Showing papers on "Multiple-scale analysis published in 2017"
TL;DR: In this paper, the authors investigated the transverse vibrations and steady-state responses of axially moving viscoelastic piezoelectric two-dimensional nanostructures.
Abstract: This work is motivated by the self-powered component of biomedical nano-robotic device which is expected to move in arterial blood vessels. The transverse vibrations and steady-state responses of axially moving viscoelastic piezoelectric two-dimensional nanostructures are investigated based on the nonlocal viscoelasticity thin plate theory. The constitutive relations of viscoelastic piezoelectric nanoplate containing the thermal effect are performed and the governing partial differential equations of the problem model are derived using the Hamilton's principle. The natural frequencies are numerically determined via the Galerkin method, the complex mode method, as well as the finite element method for comparison. Moreover, the theoretical calculations are compared with those in previous literature to verify effects in nonlocal nanoscale framework and average speed on natural frequency. Afterwards, the instable behaviors of axially non-uniformly moving viscoelastic piezoelectric nanoplate characterized as a sine variation about the constant average speed are addressed using the method of multiple scales. The analyses are mainly focused on the boundaries of instable regions in combination parametric resonance and principal parametric resonance. The non-dimensional numerical results imply the existences of nonlocal nanoscale parameter and average speed contribute to reduce the rigidity, and further produce the coupled vibrations and flutter instabilities for complex frequencies. Additionally, the instable regions of combination and principal parametric resonances decrease with increases in the biaxial compression, change of temperature, positive electric voltage and viscoelastic coefficient. The dynamic responses of axially moving viscoelastic piezoelectric nanoplate under the coupling of thermo-electro-mechanical multi-fields are expected to play significant roles in designing biomedical nano-robots.
82 citations
TL;DR: In this article, the energy-phase method was used to analyze the chaotic dynamics of a cantilevered pipe conveying pulsating fluid with a harmonic external force, and the nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of pipe material were considered.
53 citations
TL;DR: In this paper, the nonlinear vibrations of fractional viscoelastic plate with Kelvin-Voigt fractional order constitutive relationship are investigated. And the structural dynamic of the plate is modeled by using the Newton's second law.
Abstract: Nonlinear vibrations of fractional viscoelastic plate with Kelvin–Voigt fractional order constitutive relationship is investigated in this paper. Based on the Kirchhoff hypothesis for thin shells and von Karman’s assumption, the structural dynamic of the plate is modeled by using the Newton’s second law. The nonlinear coupled equations of motion are obtained by introducing the Airy stress function and the Galerkin method is used to discretize the partial differential equations. Analytical solutions for fully simply supported and clamped plate are obtained by using the method of multiple scales and finally the equations of amplitude–frequency and phase–frequency are obtained for primary, super-harmonic and sub-harmonic resonance. The obtained amplitude–frequency and phase–frequency equations are used for studying the effects of excitation, fractional parameters and nonlinearity on the frequency responses of the fractional viscoelastic plate and finally some conclusions are outlined.
47 citations
TL;DR: In this paper, the authors derived the governing equation from the generalized Hamilton's principle and discretized into a multiple-degrees-of-freedom system by the Galerkin's method, and investigated local and global resonances under the condition of 3:1 internal resonance of a super-critically axially moving beam, subjected to a harmonic exciting force.
46 citations
TL;DR: In this paper, the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions is studied.
Abstract: In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Karman strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.
45 citations
TL;DR: In this paper, a nonlinear dynamic double-cable-stayed shallow-arch model is established and the in-plane 1:1:1 internal resonance between three first modes of shallow arch and two cables under both external primary and subharmonic resonance is investigated, respectively.
Abstract: A novel nonlinear dynamic double-cable-stayed shallow-arch model of cable-stayed bridge is established and the in-plane 1:1:1 internal resonance between three first modes of shallow arch and two cables under both external primary and subharmonic resonance is investigated, respectively. The Galerkin discretization and the method of multiple scales are applied to obtain the modulation equations of the dynamic system. The stable equilibrium solutions of the modulation equations are examined by Newton-Raphson method. Numerical simulations are carried out to investigate the dynamic behavior of the new dynamic system and Runge-kutta method is also used to solve the ordinary differential equations to verify the results. The results show the rich nonlinear phenomena and some new conclusions are also drawn.
40 citations
TL;DR: In this article, the global dynamics of supercritical pipes conveying pulsating fluid considering superharmonic resonance of the second mode with 1:2 internal resonance were investigated and the governing partial differential equations in the supercritical regime were obtained based on the nontrivial equilibrium configuration of the pipe conveying fluid and then transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin's method.
