Parametric Stability and Bifurcations of Axially Moving Viscoelastic Beams with Time-Dependent Axial Speed#
28 Feb 2013-Mechanics Based Design of Structures and Machines (Taylor & Francis Group)-Vol. 41, Iss: 3, pp 359-381
TL;DR: In this article, the transverse vibration response, stability, and bifurcations of an axially moving viscoelastic beam with time-dependent axial speed are investigated.
Abstract: The subject of this article is the investigation of the transverse vibration response, stability, and bifurcations of an axially moving viscoelastic beam with time-dependent axial speed. The force and moment balances as well as constitutive relations are employed to derive the equation of motion. Due to the presence of the time-dependent axial speed and steady dissipation terms, time-dependent coefficients and nonlinear dissipation terms are generated, respectively. The equation of motion is reduced into a set of coupled nonlinear ordinary differential equations with time-dependent coefficients. The subcritical resonant response of the system is obtained using the pseudo-arclength continuation technique, while the bifurcation diagrams of Poincare maps are obtained via direct time integration of the discretized equations of motion.
Citations
More filters
TL;DR: In this paper, the vibrational analysis of functionally graded graphene platelets reinforced composite (FG-GPLRC) viscoelastic annular plate within the framework of higher order shea...
Abstract: This is the first research on the vibrational analysis of functionally graded graphene platelets reinforced composite (FG-GPLRC) viscoelastic annular plate within the framework of higher order shea...
76 citations
Cites background or methods from "Parametric Stability and Bifurcatio..."
...Ghayesh and Amabili (2013) studied the nonlinear dynamics of an axially moving viscoelastic beam made of the Kelvin-Voigt type material with time-dependent axial speed....
[...]
...There are many models to describe the viscoelastic behavior such as Maxwell (standard) (Ghorbanpour-Arani et al. 2016), KelvinVoigt (Ghayesh and Amabili 2013), Zener (Hosseini-Hashemi et al. 2015), and Boltzmann (Norouzi and Alibeigloo 2018)....
[...]
TL;DR: In this article, the Euler beam model and nonlocal theory were employed to develop the governing partial differential equations of the mathematical model for axially moving piezoelectric nanobeams, which reveal potential applications in self-powered components of biomedical nano-robot.
Abstract: This work is concerned with the thermo-electro-mechanical coupling transverse vibrations of axially moving piezoelectric nanobeams which reveal potential applications in self-powered components of biomedical nano-robot. The nonlocal theory and Euler piezoelectric beam model are employed to develop the governing partial differential equations of the mathematical model for axially moving piezoelectric nanobeams. The natural frequencies of nanobeams under simply supported and fully clamped boundary constraints are numerically determined based on the eigenvalue method. Subsequently, some detailed parametric studies are presented and it is shown that the nonlocal nanoscale effect and axial motion effect contribute to reduce the bending rigidity of axially moving piezoelectric nanobeam and hence its natural frequency decreases within the framework of nonlocal elasticity. Moreover, the natural frequency decreases with increasing the positive external voltage, axial compressive force and change of temperature, while increases with increasing the axial tensile force. The critical speed and critical axial compressive force are determined and the dynamical buckling behaviors of axially moving piezoelectric nanobeams are indicated. It is concluded the nonlocal nanoscale parameter plays a remarkable role in the size-dependent natural frequency, critical speed and critical axial compressive force.
68 citations
Cites background from "Parametric Stability and Bifurcatio..."
...There are increasing publications on axially moving strings, beams, plates and nanostructures (Wickert and Mote, 1990; Öz et al., 2001; Lim et al., 2010; Ghayesh and Amabili, 2013; Saksa and Jeronen, 2015; Kurki et al., 2016)....
[...]
TL;DR: This paper will provide a guideline to select a proper mathematical model and to analyze the dynamics of the process in advance and future research directions to enhance the technologies in this field are proposed.
Abstract: In this paper, a detailed review on the dynamics of axially moving systems is presented. Over the past 60 years, vibration control of axially moving systems has attracted considerable attention owing to the board applications including continuous material processing, roll-to-roll systems, flexible electronics, etc. Depending on the system’s flexibility and geometric parameters, axially moving systems can be categorized into four models: String, beam, belt, and plate models. We first derive a total of 33 partial differential equation (PDE) models for axially moving systems appearing in various fields. The methods to approximate the PDEs to ordinary differential equations (ODEs) are discussed; then, approximated ODE models are summarized. Also, the techniques (analytical, numerical) to solve both the PDE and ODE models are presented. The dynamic analyses including the divergence and flutter instabilities, bifurcation, and chaos are outlined. Lastly, future research directions to enhance the technologies in this field are also proposed. Considering that a continuous manufacturing process of composite and layered materials is more demanding recently, this paper will provide a guideline to select a proper mathematical model and to analyze the dynamics of the process in advance.
60 citations
TL;DR: The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates, and as the excitation frequency decreases, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinate is different, while this behavior has more similarity after this point.
Abstract: In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the oc...
21 citations
Cites background from "Parametric Stability and Bifurcatio..."
