## Abstract

A novel device, called vibrating barrier (ViBa), that aims to reduce the vibrations of adjacent structures subjected to ground motion waves is proposed. The ViBa is a structure buried in the soil and detached from surrounding buildings that is able to absorb a significant portion of the dynamic energy arising from the ground motion. The working principle exploits the dynamic interaction among vibrating structures due to the propagation of waves through the soil, namely the structure–soil–structure interaction. The underlying theoretical aspects of the novel control strategy are scrutinized along with its numerical modelling. Closed-form solutions are also derived to design the ViBa in the case of harmonic excitation. Numerical and experimental analyses are performed in order to investigate the efficiency of the device in mitigating the effects of ground motion waves on the structural response. A significant reduction in the maximum structural acceleration of 87% has been achieved experimentally.

## 1. Introduction

Control of building vibrations is crucial for structural safety and avoids the unexpected behaviours that lead to rapid deterioration or collapse of a structure. Various sources of vibrations can affect the structure, including human activities such as road traffic, high-speed trains, large machinery, rock drilling and blasting or natural disturbances such as wind gusts, ocean waves and earthquakes. Strategies for vibration control are based on the modification of the dynamic structural characteristics by: (i) increasing the dissipative properties of the structure, (ii) altering its rigidity for inducing the shift of the structural frequencies, or (iii) adding resonant devices able to absorb part of the structural vibrations. In civil engineering, devices for the passive control of ground motion waves are widely used since they do not require an external power source to operate; a few examples of passive control devices are viscous damper, tuned mass damper, tuned liquid damper, base isolation and dissipative bracing systems. These vibration control systems are successfully employed in the design of new structures; on the other hand, they are rarely used for protecting existing buildings, as they generally require substantial alteration of the original structure. In the case of heritage buildings and critical facilities or urban areas, especially in developing countries, these traditional localized solutions might become impractical. Therefore, alternative non-localized solutions represent a reliable strategy for this challenge.

In this regard, very few attempts have been made to investigate non-local strategies to ensure the safety of existing buildings, infrastructures and critical facilities. Namely, trench barriers or sheet-pile walls in the soil have been investigated for altering the displacement field based on the reflection, scattering and diffraction of dynamic surface waves (e.g. [1–5]). These attempts, even if mainly limited to the reduction of surface waves, highlighted the importance of focusing on the soil instead of the structure itself.

During the last two decades, studies on the site–city interaction [6–8] highlighted a substantial change in the ground motion wave field and the consequent dynamic response of buildings in an urban environment. Remarkably, Kham *et al.* [8] showed that the energy of ground motion at the free field in a city is reduced by up to 50% due to the perturbation induced by resonant buildings. The reasons for this phenomenon governing the site–city effects are based on the well-known structure–soil–structure interaction (SSSI) [9,10]. Warburton *et al.* [9] studied the dynamic response of two rigid masses in an elastic subspace, showing the influence of one mass with respect to the other. Luco & Contesse [10] studied the dynamic interaction between two parallel infinite shear walls placed on rigid foundations and forced by a vertically incident shear (SH) wave. Wong & Trifunac [11] extended the previous case for non-vertically incident plane SH waves by investigating the significance of the angle of incidence. A recent review of the SSSI problem can be found in Lou *et al.* [12].

The benefits arising from the presence of other buildings in reducing structural vibrations have not yet been exploited as a tool for seismic vibration control. This paper introduces for the first time a novel device, herein called the vibrating barrier (ViBa), that aims to reduce the vibrations of structures from ground motion waves by exploiting the SSSI phenomenon. Figure 1 illustrates the physical problem investigated in this paper: a cluster of buildings subjected to base excitation along with the proposed device, the ViBa, embedded in the soil for absorbing the input energy, reducing therefore damage and failures of the adjacent structures. The ViBa is a structure buried in the soil and detached from the surrounding buildings. It consists of an embedded foundation containing an internal oscillator unit that, if tuned appropriately, is able to absorb a significant part of the dynamic energy that would otherwise affect the structures. In order to study this novel vibration control strategy, a discrete model is first derived.

Discrete solutions to the SSSI problems using rigorous analytical formulations are available in the literature [13–16]. Kobori *et al.* [13] defined a multi-spring–mass system for investigating the dynamic coupling of two adjacent square superficial foundations. Mulliken & Karabalis [14] defined a simple discrete model for predicting the dynamic interaction between adjacent rigid, surface foundations supported by a homogeneous, isotropic and linear elastic half-space. Recently, Alexander *et al.* [15] developed a discrete model to study the SSSI problem of surface foundations by considering stochastic ground motion excitation; Aldaikh *et al.* [16] extended the work of Alexander *et al.* [15] to the case of three buildings with validation of the discrete theoretical model by means of experimental shake table testing.

