Blocked Forces versus Contact Forces in Transfer Path Analysis (TPA)

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There are two types of forces that can be used in Transfer Path Analysis:

  • Contact Forces (Fc) – Forces at the interface to the receiver system which were generated from the source system. The force is dependent on the interaction of the source-receiver system.
  • Blocked Forces (Fb) – Forces generated by the source system which are independent of the presence of any other component or receiver system. Sometimes referred to as invariant loads of the source.

 

This article explains the differences between these types of forces, how they are calculated, and in what circumstances they are used.

 

The article has the following sections:

   1. Background and Motivation

   2. Contact Forces

        2.1 Mount Stiffness Based Contact Force Estimation

        2.2 Matrix Inversion Contact Force Estimation

   3. Blocked Forces

        3.1 Rigid Boundary Blocked Force Estimation

        3.2 Free Boundary Blocked Force Estimation

        3.3 In-situ Blocked Force Estimation

   4. Empirical Example

        4.1 Contact Forces

        4.2 Blocked Forces

        4.3 Using Blocked Forces with New Receivers

   5. Blocked Force Application Case

   6. Conclusion

 

1. Background and Motivation

 

Contact and blocked forces are often used in Transfer Path Analysis (TPA).  Transfer Path Analysis is a technique used to understand the important dynamic paths in a source-receiver system that generates vibration or sound. 

 

In a Transfer Path Analysis (Figure 1):

  • Dynamic Forces – Forces during operating conditions which are used to characterize the source. These forces are a function of frequency or rpm (in orange). In Figure 1, these are labelled with the letter “F”, and subscript with the letter “o” for operating.
  • Transfer Functions – Dynamic structural sensitivities (usually Frequency Response Functions or FRFs) are used to characterize the receiver structure. In Figure 1, these are shown in green, and are in units of acceleration (A) over force (F). The letter H will also be used to indicate FRF transfer functions in this article.

 

TPA_Overview.pngFigure 1: Operational forces (orange, Fo) are multiplied by structural sensitivities (green) to determine vibration at a receiver location. In this case, the operational acceleration (Ao) at the receiver location is of interest.

The operating dynamic force (Fo) is multiplied by the transfer function (H) for each load path.  By evaluating the results of the multiplication, the largest contributor to sound or vibration can be determined.

 

To perform a Transfer Path Analysis, the FRFs and operational forces can originate from either simulation or test data:

  • Simulation – In Computer Aided Engineering (CAE) simulations, often an assumed load is used. For example, it might be a unit load which is equal magnitude at all frequencies. Because this assumed load is not representative of the real loads, test measurements are often done on the physical product to determine the actual operating loads.
  • Test - In the case of testing, it is often difficult to directly measure the forces acting on a source-receiver system. Instead the forces are often estimated from indirect measurements, like acceleration. Depending on the estimation method used, two different types of forces could result: contact forces or blocked forces.

 

Contact and blocked force estimation from test data are explained in the next section.  For the remainder of the article, operational forces and accelerations will not have a subscript o to indicate operational conditions, and FRFs will be referred to with a H.

 

2. Contact Forces

 

Contact forces are useful in troubleshooting noise and vibration problems.  They are the forces that are active on the receiver system.

 

Typical test methods for estimating contact forces include mount stiffness based estimation and FRF matrix inversion:

 

2.1 Mount Stiffness Based Contact Force Estimation

 

Using operational accelerations and dynamic stiffness curves, a mount-based estimation method can be used to determine the forces (F) acting within a source-receiver system. 

 

The method is based on Hooke’s law, where force equals stiffness times displacement (F=kx). In the mount method, the mount stiffness (K) is multiplied by the displacement (x) across the mount. Typically, acceleration (A) is measured on either side of the mount, double integrated, then subtracted, to get displacement.

 

A source-receiver system can have multiple inputs and multiple outputs.  A simplified diagram of such a system is illustrated in Figure 2.

 

Mount_Based_Forces.pngFigure 2: Contact forces estimated from mount stiffness use the stiffness curve (K) and relative displacement across the mount.

