Polar Orthotropic Circular Plates Research Articles

Nonlinear vibrations of a polar-orthotropic thin circular plate subjected to circularly moving point load

Here, we formulate and study nonlinear vibrations of a thin, polar-orthotropic circular plate subjected to a circularly moving point load at a fixed radius and constant angular velocity. The governing equations and boundary conditions are obtained following Kirchhoff’s plate theory, incorporating von Kármán nonlinearity, and employing extended Hamilton’s principle. The external damping is introduced via Rayleigh dissipation function. The governing equations are solved using mode summation procedure. The mode shapes and the natural frequencies of the polar-orthotropic circular plate are found using Frobenius series method. Mode summation procedure results in coupled nonlinear ordinary differential equations. These equations are solved using the Runge–Kutta method for time response and method of harmonic balance with the arc continuation method for frequency response. The spectrum of the undamped linear vibration response of isotropic and polar-orthotropic plates exhibits natural frequencies of plates and angular velocity of the rotating load. The damped response contains the frequency of the angular velocity of the rotating load only. The nonlinear transverse vibrations of the undamped and damped plate due to rotating point load reveal frequency rich spectrums. The frequency response function shows strong modal interactions.

Forced Response of Polar Orthotropic Tapered Circular Plates Resting on Elastic Foundation

Forced axisymmetric response of polar orthotropic circular plates of linearly varying thickness resting on Winkler type of elastic foundation has been studied on the basis of classical plate theory. An approximate solution of problem has been obtained by Rayleigh Ritz method, which employs functions based upon the static deflection of polar orthotropic circular plates. The effect of transverse loadings has been studied for orthotropic circular plate resting on elastic foundation. The transverse deflections and bending moments are presented for various values of taper parameter, rigidity ratio, foundation parameter, and flexibility parameter under different types of loadings. A comparison of results with those available in literature shows an excellent agreement.

Free vibration analysis of axisymmetric laminated composite circular and annular plates using Chebyshev collocation

Solutions, based on principle of collocating the equations of motion at Chebyshev zeroes, are presented for the free vibration analysis of laminated, polar orthotropic, circular and annular plates. The analysis is restricted to axisymmetric free vibration of the plates and employs first-order shear deformation theory for the displacement field, in terms of midplane displacements, u, α and w. The eigenvalue problem is defined in terms of three equations of motion in terms of the radial co-ordinate r, the radial variation of the displacements being represented in polynomial series, with appropriate boundary conditions. Numerical results are presented to show the validity and accuracy of the proposed method. Results of parametric studies for laminated polar orthotropic circular and annular plates with different boundary conditions, orthotropic ratios, lamination sequences, number of layers and shear deformation are also presented.

Polar Orthotropic Inhomogeneous Circular Plates: Vibration Tailoring

Problem of matching a desired fundamental natural frequency is solved in the closed form for the polar-orthotropic inhomogeneous circular plate, which is clamped along its circumference. The vibration tailoring is performed by posing a semi-inverse eigenvalue problem. To do this, the fundamental mode shape is postulated. Namely, the analytical expression due to Lekhnitskii, and pertaining to the static deflection of the homogeneous circular plate is demanded to serve as an exact mode shape of the inhomogeneous plate. The analytical and numerical results are reported for several ratios of orthotropic coefficient.

Design of heterogeneous clamped circular plates with specified fundamental natural frequency

The titled problem is solved by the aid of the semi-inverse formulation. At the first glance, a surprising demand is posed: Design a radially heterogeneous polar orthotropic circular plate whose fundamental mode shape coalesces with the static deflection of a homogeneous circular plate of the same radius and thickness under uniformly distributed load. Compatible polynomial variations of the radial and circumferential flexural rigidities are introduced. It turns out that these are infinite amount of polar orthotropic circular plates that solve the posed semi-inverse problem, depending on the free parameter represented by one of the coefficients in variation of flexural rigidities. This allows to provide an unique solution to the titled problem: there is a single plate that possesses the specified fundamental natural frequency.

Vibration Tailoring of a Polar Orthotropic Circular Plate With a Translational Spring

In this study, the vibration tailoring problem is analytically solved for the polar orthotropic circular plate with translational spring along its circumference. By using the semi-inverse method and postulating the mode shape as a polynomial, we derive a closed-form solution.

A Variational Formulation and Analysis of Polar Orthotropic Circular Plates Using an Energy Principle, Part II: Thick Plates

Article A Variational Formulation and Analysis of Polar Orthotropic Circular Plates Using an Energy Principle, Part II: Thick Plates was published on March 1, 2008 in the journal Science and Engineering of Composite Materials (volume 15, issue 1).

