Simcenter Testing Solutions FRF Based Substructuring

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FRF Based Substructuring (FBS) predicts the combined Frequency Response Functions (FRFs) of an assembly based on the FRFs of the individual subsystem components.  It is often used in conjunction with Transfer Path Analysis to help solve or predict noise and vibration issues in systems.

For example, a set of FRFs for a System A can be coupled to a set of FRFs for System B to predict the combined system response. A diagram illustrating the concept is shown in Figure 1.
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Figure 1: FRF Based Substructuring (FBS) predicts the behavior of system AB using the FRFs of two independent systems A and B plus their connections.  
FBS is used to predict possible noise and vibration issues when integrating dynamic systems together.  For example:
  • Components that are in different physical locations can be virtually assembled
  • Components measured via testing can be assembled with simulation models of components
Of course, if two components can easily be physically attached to measure the FRFs of the assembly, then there is no need to perform FRF Based Substructuring.

This article shows a formulation for a rigid connection, but the technique also works for systems connected by soft and/or springy connections as well.

This method of coupling two dynamic systems together is sometimes also referred to as the Lagrange Multiplier FBS method.

This article explains the theory and background of FRF Based Substructuring (FBS):
1.    The Challenge
2.    Nomenclature
   2.1 Coupling FRF
   2.2 Cross FRF
   2.3  Combined System FRF
3.    Equations for Individual Systems
   3.1 System A Equations
   3.2 System B Equations
4.    Combined Systems Equations for Rigid Connection
5.    No Force at Output 
6.    Combined System FRF
7.    Contact and Blocked Forces
8.    Measurement Considerations
   8.1 Modal Curvefitting and Noise
   8.2 Qsource Shakers
9.    Response Format

1. The Challenge

In the world of noise and vibration engineering, it is often desired to understand the combined behavior of two dynamic systems, even before they are physically assembled together.

For example, to predict the dynamic behavior of the subframe in a new vehicle body (Figure 2).
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Figure 2: An FRF Based Substructuring analysis can be used to understand the dynamic behavior of a subframe assembled together with a vehicle body
In this case, the FRFs of the two systems could be measured separately to predict the combined behavior.  This would be useful if the systems cannot be physically assembled together, for example if they are located in different countries.

For more about measuring FRFs, see the Knowledge Article: Simcenter Testlab Impact Testing.

Perhaps the subframe is to be used in a new vehicle that is yet to be physically built.  In this case, it might be desired to understand how the physical subframe and a computer simulation model of the new vehicle body will behave together (Figure 3).
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Figure 3: FRF Based Substructuring could be used to understand the dynamic behavior of a physical subframe assembled with a computer model of a vehicle body.
In this case, instead of measuring the FRFs, they could be generated from simulation.  See the Knowledge Article: Nastran and Test: Compare Mode Shapes and FRFs.

To perform FBS, the FRFs obtained from the subsystems should be acquired or simulated under free-free boundary conditions.

This type of prediction is useful for many different physical systems, not just automobiles.

2. Nomenclature

Take two connected dynamic systems (A and B) as shown in Figure 4
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Figure 4: The combined system AB (made from systems A and B) has input locations (i), output locations (o), and coupling (c) locations
Each system also has inputs (i) and output (o) locations. There are also coupling (c) locations on the two systems where they are connected. The subscripts on i, c, and o indicate which system they are located upon.

This diagram shows only one input, output, and connection per system.  However, there could be multiple locations like the components shown in Figures 1 and 2.

2.1 Coupling FRF

The dynamics of system A are captured by a set of Frequency Response Functions (represented by the letter H).  

An example of an FRF(s) at the connection location of system A is shown in Figure 5:
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Figure 5: FRF(s) located at the connection location on system A (green square), the superscript A indicates the system while the subscripts (ca/ca) indicate where the response is observed and force is applied 

The letter H can also represent multiple FRFs.  For example, if there are multiple coupling locations between systems A and B. This matrix of FRFs could contain FRFs between connection locations as well as “driving point” FRFs where the force and response are measured at the same coupling location.

This set of FRFs could also be referred to as a matrix of FRFs.