In Figure 2, there are inputs and outputs on the source and receiver systems indicated by numbered dots.  Even though a single dot is shown, each dot can represent multiple input or multiple output locations of each system:

  • 1: The input location(s) of the source system (S).
  • 2: The output location(s) of the source system (S).
  • 3: The input location(s) of the receiver system (R).
  • 4: The output location(s) of the receiver system (R).
  • 5: The indicator location(s) on the receiver system. Measurements may be taken here to indirectly calculate system forces, etc.  The indicator is not one of the input or output locations.

 

In the equation, the following frequency-based functions are used:

  • F3c: The contact forces at the input of the receiver. The subscript indicates the location of the contact force, in this case, it is at the input locations on the receiver system.
  • K: Mount stiffnesses with units of force over displacement.
  • A: Acceleration vibration under operating conditions, double integrated to get displacement. The subscript indicates the location where these accelerations are measured.

 

Key_Mount_Based_Forces.png

 

It should be noted that Figure 2 is a simplified representation of a source-receiver system.  A real-life system has multiple attachment locations between the source and receiver.  For example, a motor (source) could be attached with four mounts to a vehicle body (receiver) as shown in Figure 3.

 

Complex_Source_Receiver.pngFigure 3: Application example of source-receiver system with complex system dynamics and multiple interface locations.

Another method for estimating forces from test measurements is called the matrix inversion method.

 

Matrix Inversion Contact Force Estimation

 

In matrix inversion, the force estimation is based on multiplying an inverted matrix of local driving point FRFs (in the form of A/F) by the operational accelerations at the same locations as shown in Figure 4.

 

Matrix_Inversion_Forces.pngFigure 4: Contact forces estimated from FRF matrix inversion on the input locations of the receiver system.

The basic equation for the force estimation is (A/F)-1 * A. The operational acceleration at the input location on the receiver is measured and multiplied by the inverse of the measured transfer functions (H-1, with units F/A) at the same input locations.

 

In Figure 4, the following frequency based functions are used:

  • F3c: The contact forces at the input of the receiver. The subscript indicates the location of the contact force, in this case, it is at the input locations on the receiver system.
  • H33: Frequency Response Functions (i.e., FRFs) of the input locations on the receiver system. They are usually measured in terms of acceleration over force, and measured with only the receiver present. The subscripts indicate that these are “driving point FRFs” measured on the inputs to the receiver.  If there are multiple inputs, the cross FRFs between inputs should also be measured.  Or sometimes additional FRF measurements from the receiver inputs to receiver indicators are measured.  This can help ensure a well-conditioned set of FRF measurements for the inversion process. All the FRFs are inverted as part of the force estimation process.
  • A3: Acceleration vibration under operating conditions, measured at the input locations on the receiver.

 

Key_Matrix_Based_Forces.png

 

Contact forces are often used in troubleshooting source-receiver systems.  By calculating the multiple contact forces applied to the receiver system, the locations that cause the largest receiver responses can be identified.

 

Because the contact forces are determined at the input to the receiver, they already contain the dynamics of the source. They should only be applied on the receiver system alone to predict vibration or sound. Contact forces are calculated for a specific receiver and cannot be used when a different receiver is introduced into the system.

 

Key_Contact_Forces.png

 

Instead of calculating contact forces, blocked forces could be estimated from the same source-receiver system.

 

3. Blocked Forces

 

Blocked forces are characteristics of just the source system by itself.  To calculate a blocked force, the source system must be isolated (by either a completely rigid or completely free boundary condition).  If a source system cannot be isolated, the influence of the receiver can be removed through a FRF based inversion process called in-situ force estimation.

 

3.1 Rigid Boundary Blocked Force Estimation

 

To calculate the blocked forces of a source, the outputs must be clamped or blocked as shown in Figure 5. This is done by connecting the source to an infinitely rigid boundary.

 

Blocked_Forces_Rigid_Boundary.pngFigure 5: If the source output locations are connected to an infinitely rigid boundary, the forces at outputs are blocked forces.

In Figure 5, the following notation is used:

  • S: The source system.
  • F1: Input forces into the source.
  • F2b: Blocked forces at the source output due to reaction force at rigid boundary condition.

 

In practice, measuring operating forces from a source in a rigid boundary condition can be challenging.  An infinitely rigid boundary condition is hard to build.  An example attempt is shown in Figure 6.

 

Rigid_Fixture.pngFigure 6: Attempt to create a rigid fixture for performing force calculations on a tire.

