to 0.961 when the stud length was decreased.

The horn gain decreased by about 15% with the shorter stud. This is not reflected in figures ífÊ and ífÊ because the amplitudes have been normalized to 100 microns at the face center. Note: the gain change is only approximate, because the amplitude measurements were made at unregulated line voltage.

HRD-001

Figure ífÊ shows another 127 x 127 horn made of 2024 QQA225 plate. This horn is nominally identical to the previous horn, except that the slot is 2.6 mm closer to the face. Thus, the front and back slot webs are no longer equal    the front web is 5 mm shorter than the back web. Since this horn is not symmetric about its midplane, we should not expect the face and stud surfaces to have amplitudes.

With the standard ½   20 x 38.1 stud (per A STD 1580 1) that is bottomed in the stud hole, the face and stud-surface amplitudes are shown in figure ífÊ. (As above, all amplitudes are normalized to 100 microns with respect to the amplitude at the center of the horn face.) When this stud was replaced by a ½   20 x 19.1 stud with only 6.3 mm extending into the stud hole (.f.Figure ífÊ), the face and stud-surface amplitudes changed to that of .f.figure ífÊ.

As with HRD 002 above, reducing the stud length increased all amplitudes in relation to the face center amplitude. Also, the corner amplitude on the stud surface increased in relation to the other stud-surface amplitudes.

The overall face uniformity improved slightly from 0.887 (central) to 0.894 when the stud length was decreased. The face edge uniformity improved slightly from 0.924 to 0.946. The horn gain decreased by about 9%.

Conclusions

In most cases, the FEA generally gives good agreement with empirical results, whether or not the FEA model includes a joint. Also, as the size or depth of the stud is reduced, the face amplitudes approach the stud-surface amplitudes for horns that are symmetric about their mid-plane. This indicates that the effects of the joint on horn amplitudes can generally be neglected. This may not be true if the joint has excessive radial motion.

1. The stud has a significant effect on the horn amplitudes. For cylindrical horns up to ø125, the effect is especially pronounced on the stud surface. For rectangular horns, the stud affects the amplitudes and amplitude distributions of both the face and stud surface. Aluminum horns appear to be more affected than titanium horns. This effect of the stud is important for FEA. If the stud is not included, then the FEA-predicted uniformities, gains, and stresses may err significantly. (Also see the chapters on "Gain" and "Stress".)
2. Note possibility of adjusting face amplitude by changing masses of elements, ala riser horns -- or possibly by adding adjustable studs to elements or removing mass.
3. Presence of joint (actually, the other stack components) affects resonances of nonaxial stack resonances.

Above generally discussion ignores effect on gain and nonaxial resonances.

Define a term for a horn that is symmetric about its midplane. Define "free" (or "free-free") resonator.

(Also, see the chapter on "Loss", where very preliminary data indicates that lighter studs may cause lower joint loss than standard studs. Is this true? E. Holze tests.)

See chapters on gain and …

Effect of adjacent nonaxial resonances

HRD-456: 100 x 100 square al horn.

100 mm spool

Personal products horn

13" frito horn.

General methods for improving uniformity

Flutes (explain here and reference in section on spool horns).

Angled slots (Put in separate section on bar horns?)

Spool horn design

In this section we will look at how to design spool horns to correct the amplitude nonuniformity of unshaped solid cylindrical horns.

As we have seen in the above section, unslotted solid cylindrical horns have lower average amplitude at the edge of the horn face than at the center. This condition is called amplitude droop. When the diameter is greater than about 75 mm, the droop may cause a problem for critical welds where good uniformity is required. This problem can be corrected by modifying the horn shape to increase the amplitude toward the horn edge, thereby eliminating the droop.

Principle of operation

With unslotted solid cylindrical horns, the amplitude is highest at the center of the horn face and decreases smoothly to the edge of the horn. (We will be assuming for the time being that any amplitude asymmetry is small.) Thus, to correct the amplitude nonuniformity, we need a technique that has the greatest effect at the edge of the horn with gradually decreasing effect toward the center of the horn face.

This can be achieved by machining the horn into a spool shape. (See figure í14Ê«».) As the material behind the horn face is removed, a flange is created. Because the flange is no longer supported from behind, its amplitude will increase. Thus, the flange acts like a cantilevered beam that is being forced to vibrate at the same frequency as the horn.

The largest amplitude increase will be at the edge of the horn where the flange material is farthest from the supporting body of the horn. The influence of the flange decreases toward the center of the horn face, so that the amplitude increase will not be as large as at the edge. The least amplitude increase will be exactly at the horn centerline. This is exactly the desired effect to reduce the droop and increase the uniformity.

General design method

Figure í14Ê shows what is called the standard spool horn. (We will look later at the full spool horn.) Of the parameters shown in the figure, only D1, D2, F, and R1 will influence the face uniformity. The other parameters (D3, L1, L2, and R2) are not near the flange and therefore have little effect on face uniformity.

The first step in any horn design is to size the face to fit the part. Thus, D₁ is chosen first and is fixed. Then assuming that high gain is not needed from the horn, we can choose D₂ = D₁. L₁ will be determined by the tuned frequency. For convenience and maximum gain, we will choose L2 to be approximately ½ of L₁. (Note: L₁ and L₂ would not normally be known until the horn was tuned. Later I will give some guidelines for estimating L₁ and choosing L₂ prior to tuning.)

The next step is to choose R₁ and F. D2 is then machined until the amplitude at the edge of the horn is equal to the amplitude at the horn centerline. The lowest amplitude will then be located somewhere between the horn edge and centerline. The amount by which this lowest amplitude differs from the centerline amplitude will determine the horn uniformity. Hence, the horn uniformity is essentially fixed by the initial choice of R1 and F.