2.2 Cross FRF

Cross FRF(s) capture the dynamics across a system as shown in Figure 6.
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Figure 6: FRF set across system B (the superscript B indicates the system), the subscript indicates where the response location (ob indicated by right green square) over where the force is applied (cb indicated by left green square) 
As before, the letter H is used to represent the FRFs.  The letter H can represent a single FRF, or it can represent multiple FRFs.  Multiple FRFs would mean there is more than one coupling location or more than one output location on system B.

2.3 Combined System FRF

From the coupling and cross FRFs, it is desired to determine the FRF(s) of the combined system. An FRF set of the combined system AB is shown in Figure 7:
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Figure 7: An FRF set (H) describing the inputs (ia, left square) to the outputs (ob, right square) across the assembled system AB
FRF Based Substructuring is used to predict the FRF(s) of combined system AB.  It will be calculated from the individual FRF functions of systems A and B. 

3. Equations for Individual Systems

An FRF is used to relate a response to a force input.  For example, a force (F) is multiplied by the FRF (H) to get response (x) as shown in Figure 8.
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Figure 8: If a force (F – green, left) is applied at a given location, the response (x - black, right) can be calculated by multiplying the force times the FRF transfer function (H - blue middle) at each frequency.
The response could be in several different formats. See the last section of the article for examples.

The force (F), even though shown with constant amplitude in the figure, can have varying amplitude as a function of frequency. 

3.1 System A Equations

For example, consider system A. Potentially forces can be applied at both the input and coupling locations as shown in Figure 9.
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Figure 9: Response (x) equation of system A due to applied forces (F) at input (ia) and coupling (ca) locations
To fully describe the response at the coupling location of system A, potential forces being applied at both the input and coupling location need to be considered.

The forces, transfer functions, and responses are all functions versus frequency, and can have varying amplitudes and phases versus frequency.

Note: A similar equation could also be written for the response at the input location on system A, but this is not of interest for the FRF Based Substructuring methodology being discussed.

3.2 System B Equations

The response formulas for both the coupling and output locations of system B are shown in Figure 10.
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Figure 10: Force (F) and response (x) equations of system B
These multiplications describe the responses of system B.  The responses are influenced by forces applied at both the output and coupling locations. 

The forces, transfer functions, and responses are all functions versus frequency.  The amplitudes and phases of these functions can vary versus frequency.

4. Combined System Equations for Rigid Connection

If systems A and B are rigidly connected, the coupling points on system A and B must move the same. 

Using the equations described in the previous section, combined with the rigid connection condition, results in an equation which spans the two systems (Figure 11).
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Figure 11: Equations for systems A and B (bottom) connected using a rigid connection (middle)
Another condition of a rigid connection is that the forces at the coupling locations must sum to zero as shown in Figure 12.  
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Figure 12: At a rigid connection, the forces are equal and opposite (middle), which reduces the number of force terms in the equation for the combined system (bottom equation) via substitution (purple)

Because the forces are equal and opposite, it is possible to substitute the forces in the equations. This makes the separate force term at coupling location on system A the same (with opposite sign) as the force at coupling location on system B.

5. No Force at Output

Assuming zero force at the output location would be appropriate in many applications.  For example, if the output location is at an operator ear location (or sensitive vibration location) where the desired outcome is to reduce the noise (or vibration) levels due to the input forces (Figure 13).  
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Figure 13: If no forces are being applied at the output of the combined system, the equations can be further simplified, and then rearranged
Of course, if the operator was talking loudly (or bouncing on their seat), this would not be a good assumption (😊).

6. Combined System FRF

Continuing from the previous section, the equation can be rearranged as shown in Figure 14.
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Figure 14: Equation for the combined system relating input force on system A to coupling force on system B is created by factoring and dividing 

Using factoring and division, an equation relating input forces on system A to coupling forces on system B takes shape.

Using one more substitution, and still assuming that there is no force applied at the output, leads to the following (Figure 15):
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Figure 15: Equation which relates the output response (x) of system B to input force (F) on system A
This uses the previously covered equations for system B.