Abstract: Global dynamics of supercritical pipes conveying pulsating fluid considering superharmonic resonance of the second mode with 1:2 internal resonance are investigated. The governing partial differential equations in the supercritical regime are obtained based on the nontrivial equilibrium configuration of the pipes conveying fluid and then transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motion to the near-integrable Hamiltonian standard form. The energy-phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are subsequently interpreted in terms of the physical motion of such gyroscopic system. Two types of nonlinear normal modal motion and the chaotic pattern conversion between the locked simple bidirectional traveling wave motion an...
37 citations
TL;DR: In this paper, a continuous model and nonlinear dynamic responses of a circular mesh antenna subjected to the thermal excitation in the space environment are investigated for the first time, where a continuum cantilever circular cylindrical short shell, which is clamped at one side of the shell along the axial direction, is proposed to take place of the circular antenna composed of the repetitive beamlike lattice by the principle of equivalent effect.
34 citations
TL;DR: In this paper, a system composed of a harmonically forced single-degree-of-freedom linear oscillator coupled to a vibro-impact nonlinear energy sink (VI-NES) is experimentally investigated.
Abstract: In this paper, the dynamics of a system composed of a harmonically forced single-degree-of-freedom linear oscillator coupled to a vibro-impact nonlinear energy sink (VI-NES) is experimentally investigated. The mass ratio between the VI-NES and the primary system is about $$1\%$$
. Depending on the external force’s amplitude and frequency, either a strongly modulated response (SMR) or a constant amplitude response (CAR) is observed. In both cases, an irreversible transfer of energy occurs from the linear oscillator toward the VI-NES: process known in the literature as passive targeted energy transfer. Furthermore, the problem is analytically studied by using the method of multiple scales. The obtained slow invariant manifold shows the existence of a stable and of an unstable branch of solutions, as well as of an energy threshold (a saddle-node bifurcation) for the solutions to appear. Subsequently, the fixed points of the problem are calculated. When a stable fixed point is reached, the system is naturally drawn to it and a CAR is established, whereas when no stable point is attained, the system exhibits a SMR regime. Finally, a good correlation between the experimental and the analytical results is presented.
32 citations
TL;DR: In this article, a Lagrangian-based approach is proposed to perform a second-order analysis, which is applicable to a large class of nonlinear systems, such as saturation, jumps, hysteresis and different kinds of bifurcations.
31 citations
TL;DR: In this paper, the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions was theoretically investigated, and an effective way was proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection.
Abstract: Nonlinear modal interactions have recently become the focus of intense research in micro-resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced micromechanical devices. This article aims to theoretically investigate the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions. Under higher-order modes excitation, two nonlinear coupled flexural modes to describe microbeam-based resonators are obtained by using Hamilton’s principle and Galerkin method. Then, the Method of Multiple Scales is applied to determine the response and stability of the system for small amplitude vibration. Through Hopf bifurcation analysis, the bifurcation sets for antisymmetry mode vibration are theoretically derived, and the mechanism of energy transfer between antisymmetry mode and symmetry mode is detailed studied. The pseudo-trajectory processing method is introduced to investigate the influence of external drive on amplitude and bifurcation behavior. Results show that nonlinear modal interactions can transit vibration energy from one mode to nearby mode. In what follows, an effective way is proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection. The quantitative relationship between vibrational modes is also obtained. The displacement of one mode can be predicted by detecting another mode, which shows great potential of developing parameter design in MEMS. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.
TL;DR: In this paper, the invariant manifold method is applied to the discretized ordinary differential equations of the axially moving string and complex gyroscopic mode functions that agree well with true analytical results are obtained.
TL;DR: In this paper, a composite laminated circular cylindrical shell under the combined action of radial harmonic excitation, compressive in-plane force and aerodynamic pressure is studied.
TL;DR: In this article, the vibration and stability analysis of an unbalanced rotor mounted on high-static-low-dynamic-stiffness supports is presented. And the stiffness of the supports is modeled as symmetric of cubic order.
TL;DR: In this article, a doubly clamped viscoelastic microbeam actuated by one-sided electrode is investigated in detail, based on a modified couple stress theory, which is essentially nonlinear due to its midplane stretching effect and electrostatic force.
