Deflection of Uniformly Loaded Circular Plates

1943 ◽  
Vol 10 (4) ◽  
pp. A181-A182
Author(s):  
F. C. W. Olson
Keyword(s):  
General Theory ◽  
Circular Plate ◽  
Elastic Constants ◽  
Energy Method ◽  
Plate Thickness ◽  
Large Deflections ◽  
Bending Moments ◽  
Circular Plates ◽  
Boundary Parameter ◽  
Center Deflection

Abstract The equation for small deflections of a uniformly loaded and supported circular plate is rewritten so that the two constants of integration are interpreted as the center deflection ω0 and a boundary parameter α. Stresses and bending moments are given in terms of ω0 and α. A particular advantage of this treatment is that the nature of the support may be found experimentally even if the elastic constants, plate thickness, and load are unknown. The energy method is used to develop a more general theory of large deflections, valid for any condition of uniform support at the edge.

Large Deflections of Circular Plates of Variable Thickness

1982 ◽  
Vol 49 (1) ◽  
pp. 243-245 ◽  
Author(s):  
B. Banerjee
Keyword(s):  
Circular Plate ◽  
Variable Thickness ◽  
Large Deflection ◽  
Numerical Results ◽  
Large Deflections ◽  
Circular Plates ◽  
Uniform Load ◽  
Plate Of Variable Thickness ◽  
Plates Of Variable Thickness

The large deflection of a clamped circular plate of variable thickness under uniform load has been investigated using von Karman’s equations. Numerical results obtained for the deflections and stresses at the center of the plate have been given in tabular forms.

Large Deflections of Circular Plates

1952 ◽  
Vol 19 (3) ◽  
pp. 287-292
Author(s):  
M. Stippes ◽  
A. H. Hausrath
Keyword(s):  
Circular Plate ◽  
Critical Values ◽  
Graphical Form ◽  
Series Expansions ◽  
Large Deflections ◽  
Circular Plates ◽  
Simply Supported ◽  
Perturbation Procedure

Abstract This paper contains a solution of von Kármán’s equations for a uniformly loaded, simply supported circular plate. The method used to obtain the solution is the perturbation procedure. Series expansions for the deflection and stresses in the plate are obtained. The legitimacy of these expansions is demonstrated in the Appendix. Critical values of stress and deflection are presented in graphical form. Furthermore, tables of coefficients for the afore-mentioned series are presented if anyone desires to extend the results which are presented here.

Estimation of Charge Mass for High Speed Forming of Circular Plates Using Energy Method

2013 ◽  
Vol 845 ◽  
pp. 803-808 ◽  
Author(s):  
Roozbeh Alipour ◽  
Sudin Izman ◽  
Mohd Nasir Tamin
Keyword(s):  
Circular Plate ◽  
Energy Method ◽  
Strain Energy ◽  
High Speed ◽  
Circular Plates ◽  
Explosive Material ◽  
Charge Mass ◽  
High Speed Forming ◽  
Transmitted Energy

In this paper the explosive mass for forming a circular plate to a cone is determined using the energy method. To achieve this goal, the strain energy which is necessary to form a circular plate to a cone is calculated at first. Later the transmitted energy due to the detonation of an explosive material which is placed in a constant stand-off distance to a circular plate is specified. Equaling the strain energy and the transmitted energy, the explosive mass is figured out. Comparing the obtained results with the experiments shows the relatively acceptable compatibility.

Large Axisymmetric Deformations of Elastic/Plastic Perforated Circular Plates

2003 ◽  
Vol 125 (4) ◽  
pp. 357-364 ◽  
Author(s):  
D. Wu ◽  
J. Peddieson ◽  
G. R. Buchanan ◽  
S. G. Rochelle
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Circular Plate ◽  
Numerical Solutions ◽  
Outer Edge ◽  
Plate Bending ◽  
Large Deflections ◽  
Circular Plates ◽  
Elastic Plastic ◽  
Thickness Variations

A mathematical model of axisymmetric elastic/plastic perforated circular plate bending and stretching is developed which accounts for through thickness yielding, through thickness variations in perforation geometry, elastic outer edge restraint, and moderately large deflections. Selected numerical solutions of the resulting differential equations are presented graphically and used to illustrate interesting trends.

