NATURAL FREQUENCIES OF ORTHOTROPIC RECTANGULAR PLATES OBTAINED BY ITERATIVE REDUCTION OF THE PARTIAL DIFFERENTIAL EQUATION

1996 ◽  
Vol 189 (1) ◽  
pp. 89-101 ◽  
Author(s):  
T. Sakata ◽  
K. Takahashi ◽  
R.B. Bhat
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Rectangular Plates ◽  
Partial Differential

On Transverse Vibrations of Rectangular Plates of Unidirectional Parabolic Thickness Variation

2005 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Order Approximation ◽  
Second Order ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Partial Differential ◽  
First Order ◽  
Transverse Vibrations

This paper presents an approach for finding the solution of partial differential equation describing the motion of transverse vibrations of rectangular plates of unidirectional convex parabolic varying thickness. The partial differential equation consists of three operators: fourth-order spatial-dependent, second-order spatial-dependent, and second-order time-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. The first partial differential equation was a homogeneous equation and consisted of two operators, the fourth-order spatial-dependent and second-order time-dependent. Using the factorization method, so-called zero-order approximation of the exact solution has been found. The second partial differential equation was an inhomogeneous equation. Its solution, so-called first-order approximation of the exact solution has been found. This way the first-order approximations of the natural frequencies and mode shapes are found. Various boundary conditions can be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

Transverse Vibrations of Rectangular Plates of Linearly Varying Thickness

2007 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Ion Stroe
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Order Approximation ◽  
Second Order ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Partial Differential ◽  
First Order ◽  
Transverse Vibrations

This paper presents an approach for finding the solution of partial differential equation describing the motion of transverse vibrations of rectangular plates of unidirectional linear varying thickness. The original partial differential equation consists of three operators: fourth-order spatial-dependent, second-order spatial-dependent, and second-order time-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. The first partial differential equation was a homogeneous equation and consisted of two operators, the fourth-order spatial-dependent and second-order time-dependent. The solution of this equation was found using the factorization method. This solution was zeroth-order approximation of the exact solution. The second partial differential equation was an inhomogeneous equation. The solution of this equation was also found and led to first-order approximation of the exact solution of the original problem. This way the first-order approximations of the natural frequencies and mode shapes are found. Various boundary conditions can be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

A Transfer Matrix Technique for Evaluating the Natural Frequencies and Critical Speeds of a Rotor With Multiple Flexible Disks

1991 ◽  
Author(s):  
Fangsheng Wu ◽  
George T. Flowers
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Transfer Matrix ◽  
Natural Frequencies ◽  
Space Shuttle ◽  
Rotor System ◽  
Partial Differential ◽  
Rotor Systems ◽  
Inertial Moment ◽  
Main Engine

Abstract Modern turbomachinery is used to provide power for a wide range of applications, from steam turbines for electrical power plants to the turbopumps used in the Space Shuttle Main Engine. Such devices are subject to a variety of dynamical problems, including vibration, rotordynamical instability, and shaft whirl. In order to properly design and evaluate the performance and stability of turbomachinery, It is important that appropriate analytical tools be available that allow for the study of potentially important dynamical effects. This research effort is concerned with developing a procedure to account for disk flexibility which can readily be used for investigating how such effects might influence the natural frequencies and critical speeds of practical rotor systems. In the present work, a transfer matrix procedure is developed in which the disk flexibility effects are accounted for by means of additional terms included in the transfer matrix formulation. In this development, the shaft is treated as a discrete system while the disk is modelled as a continuous system using the governing partial differential equation. Based on this governing equation, an equivalent inertial moment Mk*, which is the generalized dynamic force coupling between shaft and disk, is then derived. Analysis shows that only the disk modes of one nodal diameter contribute to the inertial moment, Mk*, and thus influence the natural frequencies of the rotor system. To determine the Mk*, the modal expansion method is employed and the governing partial differential equation of the disk is transformed to a set of decoupled forced vibration equations in the generalized coordinates. The Mk* are then calculated in terms of modal shapes, natural frequencies, and material and geometric parameters which can be found in the literature or can be obtained from experiments. Finally the Mk* are incorporated into the point transfer matrix. By so doing, the properties of quick computational speed and ease of use are retained and the complexity of solving partial differential equations is avoided. This allows the present procedure to be easily applied to practical engineering problems. This is especially true for multiple flexible disk rotor systems. As an example, three different cases for a simplified model of the Space Shuttle Main Engine (SSME) High Pressure Oxygen Turbo-Pump (HPOTP) rotor have been studied using this procedure. Some of the more interesting results obtained in this example study are enumerated below. 1.) Disk flexibility can introduce additional natural frequency(s) to a rotor system. 2.) Disk flexibility can cause shifting of some of the natural frequencies. 3.) As disk flexibility is increased, lower natural frequencies of the rotor system will be influenced. 4.) At certain rotor speeds, disk flexibility may cause the disappearance of a natural frequency. 5.) The axial position of the disk on the rotor shaft has a significant effect on the degree of this influence.

