Need to determine the center of gravity and principal moments of inertia of an object? Simcenter Testlab can help!
The inertias of the masses contained in a system are important in understanding how they will behave when set in motion. The inertia of an object determines how much it resists or continues to move when forces are applied.
In a flexible, or deformation mode, the component is moving and twisting at different phases. Flexible modes are therefore not used to find rigid body properties. All the mass of the object participates in rigid body modes in all directions, making them suitable for the calculation of the ellipsoid of inertia.
The image below shows the rigid body modes of a disc brake (Figure 5
Figure 5: The rigid body modes of a disk brake. The top row is translational modes, the bottom row is rotational modes.
The first row shows the translational degrees of freedom, and the bottom shows the rotational. These rigid body modes occur at low natural frequencies and when found, can be used to find the center of gravity and moment of inertia for the test object. The idea is similar to the pendulum; the acceleration towards the center axis of the system can be measured because, like a pendulum, the entire body is moving in phase together. This method provides an accurate alternative to the pendulum test, without needing extra equipment than what is used in experimental modal testing.
The main principle for obtaining inertia properties from rigid body modes is utilizing its frequency response function (FRF). FRFs are the frequency response ratio of a measured response (displacement, velocity or acceleration), over an input (force or other). In vibration, typically the acceleration is measured using an accelerometer and the corresponding frequency response function is acceleration over force, also known as accelerance.
A full explanation of FRFs can be found in greater detail here: What is a Frequency Response Function (FRF)?
Frequency response curves show the expected response of single or multiple degree of freedom systems at a particular excitation frequency. The formula for a single degree of freedom system measuring acceleration over force can be written as shown in Equation 3
Equation 3: Accelerance equation
Where A is acceleration, F is force, ω is the frequency in radians/second, k is the stiffness in N/m, c is the damping value in N/(m/s), and m is the mass in kg.
Below shows an example of an FRF measuring a single degree of freedom system (Figure 6):
Figure 6: The response of a dynamic system above its mode of vibration is proportional to the inverse of the mass.
The yellow zone shows the area of the frequency response curve that is controlled by the stiffness of the system, the orange zone shows the portion controlled by damping, and the blue zone shows the mass-controlled portion of the FRF. The mass-controlled zone is what is most important concerning rigid body modes.
More about the mass line in a FRF in this knowledge article: Dynamic Stiffness, Compliance, Mobility, and more...
These mass lines correspond to 1/m, where m is the mass of the system in the measured direction. For example, if a SDOF had a weight of 10 kg, the value of the mass line would be 0.1. In the system above, the mass of the system was 25 kg, therefore, the flat portion of the mass line is at 0.04. In the mass line method, a user selects the frequency range of the FRF where the mass line is flat and unaffected by other modes (more guidance on selecting this region is offered further ahead in this document). Figure 7
below shows the highest frequency rigid body mode with a mass line that separates it from the first deformation mode.
Figure 7: With adequate frequency separation between the highest rigid body mode and lowest flexible mode, the FRF should contain a flat frequency band for determination of the mass from rigid body mode.
To use mass line methods to determine the ellipsoid of inertia of an object, the following is needed:
- Geometry of the measurement locations on test object. Should “box in” the object and have accurately measured coordinates.
- Total mass of the test object.
- FRFs with flat mass lines at frequencies higher than the rigid body modes acquired at the measurement locations.
A good reference for how these calculations are performed is “Calculation of Rigid Body Properties from FRF Data: Practical Implementation and Cases” from the International Modal Analysis Conference (IMAC) proceedings of 1997.
Also see the document attached to this article.
3. Measurement Considerations
Some considerations when performing Frequency Response Function (FRF) measurements for moment of inertia calculations.
3.1 Test Object Suspension
The test object should be suspended in a manner that simulates free-free boundary conditions (or as much as possible). An example using soft bungee cords is shown in Figure 8.
Figure 8: Powertrain test object suspended with soft bungee cords to simulate free-free boundary conditions.
In a free-free boundary condition, the test object is not affected by or attached to any other object. In other words, it is as if the test object is “floating in space”. In other words, it is suspended in a manner that does not restrict translational motion or rotational motion around any axis. In a dynamic simulation model, a free-free boundary condition results in six rigid body modes at zero Hertz.
Because a perfect free-free boundary condition is not possible in practice, there are rigid body modes in experimental conditions, but they are not located at zero Hertz. The rigid body modes are often located at frequencies just above zero Hertz.
For the rigid body calculation based on the FRFs to work properly, a flat mass line of the FRF is required. This means that there must be a large frequency separation between the flexible modes of the test object (torsion, bending, etc.) and the rigid body modes as shown in Figure 9.
Figure 9: The first flexible mode of the test object should have a frequency that is ten times (10x) greater than the highest rigid body mode.
There are multiple options to approximate a free-free suspension experimentally (bungee cords, elastomeric mounts, foam, etc.). Whatever method is used for the suspension, the resulting FRF measurements need to be evaluated to ensure there is adequate separation between rigid and flexible modes in the structure and that the mass line of the rigid body modes of the FRF are flat.
