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Mei Cheng Guang Meng Jianping Jing 《Archive of Applied Mechanics (Ingenieur Archiv)》2006,76(3-4):215-227
The non-linear dynamic behaviors of a rotor-bearing-seal coupled system are investigated by using Muszynska’s non-linear seal fluid dynamic force model and non-linear oil film force, and the result from the numerical analysis is in agreement with the one from the experiment. The bifurcation of the coupled system is analyzed under different operating conditions. It is indicated that the dynamic behavior of the rotor-bearing-seal system depends on the rotation speed, seal clearance and seal pressure of the rotor-bearing-seal system. The system state trajectory, Poincaré maps, frequency spectra and bifurcation diagrams are constructed to analyze the dynamic behavior of the rotor center. Various non-linear phenomena in the coupled system, such as periodic motion and quasi-periodic motion are investigated. The results show that the system has the potential for chaotic motion. The study may contribute to a further understanding of the non-linear dynamics of such a rotor-bearing-seal coupled system. 相似文献
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An isotropic flexible shaft, acted by nonlinear fluid-induced forces generated from oil-lubricated journal bearings and hydrodynamic
seal, is considered in this paper. Dimension reductions of the rotor system were carried out by both the standard Galerkin
method and the nonlinear Galerkin method. Numerical simulations provide bifurcation diagrams, spectrum cascade, orbits of
the disk center and Poincaré maps, to demonstrate the dynamical behaviors of the system. The results reveal transitions, or
bifurcations, of the rotor whirl from being synchronous to non-synchronous as the unstable speed is exceeded. The non-synchronous
oil/seal whirl is a quasi-periodic motion. In the regime of quasi-periodic motion, the “windows” of multi-periodic motion
were found. The investigation shows that the nonlinear Galerkin method has an advantage over the standard one with the same
order of truncations, because the influences of higher modes are considered by the former. 相似文献
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The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially
moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion
of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential
equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal
field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear
forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation
amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion,
and then the periodic motion becomes chaotic motion by period-doubling bifurcation. 相似文献
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Mergen H. Ghayesh Niloofar Moradian 《Archive of Applied Mechanics (Ingenieur Archiv)》2011,81(6):781-799
The dynamical response of axially moving, partially supported, stretched viscoelastic belts is investigated analytically in
this paper. The Kelvin–Voigt viscoelastic material model is considered and material, not partial, time derivative is employed
in the viscoelastic constitutive relation. The string is considered as a three part system: one part resting on a nonlinear
foundation and two that are free to vibrate. The tension in the belt span is assumed to vary periodically over a mean value
(as it occurs in real mechanisms), and the corresponding equation of motion is derived by applying Newton’s second law of
motion for an infinitesimal element of the string. The method of multiple scales is applied to the governing equation of motion,
and nonlinear natural frequencies and complex eigenfunctions of the system are obtained analytically. Regarding the resonance
case, the limit-cycle of response is formulated analytically. Finally, the effects of system parameters such as axial speed,
excitation characteristics, viscousity and foundation modulus on the dynamical response, natural frequencies and bifurcation
points of system are presented. 相似文献
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This work investigates the influence of structural and aerodynamic nonlinearities on the dynamic behavior of a piezoaeroelastic
system. The system is composed of a rigid airfoil supported by nonlinear torsional and flexural springs in the pitch and plunge
motions, respectively, with a piezoelectric coupling attached to the plunge degree of freedom. The analysis shows that the
effect of the electrical load resistance on the flutter speed is negligible in comparison to the effects of the linear spring
coefficients. The effects of aerodynamic nonlinearities and nonlinear plunge and pitch spring coefficients on the system’s
stability near the bifurcation are determined from the nonlinear normal form. This is useful to characterize the effects of
different parameters on the system’s output and ensure that subcritical or “catastrophic” bifurcation does not take place.
Numerical solutions of the coupled equations for two different configurations are then performed to determine the effects
of varying the load resistance and the nonlinear spring coefficients on the limit-cycle oscillations (LCO) in the pitch and
plunge motions, the voltage output and the harvested power. 相似文献
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Takashi Ikeda 《Nonlinear dynamics》2010,60(3):425-441
Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected
to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency
of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure
is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical
analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for
the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid
motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the
structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes
of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter
is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated
motions. The theoretical results are in quantitative agreement with the experimental data. 相似文献
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Geometrically nonlinear vibrations of FGM rectangular plates in thermal environments are investigated via multi-modal energy
approach. Both nonlinear first-order shear deformation theory and von Karman theory are used to model simply supported FGM
plates with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear
ordinary differential equations with quadratic and cubic nonlinearities. A pseudo-arclength continuation and collocation scheme
is used and it is revealed that, in order to obtain the accurate natural frequency in thermal environments, an analysis based
on the full nonlinear model is unavoidable since the plate loses its original flat configuration due to thermal loads. The
effect of temperature variations as well as volume fraction exponent is discussed and it is illustrated that thermally deformed
FGM plates have stronger hardening behaviour; on the other hand, the effect of volume fraction exponent is not significant,
but modal interactions may rise in thermally deformed FGM plates that could not be seen in their undeformed isotropic counterparts.
Moreover, a bifurcation analysis is carried out using Gear’s backward differentiation formula (BDF); bifurcation diagrams
of Poincaré maps and maximum Lyapunov exponents are obtained in order to detect and classify bifurcations and complex nonlinear
dynamics. 相似文献
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Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports 总被引:1,自引:0,他引:1
Cracks appearing in the shaft of a rotary system are one of the main causes of accidents for large rotary machine systems.
