Mono-coupled Periodic Systems Research Articles

On the formation of a super stop-band in finite mono-coupled periodic structures using an array of vibration absorbers: Controlling parameters and physical insight

There are two fundamental strategies in the design of metamaterials used for noise and vibration control. The first uses structural periodicity and explores the Bragg scattering to produce frequency ranges where waves do not propagate or are severely attenuated (stop-bands). This condition is usually limited by the constitutive size of the structural elements and tends to occur at higher frequencies. A second strategy consists of adding vibration absorbers to produce a resonance(s), and this can occur at much lower frequencies. The addition of periodically attached vibration absorbers alone creates a periodic medium, and Bragg stop-bands also occurs in this situation. There is a particular condition where the Bragg and local resonance stop-bands can be combined to produce an ultra-wide or a super stop-band. This paper investigates the conditions required to produce a super stop-band in a finite mono-coupled periodic system composed of symmetric cells, each with a vibration absorber attached. Using a lumped parameter system, the dynamic features and the transmissibility of a single cell are determined, and the physical mechanisms behind the super stop-band formation are investigated, including the absorber tuning frequency and the influence of damping. Two experimental setups to support the theoretical findings are described. One involves a lumped-parameter system and the other involves a rod, both with a vibration absorber attached. The results show that a smooth super stop-band only occurs when the vibration absorber is tuned to the free-free natural frequency of the single host cell, creating a union between the Bragg stop-bands due to mass and stiffness of the absorber, and that the damping in the absorber should be equal to the damping in the host cell.

Seismic Response and Vibration Transmission Characteristics of Laminated Rubber Bearings with Single Disorder

AbstractThe characteristic receptance method of analyzing the free wave propagation of a finite monocoupled periodic system with a single disorder is applied to study the vibration transmission thr...

Coupled flexural-longitudinal wave motion in a finite periodic structure with asymmetrically arranged transverse beams

A companion paper [J. Acoust. Soc. Am. 118, 3010–3020 2005] has examined the phenomena of flexural-longitudinal wave coupling in a practically undamped and semi-infinite periodic waveguide with structural side-branches. The effect of structural damping on wave coupling in such a waveguide is examined in the first part of the present paper, and the damping-dependent decrease in wave coupling is revealed for a structure with multiresonant side-branches. In the second part, the simplifying semi-infinite assumption is relaxed and general expressions for the junction responses of finite and multicoupled periodic systems are derived as a generalization of the governing expressions for finite, mono-coupled periodic systems [Ohlrich, J. Sound Vib. 107, 411–434 (1986)]. The present derivation of the general frequency response of a finite system utilizes the eigenvectors of displacement responses and wave forces that are associated with the characteristic wave-types, which can exist in a multicoupled periodic system [Mead, J. Sound Vib. 40, 19–39 (1975)]. The third part of the paper considers a finite specific test-structure with eight periodic elements and with structural terminations at the extreme ends. Audio-frequency vibration responses of this tri-coupled periodic structure are predicted numerically over a broad range of frequencies and a very good agreement is found with the measurement results obtained from an experiment with a nominally identical, periodic test-structure which is freely suspended.

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.

Damage detection of mono-coupled periodic structures based on sensitivity analysis of modal parameters

A sensitivity-based method for localization and assessment of damage in mono-coupled periodic structures is presented in this paper, in which slopes and curvatures of mode shapes are used to localize damage, and natural frequencies are then utilized to quantify the damage. The expressions of sensitivity coefficients of mode shapes, slopes, and curvatures of a mono-coupled periodic system are first derived in terms of receptances of periodic element. A mono-coupled periodic spring-mass system with 10 degrees of freedom is used to carry out a sensitivity study to compare the sensitivities of natural frequencies, mode shapes, slopes, and curvatures to damage. The results show that the sensitivities of these modal parameters in a mono-coupled periodic structure do not depend on the structural parameters, and therefore there is no need for a prior analytical model of the structure. The study also demonstrates that among these modal parameters, curvatures of modal shapes are most sensitive but slopes of mode shapes seem to be more indicative of damage location. A 20-element mono-coupled periodic spring-mass system is adopted to demonstrate the capacity of the proposed method to localize and quantify damages in the mono-coupled periodic system. Finally, a 3-storey near mono-coupled periodic experimental building is used to verify the actual application of the proposed method in consideration of the influence of measurement noise and non-perfect periodicity of actual engineering structures. Numerical and experimental results illustrate that only using a few lower modes with or without noise pollution can accurately detect damages in a mono-coupled periodic or a near mono-coupled periodic structure, either single or multiple damage locations and slight or severe damages.

