Flexural vibrations and stability of beams with variable parameters

Jaroszewicz, J.; Zoryj, Longin

doi:10.1007/bf00847086

Cited by 5 publications

(9 citation statements)

References 1 publication

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“…where P denotes the pinned (or simply supported) column and C means clamped end. Note that the buckling modes are written as polynomials while homogeneous columns necessitate transcended functions (Jaroszewicz and Zoryj, 1994). In Equations (3), the mode shapes are independent of the concentrated load.…”

Section: Introductionmentioning

confidence: 99%

Extension of Euler’s Problem to Axially Graded Columns: Two Hundred and Sixty Years Later

Elishakoff

Endres²

2005

Journal of Intelligent Material Systems and Structures

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A closed-form solution is obtained for the mode shape and the buckling load of an axially graded column that is clamped at one end and subjected to a concentrated load at the other. A semi-inverse method is employed to obtain the spatial distribution of the elastic modulus variation. A remarkable conclusion is reached on the existence of three columns with different elastic modulus variations, with the same analytical expression of the buckling load. The mode shape of one of the columns is represented by a rational expression while the other two involve irrational numbers.

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Section: Introductionmentioning

confidence: 99%

Extension of Euler’s Problem to Axially Graded Columns: Two Hundred and Sixty Years Later

Elishakoff

Endres²

2005

Journal of Intelligent Material Systems and Structures

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“…Such problems cannot be solved exactly for general function of variable cross section but in special cases, only when the equation is reduced to Euler's equation, special Bessel functions can be used to find the solution (ZORYJ 1982). The approach proposed by the author of this paper to apply the characteristic series method to the analysis of multi-parameter continuous systems seems warranted (JAROSZEWICZ, ZORYJ 1985, 1994. The literature reports analyses of this issue carried out using numerical and analytical methods including the MES, transfer matrix method and approximate methods based on energy principle such as those of Rayleigh-Ritz, Galerkin--Bubnow and Treffz (SOLECKI, SZYMKIEWICZ 1964).…”

Section: Introductionmentioning

confidence: 99%

The effect of influence of conservative and tangential axial forces on transverse vibrations of tapered vertical columns

Jaroszewicz

Radziszewski

Dragun

2017

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The Cauchy function and characteristic series were applied to solve the boundary value problem of free transverse vibrations of vertically mounted, elastically supported tapered cantilever columns. The columns can be subjected to universal axial point loads which considerate – conservative and follower /tangential/ forces, and to distributed loads along the cantilever length. The general form of characteristic equation was obtained taking into account the shape of tapered cantilever for attached and elastically secured. Bernstein-Kieropian double and higher estimators of natural frequency and critical loads were calculated based on the first few coefficients of the characteristic series. Good agreement was obtained between the calculated natural frequency and the exact values available in the literature.

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“…In their analyses, the authors used the characteristic series method and introduced formulas for subsequent series coefficients using the influence function or the Cauchy function. To calculate eigen frequency and critical forces, they used Bernstein-Kieropian double estimators, which helped find functional relationships between these values and the mass-elastic properties of the cantilever [4]. The influence function method in the analysis of the bending curve and relations of elastic supports of the beam with variable parameters was presented in [5].…”

Section: Introductionmentioning

confidence: 99%

“…Such problems cannot be solved exactly but in special cases, when the equation is reduced to Euler's equation, special Bessel functions can be used to find the solution [6]. The approach proposed by the authors of this paper to apply the characteristic series method to the analysis of multi-parameter continuous systems seems warranted [7,8]. The literature reports analyses of this issue carried out using numerical and analytical methods including the MES, transfer matrix method and approximate methods based on energy principle such as those of Rayleigh-Ritz, Galerkin-Bubnow and Treffz [9].…”

Section: Introductionmentioning

confidence: 99%

The Study of the Effect of Static Axial Loads on Vertically-Mounted Tapered Cantilever Transverse Vibrations Using the Cauchy Function

Jaroszewicz

Radziszewski

Dragun

2015

AMM

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The Cauchy function and characteristic series were applied to solve the boundary value problem of free transverse vibrations of vertically mounted, elastically supported tapered cantilever beams. A concentrated mass was attached at the same distance from the base. The beams were subjected to universal axial loads - conservative and follow wing tangential forces - and distributed loads along the cantilever length. The general form of characteristic equation was obtained taking into account the shape of the tapered cantilever, elastic foundation and nonhomogeneous material. Bernstein-Kieropian double estimators of natural frequency were calculated based on free coefficients of the characteristic series. Good agreement was obtained between the calculated natural frequency results and the exact values available in the literature.

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Flexural vibrations and stability of beams with variable parameters

Cited by 5 publications

References 1 publication

Extension of Euler’s Problem to Axially Graded Columns: Two Hundred and Sixty Years Later

Extension of Euler’s Problem to Axially Graded Columns: Two Hundred and Sixty Years Later

The effect of influence of conservative and tangential axial forces on transverse vibrations of tapered vertical columns

The Study of the Effect of Static Axial Loads on Vertically-Mounted Tapered Cantilever Transverse Vibrations Using the Cauchy Function

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