Vibration of prismatic beam on an elastic foundation by the differential quadrature element method

Chen, Chang-New

doi:10.1016/s0045-7949(99)00216-3

Cited by 48 publications

(34 citation statements)

References 5 publications

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“…It is observed from table 3 that the present results from DTM are more accurate than DQEM and RK4, and very close to analytical values. In the third natural frequency, DTM results are equal to the analytical value, while DQEM [6] and RK4 show small deviations. In table 4, the results are shown for the cantilever and fixed-fixed case, respectively.…”

Section: Resultsmentioning

confidence: 67%

“…In the first step, results are compared with DQEM [6] , 4 th order Runge-Kutta and analytical solutions. Table 3 illustrates the results for a simply supported uniform beam resting on a Winkler foundation.…”

Section: Resultsmentioning

confidence: 99%

“…Valsangkar and Pradhanang [2] and Franciosi and Masi [3] investigated Euler-Bernoulli beam on two-parameter model soil foundation, while De Rosa [4] and Ruta [5] did the same for Timoshenko beams. Chen employed the differential quadrature element method (DQEM), to investigate the vibration of beams on Winkler [6] and Pasternak [7] foundations. Hosing et al [8] have solved natural flexural vibrations of a continuous beam on discrete elastic support .…”

Section: Introductionmentioning

confidence: 99%

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Vibration analysis of a tapered beam with exponentially varying thickness resting on Winkler foundation using the differential transform method

Boreyri

Mohtat

Ketabdari

et al. 2014

International Journal of Physical Research

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In this paper, free vibration of a new type of tapered beam, with exponentially varying thickness, resting on a linear foundation is analyzed. The solution is based on a semi-analytical technique, the differential transform method (DTM). Applying DTM, nonlinear partial differential equations of the varying thickness beam are transformed into algebraic equations, which are then solved to obtain the solution. An Euler-Bernoulli beam with a number of boundary conditions and different exponential factor is taken into account. Results have been compared to the 4th order Runge-Kutta, and where possible with DQEM and analytical solution. These comparisons prove the preciseness of this method, based on which DTM can be considered as a powerful framework for eigenvalue analysis of new type of tapered beams.

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Section: Resultsmentioning

confidence: 67%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Vibration analysis of a tapered beam with exponentially varying thickness resting on Winkler foundation using the differential transform method

Boreyri

Mohtat

Ketabdari

et al. 2014

International Journal of Physical Research

View full text Add to dashboard Cite

show abstract

“…Due to the difficulty of mathematical nature of the problem, a few analytical solutions limited to special cases for vibrations of non-uniform beams resting on non-linear elastic foundations are found. Many methods are used to obtain the vibration behavior of different types of linear or nonlinear beams resting on linear or nonlinear foundations such as finite element method [1][2][3], transfer matrix method [4], Rayleigh-Ritz method [5], differential quadrature element method (DQEM) [6][7][8][9][10], Galerkin procedure [11,12] and [13][14][15]. There are various types of foundation models such as Winkler, Pasternak, Vlasov, etc.…”

Section: Introductionmentioning

confidence: 99%

Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)

Abumandour¹

2017

IJMEA

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Abstract:The natural frequencies of non-uniform beams resting on two layer elastic foundations are numerically obtained using the Generalized Differential Quadrature (GDQ) method. The Differential Quadrature (DQ) method is a numerical approach effective for solving partial differential equations. A new combination of GDQM and Newton's method is introduced to obtain the approximate solution of the governing differential equation. The GDQ procedure was used to convert the partial differential equations of non-uniform beam vibration problems into a discrete eigenvalues problem. We consider a homogeneous isotropic beam with various end conditions. The beam density, the beam inertia, the beam length, the linear (k 1 ) and nonlinear (k 2 ) Winkler (normal) parameters and the linear (k 3 ) Pasternak (shear) foundation parameter are considered as parameters. The results for various types of boundary conditions were compared with the results obtained by exact solution in case of uniform beam supported on elastic support.

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“…This is due to the intractable mathematical nature of the problem. Numerical methods such as finite element method [1][2], transfer matrix method [3], differential quadrature element method (DQM) [4][5][6], perturbation techniques [7][8] are used to obtain the vibration behavior of different types of linear or nonlinear beams resting on linear or nonlinear foundations.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Vibration Model for Initially Stressed Beam-Foundation System

Taha¹

2012

TOAMJ

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An analytical solution for nonlinear vibration of an initially stressed beam with elastic end restraints resting on a nonlinear elastic foundation is obtained. As a first step in solving nonlinear vibration equation, the linear vibration mode functions for a beam with elastic end restraints resting on a linear elastic foundation are obtained. Then, the nonlinear vibration equation is solved by employing the linear mode functions to obtain frequency equation and nonlinear response using Jacobi elliptic integral. The nonlinearity due to lateral vibrations, the nonlinearity of foundations and lateral displacement due to lateral elastic restraints at beam ends not included in previous analytical work are considered in the present work. The effects of spring stiffness at the beam ends, foundation stiffness, axial load and vibration amplitude on the frequency parameter are studied. The present solution can be used to measure the accuracy of approximate methods.

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Vibration of prismatic beam on an elastic foundation by the differential quadrature element method

Cited by 48 publications

References 5 publications

Vibration analysis of a tapered beam with exponentially varying thickness resting on Winkler foundation using the differential transform method

Vibration analysis of a tapered beam with exponentially varying thickness resting on Winkler foundation using the differential transform method

Vibration Analysis of Non-uniform Beams Resting on Two Layer Elastic Foundations Under Axial and Transverse Load Using (GDQM)

Nonlinear Vibration Model for Initially Stressed Beam-Foundation System

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