To see the effect of F, let us assume initially that R1 = 0. (Of course, this would not be a good choice for practical horns.) Now what happens if F is small?

Face reliefs

In the above discussion, we have assumed that the horn face is flat. However, face reliefs can be used without distorting the uniformity if several guidelines are followed:

1. If possible, the relief should be circular with its center located at the horn axis. Non-circular reliefs should be correctly balanced on the horn face. (See the chapter on "Contoured Horns" for a discussion of horn balance.)
2. Machine the relief into the face of the unshaped horn blank before any other dimensions are machined. Then proceed as normal if conditions 3 and 4 below are satisfied.
3. If the rellief diameter is larger than D₂, its depth should be kepth small so that the flange stiffness will not be significantly reduced by the relief.
4. If the relief diameter is much smaller than D₂, its depth can be significantly deeper without affecting the flange stiffness.
5. If the relief diameter and depth are both large, then the horn probably will not tune properly. This is a problem with all unslotted cylindrical horns. Figure í15Ê shows the effect of increasing the inside diamter of an unslotted cylindrical horn. The axial resonant frequency decreases very quickly as the relief diameter becomes large. A spool horn of similar dimensions has been tried with the same result.

Maximum spool horn size

The largest spool horn that we have successfully made is 125 mm. We have tried to design spool horns of 152 mm and 178 mm. These larger horns have presented three problems:

1. Excessive transverse face amplitude.
2. Adjacent high-side resonances.
3. Reduced uniformity. These horns could not achieve the required 0.90 uniformity, possibly because of the close adjacent resonance.

It may be possible to develop a spool size between 125 mm and 150 mm. I have heard reports of a 140 mm spool horn but I know nothing of its performance.

Problems with the 100 mm standard spool

Square versus cylindrical horns (unslotted)

Suppose you have an application that requires welding a circular part across its entire diameter. If you are concerned about the horn face uniformity, should you use an unshaped square horn or an unshaped cylindrical horn (assuming neither is slotted)?

Let's compare a 100 mm square horn with a 100 mm cylindrical horn, since this is the only case for which I have data. The face amplitudes for the 100 mm square horn (HRD 456) are shown in figure í16Ê«». You will note that amplitudes have been taken at eight locations around a circle of 100 mm diameter. The average amplitude for these eight locations is 12.05 microns. Comparing this value to the centerline amplitude of 19.4 microns, the uniformity around the circle is —

12.05 microns
¹) U_circle =              
19.4 microns


= 0.621

A repeat of this measurement on a second converter gave a face uniformity of 0.620.

Now, how does the uniformity of this square horn compare to that of a 100 mm diameter cylindrical horn? Table å1Ï lists the uniformities of 7 horns whose diameters are very close to 100 mm. The average uniformity of these seven is 0.675. Thus, when the uniformity is measured at the same diameter, the unshaped cylindrical horns gives somewhat better uniformity than the unshaped square horn.

TABLE å1Ï

Face Uniformity for Cylindrical Horns;

Face Rel. Transverse
HRD # Date Dia Uniformity Face Amplitude Asymmetry Converter Material Notes

454 8/15/80 38.1 0.979     0.018 CU90651D 3 Ti Standard

053 6/23/82 50.8 0.944     0.271 CU90224D 3 Full-wave
053 6/23/82 50.8 0.938     0.055 CU90651D 3 Converter changed

054 6/24/82 57.4 0.964     0.013 CU90224D 3 High gain

061 7/9/82 75.0 0.880 0.000 0.042 CU90215D 1
082 9/3/82 75.0 0.866 0.013 0.014 CU90215D 1
065 7/16/82 75.0 0.861     0.041 CU90215D 1

050 3/24/82 88.6 0.746 0.012 0.155          2
436 10/5/85 89.3 0.786 0.012      CU90651D 1
079 8/25/82 90.0 0.759 0.019 0.106 CU90215D 1

060 7/9/82 100.0 0.690 0.018 0.157 CU90215D 1
080         100.0 0.681                  1
064 7/14/82 100.0 0.736     0.152 CU90215D 1

051 1 6/23/82 101.3 0.666     0.533 CU90224D 2
052 1 6/23/82 101.3 0.683     0.314 CU90224D 2
052 1 6/23/82 101.3 0.707     0.302 CU90224D 2 Recheck above
052 2 6/23/82 101.3 0.655     0.559 CU90615D 2 Converter change
            101.6 0.652                     FEM analysis

081 8/25/82 110.0 0.591 0.026 0.011 CU90215D 1

073 8/26/82 125.0 0.440 0.041 0.015 CU90215D 1
059 7/9/82 125.0 0.441 0.038 0.026 CU90215D 1
156 9/17/85 124.9 0.419 0.043     CU90651D 1

1. Notes:
2. To determine the uniformity, amplitude measurements were made at two locations on the horn face:
   a) at the center of the horn face
   b) with the center of tha amplitude probe at 3.8 mm
   from the edge of the horn.
3. Materials:
   1 ==> Aerospace aluminum rod (C_o = 5084 m/sec)
   2 ==> 2024 QQA225 aluminum plate (C_o = 5167 m/sec)
   3 ==> 7 4 Titanium rod (C_o = 5037)

TABLE å2Ï

Face Uniformity for Unslotted Bar Horns;

Face Face Back
HRD # Date Width Thick Thick U_width U_thick U_overall Ŵ Converter Mat'l

