An improved finite difference analysis of uncoupled vibrations of cantilevered beams

Subrahmanyam, K.B.; Leissa, A. W.

doi:10.1016/0022-460x(85)90397-9

Cited by 16 publications

(9 citation statements)

References 7 publications

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“…The aim of the characterization of a numerical scheme is to understand how the discrete problem is able to approximate the continuous one. This alternative perspective is the numerical evaluation of approximate methods such as the Finite Difference Method or the Finite Element Method for computing the solutions of continuous problems (see for instance Greenwood, 1961;Zienkiewicz and Cheung, 1967;Cyrus and Fulton, 1968;Walz et al, 1968;Strang and Fix, 1973;Gawain and Ball, 1978;Subrahmanyam and Kaza, 1983).…”

Section: Introductionmentioning

confidence: 99%

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Challamel

Picandet

Collet

et al. 2015

European Journal of Mechanics - A/Solids

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In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pad e approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

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Section: Introductionmentioning

confidence: 99%

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Challamel

Picandet

Collet

et al. 2015

European Journal of Mechanics - A/Solids

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show abstract

“…We apply the methodology proposed in papers by Ansari et al [47,50] and Subrahmanyam and Leissa [48] by considering the fourth-order central difference approximations for the second and fourth order derivatives of ̅ ( ) in the following form By introducing the approximated relation for boundary conditions (41) into the system of equations (39), one obtains the system of algebraic equations that can be used to find the solution of the eigenvalue problem for MNBS: Eigenvalues can be obtained as non-trivial solutions of the homogeneous system of algebraic equations (42) when the determinant of the system is equal to zero…”

Section: Solution By the Finite Difference Methodsmentioning

confidence: 99%

Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium

Karličić

Kozić

Pavlović

2016

Applied Mathematical Modelling

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Highlights Multiple-nanobeam system is analyzed by Eringen nonlocal theory.  The nonlocal Euler -Bernoulli beam theory is proposed in this study.  The MNBS is simply supported and under the influence of axial load.  The natural frequencies and the critical buckling load are obtained. Abstract:The nonlocal Euler -Bernoulli beam theory is proposed for the free vibration and stability analyses of a multiple-nanobeam system (MNBS) using the Eringen nonlocal continuum theory. It is assumed that every nanobeam in the system of multiple-nanobeams is simply supported and under the influence of axial load. The MNBS is embedded in the Winkler elastic medium. The motion of the system is described by a set of m homogeneous partial differential equations, which are derived by using D'Alembert's principle. Analytical solutions of free vibrations and buckling for a multiple-nanobeam system are obtained by using the classical Bernoulli-Fourier method and trigonometric method, for the case of the identical nanobeam. To validate the accuracy of application of trigonometric methods for a determined natural frequency and buckling load of MNBS, numerical solution of the characteristic polynomial are also conducted for a different number of nanobeams. Moreover, the finite difference method is employed to solve the system of partial differential equations of motion of MNBS, where good agreement with the analytical results is achieved. Numerical studies show the effect of the nonlocal parameter, stiffness of Winkler elastic medium and number of nanobeams for the free transversal vibration and buckling of MNBS. The presented analyses are very useful for the study and design of the nanoelectromechanical systems such as nano-resonators.

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“…An improved¯nite di®erence analysis may be based on the introduction of the second-order central di®erence for the expressions of the¯rst and the second derivatives of the displacement (see Refs. 16,[28][29][30]. The higher-order¯nite di®erence formulation of the constitutive equation [Eq.…”

Section: Higher-order Finite Di®erence Methodsmentioning

confidence: 99%

“…The consideration of harmonic motion and in view of Eqs. (29) and (30) lead to the eight-order¯nite di®erence equation:…”

Section: Higher-order Finite Di®erence Methodsmentioning

confidence: 99%

See 1 more Smart Citation

On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods

Challamel

Picandet

Elishakoff

et al. 2015

Int. J. Str. Stab. Dyn.

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International audienceIn this paper, we show that two numerical methods, specifically the finite difference method and the finite element method applied to continuous beam dynamics problems, can be asymptotically investigated by some kind of enriched continuum approach (gradient elasticity or nonlocal elasticity). The analysis is restricted to the vibrations of elastic beams, and more specifically the computation of the natural frequencies for each numerical method. The analogy between the finite numerical approaches and the equivalent enriched continuum is demonstrated, using a continualization procedure, which converts the discrete numerical problem into a continuous one. It is shown that the finite element problem can be transformed into a system of finite difference equations. The convergence rate of the finite numerical approaches is quantified by an equivalent Rayleigh's quotient. We also present some exact analytical solutions for a first-order finite difference method, a higher-order finite difference method or a cubic Hermitian finite element, valid for arbitrary number of nodes or segments. The comparison between the exact numerical solution and the approximated nonlocal approaches shows the efficiency of the continualization methodology. These analogies between enriched continuum and finite numerical schemes provide a new attractive framework for potential applications of enriched continua in computational mechanics.Read More: http://www.worldscientific.com/doi/abs/10.1142/S021945541540008

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An improved finite difference analysis of uncoupled vibrations of cantilevered beams

Cited by 16 publications

References 7 publications

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium

On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods

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