Even if built rigidly, real world objects are never infinitely rigid, especially from a dynamic point of view.

 

Key_Blocked_Forces_Rigid_Fixture.png

 

Instead of trying to create a rigid connection, blocked forces can also be calculated from a free boundary condition.

 

3.2 Free Boundary Blocked Force Estimation

 

Mathematically, it can be proven that blocked forces can also be derived from measurements at the output loca­tions, if the source is suspended in free-free conditions (Figure 7).

 

Blocked_Forces_Free_Boundary.pngFigure 7: Blocked forces can be calculated at the output of the source under free boundary conditions.

In Figure 7, the following notation is used:

  • S: The source system.
  • H22: H is shorthand for a FRF. The superscript indicates the system or systems over which the FRF is measured.  In this case, the FRFs are measured on the source (S) output locations. 
  • F1: Input force to the source.
  • A2free: Response accelerations measured under free boundary conditions.
  • F2b: Calculated blocked forces, as if the source was attached to an infinitely rigid and stiff termination.

This boundary is sometimes referred to as free acceleration or free velocity conditions.

 

Key_Blocked_Forces_Free_Condition.png

 

In practice, it can be difficult to achieve a free-free boundary condition.  Imagine trying to run a source while it is suspended in mid-air!

 

3.3 In-Situ Blocked Force Estimation

 

Neither the infinitely rigid nor the free boundary condition is easy to achieve in practice.  Instead the blocked forces can also be derived from in-situ measurements where both the source and receiver are present.

 

Transfer functions between the source output locations and the receiver indicators are used to “back out” the effects of the receiver on the source output forces (Figure 8).

 

Blocked_Forces_Insitu.pngFigure 8: Blocked forces can be calculated from in-situ measurements (for example, an engine on a fixture). To do so, FRF inversion combined with measured operational data is used to calculate blocked forces at the output of the source.

In Figure 8, the following notation is used:

  • S: The source system.
  • R: The receiver system.
  • H: H is shorthand for a FRF. The superscript indicates the system (or systems) over which the FRFs are measured.  In this case, the FRFs are measured from the source (S) outputs to receiver (R) indicators.  This is done with the entire source-receiver system present, including the mounts.
  • F1: Input forces to source system.
  • K: Mount stiffnesses between source and receiver, this can also be done for rigid connections.
  • A5: Acceleration indicator measurements on the source-receiver system under operating conditions.
  • F2b: Calculated blocked forces as if the motor was attached to an infinitely rigid and stiff termination.

 

The methodology for calculating blocked forces in-situ is similar to the matrix inversion method used for contact forces.  For soft mount connections between the source and receiver, forces are calculated at the output of the source (for blocked forces) rather than the input of the receiver (for contact forces via matrix method).

 

Key_Blocked_Forces_Insitu.png

 

One challenge in the blocked forces measurement is the difficulty measuring transfer functions in assembled conditions due to the limited accessibility of many mounting points.

 

Next, examples of how and when contact forces should be used versus blocked forces will be illustrated through an empirical example.

 

4. Empirical Example

 

Consider the source-receiver system shown in Figure 9. It consists of four different masses (m1, m2, m3, and m4) connected by springs (k) and dampers (c).  Part of the system is designated the source (S) and part is the receiver (R).

 

Mass_Spring_System_Original.pngFigure 9: A dynamic mass-spring-damper system with a source (S) and receiver (R). The force input (Finput) at the source results in a velocity of the receiver (R).

In this system:

  • Mass1 (m1) = 9 kg
  • Mass2 (m2) = 4 kg
  • Mass3 (m3) = 5 kg
  • Mass4 (m4) = 7 kg
  • Stiffness1 (k12) = 4x107 N/m
  • Stiffness2 (k23) = 1x107 N/m
  • Stiffness3 (k34) = 5x106 N/m
  • Damping1 (c12) = 40 kg/s
  • Damping2 (c23) = 20 kg/s
  • Damping3 (c34) = 50 kg/s

 

A 1000 Newton dynamic force (Finput) is applied on Mass1 of the source system (S) from 0 to 1000 Hertz.  It results in a velocity (v4) of System R as shown in Figure 10.

 

Mass_Spring_Forces_Velocity_Output.pngFigure 10: Left – A 1000 Newton input force is applied to the source system (S) from 0 to 1000 Hertz. Right – The velocity output (v4) is plotted of the receiver (R) system.

