Numerical and experimental comparative of coupled neighboring buildings

abstract: In recent years a vibration control technique, called structural coupling, has been studied. This technique consists on linking two neighboring buildings, through a coupling device, with the purpose of reducing dynamic response. It is possible to control both structures response simultaneously, which is precisely the attractiveness of this technique. Given the potential of the structural coupling technique, this work evaluates numerically and experimentally the performance of the structural coupling technique in simple plane frames when subjected to an oscillatory movement in at base caused by a shaking table, designed and built in the Structure Laboratory of University of Brasilia. Initially, the model numerical dynamic responses, without and with coupling, were obtained. Then, experimentally, the plane frames were fixed to the shaking table and subjected to a base movement uncoupled and coupled in order to obtain the acceleration registers and its frequency spectra. Finally, numerical and experimental frequency spectra were compared. The results obtained showed the efficiency of the control method through coupling, which depends mainly on the mechanical properties of adjacent buildings and connecting devices.


INTRODUCTION
With the cities development there is a tendency of population increase in large urban centers, causing thus a shortage of spaces and making tall buildings a characteristic of modern cities. These types of buildings because of their height and high mass concentration characteristics are very vulnerable to collapse in the presence of one of the most destructive phenomena of nature: the earthquake. These vibrations are undesirable from the point of view of both comfort and safety.
To work avoiding this problem, medium-and high-rise structures started employing control techniques such as tuned mass dampers (TMDs) to help mitigate excessive vibrations and deformations when subjected to dynamic loads. According to Watanabe et al. [1], one of the drawbacks when a single TMD is used in flexible structures is its sensitivity to any discrepancy in the natural frequency of the structure and the TMD damping. As a solution for this problem, multiple tuned mass dampers with different dynamic characteristics are used. However, the control system becomes expensive.
Thus, a control technique called structural coupling, which consists of connecting two or more neighboring buildings through coupling devices to provide reaction control forces, showed to be a viable alternative for the protection of adjacent flexible structures. This technique can be applied to a wide variety of structural systems and can incorporate various types of control strategies including passive, active, or semiactive control [2], [3].
The application of the coupling technique to civil structures has received much attention from the 90's since it was first proposed by Klein et al. [4] and Kunieda [5]. Theoretical, numerical and experimental approaches of coupled structures were performed in order to demonstrate the viability of this technique [6]- [34]. These studies showed that the efficiency of the coupling technique depends on the mechanical and geometrical properties of the neighboring buildings, as well as on the mechanical properties of the connecting device.
In addition to these analytical and experimental research, a few full-scale applications of coupled building control have been undertaken. In 2001, the Triton Square office complex, located on the Tokyo waterfront on Harumi Island, completed construction. The complex is a cluster of three buildings, 195, 175, and 155 m high, respectively. The three buildings are coupled with two active control actuators for wind and seismic protection [35], [36]. In 2013, between the Chiesa del Sacro Cuore (Florence -Italy) and R/C bell tower, passive viscofluid shock absorbers were installed in order to reduce the pounding between them [37].
It should be noted that most researches on the coupling technique were performed in countries located in areas with a high risk of seismicity and high economic development. In Latin American countries, research related to this technique are still incipient. Thus, this work evaluates numerically and experimentally the performance of the structural coupling technique in simple plane frames when subjected to an oscillatory base movement caused by a shaking table, designed and built in the Structure Laboratory of University of Brasilia. It should also be noted that, even using steel plane frames, the results can be extended to concrete or other materials frames.

METHODOLOGY
In a previous stage, numerical preliminary models were developed to set material type and dimensions of the plane frames to be tested. Then, the shear building frames were constructed and tested to identify dynamic properties such as frequencies, vibration modes and damping factors using the ARTeMIS Modal software [38]. It should be noted that in the shear building model a degree of freedom per floor is considered, therefore, for each of these degrees a frequency value is associated.
Subsequently, numerical models were validated using the correlation of the natural frequencies and the vibrational modes, numerical and experimental, using the percentage of variation of the frequency (FER) and the Modal Assurance Criterion (MAC) [39], [40]. This validation is important to ensure that the numerical models represented the actual physical behavior of the tested plane frames.
Once validation was done, an optimization study was performed using the particle swarm optimization (PSO) [41] in a way to set the connecting device properties. Next, the device that was used in the experimental tests was built.
Then, a numerical and experimental analysis were performed to evaluate the performance of the coupling technique in reducing the amplitudes of the dynamic responses of the adjacent shear frames. For this purpose, the acceleration records and the frequency spectra of each structure were obtained and compared. All numerical analysis, as well as the treatment of experimental data, were performed using MATLAB [42].

