**By Juan Jesús García, PhD, Product Manager, Braking Systems in Applus IDIADA**

In the first part of the article, the fundamental theory of the so-called Judder Transfer Path Analysis (JPA) was introduced. This methodology allows to assess the possible ways of energy transfer from the brake to a given target location in a vehicle.

**Methodology**

*Generalized Theory
for Dynamic Systems. Automotive Applications – JPA*

Let us assume that we have a vibration system
into which we transmit vibration energy through a number of discrete connection
points and where the NVH response is analysed. When assuming *N* different possible transmission paths
over which the energy coming from the single reference source substructure can
be transmitted into the main body, the total operational acceleration [a] at a
given set of target location in the vehicle can be written as the sum of the
contribution of the partial accelerations, *a _{i
}*, related to each given

*i-th*transmission path. The determination of these partial accelerations is based upon the combination of an estimate of the operational force

*f*in the given transmission path, together with the mechanic-vibration transfer function between the target acceleration response and a force applied at the chassis side of the considered transmission path , i.e.,

_{i}*P/F*.

_{i}Then, the total acceleration can be broken down as:

We note that the terms *A _{i}/F_{i}* are the transfer functions that relate acceleration the response at the target points (vibration) and the force applied at the

*i-th*path. When considering

*M*multiple target accelerations [a] all the FRF’s can be assembled in a mechanical-vibration FRF-matrix [H]. Using matrix algebra this means,

[a] = [H] [f]

For the general case in which we have *M* target responses (acceleration) and *N* paths, equation (6) would become:

It is important to emphasize that equation (7) is a function of frequency. In practise, the frequency response functions are usually measured after disassembling the source from the structure body, eliminating source coupling of the transmission impedances. They can be measured for example by using hammer or shaker excitation techniques. The latter method in general provides the most accurate results, but at the cost of a more complex test procedure.

A technique for the force estimation is based upon the measurement of a mechanical impedance matrix [H], containing FRFs between accelerations measured at the body side of all paths on one hand and forces applied at each path on the other hand. By inverting this matrix, and multiplying it with the vector of the corresponding body side operational accelerations, estimates of the operational forces are obtained. These are:

where [a] denotes the second derivative of the displacement [x], i.e., the acceleration at the observation points. As for the static case already covered above, when solving dynamic problems in general, theoretically, it is sufficient to take into account a number of responses equal to the number of forces that has to be estimated (*N = M*). Taking into account more responses (*M > N*), the set of equations is over-determined, and better force estimates will be obtained in a least-squares sense. This is normal practise in many engineering applications. The inversion is then based upon singular value decomposition algorithms, which allow to artificially improving the conditioning of the inversion.

**TPA Applications to Brake Judder: Judder Path Analysis (JPA)**

Brake judder is highly vehicle-dependent. Since the mechanical energy travels through the structure and interacts with the steering, brake pedal, and seats, there can be major differences in how the same amount of disc thickness variation excites these systems. Thus, a certain level of DTV that is perfectly acceptable in one vehicle may lead to a major problem on another. This leads to situations where the same brake applied to different vehicles can exhibit completely different judder levels.

This situation requires a robust method to identify the reasons why certain vehicles are particularly sensitive to disc thickness variation. Current methodologies are based on an experimental trial-and-error approach implementing modifications, the brakes or the vehicle suspension or the sub-frame. This approach is time-consuming and expensive. Thus, a robust and systematic method is required to identify the vehicle paths responsible for the transmission of vibrations from the brake to those parts that affect the drive´s perception. Figure 6 shows an example of various transmission paths of judder vibration that one can encounter in vehicles. Note that in more complex suspension systems, the number of transmission paths can go up to about seven. Once the transmission paths initiated at the suspension level enter into the vehicle body, the propagation of the vibration is spread out through the vehicle body and collected back into the receiving points that define the interface between the driver and the vehicle.

The judder analysis method proposed herein must define the relative contributions originated by each input path determined by the suspension arrangement. This contribution will be determined in terms of the acceleration level that, for any given frequency, is transmitted through the path under study. This method will allow the following conclusions to be drawn (see Figure 6):

- Dominant transmission paths of detected judder
- Systems and subsystems affected by the transmission paths
- Ideal locations to act in order to ‘inhibit’ the transmission path
- Definition of optimized countermeasures for judder control in vehicles

The development of this method of judder path analysis (JPA) will materialize the integration of brake and NVH knowledge for the specific field of judder control.

Figure 6: Concept of a judder path analysis (JPA) for a judder-induced vibration

**Vehicle Instrumentation**

A transfer path analysis (JPA) requires the knowledge of operational vibration at a minimum number of points located on the vehicle parts that make up the system under study. The locations of the accelerometers are chosen so that the fundamental movements of all parts involved can be detected under operational loads. Of paramount importance is to define the acceleration matrix so that rigid body motions, bending and torsion deformation of the measured parts can be detected, since these type of movements and deformations have a big influence on the JPA results.

Figure 7 shows the array of accelerometers for JPA analysis of the suspension and chassis system used in this work for our test vehicle with a high judder sensitivity. In particular, the following points have to be emphasized when defining the tri-axial accelerometer array:

- Try detecting bending and torsion movements of the chassis and suspension system
- Accelerometers should be located at both ends of connecting joints
- Accelerometers should be located at the attachment points of the chassis to the vehicle body (chassis and body side)
- The overall extension of the array should cover from the excitation source (brake structure) to the receiving vehicle points (seat rail, steering wheel, etc).

Figure 7: (Left): layout of accelerometer locations for operational chassis vibration analysis. Additional set configuration for the operational measurements and the measurement of the FRF for the JPA analysis (right).

Figure 8 shoes the global view of the bottom part of the vehicle under study. This vehicle had a high sensitivity to brake judder and the JPA was aimed at investigating possible paths that could explain this high sensitivity.

Figure 8: The global view of the bottom view of the studied vehicle showing the area that has been investigated for JPA (Left front suspension and chassis). The array of points measured in the vehicle corresponding to the area marked with the red square is presented in Figure 7.

**Considerations about the condition number of the matrix problem**

The notion of condition number is important in all of applied mathematics. If small changes in the data of some problem always lead to reasonably small changes in the answer to the problem, the problem is said to be well-conditioned. If small changes in the data of some problem can lead to unacceptably large changes in the answer to the problem, the problem is said to be ill-conditioned. The importance of this concept is clear. In applied problems, experimental data are always inaccurate because of measuring and modelling errors, and it is important for us to know what effect such inaccuracies in the data have on the answer to the problem.

If we consider the condition of the solution [*f*] in equation [*a*] = [*H*] [*f*], in terms of the data [*a*] and [*H*], it can be
shown that the equation is well conditioned to be inverted and, thus, solved,
if the value of the expression

is not too large. Equation (9) defines what we call the
condition number of matrix [*H*], which is expressed as c([*H*]). In this case, the symbol || ||
denotes any matrix norm.

A moderate condition number guaranties that the equations are
well conditioned, i.e. small changes in the data produce reasonably small
changes in the solution. If c([*H*]) is large, however, the changes in [*F*] caused by changes in the data may be
much larger than the changes in the data.

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IDIADA**

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