Resonance Oscillations in Mechanical Systems

Evan-Iwanowski, R. M.; Dugundji, John

doi:10.1115/1.3424080

Cited by 85 publications

(56 citation statements)

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“…This means that the no linearity varies with the dynamics, one cannot that it a sine, or a cubic nonlinearity, as seen in some papers, for example. Coupled problems have a very rich system dynamics due to presence of nonlinearities arising from the mutual interaction of the coupled systems, see [6,7,8,9,10].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification of the Nonlinear Dynamics of Electromechanical Coupled Systems

Lima¹,

Sampaio²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. This paper analyzes the nonlinear stochastic dynamics of electromechanical systems with friction in the coupling mechanism and in the mechanical parts. Two different electromechanical systems were studied. The first one is composed by a cart whose motion is excited by a DC motor and in the second a new element,a pendulum, is attached to the cart. The suspension point of the pendulum is fixed in cart, so that exists a relative motion between them. The influence of the DC motor in the dynamic behavior of the system is considered. The coupling between the motor and the cart is made by a mechanism called scotch yoke, so that the motor rotational motion is transformed in horizontal cart motion over a rail. In the model of the system, it is considered the existence of friction in the coupling mechanism and in between the cart and the rail. The embarked pendulum is modeled as a mathematical pendulum (bar without mass and particle of mass m p at the end). The embarked mass introduces a new feature in the system since the motion of the pendulum acts as a reservoir of energy, i.e. energy from the electrical system is pumped to the pendulum and stored in the pendulum motion, changing the characteristics of the mechanical system. Due to the consideration of friction, the dynamic of the problem is described by a system of differential algebraic equations. One of the most important parameters of the problem is the amplitude of the cart motion that is given by the position of the pin used in the coupling mechanism. The behavior of the system is very sensitive to this parameter because it controls the nonlinearities of the problem. In the stochastic analysis, this parameter is considered uncertain and is modeled as a random variable. The Maximum Entropy Principle is used to construct its probability model. Monte Carlo simulations are employed to compute the mean motion and the 90% confidence interval of the displacements of the pendulum, of the cart and of the angular speed of the motor shaft.2026

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Section: Introductionmentioning

confidence: 99%

Uncertainty Quantification of the Nonlinear Dynamics of Electromechanical Coupled Systems

Lima¹,

Sampaio²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…For ε = 0 the eigenvalues of A 0 are ±īω n , where ω n being the natural frequencies. It is interesting to observe here as ε → 0 characteristic exponents λ n → ±īω n , and from Equation (26) we recover the condition for conventional 'combination resonance' in parametrically excited nonlinear systems with a small parameter [25,26]. The concept of 'parametric resonance' comes from the stability analysis of linear systems with time periodic coefficients.…”

Section: Order Reduction Using the Time Periodic Invariant Manifoldmentioning

confidence: 99%

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

et al. 2005

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Abstract. The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique 'reducibility condition' that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L-F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same 'reducibility conditions' obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'true combination resonances' are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

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“…Non-linear e!ects occurring either due to geometric non-linearities (e.g., the non-linear von-Karman strain}displacement relations), or due to the non-linear behavior of the material (e.g., non-linear stress}strain relations) were also included. A review and a monograph including further results and extensive bibliography was written by Evan-Iwanowski [2,3].…”

Section: Introductionmentioning

confidence: 99%

Parametric Stability of Non-Linearly Elastic Composite Plates by Lyapunov Exponents

Gilat

Aboudi

2000

Journal of Sound and Vibration

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Resonance Oscillations in Mechanical Systems

Cited by 85 publications

References 0 publications

Uncertainty Quantification of the Nonlinear Dynamics of Electromechanical Coupled Systems

Uncertainty Quantification of the Nonlinear Dynamics of Electromechanical Coupled Systems

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

Parametric Stability of Non-Linearly Elastic Composite Plates by Lyapunov Exponents

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