An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates

A three-dimensional linear, small deformation theory of elasticity solution by the direct method is developed for the free vibration of simply-supported, homogeneous, isotropic, thick rectangular plates. The solution is exact and involves determining a triply infinite sequence of eigenvalues from a doubly infinite set of closed form transcendental equations. As no restrictions are placed on the thickness variation of stresses or displacements, this formulation yields a triply infinite spectrum of frequencies, instead of only one doubly infinite spectrum by thin plate theory and three doubly infinite spectra by Mindlin's thick plate theory. Further, the present analysis yields symmetric thickness modes which neither of the approximate theories can identify. Some numerical results from the two approximate theories are compared with those from the present solution and some important conclusions regarding the effect of the assumptions made in the approximate theories are drawn. The thickness variations of stresses and displacements are also discussed. The analysis is readily extended for laminated plates of isotropic materials. Numerical results are also given for three-ply laminates, and are used to assess the accuracy of thin plate theory predictions for laminates. Extension to general lateral surface conditions and forced vibrations is indicated.

Harmonic and anharmonic oscillations of a cylindrical plasma in the presence of a uniform axial magnetic field and a steady axial current

In the present note we have studied the harmonic and anharmonic oscillations of cylindrical plasma using Lagrangian formalism. In order to study the harmonic oscillations, the equations are linearized and the resulting equation for the displacement has been numerically solved. For situations present in thermonuclear reactors, the presence of axial magnetic field is found necessary to make the periods of oscillation to become comparable with the time required for the thermonuclear reactions to set in. A detailed analysis of the anharmonic oscillations reveals that the significant interaction is between the first and the second mode. The fundamental period of anharmonic oscillation is more than the corresponding period of harmonic oscillations by 9·2%. Graphs have been drawn for the amplitudes of relative variations in density and magnetic field and of the time-varying part of anharmonic oscillation.

Vibration of simply-supported trapezoidal plates I. Symmetric trapezoids

A detailed investigation of the natural frequencies and mode shapes of simply supported symmetric trapezoidal plates is undertaken in this paper. For numerical calculations, the relationship that exists between the eigenvalue problem of a polygonal simply supported plate and the eigenvalue problem of polygonal membrane of the same shape is utilized with advantage. The deflection surface is expressed in terms of a Fourier sine series in transformed coordinates and the Galerkin method is used. Results are presented in the form of tables and graphs. Several features like the crossing of frequency curves and the metamorphosis of some of the nodal patterns are observed. By a suitable interpretation of the modes of those symmetric trapezoidal plates which have the median as the nodal line, the results for some of the modes of unsymmetrical trapezoidal plates are also deduced.

Effect of vibration on heat transfer from a horizontal cylinder to a normal air stream

The effect of vibration on heat transfer from a horizontal copper cylinder, 0.344 in. in diameter and 6 in. long, was investigated. The cylinder was placed normal to an air stream and was sinusoidally vibrated in a direction perpendicular to the direction of the air stream. The flow velocity varied from 19 ft/s to 92 ft/s; the double amplitude of vibration from 0.75 cm to 3.2 cm, and the frequency of vibration from 200 to 2800 cycles/min. A transient technique was used to determine the heat transfer coefficients. The experimental data in the absence of vibration is expressed by NNu = 0.226 NRe0.6 in the range 2500 < NRe < 15 000. By imposing vibrational velocities as high as 20 per cent of the flow velocity, no appreciable change in the heat transfer coefficient was observed. An analysis using the resultant of the vibration and the flow velocity explains the observed phenomenon.

Vibration of simply-supported trapezoidal plates II. Unsymmetric trapezoids

This paper deals with the investigation of the vibration characteristics of simply-supported unsymmetric trapezoidal plates. For numerical calculations, the relationship between the eigenvalue problems of a polygonal simply-supported plate and polygonal membrane is again effectively utilized. The Galerkin method is applied, with the deflection surface expressed in terms of a Fourier sine series in transformed coordinates. Numerical values for the first seven to eight frequencies for different geometries of the unsymmetric trapezoid are presented in the form of tables. Also the nodal patterns for a few representative configurations are presented.

