Dynamic analysis of a rotating tapered cantilever Timoshenko beam based on the power series method

The mathematical modeling of a rotating tapered Timoshenko beam with preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle descriptions. The power series method is then used to inves- tigate the natural frequencies and the corresponding complex mode functions. It is found that all the natural frequencies are increased by the centrifugal stiffening except the twist frequency, which is slightly decreased. The tapering ratio increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency. Because of the pre-twist, all the directions are gyroscopically coupled with the phase differences among the six degrees.

Nonlinear Free Vibration of a Cantilever Beam Using the Power Series Method

An analytical approach based on the power series method is used to analyze the free vibration of a cantilever beam with geometric and inertia nonlinearities.The time variable is transformed into a“harmonically oscillating time”variable which transforms the governing equation into a form well-conditioned for a power series analysis.Rayleigh’s energy principle is also used to determine the vibration frequency.Convergence of the power series solution is demonstrated and excellent agreement is seen for the vibration response with a numerical solution.

A Series Solution Approach to the Circular Restricted Gravitational Three-Body Dynamical Problem

The present manuscript examines the circular restricted gravitational three-body problem (CRGTBP) by the introduction of a new approach through the power series method. In addition, certain computational algorithms with the aid of Mathematica software are specifically designed for the problem. The algorithms or rather mathematical modules are established to determine the velocity and position of the third body’s motion. In fact, the modules led to accurate results and thus proved the new approach to be efficient.

ALGEBRAIC OPERATION OF SPECIAL MATRICES RELATED TO METHOD OF LEAST SQUARES

The following situation in using the method of least squares to solve problems often occurs.After m experiments completed and a solution of least squares obtained,the ( m+1 ) th experiment is made further in order to improve the results.A method of algebraic operation of special matrices involved in the problem is given in this paper for obtaining a new solution for the m +1 experiments based upon the old solution for the primary m experiments. This method is valid for more general matrices.

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems.

New Analytical and Numerical Results For Fractional Bogoyavlensky-Konopelchenko Equation Arising in Fluid Dynamics

In this article,(2+1)-dimensional time fractional Bogoyavlensky-Konopelchenko(BK)equation is studied,which describes the interaction of wave propagating along the x axis and y axis.To acquire the exact solutions of BK equation we employed sub equation method that is predicated on Riccati equation,and for numerical solutions the residual power series method is implemented.Some graphical results that compares the numerical and analytical solutions are given for di erent values of.Also comparative table for the obtained solutions is presented.

ANALYSIS OF COUPLED-MODE FLUTTER OF PIPES CONVEYING FLUID ON THE ELASTIC FOUNDATION

The governing equation of solid-liquid couple vibration of pipe conveying fluid on the elastic foundation was derived. The critical velocity and complex frequency of pipe conveying fluid on Winkler elastic foundation and two-parameter foundation were calculated by po,ver series method. Compared,with pipe without considering elastic foundation, the numerical results show that elastic foundation can increase the critical flow velocity of static instability and dynamic instability of pipe. And the increase of foundation parameters may increase the critical flow velocity of static instability and dynamic instability of pipe, thereby delays the occurrence of divergence and flutter instability of pipe. For higher mass ratio beta, in the combination of certain foundation parameters, pipe behaves the phenomenon of restabilization and redivergence after the occurrence of static instability, and then coupled-mode flutter takes place.

A different approach for conformable fractional biochemical reaction–diffusion models

This paper attempts to shed light on three biochemical reaction-diffusion models:conformable fractional Brusselator,conformable fractional Schnakenberg,and conformable fractional Gray-Scott.This is done using conformable residual power series(hence-form,CRPS)technique which has indeed,proved to be a useful tool for generating the solution.Interestingly,CRPS is an effective method of solving nonlinear fractional differential equations with greater accuracy and ease.

STABILITY ANALYSIS OF VISCOELASTIC CURVED PIPES CONVEYING FLUID

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.

An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative

Mathematical simulation of nonlinear physical and abstract systems is a very vital process for predicting the solution behavior of fractional partial differential equations(FPDEs)corresponding to different applications in science and engineering. In this paper, an attractive reliable analytical technique, the conformable residual power series, is implemented for constructing approximate series solutions for a class of nonlinear coupled FPDEs arising in fluid mechanics and fluid flow, which are often designed to demonstrate the behavior of weakly nonlinear and long waves and describe the interaction of shallow water waves. In the proposed technique the n-truncated representation is substituted into the original system and it is assumed the(n-1) conformable derivative of the residuum is zero. This allows us to estimate coefficients of truncation and successively add the subordinate terms in the multiple fractional power series with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some real-world applications. Finally, highlights and some closing comments are attached.

Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves

In this article,two different methods,namely sub-equation method and residual power series method,have been used to obtain new exact and approximate solutions of the generalized Hirota-Satsuma system of equations,which is a coupled KdV model.The fractional derivative is taken in the conformable sense.Each of the obtained exact solutions were checked by substituting them into the corresponding system with the help of Maple symbolic computation package.The results indicate that both methods are easy to implement,effective and reliable.They are therefore ready to apply for various partial fractional differential equations.

Analytical and approximate solutions of(2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation

In this paper,we applied the sub-equation method to obtain a new exact solution set for the extended version of the time-fractional Kadomtsev-Petviashvili equation,namely Burgers-Kadomtsev-Petviashvili equation(Burgers-K-P)that arises in shallow water waves.Furthermore,using the residual power series method(RPSM),approximate solutions of the equation were obtained with the help of the Mathematica symbolic computation package.We also presented a few graphical illustrations for some surfaces.The fractional derivatives were considered in the conformable sense.All of the obtained solutions were replaced back in the governing equation to check and ensure the reliability of the method.The numerical outcomes confirmed that both methods are simple,robust and effective to achieve exact and approximate solutions of nonlinear fractional differential equations.

Significance of thermal stress in a convectiveradiative annular fin with magnetic field and heat generation: application of DTM and MRPSM

The present paper explains the temperature attribute of a convective-radiative rectangular profiled annular fin with the impact of magnetic field.The effect of thermal radiation,convection,and magnetic field on thermal stress distribution is also studied in this investigation.The governing energy equation representing the steady-state heat conduction,convection,and radiation process is transformed into its dimensionless nonlinear ordinary differential equation(ODE)with corresponding boundary conditions using non-dimensional terms.The obtained ODE is then solved analytically by employing the Pade approximant-differential transform method(DTM)and modified residual power series method(MRPSM).Moreover,the important characteristics of the temperature field,the thermal stress,and the impact of some nondimensional parameters are inspected graphically,and a physical explanation is provided to aid in comprehension.The significant findings of the investigation reveal that temperature distribution enhances with an increase in the magnitude of the heat generation parameter and thermal conductivity parameter,but it gradually decreases with an increment of convectiveconductive parameter,Hartmann number,and radiative-conductive parameter.The thermal stress distribution of the fin varies considerably in the applied magnetic field effect.