Comparative analysis of nonlinear resonances of a mechanical system with unsymmetrical piecewise characteristic of restoring force

Bovsunovskii, A. P.

doi:10.1007/s11223-007-0021-5

Cited by 5 publications

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References 16 publications

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“…(1.3) above) allows to rewrite (1.5) as a Lipschitz system. It takes simpler form F (t, x, ẋ) = −c ẋ + M Q(t) for an impact resonator and F (t, x, ẋ) = −c ẋ + M sin ωt for a cracked-body model (see [37] and [8], where numerical experiments are performed solely). In each of these situations equation (1.5) can be rewritten as system (1.1) with Lipschitzian g provided that the constant k 2 and the amplitude of the force F are sufficiently small.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Stability of Periodic Solutions for Nonsmooth Differential Equations with Application to the Nonsmooth van der Pol Oscillator

Buică

Llibre

Makarenkov

2009

SIAM J. Math. Anal.

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Abstract. In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for the Lipschitz systemẋ = εg (t, x, ε). Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our ones. By means of the results established we construct, for small ε, the curves of dependence of the amplitude of asymptotically stable 2π-periodic solutions of the nonsmooth van der Pol oscillatorü+ε (|u| − 1)u+(1+aε)u = ελ sin t, on the detuning parameter a and the amplitude of the perturbation λ. After, we compare the resonance curves obtained, with the resonance curves of the classical van der Pol oscillatorü + ε`u 2 − 1´u + (1 + aε)u = ελ sin t, which were first constructed by Andronov and Witt.Key words. Periodic solution, asymptotic stability, averaging theory, nonsmooth differential system, nonsmooth van der Pol oscillator. AMS subject classifications. 34C29, 34C25, 47H11.1. Introduction. In the present paper we study the existence, uniqueness and asymptotic stability of the T -periodic solutions for the systeṁwhere ε > 0 is a small parameter and the function gis T -periodic in the first variable and locally Lipschitz with respect to the second one. As usual a key role will be played by the averaging functionand we shall look for those periodic solutions that starts near some v 0 ∈ g −1 0 (0). In the case that g is of class C 1 , we remind the periodic case of the second Bogolyubov's theorem ([6], Ch. 1, § 5, Theorem II) which represents a part of the averaging principle: det (g 0 ) ′ (v 0 ) = 0 assures the existence and uniqueness, for ε > 0 small, of a T -periodic solution of system (1.1) in a neighborhood of v 0 , while the fact that all the eigenvalues of the Jacobian matrix (g 0 ) ′ (v 0 ) have negative real part, provides also its asymptotic stability. This theorem has a long history and it includes results by Fatou Second Bogolubov's theorem gave a theoretical justification of resonance phenomenons in many real physical systems. The most significant example is the classical lamp oscillator whose scheme is drawn at Fig. 1.1 and whose current u is described by the following second order differential equation Fig. 3-5).where R = εR 0 , M = εM 0 , ω 2 = 1 + εb, F (t) = ελ sin t, ε > 0 is assumed to be small and the lamp characteristic is drawn at Fig 1.2a. The analysis of bifurcation

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Section: Introductionmentioning

confidence: 99%

“…A prototypic process A prototypic device (a) where a driven mass is attached to a immovable beam via a spring with piecewise linear stiffness (b), see e.g. [8], [24] (Ch.I, p.16 and Ch.IV, p.100) and [37].…”

mentioning

confidence: 99%

Asymptotic Stability of Periodic Solutions for Nonsmooth Differential Equations with Application to the Nonsmooth van der Pol Oscillator

Buică

Llibre

Makarenkov

2009

SIAM J. Math. Anal.

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“…For convenience of discussion, equations (7) have been additionally numbered as per notation of points in Fig. 1.…”

Section: Introductionmentioning

confidence: 99%

“…Reliability Assessment of the Proposed Procedure. To assess the reliability of calculated results, we compare them with the data of numerical solution as obtained by the acceleration averaging method [6,7].…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Matveev

Bovsunovskii

2008

Strength Mater

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“…Using the γ 1 and Δγ values obtained [see (13) or (11) Results Reliability Assessment. Reliability of determination of the main vibrodiagnostic parameter of crack detection (the relative amplitude A) will be assessed here by comparing the calculated results with the data of the numerical solution obtained by an acceleration averaging method [23][24][25].…”

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confidence: 99%

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 1. Weak resonance

Matveev

Bovsunovskii

2008

Strength Mater

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The paper discusses an approximate analytical method for the calculation of vibrodiagnostic parameters of an elastic body with a closing crack, which is modeled by an elastic system with a single degree of freedom at a bilinear asymmetric characteristic of the restoring force, in the region of a weak 1/2-order subharmonic resonance.Keywords: forced vibrations, nonlinear vibrations, principal and subharmonic resonances, bilinear asymmetric characteristic of restoring force, closing fatigue crack, vibrodiagnostics of fatigue damage.Introduction. The ever increasing interest in studying vibrations of elastic bodies with a fatigue crack-like damage is motivated by the necessity of assessing any possible changes in the vibration status of structural elements during their long-term operation as well as by the need to develop efficient methods of vibrodiagnostics for such damage. The related research efforts are pursued along the following three main directions: (i) conventional, i.e., the determination of changes of natural frequencies of structural elements in the presence of cracks [1-7] and tracing a variation of the mode shape [8,9], (ii) determination of parameters of nonlinear effects of vibration processes, which are due to the presence of a closing or "breathing" crack [10][11][12][13][14], and (iii) assessment of the influence of a crack on the damping characteristics of elastic bodies [15][16][17][18].Establishing relations between a closing crack parameters and vibration parameters of an elastic body in crack-induced nonlinear resonances is still one of the less explored lines of inquiry as it is difficult to analytically solve the problem and set up a correct experiment.The previously performed analysis of available methods for calculating the forced vibrations of an elastic body with a closing crack [19] suggests that there are considerable difficulties associated with finding easy-to-use solutions. The resultant sets of complex constitutive equations did eventually require a numerical solution. The available closed solutions failed to describe or distorted the system behavior in the region of nonlinear resonances and sometimes in the principal resonance region too.Our proposed approximate analytical method of finding a solution for the region of a weak second-order superharmonic resonance has demonstrated a good agreement between the calculated results and the numerical solution data. In what follows we will discuss the feasibility of approximate calculation of a stationary vibration process for the case of 1/2-order subharmonic resonance on the basis of the main concepts as used for the superharmonic resonance case.Approximate Calculation Procedure. The problem as stated above is to be solved in view of the following: 1) An elastic body with a normal-rupture crack of relatively small dimensions, which enables us to neglect any difference in mode shape between alternating deformation half-cycles, is represented, for a given mode shape, as a single-degree-of-freedom system with an asymmetric bilinear chara...

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Comparative analysis of nonlinear resonances of a mechanical system with unsymmetrical piecewise characteristic of restoring force

Cited by 5 publications

References 16 publications

Asymptotic Stability of Periodic Solutions for Nonsmooth Differential Equations with Application to the Nonsmooth van der Pol Oscillator

Asymptotic Stability of Periodic Solutions for Nonsmooth Differential Equations with Application to the Nonsmooth van der Pol Oscillator

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 2. Strong resonance

Approximate analytical determination of vibrodiagnostic parameters of a cracked elastic body under subharmonic resonance. Part 1. Weak resonance

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