By M. G. De Bruin

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**Additional resources for Pade Approximation and Its Applications, Amsterdam 1980**

**Example text**

Thus, there is essentially one solution curve passing through the critical point, put this curve may have a kink. - Non-degenerate LCP-matrix and I 9 c non-empty. Since A is non-singular we may eliminate it and Pc from equation (II) to obtain Mp 9 c +~q = where M= [N9 cO]A- 1 W, (13) [Nf], and q is vector composed from the right-most vector in (11). Equation (13) together with the complementarity condition (12) constitutes a Linear Complementarity Problem (LCP) for the unknowns w and Pgc· When M is positive definite this problem has a unique solution and again we have one solution curve, with a possible kink, passing through the pre-critical point.

However, the mode of Figures 22 and 23 has the highest growth rate and would be expected to dominate the transient process. A more plausible explanation is that clutch friction materials exhibit quite complex constitutive behaviour and it is difficult to select an appropriate elastic modulus for the analysis. The modulus given in Table 1 is the incremental modulus obtained in compression tests at the mean engagement pressure, but significant stiffening may occur under service conditions. The critical speed and the dominant eigenmode are both quite sensitive to the modulus of the friction material and plausible values could have been chosen to 'fit' the theoretical predictions to a wavenumber of 12.

Azarkhin, A. R. (1986). Thermoelastic instability for the transient contact problem of two sliding half-planes, ASME J. Appl. Mech. 53:565-572. R. (1969). Thermoelastic instabilities in the sliding of conforming solids, Proc. Soc. A312:381-394. R. (1973). Sci. 15:813819. R. (1987). Stability of thermoelastic contact, International Conference on Tribology, Institution of Mechanical Engineers, London, 981-986. R. and Pritchard, C. (1985). Implications of thermoelastic instability for the design of brakes, ASME J.