Avoidance of brake squeal by a separation of the brake ... - tuprints

2.2 Splitting of the brake rotor’s eigenfrequencies by asymmetry
as is shown in [109], where the thickness of the brake disc is denoted as h,
r p is the radius to the midpoint of the brake pad and k is its overall stiffness
simplified here as a linear spring. The generalized mass corresponding to
the double eigenfrequency f i = f j is m G , the maximal amplitude of the
corresponding shape function is ŵ i and its maximal spatial derivative ŵ j ′ is
given by ŵ j ′ = ∇ŵ j · e ϕ with e ϕ being the direction of the relative velocity
between brake disc and pad. The factor 2π in Eq. (2.17) result from the
integration of the entries of the matrix of circulatory terms over one period
for the stability analysis using Floquet theory, see Eq. (2.12). The same
holds for Eq. (2.19). For a conservative estimate, the product ŵ i ŵ j ′ can be
assessed to be
ŵ i ŵ ′ j ≤ 1, (2.18)
since the shape functions are assumed to be normalized to an amplitude of
1. For practical applications, the position of double eigenfrequencies in the
frequency band of consideration and thus also the position of instabilities is
not known before a final design is fixed. It has to be taken into account
that they can occur everywhere in the whole frequency band. Therefore, ∆f
is not evaluated for discrete frequencies f j anymore but for the continuous
frequency f. For a conservative estimate, the generalized mass can then be
chosen to be the minimal generalized mass m min
G
of all eigenmodes in the
audible frequency range. Here, the minimal generalized mass is assumed to
be calculated with eigenvectors normalized to an amplitude of 1. It can either
be gained from a simplified analytical or finite element model.
In reality, the damping of the brake system not only originates from material
damping of the disc and the brake pads, but also from the friction in the joints
and from the friction between brake pad and rotor. Thus, the damping matrix
can be a dense matrix. Still, as has been explained at the end of the previous
section, it is assumed here that the damping matrix, corresponding to the
double eigenfrequency f i = f j , is diagonal with equal elements d. Also, if the
common assumption of mass and/or stiffness proportional damping is made,
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