Complex Eigenvalue Analysis and Brake Squeal: Traps, Shortcomings and their Removal

Among many NVH problems brake squeal continues to be a difficult topic for design engineers and scientists. Both the experimental and the simulation approaches so far have failed to provide robust and reliable guidelines for the design of squeal free brakes. On the experimental side the problem clearly lies in the wide range of operating conditions which the brake encounters in its lifetime, in which it should be squeal free. From lab experiments alone, it is hardly possible to judge how far the system is from squeal, which implies that an extremely wide range of conditions is mandatory. Brake squeal simulation presents different challenges. Once a model for the brake has been formulated, including the excitation mechanism(s), it should be possible to check the robustness of a given design and system parameters against squeal. Complex eigenvalue analysis has become a standard industrial tool for squeal prediction, and is routinely applied to the simulation models. Despite many years of research and development along these lines, the reliability of squeal prediction is still not adequate. This lack of reliability of the simulation results persists, even if great care is taken to properly model the friction and damping, gyroscopic terms, etc. and a more fundamental difficulty remains in the simulations. One of the basic assumptions of complex eigenvalue analysis is that the system is time-invariant and behaves like one with constant coefficients. Brake disks however typically have ventilation channels, and therefore are not axially symmetric. This leads to periodic coefficients in the equations of motion of the brake with a rotating brake disk, to which the standard complex eigenvalue analysis is not applicable. Using basic case studies, in this paper it is shown that the conclusions drawn from complex eigenvalue analysis can be meaningless in this case, at least concerning the real part of the eigenvalue. It is shown that in situations which should be squeal free according to the standard eigenvalue analysis, the system can actually squeal, and vice versa. In order to overcome this problem, a method is outlined, which is capable to take into account the lack of axial symmetry of the brake rotors and should yield much more accurate predictions than the standard complex eigenvalue analysis.