Exact Solutions for Out-of-Plane Vibration of Curved Nonuniform Beams

2000 ◽  
Vol 68 (2) ◽  
pp. 186-191 ◽  
Author(s):  
S. Y. Lee ◽  
J. C. Chao
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Natural Frequencies ◽  
Variable Coefficients ◽  
Physical Parameters ◽  
Arc Length ◽  
Order Ordinary Differential Equation ◽  
Constant Radius ◽  
Out Of Plane ◽  
Center Angle

The governing differential equations for the out-of-plane vibrations of curved nonuniform beams of constant radius are derived. Two physical parameters are introduced to simplify the analysis, and the explicit relations between the torsional displacement, its derivative and the flexural displacement are derived. With these explicit relations, the two coupled governing characteristic differential equations can be decoupled and reduced to one sixth-order ordinary differential equation with variable coefficients in the out-of-plane flexural displacement. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the out-of-plane vibrations of the beam can be obtained. The derived explicit relations can also be used to reduce the difficulty in experimental measurement. Finally, two limiting cases are considered and the influence of taper ratio, center angle, and arc length on the first two natural frequencies of the beams are illustrated.

Free Vibration Analysis of an Inflated Toroidal Shell

2002 ◽  
Vol 124 (3) ◽  
pp. 387-396 ◽  
Author(s):  
Akhilesh K. Jha ◽  
Daniel J. Inman ◽  
Raymond H. Plaut
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Free Vibration Analysis ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Toroidal Shell

Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkin’s method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

scholarly journals Using the Differential Transform Method to Solve Non-Linear Partial Differential Equations

2020 ◽  
pp. 34-43
Author(s):  
Fadwa A. M. Madi ◽  
Fawzi Abdelwahid
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Variable Coefficients ◽  
Differential Transform Method ◽  
Partial Differential ◽  
Transform Method ◽  
Linear Partial Differential Equations ◽  
Differential Transform ◽  
Non Linear

In this work, we reviewed the two-dimensional differential transform, and introduced the differential transform method (DTM). As an application, we used this technique to find approximate and exact solutions of selected non-linear partial differential equations, with constant or variable coefficients and compared our results with the exact solutions. This shows that the introduced method is very effective, simple to apply to linear and nonlinear problems and it reduces the size of computational work comparing with other methods.

Vibrations of Suspended Cables

1983 ◽  
Vol 50 (3) ◽  
pp. 687-689
Author(s):  
J. G. Gale ◽  
C. E. Smith
Keyword(s):  
Natural Frequencies ◽  
Analytical Investigation ◽  
Plane Motion ◽  
Mode Shapes ◽  
Arc Length ◽  
Power Series Solutions ◽  
Out Of Plane ◽  
Independent Variable ◽  
Suspended Cables ◽  
Frequency Ratios

An analytical investigation of the small, normal-mode motions of a homogeneous, inextensible, perfectly flexible cable suspended in a gravitational field was made. With cable arc length as the independent variable, the differential equations that govern the mode shapes have irrational coefficients. A transformation of the independent position variable yields equations that have polynominal coefficients, which then lend themselves to power series solutions. Natural frequencies of oscillation and corresponding mode shapes are determined from these solutions. Figures showing the natural frequency ratios for a variety of cable support geometries are presented for both in-plane and out-of-plane motion.

scholarly journals Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

2020 ◽  
Vol 26 (1) ◽  
pp. 35-55
Author(s):  
Abdelkader Kehaili ◽  
Ali Hakem ◽  
Abdelkader Benali
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Variable Coefficients ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

scholarly journals Generalized multitime expansions for equations with slowly varying coefficients

1984 ◽  
Vol 7 (1) ◽  
pp. 151-158
Author(s):  
L. E. Levine ◽  
W. C. Obi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Time Scales ◽  
Constant Coefficient ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Varying Coefficients

The successive terms in a uniformly valid multitime expansion of the solutions of constant coefficient differential equations containing a small parameterϵmay be obtained without resorting to secularity conditions if the time scalesti=ϵit(i=0,1,…)are used. Similar results have been achieved in some cases for equations with variable coefficients by using nonlinear time scales generated from the equations themselves. This paper extends the latter approach to the general second order ordinary differential equation with slowly varying coefficients and examines the restrictions imposed by the method.

Natural Frequencies of the Bias Tire

1976 ◽  
Vol 4 (2) ◽  
pp. 86-114 ◽  
Author(s):  
M. Hirano ◽  
T. Akasaka
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Composite Structure ◽  
Natural Frequencies ◽  
Variable Coefficients ◽  
System Of Differential Equations ◽  
Inflation Pressure ◽  
Point Collocation ◽  
Membrane Shell ◽  
Wide Range

Abstract The lowest natural frequencies of a bias tire under inflation pressure are deduced by assuming the bias tire as a composite structure of a bias-laminated, toroidal membrane shell and rigorously taking three displacement components into consideration. The point collocation method is used to solve a derived system of differential equations with variable coefficients. It is found that the lowest natural frequencies calculated for two kinds of bias tire agree well with the corresponding experimental results in a wide range of inflation pressure.

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