Abstract: Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on quasi-static behaviors and transition mechanism of nonlinear jumping phenomena. The static and dynamic behaviors of a doubly clamped viscoelastic microbeam actuated by one sided electrode are investigated in detail, based on a modified couple stress theory. The governing equation of motion is introduced here, which is essentially nonlinear due to its midplane stretching effect and electrostatic force. Through quasi-static analysis, the equilibrium position, pull-in voltage and pull-in location of the system are obtained with differential quadrature method and finite element method. The equivalent geometric nonlinear parameter is presented to explain the influence of the scale effect on the pull-in location. Different from elastic material, there are two kinds of pull-in voltages called as instantaneous pull-in voltage and the durable pull-in voltage in viscoelastic system. Then, Galerkin discretization and the method of multiple scales are applied to determine the response and stability of the system for small vibration amplitude. A new perturbation method to deal with viscoelastic term is presented. Theoretical expressions about the parameter spaces of linear-like vibration, hardening-type vibration and softening-type vibration are then deduced. The influence of viscoelasticity and scale effect on nonlinear dynamic behavior is studied. Results show that the viscoelasticity can reduce the effective elastic modulus and make the system tend to softening-type vibration; the scale effect can increase effective elastic modulus and make the system tend to hardening-type vibration. And most of all, simulation results of case studies are used to realize parameter optimization. Then parameter conditions of linear-like vibration, which is desired for many applications, are obtained. In this paper, the results of multi-physical field coupling simulation are used to verify the theoretical analysis.
TL;DR: In this paper, the nonlinear responses and stability of double-layered nanoplate embedded in the elastic medium are investigated in the presence of 3:1 internal resonance, and the effect of small scale effect and viscous damping on nonlinear vibration is explored in details.
TL;DR: In this paper, the effects of viscosity on the natural frequency of transverse vibration of an axially moving viscoelastic beam are studied, and the qualitative difference between the natural frequencies with the material time derivative and the partial time derivative in the constitutive relation is investigated.
Abstract: One important issue in the investigation of axially moving systems is the viscoelastic constitutive relation. In the present paper, the effects of viscosity on the natural frequency of transverse vibration of an axially moving viscoelastic beam are studied. The viscoelastic material of the moving Euler-Bernoulli beam obeys the Kelvin model. For the first time, the qualitative difference between the natural frequencies with the material time derivative and the partial time derivative in the constitutive relation is investigated. The method of multiple scales with three terms is directly applied to obtain the approximate analytical solutions of the natural frequency. An interesting phenomenon is found in this study. Specifically, for an axially moving viscoelastic beam constituted by the material time derivative, the natural frequencies of transverse vibration may increase with the axial speed. Furthermore, the validity of the analytical results is examined by comparing with two numerical approaches, the di...
TL;DR: In this paper, the principal parametric resonance of axially accelerating hyperelastic beam is investigated and the effect of the material parameter (i.e., in plane Poisson's ratio) on the type of nonlinear vibration behavior and amplitude of the nontrivial solutions of steady-state have been investigated.
Abstract: This paper investigates principal parametric resonance of axially accelerating hyperelastic beam. Hyperelasticity is integrated into axially moving material for the first time. Based on the continuum mechanics theory, the coupled nonlinear partial differential equations of motion are derived from the extended Hamilton’s principle. The model equations are simplified into a single integro-differential equation, which governs the transverse vibration of the hyperelastic beam. The method of multiple scales is used to solve the integro-differential equation to obtain the nonlinear response of the principal parametric resonance. The effect of the material parameter(i.e. in plane Poisson’s ratio) on the type of the nonlinear vibration behavior and amplitude of the nontrivial solutions of steady-state have been investigated. Further, the couple nonlinear governing equations are solved by Galerkin’s method. Comparison between the analytical results and the results of the Galerkin’s method are made and good agreement are found.
TL;DR: In this paper, a simply-supported thin laminated plate subject to in-plane excitation is established based on the classic shear theory and von Karman nonlinear theory.
TL;DR: In this article, the nonlinear harmonic response of a cantilever hard-coating plate which is made of a layer of anisotropic hardcoating material and isotropic metal substrate is investigated based on the theory of high-order shear deformation of plate.
