Nonlinear vibrations of beams with various boundary conditions.

Evensen, D. A.

doi:10.2514/3.4506

Cited by 123 publications

(31 citation statements)

References 5 publications

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“…This is to be expected as the nonlinearity included here possesses hard spring characteristics as it provides additional constraints to the system. It is also interesting to note that the results here for X = 0 agree exactly with those of Evensen [14] using the perturbation method.…”

Section: Introductionsupporting

confidence: 83%

Ultraspherical polynomials applied to nonlinear vibrations of continuous media

Blotter¹,

Yen²

1976

Quart. Appl. Math.

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Introduction.In several recent papers [1-5] a method for treating discrete nonlinear vibration problems has been developed. By linearizing the nonlinear forces using a set of ultraspherical polynomials over the interval in which the motion takes place, approximate nonlinear amplitude-frequency relations with fair accuracy are obtained from the solutions of the resulting linear systems. Some more general linearizations have also been reported [6,7]. The method is restricted essentially to systems which have one degree of freedom or may be characterized by a single amplitude [8].In this note, the method of ultraspherical polynomials is extended to systems governed by nonlinear partial differential equations to obtain approximate nonlinear amplitude-frequency relations in the neighborhood of the linear eigenvibrations. It is assumed that the systems depend on one space variable and time, and the nonlinear terms in the equations depend on the displacement and its spatial derivatives but not on time explicitly. An obvious difficulty immediately arises when one attempts to follow the method developed for discrete systems because the amplitude of motion is a function of the space variable and not known in advance. To overcome this difficulty some appropriate "mode of deflection" must be assumed. In cases where a linear mode is known, it is taken to be the mode of deflection; otherwise some suitable approximation to the

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Section: Introductionsupporting

confidence: 83%

Ultraspherical polynomials applied to nonlinear vibrations of continuous media

Blotter¹,

Yen²

1976

Quart. Appl. Math.

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“…The buckling of such system appears in the perpendicular direction to the axis of the column and smaller moment of inertia. The theoretical and experimental studies were performed by many scientists: Andersen and Thomson [1], Beck [5], Evensen [13], Przybylski [20,21] and Tomski [27]. In the seventies of the previous century Roodr and Chilver [24] have proposed new method of solution of the boundary problems corresponding to slender systems.…”

Section: Fig 1 Regions Of Local and Global Instabilitymentioning

confidence: 99%

Instability and vibration of multi-member columns subjected to Euler’s load

Sokół

Uzny

2015

Arch Appl Mech

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In this paper the instability and vibration phenomena of columns composed of three elements with different bending and compression stiffnesses are presented. In the internal rod the crack is taken into consideration and is being modeled by means of a rotational spring with linear characteristic. The boundary problem has been formulated on the basis of the Hamilton's principle. The proposed general form of problem formulation allows one to create five different systems on the basis of one mathematical model. The final formulation of the boundary problem has been carried out by means of a small parameter method. The monitoring of the columns is done by the analysis of the characteristic curves and shape modes. The obtained results are compared to the ones calculated for the uncracked columns. The results of numerical calculation are concern on vibration frequency, bifurcation load magnitude and bifurcation load-crack size relationship.

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“…The authors eliminated the shear locking by a selective reduced integrations procedure. The studies of beams with immovable supports weere carried out by Woinowsky-Krieger (1950), Hsu (1960), Evensen (1968) and Lewandowski (1987). The authors used a continuum approach as well as the finite element approach proposed by Levinson and Cooke (1982), Sarma and Varadan (1983) and Kitipornchai et al (2009).…”

Section: Introductionmentioning

confidence: 99%

“…The exact solutions given in form of elliptic functions were presented by Woinowsky--Krieger (1950) and Hsu (1960). Evensen (1968) analysed beams with various boundary conditions using the perturbation method. Lewandowski (1987) applied the Ritz method to obtain the frequency-amplitude relationship for elastically supported beams.…”

Section: Introductionmentioning

confidence: 99%

Non-linear vibration of Timoshenko beams by finite element method

Rakowski

Guminiak

2015

jtam

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The paper is concerned with free vibrations of geometrically non-linear elastic Timoshenko beams with immovable supports. The equations of motion are derived by applying the Hamilton principle. The approximate solutions are based on the negligence of longitudinal inertia forces but inclusion of longitudinal deformations. The Ritz method is used to determine non-linear modes and the associated non-linear natural frequencies depending on the vibration amplitude. The beam is discretized into linear elements with independent displacement fields. Consideration of the beams divided into the regular mesh enables one to express the equilibrium conditions for an arbitrary large number of elements in form of one difference equation. Owing to this, it is possible to obtain an analytical solution of the dynamic problem although it has been formulated by the finite element method. Some numerical results are given to show the effects of vibration amplitude, shear deformation, thickness ratio, rotary inertia, mass distribution and boundary conditions on the non-linear natural frequencies of discrete Timoshenko beams.

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Nonlinear vibrations of beams with various boundary conditions.

Cited by 123 publications

References 5 publications

Ultraspherical polynomials applied to nonlinear vibrations of continuous media

Ultraspherical polynomials applied to nonlinear vibrations of continuous media

Instability and vibration of multi-member columns subjected to Euler’s load

Non-linear vibration of Timoshenko beams by finite element method

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