...…and Pratiher 2019a, 2019b; Mao, Ding, and Chen 2017a, 2017b; Pratiher and Dwivedy 2011), while most others focused on numerical results (Farokhi and Ghayesh 2018; Ghayesh 2012a, 2018; Ghayesh and Amabili 2013; Ghayesh, Amabili, and Farokhi 2013; Ghayesh and Farokhi 2015; Rehlicki et al. 2018)....
[...]
TL;DR: In this paper, the influence of viscoelastic coefficient and material length scale parameter on frequency of micro-shell vibration was investigated for accurate study of microshell vibration, which can be useful for accurate microshell analysis.
Abstract: Modeling of viscoelastic behavior can be useful to accurate study of micro-shell vibration. In this study, influence of viscoelastic coefficient and material length scale parameter on frequency of ...
19 citations
Cites background or methods from "Parametric Stability and Bifurcatio..."
...Ghayesh and Amabili (2013) studied the nonlinear dynamics of an axially moving viscoelastic beam made of the Kelvin-Voigt type material with time-dependent axial speed....
[...]
...There are many models to describe the viscoelastic behavior such as Maxwell (standard) (Ghorbanpour-Arani et al. 2016), KelvinVoigt (Ghayesh and Amabili 2013; Safarpour et al. 2020), Zener (Hosseini-Hashemi, Abaei, and Ilkhani 2015) and Boltzmann (Norouzi and Alibeigloo 2018)....
[...]
References
More filters
01 Jan 1997
TL;DR: This is a guide to the software package AUTO for continuation and bifurcation problems in ordinary differential equations and the development of HomCont has much benefitted from various pieces of help and advice from, among others, W. W. Norton.
Abstract: Preface This is a guide to the software package AUTO for continuation and bifurcation problems in ordinary differential equations. graphics program PLAUT and the pendula animation program. An earlier graphical user interface for AUTO on SGI machines was written by Taylor & Kevrekidis (1989). Special thanks are due to Sheila Shull, California Institute of Technology, for her cheerful assistance in the distribution of AUTO over a long period of time. Over the years, the development of AUTO has been supported by various agencies through the California Institute of Technology. Work on this updated version was supported by a general research grant from NSERC (Canada). The development of HomCont has much benefitted from various pieces of help and advice from, among others, W. This manual uses the following conventions. command This font is used for commands which you can type in. PAR This font is used for AUTO parameters. filename This font is used for file and directory names. variable This font is used for environment variable. site This font is used for world wide web and ftp sites. function This font is used for function names.
1,417 citations
"Parametric Stability and Bifurcatio..." refers methods in this paper
...Two numerical techniques, first one based upon the pseudoarclength continuation technique (Doedel et al., 1998), and the second on direct time integration via the variable step-size Runge–Kutta method are employed to solve the discretized first-order NODEs with time-dependent coefficients....
[...]
TL;DR: In this paper, a perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly nonlinear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky.
Abstract: Free non-linear vibration of an axially moving, elastic, tensioned beam is analyzed over the sub- and supercritical transport speed ranges. The pattern of equilibria is analogous to that of Euler column buckling and consists of the straight configuration and of non-trivial solutions that bifurcate with speed. The governing equations for finite local motion about the trivial equilibrium and for motion about each bifurcated solution are cast in the standard form of continuous gyroscopic systems. A perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly non-linear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky. The method is subsequently specialized to non-linear vibration of a traveling beam, and of a traveling string in the limit of vanishing flexural rigidity. The contribution of non-linear stiffness to the response increases with subcritical speed, grows most rapidly near the critical speed, and can be several times greater for a translating beam than for one that is not translating. In the supercritical speed range, asymmetry of the non-linear stiffness distribution biases finite-amplitude vibration toward the straight configuration and lowers the effective modal stiffness. The linear vibration theory underestimates stability in the subcritical range, overestimates it for supercritical speeds, and is most limited in the near-critical regime.
332 citations
TL;DR: In this article, the non-linear vibrations of an axially moving beam are investigated by including the stretching effect of the beam, where the beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity.
Abstract: Non-linear vibrations of an axially moving beam are investigated. The non-linearity is introduced by including stretching effect of the beam. The beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity. Approximate solutions are sought using the method of multiple scales. Depending on the variation of velocity, three distinct cases arise: (i) frequency away from zero or two times the natural frequency, (ii) frequency close to zero, (iii) frequency close to two times the natural frequency. Amplitude-dependent non-linear frequencies are derived. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.
207 citations
TL;DR: In this paper, the incremental harmonic balance (IHB) method is formulated for the nonlinear vibration analysis of axially moving beams, and the Galerkin method is used to discretize the governing equations.
Abstract: In this paper, the incremental harmonic balance (IHB) method is formulated for the nonlinear vibration analysis of axially moving beams. The Galerkin method is used to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived. The forced response of an axially moving strip with internal resonance between the first two transverse modes is studied. Particular attention is paid to the fundamental, superharmonic and subharmonic resonance as the excitation frequency is close to the first, second or one-third of the first natural frequency of the system. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear vibration of axially moving media.
150 citations
TL;DR: In this article, principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed, and closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms.
Abstract: Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partial-differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations.
148 citations