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• stability  (6)
• method of multiple scales  (4)
• 1990-1994  (10)
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Year
• 1
Electronic Resource
Springer
Nonlinear dynamics 3 (1992), S. 261-271
ISSN: 1573-269X
Keywords: Scaling behavior ; coupled nonlinear oscillator ; method of multiple scales ; Duffing equation
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales. The scaling law for coupled Duffing oscillators is that the coupling intensity should be proportional to the total energy of the system.
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• 2
Electronic Resource
Springer
Nonlinear dynamics 6 (1994), S. 49-68
ISSN: 1573-269X
Keywords: Nonlinear vibration of a beam ; three mode interaction ; mid-plane stretching ; method of multiple scales
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,ωn . Three mode interaction (ω2 ≈ 3ω1 and ω3 ≈ ω1 + 2ω2) is considered and its influence on the response is studied. The case of two mode interaction (ω2 ≈ 3ω1) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.
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• 3
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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• 4
Electronic Resource
Springer
Nonlinear dynamics 3 (1992), S. 365-373
ISSN: 1573-269X
Keywords: Beam ; gravity effect ; method of multiple scales ; nonlinear oscillations
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
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• 5
Electronic Resource
Springer
Nonlinear dynamics 4 (1993), S. 655-670
ISSN: 1573-269X
Keywords: Fluidelasticity ; stability ; nonlinear ; chaos
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract In this paper, the planar dynamics of a nonlinearly constrained pipe conveying fluid is examined numerically, by considering the full nonlinear equation of motions and a refined trilinear-spring model for the impact constraints—completing the circle of several studies on the subject. The effect of varying system parameters is investigated for the two-degree-of-freedom (N=2) model of the system, followed by less extensive similar investigations forN=3 and 4. Phase portraits, bifurcation diagrams, power spectra and Lyapunov exponents are presented for a selected set of system parameters, showing some rather interesting, and sometimes unexpected, results. The numerical results are compared with experimental ones obtained previously. It is found that in the parameter space that includesN, there exists a subspace wherein excellent qualitative, and reasonably good (N=2) to excellent (N=4) quantitative agreement with experiment. In the latter case, excellent agreement is not only obtained in the threshold flow velocities (u) for the key bifurcations, but the inclusion of the nonlinear terms improves agreement with experiment in terms of amplitudes of motion and by capturing features of behaviour not hitherto predicted by theory.
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• 6
Electronic Resource
Springer
Nonlinear dynamics 4 (1993), S. 573-603
ISSN: 1573-269X
Keywords: Slider-crank mechanism ; nonlinear resonance ; dynamic stability ; method of multiple scales
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The transverse vibrations of a flexible connecting rod in an otherwise rigid slider-crank mechanism are considered. An analytical approach using the method of multiple scales is adopted and particular emphasis is placed on nonlinear effects which arise from finite deformations. Several nonlinear resonances and instabilities are investigated, and the influences of important system parameters on these resonances are examined in detail.
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• 7
Electronic Resource
Springer
Nonlinear dynamics 5 (1994), S. 39-52
ISSN: 1573-269X
Keywords: Shaft ; stability ; bifurcation ; analysis
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The dynamic stability and self-excited posteritical whirling of rotating transversally loaded shaft made of a standard material with elastic and viscous nonlinearities are analyzed in this paper using the theory of bifurcations as a mathematical tool. Partial differential equations of motion are derived under assumption that von Karman's nonlinearity is absent but geometric curvature nonlinearity is included. Galerkin's first-mode discretization procedure is then applied and the equations of motion are transformed to two third-order nonlinear equations that are analyzed using the theory of bifurcation. Condition for nontrivial equilibrium stability is determined and a bifurcating periodic solution of the second-order approximation is derived. The effects of dimensionless stress relaxation time and cubic elastic and viscous nonlinearities as well as the role of the transverse load are studied in the exemplary numerical calculations. A strongly stabilizing influence of the relaxation time is found that may eliminate self-excited vibration at all. Transition from super- to subcritical bifurcation is observed as a result of interaction between system nonlinearities and the transverse load.
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• 8
Electronic Resource
Springer
Nonlinear dynamics 2 (1991), S. 405-417
ISSN: 1573-269X
Keywords: Collocation ; stability ; bifurcation ; modal interaction
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract A methodology is first presented for analyzing long time response of periodically exited nonlinear oscillators. Namely, a systematic procedure is employed for determining periodic steady state response, including harmonic and superharmonic components. The stability analysis of the located periodic motions is also performed, utilizing results of Froquet theory. This methodology is then applied to a special class of two degree of freedom nonlinear oscillators, subjected to harmonic excitation. The numberical results presented in the second part of this study illustrate effects caused by the interaction of the modes as well as effects of the nonlinearities on the steady state response of these oscillators. In addition, sequences of bifurcations are analyzed for softening systems, leading to unbounded response of the model examined. Finally, the importance of higher harmonics on the response of systems with strongly nonlinear characteristics is investigated.
