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.

Stress concentration factors for the torsion of curved beams of arbitrary cross-section

2006 ◽  
Vol 220 (12) ◽  
pp. 1709-1726 ◽  
Author(s):  
R E Cornwell
Keyword(s):  
Finite Element ◽  
Stress Concentration ◽  
Cross Section ◽  
Cross Sections ◽  
Shear Stresses ◽  
Curved Beams ◽  
Finite Element Implementation ◽  
Out Of Plane ◽  
Concentration Factors ◽  
Stress Concentration Factors

There are numerous situations in machine component design in which curved beams with cross-sections of arbitrary geometry are loaded in the plane of curvature, i.e. in flexure. However, there is little guidance in the technical literature concerning how the shear stresses resulting from out-of-plane loading of these same components are effected by the component's curvature. The current literature on out-of-plane loading of curved members relates almost exclusively to the circular and rectangular cross-sections used in springs. This article extends the range of applicability of stress concentration factors for curved beams with circular and rectangular cross-sections and greatly expands the types of cross-sections for which stress concentration factors are available. Wahl's stress concentration factor for circular cross-sections, usually assumed only valid for spring indices above 3.0, is shown to be applicable for spring indices as low as 1.2. The theory applicable to the torsion of curved beams and its finite-element implementation are outlined. Results developed using the finite-element implementation agree with previously available data for circular and rectangular cross-sections while providing stress concentration factors for a wider variety of cross-section geometries and spring indices.

A Finite Element Modeling Framework for Planar Curved Beam Dynamics Considering Nonlinearities and Contacts

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Tianheng Feng ◽  
Soovadeep Bakshi ◽  
Qifan Gu ◽  
Dongmei Chen
Keyword(s):  
Finite Element ◽  
Beam Dynamics ◽  
Curved Beam ◽  
Shape Functions ◽  
Curved Beams ◽  
Modeling Framework ◽  
External Wall ◽  
Beam Elements ◽  
High Compression ◽  
Assumed Strain

Motivated by modeling directional drilling dynamics where planar curved beams undergo small displacements, withstand high compression forces, and are in contact with an external wall, this paper presents an finite element method (FEM) modeling framework to describe planar curved beam dynamics under loading. The shape functions of the planar curved beam are obtained using the assumed strain field method. Based on the shape functions, the stiffness and mass matrices of a planar curved beam element are derived using the Euler–Lagrange equations, and the nonlinearities of the beam strain are modeled through a geometric stiffness matrix. The contact effects between curved beams and the external wall are also modeled, and corresponding numerical methods are discussed. Simulations are carried out using the developed element to analyze the dynamics and statics of planar curved structures under small displacements. The numerical simulation converges to the analytical solution as the number of elements increases. Modeling using curved beam elements achieves higher accuracy in both static and dynamic analyses compared to the approximation made by using straight beam elements. To show the utility of the developed FEM framework, the post-buckling condition of a directional drill string is analyzed. The drill pipe undergoes spiral buckling under high compression forces, which agrees with experiments and field observations.

In-Plane Free Vibration of Curved Beams Using Finite Element Method

2007 ◽  
Author(s):  
F. Yang ◽  
R. Sedaghati ◽  
E. Esmailzadeh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Central Line ◽  
Circular Ring ◽  
Curved Beam ◽  
The Finite Element Method ◽  
Curved Beams ◽  
Element Method

Curved beam-type structures have many applications in engineering area. Due to the initial curvature of the central line, it is complicated to develop and solve the equations of motion by taking into account the extensibility of the curve axis and the influences of the shear deformation and the rotary inertia. In this study the finite element method is utilized to study the curved beam with arbitrary geometry. The curved beam is modeled using the Timoshenko beam theory and the circular ring model. The governing equation of motion is derived using the Extended-Hamilton principle and numerically solved by the finite element method. A parametric sensitive study for the natural frequencies has been performed and compared with those reported in the literature in order to demonstrate the accuracy of the analysis.

