scholarly journals Three-Dimensional Nonlinear Global Dynamics of Axially Moving Viscoelastic Beams

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Hamed Farokhi ◽  
Mergen H. Ghayesh ◽  
Shahid Hussain
Keyword(s):  
Fourier Transforms ◽  
Dynamical Behavior ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Elastic Model ◽  
Global Dynamics ◽  
Generalized Coordinates ◽  
Time Histories ◽  
Axially Moving ◽  
Lateral Displacements

The three-dimensional nonlinear global dynamics of an axially moving viscoelastic beam is investigated numerically, retaining longitudinal, transverse, and lateral displacements and inertia. The nonlinear continuous model governing the motion of the system is obtained by means of Hamilton's principle. The Galerkin scheme along with suitable eigenfunctions is employed for model reduction. Direct time-integration is conducted upon the reduced-order model yielding the time-varying generalized coordinates. From the time histories of the generalized coordinates, the bifurcation diagrams of Poincaré sections are constructed by varying either the forcing amplitude or the axial speed as the bifurcation parameter. The results for the three-dimensional viscoelastic model are compared to those of a three-dimensional elastic model in order to better understand the effect of the internal energy dissipation mechanism on the dynamical behavior of the system. The results are also presented by means of time histories, phase-plane diagrams, and fast Fourier transforms (FFT).

scholarly journals Nonlinear Dynamical Behavior of Axially Accelerating Beams: Three-Dimensional Analysis

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Hamed Farokhi
Keyword(s):  
Differential Equations ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Global Dynamics ◽  
Nonlinear Dynamical ◽  
Time Histories ◽  
Lateral Displacements

The three-dimensional (3D) nonlinear dynamics of an axially accelerating beam is examined numerically taking into account all of the longitudinal, transverse, and lateral displacements and inertia. Hamilton’s principle is employed in order to derive the nonlinear partial differential equations governing the longitudinal, transverse, and lateral motions. These equations are transformed into a set of nonlinear ordinary differential equations by means of the Galerkin discretization technique. The nonlinear global dynamics of the system is then examined by time-integrating the discretized equations of motion. The results are presented in the form of bifurcation diagrams of Poincaré maps, time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

scholarly journals Modified stepped scheme for modelling the dynamic behaviour of 3D poroviscoelastic solids

2018 ◽  
Vol 174 ◽  
pp. 02019
Author(s):  
Leonid Igumnov ◽  
Aleksandr Ipatov ◽  
Svetlana Litvinchuk
Keyword(s):  
Dynamic Behaviour ◽  
Boundary Integral Equations ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Numerical Approach ◽  
Boundary Integral ◽  
Dynamic Responses ◽  
Elastic Model ◽  
Weakly Singular Kernel ◽  
Soil Medium

The problem of the dynamic response of a soil medium under different kinds of loads is of significant importance in various areas of engineering, especially in connection with structures. The present paper is dedicated to the modification of the numerical approach for modelling the dynamic behaviour of three dimensional poroviscoelastic solids. The basic equations for fluid-saturated porous media proposed by Biot are modified by replacing the classical linear elastic model of the solid skeleton with the viscoelastic model. Classical models of viscoelasticity are employed, such as Kelvin-Voight model, standard linear solid model and model with weakly singular kernel. Boundary integral equations method is applied to solving three-dimensional boundary-value problems. Stepped schemes modifications based on the linear and quadratic approximation of function are employed. A numerical example of poroviscoelastic rod under Heaviside type load is provided. A problem of a poroviscoelastic cube with a cavity subjected to a normal internal pressure is considered. The comparison of dynamic responses when poroviscoelastic material is described by different viscoelastic models is presented.

Quasi-Static Analysis of Beam Described by Fractional Derivative Kelvin Viscoelastic Model under Lateral Load

2011 ◽  
Vol 189-193 ◽  
pp. 3391-3394 ◽  
Author(s):  
Qing Zhao Yao ◽  
Lin Chao Liu ◽  
Qi Fang Yan
Keyword(s):  
Fractional Derivative ◽  
Mechanical Behavior ◽  
Constitutive Relations ◽  
Dynamical Behavior ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Bernoulli Beam ◽  
The Laplace Transform ◽  
Static Mechanical ◽  
Euler Bernoulli Beam

The beam is assumed to obey a three-dimensional viscoelastic fractional derivative constitutive relations, the mathematical model and governing equations of the quasi-static and dynamical behavior of a viscoelastic Euler-Bernoulli beam are established, the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative model is investigated, and the analytical solution is obtained by considering the properties of the Laplace transform of Mittag-Leffler function and the properties of fractional derivative. The result indicate that the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative viscoelastic model can reduced to the cases of classic viscoelastic and elastic, the order of fractional derivative has great effect on the quasi-static mechanical behavior of Euler-Bernoulli beam.

Nonlinear Dynamics of Axially Moving Plates With Rotational Springs

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Marco Amabili
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Plate Theory ◽  
Lagrange Equation ◽  
Global Dynamics ◽  
Nonlinear Ordinary Differential Equations ◽  
Moving Plate ◽  
Time Histories ◽  
Axially Moving

In this paper, the in-plane and out-of-plane nonlinear dynamics of an axially moving plate with distributed rotational springs at boundaries is examined numerically. The Von Kármán plate theory along with the Kirchhoff’s hypothesis are employed to construct the kinetic and potential energies of the system. The Lagrange equation is used so as to obtain the equations of motion which are in the form of a set of second-order nonlinear ordinary differential equations. This set is recast into a set of first-order nonlinear ordinary differential equations with coupled terms. Gear’s backward-differentiation-formula is employed to integrate this set of equations numerically, yielding the generalized coordinates of the system as a function of time. The bifurcation diagrams of Poincaré maps are then constructed by sectioning these time histories in every period of the external excitation force. The results are shown in the form of time histories, phase-plane portraits, and Poincaré sections. The effect of the stiffness of the rotational springs on the global dynamics of the system is also investigated.

