In engineering, the damping ratio (ζ) is a dimensionless measure describing how oscillations in a system decay after a disturbance.  One way to evaluate the damping ratio is to excite the system and then use what is called the log decrement method to evaluate the damping.  Figure 1 illustrates the log decrement method. Unfortunately, real systems don’t always behave in a way that allows clear use of the log decrement.  Figure 2 illustrates the motion of a tall (~80 m) exhaust stack that is excited and then allowed to vibrate freely. The vibration signal has been filtered to remove all but the natural frequency (~0.54 Hz) of the stack.  If the stack was viscously damped, we would expect to see an exponential decay in the motion and the log decrement method could be used to evaluate the damping ratio.  If coulomb (friction) damping controlled, we would expect to see a linear decay in the motion with time.  Looking at Figure 2, neither type of decay is evident.

In order to get a clearer measure of the damping, it is often useful to use a technique that is often used in the acoustical world.  If the vibration amplitude is plotted on a logarithmic scale, it can be shown that:

ζ = 0.183/n                  Equation 1

Where:  n = number of cycles required for the amplitude to drop by 10 dB (a factor of approximately 3.16).

The motion data in Figure 2 has been potted with a logarithmic scale in Figure 3. There are two distinct regions of decay in this portion of the signal.  A straight line passing through the first decay segment indicates that the signal would take approximately 12 cycles to decay by 10 dB.  A line passing through the second decay segment indicates that the signal would take approximately 20 cycles to decay 10 dB.  Using the Equation 1, the respective damping values would be 0.015 and 0.009.  The decreasing damping ratio as a function of amplitude is a strong indication that the damping in the stack is coulomb or frictional rather than viscous.

By: Mike Porter

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