This velocity of the receiver (v4) due to Finput will be used as a reference.  Contact and blocked forces will be applied at the interface of the source (S) and receiver (R) systems and compared to the reference in the next examples.  Initially, as shown in Figure 11, the contact force will be calculated and used.

 

4.1 Contact Forces

 

A contact force (Fc) at the interface of m3 (system R) can be calculated from the original system due to Finput.  This is often what is done in Transfer Path Analysis applications.  The Finput is rarely known in real life.  For example, it is easier to take measurements at the mount attachment locations of a motor or engine than to measure inside of it.

 

After calculating the contact forces, they are applied to the source-receiver (SR) system and to just the receiver (R) as shown in Figure 11.  While Finput is used to calculate v4 on the left side of Figure 11, it is not used in the middle and right systems. Only the contact force (Fc) is applied.  

 

Contact_Forces_Match.pngFigure 11: When contact forces (left, blue) are applied to the combined system SR, then the output velocity (v4) does not match the reference velocity (middle, red). When contact forces are applied to system R, the output velocity (v4) matches the reference velocity (right, green).

The following can be observed about the velocity outputs of the systems with only the contact force applied:

  • When Fc is applied to the combined source-receiver system (SR), the predicted v4 (red) does not match the reference v4 (blue line).
  • When Fc is applied to just system R, then the predicted v4 (green) matches the reference (blue).

 

The velocity output versus frequency is shown in Figure 12.

 

Contact_Forces_Graph.pngFigure 12: The velocity (v4) of the receiver system (R) is shown in blue. Applying contact forces to just system (R) results in a match between the reference velocity and output velocity (green). Applying the contact force to the combined source-receiver system does not match (red).

To get the correct output, the contact forces were applied to the receiver system (R) alone.

 

Key_Contact_Forces2.png

 

How would the force have to be adjusted to get v4 to match when it is applied to both systems S and R?  This is the blocked force which is discussed next.

 

4.2 Blocked Forces

 

If the source system (S) is hooked up to an infinitely stiff and massive object, blocked forces can be calculated as shown in Figure 13.

 

Mass_Spring_System_Rigid.pngFigure 13: Blocked forces from source system (S) can be calculated by attaching an infinitely stiff and massive object.

For the blocked force calculation, the source-receiver definition will be changed slightly compared to the contact force calculation.  Instead of calculating the force input on the receiver, the force output of the source will be calculated.  

 

The blocked forces can then be applied to the both source-receiver system (SR) and just the receiver system (R) as shown in Figure 14.

 

Blocked_Forces_Match.pngFigure 14: When blocked forces (left) are applied to the combined system SR, then the output velocity (v4) matches the reference velocity (middle, green). When blocked forces are applied to system R, the output velocity (v4) does not match the reference velocity (right, red).

Applying the blocked forces from the source system (S), the output velocity (v4) of the combined source-receiver (SR) system matches the reference velocity exactly, as shown in Figure 15.

 

Blocked_Forces_Graph.pngFigure 15: Using blocked forces, the output velocity (green) of the combined source-receiver system (SR) matches the reference velocity due to Finput (blue). When blocked forces are applied to system R alone, the output velocity (red) does not match the reference (blue).

Applying the block force to only system R does not yield the correct results (red line in Figure 15 above).

 

Key_Blocked_Forces.png

 

Contact forces yield the correct results when applied just to the receiver system alone.  Blocked forces yield the correct results when applied to the combined source-receiver system.  The advantage of blocked forces come about when introducing a new receiver system, and then trying to predict the new system output.  This is discussed in the next section.

 

4.3 Using Blocked Forces with New Receivers

 

A new receiver system (T) is introduced as shown in Figure 16. It has new masses: Mass 5 and Mass 6.

 

New_Receiver.pngFigure 16: The new receiver system (T) has and output velocity (v5) spectrum.

The source system (S) is the same as before.  The same Finput is applied, but a new velocity (v6) is produced on the receiver (T).

 

The new receiver system (T) is composed of:

  • Mass5 (m5) = 8 kg
  • Mass6 (m6) = 3 kg
  • Stiffness3 (k56) = 5x107 N/m
  • Damping3 (c56) = 20 kg/s

 

The previously calculated block and contact forces can be applied to the new receiver system as shown in Figure 17.  The velocity output created by these forces can be compared to the reference velocity (v6).

