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• stability  (20)
• perturbation methods  (13)
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• 1
Electronic Resource
Springer
Nonlinear dynamics 12 (1997), S. 1-37
ISSN: 1573-269X
Keywords: Chaos ; dynamic bifurcations ; matchedasymptotic expansions ; averaging ; nonlinearoscillator ; perturbation methods
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A study is made of the dynamics of oscillating systems with a slowly varying parameter. A slowly varying forcing periodically crosses a critical value corresponding to a pitchfork bifurcation. The instantaneous phase portrait exhibits a centre when the forcing does not exceed the critical value, and a saddle and two centres with an associated double homoclinic loop separatrix beyond this value. The aim of this study is to construct a Poincaré map in order to describe the dynamics of the system as it repeatedly crosses the bifurcation point. For that purpose averaging methods and asymptotic matching techniques connecting local solutions are applied. Given the initial state and the values of the parameters the properties of the Poincaré map can be studied. Both sensitive dependence on initial conditions and (quasi) periodicity are observed. Moreover, Lyapunov exponents are computed. The asymptotic expressions for the Poincaré map are compared with numerical solutions of the full system that includes small damping.
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• 2
Electronic Resource
Springer
Nonlinear dynamics 18 (1999), S. 69-87
ISSN: 1573-269X
Keywords: system identification ; nonlinear beam ; perturbation methods ; primary resonance
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A procedure is presented for using a primary resonance excitation in experimentally identifying the nonlinear parameters of a model approximating the response of a cantilevered beam by a single mode. The model accounts for cubic inertia and stiffness nonlinearities and quadratic damping. The method of multiple scales is used to determine the frequency-response function for the system. Experimental frequency- and amplitude-sweep data is compared with the prediction of the frequency-response function in a least-squares curve-fitting algorithm. The algorithm is improved by making use of experimentally known information about the location of the bifurcation points. The method is validated by using the extracted parameters to predict the force-response curves at other nearby frequencies. We then compare this technique with two other techniques that have been presented in the literature. In addition to the amplitude- and frequency-sweep technique presented, we apply a backbone curve- fitting technique and a time-domain technique to the second mode of a cantilevered beam. Differences in the parameter estimates are discussed. We conclude by discussing the limitations encountered for each technique. These include the inability to separate the nonlinear curvature and inertia effects and problems in estimating the coefficients of small terms with the time-domain technique.
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• 3
Electronic Resource
Springer
Nonlinear dynamics 19 (1999), S. 135-158
ISSN: 1573-269X
Keywords: perturbation methods ; higher-order approximations ; dynamical systems ; codimension ; stability
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Higher-order multiple-scale methods for general multiparameter mechanical systems are studied. The role played by the control and imperfection parameters in deriving the perturbative equations is highlighted. The definition of the codimension of the problem, borrowed from the bifurcation theory, is extended to general systems, excited either externally or parametrically. The concept of a reduced dynamical system is then invoked. Different approaches followed in the literature to deal with reconstituted amplitude equations are discussed, both in the search for steady-state solutions and in the analysis of stability. Four classes of methods are considered, based on the consistency or inconsistency of the approach, and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Finally, three examples are illustrated to corroborate the findings and to show the quantitative differences between the various approaches.
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• 4
Electronic Resource
Springer
Nonlinear dynamics 19 (1999), S. 173-193
ISSN: 1573-269X
Keywords: fluid conveying pipes ; high-frequency pulsating fluid ; separation of slow and fast motion ; stability ; nonlinear dynamics
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.
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• 5
Electronic Resource
Springer
Nonlinear dynamics 19 (1999), S. 313-332
ISSN: 1573-269X
Keywords: double pendulum system ; double Hopf bifurcation ; stability ; chaos
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.
