Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion

Abstract

The dynamics of flexible multibody systems (FMBSs) is governed by ordinary differential equations or differential-algebraic equations, depending on the modeling approach chosen. In both the cases, the resulting models are highly nonlinear. Thus, they are not directly suitable for the application of the modal analysis and the development of modal models, which are very useful for several advanced engineering techniques (e.g., motion planning, control, and stability analysis of flexible multibody systems). To define and solve an eigenvalue problem for FMBSs, the system dynamics has to be linearized about a selected configuration. However, as modal parameters vary nonlinearly with the system configuration, they should be recomputed for each change of the operating point. This procedure is computationally demanding. Additionally, it does not provide any numerical or analytical correlation between the eigenpairs computed in the different operating points. This paper discusses a parametric modal analysis approach for FMBSs, which allows to derive an analytical polynomial expression for the eigenpairs as function of the system configuration, by solving a single eigenvalue problem and using only matrix operations. The availability of a similar modal model, which explicitly depends on the system configuration, can be very helpful for, e.g., model-based motion planning and control strategies towards to zero residual vibration employing the system modal characteristics. Moreover, it allows for an easy sensitivity analysis of modal characteristics to parameter uncertainties. After the theoretical development, the method is applied and validated on a flexible multibody system, specifically using the Equivalent Rigid Link System dynamic formulation. Finally, numerical results are presented and discussed.

Nonlinear Control of Multibody Flexible Mechanisms: A Model-Free Approach

“…Now, considering a properly tuned robot actuated by a speed-controlled drive, with the additional feedback control of (19), Equation (20) can be written as:…”

“…As proved experimentally by Milford and Asokanthan in [17], the eigenfrequencies of a two-link flexible robot can vary up to 30% as a function of the manipulator configuration. Several sources, such as [18,19], report that linear models are accurate only for single-link mechanisms. In this sense, model-based control strategies can be very effective, as testified in several papers.…”

Optimization of Motion Planning and Control for Automatic Machines, Robots and Multibody Systems

“…To handle nonlinearities in flexible multibody systems, a paper [19] proposes a parametric modal analysis approach to obtain an analytical polynomial expression for the eigenpairs (natural frequencies and mode shapes) as a function of the system configuration. The availability of such a result can be very helpful for model-based motion planning and control strategies due to the simple analytical equations it produces.…”

An Approximation-Based Design Optimization Approach to Eigenfrequency Assignment for Flexible Multibody Systems

“…If large displacements and/or rotations of the links are considered, a nonlinear model is essential in order to account for the dependence on the multibody system configuration. Some of the FLMS modeling techniques available in the literature represent the system dynamics by means of a minimum set of second-order ordinary differential equations, which allow for a straightforward application of modal analysis [8]. One such flexible multibody formulation is the equivalent rigid-link system (ERLS) [9,10].…”

“…The former method will be addressed in this work. The use of approximation-based design optimization has been proposed for aerospace structures [14,15] and nonlinear structural mechanics [16], including crash [17] and multibody dynamics [8].…”