Parametric Vibration and Numerical Validation of Axially Moving Viscoelastic Beams With Internal Resonance, Time and Spatial Dependent Tension, and Tension Dependent Speed

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
You-Qi Tang ◽  
Zhao-Guang Ma ◽  
Shuang Liu ◽  
Lan-Yi Zhang
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Differential Quadrature Method ◽  
Parametric Vibration ◽  
Axially Moving Beam ◽  
Beam Model ◽  
Solvability Conditions ◽  
Axially Moving ◽  
External Forces

Abstract In this paper, the idea of an axially moving time-dependent beam model is briefly introduced. The nonlinear response of an axially moving beam is investigated. The effects of a time and spatial dependent tension depending on the external forces at the boundary and a tension dependent speed are highlighted, which gives a new model to study the parametric vibration of axially moving structures. This paper focuses on simultaneous resonant cases that are the principal parametric resonance of first mode and internal resonance of the first two modes. In general, the method of multiple scales can study nonlinear vibration of axially moving structures with homogeneous boundary conditions. Taking Kelvin viscoelastic constitutive relation into account, the inhomogeneous boundary conditions make the solvability conditions fail, which is also one of the highlights of this paper. In order to resolve this problem, the technique of the modified solvability conditions is employed. The influence of some parameters, such as material’s viscoelastic coefficients, viscous damping coefficients, and the axial tension fluctuation amplitudes, on the steady-state vibration responses is demonstrated by some numerical examples. Furthermore, the approximate analytical results are verified by using the differential quadrature method (DQM).

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

Analysis of the vibration of axially moving viscoelastic plate with free edges using differential quadrature method

2017 ◽  
Vol 24 (17) ◽  
pp. 3908-3919 ◽  
Author(s):  
Mouafo Teifouet Armand Robinson
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Viscoelastic Plate ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Simply Supported ◽  
System Parameters ◽  
Complex Eigenvalues ◽  
Axially Moving ◽  
Viscoelastic Rectangular Plate

The two-dimensional viscoelastic differential constitutive relation is employed in this paper, in order to establish the equation of motion of axially moving viscoelastic rectangular plate. Two boundary conditions are investigated, namely the clamped free and two opposite edges simply supported and two others free. The differential quadrature method is used to solve the resulting complex eigenvalues equation. The influence of boundary conditions on the instability of a moving viscoelastic plate is analyzed firstly, and secondly the effects of system parameters such as plate's viscosity and aspect ratio on the vibration frequencies are presented.

Transversal Vibration Analysis of Vehicle Track System

2019 ◽  
Author(s):  
Pingxin Wang ◽  
Xiaoting Rui ◽  
Jianshu Zhang ◽  
Hailong Yu
Keyword(s):  
Natural Frequency ◽  
Vibration Analysis ◽  
Stability Control ◽  
Axially Moving Beam ◽  
Natural State ◽  
Beam Model ◽  
Tracked Vehicle ◽  
Transversal Vibration ◽  
Rubber Bushing ◽  
Axially Moving

Abstract The track is mainly composed of track shoes, track pins and rubber bushing elements. In order to suppress the transversal vibration of the upper track during the smooth running process of the tracked vehicle, it is necessary to study the important factors affecting the frequency characteristics of the kinematic chain and their interaction. Unlike the conventional chain drive system, the track in the natural state has a bending rigidity due to the action of the rubber bushing. Based on the dynamic theory of axially moving beam, the differential equation of transversal vibration of a beam element is established. The entire upper track is assumed to be a continuous multi-span axially moving Euler-Bernoulli beam with an axial tension. Based on the Transfer Matrix Method of Multibody System, the transfer equation is obtained. According to the boundary conditions, the natural frequency of the system is solved. The correctness of the beam model hypothesis is verified by experiments. The results show that the first-order natural frequency of the upper track increases with the increase of the tension and the decrease of the vehicle speed. Through frequency analysis, the main excitation source for the transversal vibration of the track is the polygon effect produced by the meshing of the track and the sprocket. This study provides a theoretical basis for the vibration analysis and stability control of the upper track on the tracked vehicle.

