Abstract
In the existing reports regarding free and forced vibrations of the beams, most of them studied a uniform beam carrying various concentrated elements using Bernoulli-Euler Beam Theory (BET) but without axial force. The purpose of this paper is to utilize the numerical assembly technique to determine the exact frequency-response amplitudes of the axially-loaded Timoshenko multi-span beam carrying a number of various concentrated elements (including point masses, rotary inertias, linear springs and rotational springs) and subjected to a harmonic concentrated force and the exact natural frequencies and mode shapes of the beam for the free vibration analysis. The model allows analyzing the influence of the shear and axial force and harmonic concentrated force effects and intermediate concentrated elements on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate concentrated elements, an intermediate pinned support, applied harmonic force, left-end support and right-end support of Timoshenko beam are derived. After the derivation of the coefficient matrices, the numerical assembly technique is used to establish the overall coefficient matrix for the whole vibrating system. Finally, solving the equations associated with the last overall coefficient matrix one determines the exact dynamic response amplitudes of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force. Equating the determinant of the overall coefficient matrix to zero one determines the natural frequencies of the free vibrating system (the case of zero harmonic force) and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. The calculated vibration amplitudes of the forced vibrating systems and the natural frequencies of the free vibrating systems are given in tables for different values of the axial force. The dynamic response amplitudes and the mode shapes are presented in graphs. The effects of axial force and harmonic concentrated force on the vibration analysis of Timoshenko multi-span beam are also investigated.
Highlights
The free and forced vibration characteristics of a uniform beam carrying various concentrated elements is an important problem in engineering
The frequency parameters obtained for the Timoshenko beam without the axial force effect in this study are a little less than the values obtained for the Bernoulli-Euler beam in the reference [21], as expected, since the shear deformation is considered in Timoshenko Beam Theory
Frequency values, frequency parameters and the mode shapes for free vibration of the axially-loaded Timoshenko multi-span beam carrying a number of various concentrated elements for different values of axial compressive force
Summary
The free and forced vibration characteristics of a uniform beam carrying various concentrated elements (such as point masses, rotary inertias, linear springs, rotational springs, etc.) is an important problem in engineering. Lin and Tsai determined the exact natural frequencies together with the associated mode shapes for Bernoulli-Euler multi-span beam carrying multiple point masses [15]. Lin and Tsai investigated the free vibration characteristics of Bernoulli-Euler multiple-step beam carrying a number of intermediate lumped masses and rotary inertias [16]. The natural frequencies and mode shapes of Bernoulli-Euler multi-span beam carrying multiple spring-mass systems were determined by Lin and Tsai [17]. Wang et al studied the natural frequencies and mode shapes of a uniform Timoshenko beam carrying multiple intermediate spring-mass systems with the effects of shear deformation and rotary inertia [18]. Yesilce and Demirdag described the determination of the natural frequencies of vibration of Timoshenko multi-span beam carrying multiple spring-mass systems with axial force effect [20]. The dynamic response amplitudes of the forced vibrating systems and the natural frequencies of the free vibrating systems are calculated, the dynamic response amplitudes and the first three mode shapes are plotted and the effects of the axial force and the harmonic concentrated force and the influence of the shear are investigated by using the computer package, Matlab
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