A Re-Examination of Various Resonances in Parametrically Excited Systems1

Abstract The dynamics of parametrically excited systems are characterized by distinct types of resonances including parametric, combination, and internal. Existing resonance conditions for these instability phenomena involve natural frequencies and thus are valid when the amplitude of the parametric excitation term is zero or close to zero. In this paper, various types of resonances in parametrically excited systems are revisited and new resonance conditions are developed such that the new conditions are valid in the entire parametric space, unlike existing conditions. This is achieved by expressing resonance conditions in terms of “true characteristic exponents” which are defined using characteristic exponents and their non-uniqueness property. Since different types of resonances may arise depending upon the class of parametrically excited systems, the present study has categorized such systems into four classes: linear systems with parametric excitation, linear systems with combined parametric and external excitations, nonlinear systems with parametric excitation, and nonlinear systems with combined parametric and external excitations. Each class is investigated separately for different types of resonances, and examples are provided to establish the proof of concept. Resonances in linear systems with parametric excitation are examined using the Lyapunov–Poincaré theorem, whereas Lyapunov–Floquet transformation is utilized to generate a resonance condition for linear systems with combined excitations. In the case of nonlinear parametrically excited systems, nonlinear techniques such as “time-dependent normal forms” and “order reduction using invariant manifolds” are employed to express various resonance conditions. It is found that the forms of new resonance conditions obtained in terms of ‘true characteristic exponents’ are similar to the forms of existing resonance conditions that involve natural frequencies.

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Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

Journal of Applied Mechanics ◽

10.1115/1.3125852 ◽

1988 ◽

Vol 55 (3) ◽

pp. 702-705 ◽

Cited By ~ 71

Author(s):

Y. K. Lin ◽

Guoqiang Cai

Keyword(s):

Nonlinear System ◽

Nonlinear Systems ◽

Probability Distribution ◽

White Noise ◽

Probability Density ◽

Detailed Balance ◽

Stationary Probability ◽

Probability Flow ◽

Parametric And External Excitations ◽

White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

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The Response of Nonlinear Systems to Modulated High-Frequency Input

14th Biennial Conference on Mechanical Vibration and Noise: Dynamics and Vibration of Time-Varying Systems and Structures ◽

10.1115/detc1993-0130 ◽

1993 ◽

Author(s):

S. A. Nayfeh ◽

A. H. Nayfeh

Keyword(s):

Nonlinear Systems ◽

Natural Frequency ◽

Amplitude Modulation ◽

Carrier Frequency ◽

High Frequency ◽

Degree Of Freedom ◽

Resonant Excitation ◽

Single Degree Of Freedom ◽

Single Degree ◽

Different Types

Abstract We study the response of a single-degree-of-freedom system with cubic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system. The only restriction on the amplitude modulation is that it contain frequencies much lower than the carrier frequency of the excitation. We apply the theory to different types of amplitude modulation and find that resonant excitation of the system may occur under some conditions.

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Long-time evolution and regions of existence of parametrically excited nonlinear cross-waves in a tank

Journal of Fluid Mechanics ◽

10.1017/s0022112089003095 ◽

1989 ◽

Vol 209 ◽

pp. 249-263 ◽

Cited By ~ 10

Author(s):

Lev Shemer ◽

Eliezer Kit

Keyword(s):

Boundary Condition ◽

Parametric Excitation ◽

Wave Breaking ◽

Model Equation ◽

Numerical Study ◽

Parametrically Excited ◽

Numerical Studies ◽

Scaling Procedure ◽

Long Time ◽

Cross Waves

Results of an experimental and numerical study of parametrically excited nonlinear cross-waves in the vicinity of the cut-off frequency, are reported. Experiments are performed at three cross-wave modes and in the whole range of existence of cross-waves. Numerical studies are based on the solution of the nonlinear Schrödinger equation with a boundary condition at the wavemaker which corresponds to parametric excitation. The validity of the scaling procedure adopted in the model is verified experimentally. Dissipation is incorporated in the model equation and in the wavemaker boundary condition. The influence of the wave breaking on the range of existence of cross-waves is discussed and the relation between the maximum possible steepness of cross-waves and the limits of their existence is obtained.

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Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd2π-Periodic Functions

Complexity ◽

10.1155/2017/6020213 ◽

2017 ◽

Vol 2017 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Tamás Kalmár-Nagy ◽

Márton Kiss

Keyword(s):

Nonlinear Systems ◽

Linear Operator ◽

Linear Systems ◽

Chaotic Dynamics ◽

Periodic Points ◽

Complex Behavior ◽

Periodic Functions ◽

Infinite Dimensional ◽

Backward Shift ◽

Principal Value

Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequencesl2displays chaotic dynamics. Here we construct the corresponding operatorCon the space of2π-periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunction of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories ofC.

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Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies

Journal of Sound and Vibration ◽

10.1016/0022-460x(82)90316-9 ◽

1982 ◽

Vol 85 (4) ◽

pp. 459-472 ◽

Cited By ~ 16

Author(s):

E.G. Tezak ◽

A.H. Nayfeh ◽

D.T. Mook

Keyword(s):

Natural Frequencies ◽

Parametrically Excited ◽

Non Linear

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Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3426369 ◽

1978 ◽

Vol 100 (3) ◽

pp. 209-213 ◽

Cited By ~ 4

Author(s):

G. Langholz ◽

M. Sokolov

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Nonlinear Systems ◽

Control Theory ◽

Linear Systems ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical ◽

Modern Control Theory ◽

Open Question ◽

Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

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Equivalent linear systems of nonlinear systems

Journal of Sound and Vibration ◽

10.1016/j.jsv.2019.115126 ◽

2020 ◽

Vol 469 ◽

pp. 115126 ◽

Cited By ~ 2

Author(s):

Tobias Friis ◽

Marius Tarpø ◽

Evangelos I. Katsanos ◽

Rune Brincker

Keyword(s):

Nonlinear Systems ◽

Linear Systems ◽

Equivalent Linear

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Bursting Oscillations as well as the Mechanism in a Filippov System with Parametric and External Excitations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420501680 ◽

2020 ◽

Vol 30 (12) ◽

pp. 2050168

Author(s):

Hongfang Han ◽

Qinsheng Bi

Keyword(s):

Autonomous System ◽

Type System ◽

External Excitation ◽

Nonsmooth Boundary ◽

Bursting Oscillations ◽

Fold Bifurcation ◽

Different Types ◽

Boundary Equilibrium ◽

Parametric And External Excitations ◽

Filippov Type

The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.

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