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MOVING MASS PROBLEM: COMPLETE SOLUTION WITH THE EFFECT OF INITIAL CONDITIONS
Last modified: 2017-11-26
Abstract
In this contribution, the new semi-analytical solution for the moving mass problem from [1] is extended to account for the influence of initial conditions. It is assumed that a constant mass and a vertical force with harmonic component moves by a constant velocity along a horizontal infinite beam posted on a two-parameter visco-elastic foundation. The new semi-analytical solution identifying the beam response to such a load was presented in [1] as a closed-form formula. Apart the terms that can be obtained by the double Fourier transform, the essential new term is governed by the mass-induced frequencies.
In order to account to the initial conditions, first of all the previous solution has to be extended by a transient part of natural vibrations originated by a line discontinuity in the Laplace image of the displacement under the load. Analytical expression is derived for such a line cut. Then the transient part is determined by numerical integration and added to the response given by the closed-form formula. It is proven that in the absence of other harmonics the mass induced vibration amplitude equals approximately to the steady-state deflection of the associated force, unless the mass weight is very small compared to the applied force. This indicates that these vibrations are very severe and cannot be neglected. In summary the new contributions of this work are:
(1) Identification and derivation of transient part of natural vibrations to be added to the closed-form formula from [1].
(2) Analysis of the effect of initial conditions.
(4) Discussion of critical cases based on analytical developments.
(5) Validation of the results presented by analysis of finite beams with improved strategy according to [2].
Acknowledgements
This work was supported by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2013.
References
[1] Z. Dimitrovová, New semi-analytical solution for a uniformly moving mass on a beam on a two-parameter visco-elastic foundation, Int. J. Mech. Sci. (in press).
[3] M.E. Hassanabadi, N.K.A. Attari, A. Nikkhoo, M. Baranadan, An optimum modal superposition approach in the computation of moving mass induced vibrations of a distributed parameter system, Proc IMechE Part C: J Mechanical Engineering Science. 229 (2015) 1015–1028.
In order to account to the initial conditions, first of all the previous solution has to be extended by a transient part of natural vibrations originated by a line discontinuity in the Laplace image of the displacement under the load. Analytical expression is derived for such a line cut. Then the transient part is determined by numerical integration and added to the response given by the closed-form formula. It is proven that in the absence of other harmonics the mass induced vibration amplitude equals approximately to the steady-state deflection of the associated force, unless the mass weight is very small compared to the applied force. This indicates that these vibrations are very severe and cannot be neglected. In summary the new contributions of this work are:
(1) Identification and derivation of transient part of natural vibrations to be added to the closed-form formula from [1].
(2) Analysis of the effect of initial conditions.
(4) Discussion of critical cases based on analytical developments.
(5) Validation of the results presented by analysis of finite beams with improved strategy according to [2].
Acknowledgements
This work was supported by FCT, through IDMEC, under LAETA, project UID/EMS/50022/2013.
References
[1] Z. Dimitrovová, New semi-analytical solution for a uniformly moving mass on a beam on a two-parameter visco-elastic foundation, Int. J. Mech. Sci. (in press).
[3] M.E. Hassanabadi, N.K.A. Attari, A. Nikkhoo, M. Baranadan, An optimum modal superposition approach in the computation of moving mass induced vibrations of a distributed parameter system, Proc IMechE Part C: J Mechanical Engineering Science. 229 (2015) 1015–1028.
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