And with one more rearrangement, the formulation for the FRF(s) of the assembled system AB emerges (Figure 16):
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Figure 16: The FRF(s) of the combined system AB is composed of FRFs from subsystems A and B
This is the equation for the combined FRF(s) of system AB!

Some observations on the FRF data that are needed for FRF Based Substructuring:
  • Coupling FRFs: FRFs at the coupling locations of each subsystem A and B 
  • Input to Coupling: FRFs from the inputs on system A to the coupling locations on system A 
  • Coupling to Output: FRFs from the coupling locations on system B to the output locations on system B

With these FRFs, the combined system dynamics can be predicted.

7. Contact and Blocked Forces

Another use for FRF Based Substructuring is to transform forces.  For example, a blocked force applied at the output of system A can be transformed to the contact force at the input of system B as shown in Figure 17:
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Figure 17: Using FRF Based Substructuring, forces applied at one location can be transformed to the equivalent force applied at another location.
The exact steps to derive the force relationship from the FRFs are not shown here, but they are similar in nature to the previously explained steps for the combined system AB FRFs.  As before, the FRFs of the individual systems are obtained in free-free boundary conditions.

Transforming the forces allows the forces emerging from system A to be used to predict the forces entering system B. The forces entering system B can be used to determine the most important paths and their contributions on system B. Without the transformation, this would not be possible.

Notice that only FRFs at the coupling locations are needed for the force transformation.  This is convenient, since acquiring an FRF across some of the systems might be physically difficult.

For more information, see the knowledge article: Blocked Forces versus Contact Forces in Transfer Path Analysis (TPA)

In Simcenter Testlab, FRF Based Substructuring is used as part of “Component Based TPA”.  It can be turned on under “Tools -> Add-ins” and occupies 50 tokens while active.

More information on component assemblies in: Simcenter Testlab Virtual Prototype Assembly

8. Measurement Considerations

Because FRFs are inverted during the FBS process, small measurement errors can be amplified and generate large errors in the estimation of the stiffness coefficients on the interface. The matrix of FRFs in any operation needs to be well conditioned.  There are some techniques that can help mitigate these errors:

8.1 Modal Curvefitting and Noise

To remove noise from the FRFs, a modal curvefit could be performed and the resulting synthesized FRFs could be used for the FRF Based Substructuring calculations.  Example of a measured and synthesized FRF is shown in Figure 18.
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Figure 18: Measured FRF with noise (green) versus a synthesize FRF (red) from modal curvefitting
If done accurately, this creates an FRF with the same system dynamics without measurement noise.

More information in the knowledge article: Getting Started with Modal Curvefitting

8.2 Qsource Shakers

Not only must the FRF data contain minimal noise, they must also be of high quality.

It can be difficult to position modal impact hammers in the tight areas of the connections on systems, and equally difficult to ensure excitation over the frequency range of interest (Figure 19).
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Figure 19: Qsource structural exciters can be used in tight areas and ensure excitation over the frequency range of interest
Siemens Qsource structural exciters address these concerns.  They provide:
  • Easy access to hard to reach locations 
  • Correct shaker angle & position accuracy
  • Improved repeatability and signal to noise ratio
  • Good excitation levels in relevant frequency range of interest

More information about Qsource structural exciters in the knowledge article: Qsources: Acoustic and Structural Exciters

9. Response Format

Typical responses of these systems include displacement (x), velocity (v), or acceleration (a). These are the same quantities related by integration or differentiation.  This boils down to (respectively) division or multiplication by frequency or jω. The letter ω is the frequency (in radians).

More information in the Knowledge Article: Dynamic Stiffness, Compliance, Mobility, and more...

Questions?  Email 

Transfer Path Analysis

KB Article ID# KB000045683_EN_US



Associated Components

Simcenter Testlab Digital Image Correlation Testlab Environmental Testlab Acoustics Testlab Data Management Testlab Desktop Testlab Durability Testlab General Acquisition Testlab General Processing & Reporting Testlab Rotating Machinery & Engine Testlab Sound Designer Testlab Structural Dynamics Testlab Turbine