A Variational Formulation and Analysis of Polar Orthotropic Circular Plates Using an Energy Principle, Part I: Thin Plates

A Variational Formulation and Analysis of Polar Orthotropic Circular Plates Using an Energy Principle, Part I: Thin Plates

Free vibration of polar orthotropic circular plates with elastic growth stresses and drying stresses

This study was performed to analyze effect of transverse growth stresses and drying stresses on natural frequencies of transverse free vibration of thin tree disk with free boundary condition. Using Rayleigh–Ritz method and finite element method, explicit closed form expression was derived for homogeneous orthotropic circular plates with internal stresses, which enables to predict both the effects of growth stresses, drying stresses, and combined growth and drying stresses on the natural frequency changes. This model can be used for predicting growth stresses from known natural frequencies. The same approach can also be undertaken to validate models visco-elastic drying stresses.

Vibration of Thermally Post-Buckled Orthotropic Circular Plates

Axisymmetric vibrations of a statically buckled polar orthotropic circular plate due to uniform temperature rise have been studied numerically. Effects of geometric nonlinearities have been incorporated into the problem formulation. The problem is challenging because the buckled configuration is unknown a priori. By assuming that the amplitude of plate's vibration and the additional strains induced in it are infinitesimal, and its response harmonic, the non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations; one for the thermal post-buckling, and the other for linear vibrations of the plate superimposed upon the post-buckled configuration. The plate's boundary is taken to be either clamped or simply supported but restrained from moving in the radial direction. The two sets of coupled boundary value problems are solved numerically by a shooting method. The dependence of the first three frequencies upon the temperature rise, for both pre-buckled and post-buckled plates, have been computed, and characteristic curves of the frequency versus temperature rise for different values of material anisotropy parameters are plotted. It is found that the three lowest frequencies of the pre-buckled plate decrease with an increase in the temperature, but those of a buckled plate increase monotonically with the temperature rise. The fundamental frequency of the deformed plate approaches zero at the onset of buckling.

Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation

Buckling and vibrational behavior of polar orthotropic circular plates of linearly varying thickness is presented on the basis of classical plate theory. The plate is resting on Winkler-type foundation. An approximate solution has been obtained by Ritz method, which employs basis functions based upon static deflection of polar orthotropic plates. The effect of elastic foundation and that of orthotropy on the natural frequency of plate has been illustrated for different values of taper parameter, flexibility parameter, in-plane force and nodal diameter. The critical buckling load for clamped and simply supported plates have been obtained. A comparison of results with those available in literature has been presented.

NONLINEAR DYNAMICS OF POLAR-ORTHOTROPIC CIRCULAR PLATES

The dynamics of nonlinear polar orthotropic circular plates with simply supported boundary condition are investigated. Kirchhoff strain displacement relations for thin plates plus next higher-order nonlinear terms (von Karman type geometric nonlinearity) are considered. Lagrangian density function and Hamilton's principle are utilized to derive Lagrange's equations, from which the equations of motion and associated boundary conditions are derived. Analytical solution is obtained by the perturbation techniques and numerical solution by the Runge–Kutta method. Phase diagrams, discrete Fast Fourier Transform (FFT diagrams) and time history responses are presented for studying the forced vibration behavior. The sub-harmonic and primary resonances are studied as well as the effect of adding damping foil layers. The quadratic term in the governing equation plays a softening role on the overall behavior of the plate due to its relatively large coefficient. The increase of damping tends to smooth out the unstable region (i.e. jump phenomenon) in the system.

Lekhnitskii's classic formula serving as an exact mode shape of simply supported polar orthotropic inhomogeneous circular plates

The semi-inverse problem of free vibration of simply supported inhomogeneous polar orthotropic plates is studied. The mode shape is postulated in the form of the classic formula by Lekhnitskii, namely, the static deflection of the associated uniform polar orthotropic circular plate under uniform loading. The ratios of the circumferential rigidity to the radial rigidity are identified as integer numbers, so that the candidate mode shapes constitute the polynomial functions. The flexural rigidities themselves are also represented by polynomials. The semi-inverse problem of identifying the coefficients of the flexural rigidities is solved analytically. It appears remarkable, that for numerous cases the simple method developed in this study provides novel closed form solutions for the design of the polar orthotropic circular plates with pre-specified mode shapes.

