# Use of Non-inertial Coordinates and Implicit Integration for Efficient Solution of Multibody Systems

thesis

posted on 27.02.2015, 00:00 by Ahmed K. AboubakrDevelopment of computational methods, formulations, and algorithms to study interconnected bodies that undergo large deformation, translational, and rotational displacements is the main focus for this thesis. This thesis discusses the use of the concept of non-inertial coordinates and implicit numerical integrations methods to solve stiff MBS differential/algebraic equations. Complex MBS examples that consist of rigid and flexible bodies are used as examples in order to demonstrate the use of these developed algorithms. One of the main contributions of this thesis is to employ the concept of the inertial and non-inertial coordinates to obtain an efficient solution for practical MBS applications. Inertial coordinates have generalized inertia forces associated with them, while the non-inertial coordinates have no generalized inertia forces. In order to avoid having a singular inertia matrix and/or high frequency oscillations, the second derivatives of the non-inertial coordinates are not used when formulating the system equations of motion in this study. In this case, the system coordinates are partitioned into two distinct sets; inertial and non-inertial coordinates. The use of the principle of virtual work leads to a coupled system of differential and algebraic equations expressed in terms of the inertial and non-inertial coordinates. The differential equations are used to determine the inertial accelerations which can be integrated to determine the inertial coordinates and velocities. The non-inertial coordinates are determined by using an iterative algorithm to solve a set of nonlinear algebraic force equations obtained using quasi-static equilibrium conditions. The non-inertial velocities are determined by solving these algebraic force equations at the velocity level. The non-inertial coordinates and velocities enter into the formulation of the generalized forces associated with the inertial coordinates. Using the concept of non-inertial coordinates and the resulting differential/algebraic equations obtained in this thesis leads to significant reduction in the numbers of state equations, system inertial coordinates, and constraint equations; and allows avoiding a system of stiff differential equations that can arise because of the relatively small mass. The development of accurate nonlinear longitudinal train force models is necessary in order to better understand railroad vehicle dynamic scenarios that include braking, traction, and derailments. Car coupler forces have significant effects on the longitudinal train dynamics and stability. Using the concept of non-inertial coordinate developed in this thesis allows developing of a more detailed coupler model that captures the coupler kinematics without significantly increasing the number of state equations and the dimension of the problem. The coupler model proposed in this thesis allows for the car bodies to have arbitrary displacements, also avoids having a stiff system of differential equations that can result from the use of relatively small masses. In order to in order to examine the efficiency of using the concept of non-inertial coordinates, a comparative study of the inertial and non-inertial coordinate coupler models is conducted. The dynamics of large and complex multibody systems that include flexible bodies and contact/impact pairs is governed by stiff equations. Explicit integration methods can be very inefficient and often fail in the case of stiff problems. The use of implicit numerical integration methods is recommended in this case. To this end, the thesis presents a new and efficient implementation of the two-loop implicit sparse matrix numerical integration (TLISMNI) method proposed for the solution of constrained rigid and flexible multibody system (MBS) differential and algebraic equations. Another contribution of this thesis is to integrate the Newton-Krylov projection method in a MBS algorithm based on two-loop implicit sparse matrix numerical integration (TLISMNI) procedure with the goal to improve the efficiency and robustness of the TLISMNI method when used for the numerical solution of constrained complex rigid and flexible MBS differential and algebraic equations. The simple iterations and Jacobian-Free Newton-Krylov approaches are used in the TLISMNI implementation. The TLISMNI method does not require numerical differentiation of the forces, allows for an efficient sparse matrix implementation, and ensures that the algebraic constraint equations are satisfied at the position, velocity, and acceleration levels. In the augmented formulation and recursive method used in this investigation, the constraint equations are satisfied at all levels. Different low order integration formulas such as HHT, which includes numerical damping, Park, Trapezoidal, and BDF2 methods were used and recommendations on the appropriateness of each method for a particular problem are made. TLISMNI implementation issues including step size selection, convergence criteria, the error control, and effect of the numerical damping were discussed. Simple pendulum, complex rigid and flexible tracked vehicle, and railroad vehicle models were used to demonstrate the use of the proposed TLISMNI method. A comparison between the results obtained using the TLISMNI algorithm and the explicit Adams predictor-corrector method is presented and show good agreement. On the other hand, using TLISMNI method which does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation for solving complex and very stiff structure problems significantly improves the simulation time. For the rigid body model considered in this investigation, the TLISMNI is at least five times faster than the explicit Adams method. Using the TLISMNI algorithm with integration formulas that employ numerical damping such as HHT in the simulation of- the flexible body models considered in this study can achieve up to thirty five times faster simulation compared to Adams method. Nonetheless, it is important to mention that there are cases of non-stiff problems in which the use of explicit Adams method can be more efficient than the TLISMNI methods. The use of the Jacobian-Free Newton-Krylov approach instead of the simple iteration approach improves the convergence and accuracy of the TLSMNI method.