185 8/29/83 62.7 6.3 ø62.7 0.931           0.000          4

447 1/10/86 63.5 5.9 63.5 0.948           0.000 CU90224D 5
448 1/10/86 63.4 6.3 63.4 0.947           0.020 CU90224D 5
248 9/25/85 63.4 13.4 63.5 0.939           0.025 CU90224D 3
396 6/5/85 63.6 19.1 38.2 0.922           0.013 CU90215D 2
393 4/23/85 63.4 22.3 38.0 0.941           0.014 CU90224D 2

457 11/7/80 69.9 8.9 ø69.9 0.919           0.000          4
458 11/22/83 69.9 8.9 ø69.9 0.918           0.006 CU20595D 4
316 9/20/84 69.7 25.4 63.6 0.923           0.033 DK60033 2
315 9/20/84 69.8 25.7 63.6 0.922           0.011 DK60033 2

423 3/23/81 75.7 38.2 38.2 0.839           0.006 CU90651D 2
424 3/17/81 75.8 38.3 38.2 0.848           0.005 CU90651D 3

392 4/23/85 82.6 12.4 38.1 0.904           0.075 CU90224D 2
445 1/10/86 82.5 18.7 63.5 0.899           0.101 CU90224D 2
444 1/10/86 82.5 19.0 62.9 0.865           0.044 CU90224D 1
446 1/10/86 82.6 19.0 62.8 0.863           0.089 CU90224D 2
443 1/10/86 82.4 19.3 63.5 0.889           0.059 CU90224D 1
455 4/24/86 82.6 62.8 62.8 0.760 0.920 68.0 0.033 CU01296D 1

066 7/26/82 90.0 90.0 110 X 110 0.676 0.676 33.5 0.071 CU90215D 1

456 5/21/86 100.1 100.0 100 x 100 0.557 0.714 26.3 0.220 CU01296D 1
456 5/21/86 100.1 100.0 100 x 100 0.561 0.704 26.5 0.203 CU02207D 1

1. Notes:
2. See figures ífÊ and ífÊ for definitions of Width Uniformity and Thickness Uniformity.
3. Overall Uniformity is the worst average uniformity on the horn face.
4. Materials:
   1 ==> Aluminum [what kind?]
   2 ==> Titanium bar (stud axis in long-transverse
   material direction)
   3 ==> Titanium bar (stud axis in longitudinal
   material direction)
   4 ==> Titanium rod (machined into bar-cylindrical
   horn)
   5 ==> Titanium (unknown material direction)

[Note: 100 x 100 horn listed 2X    with 2 different converters, since data was taken to see if its asymmerty was associated with a particular converter.]

[What is correct equation of Asymmerty:

(Maximum ampl - Min ampl)
Asm =                           ?
Max ampl

where max and min ampl are measured at geometrically end similar location. If this eqn is correct, then verify that table values are calculated from this eqn. HRD 066, 456, and 455 have been calculated by this definition.

Asymmetry is the maxumum asymmetry on the horn face, not just across the horn width or thickness.

Note: Do not use Min ampl in denominator, as this could lead to division by 0 in some cases.]

TABLE å3Ï

Face Uniformity across Thickness:

Slotted Bar Horns;

Face Face Back
HRD # Date Slots Width Thickness Thickness Uniformity Asymmetry Converter Material

044 8/28/81 2 152.4 50.8 50.8 0.937 0.005         1

044 8/27/81 2 152.4 63.5 63.5 0.894 0.029         1

044 4/28/81 2 152.4 76.2 76.2 0.812 0.023         1
045 8/25/81 2 152.4 76.2 76.2 0.791 0.012         1

035 9/1/81 2 228.6 76.2 76.2 0.780 0.014         1

049 7/15/81 4 228.6 76.2 76.2 0.795     CU80791D 1

Notes:

1. Uniformity and Asymmetry are measured across the thickness dimension of the horn face, centered along the width of the horn.
2. Materials:
   1 ==> Aluminum

TABLE å4Ï

Face Uniformity Across Width:

Slotted Bar Horns;

Face Face Back
HRD # Date Slots Width Thickness Thickness Uniformity Asymmetry Converter Material

275 5/24/84 2 155.6 13.6 63.5 0.795 0.066 CU90224D 2
287 7/10/84 2 155.6 13.6 63.5 0.808 0.009 CU90224D 2
288 7/10/84 2 155.6 13.6 63.5 0.801 0.026 CU90224D 2

034 4/30/82 2 228.6 11.8 38.1 0.635 0.032 CU90215D 2

Notes:

1. Uniformity and Asymmetry are measured across the thickness dimension of the horn face, centered along the width of the horn.
2. Materials:
   1 ==> Aluminum
   2 ==> Titanium
3. HRD 255, 287, 288 are Personal Products tapered blade.

Unshaped, unslotted bar horn design

In this section we will look at uniformities of unslotted bar horns.

Figure ífÊ shows typical amplitude measurement locations for unslotted bar horns. For these horns, the highest face amplitude always occurs at the center of the horn face. The lowest amplitudes occur at the corners. The midedges have intermediate amplitudes. This assumes that there is no asymmetry and that the horn has not been modified (e.g., with flutes, etc.) to alter the face uniformity.

Table å1aÏ shows the uniformities of unshaped, unslotted bar horns. The values in this table were determined by 3-dimensional FEA for Aerospace aluminum. Because the model used brick elements, the stud was modeled as a rectangular element (11.2 x 11.2 x 19.0 deep, which is equivalent in mass to a ø12.7 x 19.0 deep stud). FEA has verified that this "brick" stud gives essentially the same results as a normal cylindrical stud.

_Must define U_thick and U_width. How close is the amplitude measured from the edge of the face?

