A finite element formulation for the large amplitude vibrations of tapered beams

Raju, K. Kanaka; Shastry, B.P.; Rao, G. Venkateswara

doi:10.1016/0022-460x(76)90887-7

Cited by 27 publications

(9 citation statements)

References 4 publications

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“…e validity of the results obtained by the present approach in the linear case has been first established by comparisons with those previously obtained in [34], summarized in Tables 1-4. A second comparison was made in the nonlinear case with results reported in [19,20].…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…e present linear results are compared in Tables 1-4 with those of [34], showing an excellent agreement since the relative difference does not exceed 0.008% for simply supported beams and 0.05% for clamped beams. In the nonlinear case, the comparisons performed, and summarized in Table 5, between the results obtained here by the second formulation and those of [19,20], correspond to tapered beams clamped at both ends and denoted by (C-C) and to beams simply supported at both ends and denoted by (SS-SS).…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…Equations ( 21) and ( 22) present the mathematical development derived, respectively, from equations ( 17) and (18), while equations ( 23) and ( 24) are derived from equations ( 19) and (20), respectively. e satisfaction of the end conditions was achieved for the two types of tapered beam considered.…”

Section: E Compatibility Conditionsmentioning

confidence: 99%

“…Raju et al [19] investigated the nonlinear behaviour of two types of Euler Bernoulli tapered beams with either a linearly varying width or depth by Galerkin's method. e same beams have been investigated in [20] by Kanaka et al, who adopted Timoshenko beams theory and used the differential quadrature method. Kumar et al [21] studied the large free vibrations amplitudes of axially functionally graded tapered beams with various taper profiles, material properties, and end conditions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multimode Analysis of Geometrically Nonlinear Transverse Free and Forced Vibrations of Tapered Beams

Hantati

Adri

Fakhreddine

et al. 2022

Shock and Vibration

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In this study, the geometrically nonlinear free and forced vibrations of tapered beams are investigated on the basis of the Euler-Bernoulli beam theory and the von Karman geometric nonlinearity assumptions. The aim of the analysis is to determine tapered beams nonlinear frequencies and modes, and the associated stress distributions by means of bending moment diagrams. The linear problem is first solved. Then, to tackle the nonlinear problem, the displacement function is expanded as a series of the linear modes and the discrete expressions for the strain and kinetic energies are derived. Assuming a point excitation of the beam, the algebraic nonlinear system obtained based on Hamilton’s principle is solved by an approximate method. The effect of the geometric nonlinearity in both the free and forced cases was illustrated and discussed and the effect of the variation of various tapered beam geometrical parameters. The effect of varying the excitation level was also examined.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: E Compatibility Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multimode Analysis of Geometrically Nonlinear Transverse Free and Forced Vibrations of Tapered Beams

Hantati

Adri

Fakhreddine

et al. 2022

Shock and Vibration

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show abstract

“…e nonlinear vibration of tapered beams similar to that of [33][34][35] is now investigated in the same way, and a comparison is made between the results obtained here, based on the second formulation and summarized in Table 9, and those of [33][34][35].…”

Section: Nonlinear Free Vibrationmentioning

confidence: 99%

A Multimode Approach to Geometrically Nonlinear Free and Forced Vibrations of Multistepped Beams

Hantati

Adri

Fakhreddine

et al. 2021

Shock and Vibration

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The scope of this study is to present a contribution to the geometrically nonlinear free and forced vibration of multiple-stepped beams, based on the theories of Euler–Bernoulli and von Karman, in order to calculate their corresponding amplitude-dependent modes and frequencies. Discrete expressions of the strain energy and kinetic energies are derived, and Hamilton’s principle is applied to reduce the problem to a solution of a nonlinear algebraic system and then solved by an approximate method. The forced vibration is then studied based on a multimode approach. The effect of nonlinearity on the dynamic behaviour of multistepped beams in the free and forced vibration is demonstrated and discussed. The effect of varying some geometrical parameters of the stepped beams in the free and forced cases is investigated and illustrated, among which is the variation in the level of excitation.

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Ritz finite element approach to nonlinear vibrations of beams

Sarma

Varadan

1984

Numerical Meth Engineering

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SUMMARYA Ritz finite element approach is used here to study the large amplitude free flexural vibrations of beams with immovable ends. The formulation is based on Lagrange's equation of motion with the definition of the time function at an instant corresponding to the point of reversal. The element displacement vector is chosen as a combination of inplane and transverse displacements. The nonlinear stiffness is written as a combination of the bending-membrane interaction and bending stiffness. The solution for nonlinear equations is sought by using an algorithm-the direct iteration technique-suitably modified for eigenvalue problems. Convergence is checked using the displacement norms on eigen-modes, and frequency norms for eigenvalues.The nonlinear frequencies, and modeshapes for transverse and longitudinal displacements, are determined for the simply-supported, clamped-clamped and simply-supported-clamped beams. Results are preented in the form of tables. In almost all the cases the nonlinear frequency values are found to be the lower bound like the earlier Galerkin finite element method.

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A finite element formulation for the large amplitude vibrations of tapered beams

Cited by 27 publications

References 4 publications

Multimode Analysis of Geometrically Nonlinear Transverse Free and Forced Vibrations of Tapered Beams

Multimode Analysis of Geometrically Nonlinear Transverse Free and Forced Vibrations of Tapered Beams

A Multimode Approach to Geometrically Nonlinear Free and Forced Vibrations of Multistepped Beams

Ritz finite element approach to nonlinear vibrations of beams

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