Free vibration of rectangular plates with continuously distributed spring-mass

Abstract

The vibratory characteristics of rectangular plates attached with continuously and uniformly distributed spring-mass in a rectangular region are studied which may represent free vibration of a human-structure system. Firstly, the governing differential equations of a plate with uniformly distributed spring-mass are developed. When the spring-mass fully occupies the plate, the natural frequencies of the coupled system are exactly solved and a relationship between the continuous system and a series of discrete two degrees-of-freedom system is provided. The degree of frequency coupling is defined. Then, the Ritz-Galerkin method is used to derive the approximate solution, when the spring-mass is distributed on a part of the plate, by using the Chebyshev polynomial series to construct the admissible functions. Comparative studies demonstrate the high accuracy and wide applicability of the proposed method. Finally, the frequency and modal characteristics of the plate partially occupied by distributed spring-mass are numerically analysed. It has been observed that both the natural frequencies and the modes appear in pairs. Moreover, a parametric study is performed for rectangular plates with three edges simply supported and one edge free. The effects of occupation size and position of the distributed spring-mass on natural frequencies of the coupled system are studied in detailed. The present investigation provides an improved understanding of human-structure interaction, such as grandstands or floors occupied by a stationary crowd.

Study on Human-structure Dynamic Interaction in Civil Engineering

“…Sachse has established the system of two degrees of freedom dynamic human-structure coupling model (Figure 3a), and the natural frequencies, mode shapes, modal mass and damping ratio of the coupled system are analyzed in detail, the results of analysis showed that damped discrete model of the human-structure system with two degrees of freedom can better explain changes across the dynamic characteristics of structures with human occupants [16]. Zhou et al uses a continuous model without damping to analyze the dynamic characteristics of a coupling system of human-structure interaction [17] (shown in Fig. 3b).…”

Effect of Various Edge Conditions on Free-Vibration Characteristics of Isotropic Square and Rectangular Plates

“…Many researchers [3][4][5][6][7][8][9][10] have worked on vibration characteristics of plates. For several decades, numerous measurements of vibrational properties and the vibrational behavior issues of plates have been studied in addition to different theories [10][11][12].…”

“…It was inferred that vibration characteristics of these types of plates can be studied for carrying any number of sprung masses and resting on the elastic foundations. Zhou and Ji [12] studied vibrations of rectangular plates associated with distributions of springs in uniform and continuous manner in a domain of rectangular nature. The Chebyshev polynomials in a series were taken to represent trial functions.…”

“…In this chapter, the WPA proposed by Zhang et al [15] for computing the vibrations is extended to plates, which is our particular motivation. Increasingly, numerical calculations have been used to investigate vibrational properties of plates through different models [5,9,12,13,21]. The method of choice is based on WPA that allows for the study of fundamental frequencies of plates over various combinations of geometric parameters, and this approach has become increasingly popular in the numerical solution of engineering applications.…”

Eigenvalue Analysis of Thin Plate With Complicated Shapes by a Novel Mesh-Free Method

“…Meshless LBIE formulations for simply supported and clamped plates under dynamic loading is studied by Sladek et al [2003] while Simkins et al [2004] employed the reproducing kernel element method in terms of globally compatible C n (n ≥ 1) triangular hierarchy. Zhou et al [2006] performed the free vibration of rectangular plates with continuously distributed spring-mass. Recently, Cui et al [2010] proposed an attractive technique based on the incorporation of various smoothing techniques and radial point interpolation method to static and free vibration analysis of thin plates without rotations, degrees of freedom (DOFs), etc.…”