GROUND MOTION DOMINANT FREQUENCY EFFECT ON THE DESIGN OF MULTIPLE TUNED MASS DAMPERS

Journal of Earthquake Engineering, Vol. 8, No. 1 (2004) 89 105 c Imperial College Press GROUND MOTION DOMINANT FREQUENCY EFFECT ON THE DESIGN OF MULTIPLE TUNED MASS DAMPERS CHUNXIANG LI School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, Shanghai 200030, PRC li-chunxiang@vip.sina.com YANXIA LIU School of Civil Engineering, Tongji University, Shanghai 200092, PRC Received 2 January 2002 Revised 14 January 2003 Accepted 17 March 2003 Utilising the Kanai Tajimi and Clough Penzien spectrums and the pseudo-excitation algorithm in the frequency domain, parametric study is performed to examine the effect of the dominant frequency of ground motion on the optimum parameters and effectiveness of multiple tuned mass dampers (MTMD) with identical stiffness and damping coefficient but with unequal mass. The examination of the optimum parameters is conducted through the minimisation of the minimum values of the maximum displacement and acceleration dynamic magnification factors of the structure with the MTMD. The optimum parameters of the MTMD include the optimum frequency spacing reflecting the robustness, the average damping ratio and the tuning frequency ratio. Minimisation of the minimum values of the maximum displacement and acceleration dynamic magnification factors, nondimensionalised respectively by the maximum displacement and acceleration dynamic magnification factors of the structure without the MTMD, is used to measure the effectiveness of the MTMD. The results indicate that in the two cases where both the total mass ratio is below 0.02 and the total mass ratio is above 0.02, but the dominant frequency ratio of ground motion is below unity (including unity), the earthquake ground motion can be modelled by a white noise. It is worth noting, however, that for the total mass ratio above 0.02, the Kanai Tajimi Spectrum or Clough Penzien spectrum needs to be employed to design the MTMD for seismic structures in situations where the dominant frequency ratio of ground motion is beyond unity. Keywords: Vibration control; multiple tuned mass dampers; dominant frequency of ground motion; performance. 1. Introduction The main disadvantage of tuned mass damper (TMD) is the sensitivity problem due to the fluctuation in tuning the TMDs frequency to the controlled frequency Correspondence to: Chunxiang Li, School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, No.1954 Huashan Road, Shanghai 200030, P. R. China. Associate Professor. PhD 89

90 C. Li & Y. Liu of a structure and/or that in the damping ratio of the TMD. The mistuning or offoptimum damping will significantly reduce the effectiveness of TMD, which means that TMD is not robust at all. As a result, the utilisation of more than one tuned mass damper, with different dynamic characteristics, has been proposed in order to improve the effectiveness and robustness of TMD. Multiple tuned mass dampers (MTMD) with distributed natural frequencies were proposed by Xu and Igusa [1992] and also investigated by many researchers such as Yamaguchi and Harnpornchai [1993]; Abe and Fujino [1994]; Igusa and Xu [1994]; Abe and Igusa [1995]; Kareem and Kline [1995]; Jangid [1995, 1999]; Li [2000]; Park and Reed [2001], Gu et al. [2001], and Chen and Wu [2001]. A multiple tuned mass damper (MTMD) is shown to be more effective in the mitigation of the oscillations of structures with respect to single TMD. Likewise, based on the various combinations available of the stiffness, mass, damping coefficient and damping ratio in the MTMD, the five MTMD models have been recently presented by Li [2002]. Through the implementation of the Min.Min.Max.displacement dynamic magnification factor (DDMF) and the Min.Min.Max.acceleration dynamic magnification factor (ADMF), it has been shown that the MTMD with the identical stiffness (k T 1 = k T 2 = = k T n = k T ) and damping coefficient (c T 1 = c T 2 = = c T n = c T ) but unequal mass (m T 1 m T 2 m T n ) provides better effectiveness and wider optimum frequency spacing (i.e. higher robustness against the change or the estimation error in the structural natural frequency) with respect to the rest of the MTMD models [2002]. Recently, the studies conducted by Li and Liu [2002] have disclosed further trends of both the optimum parameters and effectiveness and further provided suggestions on selecting the total mass ratio and the total number of the MTMD with identical stiffness and damping coefficient but with unequal mass. More recently, based on the uniform distribution of system parameters, instead of the uniform distribution of natural frequencies, the eight new MTMD models have been, for the first time, proposed in order to seek for the MTMD models without near-zero optimum average damping ratio. Found are the six MTMD models without near-zero optimum average damping. The optimum MTMD with identical damping coefficient (c T 1 = c T 2 = = c T n = c T ) and damping ratio (ξ T 1 = ξ T 2 = = ξ T n = ξ T ) but with unequal stiffness (k T 1 k T 2 k T n ) and with the uniform distribution of masses is found to be able to render better effectiveness and wider optimum frequency spacing with respect to the rest of the MTMD models [Li and Liu, 2003]. Likewise it is interesting to know that the two MTMD models mentioned above can approximately reach the same effectiveness and robustness in light of a comparison [Li and Liu, 2003]. Obviously, significant strides have been made in recent years as regards to studies on the multiple tuned mass dampers (MTMD) for the vibration control of structures. For seismic response control, however, studies for the effect of the dominant frequency of ground motion on designing the MTMD are very limited. For seismic applications, it is of interest to further study the effect of the dominant frequency of ground motion on the performance of MTMD. The present research, in which the