Based on the same principles, the effects of the soil on the structures, i.e. the soil–structure interaction (SSI) as well as the SSSI and the ViBa–SSI, are taken into account in this paper by means of linear elastic springs, as in the conventional Winkler approach for a linear elastic soil medium.

The simplified model is able to capture the main effects of the interaction phenomena of the soil, as shown in the comparison with more advanced finite-element method (FEM)/boundary element method (BEM) numerical solutions in a model of a nuclear reactor. Moreover, experimental results showed a remarkable reduction in terms of maximum acceleration of 87% of a structure controlled by the ViBa prototype.

## 2. Governing equations of the structure–soil–structure interaction

Consider the global system depicted in figure 1 under the ground excitation defined by the ground displacement *U*_{g}. The proposed ViBa is also included with the aim of reducing the vibration of the surrounding buildings. In this regard, a mechanical model able to describe the interaction phenomenon is derived first. Figure 2 shows the mechanical relations of the *i*th and *j*th structures coupled with the ViBa. Each building is modelled as a 2 d.f. system with one translational d.f. at the top of the building and one at the foundation level, i.e. *U*_{i} and *U*_{f,i} for *i*=1,…,*n* (where *n* is the number of surrounding buildings). The ViBa is modelled as an internal unit device included in a rigid box foundation and globally described by the 2 d.f., *U*_{ViBa} and *U*_{f,ViBa}. The dynamic governing equations of the global system are derived in terms of absolute displacement, as conventional in SSI, namely the dynamics of the problem take the form
**U**(*ω*) is the absolute displacement formulated in the frequency domain (*ω* is the circular frequency) in which the components are ordered as follows:
*i*th structure and of the ViBa are indicated in figure 3. In equation (2.1), **M** is the real global mass matrix and

In addition, the matrices of the global system are partitioned in the sub-matrices defined for the individual buildings and the ViBa; therefore, the global mass matrix is stated as follows:
*i*th sub-block is defined as
*m*_{i} is the mass of the *i*th structure and *m*_{f,i} is the mass of the *i*th foundation, while **M**_{V} is the mass matrix of the ViBa given by
*m*_{ViBa}, and the mass of its foundation, *m*_{f,ViBa}. Also, the global stiffness matrix *η* is the loss factor and

Furthermore, the matrix *i*th and the *j*th structures are defined as
*i*th structure and the ViBa. It is noted that in general the components

Finally, *U*_{g} is the ground motion displacement applied to the base of each foundation and **Q** is the influence matrix that depends on the soil–foundation stiffness values as follows:

The structural parameters of the ViBa represent the unknowns of the problem, as they have to be determined in order to reduce the dynamic response of the adjacent structures. Namely, all five ViBa structural parameters, *k*_{ViBa}, *m*_{ViBa}, *η*_{ViBa}, *k*_{f,ViBa}, *m*_{f,ViBa}, can be determined through an optimization procedure aimed at reducing the structural response of the adjacent buildings. However, it is noted that the relevant foundation parameters of the ViBa, namely the mass *m*_{f,ViBa} and the stiffness *k*_{ViBa}, the mass, *m*_{ViBa}, and the damping, *η*_{ViBa}, i.e. the components of the internal oscillator unit of the ViBa. These parameters are collected in the design parameters vector ** α**={

*k*

_{ViBa},

*m*

_{Viba},

*η*

_{ViBa}}.

The objective of the ViBa is to reduce the vibrations of the adjacent structures and the consequent stresses related to the relative displacements. Therefore, the optimization problem is established as
*i*th structure relative to its foundation:

## 3. Vibration control of two structures through the vibrating barrier

Consider a global system composed of two buildings protected by the ViBa, as illustrated in figure 2, with *i*=1 and *j*=2. The governing equation of motion of the system is
** α** is the design parameters vector. If the shape of the two foundations of the buildings is identical, then the interaction with the soil is identical as well and the following relations occur:

**U**and input displacement

*U*

_{g}

*ω*

_{0}. The adopted procedure consists in minimizing the transfer functions related to the structures at the input frequency

