Sensitivity Analysis of the Effectiveness of Tuned Mass Dampers to Reduce the Wind-Induced Torsional Responses 1

Use of Tuned Mass Dampers (TMDs) to reduce the wind-induced torsional response of structures has been investigated in the literature by assuming that the wind excitations can be approximated by harmonic forces or white noise random processes; however, such an assumption is not realistic. Further, wind load effects are correlated. This study is focused on the effectiveness of different linear/nonlinear TMDs configurations to reduce the wind induced response. For the analyses, the structure is modeled as a multidegree of freedom system under correlated wind load effects. The results show that the selection of optimum dampers is affected by the consideration of the correlated wind load effects.

of the TMD has been carried out as earlier as the 1950s by Den Hartog (1956) considering undamped systems.Warburton (1982) developed equations for evaluating the optimum parameters of the TMD considering the main structure as an undamped SDOF system for various combinations of response and excitation.Warburton and Ayorinde (1980) studied the effect of the damping of the structure on the optimum parameters of the TMD and showed that it is possible to represent a complex mechanical system by means of an equivalent SDOF system if its natural frequencies are well separated.In all the above-mentioned studies, the harmonic or white noise excitations are used to define the selection of the optimum parameters of the TMD, even though they may not adequately represent the characteristics of the strong wind or large earthquake induced excitations.
More recently, the use of multiple TMDs to reduce the peak responses of structures, that are sensitive to torsional effects, has been reported by Jangid and Data (1997), Pansare and Jangid (2003), Li and Qu (2006), Bakre and Jangid (2007) , and Wang et al. (2009).The excitations considered in these studies do not differ from those mentioned previously.For instance, Jangid and Data (1997), Bakre and Jangid (2007), and Wang et al. (2009) used an exciting force represented by band limited white-noise; Pansare and Jangid (2003) and Li and Qu (2006) analysed structures subjected to external harmonic excitations.The damping mechanism of the TMDs (i.e.dashpot) was modeled as linear or viscous in most of these studies.The assumption of viscous damping simplifies the analysis, even though it may not be realistic due to friction or the properties of the fluid used in the dashpot, which are temperature dependent.A state space linear mathematical model has been proposed by Mattei and Ricciardelli (2002), where a detail treatment of the wind loading is carried out; unfortunately, no guidelines on the applicability to practical problems of the proposed model is given.
Attention has also been paid to the use of nonlinear TMDs, such as the use of power law and dry-friction damping mechanisms which occur commonly in practice, to suppress the structural response.For example, the use of the power law damping mechanism has been studied by Terenzi (1999), Vickery et al. (2001) and Rüdinger (2007aRüdinger ( , 2007b)); whereas Ricciardelli and Vickery (1999) explored the use of tuned vibrations absorbers with dry friction damping and linear stiffness.
The main objective of the present study is to assess the effectiveness of different TMDs configurations with linear and nonlinear damping mechanisms (i.e.viscous damping and power law damping mechanism) with different location on a structure to reduce the wind induced torsional responses.The assessment is focused on parametric investigations by considering possible combinations of parameters and locations of the TMDs.Rather than using the harmonic or white noise excitations to represent the wind loading, a more realistic characterization of the wind excitations that is based on the power spectral density (PSD) function of the fluctuating wind forces is considered.Time history analyses were carried out to evaluate the structural peak response under the simulated wind loading that is characterized by the adopted PSD function.

