scholarly journals A General Fourier Formulation for In-Plane and Out-of-Plane Vibration Analysis of Curved Beams

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Rui Nie ◽  
Tianyun Li ◽  
Xiang Zhu ◽  
Huihui Zhou
Keyword(s):  
Potential Energy ◽  
Boundary Conditions ◽  
Calculation Model ◽  
Displacement Function ◽  
Curved Beam ◽  
Curved Beams ◽  
Concentrated Mass ◽  
Engineering Structures ◽  
Beam Structures ◽  
Out Of Plane

Based on the principle of energy variation, an improved Fourier series is introduced as an allowable displacement function. This paper constructs a calculation model that can study the in-plane and out-of-plane free and forced vibrations of curved beam structures under different boundary conditions. Firstly, based on the generalized shell theory, considering the shear and inertial effects of curved beam structures, as well as the coupling effects of displacement components, the kinetic energy and strain potential energy of the curved beam are obtained. Subsequently, an artificial spring system is introduced to satisfy the constraint condition of the displacement at the boundary of the curved beam, obtain its elastic potential energy, and add it to the system energy functional. Any concentrated mass point or concentrated external load can also be added to the energy function of the entire system with a corresponding energy term. In various situations including classical boundary conditions, the accuracy and efficiency of the method in this paper are proved by comparing with the calculation results of FEM. Besides, by accurately calculating the vibration characteristics of common engineering structures like slow curvature (whirl line), the wide application prospects of this method are shown.

Out-of-Plane Deflections of Nonprismatic Curved Beam Structures Solved by the Differential Quadrature Element Method

2002 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Boundary Conditions ◽  
Solution Procedure ◽  
Differential Quadrature ◽  
Curved Beam ◽  
Beam Structures ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Out Of Plane ◽  
Element Boundary ◽  
Element Method

The differential quadrature element method (DQEM) is used to solve the out-of-plane deflections of nonprismatic curved beam structures. The extended differential quadrature (EDQ) is used to discretize the governing differential equations defined on all elements, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions. Numerical results obtained by DQEM are presented. They demonstrate the developed numerical solution procedure.

Lateral Buckling of Cantilevered Circular Arches Under Various End Moments

2020 ◽  
Vol 20 (07) ◽  
pp. 2071005
Author(s):  
Y. B. Yang ◽  
Y. Z. Liu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Analytical Approach ◽  
Three Dimensional ◽  
Curved Beams ◽  
Lateral Buckling ◽  
Circular Arch ◽  
Out Of Plane ◽  
The Moment

Lateral buckling of cantilevered circular arches under various end moments is studied using an analytical approach. Three types of conservative moments are considered, i.e. the quasi-tangential moments of the 1st and 2nd kinds, and the semi-tangential moment. The induced moments associated with each of the moment mechanisms undergoing three-dimensional rotations are included in the Newman boundary conditions. Using the differential equations available for the out-of-plane buckling of curved beams, the analytical solutions are derived for a cantilevered circular arch, which can be used as the benchmarks for calibration of other methods of analysis.

scholarly journals Out-of-Plane Free Vibration and Forced Harmonic Response of a Curved Beam

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Hancheng Mao ◽  
Guangbin Yu ◽  
Wei Liu ◽  
Tiantian Xu
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Amplitude Ratio ◽  
Harmonic Wave ◽  
Wave Number ◽  
Curved Beam ◽  
Curved Beams ◽  
Propagation Behavior ◽  
Out Of Plane ◽  
Governing Differential Equation

Based on the governing differential equation of out-of-plane curved beam, the wave propagation behavior, free vibration, and transmission properties are presented theoretically in this paper. Firstly, harmonic wave solutions are given to investigate the dispersion relation between frequency and wave number, cut-off frequency, displacement, amplitude ratio, and phase diagram. The frequency spectrum results are obtained to verify the work by Kang and Lee. Furthermore, natural frequencies of the single and composite curved beam are calculated through solving the characteristic equation in the case of free-free, clamped-clamped, and free-clamped boundaries. Finally, the transfer matrices of the out-of-plane curved beam are derived by combining the continuity between the different interfaces. The transmissibility curves of the single and composite curved beam are compared to find the vibration attention band. This work will be valuable to extend the study of the out-of-plane vibration of curved beams.

