Thermal Stresses in Rectangular Plates

1959 ◽  
Vol 10 (1) ◽  
pp. 65-78 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Thermal Stresses ◽  
Plate Thickness ◽  
Normal Modes ◽  
Thermoelastic Stress ◽  
Characteristic Functions ◽  
Vibration Modes ◽  
Rectangular Plates ◽  
Practical Application ◽  
Flat Plates ◽  
Modes Of Vibration

SummaryThe characteristic functions for beam vibration modes are used to derive an approximate solution for the calculation of thermal stresses in rectangular isotropic flat plates subjected to arbitrary temperature distributions in the plane of the plate and constant temperatures through the plate thickness. The thermal stresses are obtained in the form of generalised Fourier expansions in terms of the characteristic functions, and their derivatives, representing normal modes of vibration of a clamped-clamped beam. Since these functions have recently been tabulated, the practical application of this new method to the thermoelastic stress analysis of plates presents no difficulty.

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.

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.

Note on Cylindrical Bending of a Heated Long Rectangular Plate

1963 ◽  
Vol 67 (625) ◽  
pp. 65-66
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Thermal Stresses ◽  
Functional Relationship ◽  
Rectangular Plates ◽  
Practical Application ◽  
Surrounding Structure ◽  
Dimensional Parameters ◽  
Design Calculations ◽  
Special Case ◽  
Excessive Number ◽  
Elastic Characteristics

In reference 1 a general analysis was presented for the calculation of thermal stresses and deflections due to temperature gradients in long rectangular plates having lengthwise edges supported elastically. The edge support was idealised as equivalent rotational and extensional stiffnesses depending on the elastic characteristics of the surrounding structure. The special case of infinite extensional stiffness and freely hinged edges (see Fig. 1) was discussed in greater detail and a number of graphs were included for the calculation of edge reactions in terms of the thermal force and thermal moment parameters. Since these graphs used an excessive number of non-dimensional parameters their practical application is unwieldy; four non-dimensional parameters were used instead of three parameters which would be sufficient to describe the required functional relationship. This note shows an alternative presentation of the results which can be used more conveniently in any design calculations.

Photothermoelastic Investigation of Thermal Stress in Flat Plates

1963 ◽  
Vol 85 (4) ◽  
pp. 566-568 ◽  
Author(s):  
Herbert Becker ◽  
Angelo Colao
Keyword(s):  
Temperature Field ◽  
Thermal Stress ◽  
Circular Plate ◽  
Thermal Stresses ◽  
Theoretical Data ◽  
Rectangular Plates ◽  
Flat Plates ◽  
Comparison With Experiments ◽  
Better Than ◽  
Good Agreement

Predictions of thermal stresses in various shape flat plates with a parabolic temperature field applied in one direction were found to yield good agreement with photothermoelastic analysis. A theory based upon an analog was found to agree with experiments on rectangular plates better than two other methods of analysis. The analog procedure was utilized for analysis of a circular plate, to the results of which an engineering modification was applied to obtain theoretical data for comparison with experiments on a hexagonal plate with and without a central circular hole.

The Determination of Vibration Modes

1949 ◽  
Vol 53 (468) ◽  
pp. 1095-1099
Author(s):  
N. F. Harpur
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Dynamic Loads ◽  
Vibration Modes ◽  
Resonance Modes ◽  
Modes Of Vibration

At some stage in the design of every aeroplane it is necessary to estimate or to measure the resonance modes of vibration. This has not always been the case, but the problems of flutter, control reversal and dynamic loads have increased in importance as speeds have risen. Nowadays, it is an airworthiness requirement that these effects be considered and the aircraft made safe for all conditions of flight. A knowledge of the normal modes of vibration is essential for all accurate estimates of these aeroelastic effects.Taking flutter as an example, the technique of flutter investigations consists of first determining which combinations of the various possible degrees of freedom are liable to excite dangerous oscillations. Typical degrees of freedom for a wing are bending and twist in each normal mode, aileron deflection and tab deflection; for a tailplane and elevator we might consider tailplane bending or twist, elevator deflection, tab deflection, fuselage bending and twist, and pitching of the whole aeroplane.

Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

1968 ◽  
Vol 72 (688) ◽  
pp. 341-344 ◽  
Author(s):  
B. Dawson
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Rotary Inertia ◽  
Characteristic Functions ◽  
Cantilever Beams ◽  
Dynamic Displacement ◽  
Modes Of Vibration ◽  
Good Agreement

Summary The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

Transverse Vibration of a Rectangularly Orthotropic Spinning Disk, Part 1: Formulation and Free Vibration

1999 ◽  
Vol 121 (3) ◽  
pp. 273-279 ◽  
Author(s):  
A. Phylactopoulos ◽  
G. G. Adams
Keyword(s):  
Transverse Vibration ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Circular Disk ◽  
Polar Coordinates ◽  
Fourier Series Expansion ◽  
Vibration Modes ◽  
Classical Formulation ◽  
Modes Of Vibration ◽  
Orthotropic Disk

The transverse vibration of a spinning circular disk with rectangular orthotropy is investigated. Two dimensionless parameters are established in order to characterize the degree of disk anisotropy and solutions are sought for a range of these parameters. The orthotropic bending stiffness is transferred into polar coordinates and is found to differ from a classical formulation for a stationary disk. A Fourier series expansion is used in the circumferential direction. Unlike the isotropic disk, the Fourier components determining the transverse vibration modes of the orthotropic disk do not separate. This condition results in an eigenvalue problem involving a coupled set of ordinary differential equations which are solved by a combination of numerical integration and iteration. Thus the natural frequencies and normal modes of vibration are determined. Because each eigenfunction contains contributions from more than one Fourier component, the normal modes do not possess distinct nodal diameters or nodal circles. Furthermore, disk orthotropy causes the natural frequencies corresponding to the sine and cosine modes to split; the degree of splitting decreases as the rotational speed increases.

A Comprehensive Study of the Free Vibration of Rectangular Plates Resting on Symmetrically-Distributed Uniform Elastic Edge Supports

1989 ◽  
Vol 56 (4) ◽  
pp. 893-899 ◽  
Author(s):  
D. J. Gorman
Keyword(s):  
Free Vibration ◽  
Free Vibration Analysis ◽  
Comprehensive Treatment ◽  
Vibration Modes ◽  
Rectangular Plates ◽  
Rapid Convergence ◽  
Linear Elastic ◽  
Modes Of Vibration ◽  
Elastic Edge ◽  
Comprehensive Study

An analytical-type solution is developed for the free vibration analysis of rectangular plates with uniform elastic edge support symmetrically distributed about the plate central axes. Both linear elastic rotational and translational support are considered to act simultaneously. Rapid convergence is encountered. Because of the symmetry of the problem, the free vibration modes fall into three distinct families. Eigenvalues are tabulated for the first four modes of vibration of a square plate with identical stiffnesses on each edge and with various ratios of translational to rotational stiffnesses. This represents, to the author’s knowledge, the first comprehensive treatment of this problem.

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

2020 ◽  
Vol 71 (7) ◽  
pp. 853-867
Author(s):  
Phuc Pham Minh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Phase Field ◽  
Deformation Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
Free Vibration Frequency

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

scholarly journals Preparation and the Microwave and Infrared Spectra of Vinyl Ketene

1979 ◽  
Vol 34 (11) ◽  
pp. 1269-1274 ◽  
Author(s):  
Erik Bjarnov
Keyword(s):  
Dipole Moment ◽  
Gas Phase ◽  
Infrared Spectra ◽  
Room Temperature ◽  
Normal Modes ◽  
Spectroscopic Constants ◽  
Electrical Dipole ◽  
Electrical Dipole Moment ◽  
Vacuum Pyrolysis ◽  
Modes Of Vibration

Vinyl ketene (1,3-butadiene-1-one) has been synthesized by vacuum pyrolysis of 3-butenoic 2-butenoic anhydride. The microwave and infrared spectra of vinyl ketene in the gas phase at room temperature have been studied. The trans-rotamer has been identified, and the spectroscopic constants were found to be Ã= 39571(48) MHz, B̃ = 2392.9252(28) MHz, C̃ = 2256.0089(28) MHz, ⊿j = 0.414(31) kHz, and ⊿JK = - 34.694(92) kHz. The electrical dipole moment was found to be 0.987(23) D with μa = 0.865(14) D and μb = 0.475(41) D. A tentative assignment has been made for 17 of the 21 normal modes of vibration

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