Nonlinear analysis of nonprismatic Timoshenko beam for different geometric nonlinearity models

Ruta, P.; Szybiński, Józef

doi:10.1016/j.ijmecsci.2015.07.020

Cited by 5 publications

(3 citation statements)

References 21 publications

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“…In another study [28], the same authors investigated tapered Timoshenko beams and presented expressions for nonlinear fundamental frequencies of simply-supported and fully-clamped beams. Torabi et al [29] used a variational iteration method to study nonlinear vibrations of a Timoshenko beam and Ruta and Szybinski [30] applied different geometrical nonlinearity models to compare the nonlinear dynamics of a non-prismatic Timoshenko beam. Palacios [31] researched nonlinear normal modes of oscillation for anisotropic beams using both intrinsic equations where velocities and strains are primary degrees-of-freedom and Cosserat's exact description of the deformed geometry.…”

Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

Navadeh

Sareh

Basovsky

et al. 2021

Journal of Computational and Nonlinear Dynamics

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Deep cantilever beams, modelled using Timoshenko beam kinematics, have numerous applications in engineering. This study deals with the nonlinear dynamic response in a non-prismatic Timoshenko beam characterized by considering the deformed configuration of the axis. The mathematical model is derived using the extended Hamilton's principle under the condition of finite deflections and angles of rotation. The discrete model of the beam motion is constructed based on the finite difference method (FDM), whose validity is examined by comparing the results for a special case with the corresponding data obtained by commercial finite element (FE) software ABAQUS 2019. The natural frequencies and vibration modes of the beam are computed. These results demonstrate decreasing eigenfrequency in the beam with increasing amplitudes of nonlinear oscillations. The numerical analyses of forced vibrations of the beam show that its points oscillate in different manners depending on their relative position along the beam. Points close to the free end of the beam are subject to almost harmonic oscillations, and the free end vibrates with a frequency equal to that of the external force. When a point approaches the clamped end of the beam, it oscillates in two-frequency mode and lags in phase from the oscillations of the free end. The analytical model allows for the study of the influence of each parameter on the eigenfrequency and the dynamic response. In all cases, a strong correlation exists between the results obtained by the analytical model and ABAQUS, nonetheless, the analytical model is computationally less expensive.

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Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

Navadeh

Sareh

Basovsky

et al. 2021

Journal of Computational and Nonlinear Dynamics

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“…Then performing the differentiation of function u x (16) over variable t, one gets (18) Substituting u x,t into formula (12) 1 defining γ tt , and taking into account the fact that t = x, ultimately, one gets (19) where (20).…”

Section: Description Of Modelmentioning

confidence: 99%

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Szybiński

Ruta

2019

Studia Geotechnica Et Mechanica

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This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

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“…In the case of constant efforts to minimize the weight of mechanical systems and maximize their strength, of great importance is the issue of optimization of such systems. Shape optimization issues can be found, among others in the works of Drazumeric and Kosel, 2012, Ruta and Szybiński, 2015, Tsiatas, 2010, Krużelecki and Barski, 2008, Bochenek and Tajs-Zielińska, 2008, Nikolic and Salinic, 2017, Szmidla and Jurczyńska, 2015and Szmidla and Wawszczak, 2008, where the application of different algorithms or proprietary solutions are proposed. This article is a response to the search for optimal shapes of slender mechanical systems subjected to a conservative load.…”

Section: Introductionmentioning

confidence: 99%

Shape Optimization of a Column under Generalized Load with a Force Directed towards Positive Pole

Szmidla

Jurczyńska

2019

Quality Production Improvement - QPI

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In this paper, the issue of shape optimization of a column subjected to the generalized load with a force directed towards the positive pole (L. Tomski’s load, specific load) was considered. Based on the Hamilton’s principle, the differential equations of movement and boundary conditions describing the system were formulated. Taking into account a kinetic criterion of stability loss and a condition of constant total volume, the scope of changes in natural frequency as a function of an external load was determined with selected geometrical and physical parameters of the loading structure. On the basis of obtained results, values of geometrical parameters of individual column segments were determined, at which the maximum critical load value was obtained. In order to find the maximum critical force, which is a function of many variables, the simulated annealing algorithm was used.

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Nonlinear analysis of nonprismatic Timoshenko beam for different geometric nonlinearity models

Cited by 5 publications

References 21 publications

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Shape Optimization of a Column under Generalized Load with a Force Directed towards Positive Pole

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