Amplitude & Motion

Introduction

Amplitude deals with the magnitude and direction of ultrasonic motion. Our main interest will be in analyzing the amplitude of resonators, although the amplitude of the medium into which they are coupled (e.g., plastic, water, air, etc.) may also be of interest.

Uses

In designing resonators, we usually start with a basic resonator design that has worked well for previous applications. This design is then modified to to fit the current customer part. After the resonator has been machined, we may find that these modifications have altered the resonator performance in undesired ways. Amplitude measurement is one method by which we can diagnose the problem and improve the resonator performance. These improvements fall into the following areas:

1. Increasing resonator life. For a given material and environment, the resonator life depends on the amplitude distribution within the resonator. If we can measure the amplitude, we will have a starting point for improving the life.
2. Reducing stack loss. Excessive stack loss (either at the joints or in the converter) often results from undesirable motion in the resonator. To diagnose the cause of the loss problem, we must find and measure the this undesirable motion.
3. Controlling undesirable modes. Complex resonators often vibrate in modes that are undesirable. To reduce the influence of these modes, we must first determine the form of the offending mode(s). This can be done by measuring the amplitude over the surface of the resonator.
4. Locating the node. Generally, boosters and converters must be acoustically isolated from their surroundings to prevent unwanted energy loss. One method of isolation is to attach these resonators at their point of least motion -- i.e., at an axial node. To find the node, the amplitude must be measured along the side of the resonator.
5. Determining gain. Boosters (and sometimes horns) must be designed for a specified gain so that a certain output amplitude is achieved. To determine if the required gain has been achieved, the output and input amplitudes must be measured.
6. Improving weld performance. From the standpoint of resonator design, weld performance depends on the resonator uniformity and asymmetry, and the maximum amplitude of the stack. These can be determined from amplitude measurements.

To support improved resonator design, amplitude measurement is also valuable in the following areas:

1. Verifiying computer results. Computers can be used to estimate certain characteristics of resonators, such as amplitude distribution, stress, and resonant frequencies. However, computer analysis is only approximate. In order to estimate the degree of approximation, the computer predicted results must be compared to the results of actual resonators.
   
   One area of relatively easy comparison is the resonator amplitude distribution. If the relative amplitude measured from an actual resonator agrees well with the computer predicted amplitude distribution, then we may have some faith that our computer analysis technique is correct. This increases our confidence in other computer predictions (e.g., resonator stress) which are much more difficult to verify in the actual resonator.
   
   If the measured resonator amplitude distribution does not agree well with the computer predictions, then we may suspect some problem with either the computer modeling or the amplitude measurements.
2. Inspecting the final product. Amplitude measurements can be used to inspect for converter or stack amplitude, uniformity, asymmetry, and gain.
3. Calibrating amplitude instruments. Amplitude measuring instruments must be calibrated to assure their accuracy. To do this, the uncalibrated instrument is compared to a reference instrument whose calibration has been established.

Fundamental concepts

When an ultrasonic resonator vibrates, each point on the resonator has motion which is essentially unique. Hence, it is not possible to define a single value that will describe the entire motion of the resonator. Instead, the motion must be separately described at each individual point.

At first glance, this might seem to be an impossible task, since there are an infinite number of points on each resonator. However, usually only a few of these points have motion that is of any interest. Also, since ultrasonic motion is continuous function (i.e., there are no sharp steps in the motion at any point on the resonator), it is often possible to deduce the entire resonator motion from amplitude measurements at just a few points. (See the chapter on "Modeshapes".)

One further aspect that simplifies the task of describing the resonator motion: every point on the resonator has motion at the same frequency as the RF voltage. (There may also be some motion at other frequencies, but we will ignore this side effect for the time being.)

Visualizing the motion

The easiest way to visualize the ultrasonic motion at a point is to think of the analogous motion of a mass attached to a spring. The mass is analogous to a point on the resonator that has been largely magnified. Its motion (ideally) is a straight-line oscillation about a "rest" position. (The "rest" position is the location of the weight after it finally slows down and stops.)

If you attached a pencil to the mass and slowly pulled a paper past the pencil point as the mass oscillated, the resulting pattern could be described as a curve that is sinusoidal in time (see figure []) -- a continuous series of rolling hills and valleys. If you could somehow attach the pencil to a resonator, you would get exactly the same type of pattern, except that the scale would be different. We now need a method to describe this pattern.

Period and amplitude

There are only two parameters that are needed to completely describe this sinusoidal curve:

1. Period. The period describes how the curve is distributed along the time axis.
2. Amplitude. The amplitude describes the "up and down" motion of the curve.

The period is the elapsed time between when the observed point reaches its maximum excursion (the top of the curve) and when it next reaches the same maximum excursion. This is just the time required for one complete cycle of vibration to be completed. Hence, there is a relation between the period and the frequency:

1) Frequency = 1 / Period

Thus, if the resonator takes 0.00005 seconds to complete one cycle (period = 0.00005 seconds), then in one second it will complete 20,000 cycles, for which the frequency is then 20,000 cycles per second.

Since every point on the resonator has exactly the same frequency (except as noted above), it follows that every point will also have exactly the same period. Thus, if we were to draw the amplitude curves for any two points on the resonator, the appearance of these two curves would differ only by the extent of the "up and down" motion. There would be no difference along the time axis.

