Seismic Energy Assessment of Buildings with Tuned Vibration Absorbers

Seismic analysis and energy assessment of building installed with distributed tuned vibration absorbers (d-TVAs) are presented. )e performance of d-TVAs is compared with single tuned vibration absorber (STVA) installed at the top of the building. )e placements of the d-TVAs are based on the modal properties of the uncontrolled and controlled buildings. )e governing equations of motion of the building with the STVA and d-TVAs are solved by employing Newmark’s integrationmethod. Various energies under earthquake ground excitations are computed to study the effectiveness of using the STVA and d-TVAs. It is concluded that the use of the d-TVAs is the most competent because it effectively dissipates the seismic energy, and they are convenient to install requiring reduced space, as are placed at various floors.


Introduction
Earthquake events caused damages in structures especially buildings.e input energy of an earthquake excitation can be dissipated by tuned vibration absorber (TVA), which causes improvement in the performance of the buildings.Several researchers had employed the TVA and reported fairly good results of dynamic response control in structures [1][2][3][4][5][6][7].Later in [8][9][10][11][12][13], researchers proposed multiple TVAs to overcome the decreased efficiency of the TVA in offtuning and occupy large floor area with its massive weight.For tall and flexible buildings, Aly et al. [14] and Aly [15,16] showed that providing the TVAs prove to be useful in mitigating their dynamic response.Dynamic response reduction in different structures achieved by using the tuned vibration absorbers was studied by Lu [17,18].ey found that the particle movements of the plug flow pattern could yield good vibration attenuation effects.e detailed literature survey on passive TVA is provided by Elias and Matsagar [19].However, in majority of studies, it is assumed that the structure vibrates in only one direction or in multiple directions independently.In addition, the fundamental modal properties of the structures were used to design the TVA or the TVAs.In reality, the structures will undergo lateral as well as torsional vibrations simultaneously under purely translational excitations.erefore, many researchers considered the three-dimensional (3D) model installed with TVA or TVAs [10,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].e common outcome of the studies shows that TVAs are more robust and effective as compared to the TVA in response mitigation of 3D structures.
In most of the studies, absorbers are placed at the top of the structures.e loss of effectiveness of the multiple TVAs is minimal if they are distributed based on mode shape of the main system as concluded by Elias et al. [37].Elias et al. [38] presented the better performance of the d-TVAs as compared to the STVA/TVAs all placed at the topmost floor.
ey had conducted study on dynamic response control of buildings wherein both (a) placement and (b) tuning of the TVAs in the main system are made in accordance with the modal proprieties of the uncontrolled and controlled buildings.Gill et al. [39] showed that the d-TVAs were quite effective in seismic response control of structures.However, these studies do not establish energy-based seismic performance assessment.erefore, the objective of this study is to investigate the efficient positioning of the TVAs based on the modal properties of the uncontrolled and controlled buildings.
e TVAs are placed where the mode shape amplitude of the building is the largest or larger in the particular mode and tuned to the corresponding modal frequency.e number of modes to be controlled is decided depending on total mass participation in the controlled modes.

Equations of Motion
Mathematical models of the buildings are developed by making the following assumptions: (1) e superstructure is considered to remain within the elastic limit during the earthquake excitation.(2) e system is subjected to a single horizontal (unidirectional) component of the earthquake ground excitation.(3) e effects of soil-structure-interaction (SSI) are not taken into consideration.(4) At least ninety percent of total mass is included in the controlled modes.
As shown in Figure 1, the total degrees of freedom (DOF) of the controlled system become (N + n) by summing up the N DOF building with n number of TVAs.For the system under consideration, the governing equations of motion for the earthquake-excited building installed with the TVA at the top and installed with the d-TVAs are obtained by considering the equilibrium of forces at the location of each degree of freedom during earthquake excitation as where  and € x n  , respectively, are the unknown relative displacement, velocity, and acceleration of the TVAs.Modal analysis is conducted to determine the natural frequencies, mode shapes, and modal mass participations of the uncontrolled/controlled building.Only the first few modes are considered for controlling in this work as they predominantly influence the total dynamic response being sum of their modal mass participations reaches ninety percent.
e locations for installation of the TVAs are identified based on the mode shapes of the uncontrolled and controlled buildings.Figure 1 also shows the placement of the five TVAs as follows: TVA-1 at 20th floor, i.e., on the topmost floor; TVA-2 at 7th floor; TMD-3 at 4th floor; TMD-4 at 14th floor; and TMD-5 at 16th floor.