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• 9
Electronic Resource
Springer
Nonlinear dynamics 3 (1992), S. 199-223
ISSN: 1573-269X
Keywords: Twin-lift helicopter ; nonlinear model ; stability ; feedback linearization
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The stability and control characteristics of two twin-lift helicopter configurations are analyzed in this paper. In order to address the issue of configuration selection from a handling qualities viewpoint, their open-and closed-loop characteristics are compared. The two twin-lift configurations considered are the twin-lift with spreader bar and twin-lift without spreader bar. The nonlinear models describing the dynamics of these two configurations in the lateral/vertical plane are derived. The open-loop characteristics of the two systems are compared by linearizing the nonlinear models about a symmetric hovering equilibrium condition. The closed-loop characteristics of the two systems are compared using nonlinear controllers based on feedback linearization schemes. The performance of the resulting closed-loop systems in carrying out a typical twin-lift mission is evaluated through nonlinear simulation. Also, the effects of helicopter performance degradation and measurement errors on the overall system performance are discussed. [B] Matrix multiplying the control vector in the nonlinear model [B1] Matrix multiplying the control vector in the linear model [C] Matrix defining vector of variables to be controlled [C1] Damping matrix CijElement of the damping matrix e Parameter used in the linear model = M 1 h 1/I 1=M 2 h 2/I 2,/ft {f} Vector independent of controls in the nonlinear model g Acceleration due to gravity, ft/sec2 h1, h2Distance of tether attachment point to the center of gravity for helicopters 1 and 2, ft h Parameter used in the linear model, =h 1=h 2, ft h′ Distance between rotor hub and the helicopter center of gravity, ft h h/l′ H Distance of the load from the spreader bar c.g., ft H1, H2Length of tethers 1 and 2, ft IRMass moment of inertia of spreader bar, slug-ft2 I1, I2Roll moments of inertia of helicopters 1 and 2, slug-ft2 k′ Non-dimensional hub control moment coefficient KDDerivative gains KIIntegral gains KPProportional gains [Ki] Stiffness matrix KijElement of the stiffness matrix l′ Parameter used in the linear model, =H 1=H 2, ft L Spreader bar length, ft
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• 10
Electronic Resource
Springer
Nonlinear dynamics 4 (1993), S. 153-181
ISSN: 1573-269X
Keywords: Dynamics ; rotors ; stability ; chaos
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract Nonlinear rotors are often considered as potential sources of chaotic vibrations. The aim of the present paper is that of studying in detail the behaviour of a nonlinear isotropic Jeffcott rotor, representing the simplest nonlinear rotor. The restoring and damping forces have been expanded in Taylor series obtaining a ‘Duffing-type’ equation. The isotropic nature of the system allows circular whirling to be a solution at all rotational speeds. However there are ranges of rotational speed in which this solution is unstable and other, more complicated, solutions exist. The conditions for stability of circular whirling are first studied from closed form solutions of the mathematical model and then the conditions for the existence of solutions of other type are studied by numerical experimentation. Although attractors of the limit cycle type are often found, chaotic attractors were identified only in few very particular cases. An attractor supposedly of the last type reported in the literature was found, after a detailed analysis, to be related to a nonchaotic polyharmonic solution. As the typical unbalance response of isotropic nonlinear rotors has been shown to be a synchronous circular whirling motion, the convergence characteristics of Newton-Raphson algorithm applied to the solution of the set of nonlinear algebraic equations obtained from the differential equations of motion are studied in some detail. c damping coefficient i imaginaty unit (i=% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbqfgBHr% xAU9gimLMBVrxEWvgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvA% Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9% vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea% 0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabe% aadaabauaaaOqaamaakaaabaGaeyOeI0IaaGymaaWcbeaaaaa!3E66!$\sqrt { - 1}$) k stiffness m mass t time x istate variables i=1, 4 z complex co-ordinate (z=x+iy) [J] Jacobian matrix Oxyz inertial co-ordinate frame Oξηz rotating co-ordinate frame δ perturbation term ε eccentricity ζ complex co-ordinate (ζ=ξ+iη) λ system eigenvalues μ nonlinearity parameter τ nondimensional time ϕ phase ω spin speed u nonrotating t rotating 0 amplitude t nondimensional terms
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