This article discusses damping behavior in dynamic systems. Methods for quantifying the damping from a measured Frequency Response Function (FRF) are also discussed.
1. What is Damping? 2. Damping Quantities 3. More versus Less Damping 4. Three dB Method 5. Cautions 5.1 Caution: Frequency Resolution 5.2 Caution: Window 5.3 Caution: Mode Spacing 6. Simcenter Testlab Damping Cursor
1. What is Damping?
Damping is the energy dissipation properties of a material or system under cyclic stress.
In Figure 1, the the response of two different dynamic systems set in motion is shown. The blue response has damping, the green system response does not.
Figure 1: Acceleration versus time of a damped vs undamped system response due to a load.
If a mechanical system had no damping, once set in motion, it would remain in motion forever. Damping causes the system to gradually stop moving over time. This is shown in the animation in Figure 2.
Figure 2: Animation of a damped system response versus a system without damping.
The more damping present in a mechanical system, the shorter the time to stop moving.
2. Damping Quantities
Damping can be expressed in several different conventions, including “loss factor”, “damping factor”, “percent critical damping”, “quality factor”, etc. Note that if the value of one of these damping forms is known, the other forms can be mathematically derived. It is just a matter of using the equations to transform the value to a different form.
For the purposes of this article, we will consider the “damping factor”, “quality factor” and “Q” to be the same as described in Equation 1 below:
Equation 1: Different damping quantities.
Why have different ways of expressing damping? Mostly, this is to make it easier to discuss differences in damping.
For example, perhaps two materials were tested, and the “percent critical damping” was 0.0123% in one case, and 0.0032% in the other case, which sounds like a small difference. In terms of “quality factor” or “damping factor”, the difference is 4065 versus 15625, which sounds much bigger!
Instead of using long numbers with small decimal differences like percent critical damping, changing to the “quality factor” form makes the differences easier to understand.
3. More versus Less Damping
As the peaks in a FRF get wider relative to the peak, the damping increases (i.e., “more” damping). This means that any vibration set in motion in the structure would decay faster due to the increased damping.
Figure 3 shows the difference between a system with Q of 10 versus a Q of 2.
Figure 3 Comparison of function with Q=2 (Red) versus Q=10 (Green). Red would be considered to have more damping than green.
Depending on the form being used to express damping, the value may be higher or lower.
For example, the “quality factor” or “damping factor” will decrease with more damping, while “loss factor” and “percent critical damping” would increase with more damping.
An example of this calculation on an FRF is shown in Figure 4.
Figure 4: Bode plot of FRF Amplitude (Top) and Phase (Bottom). The damping values "Q" and ""Damping Ratio" are shown for three different peaks in the FRF.
In a FRF, the damping is proportional to the width of the resonant peak about the peak’s center frequency. By looking at three dB down from the peak level, one can determine the associated damping as shown in Figure 5.
Figure 5: The 3 dB method diagram for calculating the damping factor Q.
The “quality factor” (also known as “damping factor”) or “Q” is found by the equation Q = f0/(f2-f1), where:
f0 = frequency of resonant peak in Hertz
f2 = frequency value, in Hertz, 3 dB down from peak value, higher than f0
f1 = frequency value, in Hertz, 3 dB down from peak value, lower than f0
When calculating damping from a FRF (especially a measured FRF), there are several items which can influence the final results:
5.1 Caution: Frequency Resolution
When using a digital FRF, the data curve is not continuous. It is broken into discrete data points at a fixed frequency interval or resolution. For example, this could be a 1.0 Hz spacing versus a 0.5 Hz spacing between data points. This will influence how the frequency values f0, f1 and f2 are determined.
A comparison of damping values calculated by measuring the same system with different frequency resolutions is shown in Figure 6.
Figure 6: Bode plot of the same system response acquired with four different frequency resolutions: 0.25 (Red), 0.5 (Green), 1.0 (Blue), 2.0 (Magenta) Hz and corresponding change in damping values (legend).
Using a very fine frequency resolution is recommended when calculating damping using the 3dB method.
5.2 Caution: Window
Sometimes, when using a modal impact hammer to calculate an experimental FRF, a user may apply an Exponential Window to avoid leakage effects to the accelerometer signal. The accelerometer response is multiplied the exponential window causing it to decay quicker in time, thus increasing the apparent damping (Figure 7).
Figure 7: When an exponential window is applied to the time response of a FRF measurement, it increases the apparent damping in the FRF.
One should be aware the resulting FRF will yield higher damping estimates when using the “3 dB method”. It is possible to back out the effects of the window and get the actual damping value. For example, in Simcenter Testlab:
Modal Curvefitting - When performing a modal analysis curvefit, the exponential window affect is removed from the modal damping estimate automatically for the user.
Damping Cursor in Display - With Simcenter Testlab Revision 18 and higher, the damping value reported by the cursor automatically adjusts for the effects of an exponential applied to the data.
If two modes are close to each other in frequency, it may be impossible to use the “3dB method” to determine the damping values as shown in Figure 8.
Figure 8: The proximity of two closely spaced modes makes the determination of Q impossible via the 3dB method.
If this is the case, there are two resonant peaks close together, which would make the peak wider than if each peak could be analyzed separately. In this case, one should use a modal curvefitter to determine the damping properly for each peak. A modal curvefitter can successfully separate the two modes influence on each other.
For more information on using a modal curvefitter to determine the damping, see the knowledge article: Getting Started with Modal Curvefitting. One advantage of a modal curvefitter is that many FRF functions can be used in the estimate of damping rather than using a single FRF function like the 3 dB method.
6. Simcenter Testlab Damping Cursor
After displaying a FRF, to calculate damping in Simcenter Testlab (formerly called LMS Test.Lab):
Right click and add an “Automatic -> Peak Cursor” (or a regular SingleX cursor if one is confident in positioning it on the actual peak) as shown in Figure 9. Note: It is not necessary for amplitude to be in dB of the FRF, this will be accounted for correctly.
Figure 9: Right click in display, and Choose "Automatic -> Peak Cursor” .
After the cursor is on the peak value, right click on the cursor and select “Calculations -> Q” or “Calculations -> Damping Ratio” as desired as shown in Figure 10.
Figure 10: Right click on the cursor and select “Calculations -> Q” or “Calculations -> Damping Ratio"
Right click on cursor and select “Automatic Peak Parameters” and add as many peaks as desired in the “Max number of extrema” field as shown in Figure 11.
Figure 11: Set "Max number of extrema" to the number of damping values to calculate.
Right click on legend and choose “Copy Values” to export the values to Excel if desired