3.2 FRF Measurements
To get the best results, FRFs need to be acquired “all around” the test object. It helps to think of creating a box around the test object as shown in Figure 10.
Figure 10: Minimal set of FRF measurement locations for a rigid body analysis.
While boxing in the object, it is also good practice to avoid the centerlines of any axis.
In the above example, a set of FRFs will be measured by applying an excitation at all eight locations in multiple directions. If using a modal impact hammer, this would mean a possible 24 separate acquisitions could be acquired. Ideally, at minimum of nine FRF sets will be acquired with multiple directions being measured at some input nodes. Acquiring greater than nine will yield more possibilities during post analysis and is recommended. The key is to excite all six rigid body modes (three translation, three rotation) with the inputs.
Some useful links on acquiring FRFs:
3.3 Accurate Coordinates of Measurement Locations
The three-dimensional coordinates of the measurement's locations are used in the calculation of the moments of inertia. The orientations of the accelerometers to the axes system are also used. Because the locations and orientations are used in the calculations, they must be measured as accurately as possible.
A two-dimensional laser level (Figure 11) can be helpful in determining the exact coordinates of the measurement locations and orientations.
Figure 11: Two dimensional laser level.
The laser site is placed next to the test object. It projects straight laser lines on the test object to make measuring distances as easy as possible in a precise manner. This can also be used to make sure the accelerometers axes are oriented as expected.
If a Computer Aided Drafting (CAD) model is available of the test object, coordinates could also be used from it.
4. Simcenter Testlab Rigid Body Calculator Software
To use the Simcenter Testlab Rigid Body Calculator, a Simcenter Testlab project containing the following is required (as described previously):
- Set of Frequency Response Functions (FRFs) on a softly suspended object needs to have been measured as shown in Figure 12.
Figure 12: A set of Frequency Response Function (FRFs) for moment of inertia determination.
A minimum of 24 output locations and 9 input locations for the FRFs are recommended.
- Geometry with precisely measured locations of the FRFs (Figure 13).
Figure 13: Schematic (left) of test object used in this example. Coordinates are entered and stored in the Simcenter Testlab geometry (right) in the project file.
- Total mass of the test object.
The Simcenter Testlab Rigid Body Calculator is used to analyze the FRFs to determine the center of gravity and moments of inertia.
4.1 Getting Started
Start Simcenter Testlab Desktop and under “Tools -> Add-ins” turn on “Modal Analysis” and “Rigid Body Calculator” as shown in Figure 14.
Figure 14: Calculating moments of inertia from FRFs requires two add-ins in Simcenter Testlab: Modal Analysis and Rigid Body Calculator.
If using token licensing, the two add-ins require 114 tokens total: 77 tokens for Modal Analysis and 37 tokens for Rigid Body Calculator.
More about Simcenter Testlab token licensing in the knowledge article: Simcenter Testlab Tokens: What are they, and how do they work?4.2 FRF Data and Modal Rigid Body Workbook
After turning on the appropriate add-ins, activate the section that contains the FRF measurements in Figure 15
Figure 15: Activate the section with the measured FRFs and then click on “Modal Data Selection” workbook.
Then click on the “Modal Data Selection” workbook.
In the “Modal Data Selection” workbook allows quick viewing of the FRFs to be used in the analysis (Figure 16).
Figure 16: View FRFs to make sure all desired are selected and then go to the Modal Rigid Body workbook.
On the middle left of the screen, green colored cells should appear. With the “Selected FRFs” checkbox on, FRFs can be interactively viewed by selecting cells.
When ready, click on the “Modal Rigid Body” workbook.
4.3 Band for Mass Line
In the “Modal Rigid Body” workbook, choose the “Change Modal Data Selection…” button in the upper left of the screen. Make sure “Active Section” is selected. Again, there should be green cells indicating FRFs are available for viewing.
Then, in the upper right of the screen, adjust the cursors to select a flat portion of the FRF curve. If necessary, zoom in to select the area between the rigid body modes and flexible modes as shown in Figure 17.
Figure 17: After selecting some FRFs to be shown (upper left), move the cursors to select a band where the FRFs are flat.
Ideally, the mass line area should be flat (as described elsewhere in this document). The number of data points in the band (i.e., spectral lines) is also shown. The more lines in the flat area, the better the calculation.
By default, the “Mass line method” setting (middle left in software interface) is set to “Unchanged” which assumes the FRFs indeed have a flat mass line. This is always the preferred method.
However, there are situations where a flat mass line can be difficult to realize: modern aircraft, for example, have the first flexible mode of the wings well below one Hertz. Designing a suspension system which would allow an adequate separation between rigid and flexible modes is expensive, if feasible at all. In this situation, dedicated methods have been developed to compensate for the lack of a flat mass line. These correction methods require a modal analysis with lower and upper residual calculations:
- Corrected FRFs: When insufficient bandwidth exists between the rigid body modes and the flexible modes, the measured FRF’s must be corrected. First the flexible modes are estimated from measured FRF’s using a modal curvefitter. Then FRF’s are synthesized from these modes (without lower residual) in the selected frequency band and subtracted from the measured FRF.
