**Article by: Juan Jesús García, PhD, Product Manager, Braking Systems in Applus IDIADA**

Brake judder investigation requires the understanding of the vibration transmission paths that define how the vibration energy generated by the fluctuating brake torque is transmitted to the vehicle body through the suspension parts and the chassis. This knowledge allows more accurate characterization of brake judder problems in vehicles and it is a general tool to identify countermeasures for reducing the vehicle body and chassis sensitivity to brake judder excitation.

Judder Transfer Path Analysis (JPA) is an analysis tool that allows us to assess the possible ways of energy transfer from the brake to a given target location in a vehicle. It supplies the tools required to locate the most important energy transfer paths for a specific problem, and to evaluate their individual effects on the target, thus providing valuable insight into the mechanisms responsible for the problem.

The essential elements in the analysis of this type of problem are:

- The excitation sources: structure and/or airborne, acoustical or vibration. Typical sources include the vibration of a car engine, road-induced vibrations or suspension loads generated during brake judder.
- The target locations: These are typically acoustical or vibration perceived by the driver in a car during operational conditions. In our case, we focus on the vibration perceived by a driver during the occurrence of brake judder. The target points to assess the vehicle performance in these circumstances normally are the brake pedal, the steering wheel and the seat rail.
- The structural transfer paths which are represented by the physical mounts and/or rigid connections whereby the noise and vibration are transferred from the source to the target location. In our case, this transfer paths will be structural paths that transmit the suspension vibration excitation into the chassis and body hard-points when the brake exhibits judder

**Introduction**

Chassis behavior during brake judder can be characterized by locating an array of tri-axial accelerometers on relevant vehicle points defining a system containing the brake, the suspension, the chassis and part of the vehicle body. These points are characterized by their 3-dimensional movement using tri-axial accelerometers. In our case, an accelerometer layout has been selected in order to determine the main flow of vibration energy from the brake caliper to the body through the knuckle, suspension elements and the sub-frame, using a Judder Transfer Path analysis method (JPA).

The method has been applied to a high-performance car with a V6 petrol engine providing 425 HP. The vehicle incorporated front and rear disc brakes with ventilated dual-cast discs and six-piston calipers (30, 34 and 38 mm of diameter, respectively). The dimensions of the rotors were:

- External diameter: 380 mm
- Thickness: 34 mm
- Effective radius: 158 mm

As it is shown herein, judder path analysis involves the generation of various matrices that contain operational data measured on the vehicle during the judder event and the frequency response functions that define the input-output relationship between the forces applied to the chassis and the acceleration produced at the vehicle target points during judder assessment (steering wheel, seta rail and brake pedal). The mathematical calculations involved in a JPA require the inversion of the matrix of frequency response functions; this step is crucial in the accuracy of the obtained results for the operational forces (the unknown values of the problem). The condition number of this matrix, as it will be defined later on, is an indicator of the accuracy of the inversion process.

The fundamental ideas of a dynamic judder path analysis can be introduced using a static model. It turns out that the static equations that are used to calculate the force contributing to the total deflection of one structure are the same as those used for a dynamic JPA. Thus we decide to introduce this static view to facilitate the understanding of more complex ideas later.

*Fundamental Theory of JPA. A Static View*

The fundamental ideas of JPA can be introduced in a quite simple way using a static model. This simple application is very useful to understand the fundamental concepts and some limitations of a transfer path analysis that can be later generalized to the more general case of a complex structure withstanding dynamic loads. Thus, this section establishes an intuitive and robust understanding of the fundamental ideas used in the JPA and partial contribution analysis in vehicles: concepts are presented for the case of a simple supported beam.

We start by introducing the ideas applied in any transfer path analysis to a simply-supported beam with various static loads. In the examples that follow, the excitation will be one or various static forces (*F*) applied to the beam and the response will be the deflection (*d*) at a set of control or observation points located on the beam. Herein, it is shown that the theory proposed for the static case can be generalized to any dynamic mechanical system, like, for example, a vehicle suspension and chassis under judder conditions.

**Basic Input – Output Relationship**

Figure 1 shows a simply supported beam loaded by a static force *F**1* (input) that produces a deformation of the structure. The ‘response’ (deflection) of this structure at a control point ‘*A*’ is denoted as *d**A1*, and can be considered as the output of the mechanical system (beam).

We can define the mathematical relationship between input and output as a real number that fulfills the equation defined in Figure 1. This real number is the ‘transfer function’ that relates the input and output values and here it is denoted as *H**A1*. It is important to realize that the reason why, in this case, the transfer function *H**A1* is just a single real number is because the load is static and therefore *H**A1* is associated to a frequency equal to zero Hz. As we will see later, if the load is dynamic (i.e. it changes its amplitude with time), then the transfer function becomes a complex number whose modulus and phase varies with the frequency of the excitation.

**Calculation of Unknown Forces **

The calculation of unknown forces is a requirement in any JPA calculation. In most cases, the direct measurement of these forces is difficult or cumbersome. Therefore, an indirect way of estimating the value of these excitation forces is required. In the case of a simple static force applied to the beam, the knowledge of the response *d**A1* and the transfer function *H**A1* allows the direct calculation of *F**1* (see Figure 2). The equation in Figure 2 shows that in order to calculate the force *F**1*, the displacement at the target point has to be divided by the term *H**A1* or multiply by its inverse. Thus if the value of *H**A1* is small or affected by electric noise, the corresponding estimations of *d**A1* will be inaccurate. This problem is related to the concept of condition number discussed later.

When more than one load are applied simultaneously, we note that the calculation of their values from the knowledge of the total deflection at a single arbitrary point *A* and the transfer functions between the applied loads and the observation point is an undetermined problem. In general, in these cases, we only have one equation and more than one unknown values of force applied to the system.

This undetermination can be easily solved increasing the number of observation points so that the number of equations is, at least, equal to the number of applied forces. This approach is explained in Figure 3, which shows that for any given number of excitation forces, their values can be calculated by measuring the deflection at a number of observation points equal (or greater) than the number of forces.

We note that, in particular, the equations shown in Figure 3 can be expressed in matrix form as

In compact form, equation (1) becomes

Therefore, from equation (2), the value of the force vector can be calculated as

Figure 4, together with equation (3), can be used to illustrate the generalisation of the procedure to calculate the load vector for any number of excitation forces. This equation is of paramount importance in the development of a judder transfer path analysis as it will be shown later.

**Definition of Partial Contributions**

Once the values of the loads applied to the structure are known, it is possible to calculate the partial contribution to the deformation produced for any given load at a particular observation point. The value of this partial contribution with respect to the total deflection will determine the relative influence of every load to the observed deflection. As shown in Figure 5 , the partial contribution of force *F**j* in the response at an arbitrary point *i* is given by the expression,

We will now extend the equations presented above to the general dynamic case, where both the forces and response (as acceleration) are not constant in time. We see that this generalization is direct and is expressed by the same equations than for the static case.

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