*ω*

_{0}. Therefore, by recalling the design parameters vector

**={**

*α**k*

_{ViBa},

*m*

_{Viba},

*η*

_{ViBa}} from equation (3.4), the optimization problem is stated as

*m*

_{ViBa}, is restrained by engineering criteria (e.g. bearing capacity of the soil, volumetric restraint, etc.). Therefore, by assigning

*m*

_{ViBa}as a known quantity, from equation (3.4), the stiffness

*k*

^{optimal}

_{ViBa}and the damping

*η*

^{optimal}

_{ViBa}are derived in closed form by determining the zeros of the transfer functions

*H*

_{1}(

**,**

*α**ω*

_{0}) and

*H*

_{2}(

**,**

*α**ω*

_{0}) according to Den Hartog [18]. Following simple algebra, the following formula is derived:

*η*

^{optimal}

_{ViBa}are derived as follows:

Therefore, after proper tuning, the ViBa is designed for mitigating the dynamic response of both buildings.

## 4. Vibration control of a single structure through the vibrating barrier

This section describes the vibration control of a structure protected by the ViBa under harmonic base excitation. The mechanical model to be analysed is shown in figure 4. The building is modelled as a simple linear oscillator characterized by the lumped mass, *m*, and the complex stiffness, *m*_{f}. The ViBa consists of the internal oscillator unit described by the lumped mass, *m*_{ViBa}, and the spring complex stiffness, *m*_{f,ViBa}, and the complex stiffness,

### (a) General solution

Equation (2.1) is rewritten in the expanded problem as follows:
**H**(** α**,

*ω*) are given by

### (b) Closed-form solution

By assigning the mass *m*_{Viba}, the optimization procedure (4.8) leads to finding the zeros of the transfer function at the frequency *ω*_{0}. After some simple algebra, the following formula is derived:
*U*(*ω*_{0}) equal to zero, for the assigned mass *m*_{ViBa}, are the following:

### (c) Particular cases

It should be noted that, for lightly damping *η*≪1, the quadratic terms involving damping are negligible; thus (*X*^{imag})^{2}→0 and equations (4.11) and (4.13) can be rewritten as
*k*_{ViBa} and *m*_{ViBa} have to be obtained since the ratio between the structural and soil stiffness becomes a relevant factor that must be taken into account.

In the case of undamped systems, i.e. every *η*=0, equation (4.9) reduces to
*U*(** α**,

*ω*

_{0})=0 for the harmonic excitation at a given frequency

*ω*

_{0}.

Both the achieved formulae of equations (4.9) and (4.18) are independent from the characteristics of the above-ground structures to be protected, such as *m*. Only *k*_{f} and *k*_{SSSI} are required for the design of the ViBa. In the case of a rigid foundation, they can be determined as a function of the foundation shape and of the stiffness of the soil. Moreover, for *m*_{ViBa}≪*m*_{f,ViBa}, equation (4.18) can be cast as
*a*=(*k*+*k*_{f}−*ω*^{2}*m*_{f}); *b*=(*k*_{ViBa}+*k*_{f,ViBaf}−*ω*^{2}*m*_{f,ViBa}) and

In this case, the ViBa is calibrated by the formula defined in equation (4.9) behaving as the classical DVA.

## 5. Numerical and experimental analyses

In this section, numerical and experimental analyses are carried out to investigate the performance of the proposed ViBa to reduce the vibrations of structures subjected to harmonic ground motion excitation. The physical model comprises one structure to be protected coupled with a ViBa, and it is represented by the prototype in figure 5. Mechanical properties are derived directly from the prototype. Structural characteristics are reported in table 1 and in the experimental section. The first fundamental frequency of the structure by considering the compliant base is *ω*_{0}=22.62 rad s^{−1}. In table 2, the known mechanical characteristics of the ViBa are determined for the manufacturing of the prototype, whereas the internal device parameters described by the mass, *m*_{Viba}, the stiffness, *k*_{ViBa}, and the loss factor, *η*_{ViBa}, are derived by the optimization procedure determined in this paper.

