An Algorithm for Solving Torsional Vibration Problems Based on the Invariant Imbedding Method

In this work the invariant imbedding method has been used to develop an algorithm to study the torsional vibration of non-uniform systems. The algorithm is based on the propagation, reflection, and transmission of waves in a stepped waveguide and is part of a procedure to transform two-point boundary value problems in initial value problems. Based on this approach, a continuous model has been developed and a simple, versatile, and robust algorithm has been constructed to solve torsional vibration problems of non-uniform shafts with circular crosssections. The proposed solution algorithm was extensively evaluated through comparisons with analytical solutions and the finite element method. The results show that the proposed method can provide the exact solution for uniform shafts with concentrated elements and accurate results for a wide variety of torsional vibration problems. Systems with continuously varying geometry may be approximated by stepped shafts. The proposed method can also be used to study the dynamic behaviour of others stepped systems.

[1]  J. M. Pouyet,et al.  Torsional vibrations of a shaft with non-uniform cross section , 1981 .

[2]  M. R. Osborne,et al.  A Riccati transformation method for solving linear BVPs. I: theoretical aspects , 1988 .

[3]  Qiusheng Li,et al.  Torsional vibration of multi-step non-uniform rods with various concentrated elements , 2003 .

[4]  Chunbiao Gan,et al.  Torsional vibration measurements on rotating shaft system using laser doppler vibrometer , 2012 .

[5]  R. Langley.  WAVE EVOLUTION, REFLECTION, AND TRANSMISSION ALONG INHOMOGENEOUS WAVEGUIDES , 1999 .

[6]  Qiusheng Li,et al.  Torsional Vibration of Non-Uniform Shafts Carrying an Arbitrary Number of Rigid Disks , 2002 .

[7]  Marianela Lentini,et al.  INVARIANT IMBEDDING, THE BOX SCHEME AND AN EQUIVALENCE BETWEEN THEM* , 1982 .

[8]  Robert D. Russell,et al.  A Riccati transformation method for solving linear BVPs. II: computational aspects , 1988 .

[9]  Paul E. Barbone,et al.  Suppression of Sound Reflected from a Piezoelectric Plate , 1992 .

[10]  Der-Wei Chen,et al.  An exact solution for free torsional vibration of a uniform circular shaft carrying multiple concentrated elements , 2006 .

[11]  A. Mioduchowski,et al.  Torsional waves and free vibrations of drive systems with stepped shafts , 1986 .

[12]  C. Bapat,et al.  General Approach for Free and Forced Vibrations of Stepped Systems Governed By the One-Dimensional Wave Equation With Non-Classical Boundary Conditions , 1994 .

[13]  Brian R. Mace,et al.  Wave propagation, reflection and transmission in non-uniform one-dimensional waveguides , 2007 .

[14]  R. C. Y. Chin,et al.  Matrix methods in synthetic seismograms , 1984 .