Note that some of the data in table å1aÏ is redundant. For example, the data for an 80 mm wide x 20 mm thick horn is exactly the same as for a 20 mm wide x 80 mm thick horn. (In fact, these are the same FEA model.) This is because the width and thickness are not physically distinguishable for unshaped, unslotted bar horns. (Note: for most bar horns, the horn width is always taken to be greater than the horn thickness. For this section, this convention does not apply.)

Figures ífÊ and .f.ífÊ show the horn uniformities across the horn width and thickness at the middle of the associated edge. Note that these two figures are identical except for axis labeling. (The reason is discussed in the previous paragraph.)

Uniformity for square double cross-slotted horns

We can draw two conclusions from figure ífÊ:

1. For a given thickness, wider horns have poorer uniformity across the width.
2. For a given width, thicker horns have better uniformity across the width. However the difference in uniformity is not significant if the horn thickness is less than 60 mm.

We can draw similar conclusions from figure ífÊ:

1. For a given width, thickner horns have poorer uniformity across the thickness.
2. For a given thickness, wider horns have better uniformity across the thickness.

Figure ífÊ shows that the corner uniformity decreases as either the the horn width and thickness increases. Note, however, that as the horn thickness increases, the horn width has much less effect on the corner uniformity.

_Show face amplitude distributions for some horns, especially HRD 456 (100x100). Compare FEA and empirical ampls. Note: amplitudes always lowest at edge for unslotted horns. Give empirical data for unshaped, unslotted bar horns: HRD-423, 424, 455.

TABLE å1aÏ

FEA Face Uniformity for Unshaped, Unslotted Bar Horns;

Face Face Tuned Tuning
Thick Width Length Rate  U_thick U_width U_corner Gain

Per Z20x20c1 below but with cylindrical ø12.7 stud, to compare with "brick" stud.
600 brick vs 48 brick for Z20x20c1.»

20.0 20.0 119.0 -154 0.992 0.992 0.985 1.137
20.0 40.0 122.0 -165 0.996 0.959 0.952 1.047
20.0 60.0 122.1 -156 0.996 0.874 0.867 1.003
20.0 80.0 120.2     0.995 0.678 0.672 0.966
20.0 90.0 117.8     1.000 0.501 0.497 0.940
20.0 100.0 113.2 -102 0.999 0.308 0.307 0.843

40.0 20.0 122.0 -165 0.959 0.996 0.952 1.047
40.0 40.0 123.6 -153 0.958 0.958 0.914 1.016
40.0 60.0 123.3 -152 0.964 0.880 0.843 0.995
40.0 80.0 121.3 -145 0.973 0.691 0.661 0.976
40.0 90.0 119.2     0.988 0.511 0.492 0.961
40.0 100.0 114.9     0.997 0.252 0.246 0.918

60.0 20.0 122.1 -156 0.874 0.996 0.867 1.003
60.0 40.0 123.3 -152 0.880 0.964 0.843 0.995
60.0 60.0 122.9 -143 0.892 0.892 0.780 0.984
60.0 80.0 121.1     0.917 0.714 0.625 0.972
60.0 90.0 119.1     0.947 0.541 0.479 0.965
60.0 100.0 115.0 -115 0.984 0.247 0.226 0.932

80.0 20.0 120.2     0.678 0.995 0.672 0.966
80.0 40.0 121.3 -145 0.691 0.973 0.661 0.976
80.0 60.0 121.1     0.714 0.917 0.625 0.972
80.0 80.0 120.0     0.771 0.771 0.535 0.966
80.0 90.0 118.3     0.832 0.610 0.430 0.962
80.0 100.0 115.2 -116 0.938 0.285 0.216 0.938

90.0 20.0 117.8     0.501 1.000 0.497 0.940
90.0 40.0 119.2     0.511 0.988 0.492 0.961
90.0 60.0 119.1     0.541 0.947 0.479 0.965
90.0 80.0 118.3     0.610 0.832 0.430 0.962
90.0 90.0 117.5     0.685 0.685 0.353 0.958
90.0 100.0 114.8     0.851 0.317 0.213 0.934

100.0 20.0 113.2 -102 0.308 0.999 0.307 0.843
100.0 40.0 114.9     0.252 0.997 0.246 0.918
100.0 60.0 115.0 -115 0.247 0.984 0.226 0.932
100.0 80.0 115.2 -116 0.285 0.938 0.216 0.938
100.0 90.0 114.8     0.371 0.851 0.213 0.934
100.0 100.0 113.8     0.585 0.585 0.154 0.945

Notes:

1. All horns are Aerospace aluminum with the equivalent of a ø12.7 x 19.0 deep stud.
2. The tuned length is at 19950 Hz.

Single-slotted bar horn design

Effects: slot length (see Tetra Pak 96 mm), earing (See E. Holze work on thin slabs)

Riser horn design

Effects: slot length, slot spacing (element width), riser design (wedge, etc), slot width. Alternate approaches to changing relative frequency between horn elements: hole in end elements, either through thickness or width; pushing nodal radius of end elements. Effects on stress?

Uniformity

A multivariable linear regression has been performed to predict the amplitudes for the axial mode. The amplitude data was taken from 13 127 x 127 horns (made of 2024 QQA225 aluminum plate material) and three 125 x 125 horns (made of Aerospace bar material). Approximately 30 data points were used. (The raw data is given in appendix E.)

The 125 x 125 horns were included because they have amplitude data at lower frequencies than were measured for the 127 x 127 horns. The small difference in horn widths should not have a significant effect on the amplitudes. However, the Aerospace aluminum tunes approximately 1/2 mm shorter than the 2024 aluminum. Thus, a tuned Aerospace horn will have approximately 1/2 mm shorter front web than an equivalent tuned 2024 horn, which may somewhat affect the horn amplitudes. However, I have chosen to neglect any error that this may cause.

