This paper investigates the flexural vibration of a finite nonuniform Rayleigh beam resting on an elastic foundation and under travelling distributed loads. For the solution of this problem, in the first instance, the generalized Galerkin method was used. The resulting Galerkin’s equations were then simplified using the modified asymptotic method of Struble. The simplified second-order ordinary differential equation was then solved using the method of integral transformation. The closed form solution obtained was analyzed and results show that, an increase in the values of foundation moduli and rotatory inertia correction factor reduces the response amplitudes of both the clamped-clamped nonuniform Rayleigh beam and the clamped-free nonuniform Rayleigh beam. Also for the same natural frequency, the critical speed for the moving distributed mass problem is smaller than that for the moving distributed force problem. Hence resonance is reached earlier in the former. Furthermore, resonance conditions for the dynamical system are attained significantly by both and for the illustrative end conditions considered. 1. Introduction This paper is sequel to an earlier one by Oni and Ayankop-Andi in [1] that considered the response of a simply supported nonuniform Rayleigh beam to travelling distributed loads. In particular, this paper is a generalization of the theory advanced in [1]. For more than a century, the analysis of continuous elastic system subjected to moving systems has been the subject of interest in many fields, from structural to mechanical to aerospace engineering. Various structures ranging from bridges and roads to space vehicles and submarines are constantly acted upon by moving masses and hence the problem of analyzing the dynamic response of these structures under the action of moving masses continues to motivate a variety of investigations. In most of the studies available in literature, such as the works of Sadiku and Leipholz in [2], Oni in [3], Gbadeyan and Oni in [4], Huang and Thambiratnam in [5], Lee and Ng in [6], Adams in [7], Chen and Li in [8], Savin in [9], Rao in [10], Shadnam et al. in [11], and Oni and Awodola in [12], the scope has been restricted to structural members having uniform cross-section whether the inertia of the moving load is considered or not and the load modelled as moving concentrated load. In practice, cross-sections of elastic structures such as plates and beams are not usually uniform and the moving loads are commonly in distributed forms. To this end, in [13] an attempt was made on the studies concerning
References
[1]
S. T. Oni and E. Ayankop-Andi, “Response of a simply supported non-uniform Rayleigh beam to travelling distributed loads,” Journal of the Nigerian Mathematical Society. In press.
[2]
S. Sadiku and H. H. E. Leipholz, “On the dynamics of elastic systems with moving concentrated masses,” Ingenieur-Archiv, vol. 57, no. 3, pp. 223–242, 1987.
[3]
S. T. Oni, “Flexural vibrations under moving loads of Isotropic rectangular plates on a non-Winkler elastic foundation,” Journal of the Nigerian Society of Engineers, vol. 35, no. 1, pp. 18–27, 2000.
[4]
J. A. Gbadeyan and S. T. Oni, “Dynamic behaviour of beams and rectangular plates under moving loads,” Journal of Sound and Vibration, vol. 182, no. 5, pp. 677–695, 1995.
[5]
M. H. Huang and D. P. Thambiratnam, “Deflection response of plate on Winkler foundation to moving accelerated loads,” Engineering Structures, vol. 23, no. 9, pp. 1134–1141, 2001.
[6]
H. P. Lee and T. Y. Ng, “Transverse vibration of a plate moving over multiple point supports,” Applied Acoustics, vol. 47, no. 4, pp. 291–301, 1996.
[7]
G. G. Adams, “Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads,” International Journal of Mechanical Sciences, vol. 37, no. 7, pp. 773–781, 1995.
[8]
Y. H. Chen and C. Y. Li, “Dynamic response of elevated high-speed railway,” Journal of Bridge Engineering, vol. 5, no. 2, pp. 124–130, 2000.
[9]
E. Savin, “Dynamic amplification factor and response spectrum for the evaluation of vibrations of beams under successive moving loads,” Journal of Sound and Vibration, vol. 248, no. 2, pp. 267–288, 2001.
[10]
G. V. Rao, “Linear dynamics of an elastic beam under moving loads,” Journal of Vibration and Acoustics, vol. 122, no. 3, pp. 281–289, 2000.
[11]
M. R. Shadnam, F. R. Rofooei, M. Mofid, and B. Mehri, “Periodicity in the response of nonlinear plate, under moving mass,” Thin-Walled Structures, vol. 40, no. 3, pp. 283–295, 2002.
[12]
S. T. Oni and T. O. Awodola, “Dynamic behaviour under moving Concentrated masses of elastically supported finite Bernoulli-Euler beam on Winkler foundation,” Journal of the Nigerian Mathematical Society, vol. 28, pp. 49–76, 2009.
[13]
S. T. Oni, “Response of a non-uniform beam resting on an elastic foundation to several masses,” Abacus, vol. 24, no. 2, 1996.
[14]
S. T. Oni and T. O. Awodola, “Vibrations under a moving load of a non-uniform Rayleigh beam on variable elastic foundation,” Journal of the Nigerian Association of Mathematical Physics, vol. 7, pp. 191–206, 2003.
[15]
S. T. Oni and B. Omolofe, “Dynamic behaviour of non-uniform Bernoulli-Euler beams subjected to concentrated loads travelling at varying velocities,” Abacus A, vol. 32, no. 2, pp. 165–191, 2005.
[16]
E. J. Sapountzakis and G. C. Tsiatas, “Elastic flexural buckling analysis of composite beams of variable cross-section by BEM,” Engineering Structures, vol. 29, no. 5, pp. 675–681, 2007.
[17]
A. Jimoh, Dynamic response to moving concentrated loads of Bernoulli-Euler beams resting on Bi-parametric subgrades [M.S. thesis], Federal University of Technology, Akure, Nigeria, 2013.
[18]
E. Esmailzadeh and M. Ghorashi, “Vibration analysis of beams traversed by uniform partially distributed moving masses,” Journal of Sound and Vibration, vol. 184, no. 1, pp. 9–17, 1995.
[19]
J. A. Gbadeyan and M. S. Dada, “Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass,” International Journal of Mechanical Sciences, vol. 48, no. 3, pp. 323–340, 2006.
[20]
R. Bogacz and W. Czyczula, “Response of beam on visco-elastic foundation to moving distributed load,” Journal of Theoretical and Applied Mechanics, vol. 46, no. 4, pp. 763–775, 2008.
[21]
L. Fryba, Vibrations of Solids and Structures under Moving Loads, Noordhoff, Groningen, The Netherlands, 1972.
[22]
S. W. Williams, Theory and Problems of Mechanical Vibrations, McGraw-Hill, New York, NY, USA, 1964.
[23]
A. H. Nayfey, Perturbation Methods, John Wiley & Sons, New York, NY, USA, 1973.
