scholarly journals Dynamics of Double-Beam System with Various Symmetric Boundary Conditions Traversed by a Moving Force: Analytical Analyses

2019 ◽  
Vol 9 (6) ◽  
pp. 1218
Author(s):  
Jing Yang ◽  
Xuhui He ◽  
Haiquan Jing ◽  
Hanfeng Wang ◽  
Sévérin Tinmitonde
Keyword(s):  
Boundary Conditions ◽  
Fourier Method ◽  
Mode Shapes ◽  
Speed Ratio ◽  
Superposition Method ◽  
Stiffness Ratio ◽  
Various Boundary Conditions ◽  
Moving Force ◽  
Beam System ◽  
Double Beam

Dynamics of the double-beam system under moving loads have been paid much attention due to its wide applications in reality from the analytical point of view but the previous studies are limited to the simply supported boundary condition. In this study, to understand the vibration mechanism of the system with various boundary conditions, the double-beam system consisted of two general beams with a variety of symmetric boundary conditions (fixed-fixed, pinned-pinned, fixed-pinned, pinned-fixed and fixed-free) under the action of a moving force is studied analytically. The closed-form frequencies and mode shapes of the system with various symmetric boundary conditions are presented by the Bernoulli-Fourier method and validated with Finite Element results. The analytical explicit solutions are derived by the Modal Superposition method, which are verified with numerical results and previous results in the literature. As found, each wavenumber of the double-beam system is corresponding to two sub-modes of the system and the two sub-modes associated with the first wavenumber of the system both contribute significantly to the vibration of the system under a moving force. The analytical solutions indicate that the mass ratio, the bending stiffness ratio, the stiffness ratio of contact springs and the speed ratio of the moving force are the factors influencing the vibrations of the system under a moving force. The relationships between these dimensionless parameters and the displacement ratio of the system are investigated and presented in the form of plots, which could be referred in the design of the double-beam system.

Exact Nonlinear Dynamic Analysis of a Beam With a Nonlinear Vibration Absorber and With Various Boundary Conditions

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Mohammad Bukhari ◽  
Oumar Barry
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Vibration ◽  
Nonlinear Dynamic ◽  
Microelectromechanical Systems ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Vibration Absorber ◽  
Nonlinear Frequency ◽  
Nonlinear Vibration Absorber ◽  
Various Boundary Conditions

Abstract We study the nonlinear vibration of a beam with an attached grounded and ungrounded nonlinear vibration absorber (NVA) using the exact natural frequencies and mode shapes of the loaded beam. The nonlinearity in the beam is due to midplane stretching and that in the NVA is of cubic stiffness nonlinearity. We consider various boundary conditions and derive their closed-form characteristic equations and mode shapes. The method of multiple scales (MMS) is directly applied to the nonlinear partial differential equations of motion to obtain explicit expressions of the nonlinear frequency, modulation, and loci of the saddle-node bifurcation equations. Our analytical approach is validated using direct numerical simulation. Parametric studies demonstrate that the performance of the NVA does not only depend on its key design variables and location, but also on the boundary conditions, midplane stretching of the beam, and type of configuration (i.e., grounded NVA versus ungrounded NVA). Our analysis also indicates that the use of common approach such as employing approximate modes in estimating the nonlinear response of a loaded beam produces significant error (i.e., up to 1200% in some case). These observations suggest that the exact modes shape and natural frequencies are required for a precise investigation of the nonlinear dynamic of loaded beams. These findings could contribute to the design improvement of NVAs, microelectromechanical systems (MEMS), energy harvesters, and metastructures.

Free and Forced Vibration of Coupled Beam Systems Resting on Variable Viscoelastic Foundations

2020 ◽  
Vol 20 (12) ◽  
pp. 2050141
Author(s):  
Jinpeng Su ◽  
Kun Zhang ◽  
Qiang Zhang ◽  
Ying Tian
Keyword(s):  
Boundary Conditions ◽  
Forced Vibration ◽  
Dynamic Properties ◽  
Least Square ◽  
Weighted Residual Method ◽  
Transient Vibration ◽  
Various Boundary Conditions ◽  
Beam System ◽  
Interface Potential ◽  
Forced Vibration Analysis

