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1.
Jerzy Warminski 《Nonlinear dynamics》2010,61(4):677-689
Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented
in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small
excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced
to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis
of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical
simulations and an essential difference between linear and nonlinear modes is pointed out. 相似文献
2.
In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der Pol–Duffing oscillator system is given. 相似文献
3.
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
4.
Alexander F. Vakakis 《Nonlinear dynamics》1992,3(2):123-143
The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a 1–1 internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given. 相似文献
5.
This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this
new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time
series, Lyapunov exponents’ spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic,
its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the
new system has strong hyperchaotic dynamics in itself. The Kaplan–Yorke dimension, Poincaré sections and the frequency spectra
are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes
an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process
reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such
four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not
only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and
then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions
of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical
simulation results show consistency with our theoretical analysis. 相似文献
6.
7.
Accuracy and precision of position control of hydraulic systems are key parameters for engineering applications in order to
set more economical and quality systems. In this context, this paper presents modeling and position control of a hydraulic
actuation system consisting of an asymmetric hydraulic cylinder driven by a four way, three position proportional valve. In
this system model, the bulk modulus is considered as a variable. In addition, the Hybrid Fuzzy-PID Controller with Coupled
Rules (HFPIDCR) is proposed for position control of the hydraulic system and its performance is tested by simulation studies.
The novel aspect of this controller is to combine fuzzy logic and PID controllers in terms of a switching condition. Simulation
results of the HFPIDCR based controller are compared with the results of classical PID, Fuzzy Logic Controller (FLC), and
Hybrid Fuzzy-PID controller (HFPID). As a result, it is demonstrated that Hybrid Fuzzy PID Controller with Coupled Rules is
more effective than other controllers. 相似文献
8.
A hyperchaotic system is often used to generate secure keys or carrier wave for secure communication and the realistic hyperchaotic
circuit often is made of capacitor, nonlinear resistor unit and induction coil. Parameters are often fixed in these hyperchaotic
circuits and the hyperchaotic property of the system can be estimated by using a scheme of synchronization and time series
analysis. In this paper, a time-varying hyperchaotic system is proposed by introducing changeable electric power source into
the circuit; the changeable electric power source is combined with induction coil or capacitor in series to generate changeable
output signals to excite the system. The diagrams of improved circuit are illustrated and critical parameters in experimental
circuits are presented; the Lyapunov exponent spectrum vs. external applied electric power source is calculated. It is confirmed
that the improved circuit always holds two positive Lyapunov exponents when the external electric power source works, and
the chaotic attractors are much too different from the original one; thus, a more changeable hyperchaotic system is constructed
in experiment. 相似文献
9.
The Stokes and Rayleigh Stokes problems for a flat plate in a viscoelastic fluid has recently been generalized to an edge
and an exact analytical solution is obtained. In this paper, the edge problem has further been extended to the case of a rectangular
pipe and exact solutions are obtained for Maxwell and second grade fluids. Also, the flow due to an oscillating edge problem
is extended to generalized Maxwell fluid. 相似文献
10.
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
11.
Resonant chaotic motions of a simply supported rectangular thin plate with parametrically and externally excitations are analyzed
using exponential dichotomies and an averaging procedure for the first time. The formulas of the rectangular thin plate are
derived by a von Karman type equation and the Galerkin’s approach. The critical condition to predict the onset of chaotic
motions for the full system is obtained by developing a Melnikov function containing terms from the non-hyperbolic mode. We
prove that the non-hyperbolic mode of the thin plate does not affect the critical condition for the occurrence of chaotic
motions in the resonant case. Simulations also show that the chaotic motions of the hyperbolic subsystem are shadowed by the
chaotic motions for the full system of the rectangular thin plate. 相似文献
12.
Nonlinear Dynamics - This paper describes the chaos behavior of an in-plane tethered satellite system induced by atmospheric drag and the Earth’s oblateness. A commonly used model, the... 相似文献
13.
The nonlinear response of a two-degree-of-freedom nonlinear oscillating system to parametric excitation is examined for the
case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple
scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes
and phases. The steadystate solutions of the modulated equations and their stability are investigated. The trivial solutions
lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study
the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe
type of chaos.
Project supported by the National Natural Science Foundation of China (19472046) 相似文献
14.