Ground Motion Dominant Frequency Effect 91 ground acceleration excitation is a stationary random process which is assumed with the spectral densities defined respectively by the Kanai Tajimi and Clough Penzien spectrums [Liu and Jhaveri, 1969; Clough and Penzien, 1993] (hereafter referred to as KTS and CPS ), will specifically address this problem. In comparison to the KTS, the CPS can significantly filter the excitation in the very low frequency region. Using the pseudo-excitation approach (in this approach, the determination of random response of a structure is converted to that of response of the structure under the harmonic pseudo-excitation) [Lin and Zhang, 1994], the optimality criteria can be defined as the minimisation of the minimum values of the maximum displacement and the acceleration dynamic magnification factors, denoted respectively by Min.Min.Max.DDMF and Min.Min.Max.ADMF. The performance of the MTMD with the identical stiffness and damping coefficient but unequal mass can then be examined through taking different values of the dominant frequency ratio of ground motion to be defined next. 2. Transfer Functions of the Structure with Multiple Tuned Mass Dampers The MTMD is considered for controlling the specific vibration mode of a structure. The structure is modelled as a SDOF system (the main system) characterised by the mode-generalised stiffness k s, damping coefficient c s and mass m s, respectively. Every TMD, with different dynamic characteristics, is also modelled as a SDOF system. As a result, the total degrees of freedom of the MTMD structure system are n + 1, as shown in Fig. 1. The analyses that follow are based on this MTMD structure system. Introducing the relative displacements of the main system (y s ) Fig. 1. Mechanical model of structure with multiple tuned mass dampers.

92 C. Li & Y. Liu and every TMD (y j ) with respect to their supports, the equations of motion of the MTMD structure system can then be formulated as follows: m s [ẍ g (t) + ÿ s ] + c s ẏ s + k s y s = n (c j ẏ j + k j y j ), (1) m j [ÿ j + ÿ s + ẍ g (t)] + c j ẏ j + k j y j = 0 (j = 1, 2, 3,..., n), (2) in which m j, c j and k j, respectively, denote the mass, damping constant and the stiffness of the jth TMD in the MTMD; ẍ g (t) is the ground acceleration excitation, which is assumed to be a stationary random process with the spectral densities defined respectively by the following KTS and CPS. [ 1 + 4ξg 2 s g (ω) = s (ω/ω ] g) 2 0 [1 (ω/ω g ) 2 ] 2 + 4ξg 2(ω/ω g) 2 = s 0 H 1 (iω) 2 (for KTS), (3) [ ] [ ] 1 + 4ξg(ω/ω 2 g ) 2 (ω/ω f ) 4 s g (ω) = s 0 [1 (ω/ω g ) 2 ] 2 + 4ξg(ω/ω 2 g ) 2 [1 (ω/ω f ) 2 ] 2 + 4ξf 2(ω/ω f) 2 = s 0 H 1 (iω) 2 H 2 (iω) 2 (for CPS), (4) in which ω g, ω f, ξ g and ξ f represent the dominant frequencies of ground motion and damping factors of two soil filters (letting ω f = 0.1ω g and ξ f = ξ g = 0.6); H 1 (iω) and H 2 (iω) are the transfer functions of the first and second filters reflecting the dynamic characteristics of the soil layers above the bedrock; s 0 is the spectral intensity which is chosen such that the RMS value of ground acceleration takes the σẍg value, i.e. /[ ] s 0 = σẍ 2 g H 1 (iω) 2 dω (for KTS), (5) 0 s 0 = σ 2 ẍ g /[ 0 ] H 1 (iω) 2 H 2 (iω) 2 dω (for CPS). (6) Taking advantage of the following introduced parameters, Eqs. (1) and (2) can be rewritten by ẍ g (t) + ÿ s + 2ξ s ω s ẏ s + ω 2 s y s = n µ j (2ξ j ω j ẏ j + ωj 2 y j), (7) ẍ g (t) + ÿ s + ÿ j + 2ξ j ω j ẏ j + ω 2 j y j = 0 (j = 1, 2, 3,..., n). (8) The parameters ξ k and ω k are, respectively, the natural frequency and damping ratio of subsystem k (k = s, j and j = 1, 2, 3,..., n); µ j is the mass ratio of the jth TMD. They are defined as ω k = k k /m k, ξ k = c k /2m k ω k, µ j = m j /m s (k = s, j and j = 1, 2, 3,..., n).

