Effect Of Modal Interactions Research Articles

Effect of Modal Interactions on Friction-Damped Self-Excited Vibrations

Abstract It is well-known that nonlinear dry friction damping has the potential to bound the otherwise unboundedly growing vibrations of self-excited structures. An important technical example are the flutter-induced friction-damped limit cycle oscillations of turbomachinery blade rows. Due to symmetries, natural frequencies are inevitably closely spaced, and they can generally be multiples of each other. Not much is known on the nonlinear dynamics of self-excited friction-damped systems in the presence of such internal resonances. In this work, we analyze this situation numerically by regarding a two degrees-of-freedom system. We demonstrate that in the case of closely-spaced natural frequencies, the self-excitation of the lower-frequency mode gives rise to non-periodic oscillations, and the occurrence of unbounded behavior well before reaching the maximum friction damping value. If the system is close to a 1:3 internal resonance, limit cycles associated with much higher frictional damping appear, however, most of these are unstable. If more than one mode is subjected to self-excitation, the maximum resistance against self-excitation is at least given by the damping capacity of the most weakly friction-damped mode. These results are of high technical relevance, as the prevailing practice is to analyze only periodic limit states and argue the stability solely by the slope of the damping-amplitude curve. Our results demonstrate that this practice leads to considerable mis- and overestimation of the resistance against self-excitation, and a more rigorous stability analysis is required.

Effect of Modal Interaction on Transient Behavior in Double-Input SEPIC DC–DC Converters

In this paper, we exploit an idea of nonlinear modal series method to investigate the effects of modal interaction in the one-cycle controlled (OCC) double-input SEPIC DC–DC converter. Based on the proposed nonlinear averaged model, the analytical approximate solutions are obtained to characterize the dynamic response characteristic of the transient behaviors in the double-input SEPIC converters. The fundamental modal analysis is utilized to identify the dominant oscillation modes and discover the relationship between parameters, fundamental modes and state variables. Furthermore, the second-order interaction indices are proposed to uncover the underlying mechanism of nonlinear interaction behaviors. In particular, the correlation between parameters and modal interaction are derived to optimize the transient process of the double-input SEPIC converters. Finally, numerical simulations are performed to verify the theoretical analysis.

Thermo-mechanical dynamics of three-dimensional axially moving beams

The thermo-mechanical nonlinear dynamics of a three-dimensional axially moving beam is examined numerically in this paper. Hamilton’s principle is employed to derive the nonlinear partial differential equations governing the motion of the system in the longitudinal, transverse, and lateral directions. The discretized equations of motion, which are in the form of second-order nonlinear ordinary differential equations, are obtained by applying the Galerkin technique to the nonlinear partial differential equations of motion. A change of variables is introduced to this set of equations, yielding a set of first-order nonlinear ordinary differential equations. The pseudo-arclength continuation technique is employed to solve this set of equations numerically so as to construct the frequency–response curves; this method allows continuation of both stable and unstable solution branches. The effect of modal interactions on the resonant dynamic behaviour of a fully symmetrical system (about the centreline) is also examined.

Flapwise non-linear dynamics of wind turbine blades with both external and internal resonances

An investigation on the non-linear aeroelastic behavior of a wind turbine blade with both external and internal resonances is presented. The external resonance is a primary resonance that appears at the first flapwise mode; it can cause severe damage to blade. The internal resonance happens at the first two flapwise modes; it can enhance the energy transfer between two modes, and change blade dynamics in primary resonance. Three aspects including blade behavior in pure primary resonance (abbr. PPR; only considering external resonance), blade behavior in combination resonance (abbr. CR; including both external and internal resonances), and the influence of internal resonance (i.e. modal interaction) on external resonance are examined. A simple Bernoulli–Euler beam model, in which geometric non-linearity and unsteady aerodynamic force are considered, is used to describe the flapwise motion of blade. The perturbation method is applied to the infinite-degree-of-freedom discrete system, which is obtained from the original continuous system via Galerkin׳s method, to get dynamic responses. Amplitude–frequency curves of resonance modes in CR and PPR are derived, and the stability of the steady state motion of blade is judged. The strongest modal interaction between two resonance modes is taken into account, and then effects of modal interaction, excitation amplitude, damping and non-linearity on non-linear vibration properties of blade are analyzed. Also, influences of three designing parameters (inflow ratio, setting angle and coning angle) and two detuning on the non-linear behavior of blade are discussed for a concrete downwind turbine.

