Nonlinear frequency response analysis of structural vibrations

In this paper we present a method for nonlinear frequency response analysis of mechanical vibrations of 3-dimensional solid structures. For computing nonlinear frequency response to periodic excitations, we employ the well-established harmonic balance method. A fundamental aspect for allowing a large-scale application of the method is model order reduction of the discretized equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. For an efficient spatial discretization of continuum mechanics nonlinear partial differential equations, including large deformations and hyperelastic material laws, we employ the concept of isogeometric analysis. Isogeometric finite element methods have already been shown to possess advantages over classical finite element discretizations in terms of higher accuracy of numerical approximations in the fields of linear vibration and static large deformation analysis. With several computational examples, we demonstrate the applicability and accuracy of the modal derivative reduction method for nonlinear static computations and vibration analysis. Thus, the presented method opens a promising perspective on application of nonlinear frequency analysis to large-scale industrial problems.

[1]  Maurice Petyt,et al.  NON-LINEAR VIBRATION OF BEAMS WITH INTERNAL RESONANCE BY THE HIERARCHICAL FINITE-ELEMENT METHOD , 1999 .

[2]  Alessandro Reali,et al.  Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems , 2014 .

[3]  Bernd Simeon,et al.  Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations , 2013 .

[4]  M. Anitescu,et al.  Polynomial Regression Approaches Using Derivative Information for Uncertainty Quantification , 2010 .

[5]  Jernej Barbic,et al.  FEM simulation of 3D deformable solids: a practitioner's guide to theory, discretization and model reduction , 2012, SIGGRAPH '12.

[6]  Zu-Qing Qu,et al.  Model Order Reduction Techniques with Applications in Finite Element Analysis , 2004 .

[7]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[8]  T. Hughes,et al.  B¯ and F¯ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements , 2008 .

[9]  J. Remke,et al.  Eine modale Reduktionsmethode zur geometrisch nichtlinearen statischen und dynamischen Finite-Element-Berechnung , 1993 .

[10]  R. Lewandowski.  Computational formulation for periodic vibration of geometrically nonlinear structures—part 2: Numerical strategy and examples , 1997 .

[11]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[12]  Pedro Ribeiro,et al.  HIERARCHICAL FINITE ELEMENT ANALYSES OF GEOMETRICALLY NON-LINEAR VIBRATION OF BEAMS AND PLANE FRAMES , 2001 .

[13]  Roman Lewandowski,et al.  Non-linear, steady-state vibration of structures by harmonic balance/finite element method , 1992 .

[14]  Sabrina Herkt,et al.  Model Reduction of Nonlinear Problems in Structural Mechanics: Towards a Finite Element Tyre Model for Multibody Simulation , 2008 .

[15]  Zu-Qing Qu,et al.  Model Order Reduction Techniques , 2004 .

[16]  Kjell Magne Mathisen,et al.  Isogeometric analysis of finite deformation nearly incompressible solids , 2011 .

[17]  F. Auricchio,et al.  The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations , 2010 .

[18]  Marco Amabili,et al.  Nonlinear normal modes for damped geometrically nonlinear systems: Application to reduced-order modelling of harmonically forced structures , 2006 .

[19]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[20]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[21]  Roman Lewandowski,et al.  Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background , 1997 .

[22]  Marco Amabili,et al.  Reduced-order models for large-amplitude vibrations of shells including in-plane inertia , 2008 .

[23]  Doug L. James,et al.  Real-time reduced large-deformation models and distributed contact for computer graphics and haptics , 2007 .

[24]  T. Hughes,et al.  ISOGEOMETRIC COLLOCATION METHODS , 2010 .

[25]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[26]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[27]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[28]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[29]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[30]  Christophe Pierre,et al.  Finite-Element-Based Nonlinear Modal Reduction of a Rotating Beam with Large-Amplitude Motion , 2003 .

[31]  Cadence Design Systems Inc. , 1993 .

[32]  U. Becker.  Efficient time integration and nonlinear model reduction for incompressible hyperelastic materials , 2013 .

[33]  Wanda Szemplińska-Stupnicka,et al.  The Behavior of Nonlinear Vibrating Systems , 1990 .

[34]  Richard Courant,et al.  Wiley Classics Library , 2011 .

[35]  P. Wriggers.  Nonlinear Finite Element Methods , 2008 .

[36]  Bernd Simeon,et al.  A finite volume method on NURBS geometries and its application in isogeometric fluid-structure interaction , 2012, Math. Comput. Simul..

[37]  Pma Paul Slaats,et al.  MODEL REDUCTION TOOLS FOR NONLINEAR STRUCTURAL DYNAMICS , 1995 .

[38]  Alberto Cardona,et al.  A reduction method for nonlinear structural dynamic analysis , 1985 .

[39]  J. Barbič.  FEM Simulation of 3 D Deformable Solids : A practitioner ’ s guide to theory , discretization and model reduction . Part 2 : Model Reduction ( version : August 4 , 2012 ) , 2012 .

[40]  R. Kolman.  ISOGEOMETRIC FREE VIBRATION OF AN ELASTIC BLOCK , 2012 .

[41]  Qinghua Zheng,et al.  Parallel harmonic balance , 1993, VLSI.

[42]  R. L. Taylor.  Isogeometric analysis of nearly incompressible solids , 2011 .

[43]  Pedro Ribeiro,et al.  Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods , 2004 .