Figure í13Ê shows typical locations for amplitude measurements.

The following equation can be used to predict the amplitudes of the axial mode at frequencies near 20 kHz:

¹) Amplitude_ratio

= B0 + B1 * Horn_length + B2 * Slot_length

+ B3 * Slot_width + B4 * Slot_spacing

+ B5 * Back_web

All dimensions are in mm. The constants B0 .. B5 depend on where the amplitude is measured on the horn face.

The Amplitude_ratio is defined as:

²) Amplitude_ratio

Average amplitude at specified location on horn face
=                                                     
Amplitude at the center of the horn face

Generally (but not always), the amplitudes were measured at four symmetric locations on the horn face and then averaged to calculate the numerator of equation ó3Â. For example, four corner amplitudes would be measured to calculate the Corner_amplitude_ratio (below).

Corner amplitude_ratio

The Corner_amplitude_ratio (Å_C) is defined as:

Average amplitude at the corners of the horn face
³) Å_C =                                                  
Amplitude at the center of the horn face

The constants for Å_C are:

B0 = -1.758E-2 [17.47]

B1 = 9.009E-3 [0.276]

B2 = -5.900E-3 [0.255]

B3 = -6.075E-2 [0.133]

B4 = 1.169E-2 [0.168]

B5 = 2.091E-2 [0.165]

As above, the numbers in brackets are the coefficients of variation (CV) for the associated equation coefficient.

Comparing this linear equation to the empirical amplitude data from which it is determined:

Average absolute error = 1.66%
Maximum absolute error = 4.81%
Multiple correlation coefficient = 0.9758

Midedge_amplitude_ratio

The Midedge_amplitude_ratio (Å_M) is defined as:

Average amplitude at midedge of the horn face
⁴) Å_M =                                              
Amplitude at the center of the horn face

The constants for Å_M are:

B0 = -9.202E-1 [0.267]

B1 = 1.340E-2 [0.145]

B2 = -3.971E-5 [32.62]

B3 = -1.769E-2 [0.362]

B4 = 8.393E-3 [0.182]

B5 = 2.898E-3 [0.964]

Comparing this linear equation to the empirical amplitude data from which it is determined:

Average absolute error = 1.30%
Maximum absolute error = 4.34%
Multiple correlation coefficient = 0.9177.

Slot_intersection_amplitude_ratio

The Slot_intersection_amplitude_ratio (Å_S) is defined as:

Average amplitude at the slot intersections
⁵) Å_S =                                            
Amplitude at the center of the horn face

Note: the numerator of this equation is the actually minimum amplitude that is found as the amplitude is scanned from the corner to the center of the horn face. This amplitude dip occurs close to the slot intersections.

The constants for Å_S are:

B0 = -1.685 [0.159]

B1 = 1.999E-2 [0.107]

B2 = -5.952E-3 [0.259]

B3 = 2.917E-3 [6.194]

B4 = 1.293E-2 [0.127]

B5 = 4.454E-3 [0.681]

Comparing this linear equation to the empirical amplitude data from which it is determined:

Average absolute error = 1.46%
Maximum absolute error = 4.66%
Multiple correlation coefficient = 0.9331.

Discussion of the regression equation

Except as noted below, the regression equation coefficients were determined over the following range:

Horn_length: 111.4   > 121.0
Slot_length: 69.9   > 88.1
Slot_width: 9.5   > 12.7
Slot_spacing: 39.2   > 51.8
Back_web: 18.3   > 25.1

Horn HRD 472 (Slot_length = 88.1) had high amplitude asymmetry (figure í14Ê), so it was not considered in calculating the regression coefficients for Å_M and Å_S. Therefore, for these regression coefficients, the maximum range of the Slot_length was reduced from 88.1 to 80.9.

Horn HRD 006 (Slot_width = 12.7) had continuously increasing amplitude from the corner to the center of the horn face (i.e., no amplitude dip over the slot intersections), so no amplitude measurements were made over the slot intersections. Therefore, this horn was not considered in calculating the regression coefficients for the Å_S. Thus, for these regression coefficients, the maximum range of the Slot_width was reduced from 12.7 to 10.0.

Looking at the signs of the regression coefficients, we see that the face amplitude ratios increase with increasing slot length, slot spacing, and back web length. The amplitude ratios increase as the horn length decreases (i.e., generally as the front web length decreases). The effect of slot width depends on which amplitude ratio is being considered.

Note that the coefficient of variation of some of the equation coefficients is quite large, especially for B[0] for Å_C (17.47), B[2] for Å_C (32.62), and B[3] for Å_S (6.194). These large coefficients of variation indicate that there we should have little confidence in the associated equation coefficient.

The majority of the horns had dimensions close to those of figure í1Ê, with slots either 9.5 or 10.0 wide. (See appendix E.) This would be considered a "normal" horn with equal element widths. Therefore, the regression equation is biased toward this "normal" horn.

The regression equations give reasonable agreement with the empirical data. Considering all amplitude ratios, the average absolute error is about 1.5%.

Optimized uniformity

If we limit the horn dimensions to those given at the beginning of the previous section and if we exclude any considerations of stress, then the regression equations show that the following horn dimensions will produce approximately optimum face uniformity at a frequency of approximately 19950 Hz:

Horn_length: 115.1
Slot_length: 74.6
Slot_width: 9.6
Slot_spacing: 51.8
Back_web: 19.0

Using these horn dimensions, the regression equations give amplitude ratios of:

Å_C = 1.001
Å_M = 0.940
Å_S = 0.954

From the lowest of these three values, the uniformity is 0.940.

