1,721,080 research outputs found
In-plane and out-of-plane motion characteristics of microbeams with modal interactions
No full textThe three-dimensional nonlinear size-dependent motion characteristics of a microbeam are investigated numerically, with special consideration to one-to-one internal resonances between the in-plane and out-of-plane transverse modes. All of the in-plane and out-of-plane displacements and inertia are taken into account and Hamilton's principle, in conjunction with the modified couple stress theory, is employed to obtain the nonlinear partial differential equations governing the motions of the system in the in-plane and out-of-plane directions. The discretization procedure is carried out by applying the Galerkin technique to the partial differential equations of motion, yielding a set of nonlinear ordinary differential equations. A linear analysis is performed upon this set of equations so as to obtain the size-dependent natural frequencies of the system. The nonlinear analysis of the discretized equations of motion is carried out by employing the pseudo-arclength continuation technique, resulting in the resonant responses of the system. It is shown that, due to the presence of one-to-one internal resonances between the in-plane and out-of-plane transverse modes, an in-plane excitation can give rise to an out-of-plane displacement; the internal resonances also cause the occurrence of extra solution branches and new bifurcation points
Global dynamics of an axially moving buckled beam
No full textA parametric study for post-buckling analysis of an axially moving beam is conducted considering four different axial speeds in the supercritical regime. At critical speed, the trivial equilibrium configuration of this conservative system becomes unstable and the system diverges to a new non-trivial equilibrium configuration via a pitchfork bifurcation. Post-buckling analysis is conducted considering the system undergoing a transverse harmonic excitation. In order to obtain the equations of motion about the buckled state, first the equation of motion about the trivial equilibrium position is obtained and then transformed to the new coordinate, i.e. post-buckling configuration. The equations are then discretized using the Galerkin scheme, resulting in a set of nonlinear ordinary differential equations. Using direct time integration, the global dynamics of the system is obtained and shown by means of bifurcation diagrams of Poincaré maps. Other plots such as time traces, phase-plane diagrams, and Poincaré sections are also presented to analyze the dynamics more precisely
Nonlinear stability and bifurcations of an axially moving beam in thermal environment
No full textThe thermo-mechanical nonlinear dynamics of an axially moving beam with coupled longitudinal and transverse displacements subjected to a distributed harmonic external force is numerically investigated. This includes a case where the system is tuned to a three-to-one internal resonance between the first two transverse modes and a case where it is not is considered. Two coupled nonlinear partial differential equations for the longitudinal and transverse motions are obtained using Hamilton’s principle and constitutive relations, as well as taking into account the thermal effects. The Galerkin method is then used to discretize these equations into a set of coupled nonlinear ordinary differential equations. Two different techniques are employed to solve the resulting equations; the pseudo-arclength continuation method and direct time integration to investigate the periodic vibrations and the global dynamics of the system, respectively. The effect of different parameters on the dynamics of the system is investigated through the frequency-response curves of the system and the bifurcation diagrams of Poincare´ maps. Furthermore, time histories, phase-plane portraits, and fast Fourier transforms are presented for a few different system parameter sets. It is illustrated that the system shows a broad variety of rich dynamics, depending on system parameters and the temperature rise.Mergen H Ghayesh and Marco Amabil
Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam
No full textBased on the modified couple stress theory, the coupled longitudinal-transverse nonlinear behaviour of an imperfect microbeam is investigated numerically. The equations governing the longitudinal and transverse motions are obtained using Hamilton's principle for the system with an initial geometric imperfection. The Galerkin scheme is employed to discretize the two partial differential equations of motion, yielding a set of second-order nonlinear ordinary differential equations with coupled terms. This set is cast into new set of first-order nonlinear ordinary differential equations and solved by means of the pseudo-arclength continuation technique. The nonlinear resonant response of the system along with bifurcations are presented via frequency-response curves. Moreover, the effect of different system parameter on the frequency-response curves is highlighted. © 2013 Elsevier Ltd. All rights reserved
Free vibrations of higher-order quasi-3D viscoelastic bidirectional functionally graded plates
No full textThis study introduces a quasi-3D shear deformation theory to analyse the coupled
eight-parameter dynamics of bi-directional functionally graded Kelvin-Voigt viscoelastic plates. By focusing on the quasi-3D formulation, this work uniquely captures the influence of continuous axial and thickness gradation in material properties, utilising a power-law rule to determine effective properties. Viscoelasticity is modelled using the Kelvin-Voigt model to incorporate the energy dissipation of the composite plate structure. Eight governing equations, which are coupled through the in-plane and out-of-plane motions including stretching, are derived via Hamilton’s variational principle. A multi-modal discretisation is conducted using a weighted-residual method as the proposed solution for the in-plane and out-of-plane displacements. Thereafter, a numerical technique is employed to solve the resultant equations, obtaining both the real and imaginary parts of the natural frequencies. The proposed model is validated for the natural frequencies with an elastic counterpart from the literature; a homogenous version of the model is also validated against literature. The results obtained not only provide a comprehensive understanding of how the bi-
directional functionally graded viscoelastic material and geometrical parameters influence the
coupled eight-parameter dynamics of composite straight plates, but also propose a reliable
benchmark for such systems through a quasi-3D model. The investigation revealed that the
difference in frequency resulting from the application of both 2D and quasi-3D theories is most dominant in the case of thick plates
Size-dependent large-amplitude oscillations of microcantilevers
