Quasi-periodicity and multi-scale resonators for the reduction of seismic vibrations in fluid-solid systems
Introduction
Mathematical modelling of earthquake mitigation in elastic multi-scale structures is an area of high practical importance. Unfortunately, real-life structures require large-scale three-dimensional transient simulations, which present a computational challenge and often lead to inconclusive results. Such problems become even more challenging when fluid-solid interaction is involved and elastic deformations are considered as non-stationary.
The purpose of the present paper is to analyse propagation of elastic waves in multi-scale systems of fluid-filled containers and to offer a design leading to suppression of undesirable vibrations. One class of important applications is in the protection of storage tanks in industrial facilities (see Fig. 1) subjected to seismic waves, which can potentially cause serious accidents (Bursi, Paolacci, Reza, Alessandri, Tondini, 2016a, Krausmann, Cruz, Affeltranger, 2010, Sezen, Whittaker, 2006). Moreover, the proposed design can be applied to reduce the vibrations of different multi-scale structural systems, induced by earthquakes or other dynamic excitations.
In many engineering systems the fluid interacts with a deformable or a moving solid, in which case the coupling between the fluid and the solid needs to be taken into account in the design process. Fluid-solid interaction problems concerning vibrations of slender structures induced by axial flow are discussed in the comprehensive treatise by Païdoussis, 1998, Païdoussis, 2004. Banerjee and Kundu (2007) used the Distributed Point Source Method to derive the ultrasonic field created by ultrasonic transducers in a solid plate immersed in a fluid, and compared the analytical results with the Lamb wave modes visualised experimentally with stroboscopic photoelasticity. Cho, Kim, Kim, Vladimir, and Choi (2015) investigated the frequency response of plate structures in contact with a fluid and subjected to an internal harmonic force. Liao and Ma (2016) calculated the resonant frequencies and the associated mode shapes of a rectangular plate lying at the bottom of a container filled with inviscid compressible fluid.
If the fluid inside a solid has a free surface, sloshing waves are generated when the system is subjected to a dynamic excitation. Housner, 1957, Housner, 1963 and other early investigators proposed to model a fluid-filled tank as a mechanical system with masses and springs, where the container flexibility and the sloshing of the liquid free surface are neglected. The frequencies of sloshing waves can be calculated analytically if the solid container is assumed to be rigid (Ibrahim, 2005). On the other hand, when the container is elastic, approximate formulations are usually employed (Haroun, 1983, Veletsos, 1984). Alternatively, numerical or experimental investigations can be conducted. Jiang, Ren, Wang, and Wang (2014) performed an experimental sweep test to determine the lower frequencies of sloshing waves in a tank with a rectangular base, considering both thick (rigid) and thin (elastic) walls, and they found that the resonant frequencies are very close to each other. Pal and Bhattacharyya (2010) proposed a meshless formulation based on the Petrov-Galerkin method to study non-linear sloshing waves in a prismatic container under harmonic base excitation, and they obtained a good agreement with the solutions given by Washizu et al. (1984) and Frandsen (2004). In engineering applications, the amplitudes of sloshing waves are usually attenuated by using baffles, as shown by Belakroum, Kadja, Mai, and Maalouf (2010) and Wang, Lo, and Zhou (2012).
Field observations during past earthquakes (Hamdan, 2000, Manos, 1991, Manos, Clough, 1985, Niwa, Clough, 1982, Steinbrugge, Rodrigo, 1963) show that storage tanks subjected to seismic loads can be damaged for different mechanisms: large lateral oscillations, buckling of the tank walls (“elephant foot” and “diamond shape” buckling), uplift of the anchorage system, collapse of the tank roof, as well as failure of the piping system. These different failure mechanisms are described in the state-of-the-art reviews by Rammerstrofer and Fischer (1990) and Ormeño, Larkin, and Chouw (2012). Many isolation techniques have been developed to prevent damages of tanks, such as linear elastomeric bearings (Shrimali, Jangid, 2002, Shrimali, Jangid, 2004).
In the present paper we focus the attention on the lateral vibrations of the fluid-filled containers. In order to mitigate these vibrations, we propose to introduce a novel system of high-contrast multi-scale resonators, made of many masses linked by light beams and attached to the fluid container. This system is designed to re-distribute vibrations in a fluid-solid system within a predefined finite frequency range. The high-contrast multi-scale resonators can be tuned to serve the required frequency interval by varying the masses or the connecting beams. The proposed design is different from the conventional Tuned Mass Dampers, which are effective only around one or two predefined frequencies.
