Axial Vibration Analysis of Stepped Bar by Differential Transformation Method

Abstract:

Stepped distributed dynamic systems are widely used in the engineering fields, and the dynamic characteristics analysis of them is very important. In this paper, the axial vibration of a stepped bar consisting of two uniform sections is studied, in order to solve the dynamic equation, the differential transformation method is used, the governing differential equation and the boundary conditions of the bar become simple algebraic equations. Doing some simple algebraic operations for these equations, the closed form solution of natural frequency, mode shape and the dynamic response can be obtained. Comparison the results obtained by the differential transformation method and finite element method, excellent agreement is achieved, and the effects of the stiffness of spring is discussed in this paper.

• Periodical: Applied Mechanics and Materials (Volume 419)
• Pages: 273-279
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.419.273
• Online since: October 2013
• Authors: Zhen Gang Wang
• Keywords: Differential Transformation Method, Mode Shape, Natural Frequency, Stepped Bar

References

[1] N. Popplewell and D. Chang: Free vibrations of a complex Euler-Bernoulli beam. Journal of sound and vibration, 190 (1996), pp.852-856.

DOI: 10.1006/jsvi.1996.0098

[2] S. Naguleswaran: Natural frequencies, sensitivity and mode shape details of an Euler-Bernoulli beam with one-step change in cross-section and with ends on classical supports. Journal of sound and vibration, 252(2002), pp.751-767.

DOI: 10.1006/jsvi.2001.3743

[3] H. P. Lin and S. Chang: Free vibration analysis of multi-span beams with intermediate flexible constraints. Journal of sound and vibration, 281(2005), pp.155-169.

DOI: 10.1016/j.jsv.2004.01.010

[4] Q. Mao and S. Pietrzko: Free vibration analysis of stepped beams by using Adomian decomposition method. Applied Mathematics and Computation, 217(2010), pp.3429-3441.

DOI: 10.1016/j.amc.2010.09.010

[5] N. Auciello, A. Erolano: Exact solution for the transverse vibration of a beam a part of which is a taper beam and other part is a uniform beam. International journal of solids and structures, 34(1997), pp.2115-2129.

DOI: 10.1016/s0020-7683(96)00136-9

[6] N. Auciello: On the transverse vibrations of non-uniform beams with axial loads and elastically restrained ends. International journal of mechanical sciences, 43(2001), pp.193-208.

DOI: 10.1016/s0020-7403(99)00110-1

[7] B. Yany: Exact transient vibration of stepped bars, shafts and strings carrying lumped masses. Journal of sound and vibration, 329(2010), pp.1191-207.

DOI: 10.1016/j.jsv.2009.10.035

[8] J. H. Zhou: Differential Transformation and its Application for Electric Circuits. China: Huazhong University Press, (1986) (in chinese).

[9] C. Bert and H. Zeng: Analysis of axial vibration of compound bars by differential transformation method. Journal of sound and vibration, 275(2004), pp.641-647.

DOI: 10.1016/j.jsv.2003.06.019

[10] B. BEDDOE: Wave theory of free torsional vibration of composite systems of shafts. Journal of mechanical engineering science, 7(1965), pp.48-56.

DOI: 10.1243/jmes_jour_1965_007_010_02

[11] M. Malik and M. Allali: Characteristic Equations of Rectangular Plates by Differential Transformation. Journal of sound and vibration, 233(2000), pp.359-366.

DOI: 10.1006/jsvi.2000.2828

[12] A. Leissa: The free vibration of rectangular plates. Journal of sound and vibration, 31(1973), pp.257-293.

DOI: 10.1016/s0022-460x(73)80371-2

[13] S. H. Ho and C. K. Chen: Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform. International journal of mechanical sciences, 48(2006), pp.1323-1331.

DOI: 10.1016/j.ijmecsci.2006.05.002

[14] S. S. Rao: Mechanical vibrations. Pearson Education, (2003).

[15] Z. F. Fu and H. X. Hua: Modal Analysis Theory and Application. China: Shanghai Jiaotong University Press, (2000) (in chinese).

[16] S. H. Ho and C. K. Chen: Analysis of general elastically end restrained non-uniform beams using differential transform . Applied Mathematical Modelling, 22(1998), pp.219-234.

DOI: 10.1016/s0307-904x(98)10002-1