Applied Asymptotic Methods in Nonlinear Oscillations

Митропольский, Ю. А.; Dao, Nguyen Van

doi:10.1007/978-94-015-8847-8

Cited by 64 publications

(45 citation statements)

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“…Engineers are generally more concerned about subcritical (hard) bifurcations as a small perturbation around the equilibrium position can lead the system to large amplitude vibration states, which the structure may not tolerate [5]. A number of authors have studied the "Mass-on-moving-Belt" model ("MB model" in the following), Tondl [6], Hetzler et al [7], Hetzler [8], Won and Chung [9], Nayfeh and Mook [10], Mitropolskii and Van Dao [11], Popp [12], Popp et al [13], Hinrichs et al [14], Andreaus and Casini [15], Awrejcewicz and Holicke [16], Awrejcewicz et al [17], which present various types of analysis of a mass-on-belt system with various kinds of friction laws, and provide in some cases, analytical expressions for the change between stick-slip and pure-slip oscillations. Many authors have attempted to use fast vibrations which in some respects seems to transform classical Coulomb friction into viscous-like damping ( [5,18]).…”

Section: Introductionmentioning

confidence: 99%

Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

2017

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The dynamical behavior of a single-degreeof-freedom system that experiences friction-induced vibrations is studied with particular interest on the possibility of the so-called hard effect of a subcritical Hopf bifurcation, using a velocity weakening-strengthening friction law. The bifurcation diagram of the system is numerically evaluated using as bifurcation parameter the velocity of the belt. Analytical results are provided using standard linear stability analysis and nonlinear stability analysis to large perturbations. The former permits to identify the lowest belt velocity (v lw ) at which the full sliding solution is stable, the latter allows to estimate a priori the highest belt velocity at which large amplitude stick-slip vibrations exist. Together the two boundaries [v lw , v up ] define the range where two equilibrium solutions coexist, i.e., a stable full sliding solution and a stable stick-slip limit cycle. The model is used to fit recent experimental observations.

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Section: Introductionmentioning

confidence: 99%

Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

2017

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“…It is supposed that v is close to the natural frequency w, namely: v2 = w2 + c:6. , (1.1) here 6. is a detuning parameter and c is a small positive one. Applying to (0.1) the asymptotic method [2] and using the amplitude and phase variables (a,()) given by we have…”

Section: Summary Of the Case Of Small Parametersmentioning

confidence: 99%

A numerical approach of chaotic motions in a Duffing-Van der Pol oscillator

Dao¹,

Định²,

Kim³

et al. 2007

Vietnam J. Mech.

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Abstract. In the present paper the influence of the excitation frequency (v) and the forcing amplitude (e) on the chaotic behaviour of the system governed by equation When e = 0, a > 0, I > 0 we have the classical Van der Pol equation which represents a self-excited oscillator with the amplitude a. = 2/ .j;;; and frequency w. Our discussion was focused upon variation of the excitation frequency v and the forcing amplitude e. The bifurcation diagrams for acquiring the overview of equation (0.1) and the Liapunov exponent method will be used [3,4,5,6,9]. For a concrete case, the parameter regions in which either periodic or chaotic motions exist were shown. In two preceding cases, the first case, when v is control parameter, it changes suddenly from periodic motion to chaotic motion , corresponding to llopf bifurcation. In the second case, it is the double-period process a nd leads to chaotic motion. Chaotic attractors illustrate the complexity of the motion of the system under consideration. SUMMARY OF THE CASE OF SMALL PARAMETERSFirst, we recall briefly some known results of deterministic motions in (0.1) for the case of smallness of the coefficients. It is supposed that v is close to the natural frequency w, namely: v2 = w2 + c:6. , (1.1) here 6. is a detuning parameter and c is a small positive one. Applying to (0.1) the asymptotic method [2] and using the amplitude and phase variables (a,()) given by we have

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“…The opposite case will be investigated in the next section. The smallness of parameters is characterized by introducing a small positive parameter E. For this case the asymptotic method of Krylov -Bogoliubov -Mitropolskii (K-B-M) [1,2] is used to seek the approximate solutions aud to study their stability.…”

Section: The Case Of Small Parametersmentioning

confidence: 99%

Van der Pol's oscillator under the parametric and forced excitations

Dao¹,

Định

Kim

2007

Vietnam J. Mech.

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Abstract.Van der Pol's oscillator under parametric an.d forced excitations is studied. The case where the system contains a small parametrer being quasilinear and the general case (without assumption on the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by means of the Krylov-Bogoliubov-Mitropolskii technique, their averaging is performed, frequencyamplitude and resonance curves are studied, on the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown.

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Applied Asymptotic Methods in Nonlinear Oscillations

Cited by 64 publications

References 1 publication

Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

Subcritical bifurcation in a self-excited single-degree-of-freedom system with velocity weakening–strengthening friction law: analytical results and comparison with experiments

A numerical approach of chaotic motions in a Duffing-Van der Pol oscillator

Van der Pol's oscillator under the parametric and forced excitations

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