Quasi-periodicity and multi-scale resonators for the reduction of seismic vibrations in fluid-solid systems

https://doi.org/10.1016/j.ijengsci.2016.09.010Get rights and content

Abstract

This paper presents a mathematical model for an industry-inspired problem of vibration isolation applied to elastic fluid-filled containers. A fundamental problem of suppression of vibrations within a finite-width frequency interval for a multi-scale fluid-solid system has been solved. We have developed a systematic approach employing full fluid-solid interaction and dispersion analysis, which can be applied to finite and periodic multi-scale systems. The analytical findings are accompanied by numerical simulations, including frequency response analyses and transient regime computations.

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 r=4m, thickness tT=0.006m and height hT=14m. It is made of steel, having Young’s modulus E¯T=190GPa, Poisson’s ratio νT=0.3 and density ρT=7870kgm3. The covering lid has thickness tL=0.08m 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

References (47)

  • LiaoC.-Y. et al.

    Vibration characteristics of rectangular plate in compressible inviscid fluid

    Journal of Sound and Vibration

    (2016)
  • D.M. Mead

    Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995

    Journal of Sound and Vibration

    (1996)
  • P. Pal et al.

    Sloshing in partially filled liquid containers - numerical and experimental study for 2-D problems

    Journal of Sound and Vibration

    (2010)
  • M.K. Shrimali et al.

    Non-linear seismic response of base-isolated liquid storage tanks to bi-directional excitation

    Nuclear Engineering and Design

    (2002)
  • M.K. Shrimali et al.

    Seismic analysis of base-isolated liquid storage tanks

    Journal of Sound and Vibration

    (2004)
  • WangJ.D. et al.

    Liquid sloshing in rigid cylindrical container with multiple rigid annular baffles: free vibration

    Journal of Fluids and Structures

    (2012)
  • M. Brun et al.

    Phononic band gap systems in structural mechanics: finite slender elastic structures and infinite periodic waveguides

    Journal of Vibration and Acoustics

    (2013)
  • O.S. Bursi et al.

    Seismic assessment of petrochemical piping systems using a performance-based approach

    Journal of Pressure Vessel Technology

    (2016)
  • Bursi, O. S. et al. (2016b). Component fragility evaluation, seismic safety assessment and design of petrochemical...
  • G. Carta et al.

    Dynamic response and localization in strongly damaged waveguides

    Proceedings of the Royal Society of London A

    (2014)
  • G. Carta et al.

    Elastic wave propagation and stop-band generation in strongly damaged solids

    Fracture and Structural Integrity

    (2014)
  • Chalhoub, M. S., & Kelly, J. M. (1988). Theoretical and experimental studies of cylindrical water tanks in base...
  • M.S. Chalhoub et al.

    Shake table test of cylindrical water tanks in base-isolated structures

    Journal of Engineering Mechanics

    (1990)
  • Cited by (32)

    • Lightweight origami isolators with deployable mechanism and quasi-zero-stiffness property

      2022, Aerospace Science and Technology
      Citation Excerpt :

      Vibration exists in engineering structures ubiquitously and vibration isolation is an important issue for many applications [1–3].

    • Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment

      2021, Journal of Sound and Vibration
      Citation Excerpt :

      Furthermore, the authors did not take into account the feedback forces from superstructures to metafoundations. In order to overcome these drawbacks, other researchers [14–15] proposed a finite lattice LRM, the so-called Metafoundation, for the seismic protection of process equipment, e.g. storage tanks. The foundation consists of standard steel columns and concrete slabs that define the primary load bearing structure, while massive concrete masses are considered as resonators.

    • Design of tunable acoustic metamaterials with periodic piezoelectric microstructure

      2020, Extreme Mechanics Letters
      Citation Excerpt :

      Within the framework of acoustic metamaterials, some of the limitations of standard materials such as negative refraction [22,39–42], superlenses [43–45], and invisibility of defects embedded into both lattice and continuous systems [46–50] can be overcome. These astonishing features, unachievable with natural materials, have found application in the fabrication of new devices such as concentration detectors, vibration dampers and also in the protection of buildings from earthquakes [51–60]. Different techniques have been proposed to accomplish the aforementioned targets.

    • Multi-stopband negative stiffness composite column design for vibration absorption

      2019, Thin-Walled Structures
      Citation Excerpt :

      The work mechanisms of these negative stiffness structures (e.g. periodic structure, acoustic metamaterial, and metastructure) are similar to those of a tuned mass damper (TMD). The local resonance of the absorber generates a concentrated inertial force to work against the internal shear force of the structure, straighten the structure, and attenuate the vibration [26,27]. However, the TMD is an individual component that is installed on structures, and the installation location of the TMD can affect the vibration-absorption performance, as shown in Fig. 1(a) [28].

    • Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

      2019, International Journal of Solids and Structures
      Citation Excerpt :

      Research on metamaterials (employed to guide and control elastic waves for applications in microstructured devices (Sigmund and Jensen, 2003; Wang et al., 2015b,a; Lim et al., 2015; Bacigalupo et al., 2017; Lepidi and Bacigalupo, 2018; Antonakakis et al., 2013) and earthquake resistant structures (Brun et al., 2012, 2013; Carta et al., 2016; Colombi et al., 2016; Achaoui et al., 2017) has focused a strong research effort to time-harmonic vibrations of periodic beam networks.

    View all citing articles on Scopus
    View full text