Symmetrically Loaded Circular Plates under the Combined Action of Lateral and End Loading

1959 ◽  
Vol 10 (3) ◽  
pp. 266-282 ◽  
Author(s):  
Raymond Hicks
Keyword(s):  
Circular Plate ◽  
Infinite Plate ◽  
Lateral Load ◽  
Simultaneous Action ◽  
Bending Moments ◽  
Circular Plates ◽  
Simply Supported ◽  
Combined Action

Expressions are obtained for the radial and tangential bending moments in a circular plate under the combined action of (a) a lateral load concentrated on the circumference of a circle and an end tension or compression, and (b) a uniformly distributed lateral load, having a diameter less than the diameter of the plate, and an end tension or compression. For both types of loading, solutions are obtained for plates which are simply-supported and for plates with an arbitrary end rotation.In addition, the following limiting cases are considered: (i) concentrated lateral load with end tension or compression, and (ii) an infinite plate under the simultaneous action of an end tension and a lateral load concentrated on the circumference of a circle of finite diameter.

Large Deflections of Elliptical Plates

1956 ◽  
Vol 23 (1) ◽  
pp. 21-26
Author(s):  
N. A. Weil ◽  
N. M. Newmark
Keyword(s):  
Circular Plate ◽  
Maximum Stress ◽  
Large Deflection ◽  
Ritz Method ◽  
Constant Thickness ◽  
Infinite Strip ◽  
Elliptical Plate ◽  
Arbitrary Dimensions ◽  
Center Deflection ◽  
Large Deflection Problem

Abstract A solution is obtained by means of the Ritz method for the “large-deflection” problem of a clamped elliptical plate of constant thickness, subjected to a uniformly distributed load. Two shapes of elliptical plate are treated, in addition to the limiting cases of the circular plate and infinite strip, for which the exact solutions are known. Center deflections as well as total stresses at the center and edge decrease as one proceeds from the infinite strip through the elliptical shapes to the circular plate, holding the width of the plates constant. The relation between edge-stress at the semiminor axis (maximum stress in the plate) and center deflection is found to be practically independent of the proportions of the elliptical plate. Hence the governing stress may be determined from a single curve for a given load on an elliptical plate of arbitrary dimensions, if the center deflection is known.

scholarly journals Thermoelastic coupling vibration and stability analysis of rotating circular plate in friction clutch

2018 ◽  
Vol 38 (2) ◽  
pp. 558-573 ◽  
Author(s):  
Yongqiang Yang ◽  
Zhongmin Wang ◽  
Yongqin Wang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Coupling Coefficient ◽  
Inertial Force ◽  
Angular Speed ◽  
Circular Plates ◽  
Thermoelastic Coupling ◽  
Coupling Vibration ◽  
Friction Clutch

Rotating friction circular plates are the main components of a friction clutch. The vibration and temperature field of these friction circular plates in high speed affect the clutch operation. This study investigates the thermoelastic coupling vibration and stability of rotating friction circular plates. Firstly, based on the middle internal forces resulting from the action of normal inertial force, the differential equation of transverse vibration with variable coefficients for an axisymmetric rotating circular plate is established by thin plate theory and thermal conduction equation considering deformation effect. Secondly, the differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Meanwhile, the thermoelastic coupling transverse vibrations with three different boundary conditions are calculated. In this case, the change curve of the first two-order dimensionless complex frequencies of the rotating circular plate with the dimensionless angular speed and thermoelastic coupling coefficient are analyzed. The effects of the critical dimensionless thermoelastic coupling coefficient and the critical angular speed on the stability of the rotating circular plate with simply supported and clamped edges are discussed. Finally, the relation between the critical divergence speed and the dimensionless thermoelastic coupling coefficient is obtained. The results provide the theoretical basis for optimizing the structure and improving the dynamic stability of friction clutches.

