MODELLING AND VIBRATION CONTROL OF PERIODIC STRUCTURES: APPLICATIONS TO RAILWAY TRACK AND PIPING SYSTEM

Abstract

The manipulation of elastic waves and vibration isolation through periodic structures which are designated as phononic crystals or metamaterials have aroused significant interest of the research community. This unique wave manipulation ability of metamaterials is due to especially designed structural configurations. The present research mainly deals with the analysis of propagation characteristics of flexural waves and vibration control in periodic structures. Here, two types of engineering structures are studied in depth: (i) periodic railway track; and (ii) pipes supported on periodic rack structure. In order to study the flexural vibration in the corresponding periodic structures, a generalized case of unit cell beam with periodic translational and rotational elastic supports carrying lumped mass at each support is considered. This beam is assumed to be undamped and is modelled based on the Euler- Bernoulli beam theory. Subsequently, the Floquet-Bloch theorem is applied to derive the analytical dispersion relations for the uncontrolled (without resonators) case of the beam in the lateral direction. This dispersion relation for a beam is accordingly used for different cases considered herein. Successively, to verify the resulting band gap and pass band characteristics, finite element (FE) models are developed. In a periodic structure it may be deemed necessary to control vibrations in the pass bands, especially when they are in the range of frequencies of interest. In this regard, lateral localized resonators (LLRs) are designed. Further, as an alternative to the LLR, a new way of vibration suppression in the periodic structures is proposed by utilizing the already existing mass in the system as lateral distributed resonators (LDRs). The numerical models are developed to conduct the relevant studies when LLRs/LDRs are employed to the analyzed structures. Primarily, the propagation behavior of lateral and vertical flexural waves and vibration control of a railway track periodically supported on rigid sleepers using fastenings are studied in depth. The dispersion properties of uncontrolled track (track without resonators) in both lateral and vertical directions are theoretically and numerically investigated first. The results show that two band gaps of narrow width and low attenuation are identified for both waves in the considered frequency range. With this basis, to control the vibrations of the examined rail in lateral and vertical directions, lateral and vertical localized resonators (LLRs/VLRs) for different mass ratios are designed. Moreover, the influence of damping of rail and resonators on band gap characteristics is also investigated. As a replacement to the conventional LLR, a novel possibility to control vibration relies on using another existing rail as a lateral distributed resonator (LDR).The genetic algorithm based optimization is performed to compute the optimal values of spring-damper units of these controllers. Efficacy of the proposed control methods is finally verified by applying a random Gaussian white noise as input. However, the performance of LDRs in terms of response attenuation is found to be lower than the case of LLRs, the former represents a novel, cost-effective and promising solution for vibration mitigation. Further, the pipe supported periodically on a rack structure is analyzed. In particular, to understand the influence of stiffness of the rack structure on wave propagation in pipe 𝑃, two models are examined; (i) 𝑃 supported on a rack and (ii) 𝑃 on simple supports. The pipe-rack system considered in this research represents a realstic scenerio while a pipe on simple supports represents a special case when the rack stiffness leads to extreme value. The band gap characteristics of the periodic piping systems are first calculated using analytical dispersion relations. The results reveal that both Bragg and resonance type band gaps coexist in piping systems due to the presence of spatial periodicity and local resonance. The pipe with rack creates a narrow locally resonant band gap in low-frequency range which is caused by the first natural mode of the rack. Conversely, the pipe on simple supports entails only Bragg type band gaps. The dependence of propagation/attenuation behavior on the number of unit cells is also investigated. On the basis of these results, to control the vibrations in a particular pass band of the pipe, a single degree of freedom (SDoF) LLR for various mass ratios is initially designed, which is then replaced with secondary pipe (i.e., LDR) available on the rack, of which each unit cell mass ratio is equivalent to the corresponding LLR. Pipe 𝑃 with one SDoF resonator at each unit cell exhibits one additional band gap around its tuned frequeny, thereby, the band gap properties are significantly get altered. Also, the influence of damping in the pipe and resonators on band gap properties is examined. Furthermore, two SDoF LLRs having identical mass are designed to control the same pass band of pipe 𝑃 similar to the case when one SDoF resonator is used with 𝑃. When two LLRs are installed at each unit cell of 𝑃, it shows two new band gaps in the frequency response function. Additionally, the effect of damping in the pipe and resonators on band gap properties is investigated. Subsequently, to replace the two LLRs, the possibility of using two existing smaller diameter pipes on the rack as LDRs is also evaluated. In this case, the effectiveness of both the LLRs and LDRs is verified using Gaussian white noise as input. Moreover, the transmission behavior of flexural vibrations in the pipe 𝑃 with rack, simple and without supports with two degree of freedom resonators (2DoF) is studied at last. To realize the effect of various resonator parameters, i.e. mass ratio, stiffness and damping, on band gaps, a detailed parametric study is performed. The results reported in this thesis can be useful to understand the propagation of flexural wave/vibration and thereby in designing efficient control mechanisms in similar periodic structures. The concept of LDRs is not limited to railway track and piping system but can be adopted in other situations where secondary systems exist in parallel to the main structure such as closely spaced bridges.