To describe the extent of this "up and down" motion (i.e., the amplitude) of the sinusoidal curve, two approaches seem possible. The first is to specify the amplitude as the distance from the "rest" position to a peak on the curve. (The "rest" position is the horizontal time line that divides the curve into upper and lower halves.) Amplitude that is specified in this manner is called peak amplitude.

The second approach is to specify the amplitude as the distance from a lower peak to an upper peak. This distance is the total excursion of the point being measured and is called the peak-to-peak amplitude. This is the method most commonly used when referring to ultrasonic amplitude. Thus:

Unless noted otherwise, any future reference to amplitude will mean PEAK-TO-PEAK amplitude.

Direction of ultrasonic motion

So far we have only described how the point on the resonator moves with time. We have said nothing about how the point moves in space -- i.e., about the direction of motion. The direction of motion is important because it determines such parameters as marking of the plastic part, stresses in the resonator, loss in the converter, etc. Also, as we will see, the direction of motion determines how we interpret the measurements from our amplitude measuring equipment.

In order to talk about direction of motion, we need some consistent frame of reference. Two reference frames are common:

1. Global system. (See figure [].) The global coordinate system has one axis that is coincident with the centerline of the resonator stud. Any motion of any of the resonator parts along this axis is called axial or longitudinal motion. The other axis is perpendicular to the first axis. Any motion of any of the resonator parts in this direction is called transverse motion. If the resonator is cylindrical, then this motion is called radial motion.
2. Local system. (See figure [].) The local system takes its reference from the surface on which the amplitude is being measured. Motion that is normal (i.e., perpendicular) to this surface is called normal motion. Motion in the plane of the surface (i.e., parallel to the surface) is called in plane motion.

For the purpose of discussing amplitude measurement, we will generally be referring to the local system.

If your eyes were good enough to actually see the ultrasonic motion, you would see that most points on the resonator move in directions that are neither entirely normal nor in plane. Instead, the motion is at some diagonal angle to the surface. (See figure [].) Technically, the actual amplitude of the point is its peak-to-peak excursion along its line of motion. The problem is: how can this amplitude be measured?

I know of no method that will measure this actual peak-to-peak amplitude along the line of motion. Instead, the amplitude measurement instrument either "looks at" only the magnitude of the normal motion or the magnitude of the in plane motion, neither of which describe the total motion of the point.

Consider figure [], which shows half of a cylindrical spool horn whose face is the flat surface closest to the bottom of the page. For computer analysis, this horn has been divided into a large number of triangular elements. The amplitudes along the edges of these elements are shown by the unconnected line segments. These line segments show both the magnitude and direction of the motion.

Let us choose a point on the face whose amplitude is 50 microns along the line of motion. If the point was looked at with a microscope, the amplitude would appear to be 30 microns, since a microscope measures only the in plane component of motion. On the other hand, if the point was "looked at" with the Branson A 450 amplitude meter, the amplitude would appear to be 40 microns, because the A 450 measures only that component of motion that is normal to the surface. Thus, two different instruments give different readings of amplitude at the same point, neither of which really describes the actual peak-to-peak amplitude.

It is even possible to use the same instrument to measure the motion at a particular point and still have two different readings. For instance, if we look at the corner of the above horn (figure []), the amplitude is 67 microns along the line of motion. If an A 450 amplitude probe is placed on the horn face at this point, the amplitude would appear to be 60 microns, which is the component of amplitude normal to the horn face. If the A 450 amplitude probe is placed at the same point but on the side of the horn, the amplitude would appear to be 30 microns, which is the component of amplitude normal to the side of the horn. Which of these measured amplitudes should be reported as the "correct" amplitude, since neither gives the true amplitude? The answer is that either may be reported, as long as you observe the following cautions:

1. Be aware that the amplitude that you measure with a particular instrument may not be the true peak-to-peak amplitude.
2. Be sure that you understand which component of motion you are actually measuring with a particular instrument.
3. In describing your amplitude measurements to someone else, give enough information (e.g., measuring instrument, direction and location measurement) so that he can properly interpret the meaning of your amplitude measurement. In some cases the meaning will be implicitly understood. For instance, if you say that the amplitude at a particular location on the horn face is 63 microns, it is understood that you mean the axial amplitude (unless you explicitly state otherwise).

Distortion of ultrasonic motion

In some cases the amplitude may be distorted. If you think about the analogy between ultrasonic motion and the vibration of a weight attached to a spring, then ideally the motion should be purely linear (i.e., the weight should move back and forth along a straight line). In some cases, however, the motion may be spatially distorted (e.g., the weight may move in an ellipse rather than in a straight line). Such spatial distortion often indicates a problem with the resonator. The amplitude instrument should allow you to see this spatial distortion.

In some cases, the motion may have frequency distortion. It is then possible for the motion to be entirely linear, but to be composed of multiple frequencies. For instance, a 20 kHz horn may have motion with a large amplitude component at 20 kHz and a smaller amplitude component at 40 kHz, both occurring at the same time. (This is believed to be quite common in 20 kHz horns.) Then, the graph showing amplitude as a function of time would no longer be a pure sine curve, but would have some distortion as shown in figure []. (The extent of the distortion depends on the relative magnitude of the 20 kHz and 40 kHz components. Often, the distortion is so small that it cannot be detected just from the appearance of the amplitude waveform.)

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.

Having thus described what is meant by amplitude, we can now describe some instruments used to measure amplitude.