scholarly journals Time Period of the Vibration of the Circular Plate with Circular Variation Both in Thickness and Density

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Reeta Bhardwaj ◽  
Amit Sharma ◽  
Sudeshna Ghosh ◽  
Naveen Mani ◽  
Kamal Kumar
Keyword(s):  
Thermal Gradient ◽  
Ritz Method ◽  
The Other ◽  
Circular Plates ◽  
Linear Quadratic ◽  
Linear Temperature ◽  
Thermally Induced ◽  
Time Period ◽  
Modes Of Vibration ◽  
Convergence Study

An analysis was carried out to investigate the time period of the thermally induced vibration of clamped and simply supported circular plates with circular variation both in thickness and density. Prior to this study, the variations considered were either linear, quadratic, parabolic, or exponential in nature. To study thermal effect, one-dimensional linear temperature variation on the plates is taken into consideration. Rayleigh–Ritz method is applied to compute the time period of the first three modes of vibration for both plates by varying tapering parameter, thermal gradient, and density. Convergence study of frequency modes for both plates conducted suggests that the convergence rate in case of circular variation is faster than the other studies done. A comparison of time period with the available published results is done. The comparison done concludes that time period obtained in the present study by varying thermal gradient and tapering parameter is found to be less than the other studies done for the same set of parameters. This study helped to establish the fact that, by using circular variation in plate parameters, we can get less time period of frequency modes in comparison to other variations considered till date.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

scholarly journals Thermal Effect on Vibration of Tapered Rectangular Plate

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Anupam Khanna ◽  
Ashish Singhal
Keyword(s):  
Rectangular Plate ◽  
Thermal Gradient ◽  
Present Model ◽  
Order Differential Equation ◽  
Ritz Method ◽  
Temperature Variations ◽  
Nonuniform Thickness ◽  
Modes Of Vibration ◽  
Aeronautical Engineering ◽  
Fourth Order Differential Equation

A mathematical model is constructed to help the engineers in designing various mechanical structures mostly used in satellite and aeronautical engineering. In the present model, vibration of rectangular plate with nonuniform thickness is discussed. Temperature variations are considered biparabolic, that is, parabolic in x-direction and parabolic in y-direction. The fourth-order differential equation of the motion is solved by Rayleigh Ritz method for three different boundary conditions around the boundary of plate. Numerical values of frequencies for the first two modes of vibration are presented in tabular form for different values of thermal gradient, taper constants, and aspect ratio.

scholarly journals Transverse Vibrations of Clamped and Simply-Supported Circular Plates with Two Dimensional Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 273-285 ◽  
Author(s):  
N. Bhardwaj ◽  
A.P. Gupta ◽  
K.K. Choong ◽  
C.M. Wang ◽  
Hiroshi Ohmori
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Two Dimensional ◽  
Circular Plates ◽  
Simply Supported ◽  
Modes Of Vibration ◽  
Dimensional Boundary ◽  
Special Case

Two dimensional boundary characteristic orthonormal polynomials are used in the Ritz method for the vibration analysis of clamped and simply-supported circular plates of varying thickness. The thickness variation in the radial direction is linear whereas in the circumferential direction the thickness varies according to coskθ, wherekis an integer. In order to verify the validity, convergence and accuracy of the results, comparison studies are made against existing results for the special case of linearly tapered thickness plates. Variations in frequencies for the first six normal modes of vibration and mode shapes for various taper parameters are presented.

scholarly journals Thermal Effect on Vibration of Clamped Visco-Elastic Rectangular Plate with Parabolic Thickness Variation in Both Directions

2010 ◽  
Vol 17 (1) ◽  
pp. 93-105
Author(s):  
A.K. Gupta ◽  
Harvinder Kaur ◽  
Sanjay Kumar
Keyword(s):  
Boundary Condition ◽  
Rectangular Plate ◽  
Thermal Gradient ◽  
Frequency Equation ◽  
Thickness Variation ◽  
Linear Temperature ◽  
Time Period ◽  
Small Deflection ◽  
Thermal Constants ◽  
Elastic Rectangular Plate

The analysis presented here is to study the effect of thermal gradient on the vibration of visco-elastic rectangular plate (having clamped boundary condition on all the four edges) of variable thickness whose thickness varies parabolically in both directions. The effect of linear temperature variation has been considered. A frequency equation of plate has been obtained by Rayleigh-Ritz technique with two terms of deflection function. The assumption of small deflection and linear visco-elastic properties of ‘Kelvin’ type are taken. Deflection and time period corresponding to the first two modes of vibrations for clamped plate have been computed for various combinations of aspect ratio, thermal constants, and taper constants. Numerical computations have been performed for an alloy ‘Duralium’ and the results obtained are depicted graphically.

