Non-linear normal modes and their applications in mechanical systems

Mikhlin, Yuri V.; Perepelkin, Nikolay V.

doi:10.1177/0954406211411254

Cited by 5 publications

(9 citation statements)

References 54 publications

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“…Vertical dynamics of a double-tracked road vehicle with sevendegrees-of-freedom is considered in Refs. [38,39]. The car is described by a rigid body.…”

Section: Parametric Vibrations Of Cylindrical Shells Under the Actionmentioning

confidence: 99%

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

Аврамов

Mikhlin

2013

Applied Mechanics Reviews

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This paper is an extension of the previous review, done by the same authors (Mikhlin, Y., and Avramov, K. V., 2010, “Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments,” ASME Appl. Mech. Rev., 63(6), p. 060802), and it is devoted to applications of nonlinear normal modes (NNMs) theory. NNMs are typical regimes of motions in wide classes of nonlinear mechanical systems. The significance of NNMs for mechanical engineering is determined by several important properties of these motions. Forced resonances motions of nonlinear systems occur close to NNMs. Nonlinear phenomena, such as nonlinear localization and transfer of energy, can be analyzed using NNMs. The NNMs analysis is an important step to study more complicated behavior of nonlinear mechanical systems.This review focuses on applications of Kauderer–Rosenberg and Shaw–Pierre concepts of nonlinear normal modes. The Kauderer–Rosenberg NNMs are applied for analysis of large amplitude dynamics of finite-degree-of-freedom nonlinear mechanical systems. Systems with cyclic symmetry, impact systems, mechanical systems with essentially nonlinear absorbers, and systems with nonlinear vibration isolation are studied using this concept. Applications of the Kauderer–Rosenberg NNMs for discretized structures are also discussed. The Shaw–Pierre NNMs are applied to analyze dynamics of finite-degree-of-freedom mechanical systems, such as floating offshore platforms, rotors, piece-wise linear systems. Studies of the Shaw–Pierre NNMs of beams, plates, and shallow shells are reviewed, too. Applications of Shaw–Pierre and King–Vakakis continuous nonlinear modes for beam structures are considered. Target energy transfer and localization of structures motions in light of NNMs theory are treated. Application of different asymptotic methods for NNMs analysis and NNMs based model reduction are reviewed.

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“…Vertical dynamics of a double-tracked road vehicle with sevendegrees-of-freedom is considered in Refs. [38,39]. The car is described by a rigid body.…”

Section: Parametric Vibrations Of Cylindrical Shells Under the Actionmentioning

confidence: 99%

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

Аврамов

Mikhlin

2013

Applied Mechanics Reviews

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“…A priori construction of the surfaces W and W0 may provide investigator with some valuable information about the dynamical system being investigated. If the surfaces (14) are obtained analytically one can use expressions (21) to define W and W0. On the other hand, if one desires to y « Q 2 study large-amplitude motions this makes evaluation of (14) via power series inappropriate while other methods of solving PDEs (16) may require some sophisticated computations.…”

Section: Investigation Via Harmonic Balance Method Construction Of Wmentioning

confidence: 99%

“…The above analysis leads to an important conclusion: ▲ Statement 3. In the state space (x, x ', y, y ') there exist unique hypersurface (14) which corresponds to forced oscillations in the initial system (9) and passes through the equilibrium position of that system in the space of variables (x, x' , y, y ') .…”

Section: Equivalent System Investigation Via Invariant Manifolds Methmentioning

confidence: 99%

“…This leads to the conclusion: ▲ Statement 4. For each point on the surface (14) which satisfy equations (16) there exist a closed trajectory in the space of variables (x,x',y,y ') which passes through that point and corresponds to periodic solution of (9) under some value of external excitation f .…”

Section: Equivalent System Investigation Via Invariant Manifolds Methmentioning

confidence: 99%

“…This expression is not changed if the phase of external load is varied by an arbitrary constant. This equivalent system of a new type can be also considered as the result of substitution (14) into (13) and division of the latter by Q2 . Due to the considerations above the system 20 The equivalent system (20) will be called the equivalent system o f 3rd type.…”

Section: Equivalent System Investigation Via Invariant Manifolds Methmentioning

confidence: 99%

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Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Perepelkin

2017

Nonlinear Dyn

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In the paper a new non-iterative variant o f Rauscher method is considered. In its current statement the method can be used in analysis o f forced harmonic oscillations in 1-DOF nonlinear system. It is shown that three different types o f equivalent authonomous dynamical systems can be built fo r a given 1-DOF non-autonomous one. Two o f them (1st and 2nd type) have wider set o f solutions than that o f the initial system. These solutions correspond to various values o f amplitude and phase o f external excitation. Solutions o f the equivalent system o f 3rd type are exclusively periodic ones. Based on the equivalent system of 3rd type such a function W(x,x') can be constructed that its level curves correspond to periodic orbits o f the initial non-autonomous system. This function can be built a priori via computation o f the invariant manifold o f the equivalent system o f 1st type. Using the same approach the Rauscher expansions cos(Qt)=C(x,x'), sin(Qt)=S(x,x') can also be constructed. It is also shown that equivalent systems can be investigated by means o f harmonic balance method which allows construction o f W(x,x'), C(x,x') andS(x,x') in semi-analytical manner.

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Invariant Manifold Representations of Nonlinear Modes of Vibration

Shaw

2014

Modal Analysis of Nonlinear Mechanical Systems

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Non-linear normal modes and their applications in mechanical systems

Cited by 5 publications

References 54 publications

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

Review of Applications of Nonlinear Normal Modes for Vibrating Mechanical Systems

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Invariant Manifold Representations of Nonlinear Modes of Vibration

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