Absence Of Internal Resonances Research Articles

Identification by means of a genetic algorithm of nonlinear damping and stiffness of continuous structures subjected to large-amplitude vibrations. Part I: single-degree-of-freedom responses

A procedure is presented to identify the nonlinear damping and stiffness parameters of a single-degree-of-freedom (SDOF) model from large-amplitude vibrations of harmonically forced continuous systems, in absence of internal resonances. Cubic nonlinear damping is introduced in the SDOF model in addition to the classical viscous one. The parameter estimation relies on the harmonic balance method. It is shown that the mean value, the first and the second harmonics are needed for a softening response, although often only the first harmonic is experimentally measured with accuracy. The identification methodology is thus split between (i) purely hardening and (ii) softening behaviour. The absence of coupling enables for an independent estimation of the nonlinear stiffness from the backbone curve, followed by an evaluation of damping at resonance. For the purely hardening case, the classical least-squares method is applied by using experimentally measured first harmonic. For the softening case, a novel cascade procedure consists in (i) a single-term harmonic balance parameter estimation used as initiation of (ii) a minimization of the distance between data and a two-term harmonic balance model by means of a genetic algorithm. These procedures are validated by parameter identification on synthetic (numerically generated) and experimental frequency-responses. In particular, the identified nonlinear damping model is compared to level-adjusted linear viscous damping with a different coefficient identified at any level of harmonic force, demonstrating its ability to account for the evolution of damping with the vibration amplitude. The identification of one-to-one internal resonances is treated in a companion paper.

Dynamics of coupled nonlinear oscillators: The formation of long-lived vibrational states in the case of molecular systems

Nonlinear vibrations in a closed system of coupled nonlinear oscillators are studied using acetylene type molecules as an example. A criterion for the stable existence of long-lived vibrational states—local modes—in one of the oscillators is obtained. It is shown that the disappearance of a local mode, as well as its appearance, proceeds abruptly, and the mechanism of stabilization of these excitations is due to the presence or absence of internal resonances of an oscillatory system such as any polyatomic molecule. Energy values needed to excite vibrations in which local modes can appear are determined. It is shown that calculation results agree with experimental data.

Nonlinear vibrations of suspended cables—Part II: Deterministic phenomena

The present work is concerned with deterministic nonlinear phenomena arising in the finite-amplitude dynamics of elastic suspended cables. The underlying theoretical framework has been addressed in Part I of this Special Issue, where the mechanical system and its mathematical modeling have been presented, and different techniques for the analysis of nonlinear dynamics have been illustrated with reference to the suspended cable. Herein, we discuss the main features of system regular and complex response, and the associated bifurcational behavior. Nonlinear phenomena are considered separately for single-degree-of-freedom and multidegree-of-freedom cable models, by distinguishing between theoretical and experimental results and comparing them with each other. Regular and nonregular vibrations are considered either in the absence of internal resonance or under various internal/external, and possibly simultaneous, resonance conditions. The most robust classes of steady periodic motions, the relevant response scenarios in control parameter space, and the main features of multimodal interaction phenomena are summarized. Bifurcation and chaos phenomena are discussed for the single-dof model by analyzing the local and global features of steady nonregular dynamics. For the experimental model, the most meaningful scenarios of transition to chaos are illustrated, together with the properties of the ensuing quasiperiodic and chaotic attractors. Finally, the important issues of determining system dimensionality and identifying properly reduced-order theoretical models of cable are addressed. There are 185 references listed in this review article.

CORRECTED SOLVABILITY CONDITIONS FOR NON-LINEAR ASYMMETRIC VIBRATIONS OF A CIRCULAR PLATE

An investigation into non-linear asymmetric vibrations of a clamped circular plate under a harmonic excitation is made. We re-examined a primary resonance studied by Sridhar, Mook and Nayfeh, in which the frequency of excitation is near the natural frequency of an asymmetric mode of the plate. We corrected their solvability conditions and found that in the absence of internal resonance, the steady state response can have not only the form of standing wave but also the form of travelling wave, which is a remarkable contrast to their conclusion.

An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes

A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canonical set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(1) is sufficient). Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems

A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal resonances.

Modal coupling in the free nonplanar finite motion of an elastic cable

In the finite motions of a suspended elastic cable the in-plane and out-of-plane oscillations are coupled, which is in contrast with what is predicted by the theory of small oscillations. To study the phenomenon of nonlinear coupling, a simple but meaningful two degree-of-freedom model is referred here, one parameter being used to describe the in-plane motion and the other the out-of-plane motion. The solution of the dynamic equilibrium equations is accomplished by an order-three perturbational expansion, which furnishes the time solution of the two displacement parameters. The modification of the free oscillations due to the exchange of energy between the two modes in absence of internal resonance is studied for different initial conditions and the effect of modal coupling is evidenced.

Combination resonances in the non-linear response of bowed structures to a harmonic excitation

A second-order uniform expansion is obtained for the response of a bowed structure (systems with quadratic and cubic non-linearities) to a combination resonance. The results show that combination resonances of the difference type can never be excited. Moreover, the results show that the steady state amplitudes depend strongly on the second-mode excitation amplitude f 2. Two threshold values ζ 1 and ζ 2 for f 2 are identified. When f 2 < ζ 1, combination resonances are never excited, when f 2 > ζ 2, combination resonances are always excited, and when ζ 2 ⩽ f 2 < ζ 2, combination resonances may or may not be excited, depending on the initial conditions. The results also show that combination resonances are very sensitive to the addition of a first-mode excitation amplitude f 1. For a certain range of values of f 1 f 2 and the phase angle τ between the two excitations, combination resonances are quenched and the response consistes essentially of the linear response. The quenching of combination resonances has been verified by numerically integrating the original differential equations in the presence as well as the absence of internal resonances. The analysis also identifies a range of values of f 1 f 2 and τ for which the response is significantly enhanced; this result has also been verified by numerically integrating the original equations in the presence as well as the absence of internal resonances.

Monofrequent oscillations of a non-linear model of a suspended cable

A two-degree-of-freedom non-linear elastic model is considered to analyze the effects of non-linearities on the free motion of a suspended cable. The discretized model is obtained by referring to simplified kinematics of the cable; the equations of motion which show quadratic and cubic non-linearities are solved through the multiple time scale perturbation technique. The monofrequent oscillations of the system are studied in order to analyze the modifications of frequency and motion amplitude of the modal oscillations due to geometric non-linearities in the absence of internal resonance. The possibility that effects arise due to non-linear coupling is examined. A numerical analysis is made for the first symmetric mode for different amplitudes of motion by parametrically varying the geometric and mechanical properties of the cable. The correction of frequency for the in-plane oscillation varies appreciably with the cable properties due to prevalence of either the quadratic or cubic term. In the out-of-plane monofrequent oscillation non-linearities establish a coupling between the two components of motion which strongly influences the frequency correction.

Non-linear resonances in the forced responses of plates, part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.

Nonlinear resonances in a class of multi-degree-of-freedom systems

An analysis is presented of the superharmonic, subharmonic, and combination resonances in a multi-degree-of-freedom system which has cubic nonlinearity and modal viscous damping and is subject to harmonic excitation. It is shown that in the absence of internal resonances, the steady-state response contains only the modes which are directly excited. It is shown that in the presence of internal resonances, modes other than those that are directly excited can appear in the response. The strong influence of internal resonances is exhibites in numerical examples involving hinged–clamped beams. It is shown that when a multimode solution exists the lowest mode can dominate the response, even when it is not directly excited. Subject Classification: 40.30, 40.20.