Ground Motion Dominant Frequency Effect 93 In terms of the pseudo-excitation approach [Lin and Zhang, 1994], the stationary random excitation with the power spectral density (PSD) s g (ω) may be replaced by the following harmonic pseudo-excitation. ẍ g (t) = s g (ω)e iωt. (9) With the hypothesis of y s =H ys ( iω)e iωt, y j =H yj ( iω)e iωt (j =1, 2, 3,..., n), and a s = ÿ s + ẍ g (t) = H as ( iω)e iωt and setting these in Eqs. (7) and (8), the displacement and acceleration transfer functions of the main system with the MTMD can then be determined by H ys ( iω) = 1 + n [µ j (ωj 2 i2ξ j ω j ω)]/(ωj 2 ω 2 i2ξ j ω j ω) ω 2 s ω2 i2ξ s ω s ω ω 2 s g (ω), H as ( iω) = s g (ω) + ω 2 s g (ω) 1 + n [µ j (ω 2 j i2ξ jω j )]/(ω 2 j ω2 i2ξ j ω j ω) n [µ j (ωj 2 i2ξ j ω j ω)]/(ωj 2 ω 2 i2ξ j ω j ω) ω 2 s ω2 i2ξ s ω s ω ω 2 n [µ j (ω 2 j i2ξ jω j )]/(ω 2 j ω2 i2ξ j ω j ω) (10). (11) 3. Evaluation Criteria of Multiple Tuned Mass Dampers The MTMD with identical stiffness and damping coefficient but with unequal mass is a preferable MTMD due to better effectiveness and robustness as well as better constructability. Thus, this MTMD model is selected to examine the ground motion dominant frequency effect on the design of multiple tuned mass dampers (MTMD). For the purposes of convenience and generality, we introduce the following definitions [Li, 2000; Li, 2002]. The average frequency of the MTMD: ω T = n ω j n. (12)

94 C. Li & Y. Liu The frequency of every TMD in the MTMD: ( ω j = ω T [1 + j n + 1 ) ] β 2 n 1 The frequency spacing reflecting the robustness of the MTMD: (j = 1, 2, 3,..., n). (13) β = (ω n ω 1 ) ω T. (14) The tuning frequency ratio of the MTMD (the ratio between the MTMD average frequency and the controlled frequency of the structure): f = ω T ω s. (15) The ratio of the frequency of the jth TMD in the MTMD to the controlled frequency of the structure: r j = ω [ ( j = f 1 + j n + 1 ) ] β. (16) ω s 2 n 1 In terms of the above definitions, the total mass ratio µ = n µ j and the average damping ratio ξ T = n ξ j/n can be expressed respectively in the following forms n ( ) 1 µ = µ j rj 2, ξ T = fξ j (j = 1, 2, 3,..., n). (17, 18) r j r 2 j Defining the external excitation frequency ratio λ = ω/ω s and the dominant frequency ratio of ground motion λ g = ω s /ω g and taking advantage of Eqs. (10) and (11), the displacement and acceleration dynamic magnification factors of the main system with the MTMD can then be calculated respectively by DDMF = ωsh 2 ys ( iλ) = [ [ Re(λ)]2 + s 0 E(λ)] [Īm(λ)]2 [Re(λ)]2 + [Im(λ)], (19) 2 in which ADMF = H as ( iλ) = [ s 0 E(λ)] For KTS: E(λ) = E KTS (λ) = [Re(λ) + λ 2 Re(λ)]2 + [Im(λ) + λ 2Īm(λ)]2 [Re(λ)]2 + [Im(λ)] 2, (20) 1 + 4ξ 2 g λ2 g λ2 (1 λ 2 gλ 2 ) 2 + 4ξ 2 gλ 2 gλ 2. For CPS: E(λ) = E CPS (λ) = E KTS (λ) 100λ 2 gλ 2 (1 100λ 2 g λ2 ) 2 + 400ξ 2 f λ2 g λ2.