TMD-structure system and equation of motion
Consider a structure that is modeled as a two-degree-of-freedom system (lateral displacement Y and rotational displacement θ) with mass M s , and mass moment of inertia I s .The structure is shown in Figure 1a.The lateral and torsional stiffness of the structure are denoted by K s and K θ , respectively; the damping coefficients of the structure for the lateral and rotational directions are represented by C s and C θ , respectively.The equation of motion of the structure under wind loading can be expressed as where F Y (t) and M(t) represent, respectively, the total horizontal force and the torsional moment due to fluctuating wind loading that will be discussed in the next section; a dot on a symbol denotes its temporal derivative (i.e., velocity); and a double-dot on a symbol denotes second temporal derivative (i.e., acceleration).Now consider that the structure is fitted with three TMDs as illustrated in Figure 1b.The use of three TMDs, especially the two that are away from the center of mass, is aimed at reducing peak torsional structural responses, while all three TMDs can be used to reduce the lateral response of the structure.The TMDs are connected to the structure by linear/nonlinear dashpots and linear elastic springs; the nonlinear damping form considered in this study is power law.The TMDs are characterized by the mass ratio µ representing the ratio of the mass of the TMDs to the mass of the main structure, and by the frequency ratio r f defined as the ratio of the frequency of the TMD to the frequency of the vibration mode of the main structure whose response is to be reduced.The stiffness of the TMD can be readily calculated if the above mentioned ratios and the properties of the main structure are known.The equation of motion of the TMD-structure system shown in Figure 1b can be expressed in matrix form as in which m i , k i and y i , i = 1,2,3, are the mass, stiffness and displacement of the i-th mass damper (see Figure 1b); and F Di (v), i = 1,2,3, represents damping force of the i-th mass damper that depend on the velocity, v,.As mentioned above, the two functional forms of damping force considered in this study are the viscous case and the power law damping mechanism.For simplicity and practical considerations, it is considered that the same type of damper is to be installed in a structure.This implies that the functional form for F Di (v), i = 1,2,3, are the same as if the viscous damping mechanism is considered, and if the power law damping mechanism is considered.In Eqs. ( 3) and (4), C L is a viscous damping coefficient; C PL is a power law viscous damping coefficient; • denotes the absolute value of its argument; β is a model parameter within the range 0.5 to 2 (Soong and Costantinou, 1994), a value of 2 is selected for the parametric analysis.
It is noteworthy that if one or two rather than three TMDs are considered, Eq. ( 2) can be reduced accordingly and used to describe the motion of the system.

