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.

Isogeometric Free Vibration Analysis of Curved Euler–Bernoulli Beams With Particular Emphasis on Accuracy Study

2020 ◽  
pp. 2150011
Author(s):  
Zhuangjing Sun ◽  
Dongdong Wang ◽  
Xiwei Li
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Convergence Rates ◽  
Harmonic Wave ◽  
Free Vibration Analysis ◽  
Curved Beam ◽  
Curved Beams ◽  
Stiffness Matrices ◽  
Accuracy Measure ◽  
Accuracy Study

An isogeometric free vibration analysis is presented for curved Euler–Bernoulli beams, where the theoretical study of frequency accuracy is particularly emphasized. Firstly, the isogeometric formulation for general curved Euler–Bernoulli beams is elaborated, which fully takes the advantages of geometry exactness and basis function smoothness provided by isogeometric analysis. Subsequently, in order to enable an analytical frequency accuracy study, the general curved beam formulation is particularized to the circular arch problem with constant radius. Under this circumstance, explicit mass and stiffness matrices are derived for quadratic and cubic isogeometric formulations. Accordingly, the coupled stencil equations associated with the axial and deflectional displacements of circular arches are established. By further invoking the harmonic wave assumption, a frequency accuracy measure is rationally attained for isogeometric free analysis of curved Euler–Bernoulli beams, which theoretically reveals that the isogeometric curved beam formulation with [Formula: see text]th degree basis functions is [Formula: see text]th order accurate regarding the frequency computation. Numerical results well confirm the proposed theoretical convergence rates for both circular arches and general curved beams.

scholarly journals Out-of-Plane Vibration of Curved Uniform and Tapered Beams with Additional Mass

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Hamdi Alper Özyiğit ◽  
Mehmet Yetmez ◽  
Utku Uzun
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Variable Cross ◽  
Additional Mass ◽  
Inertia Effects ◽  
Tapered Beam ◽  
Out Of Plane ◽  
Good Agreement

As there is a gap in literature about out-of-plane vibrations of curved and variable cross-sectioned beams, the aim of this study is to analyze the free out-of-plane vibrations of curved beams which are symmetrically and nonsymmetrically tapered. Out-of-plane free vibration of curved uniform and tapered beams with additional mass is also investigated. Finite element method is used for all analyses. Curvature type is assumed to be circular. For the different boundary conditions, natural frequencies of both symmetrical and unsymmetrical tapered beams are given together with that of uniform tapered beam. Bending, torsional, and rotary inertia effects are considered with respect to no-shear effect. Variations of natural frequencies with additional mass and the mass location are examined. Results are given in tabular form. It is concluded that (i) for the uniform tapered beam there is a good agreement between the results of this study and that of literature and (ii) for the symmetrical curved tapered beam there is also a good agreement between the results of this study and that of a finite element model by using MSC.Marc. Results of out-of-plane free vibration of symmetrically tapered beams for specified boundary conditions are addressed.

scholarly journals Free Vibration Analysis of Rings via Wave Approach

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wang Zhipeng ◽  
Liu Wei ◽  
Yuan Yunbo ◽  
Shuai Zhijun ◽  
Guo Yibin ◽  
...  
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Classical Method ◽  
Free Vibration Analysis ◽  
Material Parameters ◽  
Wave Approach ◽  
Out Of Plane ◽  
Vibration Transmissibility ◽  
Ring Structures

Free vibration of rings is presented via wave approach theoretically. Firstly, based on the solutions of out-of-plane vibration, propagation, reflection, and coordination matrices are derived for the case of a fixed boundary at inner surface and a free boundary at outer surface. Then, assembling these matrices, characteristic equation of natural frequency is obtained. Wave approach is employed to study the free vibration of these ring structures. Natural frequencies calculated by wave approach are compared with those obtained by classical method and Finite Element Method (FEM). Afterwards natural frequencies of four type boundaries are calculated. Transverse vibration transmissibility of rings propagating from outer to inner and from inner to outer is investigated. Finally, the effects of structural and material parameters on free vibration are discussed in detail.

Natural frequencies for out-of-plane vibrations of continuous curved beams

1980 ◽  
Vol 68 (3) ◽  
pp. 427-436 ◽  
Author(s):  
T.M. Wang ◽  
R.H. Nettleton ◽  
B. Keita
Keyword(s):  
Natural Frequencies ◽  
Curved Beams ◽  
Out Of Plane

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.

scholarly journals Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

2021 ◽  
Vol 39 (2) ◽  
pp. 317-325
Author(s):  
Zhaochun Teng ◽  
Pengfei Xi
Keyword(s):  
Differential Equation ◽  
Elastic Modulus ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Gradient Index ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Governing Differential Equation

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

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.

FREE VIBRATIONS OF SHEAR DEFORMABLE CIRCULAR CURVED BEAMS RESTING ON ELASTIC FOUNDATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 77-97 ◽  
Author(s):  
BYOUNG KOO LEE ◽  
SANG JIN OH ◽  
KWANG KYOU PARK
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Rotary Inertia ◽  
Transverse Shear Deformation ◽  
Curved Beams ◽  
System Parameters ◽  
Elastic Foundations ◽  
Out Of Plane ◽  
Shear Deformable

The governing differential equations for the out-of-plane, free vibration of circular curved beams resting on elastic foundations are derived and solved numerically. The formulation takes into consideration the effects of rotary inertia and transverse shear deformation. The lowest three natural frequencies are calculated for beams with hinged–hinged, hinged-clamped, and clamped–clamped end constraints. The effects of various system parameters as well as rotary inertia and shear deformation on the natural frequencies are investigated.

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