The equations of in-plane vibration in thin flat plates are solved for free vibration in circular plates clamped at the outer edge. The mode shapes are represented by trigonometric functions in the circumferential direction and by series summation of Bessel functions in the radial direction. Accuracy of the predictions of natural frequencies and mode shapes is assessed by comparisons with finite-element predictions and with previously reported results. The present solution gives very accurate predictions. The work also highlights the nature of coupling between the different circumferential and radial modes and the response of different vibrational modes at the center of the plate. It is shown that the center point of the plate vibrates only for modes with unity circumferential wave number (number of nodal diameters). Nondimensional frequency parameters are listed and the radial mode shapes of natural vibration are depicted to illustrate the free-vibration behavior in the frequency range of practical interest.
A mathematical model is developed for the prediction of the forced response of finite plates to in-plane point force excitations. The model illustrates the nature of the coupling between in-plane longitudinal and in-plane shear waves and the resonant characteristics of the in-plane vibrational behavior of finite flat plates. The predicted resonance frequencies and mode shapes are compared against the finite element results and good agreement is found. The mode shapes of the in-plane vibration are depicted for frequencies below and above the first resonance frequency. It is illustrated by example that the input power due to in-plane force excitation at the in-plane resonance frequencies is at the same level as that due to out-of-plane force excitations at the flexural resonances in the same frequency band. The participation of the longitudinal and in-plane shear waves in transmitting the vibrational power and the resulting circulatory pattern of structural intensity is also presented.
The equations of in-plane vibration of thin plates are solved for rectangular panels with two parallel edges clamped, while the other two edges are either both clamped, both free, or one clamped and one free. The propagation characteristics of in-plane waves are investigated and the frequency bands of attenuation and propagation of the waves are identified. Simple expressions for the cutoff frequencies of different wave components are derived. The nature of the coupling between in-plane longitudinal and in-plane shear waves is mathematically and physically illustrated. Although all the modes in a plate panel are coupled through Poisson’s effect, it is shown that the coupling is very weak in plate panels with all edges clamped as compared to the cases where one or two parallel edges are free. As a first order approximation, the resonant modes of in-plane waves are classified into uncoupled and coupled mode pairs. By using the coupled mode pairs approximation, it is possible to predict the dynamic response of the plate panel with a reasonable accuracy. Simple expressions are given for approximate estimation of the resonance frequencies of coupled and uncoupled modes. Mode shapes are given, for each case of edge conditions, which satisfy both the displacement and force conditions at the plate edges. A simple procedure is given for the determination of resonance frequencies and mode shapes without excessive computations. The predicted resonance frequencies and mode shapes are compared to the finite element results and good agreement is found. The mode shapes of the in-plane vibration are depicted for the first eight resonant modes for each case of edge conditions.
A mathematical model is developed for the coupling of two finite plates at an arbitrary angle for the prediction of the dynamic response and power flow at the coupling edge and at any cross section. The coupling at the joint edge considers bending, out-of-plane shear, and in-plane longitudinal (perpendicular to the joint edge) vibration. No constraint is imposed on the in-plane displacement perpendicular to the coupling edge. The exact solution for flexural mode shapes and resonance frequencies of rectangular plates with one edge free and the other edges simply supported is considered. This exact solution satisfies both the displacement and force boundary conditions, consequently it is used in the coupling of plate panels. An approximate relation is derived for prediction of the flexural resonance frequencies of higher-order modes with a reasonable accuracy. An approximate solution is presented for the in-plane response of the same plate panels when excited by in-plane forces perpendicular to the free edge. A simple expression is given for an approximate estimation of the cut-off frequencies for the different in-plane modes. The frequency response of the individual plate panels is presented as receptance functions for both flexural and in-plane vibration which are used in the plate coupling. The vibrational power injected into a plate by out-of-plane concentrated force and moment is computationally investigated. The importance of power input by a small moment excitation in some resonant bands even at low frequencies is highlighted. The spatial distribution of the components of flexural power transmitted by shear and moment across a section of the plate are investigated and the reason for the circulatory patterns of active power flow in plates is illustrated. The power flow characteristics in two plates coupled at an arbitrary angle is examined by a computational example. The effect of the coupling angle on the components of input power and power flow across the coupling edge is investigated. It is shown that the coupling of the two plates is mainly due to moment at frequencies up to the cut-off frequency of the first in-plane mode. Above this frequency, the coupling is due to out-of-plane shear and in-plane vibration with a diminishing participation of the moment in transmitting vibrational power through the coupling edge.
This paper is concerned with the prediction of dynamic response of planar coupled beam structures in the low and medium frequency ranges. Based on the receptance concept, the modal properties of the components of the coupled systems with specific boundary conditions are used to form the receptance matrix for the determination of the internal force and displacement response of the coupled structures. Only in-plane loading is considered here and so the structural response will include the longitudinal and in-plane flexural waves only and the torsional and out-of-plane flexural vibrations are not considered here. The mechanism of power flow through the joint boundaries of the in-plane coupled beam structures is investigated by calculating the power flow components of different wave types. The results from the receptance model are compared with that from the finite element analysis and the traveling wave solutions. Good agreement was found. Examples of connected beams are presented, and the characteristics of the power flow and wave interaction are highlighted.
This paper deals with the prediction of the dynamic response and vibrational power flow in general three-dimensional coupled beam structures. An arbitrary loading condition is assumed and flexural waves in two perpendicular planes and longitudinal and torsional waves, may be excited simultaneously in each beam. The interaction of different wave types at the coupling boundaries is therefore expected. Suitable for the low- and medium-frequency range, the modal receptance method used here allows the prediction of the detailed structural response and power flow components carried by the different wave types through joint boundaries and at any cross section of the beam structure. The mathematical description of the three-dimensional beam structure and the manipulation of receptance matrices are presented in such a form that the vibration of any complex beam structure may be described systematically and with clear physical interpretations. The prediction of the input and output power for each component beam quantifies the energy dissipation in the beam, which may be taken as a measure of the spatial average response level of the beam. Computational examples of the power flow components at the joint boundaries of the component beams in coupled three-dimensional structures are used to illustrate the significance of the theoretical results.
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