The Method of Finite Differences in Solutions Concerning Circular Plate

2011 ◽  
Vol 490 ◽  
pp. 305-311
Author(s):  
Henryk G. Sabiniak
Keyword(s):  
Circular Plate ◽  
Finite Differences ◽  
Machine Building ◽  
Polar Coordinates ◽  
Circular Plates ◽  
Technical Problems ◽  
Theory Of Plates ◽  
Worm Gearing ◽  
Mixed Derivatives ◽  
Plate Deflection

Finite difference method in solving classic problems in theory of plates is considered a standard one [1], [2], [3], [4]. The above refers mainly to solutions in right-angle coordinates. For circular plates, for which the use of polar coordinates is the best option, the question of classic plate deflection gets complicated. In accordance with mathematical rules the passage from partial differentials to final differences seems firm. Still final formulas both for the equation (1), as well as for border conditions of circular plate obtained in this study and in the study [3] differ considerably. The paper describes in detail necessary mathematical calculations. The final results are presented in identical form as in the study [3]. Difference of results as well as the length of arm in passage from partial differentials to finite differences for mixed derivatives are discussed. Generalizations resulting from these discussions are presented. This preliminary proceeding has the purpose of searching for solutions to technical problems in machine building and construction, in particular finding a solution to the question of distribution of load along contact line in worm gearing.

Approximated Fundamental Frequency for Thin Circular Plates Clamped or Pinned at the Edge

2007 ◽  
Author(s):  
George Weiss
Keyword(s):  
Circular Plate ◽  
Fundamental Frequency ◽  
Bessel Functions ◽  
Unit Area ◽  
Continuous System ◽  
Modified Bessel Functions ◽  
Circular Plates ◽  
Numerical Parameter ◽  
Elliptical Plate ◽  
Distributed Load

Calculating the exact solution to the differential equations that describe the motion of a circular plate clamped or pinned at the edge, is laborious. The calculations include the Bessel functions and modified Bessel functions. In this paper, we present a brief method for calculating with approximation, the fundamental frequency of a circular plate clamped or pinned at the edge. We’ll use the Dunkerley’s estimate to determine the fundamental frequency of the plates. A plate is a continuous system and will assume it is loaded with a uniform distributed load, including the weight of the plate itself. Considering the mass per unit area of the plate, and substituting it in Dunkerley’s equation rearranged, we obtain a numerical parameter K02, related to the fundamental frequency of the plate, which has to be evaluated for each particular case. In this paper, have been evaluated the values of K02 for thin circular plates clamped or pinned at edge. An elliptical plate clamped at edge is also presented for several ratios of the semi–axes, one of which is identical with a circular plate.

Circular Plates Under Hard Excitations to Include Casimir Effect: Amplitude-Voltage Response of Superharmonic Resonance of Second Order

2020 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Julio Beatriz ◽  
Jonathan Perez
Keyword(s):  
Circular Plate ◽  
Casimir Effect ◽  
Second Order ◽  
Peak Amplitude ◽  
Voltage Response ◽  
Voltage Amplitude ◽  
Circular Plates ◽  
Amplitude Response ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated

Abstract This paper deals with voltage-amplitude response of superharmonic resonance of second order of electrostatically actuated clamped MEMS circular plates. A flexible MEMS circular plate, parallel to a ground plate, and under AC voltage, constitute the structure under consideration. Hard excitations due to voltage large enough and AC frequency near one fourth of the natural frequency of the MEMS plate resonator lead the MEMS plate into superharmonic resonance of second order. These excitations produce resonance away from the primary resonance zone. No DC component is included in the voltage applied. The equation of motion of the MEMS plate is solved using two modes of vibration reduced order model (ROM), that is then solved through a continuation and bifurcation analysis using the software package AUTO 07P. This predicts the voltage-amplitude response of the electrostatically actuated MEMS plate. Also, a numerical integration of the system of differential equations using Matlab is used to produce time responses of the system. A typical MEMS silicon circular plate resonator is used to conduct numerical simulations. For this resonator the quantum dynamics effects such as Casimir effect are considered. Also, the Method of Multiple Scales (MMS) is used in this work. All methods show agreement for dimensionless voltage values less than 6. The amplitude increases with the increase of voltage, except around the dimensionless voltage value of 4, where the resonance shows two saddle-node bifurcations and a peak amplitude significantly larger than the amplitudes before and after the dimensionless voltage of 4. A light softening effect is present. The pull-in dimensionless voltage is found to be around 16. The effects of damping and frequency on the voltage response are reported. As the damping increases, the peak amplitude decreases. while the pull-in voltage is not affected. As the frequency increases, the peak amplitude is shifted to lower values and lower voltage values. However, the pull-in voltage and the behavior for large voltage values are not affected.

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