### (a) Numerical results for harmonic base excitation

In this section, numerical analyses are performed to scrutinize the efficiency of the ViBa. The goal is to reduce the vibrations of the structure subjected to harmonic excitation with circular frequency *ω*_{0} equal to its first fundamental frequency, i.e. *ω*_{str}=*ω*_{0}=22.62 rad s^{−1}, that would otherwise cause severe damage due to the induced condition of resonance. The undamped case is addressed first. Figure 6 shows the modulus of the transfer function response of the structure, |*H*(*ω*)|, and of the ViBa, |*H*_{ViBa}(*ω*)|, obtained by calibrating the mass of the ViBa, *k*_{ViBa}, by means of equation (4.18) for three mass ratios, *m*_{ViBa}/*m*={0.5;1;1.5}, with comparison with the response of the uncoupled single structure. Each curve converges at the same null value, |*H*(*ω*)|=0, at the target circular frequency, *ω*_{0}=22.62 rad s^{−1}. Therefore, in undamped systems the ViBa is able to absorb 100% of the vibrations of the structure. Figure 7 depicts the modulus of the transfer function response of the structure and the ViBa obtained for different values of the coupling interaction stiffness, *k*_{SSSI}. Three cases are reported, specifically the single structure when *k*_{SSSI}=0, the coupled structure with ViBa when *k*_{SSSI}=315 N m^{−1}, i.e. the spring value used for the prototype, and the limit case of *k*_{SSSI}=0 (e.g. very long spacing between the ViBa and the structure) is identical to the uncoupled structure.

A frequency-independent hysteretic damping model for simulating the dissipation of the energy is then considered. The loss factors applied to determine the complex stiffness springs are given in table 3 and have been determined through an identification procedure.

Optimization of the internal parameters of the ViBa is performed by the formula described in equations (4.9)–(4.14), where the stiffness *k*^{optimal}_{ViBa} is obtained once the mass *m*_{ViBa} is assigned and the optimal damping *η*^{optimal}_{ViBa} is obtained by equation (4.15). It is noted that equation (4.15) yields a negative value of the loss factor *η*^{optimal}_{ViBa} that cannot be adjusted since equation (4.13) is a function of only the soil and structural parameters, that is, it is independent of the parameters of the ViBa; thus the minimum real value *η*_{ViBa}=0 is assigned.

Figure 8*a* shows the modulus of the transfer function of the structural displacement, |*H*(*ω*)|, obtained by calibrating the stiffness of the ViBa, *k*_{ViBa}, by means of equation (4.9) for three different mass ratios, *m*_{ViBa}/*m*={0.5;1;1.5}, and compared with the response of the structure itself (referred to as a single structure). Although the steady-state responses for the three mass ratios are different because of the SSI effect, every curve converges at the same minimum value at the target circular frequency, *ω*_{0}=22.62 rad s^{−1}.

The achieved percentage reduction caused by the ViBa expressed in terms of the reduction factor (RF), which is defined as the ratio of the response between the coupled and uncoupled systems at the target circular frequency *ω*_{0}=22.62 rad s^{−1},
*b* shows, for the same case, the modulus of the transfer function of the ViBa, |*H*_{ViBa}(*ω*)|, showing an increment in the response with respect to the ViBa alone, manifesting the transfer of energy from the structure to the ViBa.

It is noted that the realization of a ViBa with zero damping might be extremely challenging if not impossible. Therefore, the effect of the ViBa in the case of damping *η*^{optimal}_{ViBa} different from zero is also investigated. Specifically, in addition to the previous analysis, the steady-state response is investigated by considering the loss factor experimentally measured in the prototype, i.e. *η*_{ViBa}=0.18. Results are reported in figure 9. The minimum values of the response obtained at the target circular frequency, *ω*_{0}=22.62 rad s^{−1}, are different for the three mass ratios and are higher than the value obtained by considering the loss factor *η*_{ViBa} as null. Nevertheless, the presence of damping is fundamental to protecting the structure for a wide-band excitation. Moreover, by increasing the mass ratio, the minimum value at the target circular frequency decreases.

The presence of damping *η*_{ViBa} different from the optimal value produces a detrimental effect for the protection of the structure subjected to harmonic loading that gets worse with an increase in the damping. This trend is illustrated in figure 10, where the RF is plotted versus the mass ratio and for various values of the loss factors *η*_{ViBa}. Note that each curve lies below the unit value; therefore, the response of the structure protected by the ViBa is always smaller than the response of the same structure without protection, manifesting the efficiency of the ViBa. Also, the RF decreases with the decrease of the loss factor *η*_{ViBa} and with the increase of the mass ratio *m*_{ViBa}/*m*, i.e. the lower the ViBa damping ratio, the higher the mass ratio *m*_{ViBa}/*m* and the higher the reduction in the dynamic response. Moreover for very low values of the ViBa damping ratio, the RF tends to become insensitive to the mass ratio *m*_{ViBa}/*m*.

