Static and Dynamic Response of a Beam on a Winkler Elastic Foundation

2005 ◽  
Author(s):  
Timour M. A. Nusirat ◽  
M. N. Hamdan
Keyword(s):  
Elastic Foundation ◽  
Governing Equation ◽  
Initial Value Problem ◽  
Nonlinear Partial Differential Equation ◽  
Static Case ◽  
Initial Value ◽  
Characteristic Curves ◽  
System Parameters ◽  
Winkler Elastic Foundation ◽  
Dynamic Case

This paper is concerned with analysis of dynamic behavior of an Euler-Bernoulli beam resting on an elastic foundation. The beam is assumed to be subjected to a uniformly distributed lateral static load, have an initial quarter-sine shape deflection. At one end, the beam is assumed to be restrained by a pin, while at the other end, the beam is assumed to be restrained by a torsional and a translational linear spring. The beam is modeled by a nonlinear partial differential equation where the nonlinearity enters the governing equation through the beam axial force. In the static case, because of a unique feature of governing equation, the analysis was carried out using the theory of linear differential equations, but takes into account the effect of actual deflection on the induced axial thrust. In the dynamic case, stability analysis of the beam is carried out by calculating the nonlinear frequencies of free vibration of the beam about its static equilibrium configuration. The assumed mode method is used to discretize and find an equivalent nonlinear initial value problem. Then the harmonic balance is used to obtain an approximate solution to the nonlinear oscillator described by the equivalent initial value problem. The analyses of results were carried out for a selected range of values of the system parameters: foundation elastic stiffness, lateral load, and maximum beam edge deflection. In the static case the results are presented as characteristic curves showing the variation of the beam static deflection and associated bending moment distribution with each of the above system parameters. In the dynamic case, the presented characteristic curves show the variation of the nonlinear natural frequency corresponding to the first and the second modes over a range of each of the above system parameters.

Computational Modeling of Micro-Fluid Flow in a Tubular Microchannel

2006 ◽  
Vol 505-507 ◽  
pp. 433-438 ◽  
Author(s):  
Chao Ming Lin ◽  
Te Hua Fang ◽  
Win Jin Chang
Keyword(s):  
Fluid Flow ◽  
Cross Section ◽  
Cross Sections ◽  
Governing Equation ◽  
Initial Value Problem ◽  
Computational Models ◽  
Flow Behavior ◽  
Viscous Force ◽  
Initial Value ◽  
The Cross

This paper develops models for micro-fluid flow in tubular microchannels with uniform or varying cross-sections. The fluid flow is governed by the pressure drop, the capillary force, and the viscous force. The governing equation developed for the flow incorporates two geometry parameters, namely the cross-section perimeter, S, and the cross-section area, A. In a microchannel with a uniform cross-section, the governing equation reduces to a linear initial value problem which can be solved analytically. However, in the case of a varying cross-section, the two geometry parameters vary as a function of the flow front position, L, and the nonlinear initial value problem must be solved numerically. The computational models developed in this study will be able to provide accurate predictions of the micro-fluid flow behavior in microchannels with uniform or varying cross-sections.

scholarly journals A characteristic initial value problem for a strictly hyperbolic system

2004 ◽  
Vol 2004 (31) ◽  
pp. 1623-1632
Author(s):  
Nezam Iraniparast
Keyword(s):  
Continuous Function ◽  
Hyperbolic System ◽  
Initial Value Problem ◽  
Real Constant ◽  
Initial Value ◽  
Characteristic Curves ◽  
Characteristic Initial Value Problem

Consider the systemAutt+Cuxx=f(x,t),(x,t)∈Tforu(x,t)inℝ2, whereAandCare real constant2×2matrices, andfis a continuous function inℝ2. We assume thatdetC≠0and that the system is strictly hyperbolic in the sense that there are four distinct characteristic curvesΓi,i=1,…,4, inxt-plane whose gradients(ξ1i,ξ2i)satisfydet[Aξ1i2+Cξ1i2]=0. We allow the characteristics of the system to be given bydt/dx=±1anddt/dx=±r,r∈(0,1). Under special conditions on the boundaries of the regionT={(x,t)≤t≤1,(−1+r+t)/r≤x≤(1+r−t)/r}, we will show that the system has a uniqueC2solution inT.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

scholarly journals A Neural Network Technique for the Derivation of Runge-Kutta Pairs Adjusted for Scalar Autonomous Problems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1842
Author(s):  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Yuri A. Khakhalev ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Neural Network ◽  
Differential Evolution ◽  
Initial Value Problem ◽  
Fitness Function ◽  
Initial Value ◽  
Runge Kutta

We consider the scalar autonomous initial value problem as solved by an explicit Runge-Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge-Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

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