Recursive formulas for design sensitivity analysis of mechanical systems

Design sensitivity analysis of a mechanical system is an essential tool for design optimization and trade-off studies. This paper presents a design sensitivity analysis method, using direct differentiation and generalized recursive formulas. The equations of motion are first generated in the Cartesian coordinate system and then transformed into the relative coordinate system by using a velocity transformation. The design-sensitivity equations are derived by directly differentiating the equations of motion. The equations of motion and of design sensitivity are discritized by using the backward difference formula (BDF) in time domain. The resulting equations constitute an overdetermined differential algebraic system (ODAS) and are treated as ordinary differential equations (ODEs) on manifolds. The computational structure of the resulting equations is examined to classify all necessary computations into several categories. The generalized recursive formula for each category is then developed and applied whenever such a category of computation is encountered in the equations of motion and of design sensitivity. Since the velocity transformation yields the equations in a compact form and computational efficiency is achieved by the generalized recursive formulas, the proposed method is not only easy to implement but also efficient. A practical example of a vehicle consisting of many joints, bushings, and tires is given to show the efficiency of the proposed method.