Vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices

This work provides a general formulation to solve vibration problems for continuous systems with damping effects, including modal, transient, harmonic and spectrum response analyses. In modal analysis, the system eigenvalues and corresponding eigenfunctions can be determined. The orthogonal relations of eigenfunctions are shown. For transient, harmonic and spectrum analyses, the generic force/actuator functions and response/sensing operators are introduced, respectively, and used to derive the system response. The time domain response is obtained for transient analysis, the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis. The solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force/actuator and response/sensor mode shape functions. In particular, one-dimensional beam and two-dimensional plate vibration analyses are illustrated by following the developed generic formulation. This work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well.

[1]  Metin Gurgoze,et al.  ON THE FREQUENCY RESPONSE FUNCTION OF A DAMPED CANTILEVER SIMPLY SUPPORTED IN-SPAN AND CARRYING A TIP MASS , 2002 .

[2]  Takeshi Sakiyama,et al.  FREE VIBRATION ANALYSIS OF RECTANGULAR PLATES WITH VARIABLE THICKNESS , 1998 .

[3]  Z. Oniszczuk,et al.  Forced transverse vibrations of an elastically connected complex rectangular simply supported double-plate system , 2004 .

[4]  D. J. Ewins,et al.  Modal Testing: Theory and Practice , 1984 .

[5]  Jerry H. Ginsberg,et al.  Mechanical and Structural Vibrations: Theory and Applications , 2001 .

[6]  N. C. Nigam.  Introduction to Random Vibrations , 1983 .

[7]  Ping-Hei Chen,et al.  Finite Element Model Verification for the Use of Piezoelectric Sensor in Structural Modal Analysis , 2006 .

[8]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

[9]  Michael J. Brennan,et al.  Mobility and impedance methods in structural dynamics , 2018, Advanced Applications in Acoustics, Noise and Vibration.

[10]  W. X. Wu,et al.  Computation of modal stress resultants for completely free vibrating plates by LSFD method , 2006 .

[11]  Z. Oniszczuk,et al.  Transverse Vibrations of Elastically Connected Double-String Complex System, Part i: Free Vibrations , 2000 .

[12]  Li Jun,et al.  Response of flexure¿torsion coupled composite thin-walled beams with closed cross-sections to random loads , 2005 .

[13]  J. Banerjee.  Explicit analytical expressions for frequency equation and mode shapes of composite beams , 2001 .

[14]  H. Saunders.  Book Reviews : Modern Formulas for Statics and Dynamics: W.D. Pilkey and Pin Yu Chang McGraw-Hill Book Co., New York, New York, 1978 , 1980 .

[15]  Metin Gurgoze,et al.  ON THE EIGENFREQUENCIES OF A CANTILEVER BEAM WITH ATTACHED TIP MASS AND A SPRING-MASS SYSTEM , 1996 .

[16]  Frank Fahy,et al.  Advanced Applications in Acoustics, Noise and Vibration , 2018 .

[17]  W. Soedel.  Vibrations of shells and plates , 1981 .

[18]  Bor-Tsuen Wang,et al.  The Use of Piezoceramic Transducers for Smart Structural Testing , 2000 .

[19]  S. Kamle,et al.  TRANSVERSE VIBRATION OF MULTI MASS LOADED VARIABLE SECTION BEAMS UNDER RANDOM EXCITATION , 1997 .

[20]  M. A. De Rosa,et al.  On the dynamic behaviour of slender beams with elastic ends carrying a concentrated mass , 1996 .

[21]  Ding Zhou,et al.  Free vibration of rectangular plates with continuously distributed spring-mass , 2006 .

[22]  Z. Oniszczuk.  TRANSVERSE VIBRATIONS OF THE ELASTICALLY CONNECTED RECTANGULAR DOUBLE-MEMBRANE COMPOUND SYSTEM , 1999 .

[23]  G. Oliveto,et al.  COMPLEX MODAL ANALYSIS OF A FLEXURAL VIBRATING BEAM WITH VISCOUS END CONDITIONS , 1997 .

[24]  L. Meirovitch.  Analytical Methods in Vibrations , 1967 .

[25]  W. Thomson.  Theory of vibration with applications , 1965 .

[26]  Tore Dahlberg,et al.  THE EFFECT OF MODAL COUPLING IN RANDOM VIBRATION ANALYSIS , 1999 .