 

New_Receiver_Match.pngFigure 17: When applying blocked forces (Fb) derived from the source system (S) to the combined source-receiver system (ST), the output velocity matches the reference velocity (left, blue). Applying the contact forces (Fc) to the new receiver system (T) does not match the reference (right, red).

Using the blocked forces from system (S), the output velocity (v6) of the combined source-receiver system (ST) matches the reference velocity.  However, when using the contact forces of original source-receiver system SR, the output velocity (v6) is not correct. This is shown in Figure 18.

 

New_Receiver_Graph.pngFigure 18: Using blocked forces, the output velocity (green) of the combined source-receiver system (ST) matches the reference velocity (v6) due to Finput (blue). When contact forces are applied to the new receiver system (T) alone, the output velocity (red) does not match the reference (blue).

In this situation, blocked forces have an advantage over contact forces.  The blocked forces can be used to predict the output of a new system, where the contact forces cannot.

 

Key_New_Receiver.png

 

In this empirical example, single attachment locations were used between masses.  In real world applications, multiple attachments are often required.  The techniques work for multiple attachment locations also.

 

An example application case is discussed below to demonstrate the use/advantage of blocked forces.

 

5. Blocked Force Application Case

 

One use for blocked force is to determine the source forces “upfront” from test rig measurements during the development process.

 

Often, a source (for example, a motor, pump, or engine) is available in the test rig before its receiver application (for example, a vehicle, train, or ship) is available.

 

If a dynamic CAE simulation model of the application is available, the test fixture forces can be combined with the dynamic FRFs to predict the vibration or sound of the assembly before it is completely built.  This allows problems to be identified and resolved before the source is ever installed in real life.

 

To illustrate this situation, a small test rig and application structure was created as shown in Figure 19.  The same source was used in both the test rig and application fixture.

 

Test_Bed_Diagrams.pngFigure 19: Left – Test rig fixture with source (black/white hash lines) mounted on top. Right – Application fixture with source (black/white hash lines).

Photos of the actual fixtures are shown in Figure 20.  An electro-dynamic shaker was attached to the source to create a force.

 

Test_Bed_Photos.pngFigure 20: Left – Photo of source attached to test fixture. Right – Photo of source with shaker attached on application fixture.

The vibration on a key area on the application fixture was predicted using contact and blocked forces derived from measurements on the test fixture.  Using the blocked forces from the test rig, the predicted acceleration matched the measured closely (Figure 21) for the application fixture.

 

Measured_Predicted_Blocked_Forces.pngFigure 21: Using blocked forces from a test bench, the calculated vibration response (green) predicts the measured vibration response (black) closely on the application fixture.

When applying the contact forces from the test fixture to the application rig, the resulting prediction is not close, as shown in Figure 22. This is because the contact forces were calculated specifically for the test rig fixture, not the application fixture.

 

Measured_Predicted_Contact_Forces.pngFigure 22: Measured acceleration response (black) on application fixture versus predicted acceleration response (blue) from test rig derived contact forces applied to application fixture.

Blocked forces have an advantage over contact forces for predicting the dynamic behavior of new receivers when they are introduced into an existing source-receiver system. They can be used to predict the new receiver response.

 

Key_New_Receiver2.png

 

The contact forces from an existing source-receiver system cannot be transferred to a new receiver.  They will not give accurate results because the forces are dependent on the original receiver system.  The contact forces are not properties of the source system alone.

 

The blocked forces are invariant with respect to the receiver.  The blocked forces are properties strictly of the source system. Blocked forces must always be applied to a combined source-receiver system for accurate predictions at receiver locations.

 

Conclusion

 

Blocked forces enable sources to be evalu­ated in different applications without having to test each situation. This allows force sources to be characterized on test benches and confidently integrated in new, untested applications. 

 

Using the blocked force approach, libraries of source and receiver components can be created and combined to build accurate virtual TPA models.  This allows original equipment manufacturers to evaluate dynamic assembly options without having to build costly prototypes.

 

Questions?  Email peter.schaldenbrand@siemens.com or download the Transfer Path Analysis White Paper.

 

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