Although these horn dimensions give optimum uniformity, the web ratio is 1.13, so the horn will have relatively poor life. (See the chapter on "Stress".)

Note that the slot spacing (51.8) is at the upper limit of values permitted by the regression equation. Therefore, if the slot spacing was permitted to increase further, it is likely that the uniformity would further improve.

Appendix E - Amplitude ratio data for 127 x 127 and 125 x 125 horns

This appendix gives the amplitude ratios for 127 x 127 and 125 x 125 aluminum horns. The stack consisted of a horn and 400 series converter. Amplitudes were measured at approximately full RF voltage using an A 455.

In the table, the amplitude ratios Å_C, Å_M, and Å_S are listed on separate lines for each HRD number. The regression amplitude estimates were determined by the regression equation:

E1) Amplitude_ratio

= B0 + B1 * Horn_length + B2 * Slot_length

+ B3 * Slot_width + B4 * Slot_spacing

+ B5 * Back_web

The listed frequencies are empirical (not regression).

Corner amplitude ratio (Å_c)

For the corner amplitude ratio, the regression constants are:

B0 = -1.758E-2 [17.47]

B1 = 9.009E-3 [0.276]

B2 = -5.900E-3 [0.255]

B3 = -6.075E-2 [0.133]

B4 = 1.169E-2 [0.168]

B5 = 2.091E-2 [0.165]

Mid-edge amplitude ratio (Å_m)

For the mid-edge amplitude ratio, the regression constants are:

B0 = -9.202E-1 [0.267]

B1 = 1.340E-2 [0.145]

B2 = -3.971E-5 [32.62]

B3 = -1.769E-2 [0.362]

B4 = 8.393E-3 [0.182]

B5 = 2.898E-3 [0.964]

slot intersection amplitude ratio (Å_S)

For the slot intersection amplitude ratio, the regression constants are:

B0 = -1.685 [0.159]

B1 = 1.999E-2 [0.107]

B2 = -5.952E-3 [0.259]

B3 = 2.917E-3 [6.194]

B4 = 1.293E-2 [0.127]

B5 = 4.454E-3 [0.681]

The immediately following data points are for the horn design shown in figure í21Ê.

Horn Slot Slot Slot Back Empirical Empirical Regression
HRD # Date Length Length Width Spacing Web Freq Å Å % Error

001-3 6/9/81 119.2 76.2 9.5 45.5 22.2 19341 1.012 1.023 1.07

0.968 0.947 -2.14

0.971 0.962 -0.97

001-4 6/10/81 118.0 76.2 9.5 45.5 22.2 19581 1.004 1.011 0.68

0.946 0.929 -1.76

0.941 0.933 -0.81

001-5 6/10/81 115.6 76.2 9.5 45.5 22.2 20019 0.979 0.989 1.00

0.904 0.897 -0.80

0.887 0.882 -0.56

002-5 6/9/81 115.6 76.2 9.6 45.5 19.6 19892 0.920 0.931 1.16

0.857 0.888 3.66

0.855 0.866 1.34

003-7 6/29/81 115.6 76.3 9.5 45.4 22.2 19906 0.979 0.993 1.41

0.904 0.904 0.04

0.881 0.894 1.42

004-2 6/30/81 115.8 76.3 9.6 45.4 22.2 19900 0.997 0.986 -1.11

0.915 0.901 -1.50

0.893 0.888 -0.53

005-5 6/5/81 114.7 69.9 9.5 45.3 22.6 19994 1.025 1.029 0.40

0.862 0.897 4.10

0.866 0.912 5.27

005-6 6/19/81 114.4 69.9 9.5 45.3 22.2 20103 1.069 1.015 -5.07

0.890 0.887 -0.34

0.911 0.895 -1.74

005-7 6/22/81 114.4 72.9 9.5 45.3 22.2 20115 1.009 0.999 -1.02

0.920 0.887 -3.59

0.907 0.881 -2.81

005-8 6/23/81 114.4 76.3 9.5 45.3 22.2 20166 0.976 0.979 0.26

0.905 0.884 -2.30

0.873 0.862 -1.30

006-4 2/11/82 111.4 76.7 12.7 45.5 18.3 19887 0.649 0.678 4.41

0.776 0.772 -0.47

                

Steadily increasing amplitude from corners to center. No amplitudes measured over slots.

007 No data

008-4 7/17/81 116.8 76.2 9.5 39.2 22.1 19696 0.923 0.928 0.54

0.855 0.866 1.34

0.806 0.835 3.57

008-5 7/20/81 115.5 76.2 9.5 39.2 22.1 19944 0.895 0.916 2.29

0.844 0.848 0.47

0.819 0.806 -1.63

009-8 9/29/81 115.5 76.2 9.6 51.8 22.4 19909 1.041 1.066 2.45

0.939 0.956 1.85

0.959 0.975 1.67

472-1 6/19/81 115.2 88.1 12.0 46.3 17.0 19706 0.690 0.659 -4.52

                

                

Severe asymmetry. See figure í14Ê.

The immediately following data points are for the horn design shown in figure í22Ê, which is the same as above but with a 12.7 radius along the corners.

010-3 7/10/81 115.7 77.7 9.6 45.5 22.1 19915 0.989 0.977 -1.22

0.921 0.901 -2.18

0.905 0.881 -2.63

011-1 8/12/81 116.0 78.0 9.6 45.6 22.1 19884 0.990 0.978 -1.20

0.922 0.904 -1.94

0.908 0.885 -2.54

012-6 10/7/81 115.5 77.9 9.6 45.3 22.4 19956 0.960 0.978 1.88

0.890 0.898 0.88

0.874 0.876 0.20

013-4 10/8/81 116.8 80.9 9.6 45.4 22.3 19804 0.957 0.969 1.28

0.898 0.910 1.37

0.872 0.882 1.12

013-5 10/9/81 116.0 80.9 9.6 45.4 22.3 19953 0.955 0.962 0.70

0.890 0.899 1.04

0.850 0.864 1.68

The following 13 data points are all 125x125 horns made from Aerospace aluminum. Otherwise, the general design is the same as in figure í21Ê.

