Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

Masjedi, Pedram Khaneh; Maheri, Alireza

doi:10.1016/j.euromechsol.2017.07.014

Cited by 23 publications

(4 citation statements)

References 41 publications

(49 reference statements)

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“…Shang et al (2017) and Pramod et al (2017) employed trigonometric equations in their analysis to define a shape function for the free vibration of planar elements, having the first one worked in the context of the Unit Partition Method. Ding, Tan, and Dowell (2017); Masjedi and Maheri (2017); and Pradhan and Chakraverty (2017) also used trigonometric equations to account for beam vibrations under different operating conditions. In the case of the vibration of a cantilever column (fixed at its base but free at its tip), the shape function in Eq.…”

Section: Mathematical Modeling Of a Practical Problemmentioning

confidence: 99%

Analytical determination of the vibration frequencies and buckling loads of slender reinforced concrete towers

Wahrhaftig

Silva

Brasil

2019

Lat. Am. j. solids struct.

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This study focused on improving the design of slender structures with reinforced concrete (RC) telecommunication towers as the main application. Analytical procedure based on Rayleigh's method to compute the first natural vibration frequency and the critical buckling load was development. All the nonlinearities present in the system were considered, in addition to the soil-structure interaction and the variation of the geometric properties along the length of the structure. The geometric nonlinearity and imperfections of the tower structure were computed as functions of the axial load using a geometric stiffness matrix. Further, the material nonlinearity was accounted for by reducing the flexural stiffness. As concrete structures exhibit viscoelasticity, creep was calculated using the Eurocode 2 model. The soilstructure interaction was modeled as a set of distributed springs. To validate the proposed method, the first frequency and critical buckling load were compared with those yielded by FEM simulations. The frequency results were in good agreement with those of the FEM simulations, indicating that the proposed method is sufficiently accurate for use in engineering design applications and easy to implement. On the other hand, the buckling load results obtained using the proposed method and FEM differed significantly, motivating further investigation.

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Section: Mathematical Modeling Of a Practical Problemmentioning

confidence: 99%

Analytical determination of the vibration frequencies and buckling loads of slender reinforced concrete towers

Wahrhaftig

Silva

Brasil

2019

Lat. Am. j. solids struct.

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“…Dan et al [13] performed the geometrically exact nonlinear model of composite box beam with non-classical effect based on Hodges' generalized Timoshenko beam theory and then showed that sectional warping has significant effects on the static deformation and natural frequencies of the composite box beam. Khaneh Masjedi and Maheri [14] presented a Chebyshev collocation method for the free vibration analysis of the fully intrinsic geometrically exact equations of motion of composite beams.…”

Section: Introductionmentioning

confidence: 99%

Formulation and free vibration analysis of spinning Timoshenko composite beams using nonlinear geometrically exact theory

Beiranvand,

Hosseini

2024

Z Angew Math Mech

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In this study, a set of generalized nonlinear equations of motion for Timoshenko composite shafts is derived using the geometrically exact approach. These partial differential equations of motion are discretized by Galerkin method and the free vibration of an orthotropic spinning shaft with hinged‐hinged boundary conditions is investigated using the multiple‐scale method. We find that in the case of two mode discretization, composite couplings can have significant linear and nonlinear effects on the behavior of the shaft and that these effects were not predictable in the classical model. Also, in the investigating of composite layering we find that the effects of the fibers angle as well as the composite couplings can be noticeable, and these effects are greater in the case of the asymmetrical layering.

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“…A nonlinear periodic structure with cycle symmetry was studied by converting it into independent nonlinear single-degree-of-freedom systems using nonlinear normal modes for frequency response analysis, and its mode localization under weak coupling was discovered by asymptotic approximation [48,49]. The nonlinear vibration of non-uniform beams and dynamic stability of sandwich beams on nonlinear elastic foundations have also been studied using the finite element and harmonic balance methods, Galerkin and iterative numerical methods, linearization and Chebyshev collocation methods, Bolotin method and Hamiltonian approach, respectively [50][51][52][53][54]. A differential quadrature method was applied to the nonlinear vibration analysis of sandwich beams where vibration modes were obtained using a numerical iterative algorithm [55,56].…”

Section: Introductionmentioning

confidence: 99%

Vibration Localization and Anti-Localization of Nonlinear Multi-Support Beams with Support Periodicity Defect

Ying

2021

Symmetry

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A response analysis method for nonlinear beams with spatial distribution parameters and non-periodic supports was developed. The proposed method is implemented in four steps: first, the nonlinear partial differential equation of the beams is transformed into linear partial differential equations with space-varying parameters by using a perturbation method; second, the space-varying parameters are separated into a periodic part and a non-periodic part describing the periodicity defect, and the linear partial differential equations are separated into equations for the periodic and non-periodic parts; third, the equations are converted into ordinary differential equations with multiple modes coupling by using the Galerkin method; fourth, the equations are solved by using a harmonic balance method to obtain vibration responses, which are used to discover dynamic characteristics including the amplitude–frequency relation and spatial mode. The proposed method considers multiple vibration modes in the response analysis of nonlinear non-periodic structures and accounts for mode-coupling effects resulting from structural nonlinearity and parametric non-periodicity. Thus, it can handle nonlinear non-periodic structures with a high parameter-varying wave in wide frequency vibration. In numerical studies, a nonlinear beam with non-periodic supports (resulting in non-periodic distribution parameters or periodicity defect) under harmonic excitations was explored using the proposed method, which revealed some new dynamic response characteristics of this kind of structure and the influences of non-periodic parameters. The characteristics include remarkable variation in frequency response and spatial mode, and in particular, vibration localization and anti-localization. The results have potential applications in vibration control and the support damage detection of nonlinear structures with non-periodic supports.

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Chebyshev collocation method for the free vibration analysis of geometrically exact beams with fully intrinsic formulation

Cited by 23 publications

References 41 publications

Analytical determination of the vibration frequencies and buckling loads of slender reinforced concrete towers

Analytical determination of the vibration frequencies and buckling loads of slender reinforced concrete towers

Formulation and free vibration analysis of spinning Timoshenko composite beams using nonlinear geometrically exact theory

Vibration Localization and Anti-Localization of Nonlinear Multi-Support Beams with Support Periodicity Defect

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