THE CHARACTERISTIC RECEPTANCE METHOD FOR DAMAGE DETECTION IN LARGE MONO-COUPLED PERIODIC STRUCTURES

An approach based on the concept of wave propagation to detect the structural damage in large mono-coupled periodic structures is presented in this paper. The free vibration analysis of a finite mono-coupled periodic structure with a single disorder has been conducted by the characteristic receptance method, and the sensitivity of the natural frequencies to the disorder in flexibility has been discussed. Based on the sensitivity analysis, the locations and magnitude of damage in large mono-coupled periodic systems have been estimated using measured changes in the natural frequencies. The paper also introduces a substructure-based method for improving the computational efficiency and the accuracy of damage detection in large mono-coupled periodic structures. Numerical results from two periodic mass-spring-structures show that the proposed method can provide good predictions of both the locations and magnitude of damage at one or more sites. Furthermore, the proposed method, in which a priori information about the nature such as stiffness of the undamaged structure is not needed, and only measurements of the change in a few of the structure's natural frequencies between the undamaged and damaged states are required, is particularly attractive in practice. However, some issues such as the role of noise in actual measurements, application to multi-coupled periodic structures with complex boundary conditions remain to be resolved before this approach becomes a truly variable method of structural damage assessment.

Wave localization and conversion phenomena in multi-coupled multi-span beams

The linear dynamics of nearly periodic disordered multi-span beams resting on flexible supports are investigated. A wave transfer matrix methodology is chosen to examine the propagation of waves and the transmission of vibration along the structure. The spans are bi-coupled through the rotation and the transverse displacement at the supports and thus the beam motion is made up of two independent wave types. While for the ordered infinite beam there exists frequency passbands for which the free harmonic waves propagate without attenuation, the introduction of a slight disorder among the span lengths results in the localization of the vibration energy to few spans and in the conversion of the energy from one type of wave to the other. The energy conversion phenomenon renders the mechanism of localization much more complex than in mono-coupled periodic systems. The contribution of each type of wave to the global beam motion is analyzed in terms of frequency. It is observed that the spatial decay of each wave type is mainly governed by an exponential envelope. The corresponding exponential decay constants define a measure of localization for each wave and are found to be equal to the Lyapunov exponents of the product of random wave transfer matrices. It is also found that at frequencies which belong to a passband for both wave types, the decay rate of an incident wave vector is bounded by the two Lyapunov exponents, while at frequencies which belong to a passband for one wave type and a stopband for the other, localization effects are best predicted by the smallest of the two Lyapunov exponents.

Individual And Interactive Mechanisms For Localization And Dissipation In A Mono-Coupled Nearly-Periodic Structure

The effect of structural damping on the dynamics of periodic and disordered nearly periodic systems is investigated. A chain of single-degree-of-freedom oscillators coupled by springs is used as a basic model of a mono-coupled periodic system. The dynamic response of the system to a harmonic excitation at the leftmost oscillator is considered. For a system with damping but no disorder (i.e., identical oscillators), an exact expression for the rate of exponential spatial amplitude decay is retrieved. It is found that the decay rate due to damping in a periodic system has a frequency dependence similar to that of the decay rate due to disorder in an undamped nearly periodic system. The magnitudes of the decay rates due to only damping or disorder are compared, and the concept of equivalent damping due to disorder is introduced. For systems with both disorder and damping, perturbation methods are used to find approximate expressions for the decay rate in the cases of strong and weak coupling between oscillators. It is shown that if the coupling is strong, the decay rates due to damping and disorder simply add up to produce the overall decay rate, and the effect of damping dominates. If the coupling is weak, however, the effects of damping and disorder on the overall decay are comparable in magnitude, and they interact in a more complicated manner.

Forced vibration and wave propagation in mono-coupled periodic structures

This paper describes the wave propagation and vibration characteristics of mono-coupled structures which are of spatially periodic nature. The receptance approach to periodic structure theory is applied to study undamped periodic systems with composite structural elements; particular emphasis is laid on investigating resonant periodic point loading and its pronounced effect on the propagation of longitudinal waves. General mono-coupled periodic systems are first assumed to be infinite in extent; thereafter reflections caused by arbitrary end terminations of finite structures are considered and a general “closed form” solution is found for the forced harmonic response at element junctions. This “junction-receptance” is used to determine discrete junction mode shapes of a finite system. Finally, the forced response of a finite structure with an internal obstruction is derived as a natural extension the determination of the junction-receptance. The influence of such a disorder is illustrated by a simple example.