Re(λ) = 1 + Īm(λ) = Ground Motion Dominant Frequency Effect 95 [ ] µ n rj 2 λ2 + 4λ 2 ξt 2 r2 j f 2 n (1/r2 j ) (rj 2 λ2 ) 2 + 4λ 2 ξt 2 r4 j f 2. [ ] µ n 2λ 3 ξ T f 1 n (1/r2 j ) (rj 2 λ2 ) 2 + 4λ 2 ξt 2 r4 j f 2. Re(λ) = 1 λ 2 Im(λ) = 2ξ s λ [ ] λ 2 µ n rj 2 λ2 + 4λ 2 ξt 2 r2 j f 2 n (1/r2 j ) (rj 2 λ2 ) 2 + 4λ 2 ξt 2 r4 j f 2. [ ] λ 2 µ n 2λ 3 ξ T f 1 n (1/r2 j ) (rj 2 λ2 ) 2 + 4λ 2 ξt 2 r4 j f 2. Now the optimality of the MTMD can be carried out through the implementation of the following two criteria R I = Min.Min.Max.DDMF, R II = Min.Min.Max.ADMF. (21a) (21b) To have a basis for comparison, the following two particular criteria measuring the effectiveness of the MTMD are introduced. R III = R I /R DKTS (R DCPS ), (22a) R IV = R II /R AKTS (R ACPS ), (22b) in which R DKTS (R DCPS ) and R AKTS (R ACPS ) denote Max.displacement dynamic magnification factor (DDMF) and Max.acceleration dynamic magnification factor (ADMF) of the structure without the MTMD, corresponding respectively to every spectrum above. 4. Performance Evaluation of Multiple Tuned Mass Dampers Displayed in Figs. 2 9 are the results of the present research, in which the damping ratio of the main system is set to be equal to 0.02. The superscript opt denotes the optimum values. Figure 2 presents the variation of the optimum frequency spacing of the MTMD with respect to the total mass ratio for various values of the dominant frequency ratio of ground motion for the six cases: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; and (f) n = 21 based on R II. Clearly, for the two cases, both the total mass ratio is below 0.035 and the total mass ratio is above 0.035 but the dominant frequency ratio of ground motion is below unity (including unity) and the optimum frequency spacing (the robustness) of the MTMD is not affected by the dominant frequency of ground motion when controlling the structural displacement response

96 C. Li & Y. Liu (a) (b) (c) (d) (e) (f) Fig. 2. Variation of the optimum frequency spacing with respect to total mass ratio for various values of the dominant frequency ratio of ground motion: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; (f) n = 21 based on R II. [Figs. 2(a), 2(c), 2(e)]. The above observation indicates that the effect of the dominant frequency of ground motion can be neglected since, under these circumstances, the earthquake ground motion can be represented by a white noise. For the total mass ratio above 0.035, however, the effect of the dominant frequency of ground

