Thomas Industrial Library

9.3 ESTIMATING DAMPING

While it is relatively easy to model the masses and springs in a cam-follower system, it is difficult to accurately estimate its damping at the design stage by any calculation method. As noted above, experimenters such as Koster [2] have measured the damping ratio in actual cam-follower systems and found it to be typically less than 0.1. This author has also measured the damping ratio in many cam-follower systems and found it to be consistent with Koster’s data. Luckily, it is a fairly simple matter to measure the damping ratio in an existing physical system. To the degree that a newly designed system is similar to one in which the damping has been experimentally measured, those data can be extrapolated to the new design. Once a prototype of a new design is built, the same simple test can be done to confirm the earlier assumption regarding its damping.

Logarithmic Decrement

Figure 9-7 shows the response of an underdamped system to an impulse (hammer blow). As shown in Figure 9-5 and defined in equation 9.5g, this response is the product of a decaying exponential function and a cosine. Figure 9-7a shows the decaying exponential as a dotted line enveloping the response. Figure 9-7b plots this exponential term by itself as was done in Figure 9-5a. These are linear plots. Figure 9-7c plots the same exponential function on a natural log (ln) axis. It plots as a straight line. The slope of this line ä is called the logarithmic decrement and is proportional to the damping in the system.

The value of the logarithmic decrement can be found from a simple test of the system. A transducer, such as an accelerometer, is placed on the output end of the system and the system is struck lightly with a hammer or other hard object. The transducer measures the transient response of the system, which will look something like the solid line in Figure 9-7a when viewed on an oscilloscope. The values of the successive peaks of the response, x 1 , x 2 , x 3 , etc., are then read from the plot. Any two successive values are sufficient to compute the logarithmic decrement from:

A potentially more accurate estimate of ä can be found by measuring a pair of peaks that are further apart in time to obtain an average value from

where x m is the m th peak measured.

The damping ratio is related to the logarithmic decrement as

For small values of damping ratio ( < 0.3), an approximate value can be found from

This technique provides a simple means to measure the damping in a system.

EXAMPLE 9-1

Measuring the Damping of a Cam-Follower System.

Given: A cam-follower system as shown on the left side of Figure 9-8

Problem: Measure the response of the system to an excitation and calculate an estimate of the system damping ratio .

Solution:

1 A small, sensitive, piezoelectric accelerometer was attached with a magnet to the output arm of the follower train at point B and the follower arm was struck with a small ball-peen hammer at point A , near where the cam contacts the roller follower. The machine power was off, but the air cylinder that closes the cam joint was pressurized.

2 Figure 9-9a shows the impulse response of the entire follower train at point B to an impact at point A . Note its general similarity to the theoretical response of a one-DOF system shown in Figure 9-5c. However, this actual response is not as “clean” as the theoretical. Moreover, it does not decay smoothly. The reason is that this is a multi-DOF system and

so has many natural frequencies. Each of these resonances is decaying with a different periodicity. These functions combine causing the irregularities in the response.

3 This can be seen more clearly in Figure 9-9b, which shows the Fourier transform* of the impulse response out to 6400 Hz. There are about 20 peaks (natural frequencies) seen here. Many of them have been tagged with their corresponding frequency in hertz. The lowest is 224 Hz, which has a period of 0.0045 sec. The spacing of the first two peaks in the impulse response of Figure 9-9a is about 0.2 ms which corresponds to a frequency of about 4800 Hz. A peak appears in the spectrum near that frequency.

4 We would like to estimate the damping in this system from the logarithmic decrement of the envelope of the impulse response in Figure 9-9a. The difficulty comes in deciding which peaks are members of the same decaying resonance. The first three peaks appear to be spaced evenly, which implies that they are of the same family. Their peak values are shown on the plot and are x 1 = 3.004, x 2 = 2.037, x 3 = 1.161. Using the first two of these in equations 9.9 a and d gives:

5 Using the first and third values in equations 9.9b and d gives:

6 We can continue this process, using any number of oscillations in the calculation. It sometimes becomes difficult to decide what the number of an oscillation is when the waveform is as irregular as this one. If we take the first peak ( x 1 = 3.004) and what appears to be the 18th peak ( x 18 = 0.018) in equations 9.9b and d we get

7 Another way to get a measure of system damping is to analyze acceleration measurements made while a machine is running. Figure 9-10 shows the system of Figure 9-8 running at 100 rpm. The accelerometer is again at point B. This cam is a double-dwell with modified trapezoid acceleration on both rise and fall. It can be seen that the measured accelerations look very little like the theoretical curves shown in Figure 3-5 (p. 35). The presence of vibration in the follower train has severely distorted the waveforms. In fact the impulse response of the system is present in the acceleration and that is why we can get a measure

* See “Spectrum Analysis” in Section 16.3 (p. 486) for an explanation of the Fourier transform.

of damping from these data as well. The values of the first three peaks during the positive acceleration pulse of the rise acceleration are noted in the figure. They are x 1 = 9.682, x 2 = 7.143, x 3 = 4.710. Using the first two of these in equations 9.9 a and d gives:

8 Using the first and third values in equations 9.9b and d gives:

9 If we average these five estimates, we conclude that this system has a damping ratio of about æ = 0.06—a very underdamped system—consistent with other data in the literature such as Koster.[2]

10 The natural frequencies of the system obtained from this measurement also will prove very useful in the analysis of vibrations in the cam-follower system, as will be discussed in a later chapter.