Vibrations of Planar Curved Beams, Rings, and Arches

1993 ◽  
Vol 46 (9) ◽  
pp. 467-483 ◽  
Author(s):  
P. Chidamparam ◽  
A. W. Leissa
Keyword(s):  
Finite Element ◽  
Static Loading ◽  
Arbitrary Shape ◽  
Nonlinear Vibrations ◽  
Curved Beam ◽  
Curved Beams ◽  
Vibratory Motion ◽  
Out Of Plane ◽  
Vibration Problems ◽  
Beam Theories

This work attempts to organize and summarize the extensive published literature on the vibrations of curved bars, beams, rings and arches of arbitrary shape which lie in a plane. In-plane, out-of-plane and coupled vibrations are considered. Various theories that have been developed to model curved beam vibration problems are examined. An overview is presented of the types of problems which are addressed in the literature. Particular attention is given to the effects of initial static loading, nonlinear vibrations and the application of finite element techniques. The significantly different frequencies arising from curved beam theories which either allow or prevent extension of the centerline during vibratory motion are shown. An extensive bibliography of 407 relevant references is included.

DQEM Analysis of In-Plane Deflection of Curved Beam Structures

2003 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Boundary Conditions ◽  
Numerical Algorithm ◽  
Differential Quadrature ◽  
Curved Beam ◽  
Analysis Model ◽  
Beam Structures ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Deflection Analysis ◽  
Element Boundary

The development of differential quadrature element method in-plane deflection analysis model of arbitrarily curved nonprismatic beam structures was carried out. The DQEM uses the extended differential quadrature to discretize the differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. The convergence of the developed DQEM analysis model is efficient.

scholarly journals Dynamic Analysis of Multi-Stepped Functionally Graded Carbon Nanotube Reinforced Composite Plate with General Boundary Condition

2021 ◽  
Vol 2021 ◽  
pp. 1-27
Author(s):  
Kwanghun Kim ◽  
Songhun Kwak ◽  
Yonguk Ri ◽  
Yongsong Paek ◽  
Wonjin Han ◽  
...  
Keyword(s):  
Carbon Nanotube ◽  
Potential Energy ◽  
Boundary Conditions ◽  
Energy Function ◽  
Forced Vibration ◽  
Potential Energy Function ◽  
Displacement Function ◽  
Functionally Graded ◽  
General Boundary Condition ◽  
Reinforced Composite

This study presents the multi-stepped functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate model for the first time, and its free and forced vibration is analyzed by employing the domain decomposition method. The segmentation technique is employed to discretize the structure along the length direction. The artificial spring technique is applied to the structural boundary and piecewise interface for satisfying the boundary conditions and the combined conditions between subplates. Based on this, the boundary conditions of subdomains could be considered as a free boundary constraint, reducing the difficulty in constructing the allowable displacement function. Since all the structures of subdomains are identical, the allowable displacement functions of them can be uniformly constructed using the two-dimensional ultraspherical polynomial expansion. The potential energy function of the plate is derived from the first-order shear deformation theory (FSDT). The allowable displacement function is substituted into the potential energy function, and then the natural frequencies and mode shapes of the multi-stepped FG-CNTRC plate are decided by using the Rayleigh–Ritz method. The accuracy and reliability of the proposed method are confirmed by the results of the previous literature and finite element method (FEM). On this basis, the influences of the geometric and material parameters on free and forced vibration of the multi-stepped FG-CNTRC plate are also studied.

OUT-OF-PLANE FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR CURVED BEAMS SUPPORTED ON ELASTIC FOUNDATION

2010 ◽  
Vol 02 (03) ◽  
pp. 635-652 ◽  
Author(s):  
P. MALEKZADEH ◽  
M. R. GOLBAHAR HAGHIGHI ◽  
M. M. ATASHI
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Elastic Foundation ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Curved Beams ◽  
Out Of Plane ◽  
Benchmark Solutions

As a first endeavor, the out-of-plane free vibration analysis of thin-to-moderately thick functionally graded (FG) circular curved beams supported on two-parameter elastic foundation is presented. The formulation is derived based on the first-order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be graded in the direction normal to the plane of the beam curvature. The differential quadrature method (DQM), as an efficient and accurate method, is employed to discretize the equations of motion and the related boundary conditions. In order to assure the accuracy of the formulation and the method of solution, convergence behavior of the nondimensional natural frequencies is examined for FG circular curved beams and comparison studies with those of isotropic curved beams, available in the literature, are performed. The effects of the elastic foundation coefficients, boundary conditions, the material graded index and different geometrical parameters on the natural frequency parameters of the FG circular curved beams are investigated. The new results can be used as benchmark solutions for future research works.

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