Abstract: The nonlinear harmonic response of a cantilever hard-coating plate which is made of a layer of anisotropic hard-coating material and isotropic metal substrate is investigated based on the theory of high-order shear deformation of plate. Firstly, based on the theories of von Karman and Reddy’s three-order shear deformation, the nonlinear dynamic equations of hard-coating plate are built by Hamilton variation principle. Secondly, to obtain nonlinear governing equation of hard-coating plate under transverse load, these equations are discretized in Galerkin method. The system averaged equations with 1:3 internal resonances are obtained by the method of multiple scales, and the multi-periodic responses behavior of cantilever hard-coating plate under transverse loading could be presented. Finally, the vibration response experiment of hard-coating plate is conducted, and the multi-periodic responses are also present for the hard-coating plate with three-to-one internal resonance. Besides, through the vibration response experiment of uncoated titanium alloy plate, the damping characteristic of hard coating is further analyzed.
TL;DR: The experimental results demonstrated that the subharmonic component of the sensing signal could be used to detect the fatigue crack and further to distinguish it from inherent nonlinear boundary conditions.
TL;DR: In this paper, an improved single-degree-of-freedom model to describe microbeam-based resonators is obtained by using fractional Kelvin constitutive model, Hamilton's principle and Galerkin method.
Abstract: Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on transition mechanism of nonlinear jumping phenomena and transient chaotic behaviors. This article aims to theoretically investigate the static and dynamic characteristics of electrically actuated viscoelastic bistable microbeam via a low-dimensional model. An improved single-degree-of-freedom model to describe microbeam-based resonators is obtained by using Fractional Kelvin constitutive model, Hamilton’s principle and Galerkin method. Through static bifurcation analysis, three kinds of parameter conditions of the bistable system are obtained, and potential energy function of the Hamiltonian system is theoretically derived. The influence of fractional viscoelasticity on dynamic pull-in phenomena is distinguished from the viewpoint of energy. Then, the method of multiple scales is applied to determine the response and stability of the system for small vibration amplitude and AC voltage. The influence of fractional viscoelasticity on amplitude, frequency and bifurcation behavior is investigated. Results show that compared with the elastic material, nonlinear phenomenon becomes weak, resonance frequency increases and amplitude decreases in the viscoelastic system. Besides, the numerical discretization method of fractional derivative is given to verify theoretical results. To study the influence of fractional viscoelasticity on complicated vibration, Melnikov method is applied to predict the existence of chaos, and numerical simulation is carried out to find the stable regions, chaotic regions and dynamic pull-in regions by using bifurcation diagrams with local maximum method. Rational increase in material modulus ratio parameter and fractional order is effective to reduce the possibility of chaos and dynamic pull-in. This analysis has the potential of developing parameter design in MEMS.
TL;DR: In this article, a rotating elastic ring is modeled using thin-ring theory with radial and tangential deformations, and the frequency and vibration modes are determined by discretizing the governing equations using Galerkin's method.
TL;DR: In this article, the Routh-Hurwitz criterion is used to determine the instability boundaries of axially accelerating beams with 1:3 internal resonance and the effects of viscoelastic coefficient and the viscous damping coefficient are examined on the instability boundary.
TL;DR: In this article, a closed-form solution to the linear eigenvalue problem in the form of a general Meijer-G differential equation for which a solution is readily available in the shape of the MEJER-G functions is presented and used in the discretization of the nonlinear partial-differential equation describing the dynamics of the system.
TL;DR: In this article, a two-degree-of-freedom model of a primary nonlinear structure endowed with the hysteretic vibration absorber is investigated to explore transfers of energy from the structure to the absorber resulting into optimal vibration amplitude reduction.
Abstract: The method of multiple scales is adopted to investigate the dynamic response of a nonlinear Vibration Absorber (VA) whose constitutive behavior is governed by hysteresis with pinching. The asymptotic analysis is first devoted to study the response of the absorber to harmonic excitations and to evaluate its sensitivity to the main constitutive parameters. The frequency response obtained in closed form allows to carry out the stability analysis together with a parametric study leading to behavior charts characterizing multi-valued softening/hardening responses or single-valued, quasi-linear responses. A two-degree-of-freedom model of a primary nonlinear structure endowed with the hysteretic vibration absorber is investigated to explore transfers of energy from the structure to the absorber resulting into optimal vibration amplitude reduction. The asymptotic solution is proved to be in good agreement with the numerical solution obtained via continuation. The asymptotic approach is embedded into a differential evolutionary algorithm to obtain a multi-parameter optimization procedure by which the optimal hysteresis parameters are found.
11 Jan 2017-Microsystem Technologies-micro-and Nanosystems-information Storage and Processing Systems
TL;DR: In this paper, the effects of non-ideal boundary conditions (BCs) on fundamental parametric resonance behavior of fluid conveying clamped microbeams are investigated by considering different system parameters.
