Vibration suppression for strings with distributed loading using spatial cross-section modulation

Abstract

Document VersionPeer reviewed version Link back to DTU Orbit Citation (APA): Sorokin, V., & Thomsen, J. J. (2015). Vibration suppression for strings with distributed loading using spatial cross-section modulation. Journal of Sound and Vibration, 335, 66-77. DOI: 10.1016/j.jsv.2014.09.028 Vibration suppression for strings with distributed loading using spatial cross-section modulation Abstract A problem of vibration suppression in any preassigned region of a bounded structure subjected to action of an external time-periodic load which is distributed over its domain is considered. A passive control is applied, in that continuous spatially periodic modulations of structural parameters are used as a means for vibration suppression. As an example, stationary vibrations of a string under action of a distributed time-periodic load are studied. This system in a simplified form models such processes as interaction between membranes and colloids, oscillations of transmission lines under action of rain and wind, and dynamics of suspension bridges and stay cables. Suppression of vibration in predefined regions of the string is performed by continuous spatial modulation of its cross-section. For analyzing the problem considered a novel approach named the method of varying amplitudes is employed. This approach is applicable for solving differential equations without a small parameter, and may be considered as a natural continuation of the classical methods of harmonic balance and averaging. As a result, optimal parameters for the string cross-sectional area modulation are determined for the cases of harmonically, uniformly and arbitrarily distributed load, which allows for completely suppressing or considerably reducing vibration in the prescribed part of the string (compared to the case without modulation). vibration suppression; bounded structure; distributed loading; continuous spatially periodic modulation; the method of varying amplitudes.

Frequency Detuning Effects for Parametrically and Directly Excited Elastic Structures

“…Frequency detuning, by contrast, generally yields quasi-periodic steady-state responses. To solve (1) for the quasi-periodic response, we use the method of varying amplitudes (MVA) [5], and compare quantitatively with results of direct numerical integration and qualitatively with experimental observations. First a frequency detuning σ pd between the parametric and direct excitation is introduced, defined by:…”

“…which by contrast to perturbation methods such as multiple scales or averaging is without restrictions such as slowly varying amplitudes, or the presence of small parameters [5,6]. Also, by contrast to the method of harmonic balance, where the coefficients X m1 and X m2 would be constants and (3) an approximation, the allowed time dependency of X m1 and X m2 means that (3) merely represents a shift of variables, which is exact for all values of n. The shift from the original dependent variable x to 2n new variables, X m1 and X m2 , implies that 2n equations are needed.…”

Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties

“…In particular, the paper addresses the prediction of bandgaps [22] and their influence on the distribution of eigenfrequencies. For analyzing the problem considered a novel approach named the method of varying amplitudes (MVA), proposed in [23], is employed. In [23] the dynamics of a string under the action of an external distributed time-periodic load was studied, and optimal parameters of the strings cross-sectional area were determined so as to suppress vibration in predefined regions of the string.…”

“…For analyzing the problem considered a novel approach named the method of varying amplitudes (MVA), proposed in [23], is employed. In [23] the dynamics of a string under the action of an external distributed time-periodic load was studied, and optimal parameters of the strings cross-sectional area were determined so as to suppress vibration in predefined regions of the string. By contrast to [23], the present work considers free oscillations of a Bernoulli-Euler beam, implying solving a fourth order differential equation, with the aim to reveal the effect of modulation on the dispersion relation and the eigenproperties.…”

“…In [23] the dynamics of a string under the action of an external distributed time-periodic load was studied, and optimal parameters of the strings cross-sectional area were determined so as to suppress vibration in predefined regions of the string. By contrast to [23], the present work considers free oscillations of a Bernoulli-Euler beam, implying solving a fourth order differential equation, with the aim to reveal the effect of modulation on the dispersion relation and the eigenproperties. The MVA is inspired by the method of direct separation of motions (MDSM) [24,25], and may be considered a natural continuation of the classical methods of harmonic balance [26] and averaging [27][28][29].…”