017-14 2/18/82 118.9 76.0 9.7 45.1 25.0 19442 1.024 1.062 3.66

0.939 0.945 0.60

0.975 0.960 -1.58

017-15 2/18/82 117.9 76.0 9.7 45.1 25.0 19605 1.052 1.053 0.13

0.925 0.933 0.82

0.942 0.940 -0.16

017-16 2/18/82 116.8 76.0 9.7 45.1 25.0 19781 1.072 1.044 -2.56

0.918 0.919 0.16

0.915 0.920 0.52

017-17 2/18/82 115.9 76.0 9.7 45.1 25.0 19959 1.063 1.036 -2.58

0.905 0.906 0.13

0.887 8.989 1.34

018-5 1/25/82 117.0 78.2 10.0 45.0 22.0 19463 0.969 0.960 -0.90

0.920 0.916 -0.46

0.909 0.901 -0.92

018-6 1/25/82 115.7 78.2 10.0 45.0 22.0 19692 0.937 0.949 1.25

0.890 0.899 0.98

0.869 0.874 0.54

018-7 1/25/82 114.9 78.2 10.0 45.0 22.0 19834 0.922 0.942 2.13

0.873 0.888 1.73

0.846 0.857 1.31

018-8 2/4/82 114.9 80.9 10.0 45.0 22.0 19834 0.938 0.928 -1.10

0.880 0.889 1.01

0.824 0.846 2.68

023-1 4/5/82 121.0 78.0 10.0 45.0 21.9 18821 0.996 0.991 -0.48

0.956 0.963 0.72

0.957 0.976 1.97

023-2 4/7/82 119.0 78.0 10.0 45.0 21.9 19144 0.992 0.975 -1.71

0.945 0.939 -0.65

0.949 0.938 -1.17

023-3 4/8/82 117.0 78.0 10.0 45.0 21.9 19467 0.983 0.959 -2.46

0.928 0.915 -1.42

0.922 0.900 -2.39

023-4 4/8/82 115.1 78.0 10.0 45.0 21.9 19819 0.939 0.941 0.24

0.889 0.889 -0.04

0.872 0.859 -1.53

023-5 4/8/82 114.4 78.0 10.0 45.0 21.9 19945 0.932 0.935 0.31

0.883 0.879 -0.42

0.846 0.844 -0.25

Amplitude asymmetry

HRD 456 3, 100 x 100 Al horn:

The asymmetry, especially among the mid-edge amplitudes of the face and back, is probably caused either by an asymmetrical resonance at 21164 Hz (figure ífÊ«») or by an asymmetric resonance at 19620 (.f.figure ífÊ«»). (Note: the axial resonance of this horn is 19957 Hz.) The asymmetric resonance of figure ífÊ is most probably responsible, because corresponding locations of the face and back have the same phase. This corresponds to the mid-edge amplitudes of the axial mode, which are either high or low at corresponding locations on the face or stud surface. (Note that the asymmetric resonance of figure ífÊ has opposite phase on the face and back.) Must distinguish between phase measured with A 450 and phase of FEA. (For further discussion, see the section on _.)

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General notes

1. Uniformity does not depend on material or material direction. However, see Zemanek "An Experimental and Theoretical Investigation of Elastic Wave Propagation in a Cylinder", The Journal of the Acoustical Society of America, vol. 51, number 1 (part 2), 1972. pp. 265   283. Equation 2 shows dependence on wave speed and Poisson's ratio. Also, see equations 18.
2. Uniformity does not depend on shape of horn back quarter wave section. (Compare bar-rectangular U to bar horn U.) For unslotted bar horns, does not seem to depend on gain.
3. Asymmetry does not depend on particular converter? Possibly depends on converter series? Seems random for bar horns. Depends on other nonaxial resonance for cyl horns.
4. Note low uniformity of HRD 423 and 424.

[Asymmetry of 100 mm spool    asm depends on freq. separation].

This can cause severe scrubbing of the horn on the plastic part. This can result in marking and power loss without any increased welding. Of course, if you have an application where shear welding is desirable, then a horn of this diameter might be effective.

[Limitations on Pochhammer equation: (Note: a cylinder of finite length will have certain end effects that distinguish its performance from Zemanek's infinitely long cylinder. However, based on the work of Zemanek and others, Thurston (p. 16) concludes that no appreciable end corrections are needed for finite length cylinders as long as õ is less than 2.6. Hutchinson gives a theoretical analysis of a cylinder of finite length, but the result is complex and suitable only for computer analysis. Also see van Randeraat figure 8.4, p. 130.)

Discussion of stud-surface uniformity for P&G 114 wide horn (from SN_PROJ\LOVETT4.89):

Also note that the empirical data indicates much worse stud-surface uniformity than does the FEA model. This was probably caused by an error in measuring the horn input amplitude. Table å1Ï shows that the empirical gain is 7.3% higher than the FEA result and the empirical stud-surface uniformity is 9.4% higher than the FEA result. (See figure í7Ê for further gain information about HRD 283.) This indicates that the empirically measured input amplitude (using a converter and Titanium Standard) was about 8% too low. (Note: FEA gives good agreement with empirical data for face uniformity (figures ífÊ and í11Ê) and frequency (í2Ê).)

