A New Numerical Technique for Index-3 DAEs Arising from Constrained Multibody Mechanical Systems

Brahim Benhammouda,Charalampos Skokos

doi:10.1155/2022/2712196

Brahim Benhammouda, Charalampos Skokos

Open Access

https://doi.org/10.1155/2022/2712196

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Abstract

Index-3 differential-algebraic equations (DAEs) are mathematical models for the dynamics of constrained multibody mechanical systems that arise in many applications. These DAEs are known to pose a challenge to numerical methods. The purpose of this paper is to propose a new approach to efficiently solve these equations. This approach relies on an effective combination of the power series method (PSM) with the Adomian polynomials. Here, the PSM is directly applied to these DAEs without using the usual index reduction techniques, which are costly and often lead to nonphysical solutions. We expand the nonlinear terms in a series form using the Adomian polynomials to overcome the limitation of the PSM in collecting the coefficients of the power series solution. This technique has led to a simple and efficient algorithm. The domain of convergence of the power series solution is expanded by developing a multistage PSM (MSPSM). To demonstrate the efficiency of the MSPSM and show its applicability, an index-3 nonlinear DAE problem describing a two-link planar robot arm is solved. The numerical results show that the MSPSM is a powerful tool for solving the index-3 DAEs arising from constrained multibody mechanical systems.

Highlights

Constrained multibody systems [1] have applications in many important applications like rocket-towed systems [2], aerospace engineering [3], vehicle industry [4, 5], and robotics [6]
Constrained mechanical multibody systems are collections of rigid and/or flexible bodies that are interconnected by kinematic joints. e role of these kinematic joints is to impose some restrictions on the relative motion of the bodies of the system
Some constraint forces are present in the system. e forces that act on the multibody system components can come from springs, dampers, and actuator or external forces. e dynamics of these complex mechanical systems is described by nonlinear index-3 differential-algebraic equations (DAEs) systems of the form u′ v, M(u)v′ f(u, 0 g(u, t), v, t)

Summary

Introduction

Constrained multibody systems [1] have applications in many important applications like rocket-towed systems [2], aerospace engineering [3], vehicle industry [4, 5], and robotics [6]. E reason is that the method may converge only over smaller intervals and the collection of the coefficients of the power series solution can be a difficult task We can overcome these drawbacks using the Adomian polynomials and the multistage technique. We present a new technique to solve general strongly nonlinear index-3 DAEs (1) arising from constrained mechanical systems efficiently. E proposed technique has led to a simple and efficient algorithm that can be implemented to solve general nonlinear index-3 DAEs. Our technique can be used in designing software for industrial applications and provides a powerful tool for mechanical engineers in particular.

Adomian Polynomials

Derivation of the Proposed Technique

Numerical Simulation and Discussion