Effects of Some Structural Parameters on the Vibration of a Simply Supported Non-prismatic Double-beam System

The aim of this work is to examine the influence of some structural parameters, namely, the mass per unit length and flexural rigidity of the upper beam on the natural frequencies of a symmetric non-prismatic double-beam system elastically connected by a Pasternak-type layer. A semianalytical technique known as differential transform method was used to carry out the analysis of the vibration problem in this paper. The results of the analysis revealed that there is tendency to lower the vibration frequency of the double-beam system by increasing the mass of the upper beam. It was also found that the natural frequencies of the double-system generally increase with an increase in the flexural rigidity of the upper beam of the double-beam system. It can be concluded that both the mass per unit length and the flexural rigidity of the upper beam generally have influence on the natural frequencies of a non-prismatic double-beam system elastically coupled by a Pasternak-type elastic mediu

Convection Flow of MHD Couple Stress Fluid in Vertical Microchannel with Entropy Generation

Entropy generation of fully developed steady, viscous, incompressible couple stress fluid in a vertical micro-porous-channel in the presence of horizontal magnetic field is analysed in this work. The governing equations for the flow are derived, and nondimensionalised and the resulting nonlinear ordinary differential equations are solved via a rapidly convergent technique developed by Zhou. The solution of the velocity and temperature profiles are utilised to obtain the flow irreversibility and Bejan number. The effects of couple stresses, fluid wall interaction parameter (FSIP), effective temperature ratio (ETR), rarefaction and magnetic parameter on the velocity profile, temperature profile, entropy generation and Bejan number are presented and discussed graphically

Vibration of an Elastically Connected Nonprismatic Double-beam System Using Differential Transform Method

In this paper, the free vibration characteristics of an elastically connected non-prismatic double-beam system based on Euler-Bernoulli beam theory are determined using differential transform method. The double-beam system is composed of two parallel non-uniform cantilever beams which are attached to each other by a Pasternak elastic medium. Numerical results of the method used are validated by comparing with the ones available in the published literature. The effect of the taper ratio on the natural frequency of the double-beam system is also studied

COMPUTING OSCILLATING VIBRATIONS EMPLOYING EXPONENTIALLY FITTED BLOCK MILNE’S DEVICE

Background and Objectives: The idea of estimating oscillating vibration problems via multinomial basis function haven been seen by some authors as a convenient approach but not appropriate. This is as result of the behavior of the problem and as such depends largely on the step size and frequency. This research article is geared towards computing oscillating vibrations employing exponentially fitted block Milne’s device (COVEFBMD). Materials and Methods: This is specifically designed using interpolation and collocation via exponentially fitted method as the approximate solution to generate COVEFBMD, thereby finding the tolerance level of the method. Results: Some numerical examples were selected and implemented on Mathematica kernel 9 to show speed, technicality and accuracy. Conclusion: The completed solutions show that COVEFBMD performs better than the existing methods because of its ability to design a worthy step size; decide the tolerance level resulting to maximized errors

Effect of Mass per Unit Length on freely vibrating Simply Supported Rayleigh Beam

In this paper, free vibration characteristics of a uniform Rayleigh beam are studied using the differential transform method. The procedure entails transforming the partial differential equation governing the motion of the beam under consideration and the associated boundary conditions. The transformation yields a set of difference equations. Some simple algebraic operations are performed on the resulting difference equations to determine any ith natural frequency and the closed-form series function for any ith mode shape. Finally, one problem is presented to illustrate the implementation of the present method and analyse the effect of mass per length on the natural frequencies of the beam

Variational Iteration Method for Natural Frequencies of a Cantilever Beam with Special Attention to the Higher Modes

In this work, the variational iteration method (VIM) is used to calculate the natural frequencies of a cantilever prismatic beam especially for the higher modes of vibration. The solutions yielded by VIM are validated by comparing with the natural frequencies of the said beam for lower modes earlier obtained using analytical method and the differential transform metho

Parallel Solver for Oscillatory Stiff Systems of ODEs

The aim of this study will be to design Parallel solver (PS) for oscillatory stiff systems of ordinary differential equations (ODEs). PS will be constructed via a type of specially transformed exponentially fitted multinomial approximant in accordance with the behaviour of the solution. The method of interpolation and collocation will be utilized. The principal local truncation errors of PS will be used to derive a suitable step size and decide the error tolerance criteria for establishing the convergence of PS. Some examples of stiff ODEs will be examined and compared with existing methods to show case the efficiency and accuracy of the scheme. Parallel solver will be seen as a unique model for solving stiff ODEs without dependent on absolute stability as required b

Free Vibration Analysis of Tapered Rayleigh Beams resting on Variable Two-Parameter Elastic Foundation

This study aims at analyzing the effect of variable foundation parameters on the natural frequencies of a prestressed tapered Rayleigh beam having general elastically restrained ends. In this work, the elastic coefficients of the foundations are assumed varying along the beam major axis. In particular, the constant, linear and parabolic variations of the Pasternak foundation are considered. A semi-analytical approach known as differential transform method (DTM) is applied to the non-dimensional form of the governing equations of motion of the prestressed tapered Rayleigh beam and a set of recurrence algebraic equations are determined. Performing some direct algebraic operations on these derived equations and using some computer codes developed and implemented in MAPLE 18, the dimensionless natural frequencies and the associated mode shapes of the beam are obtained, the effects of these Pasternak foundation variations for various values of the slenderness ratio on the natural frequencies are investigated. It is found among others that : (i) an increase in foundation stiffness led generally to an increase in the natural frequencies; (ii) the constant elastic variations of Pasternak foundation produced highest values of natural frequencies; and (iii) the natural frequencies of tapered Rayleigh beam resting on Pasternak foundation are higher than those from the same beam on Winkler foundation. Finally, the efficiency and accuracy of differential transform method are illustrated by solving two numerical examples of vibration problems and validating the results obtained with those in the open literature, and are found to compare favorably well

CLOSED FORM EXPRESSIONS OBTAINED FROM THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS OF THE PROBABILITY DENSITY FUNCTION OF THE BETA DISTRIBUTION

In this paper, some closed form expressions for selected parameters for the probability density function (PDF) of the beta distribution are obtained. The closed form expressions are recovered from the solution of the ordinary differential equations (ODEs), obtained from the differentiation of the PDF of the distribution. The paper shows that the shape of the distributions also determines the nature of the resulting ODE which has shown how distributions related to the beta distribution can be traced via the solutions of the ODEs. Numerical methods are unnecessary because the closed form expressions are the same with the values obtained from the standard statistical software

Influence of inclined magnetic field and chemical reaction on the entropy generation of Blasius and Sakiadis flows

This work considers the well-known laminar boundary layer flows; about a flat-plate in a uniform stream of fluid (Blasius flow) and about a moving plate in a quiescent am�bient fluid (Sakiadis flow) both under a convective surface boundary condition. Entropy generation due to the effect of angle of inclination, magnetic parameter, chemical reaction parameter and Schmidt number on the flows is investigated. The third order partial dif�ferential equations governing the flows are reduced to ordinary differential equations by suitable similarity variables. The obtained equations are tackled by the Runge-Kutta fourth order method with shooting technique and the results are employed to calculate entropy generation. The solution of Blasius flow is compared with the works in literature and are found to be in excellent agreement. Entropy generation can be minimized by increasing the magnetic parameter (M), chemical reaction parameter (R) and Schmidt number (Sc) for Blasius flow. Magnetic parameter reduces entropy generation for Sakiadis flow while other parameters such as angle of inclination, chemical reaction parameter and Schmidt number boost fluid irreversibility