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The objective of this study is to develop, simulate and verify experimentally a model of a nonlinear spring, based on the principle of a cantilevered beam with a mass on its tip, and whose overall lateral vibration is constrained ...
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The objective of this study is to develop, simulate and verify experimentally a model of a nonlinear spring, based on the principle of a cantilevered beam with a mass on its tip, and whose overall lateral vibration is constrained by a specially shaped rigid boundary. The focus here is the use of this spring for vibration reduction applications. The modeling approach uses concepts of plane kinematics of rigid bodies, combined with quasi-static analysis to develop suitable equations of motion for a base-excited spring with a ninth-order geometric nonlinearity. In addition, a parametric identification procedure is implemented for obtaining the required coefficients for computational simulations. An approximated analytical solution to the model is completed with the aid of the method of harmonic balance and its stability is assessed through Floquet theory. Finally, the model is experimentally verified, with the use of two specimens, fabricated specifically for this study. The model, simulations and experimental measurements show the hardening and broadband behavior of the nonlinear spring.
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The non-linear natural frequencies of the first three modes of a clamped tapered beam are investigated. The mathematical model is derived using the Euler-Lagrange method and the continuous system is discretized using the assumed m...
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The non-linear natural frequencies of the first three modes of a clamped tapered beam are investigated. The mathematical model is derived using the Euler-Lagrange method and the continuous system is discretized using the assumed mode method. The resulted uni-modal nonlinear equation of motion was solved using the harmonic balance (HB) to obtain approximate analytical expressions for the nonlinear natural frequencies. Results were obtained for two types of taper; double taper, i.e. the beam width and thickness are varied linearly along the beam axis and single taper "wedge shaped beams", i.e. the variation is in thickness only. The effects of vibration amplitude and taper ratio on the nonlinear natural frequencies for the first three modes are obtained and presented in non-dimensional form.
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Abstract Subjected to high level forcing, flexible and curved beams exhibit pronounced geometrical nonlinearities. In particular, intrinsic nonlinearities of cantilevers are different from their counterparts with end-constrained b...
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Abstract Subjected to high level forcing, flexible and curved beams exhibit pronounced geometrical nonlinearities. In particular, intrinsic nonlinearities of cantilevers are different from their counterparts with end-constrained boundaries and the combination of the enhanced nonlinear-inertia effects with initial curvature creates harsh demand on the modeling, numerical simulation and understanding of associated physical phenomena. This paper investigates the salient nonlinear features in a curved cantilever beam, with particular attention paid to the inertia-induced effects through both linear and nonlinear analyses. An inextensible condensation model, with the consideration of the initial curvature, is proposed based on a geometrically exact model for an Euler–Bernoulli cantilever beam. The free boundary of the cantilever gives rise to more significant longitudinal motion, which increases the inertia effects in the beam vibration which is in turn enhanced by the initial curvature. Specific techniques are proposed to numerically implement the developed model with increased accuracy and robustness. Numerical simulations are then conducted to validate the proposed model through comparisons with the finite element method, examine the assumptions underpinning the model and explore the salient physical features, in particular the inertia-induced effects in both linear and nonlinear cases. Results show a decrease in the natural frequencies due to the initial curvature effect, a transition of the first mode from hardening to softening caused by enhanced curvature-induced inertia effect, and a pronounced asymmetry of the higher order modes with respect to frequencies.
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Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two ...
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Bifurcation analysis of the nonlinear vibration of an inextensible cantilever beam is analyzed by using the nonlinear normal mode concept. Two flexural modes of the cantilever beam, one in each transverse plane is considered. Two degrees-of-freedom nonlinear model for the vibration in the transverse direction is obtained by the discretization of the governing equation using Galerkins method based on the eigenmodes in each direction. The method of multiple scales is used to derive two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of the dominant mode for the case of 1:1 internal resonance. The bifurcation diagrams are computed considering the frequency of excitation and the magnitude of the excitation as the control parameters. The stability of the fixed point is determined by examining the eigenvalues of the Jacobian matrix. The results show that a saddle-node-type bifurcation of the solution can occur under certain parameter conditions.
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A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam's inextensibility - local arc length preservation - rather than traditional extensible effects attributed to fully res...
