Hello everyone!
We often hear the word damping in context of dynamic analysis.
What does it mean?
Damping is an energy-dissipation mechanism that causes vibrations to diminish over time.
For example, shock absorbers absorb the shocks/vibrations to maintain a comfortable ride.
Another example where we can see damping is the swings in park.
We can see in the video that the motion of swing is decaying because of presence of damping at the joints and material as well.
Remember that damping, unlike elastic modulus, is not something we can often directly measure since energy dissipation can come from joints and part interactions, fluid interactions, material behavior, and other sources.
Damping can be estimated via testing or is specified by industry practice.
Damping is present in most systems and should be specified in a dynamic analysis.
In this video, we will demonstrate damping and its usage for modal-based analysis types which are mode-superposition Harmonic, mode-superposition Transient, Random Vibration and Response Spectrum analysis.
Ready? Let’s get started!
Before covering damping in modal-based analyses, let's first have a brief discussion on the mode-superposition, or MSUP, method.
In the general equation of motion: The first term represents inertial forces, which is mass times acceleration.
The second term represents damping forces, which is product of damping coefficient and velocity.
The third term represents the force due to stiffness of structure, which is product of stiffness and displacement.
F represents the applied forces.
In a modal-based analysis, the displacement ‘u’ can be expressed in terms of modal coordinates ‘yi’:
The term ‘phi i’ is the ith mode shape and ‘n’ is the total number of modes obtained from the modal analysis.
We can see this is where the mode-superposition method gets its name since the final response {u} is a linear combination of the mode shapes.
When the displacement is expressed in terms of modal coordinates and is substituted in the equation of motion, we get the equation of motion in modal coordinates.
‘Omega j’ represents the natural circular frequency of mode j and ‘zai j’ represents fraction of critical damping for mode j.
‘Omega j’ and ‘zai j’ can be represented in terms of mass, stiffness, and damping terms:
Let us now investigate the damping term ‘zai j.’
In general, there are three types of damping typically used in mode-superposition-based analyses.
They are: Constant damping, Alpha damping, Beta damping
Damping can be defined using any or all the above damping types.
We can express damping as sum of constant damping ratio, alpha and beta damping terms, as indicated below.
On the right side of the equation, we have ‘zai’ which is the constant damping ratio.
The second term represents the alpha damping, where alpha is the mass coefficient, and the third term represents the beta damping where beta is the stiffness coefficient.
Alpha and beta damping together are called Rayleigh damping.
The constant damping ratio, as its name suggests, is constant for each frequency.
It is the ratio of damping to critical damping at a given frequency (mode).
Alpha damping results in a damping ratio which is inversely proportional to frequency.
Because of this, we usually do not use alpha damping since it has adverse effects on very low frequencies.
Hence one should be careful when using alpha damping.
Beta damping provides a damping ratio that is proportional to frequency.
Consequently, it tends to affect higher-frequency content.
We can plot these components of damping ratio vs. frequency.
These damping values may be defined on a system level (“global”), affecting all parts – done via “Analysis Settings”
or the damping ratio may be defined on a material basis (“material”), affecting all parts that have that material assigned – done via Engineering Data.
One can refer to this table to see the damping type supported in modal-based analysis:
We can see that all the three damping types discussed are supported for these modal based analyses.
There are some important points that should be kept in mind:
Mode-superposition analyses expect lightly-damped systems.
Mode superposition methods typically use the undamped modes, but heavy damping can cause modes and frequencies to change.
Thus, the methods described here assume light damping.
For harmonic and transient analyses, we do have the “full” method that can be used for cases with significant damping.
Effects of damping are cumulative, so if you specify global damping and one part has material damping, then that part sees the effect of the sum of material and global damping.
For transient analyses, there is also Numerical Damping used to minimize noise produced by high frequency content of the structure, and this is not damping that is reflective of a physical energy dissipation process.
This parameter usually does not need to be changed and can be kept as the default value.
Response spectrum analysis only incorporates damping for the CQC and Rosenblueth mode combination methods.
Damping is not used in the SRSS combination method.
For random vibration analysis, a global damping ratio of 0.01 is used by default, if no damping is specified.
Note that the discussion in this lesson is related to using modes from an undamped modal analysis, and damping is included in the downstream mode-superposition analysis.
There is a damped modal analysis solver available in Mechanical that supports additional functionality, but its discussion is outside of the scope of this lesson.
Now let us move to a walkthrough example, the harmonic response analysis of a system under the effect of damping.
In this walkthrough example we will show you the effect of the constant and beta damping on an electronic enclosure exposed to harmonic base excitation within the 200-350Hz frequency range.
The electronic components inside the enclosure are modeled as point and distributed masses.
The enclosure is modeled as a surface body, meshed with shell elements, and made of structural steel.
Ready, let’s get started.
Open “Electronic_Enclosure. wbpz” file.
The Modal analysis system is already set up on the project page.
Open Mechanical.
Expand Geometry branch to see point and distributed masses representing electronic components inside the electronic enclosure.
Under Mesh set the Elements size to 5mm to get finer mesh of the enclosure.
Apply a Fixed support to five holes of the enclosure.
Under Analysis Settings of Modal analysis, set the maximum number of modes to find to 10 to examine natural frequencies and determine any prominent modes.
Define Future Analysis as MSUP Analyses under Analysis Data Management to save the necessary files for the downstream MSUP Analysis.