The possibility to reduce the vibrations in a finite frequency interval is crucial when the spectrum of the system depends on a random parameter, such as the level of fluid inside the container. Furthermore, earthquake accelerograms are characterised by a wide Fourier amplitude spectrum in the range [0, 30] Hz. The main advantage of the isolation system devised in this paper is that it is effective in a wide frequency interval, hence it can be applied to a large range of structures (with or without fluid) that can be excited by different frequencies of vibrations within a predefined range.
The example illustrating the efficiency of the proposed design is shown in Fig. 2, which presents relative displacements of fuel tanks with resonators and without resonators in a real-life earthquake scenario (the seismographic record is taken from the Northridge earthquake of 1994, discussed in Section 3.3). It is demonstrated that the modulated vibrations of containers without resonators attain much higher amplitudes than those with the multi-scale resonators proposed in this paper.
We begin by illustrating the use of the multi-scale resonators in the reduction of the vibrations of a three-dimensional cylindrical fuel tank used in a petrochemical plant. The fluid-solid system and the design of the resonators are described in Section 2. In Section 3 we analyse the response of the fuel tank in the frequency domain under a harmonic excitation, as well as the response in the transient regime under real seismic excitations. We continue by taking a large cluster of connected fluid-filled elastic containers, subjected to externally induced vibrations, as discussed in Section 4. A set of many containers is an interesting scenario, considering that in an industrial plant there are areas covered by tank farms (see Fig. 1a). We study the large cluster of containers as a periodic structure and we construct dispersion diagrams, which clearly show existence of stop-bands as well as standing waves in this multi-scale structure. In Section 5 we assess the effect of the multi-scale resonators on the fluid sloshing waves in the transient domain. Finally, in Appendix A we discuss several approximations to estimate the resonant frequencies of the combined fluid-solid system, while in Appendix B we present the analytical calculations of the frequencies of sloshing waves.
Section snippets
Multi-scale high-contrast resonators for the seismic protection of fluid-filled tanks
Tanks in an industrial plant are used to store flammable gases or liquids. If the plant is located in a region of high seismicity, they need to resist strong earthquakes without undergoing serious damage.
Fluid-filled tank
We consider a slender storage tank containing petrol, typically found in a petrochemical plant, which represents one of the case studies of the European project INDUSE-2-SAFETY (Bursi et al., 2016b).
The tank has radius thickness and height . It is made of steel, having Young’s modulus Poisson’s ratio and density . The covering lid has thickness and has the same constitutive properties as the tank.
The foundation is a square block of
Waves in a large cluster of fluid-filled containers
In an industrial facility some areas are usually covered by sets of fuel storage tanks, connected to each other by the foundation. In regions of high seismic hazard, it is essential to study how these sets of tanks behave when they are subjected to an earthquake and how their vibrations can be reduced in order to avoid serious accidents due to structural failure.
For simplicity we look at two-dimensional systems, because numerical simulations with several three-dimensional cylindrical tanks
Sloshing waves in the transient regime
Mathematical modelling of waves in fluids is a classical subject, which has generated a lot of interest among applied mathematicians, physicists and engineers. In particular, the classical texts by Ursell, 1958, Ursell, 1994 and Kuznetsov, Maz’ya, and Vainberg (2002) present an excellent theoretical framework of the theory of water waves in the context of partial differential equations. The dynamics of sloshing is well described in Ibrahim (2005), which provides elegant estimates of
Conclusions
The paper has presented an innovative design for isolation of vibrations of multi-scale structures consisting of fluid-filled containers. The idea of employing high-contrast multi-scale resonators has proved to be elegant and efficient to reduce vibrations of elastic containers filled with fluid within a predefined interval of frequencies.
The analytical estimates for the choice of parameters of the resonators have been based on the Floquet-Bloch approach, which provides a constructive guidance
Acknowledgements
G.C. and O.S.B. acknowledge the support from the Research Fund for Coal and Steel of the European Commission, INDUSE-2-SAFETY project, grant number RFSR-CT-2014-00025. A.B.M. would like to thank the EPSRC (UK) for its support through Programme grant no. EP/L024926/1. L.P.A. would like to thank the University of Liverpool for financial support and provision of excellent research facilities. The final part of the work was completed while A.B.M. was visiting the University of Trento, with the
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