Comparison of Different Approaches to Include Connection Elements into Frequency-Based Substructuring

Abstract

Dynamic substructuring (DS) is a research field that has gained a great deal of attention in both science and industry. The aim of DS techniques is to provide engineers in structural vibrations and sound practical solutions for analyzing the dynamic behavior of complex systems. This paper addresses the singularity problem that occurs when flexible joints are implemented as substructures into the Lagrange Multiplier Frequency-Based Substructuring (LM-FBS) coupling process. For illustration, we use rubber bushings from an automotive application. Considering the rubber isolators to exhibit hysteretic damping, we assume that only the property of the dynamic stiffness of material is given. To avoid singularity appearing in the admittance when inverting the impedance of a massless joint, we compare three different approaches to include rubber bushings in the framework of LM-FBS. One method consists in including the dynamic stiffness of material directly in the space of the interface constraints and add it to the assembled interface flexibility of the LM-FBS equation. This corresponds to a relaxation of the interface compatibility condition. In the second method, the rubber bushing is treated as a substructure by adding small masses to the equation of the joint. As a result, we obtain a nonsingular total dynamic stiffness matrix that can be included in the coupling process. The third method describes a novel extension of the LM-FBS approach, based on a solution for singular problems. If the applied forces are self-equilibrated with respect to the rigid body modes, a solution for the singular dynamic stiffness matrix exists. The methods are outlined, both mathematically and conceptually, based on a notation commonly used in LM-FBS. They facilitate the integration of connecting elements together with experimental or numerical determined system dynamics of substructures in order to predict the assembled system behavior.

On the prediction of the time-varying behaviour of dynamic systems by interpolating state-space models

“…This is due to the fact that many mechanical systems are composed by parts, whose dynamics can significantly change over time when these mechanical systems are running under normal operating conditions. A classical example of these parts are the rubber mounts (which are commonly used in the automotive industry [1]), whose dynamic stiffness is highly dependent on many factors, for instance, on temperature and on the applied static pre-load (see [2], [3], [4]). To analyze the dynamic behaviour of components presenting time-varying mechanical behaviour, the use of time domain approaches is advantageous, because, by working on this domain, we can analyze components presenting time-domain variations on their dynamics and submitted to any kind of excitation.…”

Estimation of Joint Parameters Using Frequency-Based Substructuring Techniques

“…De Klerk et al [15] introduced the LM-FBS method to improve the classical FBS method for assembling the dynamic admittance of substructures. Mahmoudi et al [16] used the LM-FBS algorithm to treat flexible joints as substructures, presenting three approaches for avoiding the singularity problem in coupling substructures. Bouslema et al [17] investigated the global dynamic behavior of a coupled transmission system based on a double-reducer stage.…”

On the use of Lagrange Multiplier State-Space Substructuring in dynamic substructuring analysis

“…Dynamic Substructuring (DS) was introduced in the last century with the belief that complex structures can be analyzed in a more efficient way if they are considered as assemblies of several simpler components. From its first introduction, this idea pushed the scientific community to an intense activity in this field, and several DS approaches have been released [1]. Examples of those techniques are the ones clustered under the labels Component Mode Synthesis (CMS) and Frequency Based Substructuring (FBS).…”