Ground Motion Dominant Frequency Effect 97 motion needs to be accounted for in situations where the dominant frequency ratio of ground motion is larger than unity. In other words, in this case the earthquake ground motion cannot be modelled by a white noise. With reference to the case of controlling the structural displacement response, the effect of the dominant frequency of ground motion is slightly smaller on the optimum frequency spacing (the robustness) of the MTMD in the case of suppressing the structural acceleration response [Figs. 2(b), 2(d), 2(f)]. In the concrete, the earthquake ground motion can be represented by a white noise for the two cases where both the total mass ratio is below 0.055 and the total mass ratio is above 0.055 but the dominant frequency ratio of ground motion lies below unity (including unity). Figure 3 shows the variation of the optimum average damping ratio of the MTMD with respect to the total mass ratio for various values of the dominant frequency ratio of ground motion for the six cases: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; and (f) n = 21 based on R II. It is worth noting that for the two cases both the total mass ratio is below 0.04 and the total mass ratio is above 0.04, but the dominant frequency ratio of ground motion is below unity (including unity), and the optimum average damping ratio of the MTMD obtained using KTS and CPS practically equals to that obtained on the supposition that the earthquake ground motion is a white noise process, when controlling the structural displacement response [Figs. 3(a), 3(c), 3(e)]. It is seen however, that for the total mass ratio to be above 0.04 the higher optimum average damping ratio is needed with respect to that obtained based on the white noise hypothesis of earthquake ground motion in situations where the dominant frequency ratio of ground motion is larger than unity. With regard to the case of controlling the structural displacement response, the dominant frequency of ground motion has lesser effect on the optimum average damping ratio of the MTMD in the case of suppressing the structural acceleration response. In physical terms, in this situation, the results in terms of the white noise hypothesis of earthquake ground motion are usable. Furthermore, the MTMD with n = 21 and µ = 0.01 possesses the near-zero optimum average damping ratio [Figs. 3(e), 3(f)]. Despite higher robustness and better effectiveness, the MTMD with the near-zero optimum average damping ratio cannot be used due to the large stroke displacement [Li and Liu, 2002]. Figure 4 shows the variation of the optimum tuning frequency ratio of the MTMD with respect to the total mass ratio for various values of the dominant frequency ratio of ground motion for the six cases: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; and (f) n = 21 based on R II. It is seen that the optimum tuning frequency ratio of the MTMD obtained utilising both the KTS and CPS practically equals to that obtained in terms of the white noise hypothesis of earthquake ground motion in situations where the dominant frequency ratio of ground motion is below unity (including unity), irrespective of both the structural displacement and acceleration response suppressions. However, the dominant frequency of ground motion has large

98 C. Li & Y. Liu (a) (b) (c) (d) (e) (f) Fig. 3. Variation of the optimum average damping ratio with respect to total mass ratio for various values of the dominant frequency ratio of ground motion: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; (f) n = 21 based on R II. influence upon the optimum tuning frequency ratio of the MTMD in situations where the dominant frequency ratio of ground motion is above unity. This means that the earthquake ground motion cannot be represented by a white noise in situations where the dominant frequency ratio of ground motion is beyond unity.

Ground Motion Dominant Frequency Effect 99 (a) (b) (c) (d) (e) (f) Fig. 4. Variation of the optimum tuning frequency ratio with respect to total mass ratio for various values of the dominant frequency ratio of ground motion: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; (f) n = 21 based on R II. Likewise in this situation the effect of the dominant frequency of ground motion is getting more significant on the optimum tuning frequency ratio of the MTMD with the increase of the total mass ratio. Thus, as far as the effect on the optimum tuning frequency ratio of the MTMD is concerned, the KTS or CPS needs to be employed to

100 C. Li & Y. Liu design the MTMD for seismic structures in situations where the dominant frequency ratio of ground motion is beyond unity. Additionally, it is noteworthy that the common unitary tuning may, in this situation, lead to an inefficient MTMD due to the large offset frequency (i.e. ω s ω T ). (a) (b) (c) (d) (e) (f) Fig. 5. Variation of both the R III and R IV values with respect to total mass ratio for various values of the dominant frequency ratio of ground motion: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; (f) n = 21 based on R II.

Ground Motion Dominant Frequency Effect 101 Figure 5 depicts the variations of both the R III and R IV values reflecting the effectiveness in reducing the structural displacement and acceleration responses with respect to the total mass ratio for various values of the dominant frequency ratio of ground motion for the six cases: (a) n = 3 based on R I ; (b) n = 3 based on R II ; (c) n = 11 based on R I ; (d) n = 11 based on R II ; (e) n = 21 based on R I ; and (f) n = 21 based on R II. It is clearly seen that for the two cases where both the total mass ratio is below 0.02 and the total mass ratio is above 0.02 but the dominant frequency ratio of ground motion is below unity (including unity), the R III and R IV values (the effectiveness) of the MTMD are not affected by the dominant frequency of ground motion, regardless of both the structural displacement and acceleration response suppressions. This implies that under these circumstances, the earthquake ground motion can be modelled by a white noise. However, for higher total mass ratio above 0.02, in order to gain an efficient MTMD for seismic structures, it is of vital importance to utilise the KTS or CPS in the design in situations where the dominant frequency ratio of ground motion is beyond unity. Likewise with the further increasing of the total mass ratio from 0.02 the effect of the dominant frequency of ground motion is getting more significant on the effec- (a) (b) (c) (d) Fig. 6. Variation of the optimum frequency spacing with respect to total number for various values of the dominant frequency ratio of ground motion: (a) µ = 0.02 based on R I ; (b) µ = 0.02 based on R II ; (c) µ = 0.04 based on R I ; (d) µ = 0.04 based on R II.