A Finite Element Approach to Plate Vibration Problems

1965 ◽  
Vol 7 (1) ◽  
pp. 28-32 ◽  
Author(s):  
D. J. Dawe
Keyword(s):  
Finite Element ◽  
Arbitrary Shape ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Uniform Thickness ◽  
Flat Plates ◽  
Finite Element Approach ◽  
Various Boundary Conditions ◽  
Element Approach ◽  
Vibration Problems

A method of computing the natural frequencies of vibration of flat plates of arbitrary shape is outlined in which the plate is considered as an assemblage of elements. Both stiffness and inertia matrices are derived for a rectangular isotropic plate element of uniform thickness, and these matrices are used to find the natural frequencies of square plates subject to various boundary conditions. Comparison of finite element frequencies with known exact, experimental and energy solutions shows the method to give good results even for relatively few elements.

scholarly journals Experimental Study on Nonlinear Vibrations of Fixed-Fixed Curved Beams

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Ajay Kumar ◽  
B. P. Patel
Keyword(s):  
Experimental Study ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Excitation Amplitude ◽  
Curved Beam ◽  
Symmetric Mode ◽  
Curved Beams ◽  
Direct Excitation ◽  
First Time ◽  
Good Agreement

AbstractNonlinear dynamic behavior of fixed-fixed shallow and deep curved beams is studied experimentally using non-contact type of electromagnetic shaker and acceleration measurements. The frequency response obtained from acceleration measurements is found to be in fairly good agreement with the computational response. The travellingwave phenomenon along with participation of higher harmonics and softening nonlinearity are observed. The experimental results on the internal resonance of curved beams due to direct excitation of anti-symmetric mode are reported for the first time. The deep curved beam depicts chaotic response at higher excitation amplitude.

scholarly journals The Analysis of Curved Beam Using B-Spline Wavelet on Interval Finite Element Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhibo Yang ◽  
Xuefeng Chen ◽  
Yumin He ◽  
Zhengjia He ◽  
Jie Zhang
Keyword(s):  
Finite Element ◽  
Polynomial Interpolation ◽  
Physical Space ◽  
Curved Beam ◽  
Approximation Properties ◽  
Curved Beams ◽  
Spline Wavelet ◽  
B Spline ◽  
Wavelet Space ◽  
Interval Finite Element Method

A B-spline wavelet on interval (BSWI) finite element is developed for curved beams, and the static and free vibration behaviors of curved beam (arch) are investigated in this paper. Instead of the traditional polynomial interpolation, scaling functions at a certain scale have been adopted to form the shape functions and construct wavelet-based elements. Different from the process of the direct wavelet addition in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of the corresponding transformation matrix. Furthermore, compared with the commonly used Daubechies wavelet, BSWI has explicit expressions and excellent approximation properties, which guarantee satisfactory results. Numerical examples are performed to demonstrate the accuracy and efficiency with respect to previously published formulations for curved beams.

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 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.

Free in-plane vibration of curved beam structures: A tutorial and the state of the art

2017 ◽  
Vol 24 (12) ◽  
pp. 2400-2417 ◽  
Author(s):  
F Yang ◽  
R. Sedaghati ◽  
E. Esmailzadeh
Keyword(s):  
State Of The Art ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
The State ◽  
Tangential Displacement ◽  
Curved Beam ◽  
Curved Beams ◽  
Plane Vibration ◽  
Structural Deformations ◽  
Beam Theories

The study of free in-plane vibration of curved beams, using different beam theories, is more challenging than that of straight beams, since the structural deformations in curved beams depend not only on the rotation and radial displacements, but also on the coupled tangential displacement caused by the curvature of structures. A critical review of the publications on the free in-plane vibration of curved beams to demonstrate the state of the art has been presented. The governing differential equations of motion for the curved beams, based on different hypotheses (including and excluding the axial extensity, rotary inertia and the shear deformation), were discussed and different approaches to solve the developed equations of motion have been identified. Finally, a systematic comparison of the dynamic properties of curved beams evaluated with various forms of curvatures based on different hypotheses were presented.

Stability and Large Deformation of 2-D Laminate Circular Thin Curved Beams

2014 ◽  
Vol 644-650 ◽  
pp. 5146-5150
Author(s):  
Chiu Wen Lin ◽  
Han Ming Tseng ◽  
Tso Liang Teng
Keyword(s):  
Stability Analysis ◽  
Large Deformation ◽  
Static Loading ◽  
Shear Force ◽  
Radius Of Curvature ◽  
Curved Beam ◽  
Curved Beams ◽  
Governing Equations ◽  
Laminate Theory ◽  
Tangent Slope

In this research, both un-deformed or Lagrangian state and deformed or Eulerian state are used to derive for stability analysis and large deformation. By choosing the deformed radius of curvature and deformed angle of tangent slope as parameters, the governing equations of laminated curved beam under static loading are transformed into a set of equations in terms of angle of tangent slope. All the quantities of axial force, shear force, radial and tangential displacements of circular thin curved beam are expressed as functions of angle of tangent slope by using laminate theory. The buckling load and large deformation analytical solutions of circular thin curved beam under a pair of forces are presented as well.

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