Design sensitivity analysis of elastic multibody mechanisms

The purpose of this work is to formulate a system of equations for evaluating the design sensitivity coefficients of elastic multibody mechanisms with a large number of design variables. The direct differentiation or the Design of Experiment (DOE) methodology would be far more costly than the adjoint approach developed in this research. A Hamiltonian formulation of the equations of motion is presented first to provide a foundation for the subsequent development of both first and second order adjoint design sensitivity equations derived from a second order variational identity. The resulting costate (adjoint) equation is a system of first order differential-algebraic equations with the initial conditions at the final stage of the multibody dynamics simulation. A general formulation is developed for linear elastic multibody mechanisms with the sectional properties of individual finite elements as the design variables. A slider-crank with elastic connecting rod example is utilized to verify the sensitivity coefficients with the results from design experiments.; The mathematical model presented in this work is suitable for stiff multibody mechanisms with two sets of characteristically different generalized coordinates. One represents the overall rigid body motion, the other the total nodal degrees of freedom in the elastic components. Using an appropriate coordinate transformation on the nodal coordinates, it is shown that the equations of motion can be expressed in a state space form, and the system matrix associated with the transformed elastic generalized coordinates is not a function of the elastic coordinates. This special property is exploited in two ways: (1) The final form of the adjoint sensitivity equations can be formulated and significantly simplified. (2) This matrix can be approximated by a sequence of piecewise constant matrices. The elastic generalized coordinates are integrated by evaluating the exponential of the system matrix and the convolution integral in each integration step. The stepsize is determined by the speed of the overall rigid body motion to satisfy the piecewise assumption on the system matrix and is independent of the frequencies of the elastic motion. This technique is developed in a general form and demonstrated by an example.