Deformation and Moments in Elastically Restrained Circular Plates Under Arbitrary Load or Linear Thermal Gradient

1960 ◽  
Vol 82 (4) ◽  
pp. 423-438
Author(s):  
M. Zaid ◽  
M. Forray
Keyword(s):  
Thermal Gradient ◽  
Load Distribution ◽  
Loading Condition ◽  
Stress Pattern ◽  
Algebraic Equations ◽  
Circular Plates ◽  
Linear Temperature ◽  
Linear Temperature Gradient ◽  
Elastically Restrained ◽  
Elastic Constraints

The problem of a circular plate, with a central hole, elastically constrained against rotation and deflection, acted upon by a transverse linear temperature gradient or a general axisymmetrical loading condition is considered in this paper. With the aid of presented graphs and simple algebraic equations, it is a relatively simple matter to construct the desired deflection and stress pattern for any combination of elastic constraints and load distribution.

scholarly journals A Study on Vibration of Tapered Non-Homogeneous Rectangular Plate with Structural Parameters

2018 ◽  
Vol 23 (4) ◽  
pp. 873-884 ◽  
Author(s):  
N. Kaur ◽  
A. Singhal ◽  
A. Khanna
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Thermal Gradient ◽  
Structural Parameters ◽  
Ritz Method ◽  
Frequency Parameter ◽  
Tabular Form ◽  
Plate Material ◽  
Simply Supported ◽  
Modes Of Vibration

Abstract Effects of structural parameters on the vibration of a tapered non-homogeneous rectangular plate with different combinations of boundary conditions are discussed. Tapering in the plate is assumed to be sinusoidal in the x-direction. Here, temperature variation and non-homogeneity in the plate material are also considered sinusoidal in the x-direction. The Rayleigh-Ritz method is used to calculate the frequency parameter for the first two modes of vibration for different values of the structural parameters, i.e. the taper parameter, thermal gradient, aspect ratio and non-homogeneity constant. Results are obtained for three boundary conditions, i.e. clamped boundary (C-C-C-C), simply supported boundary (SS-SS-SS-SS) and clamped-simply supported boundary (CSS-C-SS). Numerical values of the frequency parameter are given in a compact tabular form.

scholarly journals Free Transverse Vibration of Orthotropic Thin Trapezoidal Plate of Parabolically Varying Thickness Subjected to Linear Temperature Distribution

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Arun Kumar Gupta ◽  
Shanu Sharma
Keyword(s):  
Temperature Distribution ◽  
Aspect Ratio ◽  
Transverse Vibration ◽  
Thermal Gradient ◽  
Ritz Method ◽  
Linear Temperature ◽  
Varying Thickness ◽  
Free Transverse Vibration ◽  
Trapezoidal Plate ◽  
Taper Constant

The present paper deals with the free transverse vibration of orthotropic thin trapezoidal plate of parabolically varying thickness inx-direction subjected to linear temperature distribution inx-direction through a numerical method. The deflection function is defined by the product of the equations of the prescribed continuous piecewise boundary shape. Rayleigh-Ritz method is used to evaluate the fundamental frequencies. The equations of motion, governing the free transverse vibrations of orthotropic thin trapezoidal plates, are derived with boundary condition CSCS. Frequency corresponding to the first two modes of vibration is calculated for the orthotropic thin trapezoidal plate having CSCS edges for different values of thermal gradient, taper constant, and aspect ratio. The proposed method is applied to solve orthotropic thin trapezoidal plate of variable thickness with C-S-C-S boundary conditions. Results are shown by figures for different values of thermal gradient, taper constant, and aspect ratio for the first two modes of vibrations.

The incumbent, challenger, and population during revolution and civil war

2019 ◽  
Vol 14 (2) ◽  
Author(s):  
Kjell Hausken ◽  
Mthuli Ncube
Keyword(s):  
Civil War ◽  
Public Goods ◽  
Arab Spring ◽  
The Other ◽  
Time Period ◽  
Geographic Regions

We consider revolutions and civil war involving an incumbent, a challenger, and the population. Revolutions are classified into eight outcomes. In four outcomes incumbent repression occurs (viewed as providing sub-threshold benefits such as public goods to the population). Accommodation occurs in the other four outcomes (benefits provision above a threshold). The incumbent and challenger fight each other. The incumbent may win and retain power or else lose, thereby causing standoff or coalition. In a standoff, which is costly, no one backs down and uncertainty exists about who is in power. In a coalition, which is less costly, the incumbent and challenger cooperate, compromise, and negotiate their differences. If the population successfully revolts against the incumbent, the challenger replaces the incumbent. Eighty-seven revolutions during 1961–2011, including the recent Arab spring revolutions, are classified into the eight outcomes. When repressive, the incumbent loses 46 revolutions, remains in power through 21 revolutions, and builds a coalition after 12 revolutions. When accommodative, the incumbent loses seven revolutions and builds a coalition after one revolution. The 87 revolutions are classified across geographic regions and by time-period.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

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