### (b) Experimental test

Experimental shake table tests are performed in order to validate the analytical formulae and numerical results determined in the previous sections. The prototype as depicted in figure 5 is designed to reproduce the system of figure 4 and is tested on a 45 × 45 cm shake table. The purpose is to show the efficacy of the ViBa, calibrated by equation (4.9), in reducing the structural accelerations. The structure to be protected is realized as a one-storey shear-type building made up of acrylic for the base foundation as well as for the storey (80.5×80.5×10 mm) while aluminium sheets (80.5×141.6×0.6 mm) are used for the walls. Additional masses are placed on the top of the structures in conjunction with an accelerometer of mass 4.2 g on the top of the structure in order to record the structural response. The interaction effects with the soil are captured by linear elastic springs leading to a total stiffness *k*_{f}=640 N m^{−1} and *k*_{f,ViBa}=760 N m^{−1}. The SSSI is represented by a linear spring *k*_{SSSI}=315 N m^{−1} coupling the structure with the device. The prototype is set up on a Quanser Shake Table II for performing dynamic tests. Harmonic tests are carried out at several frequencies ranging from 2.0 to 10.0 Hz considering first the structure on its own and then coupled with the ViBa. The inherent damping is quantified through a best fit of transfer functions evaluated numerically and experimentally. Moreover, the ViBa is modelled as a rigid container with an internal oscillator made up of a spring connected to a mass and placed adjacent to the structure as shown in figure 5. The ViBa has been calibrated in order to absorb the energy affecting the structure at a selected frequency, i.e. the resonant frequency in the uncoupled system, that is, *ω*_{str}=*ω*_{0}=22.62 rad s^{−1}. However, the device is efficient for whichever desired frequency, i.e. the device can be calibrated to absorb any target frequency. The optimal design is accomplished by means of equation (4.9). Namely, the optimal stiffness of the ViBa, calculated by means of equation (4.9), is *m*_{ViBa} equal to 0.629 kg. The transfer function is derived as the ratio between the maximum acceleration recorded at the top of the structure and that applied to the shake table. Figure 11 shows the comparison of the calculated transfer function of the structure uncoupled and coupled with the ViBa. The experimental curves show the efficiency of the ViBa in altering the structural response. Moreover, numerical transfer function curves are drawn by means of equation (4.3), and show a good match with the experimental results. Finally, the RF calculated as the ratio between the maximum acceleration recorded in the coupled system and uncoupled system provides
*ω*_{str}=*ω*_{0}=22.62 rad s^{−1} with maximum acceleration of 1 m s^{−2} is applied.

### (c) Numerical application on a reactor building

In this section, the proposed optimization procedure for tuning the ViBa parameters is applied to investigate the response of the model of a reactor building as depicted in figure 13. The characteristics of the model of the reactor building are derived from the Electric Power Research Institute [19]. The relevant dimensions are summarized in table 4. The model of the reactor building is founded on a 30 m thick soil deposit (which is the conventional soil depth for the soil classification provided by Eurocode) characterized by shear wave velocity *V*_{s}=400 m s^{−2}, mass density *ρ*_{s}=2.1 KN m^{−3}, Poisson ratio *v*_{s}=0.45 and hysteretic damping *η*_{s}=0.1, resting on bedrock with a shear wave velocity of *V*_{s}=800 m s^{−2} and hysteretic damping *η*_{bed}=0.05.

The ViBa is placed at a net distance of 6.45 m in order to mitigate the response to the harmonic input with frequency *ω*_{0} equal to the first natural frequency *ω*_{str}=21.05 rad s^{−1}. The ViBa is modelled externally as a circular embedded foundation characterized by dimensions reported in table 5. The internal structure of the ViBa is a single oscillator characterized by the internal mass, *m*_{ViBa}, the stiffness, *k*_{ViBa}, and the hysteretic damping, *η*_{ViBa}. The study has been undertaken by assuming linear behaviour of the structure, soil and ViBa. The substructure methodology proposed by Kausel *et al*. [20] is applied in order to solve the dynamic problem in which the reactor building is modelled according to the finite-element approach by means of the Code_Aster open source finite-element software [21], whereas the BEM formulation is used to derive the soil dynamic impedances by means of Miss3D [22]. As the BEM approach is used no internal mesh is required for the soil. The overall damping of the soil is the result of two dissipation phenomena: the first is the material damping defined by means of the complex hysteretic coefficient, the second is the radiation damping that is internally accounted for by the solution of the BEM, which applies the condition of convergence at infinity. It has to be emphasized that this approach has been validated through comparison with the experimental results of Clouteau *et al.* [23].

