Article by: Juan Jesús García, PhD, Product Manager, Braking Systems in Applus IDIADA
Read Part 1 of this article
In the first part of the article, the objective of this investigation has been introduced: to present an experimental methodology that allows a holistic interpretation of the in-service movement of a brake shoe under squeal conditions based on fluctuating acceleration data.
In this section we present the fundamental idea that justifies the movement decomposition method applied in this work. To apply this method, we assume that the body under study is moving in space with an arbitrary displacement and rotations about an arbitrary system of reference X, Y and Z (see figure 4). The theory of the method is covered in this section for convenience of the reader.
For any value of time, the instantaneous displacement of the i-th point in the body depicted in figure 4 can be expressed as the sum of a pure translation of the rigid body plus a rotation about the reference axis of the system of reference used in the analysis. Therefore we can write that
- ui, vi and wi are the displacements of the i-th point on the body shown in figure 1 (see previous article part).
- uo, vo and wo are the rigid body translations of the body assumed rigid.
- p, q and r are the rotations of the body (assumed rigid) about the axes x, y and z respectively.
- xi, yi and zi are the coordinates of the i-th point about the axes X, Y and Z respectively.
If we consider N measurement points on the brake shoe, then equation (1) can be expanded as
We note that uo, vo, wo, p, q and r are unknown parameters for each value of time. Using matrix notation, equation (2) can be condensed as
where the meaning of [D]i, [G] and [D]o can be concluded by direct comparison of equations (2) and (3) above.
We note that equation (3) represents an over-determined system of linear equations in which vector [D]o is equivalent to the least square solution of the body translations and rotations that minimize the squared error between the real measured displacements of the N points of the studied brake shoe and that part of those displacements that can be explained as the results of the combination of translations and rotations of the body assumed completely rigid. The optimum solution (in a least square sense) of displacements and rotations that partially explain the observed displacements on the body is given by the equation,
Therefore, the part of the movement of the N points on the brake shoe that cannot be explained by a combination of rigid body movements is given by
where [D]i,def represents the ‘deformation vector’ of the N points on the body. The values in this vector are not caused by rigid body movements and, therefore, represent a vector of displacements caused by vibrational deformation. Equation (5) can be solved for each value of time of an ensemble of time histories of displacements obtained from an array of accelerometers on the shoes. This will allow the calculation of the time histories of the variables uo, vo, wo, p, q and r. For the case studied here, equation (3) becomes
In this case, for each value of time, the last column vector in equation (6) defines the position of point 2 in figure 1 (see previous article part), (uo, vo), and the rotation of the brake shoe about a vertical the axis passing through point 2. The positive direction for this angle is anti-clockwise.
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