102 C. Li & Y. Liu (a) (b) (c) (d) Fig. 7. Variation of the optimum average damping ratio with respect to total number for various values of the dominant frequency ratio of ground motion: (a) µ = 0.02 based on R I ; (b) µ = 0.02 based on R II ; (c) µ = 0.04 based on R I ; (d) µ = 0.04 based on R II. tiveness of the MTMD in situation where the dominant frequency ratio of ground motion is beyond unity. Meanwhile, it is also noted that with reference to the case of controlling the structural displacement response, the effect of the dominant frequency of ground motion is slightly smaller on the effectiveness of the MTMD in the case of attenuating the structural acceleration response. In order to investigate whether the total number of the MTMD amplifies the effect of the dominant frequency of ground motion on the performance of the MTMD, Figs. 6 9 give, respectively, the plots of the optimum frequency spacing reflecting the robustness, optimum average damping ratio, optimum tuning frequency ratio and R III and R IV values reflecting the effectiveness in suppressing the structural displacement and acceleration responses with respect to the total number for various values of the dominant frequency ratio of ground motion for the four cases: (a) µ = 0.02 based on R I ; (b) µ = 0.02 based on R II ; (c) µ = 0.04 based on R I ; and (d) µ = 0.04 based on R II. In terms of the results shown in Figs. 6 9, increasing the total number does not, in general, amplify the effect of the dominant frequency of ground motion on the optimum frequency spacing (the robustness),

Ground Motion Dominant Frequency Effect 103 (a) (b) (c) (d) Fig. 8. Variation of the optimum tuning frequency ratio with respect to total number for various values of the dominant frequency ratio of ground motion: (a) µ = 0.02 based on R I ; (b) µ = 0.02 based on R II ; (c) µ = 0.04 based on R I ; (d) µ = 0.04 based on R II. (a) (b) Fig. 9. Variation of both the R III and R IV values with respect to total number for various values of the dominant frequency ratio of ground motion: (a) µ = 0.02 based on R I ; (b) µ = 0.02 based on R II ; (c) µ = 0.04 based on R I ; (d) µ = 0.04 based on R II.

104 C. Li & Y. Liu (c) (d) Fig. 9. (Continued) optimum average damping ratio, optimum tuning frequency ratio and R III and R IV values (the effectiveness), irregardless of both the structural displacement and acceleration response suppressions. From the above presentation, the dominant frequency ratio of ground motion is divided into the two regimes below and above unity. Corresponding respectively to every dominant frequency ratio regime of ground motion, the deviation of the dominant frequency ratio of ground motion will not amplify its effect on the performance of the MTMD in terms of further numerical results. 5. Conclusions From the results presented, the following conclusions can be drawn: (i) The earthquake ground motion can be represented by a white noise in the two cases where both the total mass ratio is below 0.02 and the total mass ratio is above 0.02 but the dominant frequency ratio of ground motion is below unity (including unity). (ii) For the higher total mass ratio above 0.02, the KTS or CPS needs to be used to design the MTMD for seismic structures in situations where the dominant frequency ratio of ground motion is beyond unity. (iii) Increasing the total number of the MTMD does not, in general, amplify the effect of the dominant frequency ratio of ground motion on the performance of the MTMD. (iv) With reference to the case of controlling the structural displacement response, the dominant frequency ratio of ground motion has slightly lesser influence on the performance of the MTMD in the case of suppressing the structural acceleration response. (v) Corresponding to the respective dominant frequency ratio regime of ground motion, the deviation of the dominant frequency ratio of ground motion will not amplify its effect on the performance of the MTMD.

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