Thermoelastic Coupling Vibration Characteristics of the Axially Moving Beam With Frictional Contact

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Guo Xu-Xia ◽  
Wang Zhong-Min
Keyword(s):  
Frictional Contact ◽  
Differential Quadrature Method ◽  
Contact Point ◽  
Normal Pressure ◽  
Vibration Characteristics ◽  
Axially Moving Beam ◽  
Thermoelastic Coupling ◽  
Coupling Vibration ◽  
Moving Beam ◽  
Axially Moving

The thermoelastic coupling vibration characteristics of the axially moving beam with frictional contact are investigated. The piecewise differential equation of motion for the axially moving beam in the thermoelastic coupling case and the continuous conditions at the contact point are established. The eigenequation is derived by the differential quadrature method, and the first order dimensionless complex frequencies of the simply supported axially moving beam under the coupled thermoelastic case are calculated. The effects of the dimensionless thermoelastic coupling factor, the dimensionless moving speed, the spring stiffness, the friction coefficient, and the normal pressure on the thermoelastic coupling vibration characteristics of the axially moving beam with frictional contact are discussed.

Equilibrium and Forced Vibration of an Axially Moving Belt with Belt-Pulley Contact Boundaries

2019 ◽  
Vol 24 (3) ◽  
pp. 600-607 ◽  
Author(s):  
Hu Ding
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Forced Vibration ◽  
Equilibrium Configuration ◽  
Differential Quadrature Method ◽  
Vertical Motion ◽  
Contact Boundary ◽  
Parametric Effects ◽  
Steady State Responses ◽  
Axially Moving

Axially moving materials have usually dealt with classic boundary conditions, i.e. zero boundaries, such as the simply supported and the fixed ends. In this paper, the dynamics responses of the axially moving belt with beltpulley contact boundary conditions are studied for the first time. Therefore, due to the fact that non-homogeneous terms are included in the boundary conditions, the traditional generalized eigenvalue method is no longer applicable. In this work, the belt is numerically discretized by using the differential quadrature method (DQM). Iterative schemes are proposed for determining the equilibrium configuration. Harmonic inertia excitation is considered to be the vertical motion of the whole system. The steady-state responses of the forced vibration are also numerically solved by applying the DQM. The parametric effects on the equilibrium configuration and the steady-state response are investigated. The numerical investigations reveal that the radius of the support pulley has significant effects on both the equilibrium configuration and the transition phase of the transverse vibration of the axially moving belt under inertia excitation.

scholarly journals Free Vibration Analysis of Triclinic Nanobeams Based on the Differential Quadrature Method

2019 ◽  
Vol 9 (17) ◽  
pp. 3517 ◽  
Author(s):  
Behrouz Karami ◽  
Maziar Janghorban ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Strain Gradient ◽  
Differential Quadrature Method ◽  
Free Vibration Analysis ◽  
Differential Quadrature ◽  
Vibration Response ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Beam Model

In this work, the nonlocal strain gradient theory is applied to study the free vibration response of a Timoshenko beam made of triclinic material. The governing equations of the problem and the associated boundary conditions are obtained by means of the Hamiltonian principle, whereby the generalized differential quadrature (GDQ) method is implemented as numerical tool to solve the eigenvalue problem in a discrete form. Different combinations of boundary conditions are also considered, which include simply-supports, clamped supports and free edges. Starting with some pioneering works from the literature about isotropic nanobeams, a convergence analysis is first performed, and the accuracy of the proposed size-dependent anisotropic beam model is checked. A large parametric investigation studies the effect of the nonlocal, geometry, and strain gradient parameters, together with the boundary conditions, on the vibration response of the anisotropic nanobeams, as useful for practical engineering applications.

Sign in / Sign up

Export Citation Format

Share Document