Numerical integration of stiff differential/algebraic equations of flexible multibody systems

In the beam models based on the absolute nodal coordinate formulation (ANCF), the cross-section is allowed to deform and it is no longer treated as a rigid surface as in some classical formulations. These more general models lead to new geometric terms that do not appear in the classical formulations of beams. Some of these geometric terms are the result of the coupling between the deformation of the cross-section and other modes of deformations such as bending and they lead to a new set of modes called the ANCF-coupled deformation modes. The effect of the ANCF-coupled deformation modes can be significant in the case of very flexible structures. Nonetheless, in the case of stiff and thin structures, these ANCF modes can be associated with very high frequencies that negatively impact the element performance and its efficiency.;In this thesis the effect of the ANCF coupled deformation modes in planar and spatial multibody system applications is demonstrated. In order to address the problem of the high stiffness associated with ANCF coupled deformation modes, implicit integration methods with adjustable numerical damping properties are considered in this thesis. One of the implicit integration methods used in this investigation is the Hilber-Hughes-Taylor method applied in the context of Index 3 differential-algebraic equations (HHT-I3). In order to develop an efficient procedure for solving the differential/algebraic equations of multibody systems, a Two-Loop Implicit Sparse Matrix Numerical Integration procedure (TLISMNI) is proposed in this study. The proposed method ensures that the kinematic constraint equations are satisfied at the position, velocity and acceleration levels and does not require the numerical or analytical differentiation of the forces. The proposed TLISMNI method is ideally suited for the use with ANCF finite elements that lead to a constant mass matrix, and therefore, an optimum sparse matrix structure can be obtained for the multibody system dynamic equations. TLISMNI implementation issues including the step-size selection, the error control, and the effect of the numerical damping are discussed in this investigation. The relation between the step-size selection and the structure stiffness is also discussed. The use of the computer implementation proposed in this thesis is demonstrated by solving very stiff structure problems using the Hilber---Hughes---Taylor (HHT) method, which includes numerical damping. An eigenvalue analysis and Fast Fourier Transform (FFT) are performed in order to identify the fundamental modes of deformation and demonstrate that the contributions of these fundamental modes can be erroneously damped out when some other implicit integration methods are used. The TLISMNI method, on the other hand, captures the contributions of these fundamental modes. The results obtained using the TLISMNI method are compared with the results obtained using other methods including the implicit HHT-I3 and the explicit Adams integration methods.