This research focuses on investigating the bifurcation and chaotic behavior of a rotating system with considerations of various
crack depth and rotating speed of the system’s shaft. An equivalent linear-spring model is utilized to describe the cracks
on the shaft. The breathing of the cracks due to the rotation of the shaft is represented with a series truncated time-varying
cosine series. The geometric nonlinearity of the shaft, the masses of the shaft and a disc mounted on the shaft, and the viscoelasticity
of the supports are taken into account in modeling the nonlinear dynamic rotor system. Numerical simulations are performed
to study the bifurcation and chaos of the system. Effects of the shaft’s rotational speed, various crack depths and viscosity
coefficients on the nonlinear dynamic properties of the system are investigated in detail. The system shows the existence
of rich bifurcation and chaos characteristics with various system parameters. The results of this research may provide guidance
for rotary machine design, machining on rotary machines, and monitoring or diagnosing of rotor system cracks. 相似文献
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In this paper, the feedback linearization scheme is applied to the control of vehicle’s lateral dynamics. Based on the assumption
of constant driving speed, a second-order nonlinear lateral dynamical model is adopted for controller design. It was observed
in (Liaw, D.C., Chung, W.-C. in 2006 IEEE International Conference on Systems, Man, and Cybernetics, 2006) that the saddle-node
bifurcation would appear in vehicle dynamics with respect to the variation of the front wheel steering angle, which might
result in spin and/or system instability. The vehicle dynamics at the saddle node bifurcation point is derived and then decomposed
as an affine nominal model plus the remaining term of the overall system dynamics. Feedback linearization scheme is employed
to construct the stabilizing control laws for the nominal model. The stability of the overall vehicle dynamics at the saddle-node
bifurcation is then guaranteed by applying Lyapunov stability criteria. Since the remaining term of the vehicle dynamics contains
the steering control input, which might change system equilibrium except the designed one. Parametric analysis of system equilibrium
for an example vehicle model is also obtained to classify the regime of control gains for potential behavior of vehicle’s
dynamical behavior. 相似文献
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Yong-Gang Wang Xiao-Meng Li Dan Li Xin-Zhi Wang 《Archive of Applied Mechanics (Ingenieur Archiv)》2011,81(12):1925-1933
An integrated mathematic model and an efficient algorithm on the dynamical behavior of homogeneous viscoelastic corrugated
circular plates with shallow sinusoidal corrugations are suggested. Based on the nonlinear bending theory of thin shallow
shells, a set of integro-partial differential equations governing the motion of the plates is established from extended Hamilton’s
principle. The material behavior is given in terms of the Boltzmann superposition principle. The variational method is applied
following an assumed spatial mode to simplify the governing equations to a nonlinear integro-differential variation of the
Duffing equation in the temporal domain, which is further reduced to an autonomic system with four coupled first-order ordinary
differential equation by introducing an auxiliary variable. These measurements make the numerical simulation performs easily.
The classical tools of nonlinear dynamics, such as Poincaré map, phase portrait, Lyapunov exponent, and bifurcation diagrams,
are illustrated. The influences of geometric and physical parameters of the plate on its dynamic characteristics are examined.
The present mathematic model can easily be used to the similar problems related to other dynamical system for viscoelastic
thin plates and shallow shells. 相似文献
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Mergen H. Ghayesh Siavash Kazemirad Mohammad A. Darabi Pamela Woo 《Archive of Applied Mechanics (Ingenieur Archiv)》2012,82(3):317-331
Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking
into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived
through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal
motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations
with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode
functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on
the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric
analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass,
and temperature change. 相似文献
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A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation. 相似文献
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Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming
nonplanar motion of the beam, we introduce a nonlinear two-degree-of-freedom model by using Galerkin’s method based on the
first mode in each direction. The system turns out to have two normal modes. Using Synge’s stability concept, we examine the
stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré
map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion
obtained analytically. 相似文献
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The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence
their performance. To date, most widely used theoretical approaches, although opposite extremes with respect to complexity,
are nonlinear lumped-mass and finite-element models. While a lumped-mass approach is useful for a qualitative understanding
of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Finite-element
simulations, on the other hand, are adequate for static analysis, but inadequate for dynamic simulations. A third approach
is that of a reduced-order modeling which has gained significant attention for single-element micro-electromechanical systems
(MEMS), yet leaves an open amount of fundamental questions when applied to MEMS arrays. In this work, we employ a nonlinear
continuum-based model to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the array’s behavior in regions of its internal one-to-one, parametric,
and several internal three-to-one and combination resonances, which correspond to low, medium and large DC-voltage inputs,
respectively. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method
for the weakly nonlinear system in the afore mentioned resonance regions, respectively. Analytically obtained results of a
two-element system are verified numerically and complemented by a numerical analysis of a three-beam array. The dynamic behavior
of the two- and three-beam systems reveal several in- and out-of-phase co-existing periodic and aperiodic solutions. Stability
analysis of such co-existing solutions enables construction of a detailed bifurcation structure. This study of small-size
microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and
large-size arrays. Furthermore, the results of this present work motivate future experimental work and can serve as a guideline
to investigate the feasibility of new MEMS array applications. 相似文献
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Turbocharger rotor dynamics with foundation excitation 总被引:1,自引:0,他引:1
Guangchi Ying Guang Meng Jianping Jing 《Archive of Applied Mechanics (Ingenieur Archiv)》2009,79(4):287-299
To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger
rotor-bearing system is established which includes the engine’s foundation excitation and nonlinear lubricant force. The rotor
vibration response of eccentricity is simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear
rotor dynamics with various rotational speeds are studied. The results obtained by numerical simulation show that the differences
of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation
excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational
speed. 相似文献
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Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters 总被引:1,自引:0,他引:1
A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed.
The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric,
inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle
and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions
for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of
the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested
electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient
to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence
on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that,
for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation. 相似文献