Effect of Axial Force on the Free Vibration of Reddy—Bickford Multi-span Beam Carrying Multiple Spring

Abstract

Structural elements supporting motors or engines are frequently seen in technological applications. The operation of a machine may introduce additional dynamic stresses on the beam. It is important to know the natural frequencies of the coupled beam—mass system for a proper design of the structural elements. There is plenty of literature regarding the free vibration analysis of Bernoulli—Euler single-span beams carrying a number of spring—mass system and Bernoulli—Euler multi-span beams carrying multiple spring—mass systems, but less is available regarding Reddy—Bickford multi-span beam carrying multiple spring—mass systems with/without axial force effect. This paper aims at determining the exact solutions for the first five natural frequencies and mode shapes of Reddy—Bickford beams. The model allows the influence of the shear effect and spring—mass systems on the dynamic behavior of the beams to be analyzed by using Reddy—Bickford beam theory. The effects of the attached spring—mass systems on the free vibration characteristics of one to four span beams are studied. The calculated natural frequencies of Reddy—Bickford and Timoshenko multi-span beams using a secant method for nontrivial solution of the different values of axial force are given. The mode shapes of Reddy—Bickford multi-span beams are also presented.

Keywords

Axial force effect free vibration Reddy—Bickford multi-span beam spring—mass system.

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References

Bickford, W.B. , 1982, "A consistent higher order beam theory," Development in Theoretical and Applied Mechanics 11, 137-150.

Chang, C.H. , 2000, "Free vibration of a simply supported beam carrying a rigid mass at the middle," Journal of Sound and Vibration 237(4), 733-744.

Chen, Y. , 1963, "On the vibration of beams or rods carrying a concentrated mass," Journal of Applied Mechanics 30, 310-311.

Cowper, G.R. , 1966, "The shear coefficient in Timoshenko’s beam theory ," Journal of Applied Mechanics 33(2), 335-340.

Dowell, E.H. , 1979, "On some general properties of combined dynamical systems ," Transactions of the ASME 46, 206-209.

Gruttmann, F. and Wagner, W. , 2001, "Shear coefficient factors in Timoshenko’s beam theory for arbitrary shaped cross-section," Computational Mechanics 27, 199-207.

Gürgöze, M. , 1984, "A note on the vibrations of restrained beams and rods with point masses," Journal of Sound and Vibration 96(4), 461-468.

Gürgöze, M. , 1985, "On the vibration of restrained beams and rods with heavy masses," Journal of Sound and Vibration 100(4), 588-589.

Han, S.M. , Benaroya, H. , and Wei, T. , 1999, "Dynamics of transversely vibrating beams using four engineering theories," Journal of Sound and Vibration 225, 935-988.

10.

Heyliger, P.R. and Reddy, J.N. , 1988, "A higher-order beam finite element for bending and vibration problems," Journal of Sound and Vibration 126(2), 309-326.

11.

Kukla, S. and Posiadala, B. , 1994, "Free vibrations of beams with elastically mounted masses ," Journal of Sound and Vibration 175(4), 557-564.

12.

Levinson, M. , 1984, "A new rectangular beam theory," Journal of Sound and Vibration 74, 81-87.

13.

Lin, H.Y. and Tsai, Y.C. , 2007, "Free vibration analysis of a uniform multi-span beam carrying multiple spring- mass systems," Journal of Sound and Vibration 302, 442-456.

14.

Liu, W.H. , Wu, J.R. , and Huang, C.C. , 1998, "Free vibration of beams with elastically restrained edges and intermediate concentrated masses," Journal of Sound and Vibration 122(2), 193-207.

15.

Murthy, A.V. , 1970, "Vibration of short beams," AIAA Journal 8, 34-38.

16.

Naguleswaran, S. , 2002, "Transverse vibrations of an Euler-Bernoulli uniform beam carrying several particles," International Journal of Mechanical Science 44, 2463-2478.

17.

Naguleswaran, S. , 2003, "Transverse vibration of an Euler-Bernoulli uniform beam on up o five resilient supports including ends," Journal of Sound and Vibration 261, 372-384.

18.

Su, H. and Banerjee, J.R. , 2005, "Exact natural frequencies of structures consisting of two parts beam-mass systems," Structural Engineering and Mechanics 19(5), 551-566.

19.

Timoshenko, S.P. , 1921, "On the correction for shear of the differential equation for transverse vibrations of prismatic bars," Philosophical Magazine 41, 744-746.

20.

Tuma, J.J. and Cheng, F.Y. , 1983, Theory and Problems of Dynamic Structural Analysis (Schaum’s Outline Series), McGraw-Hill, New York.

21.

Wang, C.M. , Reddy, J.N. , and Lee, K.H. , 2000, Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier, Amsterdam.

22.

Wu, J.S. and Chou, H.M. , 1999, "A new approach for determining the natural frequencies and mode shape of a uniform beam carrying any number of spring masses," Journal of Sound and Vibration 220, 451-468.

Effect of Axial Force on the Free Vibration of Reddy—Bickford Multi-span Beam Carrying Multiple Spring—mass Systems

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