Semi-analytical analysis of vibrations induced by a mass traversing a beam supported by a finite depth foundation with s
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RECENT ADVANCES IN NONLINEAR DYNAMICS AND VIBRATIONS
Semi-analytical analysis of vibrations induced by a mass traversing a beam supported by a finite depth foundation with simplified shear resistance Zuzana Dimitrovova´
Received: 29 December 2019 / Accepted: 6 October 2020 Ó Springer Nature B.V. 2020
Abstract In this paper the new semi-analytical solution for the moving mass problem on massless foundation, published by the author of this paper, is extended to account for inertial foundation modelled by a continuous homogeneous finite depth foundation with simplified shear resistance. Derivations are presented for infinite as well as finite homogeneous beams. Mode expansion method is used to solve the problem on finite beams, thus vibration modes, the corresponding orthogonality condition, reengagement of coupled equations to ensure significant calculation time savings are derived. Methods of integral transforms and contour integration are exploited to obtain the solution on infinite beams. Resulting vibrations are derived as a sum of the steady and unsteady harmonic vibrations and a transient contribution. The unsteady harmonic vibration is proven to be a useful indicator of unstable behaviour through the mass induced frequencies. Besides frequency lines also discontinuity lines are determined and their influence on the proximity of harmonic and full solutions is discussed. Even if the differences between these two versions are larger than Z. Dimitrovova´ (&) Departamento de Engenharia Civil, Faculdade de Cieˆncias e Tecnologia, Universidade Nova de Lisboa, Lisbon, Portugal e-mail: [email protected] Z. Dimitrovova´ IDMEC, Instituto Superior Te´cnico, Universidade de Lisboa, Lisbon, Portugal
for the massless foundation, it is shown that the harmonic solution provides a very good estimate of the full solution (in several cases perfect match is achieved) with the advantage to be obtainable by a simple evaluation of the derived closed-form results. Like for the massless foundation, also here, vibrations on infinite beams can be obtained on long finite beams with eliminated effect of its supports. All mentioned approaches are also validated by the finite element method. Keywords Transverse vibrations Moving mass Constant and harmonic load Normal force Massinduced frequency Semi-analytical solution Foundation with simplified shear resistance
1 Introduction Several fields of engineering applications have to deal with moving load problems. These problems are specified by a guiding structure, usually in form of a beam, the foundation, which the guiding structure is placed on, and the moving load. Although this kind of applications generally uses the term ‘‘moving loads’’, it should be clearly stated whether mass and thus inertial forces are included in the moving object or not, in other words, if moving force(s) or moving mass(es), oscillator(s), etc. are being considered.
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Meccanica
This paper deals with the moving mass problem, which is inherently a non-
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