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Research Papers

Dynamic Response of Prestressed Rayleigh Beam Resting on Elastic Foundation and Subjected to Masses Traveling at Varying Velocity

[+] Author and Article Information
S. T. Oni

Department of Mathematical Sciences, School of Sciences, Federal University of Technology, PMB 704, Ondo State, Akure 234000, Nigeria

B. Omolofe

Department of Mathematical Sciences, School of Sciences, Federal University of Technology, PMB 704, Ondo State, Akure 234000, Nigeriababatope_omolofe@yahoo.com

J. Vib. Acoust 133(4), 041005 (Apr 07, 2011) (15 pages) doi:10.1115/1.4003405 History: Received October 03, 2009; Revised December 08, 2010; Published April 07, 2011; Online April 07, 2011

Abstract

In this study, the dynamic response of axially prestressed Rayleigh beam resting on elastic foundation and subjected to concentrated masses traveling at varying velocity has been investigated. Analytical solutions representing the transverse-displacement response of the beam under both concentrated forces and masses traveling at nonuniform velocities have been obtained. Influence of various parameters, namely, axial force, rotatory inertia correction factor, and foundation modulus on the dynamic response of the dynamical system, is investigated for both moving force and moving mass models. Effects of variable velocity on the vibrating system have been established. Furthermore, the conditions under which the vibrating systems will experience resonance effect have been established. Results arrived at in this paper are in perfect agreement with existing results.

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Figures

Figure 1

Schematic diagram of elastic beam carrying moving mass

Figure 2

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated forces moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———K=0, – – – – – ⋅K=4000, – ⋅ – ⋅ – ⋅ –K=40,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ K=400,000

Figure 3

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated forces moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———N=0, – – – –N=2,000,000, – ⋅ – ⋅ȉ– ⋅N=20,000,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ N=200,000,000

Figure 4

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated forces moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———R0=150, – – – –R0=350, – ⋅ – ⋅ – ⋅R0=550, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ R0=850

Figure 5

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated masses moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ————K=0, – – – – – ⋅K=4000, – ⋅ – ⋅ – ⋅ –K=40,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ K=400,000

Figure 6

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated masses moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———N=0, – – – –N=2,000,000, – ⋅ – ⋅ – ⋅N=20,000,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ N=200,000,000

Figure 7

Comparison of the moving force and moving mass model of the simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated masses moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———— moving force and ……… moving mass

Figure 8

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated forces moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———a=3 m/s2, – – – –a=6 m/s2, – ⋅ – ⋅ – ⋅a=9 m/s2, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 12 m/s2

Figure 9

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated masses moving with variable velocity; ((a)–(c)) accelerated motion: (a) α=0.5, (b) α=1.5, and (c) α=2.5 and ((d)–(f)) decelerated motion: (d) α=0.5, (e) α=1.5, and (f) α=2.5; ———a=−3 m/s2, – – – –a=−6 m/s2, – ⋅ – ⋅ – ⋅a=−9 m/s2, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ −12 m/s2

Figure 10

Transverse-displacement response of a simply supported Rayleigh beam resting on elastic foundation and under the actions of concentrated masses moving with variable velocity for fixed values of axial force N=20,000, foundation moduli K=40,000, rotatory inertia R0=50, acceleration a=3 m/s2, and α=0.5; ⋯⋯⋯⋯⋯⋯⋯⋯⋯ε∗=0.2, – ⋅ – ⋅ – ⋅ –ε∗=0.3, – – – – – – – –ε∗=0.4, and ————ε∗=0.5

Figure 11

Transverse displacement of a simply supported uniform Rayleigh beam under the actions of concentrated forces traveling at variable velocity for various values of axial force N and for fixed values of foundation modulus K=40,000 and rotatory inertia R0=50; ————N=0, – – – – –N=2,000,000, – ⋅ – ⋅ – ⋅N=20,000,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ N=200,000,000

Figure 12

Transverse displacement of a simply supported uniform Rayleigh beam under the actions of concentrated masses traveling at variable velocity for various values of axial force N and for fixed values of foundation modulus K=40,000 and rotatory inertia R0=50; ———N=0, – – – – –N=2,000,000, – ⋅ – ⋅ – ⋅N=20,000,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ N=200,000,000

Figure 13

Deflection profile of a simply supported uniform Rayleigh beam under the actions of concentrated forces traveling at variable velocity for various values of foundation modulus K and for fixed values of axial force N=20,000 and rotatory inertia R0=50; —————K=0, – – – – – ⋅K=4000, – ⋅ – ⋅ – ⋅ –K=40,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ K=400,000

Figure 14

Deflection profile of a simply supported uniform Rayleigh beam under the actions of concentrated masses traveling at variable velocity for various values of foundation modulus K and for fixed values of axial force N=200,000 and rotatory inertia R0=50; ————K=0, – – – – – ⋅K=4000, – ⋅ – ⋅ – ⋅ –K=40,000, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ K=400,000

Figure 15

Response amplitude of a simply supported uniform Rayleigh beam under the actions of concentrated forces traveling at variable velocity for various values of rotatory inertia R0 and for fixed values of foundation modulus K=40,000 and axial force N=20,000; ————R0=150, – – – –R0=350, – ⋅ – ⋅ – ⋅R0=550, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ R0=850

Figure 16

Response amplitude of a simply supported uniform Rayleigh beam under the actions of concentrated masses traveling at variable velocity for various values of rotatory inertia R0 and for fixed values of foundation modulus K=40,000 and axial force N=20,000; ————R0=150, – – – –R0=350, – ⋅ – ⋅ – ⋅R0=550, and ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ R0=850

Figure 17

Comparison of the displacement response of moving force and moving mass cases of a uniform simply supported Rayleigh beam for fixed values of N=20,000, K=40,000, and R0=50; ———————— moving force and ………… moving mass

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