Vibration and Sensitivity Analysis of Spatial Multibody Systems Based on Constraint Topology Transformation

Abstract

Many kinds of mechanical systems are often modeled as spatial multibody systems, such as robots, machine tools, automobiles and aircrafts. A multibody system typically consists of a set of rigid bodies interconnected by kinematic constraints and force elements in spatial configuration (Flores et al., 2008). Each flexible body can be further modeled as a set of rigid bodies interconnected by kinematic constraints and force elements (Wittbrodt et al., 2006). Dynamic modeling and vibration analysis based on multibody dynamics are essential to design, optimization and control of these systems (Wittenburg, 2008 ; Schiehlen et al., 2006). Vibration calculation of multibody systems is usually started by solving large-scale nonlinear equations of motion combined with constraint equations (Laulusa & Bauchau, 2008), and then linearization is carried out to obtain a set of linearized differential-algebraic equations (DAEs) or second-order ordinary differential equations (ODEs) (Cruz et al., 2007; Minaker & Frise, 2005; Negrut & Ortiz, 2006; Pott et al., 2007; Roy & Kumar, 2005). This kind of method is necessary for solving the dynamics of nonlinear systems with large deformation. However, there are two major disadvantages for vibration calculation of multibody systems by using the conventional methods. On one hand, the computational efficiency is very low due to a large amount of efforts usually required for computation of trigonometric functions, derivation and linearization. Many approaches have been proposed to simplify the formulation, such as proper selection of reference frames (Wasfy & Noor, 2003), generalized coordinates (Attia, 2006; Liu et al., 2007; McPhee & Redmond, 2006; Valasek et al., 2007), mechanics principles (Amirouche, 2006; Eberhard & Schiehlen, 2006), and other methods (Richard et al., 2007; Rui et al., 2008). On the other hand, despite sensitivity analysis of multibody systems based on the conventional methods are well documented (Anderson & Hsu, 2002; Choi et al., 2004; Ding et al., 2007; Sliva et al. 2010; Sohl & Bobrow, 2001; Van Keulen et al. 2005; Xu et al., 2009), the formulation is quite complicated because the resulting equations are implicit functions of the design parameters. Actually, what people concern, for many kinds of mechanical systems under working conditions, are eigenvalue problems and the relationship between the modal parameters and the design parameters. And the designer needs to know the results as quickly as possible so as to perform optimal design. From this point of view, fast algorithm for