- Lower residual: In the situation where accurate measured FRF’s are not available in the frequency range between the rigid body and the flexible modes, lower residual terms can be used. Lower residual terms are estimated from the measured FRF’s using a modal curvefitter. The lower residual represents the influence of the rigid body modes and is used to approximate the mass line.
Next some automated checks on data quality can be run.4.4 Data Quality Check
Press the “Check” button in lower left to perform an automated assessment of the data quality to be used for the moment of inertia calculation (Figure 18
Figure 18: Press “Check” in the lower left to initialize the automated data check. Then press “Logging” to see the results of the check.
Checks on the data include:
- The recommended minimum of inputs and outputs for the FRFs are met.
- Data quality expressed in correlation percentage (100% being the best).
In this example, the data check is “orange” indicating that an improvement in the FRF data quality (69.02%) is desirable.
The FRFs in question and their correlation can be identified by pressing the “List” button and scrolling left to right to find the low percentage FRF as shown in Figure 19
Figure 19: In “Modal Data Selection” right click and exclude the previously identified low percentage FRFs.
The excluded FRFs will turn red to indicate they are removed from the analysis.
Returning to the “Modal Rigid Body” workbook, the data quality check can be run again as shown in Figure 20.
Figure 20: Performing the automated data quality check should ideally result in all criteria being marked green.
After excluding the problem data, the status of the quality check should turn green, indicating that all criteria were met. It is helpful to acquire more than the minimum nine inputs in case data needs exclusion.
Now the Moments of Inertia can be calculated.4.5 Calculate Moment of Inertia
To calculate the moments of inertia from the FRF data, click on the “Calculate” minor sheet (located near the top) of the Modal Rigid Body workbook (Figure 21
Figure 21: With the data quality status green and having entered the mass of the test object, the moments of inertia can be calculated.
In the “Calculate” minor workbook, the “Calculate” button (middle left) can be pressed. The moments of inertia will be shown in the upper right section of the screen, and the center of gravity location will be shown on the geometry in the lower right.
The output includes:
- Coordinates of center of gravity (cog)
- Moments and products of inertia about the center of gravity and user defined reference point
- Principal moments of inertia and their direction
- Synthesis of six scaled rigid body modes with user defined frequency and damping for use in simulation models
The calculated results can be copied to Microsoft Excel. Highlight the cells, right click, choose “Copy” as shown in Figure 22
Figure 22: Rigid Body Calculator results can be copied to Excel by highlighting all cells (upper left corner) and selecting “Copy”.
Open Excel and then choose “Paste”.
In the “Advanced...” button of the workbook, it is possible to calculate the inertia with respect to any origin.4.6 Simcenter Testlab Output Illustrated
There are several items output in the Rigid Body Calculator results. Some of the results are oriented to the original measurement coordinates, while others are oriented to the principal axes.
The origin used to describe the original measurement locations is listed (also called the “reference”), and the location of the calculated center of gravity (Figure 23
Figure 23: Measurement reference origin (blue dot) and center of gravity (black dot).
The reference coordinate system are the coordinates used to do the measurements. Because the location of the center of gravity and the orientation of the principal axes were not known when beginning the measurements, there is a good chance that the two coordinate systems (reference and principal) will not align.
The moment of inertias about the reference origin are also listed as shown in Figure 24
Figure 24: Moments of inertia around the reference origin and reference axes (light blue).
Because these moments (Ixx, Iyy, Izz) are not calculated around the center of gravity and in the principal axes, there are products of inertia (Ixy, Ixz, Iyz). As mentioned at the beginning of the document, this means that if the object were sent into a rotation around the reference origin, it would move away from the reference origin due to imbalance. In practice, it is even possible that the reference origin could be located outside the ellipsoid of inertia.
The moments of inertia are also calculated around the center of gravity (cog) and in the original reference axes (Figure 25
Figure 25: Moments of inertia around the center of gravity (cog) and reference axes (green).
Just like the moments of inertia (Ixx, Iyy, Izz) calculated around the reference origin, these also have products of inertia (Ixy, Ixz, Iyz).
Finally, the principal moment of inertia around the center of gravity and in the principal axis system is also calculated and displayed (Figure 26
Figure 26: Principal moments of inertia (red) around the center of gravity (cog) and principal axes.
The three principal moments (I11, I22, I33) are listed in numerical order, from highest to lowest. There are no products of inertia because they have a value of zero when using the principal axis system.
Lastly, the rotation between the reference axis system orientation and the principal axis orientation is listed (Figure 27
Figure 27: Translations and rotations between reference and principal axes systems (magenta).
The rotation is expressed two different ways: angles (to be performed in order from left to right) and as a rotation matrix.5. Conclusion
Experimental determination of the moments of inertia can be useful in verifying a simulation model. To get the proper motion (whether vibration or other) in a model, the mass properties of each component must be correct. This can be particularly useful when the Computer Aided Drawing (CAD) model of some components do not exist, but physical parts are available. These components can be represented by their experimentally verified mass inertial properties.
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