[27]  H. Saunders,et al.  Literature Review : Books: THEORY AND ANALYSIS OF PLATES -- C LASSICA L AND NUMERICAL METHODS Rudolph Szilard Prentice Hall, Inc., Englewood Cliffs, N.J. (1974) , 1974 .

[28]  Bor-Tsuen Wang,et al.  The use of polyvinylidene fluoride films as sensors for the experimental modal analysis of structures , 2004 .

[29]  K. Liew,et al.  Vibration analysis of rectangular mindlin plates resting on elastic edge supports , 1997 .

[30]  D. J. Gorman.  Accurate free vibration analysis of the orthotropic cantilever plate , 1995 .

[31]  J. Bendat,et al.  Random Data: Analysis and Measurement Procedures , 1971 .

[32]  Jong-Shyong Wu,et al.  DYNAMIC ANALYSIS OF A UNIFORM CANTILEVER BEAM CARRYING A NUMBER OF ELASTICALLY MOUNTED POINT MASSES WITH DAMPERS , 2000 .

[33]  Thomas L. Paez,et al.  Random Vibrations: Theory and Practice , 1995 .

[34]  Arthur W. Leissa,et al.  The free vibration of rectangular plates , 1973 .

[35]  Osman Kopmaz,et al.  FREE VIBRATIONS OF A RECTANGULAR PLATE CARRYING A DISTRIBUTED MASS , 2002 .

[36]  Bor-Tsuen Wang.  STRUCTURAL MODAL TESTING WITH VARIOUS ACTUATORS AND SENSORS , 1998 .

[37]  R. Szilard,et al.  Theory and Analysis of Plates, Classical and Numberical Methods , 1974 .

[38]  H. Tiersten,et al.  Free vibrations of rectangular cantilever plates. Part 1: out-of-plane motion , 2004 .

[39]  Bor-Tsuen Wang,et al.  Feasibility analysis of using piezoceramic transducers for cantilever beam modal testing , 1997 .

[40]  Eric E. Ungar,et al.  Mechanical Vibration Analysis and Computation , 1989 .

[41]  Jong-Shyong Wu,et al.  USE OF THE ANALYTICAL-AND-NUMERICAL-COMBINED METHOD IN THE FREE VIBRATION ANALYSIS OF A RECTANGULAR PLATE WITH ANY NUMBER OF POINT MASSES AND TRANSLATIONAL SPRINGS , 1997 .

[42]  R. G. White,et al.  Analysis and control of vibrational power transmission to machinery supporting structures subjected to a multi-excitation system, Part I: Driving point mobility matrix of beams and rectangular plates , 1996 .

[43]  N. Kessissoglou,et al.  Dynamic response analysis of stochastic truss structures under non-stationary random excitation using the random factor method , 2007 .

[44]  T. Kocatürk.  DETERMINATION OF THE STEADY STATE RESPONSE OF VISCOELASTICALLY POINT-SUPPORTED RECTANGULAR ANISOTROPIC (ORTHOTROPIC) PLATES , 1998 .

[45]  M. J. Maurizi,et al.  Damping in exact analysis of tapered beams , 2005 .

[46]  L. Perotin,et al.  An Inverse Method for the Identification of a Distributed Random Excitation Acting on a Vibrating Structure: Part 1 — Theory , 1997, 4th International Symposium on Fluid-Structure Interactions, Aeroelasticity, Flow-Induced Vibration and Noise: Volume I.

[47]  Stephen J. Elliott,et al.  A WAVENUMBER APPROACH TO MODELLING THE RESPONSE OF A RANDOMLY EXCITED PANEL, PART I: GENERAL THEORY , 2002 .

[48]  Demeter G. Fertis,et al.  Mechanical And Structural Vibrations , 1995 .

[49]  L. Meirovitch.  Principles and techniques of vibrations , 1996 .

[50]  D. Cho,et al.  Modal Interactions of a Randomly Excited HINGED-CLAMPED Beam , 2000 .

[51]  N. Ferguson,et al.  Variations in steepness of the probability density function of beam random vibration , 2000 .

[52]  M. Gurgoze,et al.  ON THE EIGENVALUES OF A VISCOUSLY DAMPED CANTILEVER CARRYING A TIP MASS , 1998 .