Solve the Modal analysis.
We can see that the natural frequencies range from about 240-960Hz for the 10 extracted modes.
We are interested in the harmonic response in the 200-350Hz frequency range, hence enough modes are extracted.
Highlight the first two frequency modes in tabular data and create modes shapes.
Under Solution Information set Solution Output to ‘Participation Factor Summary”,
we can see that first two modes have higher participation factors in Y direction for the first two modes, within the frequency range of 200-350Hz.
Go back to the project page and drag and drop harmonic response analysis system over Engineering Data, Geometry, Model and Solution of the Modal analysis to create MSUP workflow.
Go back to Mechanical, the Harmonic Response system is automatically added in the tree.
Under the Analysis settings of the harmonic response analysis set the frequency range from 200-350Hz.
Set the Cluster Results to “Yes” and specify the Cluster number to 4, to automatically cluster solution points near the structure’s natural frequencies ensuring the capture of behavior near the peak responses.
Under Damping controls set the Damping Ratio to 0.03 to define constant damping of 3% within the frequency range of 200-350Hz.
Right mouse click on Harmonic Response > Insert acceleration.
Set Base excitation to “Yes.”
Set boundary condition to Fixed Support, Magnitude to 1000 mm/s^2 and Direction to Y axis.
Solve Harmonic Response analysis.
Right mouse click on the Solution and insert frequency response > deformation.
Select 8 faces of the enclosure base and apply.
Set Spatial Resolution to “Use Maximum” which will report results from the location with the largest amplitude.
Set the Orientation to Y-axis.
Right mouse click on the Frequency response result and duplicate without results, new frequency response result is inserted in the tree.
Select cover face and apply under Geometry.
Evaluate All Results.
We can see that the peak amplitude is around 246Hz and 315Hz which are the frequencies of the first and second mode respectively.
Similarly for the frequency response of the cover we can see the peak amplitude is around 315 Hz.
Keep in mind that in this case the constant damping ratio of 3% is applied throughout the frequency range.
Go back to the project page and drag and drop another harmonic response system over Engineering Data, Geometry, Model and Solution of the Modal analysis to create MSUP Harmonic Response workflow.
In Mechanical Under Analysis Settings specify the same frequency range from 200-350Hz, Cluster results to “Yes”, and Cluster Number to 4.
Under damping controls we will specify Stiffness Coefficient, which is beta damping, by Damping vs. Frequency.
Set Frequency to 350 Hz and Damping ratio 0.03.
Keep in mind that beta damping is linearly proportional to frequency.
In our example at 350Hz damping is 3%, so at the frequency of 175Hz damping will be 1.5%.
On the other hand, if we had to extend our frequency range to 700 Hz, the damping would be double, or 6%, so please be careful of your frequency range when using beta damping.
Drag and drop acceleration from the first harmonic response branch to the second harmonic response branch.
Solve the second harmonic response analysis.
Drag and drop Frequency Response of the base from the first harmonic response branch to the second harmonic response branch.
Evaluate All Results.
We can see that the peak amplitude is around 246Hz and 315Hz which is the frequency of the first and second mode respectively.
Similarly for the frequency response of the cover we can see the peak amplitude is around 315 Hz.
Let’s now compare the frequency responses of these two cases using Chart feature in Mechanical.
Select Frequency response of the base for the damping ratio case and Frequency response of the base for the beta damping case.
Click on Chart in the context tool bar.
Chart will be inserted in the tree with 2 objects under outline selection.
Display Frequency response amplitudes of these two cases and omit phase angle.
On the Chart we can see the Y deformation frequency response, amplitude vs frequency of the base for the 200-350Hz frequency range.
Create another Chart with outline selections of Frequency response of the cover for the damping ratio case and Frequency response of the cover for beta damping case.
Display Frequency response amplitudes of these two cases and omit phase angle.
The red line represents the response with damping ratio defined, while the green line represents the response with beta damping defined.
As we mentioned earlier, damping ratio is constant within frequency range while beta damping is linearly proportional to the frequency.
At 350Hz damping ratio and beta damping are both 3% in our example.
So, if we look at the frequency response graph of the base, for the frequency of 246 Hz which is the frequency of the first mode, the frequency response is higher for the case where the beta damping is used compared to the case where damping ratio is used.
This is because the beta damping is less than 3% for frequencies lower than 350 Hz while damping ratio is constant at 3% within the frequency range.
If we look at the frequency response graph of the cover, we can see that at the frequency of 315 HZ which is the frequency of the second mode,
the frequency response for the case where beta damping is used is slightly higher than the frequency response when the damping ratio is used, but the difference is not as pronounced.
We can also see that for the cover, frequency response values are closer since peak response is at higher frequency than for the base,
so the damping ratio for beta damping case is closer to 3% , but still below 3%.
On the other hand, for the base, difference in frequency response is larger since damping for beta case is smaller at the lower frequencies.
This completes the demo,let’s summarize.
Damping results in energy loss, which minimizes vibrations and thus crucial for protecting the system in any dynamic analysis.
In Ansys Mechanical, damping can be defined in different ways, and we estimate it based on testing or other industry methods.
In both Full and MSUP methods, the damping is cumulative, meaning that one form of damping does not override the other but rather adds to it.
I hope you have found this video informative.
Thank you for watching and do check out our other courses to discover more useful learning resources.