Shock and Vibration

Volume 2015, Article ID 348106, 10 pages

http://dx.doi.org/10.1155/2015/348106

## Energy-Based Optimal Ranking of the Interior Modes for Reduced-Order Models under Periodic Excitation

Department of Management and Engineering (DTG), University of Padova, 36100 Vicenza, Italy

Received 14 May 2015; Accepted 2 August 2015

Academic Editor: Georges Kouroussis

Copyright © 2015 Ilaria Palomba et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper introduces a novel method for ranking and selecting the interior modes to be retained in the Craig-Bampton model reduction, in the case of linear vibrating systems under periodic excitation. The aim of the method is to provide an effective ranking of such modes and hence an optimal sequence according to which the interior modes should be progressively included to achieve a desired accuracy of the reduced-order model at the frequencies of interest, while keeping model dimensions to a minimum. An energy-based ranking (EBR) method is proposed, which exploits analytical coefficients to evaluate the contribution of each interior mode to the forced response of the full-order system. The application of the method to two representative systems is discussed: an ultrasonic horn and a vibratory feeder. The results show that the EBR method provides a very effective ranking of the most important interior modes and that it outperforms other state-of-the-art benchmark techniques.

#### 1. Introduction

The synthesis of accurate finite element (FE) models of complex vibrating mechanisms and structures usually imposes the use of fine meshes, leading to large dimensional models. Unfortunately, large dimensional models are often undesirable since they are difficult to handle and often numerically ill-conditioned because of the presence of large condition numbers (i.e., the ratio between the maximum and the minimum singular values of a matrix). Therefore, they can be too cumbersome for use in simulations, synthesis of controllers [1] (mainly of real-time model-based controllers), model-based design [2], and optimization techniques [3–6]. In order to cope with such an issue and to optimally trade off model accuracy and dimension, several model reduction techniques have been developed in the last decades [7]. In the field of structural dynamics and multibody system dynamics one of the most implemented reduction techniques is the Craig-Bampton (CB) method [8] because of its effectiveness and easiness of implementation.

Two peculiar reasons make the use of the CB method particularly advantageous. First of all, it allows retaining physical displacements in the reduced-order coordinate vector. Physical coordinates may be of interest, for instance, for coupling a system with other systems designed either in the mechanical domain or in other domains (consider not only the CMS [8, 9] but also the coupling between electromechanical systems in multiphysics simulations [10]). Physical coordinates are also of interest when modifications of physical parameters should be computed through inverse structural modification techniques [5]. Secondly, the CB method makes direct use of second-order model formulations (i.e., models with mass and stiffness matrices) rather than first-order ones: second-order models are usually preferred in structural dynamics [7]. These features are not met by many other model reduction techniques, such as, for instance, Krylov subspace and the balanced truncation [9].

Basically, the CB method is a combination of Guyan’s static condensation and modal truncation. Indeed, it uses the static deformation shapes of some degrees of freedom (dofs) of the system (the so-called master dofs), as in Guyan’s condensation, and then enriches this space with a reduced set of coordinates referred to as the interior modes of the system. The practical implementation of the method imposes, as a first step, partitioning the displacement vector into a subset of master dofs , named the external dofs, and a subset of slave dofs , the interior dofs, with , and then transforming it into a set of hybrid coordinates through the CB basis (the transformation matrix):The definition of the transformed hybrid coordinates introduces vector , which includes the aforementioned interior modal coordinates (often referred to as the fixed interface modal coordinates) which replace the slave dof coordinates . In contrast, the master dof coordinates are entirely retained in and are, therefore, usually chosen as those lying at the interfaces that are to be coupled to other systems and/or as those where external forces are applied. In (2), is the nonsingular CB transformation matrix, is a Guyan’s reduction basis, and is the eigenvector matrix obtained by solving the eigenvalue problem of the system with all the external dofs constrained (which will henceforth be referred to as the constrained subsystem). The columns of are the interior normal mode shapes, henceforth referred to as , associated with the interior modal coordinates collected in vector . Finally, and represent, respectively, the identity and the null matrices.

In order to reduce model dimensions, the interior modal coordinate vector is truncated to a smaller vector , , by leading to the reduced coordinate vector , which is expected to provide an effective approximation of :In (3) is the rectangular CB reduction matrix, obtained from (2) by removing the columns of associated with the interior neglected modes.

The crux in the practical implementation of the CB method is performing an optimal selection of the interior modes to be retained in reduced models. Typically, as a widespread rule of thumb, model reduction relies on retaining the interior modes whose eigenfrequencies are not greater than about twice the highest operating frequency [11]. Such a sorting rule based on the eigenfrequencies of the interior modes (henceforth referred to as SBE, for brevity), however, is not based on rigorous principles and neglects the characteristic of the external excitation influencing the system response. As a consequence, it may lead to rough approximations of the full-order model by discarding high-frequency interior modes whose participation in the system dynamics is important, or, in contrast, may lead to large dimensional reduced models including low-frequency modes providing negligible contributions.

In order to improve the effectiveness of the reduction, some methods have been proposed in literature to rank and select the interior modes. These techniques can be seen as “auxiliary” methods for CB reduction, since they operate between the two steps of the CB method. This idea is schematically depicted in Figure 1. Among the ranking methods proposed in literature one should at least recall the “Component Mode Synthesis *χ*” (CMS *χ*) [12], the “Effective Interface Mass” (EIM) [13, 14], and the “Optimal Modal Reduction” (OMR) [15], which have been proved to be effective techniques that allow reducing model dimensions while preserving accuracy. Their effectiveness is certainly higher than that of the traditional SBE approach. Basically, these approaches rank the interior modes on the basis of some terms representing the contributions of each mode to the dynamic loads at the interface. They are general purpose methods that can be applied to several applications but can lead to approximate and less effective results in some particular cases. Indeed, they cannot handle requirements on the frequencies at which the reduced model should be accurate or on the external forces in terms of both spatial distribution and frequencies. Such requirements are often known, mainly in those systems which are designed to operate excited at a specific frequency, for instance, in the neighborhood of a resonance frequency, with a prescribed shape and in the presence of a known external force. Typical examples are open-loop resonators, such as ultrasonic horns or vibratory feeders, or systems in the presence of a single-point harmonic force excitation. For these systems, the availability of reduced models is an essential need for performing numerical simulations, such as multiphysics simulation coupling mechanical and electromechanical models or when simulating complex plants (e.g., manufacturing plants), and for developing model-based design and optimization techniques (mainly when inverse structural modification is employed, see, e.g., [4, 5]). Indeed, although the accuracy of a model might decrease when large modifications of some critical physical parameters are made, it has been widely demonstrated that using simplified, though approximate, models increases the numerical reliability and solvability of the design techniques and also allows applying some design methods that do not work, or may fail, for large scale models. For instance, using smaller problems improves the conditioning number of matrices and allows solving least-squares problems, which are often ill-posed by nature [5]. As for the computational effort, reducing models reduces more than linearly the analysis time. Therefore, the computational cost for reducing models is balanced out by the improvement in the application of the techniques, whenever the reduction is capable of preserving model accuracy.