Free vibration of a beam subjected to large static deflection

Cornil, Marie-Blanche; Capolungo, Laurent; Qu, Jianmin; Jairazbhoy, Vivek A.

doi:10.1016/j.jsv.2007.02.016

Cited by 18 publications

(11 citation statements)

References 13 publications

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“…The first approach neglects static strain in governing equation and effect of large static deformation is incorporated on free vibration behavior only through statically deformed geometry. 15,16 Whereas in the more rigorous second approach, large static strain is considered in derivation of dynamic governing equation taking statically deformed beam geometry as reference. 17–19 As governing equation derived through either of the two approaches involves geometric nonlinearity, analytical solution is very few.…”

Section: Introductionmentioning

confidence: 99%

Free vibration analysis of large deformed curved beam incorporating rotary inertia and shear deformation effects and its experimental validation

Ghuku

Saha

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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The paper theoretically and experimentally analyzes free vibration characteristics of statically loaded moving boundary type curved beam considering rotary inertia and shear deformation effects. Effects of rotary inertia and shear deformation are observed for different thickness to span ratios of curved beam. The subject problem is decoupled into two interrelated problems: determining equilibrium configuration under static load and finding the corresponding free vibration frequency. The static problem is analyzed incrementally in body fitted curvilinear frame as it involves geometric nonlinearity due to generalized curvature, large deformation, and moving boundaries. Variational energy principle is employed to derive governing equation. The nonlinear governing equation associated with complicated boundary conditions is solved through iterative geometry updation. Once static problem is solved for current load step, governing equation for dynamic characteristics is derived using Hamilton’s principle. The governing equation gets linearized by using the static configuration, which finally yields a linear eigenvalue problem. Experiment is performed in a dedicated setup with two master leafs having different thickness to span ratios. The roller supported specimens are excited with an instrumented hammer and response signals are captured by accelerometers. The excitation and response signals are recorded using HBM-MX840B data acquisition system. Frequency response functions of the curved beam systems under different static loads are obtained from postprocessing of the dynamic signals in MATLAB®. First two natural frequencies of the specimens are noted from the experimental results and the corresponding theoretical results are generated. The specimens are also modeled in ABAQUS® CAE and finite element results are computed. Comparison between the theoretical, experimental, and finite element results validates the present model. The study also provides some meaningful observations on effects of rotary inertia and shear deformation. Based on the observations, more results are generated for different thickness to span ratios and findings are reported suitably.

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Section: Introductionmentioning

confidence: 99%

Free vibration analysis of large deformed curved beam incorporating rotary inertia and shear deformation effects and its experimental validation

Ghuku

Saha

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…They solved the problem using the differential transform method and studied the effects of the spinning speed, twist angle and axial force. Cornil et al. (2007) carried out the free vibration analysis of beams under large static deflection.…”

Section: Introductionmentioning

confidence: 99%

A new tangent stiffness-based formulation to study the free vibration behavior of a transversely loaded Timoshenko beam with geometric nonlinearity

Das

2016

Journal of Vibration and Control

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A new formulation is introduced to study the free vibration behavior of a statically loaded beam with geometric nonlinearity. The tangent stiffness of the statically loaded beam is used to investigate the free vibration behavior of the beam about its loaded configuration. The problem is formulated for a linearly tapered beam, and a uniform beam is obtained as a special case. Energy principles based on the variational approach are used to derive the governing equations for the static and dynamic problems. The Ritz method of approximate displacement field is followed to solve the governing equations. The Ritz coefficients are used to derive the tangent stiffness of the loaded beam. Components of the tangent stiffness matrix are derived for a Timoshenko beam with von Kármán-type nonlinearity. Illustrative results are presented for four different classical boundary conditions having in-plane restraint. Results for the first two modes of transverse vibration are presented in the nondimensional deflection-frequency plane. Validation of the work is carried out using finite element software ANSYS. The formulation is new of its kind and can be used for any displacement-based problem following the Ritz method.

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“…Agarwal et al (2006) studied the geometric nonlinear effects of the static and dynamic behaviour of beams made of isotropic, composite and functionally graded materials, using first order shear deformation theory (FSDT). The free vibration problem of a beam under a large static deflection was investigated by Cornil et al (2007), using the nonlinear equations of motion. To obtain a solution, these equations were decomposed into a set of nonlinear differential equations for static deflection and a set of linear differential equations for the dynamic problem.…”

Section: Introductionmentioning

confidence: 99%

Large Displacement of Crossbeam Structure through Energy Method

Mitra¹,

Sahoo²,

Saha³

2012

Int J Automot Mech Eng

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This paper undertakes a geometric nonlinear large displacement static analysis of crossbeam structure. A crossbeam structure comprises two beams in contact with their longitudinal axes perpendicular to each other, used effectively in civil and mechanical engineering, marine and aerospace structures. The energy method forms the basis for the mathematical formulation and the governing set of equations are obtained using the principle of extremisation of the total energy of the system in its equilibrium state. To obtain the solution, an iterative procedure is developed based on the reaction force generated between the two beams of the system. The method is validated by experiment and simulation through ANSYS v11. Results are presented in terms of plots of reaction force and the displacement of the interaction point versus load in dimensional form and additionally, the deflected shapes of the crossbeam structure at static equilibrium condition, under a particular load, are provided. The static response of the system has been studied for variation of beam thickness, loading pattern and position of the supporting beam.

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Free vibration of a beam subjected to large static deflection

Cited by 18 publications

References 13 publications

Free vibration analysis of large deformed curved beam incorporating rotary inertia and shear deformation effects and its experimental validation

Free vibration analysis of large deformed curved beam incorporating rotary inertia and shear deformation effects and its experimental validation

A new tangent stiffness-based formulation to study the free vibration behavior of a transversely loaded Timoshenko beam with geometric nonlinearity

Large Displacement of Crossbeam Structure through Energy Method

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