1. The effect of the converter. The FEA horn was modeled without a converter attached. It is possible that the converter (used for empirical measurements) affects the stud-surface uniformity. (See the chapter on "Uniformity".) Note, however, that FEA gives good agreement with empirical data on face uniformity (figure ífÊ). Of course, it is possible that the converter affects the stud-surface uniformity with affecting the face uniformity, although this has not been found in other FEA studies where horn was modeled with an attached converter.
2. An error in choosing the Ti 7Al 4Mo properties for FEA. Ti 7Al 4Mo is orthotropic, for which three different Young's moduli and Poisson's ratios should have been used. (See the chapter on "Elastic Material Properties".) However, for lack of well-defined values, only a single value for each was used. The modulus (120.7 MPa, 17.5 lbf/in²) was chosen to give the correct FEA axial frequency. Poisson's ratio was chosen as 0.30. It is known that Poisson's ratio can significantly affect uniformities. (See the chapter on "Uniformity".)
3. An error in measuring the horn input amplitude. Table å1Ï shows comparative data for the above empirical horns in their final configuration and for the FEA horn. The frequencies and face uniformities agree well. (For frequencies of HRD 283 and FEA, see figure í2Ê. For the face uniformities, see the right-most data points of figure í11Ê and the left-most data points of figure ífÊ.) However, the empirical gain is 7.3% higher than the FEA result and the empirical stud-surface uniformity is 9.4% higher than the FEA result. (See figure í7Ê for further gain information about HRD 283.) This most likely indicates that the empirically measured input amplitude (using a converter and Titanium Standard) was about 8% too low.

Note: "edge gain" gives good agreement between FEA and empirical, but stud-centerline gain does not. Probably indicates error in measuring converter input amplitude.

TABLE å1Ï

Comparison of Empirical Data and FEA Results

for 114 X 48 X 67 Ti 7Al 4Mo Bar Horns;


° Frequency ° Uniformity °
HRD # ° Low-side Axial High-side ° Stud surface Face ° Gain
° ° °
HRD 284 5 ° 16429 19654 20568 ° 1.266 1.153 ° 1.41
° ° °
HRD 283 ° 16270 19840 20510 ° 1.311 1.160 ° 1.40
° ° °
FEA ° 16203 19816 20429 ° 1.177 1.162 ° 1.31

4/7/89: originally from the section on effect of joint and stud on U:

ø125 unshaped aluminum horn

Since the effect of stud and joint are largest for the larger diameter horns, we will look more closely at a ø125 unshaped Aerospace aluminum horn. We will look at amplitudes at corresponding locations on the face and stud surface. The ratio of these amplitudes will be called the "amplitude rato".

Effect of the stud. Let us exclude (for the moment) the effect of the joint. Without any stud (the horn is symmetric in all respects), we should expect the amplitude ratio to be exactly 1.0. FEA shows that this is true. (See the squares of figure ífÊ.)

However, as the stud diameter increases, figure ífÊ shows that the amplitude ratio deviates significantly from 1.0. (All studs of figure ífÊ have a depth of 19.0 mm into the horn.)

The empirical data of figure ífÊ (ø12.7 x 19.0 deep stud) agrees reasonably well with the FEA. For HRD 156 4, the amplitudes were measured at four locations at each of three diameters on the face and stud surface. (See figure ífÊ.) For HRD 059 2 and HRD 073 4, the amplitudes were measured at eight locations at a single diameter on the face and stud surface.

Figures ífÊ and .f.ífÊ show the actual amplitude distributions for the face and stud surface. Note that the stud mainly affects the stud-surface amplitudes. It's effect on the face amplitudes is much smaller.

Effect of the joint. Figures ífÊ and ífÊ show the FEA model of the ø125 horn, without and with an attached Aluminum Standard.

For this ø125 horn, the effect of the joint is relatively small. Figure ífÊ shows the amplitude ratios, with and without a joint. The Aluminum Standard joint somewhat changes the horn's amplitude over the region of the joint and also toward the periphery of the horn. These changes occur mainly on the stud surface. The addition of the Aluminum Standard to the ø125 horn decreases the horn's face uniformity from 0.372 to 0.368 (a 1.1% decrease) and increases the stud-surface uniformity from 0.278 to 0.298 (a 7.2% increase).

ø125 unshaped titanium horn

The effect of the stud has also been analyzed for a ø125 unshaped Ti 7Al 4Mo horn (no joint). Figure ífÊ shows a comparison of the amplitude ratios for these aluminum and titanium horns with ø12.7 studs. Figure ífÊ shows that the titanium horn is much less affected by the stud than is the aluminum horn. This is reasonable, since the stud is inserted to a shallower depth in the titanium horn than in the aluminum horn (12.7 mm versus 19.0 mm, per normal usage of ø12.7 studs) and since the density of the steel stud is closer to that of titanium than aluminum. Thus, the stud appears less foreign to the titanium horn than to the aluminum horn.

References and notes

2) Piezoelectric Ceramics Application Handbook, Ferroxcube Corp., 1974.

6) Edward J. Kubel, Jr., "Al Li Alloys: New Hope for Weight Watchers", Materials Engineering, April 1985, pp. 41 44.

7) Richard K. Case, Darrel D. Lemon, David W. Paule (Ball Aerospace Systems Div., Boulder CO), "Beryllium: The Lightweight Contender", Machine Design, June 7, 1984, pp. 59 61.

8) Designing with Beryllium, Brush Wellman, Inc., 1200 Hanna Building, Cleveland, OH 06443

9) Beryllium Science and Technology, Dennis R. Floyd and John N. Lowe, Vol 2, Plenum Press, New York, 1979