Mode selection for reduced order modeling of mechanical systems excited at resonance

Abstract Welding, food cutting, atomizing, cleaning and deagglomerating are just a few common uses for ultrasonic resonators, which are carefully designed to operate excited in resonance. Finite element analysis is nearly always adopted for predicting and improving resonator performances. Large number of small elements are usually needed to guarantee accuracy. As a consequence, models have typically very large dimensions, and hence considerable computational and ill conditioning problems arise. Model reduction techniques can be extremely useful to keep model dimensions to a minimum. In this paper a new ranking method, called Interior Mode Ranking (IMR), is introduced for the selection of the interior normal modes in the Craig Bampton reduction technique, which is one of the most popular model reduction methods, often available in commercial finite element software packages. The IMR method allows ranking the interior modes analytically by comparing the contributions provided by the interior modes of the subsystem with constrained boundary conditions to the dynamics of interest of the complete system (with actual boundary conditions). The method is general and can be applied to any resonator in the reduction at the system level. Here it is employed to obtain an accurate reduced-order model of an ultrasonic welding bar horn. The results achieved by the method are compared with those yielded by other ranking techniques. The comparison shows that the IMR method outperforms the other ranking techniques and leads to accurate representations of the excited modes.

[1]  C. C. Flanigan.  Model reduction using Guyan, IRS, and dynamic methods , 1998 .

[2]  Alessandro Gasparetto,et al.  Design and implementation of an ERLS-based 3-D dynamic formulation for flexible-link robots , 2013 .

[3]  Wim Desmet,et al.  Use of Sub-System Global Modal Parameterization Models in Extended Kalman Filtering for Online Coupled State/Force Estimation , 2012 .

[4]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[5]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[6]  Dan Givoli,et al.  Optimal modal reduction of vibrating substructures , 2003 .

[7]  Rémi Lemaire,et al.  Using component modes in a system design process , 2011 .

[8]  Jun-Yeob Song,et al.  Design of highly uniform spool and bar horns for ultrasonic bonding , 2011, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[9]  Daniel C. Kammer,et al.  Selection of Component Modes for Craig-Bampton Substructure Representations , 1996 .

[10]  R. L. Goldman,et al.  Vibration analysis by dynamic partitioning. , 1969 .

[11]  Dan Givoli,et al.  Which are the important modes of a subsystem? , 2004 .

[12]  Zhaojun Bai,et al.  The important modes of subsystems: A moment‐matching approach , 2007 .

[13]  Eric L. Blades,et al.  A craig-bampton test-analysis model , 1997 .

[14]  M. Friswell,et al.  THE CONVERGENCE OF THE ITERATED IRS METHOD , 1998 .

[15]  Roberto Caracciolo,et al.  Design and experimental validation of piecewise-linear state observers for flexible link mechanisms , 2006 .

[16]  D. Rixen.  A dual Craig-Bampton method for dynamic substructuring , 2004 .

[17]  R. Macneal.  A hybrid method of component mode synthesis , 1971 .

[18]  Roy R. Craig,et al.  A review of substructure coupling methods for dynamic analysis , 1976 .

[19]  Klaus Zeman,et al.  Model Reduction of a Parametrically Excited Drivetrain , 2012 .

[20]  Michiel E. Hochstenbach,et al.  A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control , 2013 .

[21]  Dario Richiedei,et al.  Discrete mass and stiffness modifications for the inverse eigenstructure assignment in vibrating systems: Theory and experimental validation , 2012 .

[22]  R. Guyan.  Reduction of stiffness and mass matrices , 1965 .

[23]  Nong Zhang,et al.  DYNAMIC CONDENSATION OF MASS AND STIFFNESS MATRICES , 1995 .

[24]  R. Hintz.  Analytical Methods in Component Modal Synthesis , 1975 .

[25]  Dario Richiedei,et al.  A Constrained Convex Approach to Modal Design Optimization of Vibrating Systems , 2011 .

[26]  S. Rubin.  Improved Component-Mode Representation for Structural Dynamic Analysis , 1975 .

[27]  Luc Gaudiller,et al.  Hybrid modal nodal method for multibody smart structure model reduction: application to modal feedback control , 2006 .

[28]  Xavier Fischer,et al.  Fast reduced model of non-linear dynamic Euler–Bernoulli beam behaviour , 2008 .

[29]  Daniel Rixen,et al.  A general mixed boundary model reduction method for component mode synthesis , 2010 .

[30]  W. A. Benfield,et al.  Vibration Analysis of Structures by Component Mode Substitution , 1971 .

[31]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[32]  Jean-Louis Guyader,et al.  Identification of vibration excitation using a regularized Finite Element operator and a deconvolution post-process , 2012 .

[33]  Stefano Miani,et al.  Dynamic Model of a Rotating Channel Used in the Steel Industry and Implementation of a Controller , 2004 .

[34]  W. Hurty.  Dynamic Analysis of Structural Systems Using Component Modes , 1965 .