Periodic and Chaotic Oscillations of an Axially Moving Viscoelastic Belt in Three-Dimensional Space

2007 ◽  
Author(s):  
Wei Zhang ◽  
Yan-Qi Liu ◽  
Li-Hua Chen ◽  
Ming-Hui Yao
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Out Of Plane ◽  
Axially Moving

Periodic and chaotic space oscillations of an axially moving viscoelastic belt with one-to-one internal resonance are investigated for the first time. The Kelvin viscoelastic model is introduced to describe the viscoelastic property of the belt material. The external damping and internal damping of the material for the axially moving viscoelastic belt are considered simultaneously. The nonlinear governing equations of motion of the axially moving viscoelastic belt for the in-plane and out-of-plane are derived by the extended Hamilton’s principle. The method of multiple scales and Galerkin’s approach are applied directly to the partial differential governing equations of motion to obtain four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance of the first order modes for the in-plane and out-of-plane oscillations. Numerical method is used to investigate periodic and chaotic space motions of the axially moving viscoelastic belt. The results of numerical simulation demonstrate that there exist periodic, period-2, period-3, period-4, period-6, quasiperiodic and chaotic motions of the axially moving viscoelastic belt.

Stability and Bifurcations in Three-Dimensional Analysis of Axially Moving Beams

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Marco Amabili ◽  
Hamed Farokhi
Keyword(s):  
Differential Equations ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Three Dimensional ◽  
Axially Moving Beam ◽  
Bifurcation Diagrams ◽  
Response Curves ◽  
Axially Moving

The geometrically nonlinear dynamics of a three-dimensional axially moving beam is investigated numerically for both sub and supercritical regimes. Hamilton’s principle is employed to derive the equations of motion for in-plane and out-of plane displacements. The Galerkin scheme is applied to the nonlinear partial differential equations of motion yielding a set of second-order nonlinear ordinary differential equations with coupled terms. The pseudo-arclength continuation technique is employed to solve the discretized equations numerically so as to obtain the nonlinear resonant responses; direct time integration is conducted to obtain the bifurcation diagrams of the system. The results are presented in the form of the frequency-response curves, bifurcation diagrams, time histories, phase-plane portraits, and fast Fourier transforms for different sets of system parameters.

Experimental Investigation vs Numerical Simulation of the Dynamic Response of a Moored Floating Structure to Waves

2014 ◽  
Author(s):  
Daniele Dessi ◽  
Sara Siniscalchi Minna
Keyword(s):  
Dynamical Behavior ◽  
Irregular Waves ◽  
Mooring Line ◽  
Floating Structure ◽  
Mooring Lines ◽  
Wave Energy Converters ◽  
Time Histories ◽  
Actual Configuration ◽  
Proper Orthogonal ◽  
Dynamic Wave

A combined numerical/theoretical investigation of a moored floating structure response to incoming waves is presented. The floating structure consists of three bodies, equipped with fenders, joined by elastic cables. The system is also moored to the seabed with eight mooring lines. This corresponds to an actual configuration of a floating structure used as a multipurpose platform for hosting wind-turbines, aquaculture farms or wave-energy converters. The dynamic wave response is investigated with numerical simulations in regular and irregular waves, showing a good agreement with experiments in terms of time histories of pitch, heave and surge motions as well as of the mooring line forces. To highlight the dynamical behavior of this complex configuration, the proper orthogonal decomposition is used for extracting the principal modes by which the moored structure oscillates in waves giving further insights about the way waves excites the structure.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

scholarly journals Microtubules soften due to cross-sectional flattening

eLife ◽  
2018 ◽  
Vol 7 ◽  
Author(s):  
Edvin Memet ◽  
Feodor Hilitski ◽  
Margaret A Morris ◽  
Walter J Schwenger ◽  
Zvonimir Dogic ◽  
...  
Keyword(s):  
High Strain ◽  
Three Dimensional ◽  
Elastic Model ◽  
Softening Effect ◽  
Cross Sectional ◽  
Mechanical Compliance ◽  
Wide Range ◽  
Elastic Filament ◽  
New Interpretation ◽  
Effective Mechanical Properties

We use optical trapping to continuously bend an isolated microtubule while simultaneously measuring the applied force and the resulting filament strain, thus allowing us to determine its elastic properties over a wide range of applied strains. We find that, while in the low-strain regime, microtubules may be quantitatively described in terms of the classical Euler-Bernoulli elastic filament, above a critical strain they deviate from this simple elastic model, showing a softening response with increasing deformations. A three-dimensional thin-shell model, in which the increased mechanical compliance is caused by flattening and eventual buckling of the filament cross-section, captures this softening effect in the high strain regime and yields quantitative values of the effective mechanical properties of microtubules. Our results demonstrate that properties of microtubules are highly dependent on the magnitude of the applied strain and offer a new interpretation for the large variety in microtubule mechanical data measured by different methods.

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