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• 6
Electronic Resource
Springer
Nonlinear dynamics 3 (1992), S. 225-243
ISSN: 1573-269X
Keywords: Nonlinear oscillations ; chaos ; escape ; perturbation methods
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The paper is devoted to the study of common features in regular and strange behavior of the three classic dissipative softening type driven oscillators: (a) twin-well potential system, (b) single-well potential unsymmetric system and (c) single-well potential symmetric system. Computer simulations are followed by analytical approximations. It is shown that the mathematical techniques and physical concepts related to the theory of nonlinear oscillations are very useful in predicting bifurcations from regular, periodic responses to cross-well chaotic motions or to escape phenomena. The approximate analysis of periodic, resonant solutions and of period doubling or symmetry breaking instabilities in the Hill's type variational equation provides us with closed-form algebraic simple formulae; that is, the relationship between critical system parameter values, for which strange phenomena can be expected.
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• 7
Electronic Resource
Springer
Nonlinear dynamics 7 (1995), S. 11-35
ISSN: 1573-269X
Keywords: Periodic solutions ; stability ; local bifurcations ; Fourier series
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract This paper explores the application of the method of variable-coefficient harmonic balance to nonautonomous nonlinear equations of the form XsF(X, t:λ), and in particular, a one-degree-of-freedom nonlinear oscillator equation describing escape from a cubic potential well. Each component of the solution, X(t), is expressed as a truncated Fourier series of superharmonics, subharmonics and ultrasubharmonics. Use is then made of symbolic manipulation in order to arrange the oscillator equation as a Fourier series and its coefficient are evaluated in the traditional way. The time-dependent coefficients permit the construction of a set of amplitude evolution equations with corresponding stability criteria. The technique enables detection of local bifurcations, such as saddle-node folds, period doubling flips, and parts of the Feigenbaum cascade. This representation of the periodic solution leads to local bifurcations being associated with a term in the Fourier series and, in particular, the onset of a period doubled solution can be detected by a series of superharmonics only. Its validity is such that control space bifurcation diagrams can be obtained with reasonable accuracy and large reductions in computational expense.
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• 8
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 401-420
ISSN: 1573-269X
Keywords: Chaos ; perturbation methods ; elliptic functions ; differential equations
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract We investigate the system $$\ddot x - x\cos \varepsilon 1 + x^3 = 0$$ in which ε≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the $$x - \dot x$$ phase plane which exhibits a center at the origin when cos ε1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 〉 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.
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• 9
Electronic Resource
Springer
Nonlinear dynamics 1 (1990), S. 459-475
ISSN: 1573-269X
Keywords: Regularization ; stability ; constrained multibody systems ; dynamics of multibody systems ; non-holonomic ; singularity
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract In the analysis of multibody dynamics, we are often required to deal with singularity problems where the constraint Jacobian matrix may become less than full rank at some instantancous configurations. This creates numerical instability which will affect the performance of the mechanical system. A modification procedure of the constraints when they vanish or become linearly dependent is proposed to regularize the dynamics of the system. A distinction between the asymptotic stability due to the representation of the constraints (at the velocity and acceleration level), and the one due to the singularity is discussed in full in this paper. It is shown that Baumgarte technique could be extended to accommodate the representation of the constraints in the neighborhood of singularity. A two link planar manipulator undergoing large motion and passing through a singular configuration is used to illustrate the proposed stability technique.
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• 10
Electronic Resource
Springer
Nonlinear dynamics 10 (1996), S. 333-357
ISSN: 1573-269X
Keywords: Pipes ; parametric excitation ; nonlinearity ; chaos ; multiple time scales ; harmonic balancing ; stability
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Chaotic motions of a simply supported nonlinear pipe conveying fluid with harmonie velocity fluetuations are investigated. The motions are investigated in two flow velocity regimes, one below and above the critical velocity for divergence. Analyses are carried out taking into account single mode and two mode approximations in the neighbourhood of fundamental resonance. The amplitude of the harmonic velocity perturbation is considered as the control parameter. Both period doubling sequence and a sudden transition to chaos of an asymmetric period 2 motion are observed. Above the critical velocity chaos is explained in terms of periodic motion about the equilibrium point shifting to another equilibrium point through a saddle point. Phase plane trajectories, Poincaré maps and time histories are plotted giving the nature of motion. Both single and two mode approximations essentially give the same qualitative behaviour. The stability limits of trivial and nontrivial solutions are obtained by the multiple time scale method and harmonic balance method which are in very good agreement with the numerical results.
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