Equation (4.9) is used for obtaining the ViBa parameters; owing to the frequency dependence of the soil dynamic impedances, the stiffness and damping values of each impedance are calculated at the circular frequency *ω*_{0}. A null damping, *η*_{ViBa}=0, is assigned as the optimization problem also in this case leads to an optimal negative value.

Figure 14 shows the transfer function responses of the nodal displacement recorded at the top of the dome by varying the mass ratio between the mass, *m*_{ViBa}, of the ViBa and the mass of the reactor building, *m*. Note that the reduction of the response at the circular frequency *ω*_{0} due to the effect of coupling with the ViBa is independent from the mass ratio, as already shown in figure 8 in the case of null ViBa damping. The calculated RF is 0.25; therefore, a relevant reduction of 75% of the absolute displacement is achieved. This shows the efficiency of the ViBa in realistic cases by designing the device with formulae determined through simplified structural models.

Finally, the performance of the ViBa is tested for a wide-band signal. The real ground motion recorded in the 1989 Loma Prieta earthquake event is thus applied to the system. The ViBa is designed by equation (4.9) by considering the mass ratio *m*_{ViBa}/*m*=1.5. Figure 15 shows the comparison of the responses in terms of the acceleration at the top of the dome for both cases of a single structure and a structure protected by the ViBa. The comparison of the time-history accelerations show a beneficial effect due to the ViBa achieving a reduction in the maximum acceleration of 43.2% (RF=0.568).

## 6. Conclusion

A novel passive control device, called ViBa, is herein presented. The ViBa aims to reduce the vibrations of a cluster of neighbouring structures, subjected to ground motion waves, by exploiting the SSSI effect. The proposed vibration control strategy presents a novel approach moving towards a global solution for the passive control of structures under ground motion as an alternative to the conventional localized solutions.

A simplified discrete model has been determined to analyse the performance of the proposed device on the structural dynamic response. Closed-form solutions have been derived for designing the ViBa in order to protect surrounding buildings subjected to harmonic ground motion waves.

Numerical analyses are performed to evaluate the performance of the ViBa calibrated through the proposed procedure. Results have confirmed the efficiency of the proposed closed-form solutions. The ideal condition of zero damping has been considered first to study the performance of the ViBa, and for this academic case a reduction of 100% of the relative structural displacement has been shown. For the damped condition in all the cases analysed, a reduction greater than 75% has been found.

Shake table experimental tests have been performed showing a significant reduction in the structural acceleration of 87%.

It has to be emphasized that, even if the ViBa has been designed to control harmonic excitations, it was able to significantly reduce the response of a nuclear reactor to broad-band excitation as well.

Clearly, it has to be pointed out that all the three seismic wave components cannot be damped by a single one-dimensional ViBa. To this aim, a three-dimensional ViBa should be used. Also in analogy with the tuned mass damper (TMD) technology, multiple ViBas might also be embedded in the soil to take account of a large variety of inputs.

In addition, the paper addresses the simplistic hypothesis of linear soil behaviour and nonlinearity might induce issues in the tuning of the ViBa. A possible approach to consider nonlinear soil behaviour in the study is to use the formulation presented in this paper by considering nonlinear springs as proposed in Allotey & El Naggar [24] and then calibrate the ViBa numerically.

The major criticism that can be raised for the novel device is the large mass of the ViBa. Although the basic working principle is similar to TMD systems, the mass of the ViBa needs to be large in comparison with the TMD. It has to be emphasized, on the other hand, that the ViBa can be designed to reduce the vibrations of more than one building and/or for buildings of historical relevance, for which it might still be considered a viable solution.

## Data accessibility

Supporting data are available from the University of Brighton data archive at http://dx.doi.org/10.17033/DATA.00000013.

## Author' contributions

Both authors had (1) substantial contributions to the conception and design, acquisition of data, analysis and interpretation of data; (2) drafting the article and revising it critically for important intellectual content; and (3) final approval of the version to be published. P.C. conceived the basic idea of the ViBa, organizing the research to be carried out and drafting the paper and the analytical formulation; A.T. contributed significantly to the analytical, numerical and experimental studies as well as in the writing of the paper.

## Competing interests

We declare we have no competing interests.

## Funding

This research was supported by the EPSRC First Grant EP/K004867/1 ‘Vibrating barriers for the control of seismic waves (ViBa)’.

## Acknowledgements

The authors wish to thank Dr I. Zentner, LaMSID UMR EDF-CNRS-CEA, France, for her valuable support for the analyses of the nuclear reactor.

- Received February 4, 2015.
- Accepted May 27, 2015.

© 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.