Coupled models
In Figure 1, two plane frames are shown, the structure 1 with n + m floors and the structure 2 with n floors. Each of these buildings has mass, stiffness, and damping properties mi j , ci j , ki j known beforehand, where i indicates the floor number and j the building number. The strategy of passive coupling control involves the placement of springs and dampers between the two buildings, whose mechanical properties are kn 3 and cn 3 , here n indicates the position of these elements.
where Mss, Css and Kss are the mass, damping and stiffness matrices of the coupled system respectively; C 3 and K 3 are the matrices that contains the damping and stiffness coefficients of the linking system kn 3 and cn 3 ; xss(t), ẋss(t) and ẍss(t) are the vectors containing the displacements, velocities and accelerations of the two structures relative to the ground. Mass, stiffness and damping matrices, Mss, Kss and Css, of the coupled structure are defined as follows: , , where j indicates the structure number, mi j , ci j and ki j are the mass, damping and stiffness value of the ith story in the jth building. The matrix that contains the damping coefficients of the linking system C 3 is: n m n m n m n 3 2 n n m n n where cn 3 is the damping coefficient of the nth passive linking element. The matrix that contains the stiffness coefficients of the linking system K 3 can be obtained from Equation 8, replacing the damping coefficients cn 3 by the corresponding stiffness coefficients kn 3 . From the second-order model (Equation 1), a first-order state-space model can be derived: Az E x y C z (11) with state vector: The state and disturbance input matrices, A and E, in Equation 11 have, respectively, the following form: Obtained the state vector z(t) and using the output matrix Cy shown in Equation 14, the vector y(t) (Equation 15) can be obtained, which contains the responses in terms of displacement and velocity relative to the ground, as well as the absolute acceleration of the two buildings.

Correlation between numerical and experimental data
Correlation methods are some techniques to compare the analytical modal data with the experimental modal data. Among the most used correlation techniques are the FER frequency variation and the Modal Assurance Criterion (MAC) [39], [40].
The correlation between natural frequencies is calculated using the FER frequency variation. The FER index for the g experimental mode is calculated using Equation 16, where λh and λg are the numerical and experimental natural frequency of the h and g mode, respectively.
The agreement between numerical and experimental modes of vibration is obtained through the Modal Assurance Criterion (MAC) index [39], [40]: where Φg and Φh are the test modal and analytical modal vectors, respectively. The MAC index takes values between 0 (representing no consistent correspondence) and 1 (representing a consistent correspondence). Values larger than 0.9 indicate consistent correspondence whereas small values indicate poor resemblance of the two shapes [43].

Numerical and experimental models
Six modules were constructed as shown in Figure 2a and represent a common floor of a residential building. Each of these modules consists of two stainless steel columns (ferritic alloy 430 -E = 210 GPa) with a rectangular section (1.5×25.4 mm) and rigid aluminum beams with a rectangular section (25.4×9.8 mm). The height of the columns was considered of 200 mm [31]. By joining several of these modules, it is possible to simulate more pavement structures as if they were a shear building (Figure 2b), since the stiffness of the aluminum bars, which constitute the floors, is much higher in relation to the stainless-steel plates stiffness, which represent the columns of the experimental models. In Figure 2c, the numerical models of adjacent uncoupled plane frames are presented.
The mass and stiffness matrices of the models showed in the Figure 2c are defined in the following Equations 18 and 19. In the mass matrices, besides the mass of the elements that make up the modules, the masses of the screws, nuts and washers used to join them were considered. In the case of stiffness matrices, it was considered that the stiffness of the modules is given by the columns fixed at the base.
, , , To assemble the stiffness matrices, the Caughey or extended Rayleigh method [44] (Equation 20) was used to construct these matrices from data obtained from experimental modal analysis of both buildings, that is, from a natural frequency ωi and its corresponding modal damping ξi of the ith mode.
where ξn e ωn are the natural frequencies and damping factors for the n modes of the system. Thus, the equation above becomes a set of equations, one for each ωn and ξn, where αi is the constant to be calculated.

