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CIVIL AND STRUCTURAL ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping and Y. Tsompanakis
Dynamic Analysis of Beam Structures under Moving Loads: A Review of the Modal Expansion Method
UNIC, Department of Civil Engineering, Universidade Nova de Lisboa, Portugal
Z. Dimitrovová, "Dynamic Analysis of Beam Structures under Moving Loads: A Review of the Modal Expansion Method", in B.H.V. Topping and Y. Tsompanakis, (Editor), "Civil and Structural Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 4, pp 99-130, 2011. doi:10.4203/csets.28.4
Keywords: dynamic stiffness matrix, state-space formulation, moving load, moving mass, modal expansion, localized disturbances.
The transient behaviour of general one-dimensional distributed dynamic systems is often studied by implementing variable separation and assuming the existence of free harmonic vibrations. The frequency of these vibrations is named as the natural frequency and it is determined from the eigenvalue problem obtained from the homogeneous governing equation. Then the transient response in the time domain is expressed as an infinite series of these modes, where each vibration mode (function of the spatial coordinate) is multiplied by a generalized displacement (modal coordinate, amplitude function) that is a function of time. This solution method is called the eigenvalues expansion, the modal expansion, the modal superposition or the normal mode analysis.
In this chapter, the dynamic analysis of beam structures under moving loads with emphasis on the modal expansion method is presented. First of all, some numerical issues related to the standard finite element method employing polynomial shape functions and to the exact shape functions of distributed mass elements are discussed. The level of error in natural frequencies, inherent to the finite element method obtained analytically is discussed and compared with results of standard finite element software.
Hamilton's principle is implemented as an efficient tool for obtaining governing equations for transversal vibrations in Timoshenko-Rayleigh beam structures induced by moving loads in a general form, including several sub-domains and involving the effects of localized springs, dampers and masses. A determination of elastic constants representing the effect of an elastic half space is presented. The foundation reaction is introduced by Winkler and Pasternak visco-elastic contributions.
The concept of the dynamic stiffness matrix is posted as a general principle for finite, semi-infinite and infinite beams. Its implementation in beam structures composed of several sub-domains is developed. A connection to semi-infinite and infinite beams is emphasized. Semi-infinite beams could be an efficient tool for preventing reflected waves, however, its implementation raises several difficulties, as explained.
Advantages and disadvantages of natural modes calculation including the concentrated disturbances or not are summarized. Further implications of this fact on generalized displacements determination are summarized. The importance of modal equations uncoupling is pointed out and the concept of the Rayleigh damping is invoked. A state–space formulation of governing equations is presented for the Timoshenko-Rayleigh beam structure in a self-adjoint form and advantages with respect to the standard uncoupled governing equation are summarized. Orthogonality conditions are established and differences of expansions in series governed by damped or undamped modes are indicated.
In conclusion a formulation improving the series convergence for internal forces determination is given. A possibility of the load inertia inclusion is shown. Some developments presented are new; a review and summary of published works is far from complete due to the considerable amount of studies that have been published on this subject.
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