Tuning of TMDs.
Figure 2 shows the procedure followed for (a) placement of the d-TVAs and (b) optimization of parameters of the d-TVAs.e modal analysis is conducted to find the natural frequencies [Ω i′ , ω j′ ], mode shapes {ϕ i′,j′ }, and modal mass contribution [M r ] of the uncontrolled and controlled building using its stiffness [K s ] and mass [M s ] matrices for (N + n) degrees of freedom (DOF).Based on an assumption that 90% of the modal mass participates in the response, the first five modal responses (n � 5) are controlled, and subsequently, the optimum tuning frequency of each TVA [40] is calculated as follows: where μ i � μ/n, ϕ and ζ N , are, respectively, the mode shape and damping ratio of the Nth mode.Moreover, ω i � ω 1 ...ω 5 are the frequencies of the five TVAs, respectively, from the lowest to the highest, and Ω i � Ω 1 ...Ω 5 are the first five natural frequencies of the building.For the TVAs, while keeping the masses, , equal for all the TVAs, the mass (m i ) of each TVA is calculated by e stiffness is used for frequency tuning of each TVA and the stiffness of the TVAs (k i ) installed on different floors (i) are then determined by e optimum damping ratio ( e optimum damping ratio is calculated for base excited structure as per the formula given by Sadek et al. [40]. 2 Shock and Vibration and the damping coefficients (c i ) of the TMDs are calculated by

Energy Formulation.
Energy is input to the building by the ground motion during an earthquake excitation.Nevertheless, entire input energy is not dissipated at any time during the earthquake excitation, and remainder of the energy is stored in the structure in the form of interchangeable kinetic and strain energies.e input energy (E i ) is the work done by the ground motion on the building. where is the duration of the earthquake excitation, and t d is the small time increment.It is desirable that the chimney installed with the TVAs be able to dissipate more damping energy than without installation of the TVAs, as the tuned mass dampers help in maximizing the damping energy dissipation.In this study, E d is defined as the damping energy per mass of the building and dampers.e damping energy per mass for uncontrolled (NC) and controlled building is calculated by In these expressions, at any point of time, the input energy and damping energy dissipated by the building and TVAs is the summation of the total damping energy dissipated till that time [41].
e energy stored in the 25m Shock and Vibration 3 building due to its deformations is strain energy (E s ).In this study, it is assumed to model shear building; therefore, E s can be Similarly, the kinetic energy (E k ) is stored in the structure in the form of its motion.

Numerical Study
e seismic energy assessment of the 20-storey concrete building installed with the STVA and d-TVAs is investigated.e 20-storey building is modeled as shear-type lumped masses.
e mass each floor is 252 tonnes, and stiffness of each floor is calculated from the columns.In this study, the beam and column sections are assumed to be 0.6 × 0.6 m 2 and 0.7 × 0.7 m 2 , respectively.
e modulus of elasticity (E c ) of the concrete is assumed to be 2.5 × 10 13 N/m 2 and 2.9 × 10 10 N/m 2 , respectively, for the beam and the column, and density of the concrete is considered 2,400 kg/m 3 .e floor-to-floor height is 4 m, and the floors are assumed to be rigid in the horizontal plane because they provide diaphragm action in its own plane.e damping matrix is not explicitly known; hence, it is defined with the help of the Rayleigh's approach using damping ratio (ζ s � 5%) in all modes of vibration.
e mass participation factors (Γ i ) for the first, second, third, fourth, and fifth vibration modes are about 0.7300, 0.0829, 0.0502, 0.0306, and 0.0294, respectively.e present energy study contains three parts as follows: (i) the uncontrolled building (NC); (ii) the STVA is installed at the topmost floor of the building; (iii) in case of the d-TVAs, the TVAs are placed at the locations where the mode shape amplitude of the

Location optimizer
Parameter (m i ,k i ,c i ) optimizer Locate TMD based on mode shapes Modal analysis to find [Ω i′ ,ω j′ ], {ϕ (i′,j′) }, [M r ], j′ = 0:n buildings is the largest/larger in the particular mode, and each TVA is tuned to the corresponding modal frequency, while controlling first five modes having total mass participation of greater than 90%.e earthquake motions selected for the study are as follows: (i) 1940 Imperial Valley earthquake recorded at Elcentro; (ii) 1994 Northridge earthquake recorded at Sylmar station; and (iii) 1995 Kobe earthquake recorded at Japan meteorological agency (JMA).e peak ground acceleration (PGA) of the Imperial Valley, Northridge, and Kobe earthquakes are 0.34 g, 0.6 g, and 0.86 g, respectively; where, g denotes the gravitational acceleration.

