Resumen del tema

In order to carry out CEA on a FE brake model, Abaqus/Standard is widely used in industry. The reason is this software provides an option to define frictional contact interfaces and incorporates the contribution of friction to the stiffness matrix. To do CEA in Abaqus, a few steps must be followed (Kung et al., 2003):

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Nonlinear static analysis for applying brake pressure

Nonlinear static analysis for spinning the disc

Normal mode analysis of the full brake assembly

Complex eigenvalue analysis of the full brake assembly.

As there are a number of contact interfaces in the full brake assembly in addition to pads-to- disc contact interface, before pressurising the pistons, a few pre-loading steps are usually taken for establishing these contacts. Then, through pressurising the pistons, the pads-to-disc contact interface is established and the effect of nonlinear geometry is incorporated in the analysis. According to chapter 3, three-dimensional, 3-node hydrostatic fluid elements are employed in this study to model the brake fluid. For the brake squeal problem, various brake pressures are usually considered in order to include different loading cases in the results. The brake pressure is commonly selected as 2,5,10 or 20 bar. Figure 5.5 shows the results of the nonlinear static analysis for 10-bar brake pressure on the backplate and friction material.

Figure ‎5.5. Pressure distribution on the pad and backplate

The second step is to spin the disc virtually. For this step all nodes of the disc are assigned to a node set (*NSET) and spun around the disc centre. The friction coefficient at the pads-to- disc contact interface is also set in this step. This coefficient is usually selected as 0.3, 0.4, 0.5, 0.6 or sometimes 0.7. Although the nominal value of the friction coefficient is mostly about 0.4 to 0.44, a variety of reasons such as humidity, thermal effects, ambient temperature, etc., can increase or decrease the friction coefficient. Imposing the rotational speed in this way leads to the formation of shear forces at pads-to-disc contact interface. Moreover, if someone wants to include a velocity-dependent friction coefficient, its definition can be tabulated in this step. However, such a friction regime is not considered in the current investigation.

The third step is normal mode analysis of the full brake assembly. This analysis was done in chapter 3 for the individual brake components. This time the full brake assembly must be analysed. In Abaqus, the effects of damping and also the contribution of friction into the stiffness matrix are neglected during normal mode analysis. Therefore, there is no real part

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for the eigenvalues of the system and they become purely imaginary. The symmetrised eigenvalue problem then may be written as (AbuBakar and Ouyang, 2006):

( ) (5.8)

where represents the symmetrised stiffness matrix. One of Abaqus eigensolvers, mostly Lanczos, is used to derive the eigenvalues and eigenvectors of the assembly.

The last step is to calculate the complex eigenvalues and eigenvectors of the system. For this purpose, the original mass, damping and stiffness matrices are projected onto a subspace determined by the eigenvectors of the symmetrised system (Abaqus documentation):

, - , - , - , - , - , -

(5.9)

As there is no damping in the FE model of the brake under this study, . The reduced eigenvalue problem is then solved via the standard QZ method:

, - . (5.10)

Briefly speaking, the QZ method is a well-known algorithm in linear algebra for matrix decomposition and determining its eigenvalues and eigenvectors. Eventually, the original eigenvectors of the system must be recovered:

, - . (5.11)

Note that Abaqus reports the approximation of the right eigenvector due to the fact that the left and right eigenvectors are different for a system with asymmetric stiffness matrix. Running CEA produces complex eigenvalues ( ) and eigenvectors of the system. As long as all real parts of the system are negative or zero, the steady-state solution is stable. However, a positive real part causes instability and may result in squeal noise. Many studies in the brake squeal research community take the damping ratio ⁄ as an index to evaluate the quality of a brake in terms of noise. Each car manufacturer accordingly sets a target value for the „squeal index‟. If the absolute values of this index remain smaller than the target level from different CEA runs, this brake design will be considered acceptable. Otherwise, some structural modifications must be made to avoid squeal. Although it is not guaranteed that smaller real parts lead to a quieter brake and larger ones causes squeal, this index is taken as a criterion to evaluate squeal propensity in this study.

What is common in industry for the brake squeal problem is to run a number of analyses with different friction coefficients at the pads-to-disc contact interface and also different

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brake pressures. In spite of the fact that this study aims to propose a method for covering the effects of variability and uncertainty including the friction coefficient and brake pressure, for now the common approach in industry is followed here. Figure 5.6 shows the results of CEA on the brake under this study for different friction coefficients from 0.3 to 0.6 and brake pressure form 2 to 20 bar. An example target level is shown in Figure 5.6. This threshold is set by brake engineers based on their industrial experience.

Figure ‎5.6. CEA of the brake model with different friction coefficients and brake pressures In the range of 2.0 to 5.0 kHz (low-frequency squeal range), there are two unstable modes for the brake under this study. The mode shapes of the first and second unstable mode are displayed in Figure 5.7. This study is focused on the first unstable mode, which was considered a priority for the car manufacturer. This mode occurs in the form of calliper‟s shear mode and disc‟s 4-ND out-of-plane mode (ND represents “nodal diameter”). In the second unstable mode, only pads‟ ears and guide-pins are deformed (effectively there is no deformation elsewhere). Therefore, the calliper has been removed from Figure 5.7b in order to better display the second unstable mode.

In industry, brake engineers usually identify the critical mode and the components that have a significant amount of strain energy within that mode. Structural modifications will be proposed and if they bring the „squeal index‟ within acceptable levels, the recommendations are discussed for feasibility with the suppliers. If they are practical, they will be incorporated

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into the design ready for validation on a test rig. An example of structural modification on this brake will be presented later in section ‎5.9.

Figure ‎5.7. The unstable modes of the brake: a) the 1st unstable mode at about 2.5 kHz; b) the 2nd unstable mode at about 4.8 kHz

As in CEA the nonlinear terms leading to a limit is neglected, some Dyno tests must be carried out in parallel with the numerical simulations in order to identify which one of the predicted unstable mode causes squeal. This matter will be discussed in the next section.