Thin Clamped Circular Plate Research Articles

Integrated analytical, experimental, and numerical study of vibrational response in clamped circular plates with pressurised air cavities

This paper presents a comprehensive investigation, comprised of analytical, experimental and numerical approaches, into the interaction between the sound field produced within a pressurised air-filled back cavity and the vibration of a thin clamped circular plate subjected to its pressure. Previous investigations into the interaction of sound fields with vibrating boundaries have primarily been restricted to non-pressurised cavities. We extend the application of the weak modal coupling method to derive a solution for this unexplored physical problem by utilising classic plate theory, Fourier-Bessel series formulation and the uncoupled solutions for a pressurised cylindrical sound field and a thin clamped circular plate with uniform radial tension. We describe the vibration response of the coupled system in terms of the uncoupled cavity and plate modes. Using a set of geometric and physical properties for the system, the resonance frequencies and response functions are calculated and the coupling between the sound field and the plate is investigated as a function of cavity pressure, providing novel fundamental insights into the physical system. We construct an experimental rig to embody the conditions of the analytical investigation for the purpose of validation. Finite element analysis (FEA), employing modal analysis and transient dynamic analysis, is used to further validate and extend the analytical insights while more accurately mirroring the experimental conditions. The response functions, mode shapes, resonance frequencies and their dependence on the cavity pressure were experimentally measured and numerically simulated, with direct comparisons made with the analytical model. A high degree of accuracy for the analytical model is validated along with its ability to describe the underlying physical phenomena. The validated analytical model is then leveraged to perform fundamental explorations into the modal contributions as a function of cavity pressure and a parametric sensitivity analysis on the effects of plate radii and plate thickness on the coupled system resonance frequencies.

Sound Radiation of the Resonator in the Form of a Vibrating Circular Plate Embedded in the Outlet of the Circular Cylindrical Cavity

The Neumann axisymmetric boundary value problem is considered for a vibrating thin clamped circular plate embedded in the flat rigid screen in the outlet of the circular cylindrical cavity. It is assumed that the two pistons, one cylindrical and the other one annular/circular, are vibrating inside the cavity with the same single frequency and different initial phases. The pistons are the only sources of excitation of the fluid. The acoustic pressure difference on both sides of the plate forces its vibrations. The acoustic waves are radiated into the half-space above it. A rigorous theoretical analysis of sound radiation has been performed based on the exact solution of the problem of free vibrations of the plate. The system of three coupled partial differential equations is solved. They are the two Helmholtz equations for the cavity and for the half-space, and the equation of motion of the plate. Consequently, the acoustic pressure distribution in both spaces is presented as well as the acoustic power radiated.

Quasi-Static Thermal Deflection of a Thin Clamped Circular Plate Due to Heat Generation

This paper deals with the determination of thermal deflection in a thin clamped circular plate defined as 0 ≤r ≤ b; 0 ≤z ≤ h due to internal heat generation within it. A thin clamped circular plate is considered having arbitrary initial temperature and subjected to time dependent heat flux at the fixed circular boundary (r = b). The lower surface (z = 0) is at zero temperature whereas upper surface (z = h) is thermally insulated. The governing heat conduction equation has been solved by using integral transform technique. The edge of the circular plate is fixed and clamped at r = b. The results are obtained in series form in terms of Bessel's functions. As a special case different metallic plates have been considered and the results for thermal deflection have been computed numerically and are illustrated graphically.

AN INVERSE QUASI-STATIC THERMAL DEFLECTION PROBLEM FOR A THIN CLAMPED CIRCULAR PLATE

ABSTRACT An inverse axially symmetric quasi-static problem of thermoelasticity for a thin clamped circular plate in which a heat flux is prescribed on an internal cylindrical surface of the plate and suitable heat exchange conditions are met on the upper and lower surfaces of the plate is solved with the help of a generalized integral transform technique. The results, obtained in a series form involving Bessel functions, are illustrated numerically.

