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

4.3 Automotive brake disc with cooling channels
between two cooling ribs and thus give the minimal and maximal distances
between them, which have to be kept due to cooling and material strength
requirements. In contrast to the case with radial holes, the maximal and
minimal distance constraints can be written in linear form, since the variation
in radial direction is much smaller. The brake disc has to be statically and
dynamically balanced leading to the nonlinear equality constraints
c eq
1 (r,ϕ) = ∑ N
n=1 r ncosϕ n ,
c eq
2 (r,ϕ) = ∑ N
n=1 r nsinϕ n ,
(4.10)
which can easily be derived, since the brake disc’s friction rings and each of
the cooling ribs are assumed to be homogeneous bodies of constant geometry
rotating at a constant angular velocity. These constraints for static balancing
represent the projection of the excentricity of the brake disc induced by the
placement of the cooling ribs onto x-direction (c eq
1 ) and y-direction (ceq 2 ),
respectively. Also, the dynamic balancing is automatically guaranteed due to
the symmetry with respect to the middle plane of the disc.
As has been introduced in section 4.1, the presented optimization problem
is nonconvex and nonlinear with linear and nonlinear constraints and many
local optima to be expected. The brake disc to be optimized is modeled using
the method proposed in chapter 3. The underlying FE meshes of the friction
rings and each cooling rib have been assembled using Abaqus with 3D solid
isoparametric elements of quadratic shape function order (C3D20) leading to
65250 DOF in the friction ring mesh and 4860 DOF for each cooling rib, fulfilling
convergence requirements. The parameters of the filling material between
the friction rings were chosen as Ẽ = 1.4 MPa and ˜ρ = 1 kg/m3 . Since the
limit frequency f lim is estimated to be 9 kHz, the first 21 eigenfrequencies
of the brake disc with N = 39 cooling ribs are considered in the calculation.
Solutions are treated as infeasible if one of the constraints c eq
1
and ceq 2
is violated
by an absolute value of 0.005 m. The parameters for the upper and
lower bounds and the linear constraints can be found in Tab. 4.6. Due to the
properties of the optimization problem, deterministic optimization approaches
cannot guarantee the finding of the global optimum, therefore, heuristic algorithms
are indicated. In the following, these two types of algorithms will be
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