Free vibrations of a mono-coupled periodic system

Vibration problems of periodic systems can be analyzed efficiently by means of the transfer matrix method. The frequency equation for the whole system is shown to be obtained in terms of the eigenvalues, or their natural logarithms, which are often called “propagation constants”, of the transfer matrix for a single periodic subsystem. In case of a mono-coupled system this frequency equation may be solved graphically by using the propagation constant curve, thereby saving a great deal of computational effort. Two types of mono-coupled systems are considered as numerical examples: a spring-mass oscillating system and a continuous Timoshenko beam resting on regularly spaced knife-edge supports. Depending on whether the transfer matrix is derived by an analytical procedure or by the finite element method, the numerical solutions become either exact or approximate.

Mono-coupled periodic systems with a single disorder: Free wave propagation

A general method has been presented for analyzing free harmonic wave propagation through a mono-coupled periodic system with a single disorder. Expressions have been derived for the magnitudes of the waves transmitted and reflected by the disorder. These general expressions have then been used to study flexural wave motion through a periodic beam system into which three different types of disorder have been introduced: (i) a beam element of non-periodic length; (ii) a rotary mass at a support; (iii) a rotary spring at a support. The disorder always results in reduced transmission of the flexural wave when the frequency is in a frequency propagation zone of the periodic system but the first two disorders may lead to increased transmission in a frequency attenuation zone. Conditions have been identified under which the combined disorder plus periodic system can behave like a resonating spring-mass system, or as a spring-mass damper system. The adverse effects of resonating disorders are pointed out. Qualitative and quantitative analysis based upon computer studies have indicated how disorders can be used most effectively for vibration isolation in existing periodic systems or when designing new systems.

Mono-coupled periodic systems with a single disorder: Response to convected loadings

This paper presents a general theory of the forced response under convected loading of mono-coupled periodic systems with a single disorder. The general expressions derived have been used to study the response of an infinite periodic beam on simple supports with one of the support spacings different from all the others. Convected harmonic pressure fields and frozen random pressure fields have been considered. Computer studies are presented showing the moment response at supports and the space-time-averaged responses in the disorder and in the nearby periodic beam elements. High response levels can occur due to (i) resonances of the beam length disorder against the stiffness of the attached periodic systems and (ii) hydrodynamic coincidence vibration occurring in the periodic beam. The frequency zones in which these high responses may occur are identified. The high response due to the resonance (ii) is restricted to the vicinity of the disorder, whereas that due to coincidence occurs throughout the system. Computed results show that the highest response levels do not necessarily occur in the beam length disorder, but may occur in one of the nearby periodic beam elements. The dependence of the maximum response levels on the magnitude of the disorder has been investigated. The conditions under which small disorders may be neglected have been pointed out.

Free wave motion in periodic systems with multiple disorders

This paper presents a general method for the analyses of free wave propagation through mono-coupled periodic systems with multiple disorders. The method is applied to investigate the free flexural wave motion in infinite beam type systems on simple supports, comprising a large number of identically disordered repeating units. A more practical case of disorder due to unequal support spacings is considered. Flexural wave propagation in disordered periodic beams is compared to that in periodic beams. The effect of the disorder on the attenuation of free flexural waves in undamped and damped periodic beams is discussed.

Wave propagation and natural modes in periodic systems: I. Mono-coupled systems

In this paper the relationship is studied between the bounding frequencies of the propagation zones of mono-coupled periodic systems and the natural frequencies of the individual elements of which the system is composed. It is also shown how these relate to the natural frequencies of finite mono-coupled periodic systems. The concept of the characteristic receptance of a free wave in the periodic system is developed, and the manner in which this receptance varies with frequency is studied. It is used to examine the attenuation and phasechange undergone by a free wave when it impinges on and is reflected from a non-dissipative boundary. This leads on to a simple physical explanation for the occurrence of natural frequencies of finite systems in either the attenuation zones or propagation zones of the infinite system. Distinction is drawn between systems whose elements are symmetric or unsymmetric, positive-definite or semi-definite.