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How to calculate damping from a FRF?

by Siemens Genius Siemens Genius ‎07-11-2016 10:44 AM - edited ‎04-20-2017 08:33 PM

How does one calculate damping values from a Frequency Response Function (FRF)?

 

 A classical method of determining the damping at a resonance in a Frequency Response Function (FRF) is to use the “3 dB method” (also called “half power method”).  

 

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.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 the 3 dB down from the peak level, one can determine the associated damping.

 

 

3DB.png

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

 

What is damping?

 

Damping is the energy dissipation properties of a material or system under cyclic stress.

 

Acceleration versus time of a damped vs undamped system response due to a loadAcceleration 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.  The more damping present in a mechanical system, the shorter the time to stop moving.

 

Forms of Damping

 

Damping can be expressed in several different forms, 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 below:

 

damping_values.png

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.

 

More vs 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.

 

Comparison of function with Q=2 (Red) versus Q=10 (Green).  Red would be considered to have more damping than green.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.

 

Cautions

 

When calculating damping from a FRF, there are several items which can influence the final results:

 

Caution #1: 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.  

 

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.  Change in Q value, Damping Ratio and Loss Factor also shownBode 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. Change in Q value, Damping Ratio and Loss Factor also shown

Using a very fine frequency resolution is recommended when calculating damping using the 3dB method.

 

Caution #2: 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.

 

When an exponential window is applied to the time response of a FRF measurement, it increases the apparent damping in the FRF.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 LMS Test.Lab, when performing a modal analysis curvefit, the exponential window affect is removed from the modal damping estimate automatically for the user.

 

Caution #3: Mode Spacing

 

If two modes are close to each other in frequency, it may be impossible to use the “3dB method” to determine the damping values.

 

The proximity of two closely spaced modes makes the determination of Q impossible via the 3dB method.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.

 

LMS Test.Lab Damping Cursor

 

After displaying a FRF, to calculate damping in 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). Note: It is not necessary for amplitude to be in dB of the FRF, this will be accounted for correctly.

auto_cursor.png

  • After the cursor is on the peak value, right click on the cursor and select “Calculations -> Q” or “Calculations -> Damping Ratio” as desired

rightclickQ.png

 

  • Right click on cursor and select “Automatic Peak Parameters” and add as many peaks as desired in the “Max number of extrema” field.

 

extrema.png

  • Right click on legend and choose “Copy Values” to export the values to Excel if desired

 

Have more questions on using the half power method to calculate damping from a FRF?  Feel free to contact us!

 

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