A segmenting scheme for evaluating exact high-order modes of uniform Timoshenko beams

Abstract Numerical evaluation of exact high-order modes of uniform Timoshenko beams is a classic problem in the field of acoustics and vibration. In the classic method, however, the determinant of the frequency matrix dominated by hyperbolic functions can increase exponentially and quickly reach the precision limitation of a computer. As a result, a bottleneck occurs in the evaluation of high-order modes due to round-off errors. Addressing this bottleneck, this study proposes an enhanced classic method relying on a segmenting scheme for evaluating exact high-order modes of uniform Timoshenko beams. With the segmenting scheme, a uniform Timoshenko beam is uniformly segmented into several segments, whereby the hyperbolic functions involved in the determinant of the frequency matrix become much smaller for the same given frequency. Accordingly, under the fixed precision limitation in a computer, higher-order modes can be obtained. The capacity of this enhanced classic method for evaluating exact high-order modes is validated by scenarios of a uniform Timoshenko beam with different numbers of segments. The results show that the high-order modal frequencies and mode shapes can be properly obtained. The accuracy of the modal frequencies is verified by the well-established exact dynamic stiffness method with the Wittrick-Williams algorithm. The results show that the modal frequencies are highly accurate. Applied to beams of different materials, the artificial segmenting is more suitable than the natural segmenting in evaluating high-order modes.

[1]  J. R. Banerjee,et al.  Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened timoshenko beams , 2001 .

[2]  Hongxing Hua,et al.  On the transverse vibration of Timoshenko double-beam systems coupled with various discontinuities , 2014 .

[3]  T. Huang,et al.  The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams With Simple End Conditions , 1961 .

[4]  Mohammed F. Daqaq,et al.  On Using a Strong High-Frequency Excitation for Parametric Identification of Nonlinear Systems , 2017 .

[5]  Chih-Peng Yu,et al.  Dynamic Stiffness Matrices for Linear Members with Distributed Mass , 2001 .

[6]  Michael J. Brennan,et al.  Exact expressions for numerical evaluation of high order modes of vibration in uniform Euler-Bernoulli beams , 2018, Applied Acoustics.

[7]  Arthur W. Leissa.  Vibration of Continuous Systems , 2011 .

[8]  Robert J. Bernhard,et al.  Structural-acoustic design at high frequency using the energy finite element method , 1999 .

[9]  W. H. Wittrick,et al.  An automatic computational procedure for calculating natural frequencies of skeletal structures , 1970 .

[10]  Michael J. Brennan,et al.  Numerical evaluation of high-order modes of vibration in uniform Euler–Bernoulli beams , 2007 .

[11]  Singiresu S. Rao.  Vibration of Continuous Systems , 2019 .

[12]  Yu Tang,et al.  Numerical Evaluation of Uniform Beam Modes , 2003 .

[13]  M. Cao,et al.  Numerical Evaluation of High-Order Modes for Stepped Beam , 2014 .

[14]  L. C. Schmidt,et al.  Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package , 1995 .

[15]  Hai-Ping Lin,et al.  Direct and inverse methods on free vibration analysis of simply supported beams with a crack , 2004 .

[16]  Wei Xu,et al.  Identification of Incipient Damage Using High-Frequency Vibrational Responses , 2015 .

[17]  K. H. Low.  A reliable algorithm for solving frequency equations involving transcendental functions , 1993 .

[18]  N. Khaji,et al.  Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions , 2009 .

[19]  J. R. Banerjee,et al.  Free vibration of sandwich beams using the dynamic stiffness method , 2001 .

[20]  Frederic W. Williams,et al.  Exact Buckling and Frequency Calculations Surveyed , 1983 .

[21]  Seyed M. Hashemi,et al.  A dynamic stiffness element for free vibration analysis of delaminated layered beams , 2012 .

[22]  H. Benaroya,et al.  DYNAMICS OF TRANSVERSELY VIBRATING BEAMS USING FOUR ENGINEERING THEORIES , 1999 .

[23]  J. R. Banerjee,et al.  Dynamic stiffness formulation for structural elements: A general approach , 1997 .

[24]  J. R. Banerjee,et al.  Free vibration of a three-layered sandwich beam using the dynamic stiffness method and experiment , 2007 .

[25]  S. Timoshenko,et al.  X. On the transverse vibrations of bars of uniform cross-section , 1922 .

[26]  F. W. Williams,et al.  Natural frequencies of frames with axially loaded Timoshenko Members , 1973 .

[27]  Amin Zare,et al.  Exact dynamic stiffness matrix for flexural vibration of three-layered sandwich beams , 2005 .