scholarly journals Vibration analysis of irregular-shaped plates on simple supports

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210184
Author(s):  
Abhijit Ghosh ◽  
Anirvan DasGupta
Keyword(s):  
Boundary Conditions ◽  
Modal Analysis ◽  
Circular Plate ◽  
Solution Structure ◽  
Circular Plates ◽  
Perturbative Approach ◽  
Irregular Boundary ◽  
Generalized Fourier Series ◽  
Perturbative Solution ◽  
Smallness Parameter

In this work, we propose a general perturbative approach for modal analysis of irregular-shaped plates of uniform thickness with uniform boundary conditions. Given a plate of irregular boundary, first, a uniform circular plate of identical thickness and area, centred at the centroid, is determined. The irregular boundary is then treated as a perturbation with a suitable smallness parameter, and is expressed as a generalized Fourier series. The frequency parameter, shape function and boundary conditions are then perturbed in terms of the smallness parameter. The homogeneous zeroth-order equation corresponds to the circular plate, which is exactly solvable. We show that the inhomogeneous equations in the higher orders can also be solved exactly using a particular solution structure. We can then construct the exact perturbative solution up to any order. The proposed method is demonstrated through the modal analysis of simply supported super-circular plates. The results are validated using the numerical results obtained from ANSYS ® , which are an excellent match. Interestingly, the supposedly degenerate modes with an even number of nodal diameters of super-circular plates are found to split naturally.

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.

scholarly journals Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 429 ◽  
Author(s):  
Krzysztof Żur ◽  
Piotr Jankowski
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Special Functions ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Circular Plates ◽  
Classical Plate Theory ◽  
Coupled Equations

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as a linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the neglected effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Bending of Circular Plates Under a Uniform Load on a Concentric Circle—Part II

1968 ◽  
Vol 90 (2) ◽  
pp. 279-293
Author(s):  
J. C. Heap
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Concentric Circle ◽  
Outer Edge ◽  
Circular Plates ◽  
Uniform Load ◽  
Simply Supported ◽  
Basic Equations

The basic equations of deflection, slope, and moments for a thin, flat, circular plate subjected to a uniform load on a concentric circle were derived for four generalized cases. From these generalized cases, six simplified cases were deduced. The four generalized cases have the uniform load acting on a concentric circle of the plate between the inner and outer edges, with the following boundary conditions: (a) Outer edge supported and fixed, inner edge fixed; (b) outer edge simply supported, inner edge free; (c) outer edge simply supported, inner edge fixed; and (d) outer edge supported and fixed, inner edge free.

scholarly journals An Analytical Closed-Form Solution for Free Vibration of Functionally Graded Circular Plates with Even and Uneven Porosity Distributions

2018 ◽  
Author(s):  
Krzysztof Kamil Żur
Keyword(s):  
General Solution ◽  
Boundary Conditions ◽  
Circular Plate ◽  
Volume Fraction ◽  
Functionally Graded ◽  
Numerical Results ◽  
Axisymmetric Vibration ◽  
Gradient Index ◽  
Circular Plates ◽  
Coupled Equations

Free axisymmetric and non-axisymmetric vibration analysis of the unsaturated porous functionally graded circular plates has been presented on the basis of classical plate theory. The defined coupled equations of motion for the porous functionally graded circular plate were decoupled based on the properties of the physical neutral surface. The one general solution of the decoupled equation of motion was obtained as linear combinations of the multiparametric special Bessel functions for the functionally graded circular plate with even and uneven porosity distributions. The influences of the even and uneven distributions of porosity, gradient index, diverse boundary conditions and the negligible effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied. The obtained numerical results show the differences and significant effect of considered types of distributions of porosities, values of the gradient index and the porosity volume fraction on the distribution of eigenfrequencies of the circular plates. Additionally, the obtained multiparametric general solution of the defined differential equation will allow to study the influences of diverse additional complicating effects such as stepped thickness, cracks, additional mounted elements expressed by only additional boundary conditions on the dynamic behavior of the porous functionally graded circular/annular plates. The formulated boundary value problem, the method of solution and the obtained numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported. The present paper fills this void in the literature.

scholarly journals Exact Analytical Solution for Free Axisymmetric and Non-Axisymmetric Vibrations of FGM Porous Circular Plates

2018 ◽  
Author(s):  
Krzysztof Kamil Żur ◽  
Piotr Jankowski
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Special Functions ◽  
Plate Theory ◽  
Exact Analytical Solution ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Circular Plates ◽  
Classical Plate Theory ◽  
Coupled Equations

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the negligibled effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Transient Interaction of a Circular Plate and a Fluid Medium

1979 ◽  
Vol 46 (1) ◽  
pp. 26-30 ◽  
Author(s):  
J. W. Berglund
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Applied Pressure ◽  
Plate Boundary ◽  
Fluid Medium ◽  
Fluid Field ◽  
Elastic Circular Plate ◽  
Unstrained State ◽  
State Of Rest ◽  
Acoustic Fluid

The transient dynamic response of an elastic circular plate subjected to a suddenly applied pressure is determined for several edge boundary conditions. The plate boundary is attached to a semi-infinite, radially rigid tube which is filled with an acoustic fluid, and pressure is applied to the in-vacuo side of the plate. The transient solution is determined by using a technique in which the plate is subjected to a periodic pressure function constructed of appropriately signed and time-shifted Heaviside step functions, and by relying on a physical mechanism which returns the plate and fluid near the plate to an unstrained state of rest between pulses. The plate response is presented for a number of radius-to-thickness ratios and edge boundary conditions when interacting with water. Comparisons are also made with solutions obtained using a plane wave approximation to the fluid field.

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.

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