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Research Papers

The Response of Infinite Strings and Beams to an Initially Applied Moving Force: Analytical Solution

[+] Author and Article Information
André Langlet1

Laboratoire PRISME/Dynamique des Matériaux et des Structures, 63 avenue du Maréchal, de Lattre de Tassigny, 18020 Bourges Cedex, Franceandre.langlet@bourges.univ-orleans.fr

Ophélie Safont

Laboratoire PRISME/Dynamique des Matériaux et des Structures, 63 avenue du Maréchal, de Lattre de Tassigny, 18020 Bourges Cedex, France; Nexter Systems, 7 route de Guerry, 18000 Bourges France

Jérôme Renard

Laboratoire PRISME/Dynamique des Matériaux et des Structures, 63 avenue du Maréchal, de Lattre de Tassigny, 18020 Bourges Cedex, France

1

Corresponding author.

J. Vib. Acoust 134(4), 041005 (May 31, 2012) (16 pages) doi:10.1115/1.4005847 History: Received April 07, 2011; Revised November 22, 2011; Published May 29, 2012; Online May 31, 2012

This paper presents the analytical solutions for bilaterally infinite strings and infinite beams on which a point force is initially applied, which then moves on the structure at a constant velocity. The solutions are sought by first applying the Fourier transform to the spatial coordinate dependence, and then the Laplace transform to the time variable of dependence, of the governing equations of motion. For the strings, it is necessary to distinguish between the case of a sonic load (a force moving at the phase velocity of transverse waves) and the cases of subsonic and supersonic loads. This is achieved by a suitable expansion in polynomial ratios of the Laplace transform, before going back to the original Fourier transform, whose inverse is obtained by exact calculations of the integrals over the complex infinite domain. For the Euler-Bernoulli beam, the same process leads to the closed-form (exact) formula for the displacement, from which the stress can be deduced. The displacement consists of the sum of two integrals: one representing the transient part, and the other, the stationary part of the solution. The stationary part is observed in the vicinity of the force for a very long travel time. The transient part is observed at a finite position coordinate, in relative proximity to the starting point of the moving force. For the Timoshenko beam, the final step in the calculation of the displacement and rotation, which requires a numerical evaluation of the integrals, leads to Fourier cosine and sine transforms. The response of the beam depends on the load velocity, relative to the two characteristic velocities: those of shear waves and longitudinal waves. This demonstrates that the transient parts of the solutions, in the Euler-Bernoulli beam or in the Timoshenko beam, are quasi identical. However, classical theory fails to forecast high frequency responses, occurring with velocities of the load exceeding twenty per cent of the bar velocity. For a velocity greater than the velocity of the shear waves, classical theory wrongly forecasts the response. In addition, according to the Euler-Bernoulli beam theory, the flexural waves are able to exceed the bar velocity, which is not realistic. If the load moves for a long period, the solution in the vicinity of the load tends towards a stationary solution. It is important to note that the solution to the stationary problem must be completed by the solution to the associated homogeneous system to represent the physical stationary solution.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

The string and its loading

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Figure 2

Transverse displacements of a long string subject to a moving force

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Figure 3

The beam description

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Figure 4

Flexural displacements in an Euler-Bernoulli beam after the application of a moving force

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Figure 5

Typical form of the flexural displacement W for large values of time (Euler-Bernoulli beam)

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Figure 6

Typical flexural stresses in the Euler-Bernoulli beam

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Figure 7

Transverse displacements in the Timoshenko beam

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Figure 8

Evolution with time of the cross section rotations in a Timoshenko beam

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Figure 9

Comparison of the analytical solutions, Eqs. 91,92,93,94, with numerical solutions and stationary solutions for the particular case V  =  θ

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Figure 10

Flexural stesses in the Timoshenko beam

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Figure 11

Maximum flexural stresses in the Timoshenko beam

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