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 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 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.

Vibration of Triangular Cantilever Plates by the Ritz Method

1954 ◽  
Vol 21 (4) ◽  
pp. 365-370
Author(s):  
B. W. Andersen
Keyword(s):  
Cantilever Beam ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Plan View ◽  
Modes Of Vibration ◽  
Accuracy Of Results ◽  
Isosceles Triangles

Abstract Using the method published by Ritz in 1909, natural frequencies and corresponding node lines have been determined for two symmetric and two antisymmetric modes of vibration of isosceles triangular plates clamped at the base and having length-to-base ratios of 1, 2, 4, and 7 and for the two lowest modes of right triangular plates clamped along one leg and having ratios of the length of the free leg to that of the clamped one of 2, 4, and 7. A nonorthogonal co-ordinate system was used which gave constant limits of integration over the area of the triangle. The co-ordinate transformation made it necessary to modify the functions used by Ritz in approximating deflections and to consider cross products in the integration. The integration was done numerically, using tables compiled by Young and Felgar in 1949. To check the accuracy of results, a solution was obtained to the problem of a vibrating cantilever beam of uniform depth and triangular plan view. The results obtained were found to be consistent with those obtained for the plates by using an eight-term series to approximate the deflections of the symmetric plates (isosceles triangles) and a six-term series to approximate the deflections of the unsymmetric plates (right triangles).

scholarly journals On Unique and Nonunique Fixed Points and Fixed Circles in M v b -Metric Space and Application to Cantilever Beam Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Meena Joshi ◽  
Anita Tomar ◽  
Hossam A. Nabwey ◽  
Reny George
Keyword(s):  
Differential Equation ◽  
Fixed Points ◽  
Metric Space ◽  
Order Differential Equation ◽  
Unique Fixed Point ◽  
Elastic Beam ◽  
Closed Ball ◽  
Open Ball ◽  
Ball Convergence ◽  
Fourth Order Differential Equation

We introduce M v b -metric to generalize and improve M v -metric and unify numerous existing distance notions. Further, we define topological notions like open ball, closed ball, convergence of a sequence, Cauchy sequence, and completeness of the space to discuss topology on M v b -metric space and to create an environment for the survival of a unique fixed point. Also, we introduce a notion of a fixed circle and a fixed disc to study the geometry of the set of nonunique fixed points of a discontinuous self-map and establish fixed circle and fixed disc theorems. Further, we verify all these results by illustrative examples to demonstrate the authenticity of the postulates. Towards the end, we solve a fourth order differential equation arising in the bending of an elastic beam.

Metamorphism of the MacLean Lake and Central Metavolcanic belts, La Ronge domain, Trans-Hudson Orogen: pressure-temperature variations and tectonic implications

1998 ◽  
Vol 35 (8) ◽  
pp. 905-922 ◽  
Author(s):  
Haiming Yang ◽  
Kurt Kyser ◽  
Kevin Ansdell
Keyword(s):  
High Temperature ◽  
Thermal Gradient ◽  
Amphibolite Facies ◽  
Tectonic Zone ◽  
Thrust Sheet ◽  
Temperature Variations ◽  
High Thermal Gradient ◽  
Tectonic Implications ◽  
Peak Metamorphism ◽  
Peak Pressures

Metamorphic assemblages differ between the metasedimentary MacLean Lake belt and the adjacent Central Metavolcanic belt in the La Ronge domain, Trans-Hudson Orogen. The former consists of meta-arkoses, psammitic gneisses, metaconglomerates, and calc-silicate gneisses of upper amphibolite facies (600-740°C, 440-660 MPa) with local migmatization, whereas the latter is comprised mainly of metavolcanic and plutonic rocks, with minor metasedimentary schists of greenschist to lower amphibolite facies (480-630°C, 520-560 MPa). Petrographic evidence indicates that peak metamorphic conditions were reached towards the end of D1 deformation during which the Central Metavolcanic belt was thrust onto the MacLean Lake belt along the McLennan Lake tectonic zone, which separates the two belts. Peak metamorphic assemblages did not undergo retrograde alteration during D2 deformation, indicating that high temperature was maintained during D2 deformation. Differences in pressure (P) and temperature (T) between the northeastern and southwestern parts of the Central Metavolcanic belt may have resulted from tilting along strike after peak metamorphism. Peak temperatures increase gradually from the Central Metavolcanic belt to MacLean Lake belt across the McLennan Lake tectonic zone. Peak pressures in the two belts are similar, implying that the Central Metavolcanic belt thrust sheet was probably thin. The P-T data for the MacLean Lake belt indicate a relatively high thermal gradient (40-50°C/km), similar to that in the metasedimentary Kisseynew domain in the orogen.

scholarly journals Asymptotic theory for a critical case for a general fourth-order differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 479-488
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
Fourth Order Equations ◽  
Fourth Order Differential Equation

In this paper we identify a relation between the coefficients that represents a critical case for general fourth-order equations. We obtained the forms of solutions under this critical case.

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