Experiments on Chaotic Vibrations of an Outer-Free and Inner-Clamped Annular Plate with Initial Deflection

This paper presents experimental results on chaotic vibrations of an annular plate with initial deflection. The effect of modal interaction on chaotic vibrations is main scope in this paper. The plate is free on the outer edge and clamped on the inner edge. The initial deflection of the plate is nine thousandth times to the difference between the outer radius and the inner radius. The lowest and the third modes with one nodal-diameter have different natural frequencies due to the initial deflection. The second mode corresponds to the mode without nodal-diameter. The fourth and fifth modes have two cross nodal-diameters. The linear natural frequencies of the second and the third modes coincide with a slight deviation. The plate is subjected to lateral periodic acceleration. Sweeping the excitation frequency, nonlinear response curves of the plate are recorded. In the wide frequency ranges, non-periodic responses are generated. The responses are inspected by the Fourier spectrum, the Poincaré projection, the maximum Lyapunov exponent and the principal component analysis. The predominant non-periodic response is confirmed as the chaotic response generated from one-to-one and two-to-three internal resonances. The lowest, the second and the third modes are coupled in the internal resonances. The principal component analysis is applied to the chaotic response selected with short-time intervals sequentially collected from the long-time response. It is found that exchanges of the contribution ratio are observed among the lowest, the second and the third modes. Furthermore, the nodal diameters of the fourth and the fifth modes are fluctuated along the circumferential direction in the chaotic response.

A MODAL METHOD FOR COUPLED FLUID-STRUCTURE INTERACTION ANALYSIS

A modal method is developed for solving, analyzing, and controlling vibration and sound radiation of coupled fluid-structure systems. The method recasts the coupled equation of a coupled fluid-structure system in the classical matrix structural dynamic equation by modeling the acoustic load vector as direct linear function of the acceleration, velocity, and displacement vector. With the Rayleigh damping assumption of the coupled fluid-structure system the resulting equation can be uncoupled via a transformation to modal coordinates and analyzed by solving independent equations of single degree of freedom system. The modal radiation efficiencies, effect of modal interaction on sound radiation, mode shapes, and modal control of the coupled fluid-structure system are presented and discussed. Numerical example of the vibration and sound radiation of a fluid-loaded stiffened plate is presented solely as a vehicle to demonstrate the method. The comparisons in terms of computed sound power of the present method with the standard coupling method and available published results show a very good agreement. The mode shapes and the self- and mutual-radiation efficiencies of modes of the fluid-loaded stiffened plate are given and discussed. The study of the effect of modal interaction on sound power shows that the power radiated by a single mode is to increase total radiated power and the interaction of modes may lead to an increase or a decrease or no change in the total radiated power. Numerical results also show that the modal control achieves good reductions in the mean square velocity and the sound power of the fluid-loaded stiffened plate.

Effects of Modal Interactions on Vibration Performance in Ultrasonic Cutting

The effects of modal interactions and nonlinear response characteristics can hamper implementation of high power ultrasonic technologies, due to the resulting modal coupling, increased stress, audible noise levels and poor control of the operating response. These different adverse responses are illustrated by characterising the vibration behaviour of single-blade and multi-blade ultrasonic cutting systems. This paper proposes design strategies to eliminate the effects of modal interactions, by focusing on reducing the number of vibration modes, and to reduce the effects of nonlinear responses, by serial coupling of tuned components with appropriate cubic softening and hardening response characteristics.

Theory of Combined Sway and Nonsway Frames Buckling

Conventional design procedures for rigid jointed frames encourage a situation in which more than one buckling mode could occur simultaneously. Even though this practice is known to often give rise to increased sensitivity of buckling loads to small initial imperfections, current design practice is usually lacking in explicit design guidance. This paper seeks to explore the extent to which buckling loads in framed structures are reduced by the effects of modal interactions arising when designs are optimized. It takes, as a specific example, the situation where the buckling and bending planes for a rigid-jointed frame coincide and for which sway and nonsway buckling modes occur at similar load levels. It explores the extent to which elastic-plastic buckling loads may be affected by interactions of sway and nonsway modes and compares predictions with results from a recently conducted test program. It is suggested that the theoretical approach described has the potential to provide an extended and improved alternative to existing design practice.

Effect of modal interaction on sound radiation from vibrating structures

This technique permits the extraction of acoustic intermodal coupling information from numerically modeled, baffled planar structures. The technique generates results in close agreement to a parallel analytical development. Below the coincidence wave number ratio, the intermodal coupling effects cannot be neglected with respecty to the total radiated power.

Non-linear dynamic analysis of orthotropic elliptical plates

Von Kármán-type field equations are used in this paper to investigate the non-linear static and dynamic response of rectilinearly orthotropic elliptical plates. A multiple-mode approach has been used to find solutions to this plate geometry where the boundaries are taken to be clamped and immovable. An exact solution is reported for one of the two non-linear governing equations. Numerical results are graphically presented and compared with those of a single-mode approach. A close agreement was found between these values. The analysis presented here provides improved results for large deflections and non-linear periods and an insight into the effects of modal interaction.

The Effect of Modal Interaction in the Xenon Instability Problem

The space and time dependent reactor stability problem of neutron flux shape variations due to Xe/sup 135/ is examined. The effect of modal interaction on stability is investigated for a simple reactor model when the characteristic functions of the wave equation are used in a modal expansion. It is shown that modal interaction may contribute to or detract from stability depending on the circumstances, and that in the case of a very large reactor, stability can be seriously overestimated if modal interaction is neglected. (auth)