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

scholarly journals Flexural Vibration Analysis of Nonuniform Double-Beam System with General Boundary and Coupling Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Lujun Chen ◽  
Deshui Xu ◽  
Jingtao Du ◽  
Chengwen Zhong
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Admissible Function ◽  
Flexural Vibration ◽  
Vibration Characteristics ◽  
Modeling Framework ◽  
Varying Thickness ◽  
Beam System ◽  
Double Beam

In this paper, an analytical modeling approach for the flexural vibration analysis of the nonuniform double-beam system is proposed via an improved Fourier series method, in which both types of translational and rotational springs are introduced to account for the mechanical coupling on the interface as well as boundary restraints. Energy formulation is employed for the dynamic description of the coupling system. With the aim to treat the varying thickness across the beam in a unified pattern, the relevant variables are all expanded into Fourier series. Supplementary terms with the smoothed characteristics are introduced to the standard Fourier series for the construction of displacement admissible function for each beam. In conjunction with the Rayleigh–Ritz procedure, the transverse modal characteristics of nonuniform double-beam system can be obtained by solving a standard eigenvalue problem. Instead of solving the certain value of nonideal boundary conditions, the continuous spring stiffnesses of the boundary conditions are considered, and the rotational restrains are introduced in the coupling beam interface. Numerical results are then presented to demonstrate the reliability of the current model and study the influence of various parameters, such as taper ratio, boundary, and coupling strength on the free vibration characteristics, with the emphasis put on the rotational restraining coefficients on the beam interface. This work can provide an efficient modeling framework for the vibration characteristics study of the complex double-beam system, especially with arbitrary varying thickness and coupling stiffness.

scholarly journals Free Vibration Analysis of L-Shaped Folded Thin Plates

2019 ◽  
Vol 2 (1) ◽  
pp. 67-73
Author(s):  
Koji Sekine
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Thin Plates ◽  
Mode Shapes ◽  
Width Ratio ◽  
Various Boundary Conditions

Free vibration analysis of L-shaped folded thin plates having various boundary conditions is presented. Vibration properties of the folded plates are analyzed by means of the Ritz method. Displacement functions satisfying the geometric boundary conditions are assumed in the form of double power series. The interconnection of plate elements of the folded plates is defined by translational and rotational coupling springs. The generalized eigenvalue problem, which is derived by means of minimizing the energy functional, is solved to determine the natural frequencies and mode shapes. The accuracy and validity of the present solutions are demonstrated through convergence studies and comparisons with the results from the literature and FEM (finite element method) analysis solutions. Numerical results are presented for different conditions, such as width ratio, length ratio and the four types of boundary condition.

scholarly journals Effects of Cable on the Dynamics of a Cantilever Beam with Tip Mass

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Yi-Xin Huang ◽  
Hao Tian ◽  
Yang Zhao
Keyword(s):  
Cantilever Beam ◽  
Spectral Element Method ◽  
Natural Frequencies ◽  
Spectral Element ◽  
Mode Shapes ◽  
Beam System ◽  
Double Beam ◽  
Tip Mass ◽  
Beam Mode ◽  
Element Method

The dynamic effects of cable attachment on a cantilever beam with tip mass are investigated by an improved Chebyshev spectral element method. The cabled beam is modeled as a double-beam system connected by springs at several discrete locations. By utilizing high order Chebyshev polynomials as basis functions and meshing the system at the locations of connections, precise numerical results of the natural frequencies and mode shapes can be obtained using only a few elements. The accuracy of this method is validated through comparing the results of finite element method and those of spectral element method in literature. The validated method is implemented to investigate the effects of parameters, including spring stiffness, number of connections, density, and Young’s modulus of cable. The results show that the mode shapes of the cabled beam system can be classified into two types: beam mode shapes and cable mode shapes, according to their main deformation. Their corresponding natural frequencies change in very different ways with the variation of system parameters. This work can be applied to optimize the dynamic characteristics of precise spacecraft structures with cable attachments.

scholarly journals Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Qibo Mao
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Decomposition Method ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Rotating Beams ◽  
Arbitrary Boundary ◽  
Various Boundary Conditions ◽  
Governing Differential Equation

The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparison published results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beams with arbitrary boundary conditions.

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