Shaking Table
In order to simulate the movement at the base of the small-scale models, the unidirectional shaking table shown in  The oscillating motion of the table is generated by a rod-crank system, which can transform the circular movement into a translational movement. The crank describes the flat rotational movement, the rod pushes the main platform which, by means of the four linear bearings located in its corners, has rectilinear translation motion in the orthogonal direction X. The rotation movement of the crank is promoted by a motor whose angular rotation speed ω is controlled by a frequency inverter. The constructed model and mechanical and electronic components of the Shaking table can be observed in Figure 4.

Instrumentation
To obtain the accelerations in the plane frames, as well as their modal identification, PCB Piezotronics model 353B01 accelerometers were used, which have an approximate mass of 10g and sensitivity of ±5% 20 mV/g (2.04 mV/(m/s 2 )). These accelerometers were connected to the IPC Model 482A22 unit gain signal conditioner from the same company and which translates the vibration of the structure into electrical pulses and sends them to the ADS2000 signal receptor, manufactured by Lynx Tecnologia Eletrônica. Finally, the acceleration registers are monitored and recorded using AqDados 7 software [45].
The acquisition equipment was configured to capture records from 1 to 3 channels, at time instants of 0.005s, resulting in a 200 Hz sampling frequency. Consequently, the Nyquist frequency was half the sampling frequency, in this case, 100Hz.  Table 1 shows the natural frequencies obtained from the acceleration records acquired in the laboratory for each plane frame model (Figure 2b), as well as the FER frequency variation and the damping factor ξ for each vibration mode. The comparison of the numerical and experimental vibration modes of Structure 1 (Figure 2b), as well as the matrix of the MAC index correlation, are shown in Figure 5.

Plane frame modal identification
It can be seen in the Table 1 that the first frequencies in both plane frames presented a FER index lower than 1%. However, there was a low value for the second frequency, with differences of more than 7%. On the other hand, it is observed in Figure 5 that the MAC indexes presented diagonal values approximately equal to 1, which indicates a good agreement between the modal forms.
It is interesting to note that, as expected, the damping factors presented very low values as seen in the Table 1 [46]. Thus, having the frequencies and damping factors and using Equations 20 and 21, the damping matrices of the structures shown in Figure 2b can be assembled.
Finally, it is worth noting that the 1% error between the numerical and experimental frequencies for the first mode (fundamental mode) is acceptable due to the possible differences that the shear building model or plane frame are unable to represent, besides the possible instrumental errors. Thus, it was decided to not be updating the models.

Connection Device
An optimization study was performed using the particle swarm optimization (PSO) [41] in a way to set the mechanical properties kn 3 and cn 3 of the connection device ( Figure 1). The summation of coupled model rms (Root Mean Square) accelerations was used as an objective function. Once these mechanical properties were obtained, the damper that was used in the experimental tests was built. Table 2 presents the results of the optimization.
It can be observed in Table 2 that k 3 and c 3 had a very little variation. It also can be noticed that to control dynamic response on both structures, it is necessary to use a connection device with c 3 = 9.48 Ns/m and k 3 = 0. It can be said that the best connection device to the analyzed model is using only a viscofluid damper

Viscofluid damper construction
The connecting element used in this work was of the viscofluid type (c 3 ≠ 0, k 3 = 0) as shown in Figure 6. In this type of shock absorber, the energy dissipation occurs due to the rub of the piston in crossing the fluid generated by the viscous fluid, defined as viscous friction. The piston restricts the flow of oil through holes when it moves, causing a dissipation of mechanical energy in the form of heat. As the piston rod moves within the cylinder (Figure 6a), the fluid exerts a resistive force proportional to the velocity. The simplest mathematical modeling of the damping force, adopted in this work, is given by Equation 22, where Fd is the viscous resistance or damper force, c 3 is the viscous damping coefficient of the connecting element and ẋ is the relative velocity.
For the calculation of the coefficient c 3 , it is generally used Equation 23 proposed by Rao [47]: where: μ is the dynamic viscosity of the fluid; D and h are the diameter and height of the fistor head, respectively; d is the diameter of the orifice in the fistor head. For the construction of this device, the measurements listed in Table 3 were employed. Table 3. Geometric properties of the fluid viscous damper.