Vibroacoustic behavior of a plate surrounded by a cavity containing an inclined part–through surface crack with arbitrary position

2019 ◽  
Vol 25 (17) ◽  
pp. 2365-2379 ◽  
Author(s):  
F. Motaharifar ◽  
M. Ghassabi ◽  
R. Talebitooti
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Surface Crack ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Flexible Plate ◽  
Partial Differential ◽  
Cavity Depth ◽  
Cracked Plate ◽  
Arbitrary Position

Acoustical investigation of a thin plate having a part–through surface crack surrounded by an air enclosure is studied in this paper. The enclosure comprises five rigid walls and a flexible plate based on the Kirchhoff plate theory. It is also assumed that the crack is located at an arbitrary position and orientation with a specific length. Accordingly, partial differential equation related to the coupled cracked plate–cavity system is presented. In order for the partial differential equation (PDE) to be solved, firstly, the sound pressure inside the cavity is estimated by a suitable number of the plate modes. Then, the coupled PDE decomposes to some ordinary differential equations in the time-domain by employing the Galerkin method for three different boundary conditions. In addition, the linear natural frequencies are obtained in vacuo and coupled conditions for an uncracked plate and then, a similar procedure is performed for a cracked plate. Furthermore, comparing the results with available data in the literature shows the reliability and accuracy of the present work. Finally, the influences of the crack angle, crack length, crack position, and cavity depth on the natural frequencies are investigated.

Plate Characteristic Functions and Natural Frequencies of Vibration of Plates by Iterative Reduction of Partial Differential Equation

1993 ◽  
Vol 115 (2) ◽  
pp. 177-181 ◽  
Author(s):  
R. B. Bhat ◽  
J. Singh ◽  
G. Mundkur
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequency ◽  
Normal Modes ◽  
Free Edge ◽  
Characteristic Functions ◽  
Rectangular Plates ◽  
Kantorovich Method ◽  
Partial Differential ◽  
Edge Conditions

Natural frequency coefficients of rectangular plates and the corresponding plate characteristic functions are obtained by reduction of plate partial differential equation to an ordinary differential equation and solving it exactly. The reduction is carried out by assuming a deflection shape in one direction consistent with the boundary conditions and applying Galerkin’s averaging technique to eliminate the variable. The reduction method, commonly known as Kantorovich method, is applied sequentially on either directions of the plate and iterated until convergence is achieved for the natural frequency coefficients. The resulting plate characteristic functions are very good approximations to the normal modes of the plate. The results are tabulated for plates with combination of clamped, simply-supported, and free edge conditions.

On Non-Axisymmetrical Transverse Vibrations of Circular Plates of Convex Parabolic Thickness Variation

2004 ◽  
Author(s):  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Factorization Method ◽  
Mode Shapes ◽  
Circular Plates ◽  
Partial Differential ◽  
First Order

This paper presents an approach for finding the solution of the partial differential equation of motion of the non-axisymmetrical transverse vibrations of axisymmetrical circular plates of convex parabolical thickness. This approach employed both the method of multiple scales and the factorization method for solving the governing partial differential equation. The solution has been assumed to be harmonic angular-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. Solving them, the solution resulted as first-order approximation of the exact solution. Using the factorization method, the first differential equation, homogeneous and consisting of fourth-order spatial-dependent and second-order time-dependent operators, led to a general solution in terms of hypergeometric functions. Along with given boundary conditions, the first differential equation and the second differential equation, which was nonhomogeneous, gave respectively so-called zero-order and first-order approximations of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

scholarly journals Behaviour under Moving Distributed Masses of Simply Supported Orthotropic Rectangular Plate Resting on a Constant Elastic Bi-Parametric Foundation

2020 ◽  
pp. 64-92
Author(s):  
T. O. Awodola ◽  
S. Adeoye
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourth Order ◽  
Closed Form Solution ◽  
Form Solution ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Simply Supported ◽  
Orthotropic Rectangular Plate

This work investigates the behavior under Moving distributed masses of orthotropic rectangular plates resting on bi-parametric elastic foundation. The governing equation is a fourth order partial differential equation with variable and singular co-efficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al[1]. This is then simplified using modified asymptotic method of Struble. The closed form solution is analyzed, resonance conditions are obtained and the results are presented in plotted curves for both cases of moving distributed mass and moving distributed force.

scholarly journals Nonlinear Frequencies for Transverse Free Oscillations of a Transporting Tensioned Beam

2011 ◽  
Vol 2-3 ◽  
pp. 807-816
Author(s):  
Hu Ding ◽  
Li Qun Chen ◽  
Hai Yan Jiang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Natural Frequencies ◽  
Nonlinear Oscillations ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Coefficient ◽  
Governing Equations ◽  
Partial Differential ◽  
Transverse Oscillations

This study focuses on the natural frequencies of nonlinear free transverse oscillations of transporting tensioned beams via the standard fast Fourier transform (FFT). The transverse oscillations of transporting tensioned beams can be governed by a nonlinear partial-differential equation or a nonlinear integro-partial-differential equation. Numerical schemes are respectively presented for the two governing equations via the differential quadrature method. For each nonlinear equation, the nonlinear oscillations frequencies are investigated via FFT with the time responses histories. The numerical results depict the tendencies of the frequencies of nonlinear free transverse oscillations of transporting tensioned beams with the changing oscillations amplitude, transporting speed, the nonlinear coefficient and the flexural stiffness.

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