Two methods of sensitivity analysis for multibody systems with collisions

In this paper, two semi-analytical methods of sensitivity analysis, the Direct Differentiation Method and the Adjoint Variable Method, are extended to multibody systems with collisions. Elementary impact theory is used for the description of collisions, i.e. the duration of the impact, configuration changes during the impact, wave propagation and so on are neglected. The variables of state are discontinuous at the time of impact. As a consequence, the design derivatives of the variables of state and hence the adjoint variables are also discontinuous. An algebraic relationship between the adjoint variables before and after the impact is derived. The number of adjoint equations are not increasing with the number of design variables whereas the number of equations in the Direct Differentiation Method depends linearly on the dimension of the design space. This is not only true for the two methods as they are known from the literature but also for the extended methods. Although the implementation of the extended Adjoint Variable Method in a numerical simulation package is rather elaborate and cumbersome, it has been done successfully as is shown with the aid of a circuit breaker mechanism with three contacts. The numerical efficiency of the new approach is discussed in case of this example system.