A method for the numerical integration of mechanical systems with unilateral constraints: study of impact in multibody systems

1987 ◽  
Vol 29 (5) ◽  
pp. 413-420 ◽  
Author(s):  
Vojin Drenovac
Keyword(s):  
Numerical Integration ◽  
Multibody Systems ◽  
Mechanical Systems ◽  
Unilateral Constraints

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

scholarly journals Galerkin–Ivanov transformation for nonsmooth modeling of vibro-impacts in continuous structures

2020 ◽  
pp. 107754632094544
Author(s):  
Surya Samukham ◽  
S. N. Khaderi ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Event Detection ◽  
Applied Mathematics ◽  
Detection Algorithm ◽  
Jump Conditions ◽  
Unilateral Constraints ◽  
Velocity Jump ◽  
Impact Motion

This work deals with the modeling of nonsmooth vibro-impact motion of a continuous structure against a rigid distributed obstacle. Galerkin’s approach is used to approximate the solutions of the governing partial differential equations of the structure, which results in a system of ordinary differential equations. When these ordinary differential equations are subjected to unilateral constraints and velocity jump conditions, one must use an event detection algorithm to calculate the time of impact accurately. Event detection in the presence of multiple simultaneous impacts is a computationally demanding task. Ivanov (Ivanov A 1993 “Analytical methods in the theory of vibro-impact systems”. Journal of Applied Mathematics and Mechanics 57(2): pp. 221–236.) proposed a nonsmooth transformation for a vibro-impacting multi-degree-of-freedom system subjected to a single unilateral constraint. This transformation eliminates the unilateral constraints from the problem and, therefore, no event detection is required during numerical integration. This nonsmooth transformation leads to sign function nonlinearities in the equations of motion. However, they can be easily accounted for during numerical integration. Ivanov used his transformation to make analytical calculations for the stability and bifurcations of vibro-impacting motions; however, he did not explore its application for simulating distributed collisions in spatially continuous structures. We adopt Ivanov’s transformation to deal with multiple unilateral constraints in spatially continuous structures. Also, imposing the velocity jump conditions exactly in the modal coordinates is nontrivial and challenging. Therefore, in this work, we use a modal-physical transformation to convert the system from modal to physical coordinates on a spatially discretized grid. We then apply Ivanov’s transformation on the physical system to simulate the vibro-impact motion of the structure. The developed method is demonstrated by modeling the distributed collision of a nonlinear string against a rigid distributed surface. For validation, we compare our results with the well-known penalty approach.

Dynamic Equilibrium Configurations of Multibody Systems

2014 ◽  
Vol 619 ◽  
pp. 8-12
Author(s):  
Ju Seok Kang
Keyword(s):  
Iteration Method ◽  
Multibody Systems ◽  
Equilibrium Configuration ◽  
Dynamic Equilibrium ◽  
Mechanical Systems ◽  
Railway Vehicle ◽  
Algebraic Form ◽  
Constrained Mechanical Systems ◽  
Speed Governor ◽  
Equilibrium Configurations

It is difficult to calculate dynamic equilibrium configuration in the mechanical systems, especially with the constraint conditions. In this paper, a method to calculate the dynamic equilibrium positions in the constrained mechanical systems is proposed. The accelerations of independent coordinates are derived in the algebraic form so that the numerical solution is easily obtained by the iteration method. The proposed method has been applied to calculate the dynamic equilibrium configuration for speed governor and the wheelset of railway vehicle.

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