AXISYMMETRIC VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF QUADRATICALLY VARYING THICKNESS RESTING ON ELASTIC FOUNDATION

This paper is concerned with the axisymmetric vibration problem of polar orthotropic circular plates of quadratically varying thickness and resting on an elastic foundation. The problem is solved by using the Rayleigh–Ritz method with boundary characteristic orthonormal polynomials for approximating the deflection function. Numerical results are computed for frequencies, nodal radii and mode shapes. Three-dimensional graphs are also plotted for the first four normal modes of axisymmetric vibration of plates with free, simply-supported and clamped edge conditions for various values of taper, orthotropy and foundation parameters.

Approximate closed form solutions for free vibration of polar orthotropic circular plates

For free vibrations of polar orthotropic plate, simple approximate closed form solutions for mode shapes and its natural frequencies were obtained using the Rayleigh–Ritz method. Coordinate function satisfying the natural boundary conditions and the predetermined coefficients was adapted, which results in compact expressions and enables to readily calculate symmetric and nonsymmetric natural frequencies for arbitrary values of the elastic constants. The derived formulation can be used in designing of circular plates such as wood disk, which are naturally endowed with material orthotropy as well as fiber reinforced composite materials. The model can easily be used for the evaluation of parametric studies on dynamic behaviors and nondestructive methods during the initial design process.

Free vibration of polar orthotropic circular plates of quadratically varying thickness resting on elastic foundation

Classical plate theory is used to study free vibration of polar orthotropic circular plates of quadratically varying thickness resting on Winkler elastic foundation. Boundary characteristic orthonormal polynomials are used in the Rayleigh–Ritz method to analyze the plate for free, simply supported and clamped edge conditions. The numerical convergence of the method is tested and comparisons are made in particular case with the results already available in the literature. First twelve frequencies for various values of parameters describing the elastic foundation, thickness profile and orthotropy of the plate are given in tabular form. Two dimensional graphs for nodal lines and three dimensional graphs for mode shapes are plotted.

EFFECT OF ELASTIC FOUNDATION ON ASYMMETRIC VIBRATION OF POLAR ORTHOTROPIC LINEARLY TAPERED CIRCULAR PLATES

Asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation of Winkler type are discussed on the basis of the classical plate theory. Ritz method has been employed to obtain the natural frequencies of vibration using the functions based upon the static deflection of polar orthotropic plates, which has faster rate of convergence as compared to the polynomial co-ordinate functions. Frequency parameter of the plate with elastically restrained edge conditions are presented for various values of taper parameter, rigidity ratio and foundation parameter. A comparison of the results with those available in the literature obtained by finite element method, receptance method and polynomial co-ordinate functions shows an excellent agreement.

ASYMMETRIC VIBRATIONS AND ELASTIC STABILITY OF POLAR ORTHOTROPIC CIRCULAR PLATES OF LINEARLY VARYING PROFILE

Asymmetric vibration of polar orthotropic circular plates of linearly varying thickness subjected to hydrostatic in-plane force are discussed on the basis of classical plate theory. An approximate solution of the problem has been obtained by the Ritz method, which employs functions based upon the static deflection of polar orthotropic plates. This method has a faster rate of convergence as compared to the polynomial co-ordinate functions. Frequency parameters of the plate with elastically restrained edge conditions are presented for the three modes of vibrations for various values of taper parameter, rigidity ratio, flexibility parameter and buckling load parameter. The critical buckling loads for elastically restrained edge conditions have been obtained. A comparison of results with those available in the literature shows an excellent agreement.

Closed-form fundamental-frequency estimates for polar orthotropic circular plates

This paper provides simple, yet accurate, estimates of the fundamental natural frequencies of vibration of simply supported and clamped polar orthotropic circular plates with and without internal (central) point supports. Such estimates are obtained by means of Rayleigh’s method coupled with the deflection functions for the corresponding statically loaded systems. As will be seen, the present formulation, which generalizes an earlier work dealing with plates composed of isotropic materials, yields compact, closed-form expressions, enabling the ready calculation of (approximate) natural frequencies for arbitrary values of the elastic constants. Furthermore, remarkable simplifications in the derivations for, and form of, some of these expressions are achieved through the application of a computerized symbolic algebra system.

FREE VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF VARIABLE THICKNESS WITH ELASTICALLY RESTRAINED EDGE

Asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness with elastic/rigid support are discussed on the basis of the classical plate theory. An approximate solution of the problem is obtained by the Rayleigh–Ritz method using functions based on static deflection of polar orthotropic circular plates. This type of approximating functions has a faster rate of convergence as compared to the polynomial co-ordinate functions. Frequency parameters of the plate have been obtained for different values of taper parameter, flexibility parameter and rigidity ratio in the first three modes of vibration. A comparison of results for special cases of isotropy, uniform thickness and of axisymmetric vibrations obtained by other techniques such as the finite element method and the Receptance method shows excellent agreement.