The general fundamental solution of the sixth-order Reissner and Mindlin plate bending models revisited

The sixth-order differential equation system of the Reissner and Mindlin plate bending models describe mathematically the plate problem, where lines normal to the mid-plane before deformation remain straight and inextensible in the deformed configuration, but not necessarily normal to the reference plane anymore. The non-normality condition is due to the consideration of transverse shearing strains, disregarded in the classical bi-harmonic Kirchhoff plate model. As the plate thickness is reduced and/or the transverse shear modulus is increased, the deformed configuration is less dependent of the transverse shear strains and, in the limit as the plate thickness approaches zero and/or the transverse shear modulus approaches ∞, the transverse shear strain effects vanish and the problem is exactly that described by the classical plate model. In this paper, we investigate the fundamental solutions of both the fourth-order Kirchhoff and the sixth-order Reissner and Mindlin plate models. We consider a transversely isotropic material and show that the fundamental solution of the bi-harmonic problem can be obtained directly from the general fundamental solution of the sixth-order plate problem, in the limit as the plate thickness approaches zero and/or the transverse shear modulus approaches ∞. This solution is in agreement with the analytical solution of an infinite thin clamped circular plate submitted to a unitary concentrated load acting at its center.

Application of the modified method of multiple scales to solving the problem of a thin clamped circular plate of a very large deflection

In this paper, asymptotic behaviour of the solution to the problem of a thin clamped circular plate under uniform normal pressure at very large deflection is restudied by means of the modified method of multiple scales given in [1–2]. The result presented herein is in good agreement with one obtained by professor Chien Wei-zang who first proposed the method of composite expansions to solve this problem in [3]. However, by contrast, the advantage of the modified method of multiple scales it seems to be relatively simpler than the method used in [3]. It is also shown that the restriction of the method of paper [1–2] pointed out in paper [4] is not essential, and several computation errors in [3] are corrected as well.

A note on quasi-static thermal deflection of a thin clamped circular plate due to ramp-type heating of a concentric circular region of the upper face

A note on quasi-static thermal deflection of a thin clamped circular plate due to ramp-type heating of a concentric circular region of the upper face

A thin clamped circular plate under uniform or uniformly varying load on an eccentric ellipse

Exact and closed expressions are constructed for the complex potentials which solve the problem of a thin clamped isotropic circular plate bent by uniform or uniformly varying pressure distributed over the area of an eccentric elliptic patch.

Thin circular plates under certain distributions of normal loading

The problem of a concentrated normal force at any point of a thin clamped circular plate was treated in terms of infinite series by Clebsch [1], who gave the general solution of the biharmonic equation D ∇ 4 w = p . Using the method of inversion Michell [2] found a solution for the same problem in finite terms. The method of complex potentials was used by Dawoud [3] to solve the problem of an isolated load on a circular plate under certain boundary conditions. Applying Muskhelishvili's method Washizu [4] obtained the same results for clamped and hinged boundaries. The complex variable method was applied by the authors [5] to obtain solutions for a thin circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under a particular form of boundary condition defining certain types of boundary constraints which include the usual clamped and hinged boundaries as well as other special cases. Flugge [6] gave the solution for a linearly varying load over the complete simply supported circular plate. Using complex variable methods Bassali [7] found the solution for the same load distributed over the area of an eccentric circle under the boundary conditions mentioned before [5], and the authors [8] obtained the solutions for general loads of the type cos nϑ(or sin nϑ ), spread over the area of a circle concentric with the plate. In this paper the solutions for a circular plate subjected to the same boundary conditions are obtained when the plate is acted upon by the following types of loading: (a) a concentrated load at an arbitrary point; (b) a line load spread on any part of a diameter; (c) a load distributed over the area of a sector of the plate; (d) a concentrated couple at an arbitrary point of the plate. As a limiting case we find the deflexion at any point of a thin elastic plate having the form of a half plane clamped along the straight edge and subject to an isolated couple at any point.