L [mm] D [mm] d [mm] h [mm] 200 10 1 3
Having calculated the optimal c 3 value of the connection element and using the geometric properties of the fluid viscous damper (Table 3) as well as Equation 23, it is possible to obtain the dynamic viscosity μ of the fluid to be placed in the device. Thus, µ = 0.279 Pa.s was obtained.
In this work, silicone oils were used, which are common in the manufacture of shock absorbers for RC cars. However, an oil with dynamic viscosity slightly higher than that calculated was used due to the unavailability on the market for the purchase of oils. Therefore, it was used a 47V350 silicone oil [48] of 0.97 g/cm 3 density and a dynamic viscosity μ = 0.350 Pa.s. Table 4 shows the mechanical properties, ideal and used, of the viscofluid damper.

Dynamic responses
This section presents the comparison between the experimental and numerical responses of the plane frames shown in Figure 7, without and with coupling, when submitted to base movement. For each case, ten tests were performed, five for the uncoupled system and five for the coupled system, in order to calculate an average of the responses and decrease the error tendency. Generally, in the study of the coupling technique, the acceleration records of El Centro, Kobe and Northridge earthquakes are used, which present the highest values of peak ground acceleration (PGA) in a frequency range between 2 and 5 Hz. Thus, in this work, a movement in the base with an oscillation frequency of 3 Hz was used (Figure 8), which represents the average of these frequency range. This simplification is necessary to be able to compare numerical and experimental results.  (Table 5), which, when compared, gives an indication of the effectiveness of the coupling technique in reducing dynamic responses [10]. Likewise, using these acceleration registers it was possible to obtain the frequency spectra, shown in Figures 11 and 12. It should be noted that these frequency spectra and the rms values listed in Table 5 show the mean of the data collected from all tests.
It can be seen from Figures 9 and 10 and Table 5 that, by coupling the structures with the viscofluid damper, it was possible to considerably decrease the accelerations in Structure 1 (red lines) compared to the uncoupled structure (gray lines). On the other hand, in Structure 2, even having a fundamental frequency (27.34 Hz) well above the value of the excitation frequency (3 Hz), the coupling technique managed to slightly decrease the accelerations compared to the uncoupled structure.
It can be seen in Figures 11 and 12 that the numerical and experimental spectra were almost identical, displaying differences only in the amplitude values at the peak frequency of the structures. Similarly, it is observed that the frequency spectra of the coupled system (red lines) decreased in the peaks where the fundamental frequency of each structure appears without increasing or decreasing the frequency of the structures.
In the case of the frequency spectra of Structure 2 (Figures 11 and 12), it is observed that in some ranges of the frequency spectrum there is an increase of the amplitudes when this building is coupled, however, these peaks correspond to harmonics associated to the movement of the shake table.
Finally, it is noteworthy that even without updating the numerical models, it was obtained response values close to those reached experimentally, and the structure 1 presented the closest approximation between the results.

CONCLUSION
The structural coupling technique consists of connecting two or more neighboring buildings by means of coupling devices in order to provide reaction control forces and thus, reduce the dynamic effects. Doing that, at first, it is possible to control the response of both structures simultaneously, which is precisely the attractiveness of the technique.
It is important to note that in this work, a base movement with an oscillation with only one frequency component (3 Hz) was used that represents the average of the frequencies range of the earthquakes most considered in research on the coupling technique.
Based on the results obtained in this work, several conclusions can be drawn. The optimization results showed that the fluid viscous damper is the ideal connection device, since increases the damping of the coupled structures and did not interfere in the values of the frequencies of the coupled system.
Another aspect to mention is that, even though the oil with the optimum dynamic viscosity was not used due to the unavailability of the material (common situation in civil construction), there was a decrease in the peak values of the acceleration records as well as in the amplitude peaks in the frequency spectra.
Finally, it is important to note that this research can be adapted for concrete or steel buildings [1], [15], [36]. The mathematical formulation would be the same, however, it would change the values of the mechanical properties of the structures.