Variation of Responses for STVA and d-TVAs.
In this section, a comparison is made between the seismic responses of the building controlled with the STVA and d-TVAs.For STVA, the damping ratio (ζ dopt ) is calculated by Equation ( 5) and mass ratio (µ) is assumed to be 0.05.e fair comparison requirement is to install the d-TVA system having same total mass ratio (µ) of 0.05.Equations (2a) through ( 6) are used to calculate the optimum parameters for TVA system.e focus of the present is not optimization of the parameters of the TVA or TVA schemes.In this section, spectral displacements and pseudospectral accelerations are calculated for the NC, STVA, and d-TVAs.e 20-storey building is considered while increasing the rigidity that the fundamental time period reaches 0.03 sec.Similarly, the rigidity of the building is reduced that the fundamental time period reaches 5 sec.e variation in both spectral displacements and pseudospectral accelerations by varying the rigidity is shown in Figure 3. Generally, it is seen that, by making the structure more flexible, the spectral displacement increases.However, the pseudospectral acceleration is decreased significantly.In addition, it is observed that both controller schemes are quite effective in controlling the spectral displacements and pseudospectral accelerations.In case spectral displacement response is concerned, it is seen that up to 40% and 50% reductions achieved, respectively, for the STVA and d-TVAs.Further, it is seen that both schemes are less effective for rigid structures.Best performances are achieved for both schemes while the time period is between 2 sec and 4 sec.erefore, it is concluded

Shock and Vibration 5
that the STVA and d-TVAs are effective in displacement response control of flexible structures as compared to rigid structures.Also, better performance of d-TVAs as compared to STVA is evident based on results presented.Further, the installation of d-TVAs is practically easier as compared to STVA because of its reduced size.
In case pseudospectral acceleration response is concerned, the response is reduced for flexible structures for NC, STVA, and d-TVAs.
e STVA is generally more effective in structure with time period higher than 1 sec.It is seen that up to 35% reduction is achieved after installation of the STVA.It is also clearly found that the d-TVAs are better controllers as compared to the STVA for the range of the structure from rigid to flexible.e reduction in pseudospectral acceleration is up to around 50% for the case where the d-TVAs are installed.It is concluded that the d-TVAs is more effective in controlling the pseudospectral acceleration response of the structures.

Variation of Energies for NC/STVA/d-TVAs.
Based on the results presented in Figure 3, the building with fundamental time period of 2.91 sec is selected as most vulnerable.
erefore, for the energy assessment, the building with fundamental time period of 2.91 sec is selected.Figure 4 shows the time histories of the total energy, damping energy, kinetic energy, and strain energy for the building without (NC) and with the TVAs used in different schemes defined earlier; STVA and d-TVAs and corresponding peak values are given in Table 1.e mass ratio of 5%, tuning frequency ratio by Equation (2a), and damping ratio by Equation ( 5) are considered for both the schemes.It is generally seen from the energy plots that, as compared to the NC case, higher damping energies are dissipated when the TVAs are installed.e relative increase in the damping energy is significantly more in the d-TVAs than the STVA and NC.Increase in total energy and damping energy are, respectively, 13% and 12.5% by installing STVA, whereas it is 30% and 28% by installing the d-TVAs.us, the effectiveness of the multimode control strategy using the d-TVAs in enhancing damping energy in the system is confirmed.Shock and Vibration Further, it is evident that the kinetic energy and strain energy are reduced by installing the STVA and d-TVAs as compared to NC.It is generally due to reduction of the displacement and acceleration responses of the structure.e improved performance of d-TVAs is observed as compared to the STVA.Similar trend of response control and observations can be made from Figures 5 and 6 for different earthquake excitations.It is concluded that increase in the total energy and damping energy is significantly more in the d-TVAs than the STVA and NC.It is also concluded that the decrease in the kinetic energy and strain energy is significantly less in the d-TVAs than the STVA and NC.Generally, it is concluded that the use of the d-TVAs is the most competent because it effectively dissipates the seismic energy, and they are convenient to install requiring reduced space, as are placed at various floors.e procedure adopted here is based on ignoring the torsional degrees of freedom; whereas practically structures will undergo lateral as well as torsional vibrations simultaneously under purely translational excitations.e TVAs for seismic response control of the irregular structures using the multimode control approach may require separate complex analysis under other  Shock and Vibration parametrics involved in the system, which can be investigated in the future studies.

Conclusions
Seismic response control of buildings installed with a single tuned vibration absorber (STVA) and distributed tuned vibration absorbers (d-TVAs) is investigated.An assessment is made on the response of the buildings installed with the STVA and TVAs distributed along the height of the buildings.From the trends of the results of the present study, the following conclusions are drawn:  Shock and Vibration

Figure 2 :
Figure 2: Flowchart for location and design parameters of d-TVAs for seismic response control of buildings.

Figure 3 :
Figure 3: Spectral displacement and pseudospectral acceleration for the NC, STVA, and d-TVAs subject to different ground excitations.

Table 1 :
Peak energy for the uncontrolled building and controlled with STVA and d-TVAs having mass ratio of 0.05.