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A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam's inextensibility - local arc length preservation - rather than traditional extensible effects attributed to fully restricted boundary conditions. Enforcing inextensibility leads to: nonlinear stiffness terms, which appear as quasilinear and semilinear effects, as well as nonlinear inertia effects, appearing as nonlocal terms that make the beam implicit in the acceleration. In this paper we discuss the derivation of the equations of motion via Hamilton's principle with a Lagrange multiplier to enforce the effective inextensibility constraint. We then provide the functional framework for weak and strong solutions before presenting novel results on the existence and uniqueness of strong solutions. A distinguishing feature is that the two types of nonlinear terms present independent challenges: the quasilinear nature of the stiffness forces higher topologies for solutions, while the nonlocal inertia requires the consideration of Kelvin-Voigt type damping to close estimates. Finally, a modal approach is used to produce mathematically-oriented numerical simulations that provide insight into the features and limitations of the inextensible model.
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In this paper, we compare the experimentally and theoretically obtained single-mode responses of a cantilever beam. The analytical portion involves solving an integro-differential equation via the method of multiple scales. For th...
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In this paper, we compare the experimentally and theoretically obtained single-mode responses of a cantilever beam. The analytical portion involves solving an integro-differential equation via the method of multiple scales. For the single-mode response, a large discrepancy is found between theory and experiment for an assumed ideal clamp model. Through some experimental detective work, it was found, and later shown through analysis, that the substitution of a torsionally elastic end for the fixed support brought the theoretical and experimental results into excellent agreement. The torsional spring has both linear and nonlinear (cubic) stiffness components. (C) 2000 Elsevier Science Ltd. All rights reserved. [References: 34]
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Grazing behavior in soft impact dynamics of a harmonically based excited flexible cantilever beam is investigated. Numerical and experimental methods are employed to study the dynamic behavior of macro- and micro-scale cantilever ...
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Grazing behavior in soft impact dynamics of a harmonically based excited flexible cantilever beam is investigated. Numerical and experimental methods are employed to study the dynamic behavior of macro- and micro-scale cantilever beam-impactor systems. For off-resonance excitation at two and a half times the fundamental frequency, the response of the oscillating cantilever experiences period doubling as the separation distance or clearance between the beam axis and the contact surface is decreased. The nonlinear phenomenon is studied by using phase portraits, Poincar, sections, and spectral analysis. Motivated by atomic force microscopy, this general dynamic behavior is studied as a means to locating the separation distance corresponding to grazing where the contact force is minimized.
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The present work is concerned with the nonlinear dynamic analysis of a vibrating beam microgyroscope composed of a rotating cantilever beam with a tip mass at its end. The rigid mass is coupled to two orthogonal electrodes in the ...
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The present work is concerned with the nonlinear dynamic analysis of a vibrating beam microgyroscope composed of a rotating cantilever beam with a tip mass at its end. The rigid mass is coupled to two orthogonal electrodes in the drive and sense directions, which are attached to the rotating base. The microbeam is driven by an AC voltage in the drive direction, which induces vibrations in the orthogonal sense direction due to rotation about the microbeam axis. The electrode placed in the sense direction is used to measure the induced motions and extract the underlying angular speed. A reduced-order model of the gyroscope is developed using the method of multiple scales and used to examine its dynamic behavior.
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Piezoelectric wind energy harvesters consisting a bluff body and a piezoelectric cantilever beam have great potential for powering small-sized wireless devices. To achieve a higher energy output, the beam is designed for large def...
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Piezoelectric wind energy harvesters consisting a bluff body and a piezoelectric cantilever beam have great potential for powering small-sized wireless devices. To achieve a higher energy output, the beam is designed for large deformation. This results in the nonlinear nature of the energy harvesters. In this paper, a nonlinear model of a piezoelectric wind harvester with different geometrical parameters is developed. A comparison of the energy harvesting performance of these energy harvesters with different geometrical parameters is provided. Results show that the onset speeds of galloping for trapezoidal and exponential piezoelectric energy harvesters are significantly lower than those of rectangular beam. The average power output density of the beam with exponential shape is larger than trapezoidal and rectangular beams. Therefore, designing a beam with exponentially varying shape can obtain the largest power density and therefore can reduce the cost of piezoelectric wind energy harvester.
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The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are o...
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The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam. (C) 2017 Elsevier Ltd. All rights reserved.
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