Mathematic modeling and characteristic analysis for dynamic system with asymmetrical hysteresis in vibratory compaction 总被引:3,自引:0,他引:3
We investigate dynamic characteristics of vibratory compaction system with asymmetrical hysteresis. An asymmetrical model
derived from Bouc-Wen differential equation is employed to describe hysteretic behavior of vibration engineering. A practical
polynomial expression for hysteretic restoring force is deduced to be substituted into standard equation of the system, assuming
that the non-linearity of the restoring force is weak. An asymptotic method, which combines Krylov-Bogolyubov-Mitropolsky
(KBM) method with harmonic balance (HB) method, is applied to analyze steady-state responses of the asymmetrical hysteretic
system subjected to harmonic excitation. Dynamic responses, such as the restoring force time histories and frequency responses
of the system for the first order approximate, are obtained. Furthermore, numerical solution obtained using Runge-Kutta method
as well as results of experiments (asphalt compaction on the Beijing-Fuzhou highway) are compared with the asymptotic solution.
These results investigated that asymmetrical hysteretic model and the methods applied in this paper are quite appropriate
for engineering applications. 相似文献
15.
Necessary and sufficient conditions for the linearization of the one-dimensional Itô stochastic differential equations driven by fractional Brownian motion (fBm) are given. Stochastic integrating factor has been introduced. A modified Milstein method has been developed to obtain numerical solutions. Analytical solutions have been compared with the numerical solutions for linearizable equations. 相似文献
16.
A Predator–Prey type of dynamical systems with non-monotonic response function and time-periodic perturbation is considered in this paper. We present a proof for the number of equilibria in the unperturbed system at some parts of the parameter space. The perturbed system is a dynamical system defined by a periodic vector field. We present an alternative proof for a classical result on the period of the periodic solution. By using a numerical continuation method AUTO (Doedel et al., 1986 [9]), we present a bifurcation analysis for periodic solution of the perturbed system where we found fold, cusp and Swallowtail bifurcations. 相似文献
17.
Nonlinear Dynamics - This paper investigates the problem of attitude fault tolerant control (FTC) for satellites subject to multiple disturbances and actuator faults. Due to the limitation of the... 相似文献
18.
We investigate some of the instabilities in a single-machine quasi-infinite busbar system. The system's behavior is described by the so-called swing equation, which is a nonlinear second-order ordinary-differential equation with additive and multiplicative harmonic terms having the frequency . When 0, where 0 is the linear natural frequency of the machine, we use digital-computer simulations to exhibit some of the complicated responses of the machine, including period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). To predict the onset of these complicated behaviors, we use the method of multiple scales to develop an approximate first-order closed-form expression for the period-one responses of the machine. Then, we use various techniques to determine the stability of the analytical solutions. The analytically predicted period-one solutions and conditions for its instability are in good agreement with the digital-computer results. 相似文献
19.
The unsteady laminar flow of an incompressible micropolar fluid over a stretching sheet is investigated. The unsteadiness
in the flow and temperature fields is caused by the time-dependence of the stretching velocity and the surface temperature.
Effects of the unsteadiness parameter, material parameter and Prandtl number on the flow and heat transfer characteristics
are thoroughly examined. 相似文献
20.
Auxiliary bearings are utilized in practical installations of magnetically suspended rotating machines with the main functions
to provide support to the rotating machines during their non-operational period, and to protect the magnetic bearings and
the rotating assembly from being damaged due to power loss during operation. The relatively small clearances of these bearings,
which are typically half of that of the magnetic bearings, may at time cause contact between the rotor and these bearings
to occur even during normal operation of the rotating machines. The work presented herein examines the bifurcations in the
response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, which occurs during contact between
the rotor and the auxiliary bearings. Numerical results revealed the occurrence of period-doubling bifurcation resulting in
vibrations of period-2, -4, -8, -16 and -32, as well as quasi-periodic and chaotic vibrations. The results further showed
that for a relatively small rotor imbalance magnitude, which is within the prescribed level of certain classes of practical
rotating machinery, such nonlinear dynamical phenomena would not have been discovered had the auxiliary bearings forces been
omitted in the model of the rotor-bearing system. As these bearings are essential elements in practical installations of magnetically
supported rotating machines, failure to include them in the rotor-bearing model may result in incorrect prediction of the
rotor’s vibration response. 相似文献