No full textThe size-dependent oscillations of a microcantilever with a tip (end) mass and a spring support undergoing a large-amplitude motion is analysed theoretically, taking into account curvature-related nonlinearities. Modelling small-size effects via use of the modified couple stress theory, the size-dependent potential and kinetic energies of the system are obtained. The continuous models for the motion behaviour of the microcantilever are developed via use of an energy method on the basis of Hamilton’s principle. Application of the centreline-inextensibility in oscillation course of the microcantilever results in a continuous model of the system with nonlinear inertial terms, which when coupled with curvature nonlinearities produces a highly nonlinear system. A weighted-residual method is then employed to truncate the continuous model, yielding the reduced-order model of the microcantilever motion with a generalised-coordinate-dependent mass matrix (due to inertial nonlinearities); a coupled continuation-time-integration method is then employed for the numerical simulations. The large-amplitude oscillation behaviour of the system is examined by constructing the frequency–responses and force-responses. The effect of the size of the end-mass on the nonlinear oscillation behaviour of the microcantilever is analysed. The importance of taking into account different nonlinearity sources is discussed. It is shown that the modified couple stress theory results in a stronger softening behaviour when compared to the classical continuum mechanics
A constrained broadband nonlinear energy harvester
Full text linkThis paper investigates the nonlinear electromechanical behaviour of a constrained bimorph piezoelectric energy harvester utilising a coupled multi-modal fully nonlinear model, for the first time. The electromechanical system is modelled via using the nonlinear beam theory of Euler-Bernoulli, while assuming an inextensible centreline, and the piezoelectric constitutive equations. The motion constraints are modelled as nonlinear springs, consisting of linear and cubic terms. The coupled electromechanical model, including the electrical circuit equation, is obtained for both parallel and series connections of the piezoelectric layers. The Galerkin scheme is used to discretise the partial differential-type model while retaining a large number of modes to guarantee converged results. Next, the electromechanical model consisting of geometric, inertial, impact, and piezoelectric-type nonlinearities is solved using a well-optimised continuation code. The proposed model and numerical method are verified through comparison to experimental data in the literature. It is shown that adding a tip mass of 50% of the bimorph cantilever mass increase the power output by 263%. Additionally, it is shown that for a power output threshold of 15 mW, the addition of motion constraints results in a 66.8% resonance bandwidth (normalised with respect to first natural frequency) which is almost 10 times the bandwidth of the unconstrained system
Parametric instability of microbeams in supercritical regime
No full textThe supercritical parametric instability of a microbeam subject to a time-dependent axial load is examined. The axial load is comprised of a constant mean value along with harmonic fluctuations. The mean value is increased from zero and is set to a value in the supercritical regime; the nonlinear parametric instability over the buckled state, due to the axial load variations, is examined. From the modelling perspective, based on the modified couple stress theory, the potential energy of the system is obtained in terms of the system parameters and displacement field. Moreover, the kinetic energy is formulated as a function of system parameters and the displacement field. The continuous model is developed by means of Hamilton’s principle and then truncated into a reduced-order model via a weighted-residual technique. Three different numerical techniques, i.e. the pseudo-arclength continuation method, a direct time-integration scheme, and an eigenvalue extraction, are employed to solve the high-dimensional reduced-order model. A stability analysis is also conducted via the Floquet theory. The nonlinear size-dependent parametric response of the system over the buckled configuration is presented in the form of frequency–response diagrams, force–response curves, time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections
Size-dependent behaviour of electrically actuated microcantilever-based MEMS
No full textIn this paper, the nonlinear size-dependent static and dynamic behaviours of a microelectromechanical system under an electric excitation are investigated. A microcantilever is considered for the modelling of the deformable electrode of the MEMS. The governing equation of motion is derived based on the modified couple stress theory (MCST), a non-classical model capable of capturing small-size effects. With the aid of a high-dimensional Galerkin scheme, the nonlinear partial differential equation governing the motion of the deformable electrode is converted into a reduced-order model of the system. Then, the pseudo-arclength continuation technique is used to solve the governing equations. In order to investigate the static behaviour and static pull-in instabilities, the system is excited only by the electrostatic actuation (i.e., a DC voltage). The results obtained for the static pull-in instability predicted by both the classical theory and MCST are compared. In the second stage of analysis, the nonlinear dynamic behaviour of the deformable electrode due to the AC harmonic actuation is investigated around the deflected configuration, incorporating size dependence
Motion limiting nonlinear dynamics of initially curved beams
Full text linkAn initially curved beam is considered and its motion is constrained using two elastic constraints; the corresponding non-smooth nonlinear transverse dynamics is investigated for the first time. A clamped-clamped beam with one axially movable end is modelled via Bernoulli-Euler beam theory together with the inextensibility condition, giving rise to nonlinear inertial terms along with nonlinear geometric terms. Furthermore, the damping is modelled via Kelvin-Voigt internal damping model. The proposed model is verified for linear and nonlinear behaviours via comparison to a finite element model. The impact between beam and constraints is incorporated via calculating its work contribution. The nonlinear equation of motion is derived while incorporating geometric, damping, inertial, and constraints nonlinearities. A series of spatial basis functions together with corresponding vibration modes are used as the proposed solution of the transverse displacement. A modal discretisation is performed via the weighted-residual method of Galerkin and the corresponding non-smooth terms are kept intact while conducting numerical integration. A numerical continuation technique is utilised to solve the resultant equations. The non-smooth response is obtained for various cases and the effects of several parameters are studied thoroughly
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