Abstract: This study aimed to present the effects of non-ideal boundary conditions (BCs) on fundamental parametric resonance behavior of fluid conveying clamped microbeams. Non-ideal BCs are modelled by using the weighting factor (k). Equations of motion are obtained by using the Hamilton’s Principle. A perturbation technique, method of multiple scales, is applied to solve the non-linear equations of motions. In this study, frequency-response curves of fundamental parametric resonance are plotted and the effects of non-ideal BCs are shown. Besides, instability areas of microbeams under ideal and non-ideal BCs are investigated by considering different system parameters. Numerical results show that instability areas significantly changed by the effect of non-ideal BCs.
TL;DR: In this article, the global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated.
Abstract: Global dynamics of an autoparametric beam structure derived from a flexible L-shaped beam subjected to base excitation with one-to-two internal resonance and principal resonance are investigated. Hamilton’s principle is employed to obtain the nonlinear partial differential governing equations of the multi-beam structure. A linear theoretical analysis is implemented to derive the modal functions, and the orthogonality conditions are established. The analytical modal functions obtained are then adopted to truncate the partial differential governing equations into a set of coupled nonlinear ordinary differential equations via the Galerkin’s procedure. The method of multiple scales is applied to yield a set of autonomous equations of the first-order approximations to the response of the dynamical system. The Energy-Phase method is used to study the global bifurcation and multi-pulse chaotic dynamics of such autoparametric system. The present analysis indicates that the chaotic dynamics results from the existence of Silnikov’s type of homoclinic orbits and the parameter set for which the system may exhibit chaotic motions in the sense of Smale horseshoes are predicted analytically. Numerical simulations are performed to validate the theoretical results.
TL;DR: In this paper, a theoretical solution is formulated to analyze the vibration behaviors of circular diaphragm-type piezoactuators based on the Hamilton's principle and Rayleigh-Ritz method, which are particular suitable for modeling the deflection of multilayer structures.
Abstract: A theoretical solution is formulated to analyze the vibration behaviors of circular diaphragm-type piezoactuators based on the Hamilton’s principle and Rayleigh-Ritz method, which are particular suitable for modeling the deflection of multilayer structures. Each of the actuator three layers is considered as an individual layer in the modeling. The energy associated with the solution includes the kinetic energy of the actuator, the elastic potential energy of the various layers, the electric potential energy in the piezodisc, and the work done by the force of electric filed. The transverse displacement is separated into a time dependence term and a mode shape term, then the vibrational governing equation is derived using the functional variation, and is approximately solved through the method of multiple scales. Moreover, added mass loads are introduced to the diaphragm center for the sake of decreasing the resonant frequency, where many MEMS devices, such as gas micropumps and ejectors, have a higher working efficiency. The proposed analytical solution is validated numerically via the finite element method (FEM) and experimentally via measurements; the theoretical results are found to be in good agreement with the FEM results as well as with the experimental results. Furthermore, the effects of mass loads, geometric dimensions and material properties of the piezoactuator on the resonant frequency are discussed.
TL;DR: In this paper, an analytical expression as a function of the system parameters describing the forced vibration of a spinning composite shaft in the neighborhood of the primary resonance is obtained, and the results are compared with both one and two modes, and it is shown that although the excitation is tuned in the neighbourhood of the first mode, one-mode discretization is not sufficient and it leads to inaccurate results.
Abstract: In this paper, nonlinear dynamics of an unbalanced composite spinning shaft are studied. Extensional–flexural–flexural–torsional equations of motion are derived via utilizing the three-dimensional constitutive relations of the material and Hamilton’s principle. The gyroscopic effects, rotary inertia and coupling due to material anisotropy are included, while the shear deformation is neglected. To analyze the rotor dynamic behavior, the full form of the equations without any simplification assumption (e.g., stretching or shortening assumption) is used. The method of multiple scales is applied to the discretized equations. An analytical expression as a function of the system parameters describing the forced vibration of a spinning composite shaft in the neighborhood of the primary resonance is obtained. The discretization is done with both one and two modes, and the results are compared. It is shown that although the excitation is tuned in the neighborhood of the first mode, one-mode discretization is not sufficient and it leads to inaccurate results. It shows the necessity of employing at least two modes in discretization due to the coupling in the equations. The effects of the external damping, eccentricity and the lamination angle on the vibration amplitude are investigated. In addition, the effect of the extensional–torsional coupling on the frequency response curves is investigated. To validate the perturbation results, numerical simulation is used.