First passage probability during random vibration

The probability that a random process crosses an arbitrary level for the first time is expressed as a Gram—Charlier series, the leading term of which is the Poisson approximation. The coefficients of this series are related to the moments of the number of level crossings. The results are applicable to both stationary and non-stationary processes. Some numerical results are presented for the response process of a linear single-degree-of-freedom oscillator under Gaussian white noise excitation.

Nonlinear vibration analysis of composite laminated and sandwich plates with random material properties

Nonlinear vibration analysis is performed using a C-0 assumed strain interpolated finite element plate model based on Reddy's third order theory. An earlier model is modified to include the effect of transverse shear variation along the plate thickness and Von-Karman nonlinear strain terms. Monte Carlo Simulation with Latin Hypercube Sampling technique is used to obtain the variance of linear and nonlinear natural frequencies of the plate due to randomness in its material properties. Numerical results are obtained for composite plates with different aspect ratio, stacking sequence and oscillation amplitude ratio. The numerical results are validated with the available literature. It is found that the nonlinear frequencies show increasing non-Gaussian probability density function with increasing amplitude of vibration and show dual peaks at high amplitude ratios. This chaotic nature of the dispersion of nonlinear eigenvalues is also r

On the use of a coordinate transformation for analysis of axisymmetric vibration of polar orthotropic annular plates

Estimates of natural frequencies corresponding to axisymmetric modes of flexural vibration of polar orthotropic annular plates have been obtained for various combinations of clamped, simply supported and free edge conditions. A coordinate transformation in the radial direction has been used to obtain effective solutions by the classical Rayleigh-Ritz method. The analysis with this transformation has been found to be advantageous in computations, particularly for large hole sizes, over direct analysis. Numerical results have been obtained for various values of hole sizes and rigidity ratio. The eigenvalue parameter has been found to vary more or less linearly with the rigidity ratio. A comparison with the results for isotropic plates has brought out some interesting features.

Active vibration suppression of beams with discrete actuators using optimal dynamic inversion

Most of the structural elements like beams, cables etc. are flexible and should be modeled as distributed parameter systems (DPS) to represent the reality better. For large structures, the usual approach of 'modal representation' is not an accurate representation. Moreover, for excessive vibrations (possibly due to strong wind, earthquake etc.), external power source (controller) is needed to suppress it, as the natural damping of these structures is usually small. In this paper, we propose to use a recently developed optinial dynamic inversion technique to design a set of discrete controllers for this purpose. We assume that the control force to the structure is applied through finite number of actuators, which are located at predefined locations in the spatial domain. The method used in this paper determines control forces directly from the partial differential equation (PDE) model of the system. The formulation has better practical significance, both because it leads to a closed form solution of the controller (hence avoids computational issues) as well as because a set of discrete actuators along the spatial domain can be implemented with relative ease (as compared to a continuous actuator).

Mixed-mode stress intensity factors for arbitrarily oriented cracks in cylindrical shells under axial pull and torsion

Results of photoelastic investigations conducted on cylindrical tubes (made of Araldite material) containing cracks oriented at 0°, 30°, 45°, 60° and 90° to the axis of the tube and subjected to axial and torsional loads are reported. The stress-intensity factors (SIFs) were determined by analysing the crack-tip stress fields. Smith and Smith's method [Engng Fracture Mech.4, 357–366 (1972)] and a new method developed by the authors by modifying Rakesh et al.'s method [Proc. 26th Congress of ISTAM, India (1981)] were employed to evaluate the mixed-mode SIFs.

Isoscalar axial-vector renormalization constant and polarized proton structure function

The isoscalar axial-vector renormalization constant is reevaluated using the QCD sum-rule method. It is found to be substantially different from the anomaly-free octet axial-vector u¯γμγ5+d¯γμγ5-2s¯γμγ5 coupling. Combining this determination with the known values of the isovector coupling GA and the F/D ratio for the octet current, we find the integral of the polarized proton structure function to be Gp=Fgp1(x)dx=0.135, in agreement with recent measurement by the European Muon Collaboration.