Wind Loading and Analysis Procedure
Wind speed is often represented by a mean wind component and a fluctuating wind component.The fluctuating component varies in time, is stochastic and characterized by its PSD function.The well-known PSD functions for the turbulent wind speed include the ones given by Kolgomorov, von Karman, Davenport, Harris, Kaimal (Simiu and Scanlan, 1996).The forces or pressures induced by wind can be viewed as the sum of the mean wind force and the fluctuating wind force which is directly proportional to the fluctuating wind speed, if the quadratic term of the fluctuating wind speed is ignored.In this study, it is considered that the PSD function of the normalized fluctuating wind force, S(f ), can be expressed as where f L and f U are upper and lower bounds of f, α is an exponent that depends on the direction of the wind velocity, f is frequency in Hz, and A is a normalization constant such that the integration of the PSD function equals one.The adoption of this PSD function is justified since boundary layer wind tunnel test results (Pozos-Estrada et al., 2011) indicate that such a PSD function provides sufficient accurate characterization of the fluctuating wind force in the inertial subrange, which is of interest in the present study.It is noted that the value of f that most of the structures are sensitive to, especially tall buildings, falls within 0.05 to 0.5 (Hz).
Since structural geometric shapes vary and wind pressure acting on the structures are not necessarily symmetric, the spatially distributed wind pressure could induce both along wind vibration and torsional responses around the vertical axis.To simplify the parametric study, it is considered that the overall wind force acting on the structural model considered can be modeled as two uncorrelated horizontal along wind loads.These loads are proportional to F 1 (t) and F 2 (t), where their PSD functions are given by Eq. ( 5), and are acting at different points as shown in Figure 1a.In such a case, the total horizontal force F Y (t) and the torsional moment M(t) are given by ) where c, R and ) Given the PSD function of the wind force, its time history can be simulated using, for example, the spectral representation method (Shinozuka, 1972), although more sophisticated algorithms can also be used (Di Paola, 1998).The method consists of using sum of a series of sine functions with random phase angles.More specifically, if a stationary stochastic process Z(t) is characterized by its (one sided) PSD function S ZZ (f), a sample of the stochastic process, z(t), can be expressed as where f i = i×∆f, i = 1, 2, …, N, ∆f = 1/(N∆t), ∆t is the time increment and i ϕ is a uniformly dis- tributed random variable within 0 to 2π.Therefore, by using this simulation approach, samples of F 1 (t) and F 2 (t) can be simulated if the parameters of the PSD function shown in Eq. ( 5), α, f L and f U are given.The simulated time series are then used in Eq. ( 6) to evaluate F Y (t) and M(t) for given values of , and e σ .The samples of F Y (t) and M(t) can then be used in Eqs. ( 1) and (2) to calculate the response time history of the structure with and without the TMD system using the Newmark's method, and to assess the benefit of using the TMD system.However, before providing the detailed analysis procedure to investigate the effectiveness of the TMD system, it is noteworthy that most of the existing studies mentioned in the introduction section considered that the use of the TMD systems is aimed at controlling the wind induced peak translational structural response, Y p .It is known that excessive responses such as the peak rotational velocity, p θ ɺ , and the translational acceleration at a point within the structural footprint, pr Y ɺɺ , could also affect significantly the building occupants' comfort (Chen and Robertson, 1972;Irwin, 1978;Melbourne and Palmer, 1992;Isyumov, 1993;Isyumov, 1995;Pozos-Estrada et al., 2010).Therefore, one could also be interested in using the TMD systems to reduce these peak responses.
Given the time history of the responses of duration T d , a sample of the mentioned peak responses can be obtained from where ( ) Y t is the translational displacement; ( ) are the lateral and rotational accelerations, respectively; and r is the distance from the point of interest within the structural footprint to the center of rotation of the main structure.
The effectiveness of the TMD system to reduce the peak responses can be measured using the ratios of the peak responses of the structure without and with a TMD system.These ratios are defined as where the symbols Y p , p θ ɺ and pr Y ɺɺ with an additional subscript O denote the (maximum) peak responses of the structure without a TMD system shown in Figure 1a, while those with an additional subscript T denote the (maximum) peak response of the structure with a TMD system shown in Figure 1b.For a given structure and parameters defining the wind loading (i.e., parameters for Eqs. ( 6) and ( 7)), based on the above, the analysis procedure to assess the effectiveness of the TMD in suppressing the peak responses can be summarized as follows: 1) Sample a wind load time history by using the wind load model defined in Eq. ( 6) for a duration greater than Td, such that the useful time of the time history of the responses is equal to Td; 2) Evaluate the time history of the responses of the structure without and with a TMD system by solving, respectively, Eq. (1) and Eq. ( 2) using Newmark's method; 3) Evaluate the peak responses Y pO , pO θ ɺ and prO Y ɺɺ of the structure without a TMD system and Y pT , pT θ ɺ and prT Y ɺɺ of the structure with a TMD system.Evaluate R Y , R θ ɺ and Y R ɺɺ , defined in Eq. ( 9), employing the obtained peak responses; The above procedure can also be used to calculate samples and statistics of the responses of the structure without or with TMD systems alone by simply solving Eq. (1) or Eq.(2) only in Step 2), and ignoring the evaluation of the ratios in Step 4).

Identification of optimum TMD system
For a given structure, the optimum TMD system is often defined as the one that minimizes the expected response ratio such as m RY .In practice, the design of a TMD starts by setting the value of the mass ratio µ, which was defined previously.The value of µ is usually set within 1% to 10% (Warburtun and Ayorinde, 1980).One of the considerations for selecting this parameter is the available space to place the TMD.Although the use of an increased mass ratio value helps to further reduce the response, it may not be feasible or economic since other structural elements may need to be strengthened due to the additional weight of the TMD.Once the value of the mass ratio has been selected, other parameters such as frequency ratio r f and damping coefficients of the TMDs need to be selected for the TMDs.Several recommendations have been given to determine the optimum parameters of the TMD as a function of the mass ratio (Den Hartog, 1956;Tsai and Lin, 1983;Fujino and Abe, 1993;Vickery et al., 2001).These recommendations were developed based on harmonic excitations and white noise excitations for specific target translational responses only.
In this section, steps for identifying the optimum TMD system for specified mass ratios considering both translational and rotational responses are outlined.The identification of the optimum TMD system only needs to consider a combination of suitable ranges of the damping coefficient C (i.e., C L or C PL ) of the TMD and r f values.In order to identify the parameters leading to the optimum TMD system, numerical search algorithms can be used (Vanderplaats, 1984;Haupt and Haupt, 2004).The use of numerical algorithms for the identification is justified since no closed form solution is available for the considered types of TMD system under stochastic wind loads.Since only two parameters (C and r f ) need to be identified, a simple grid search procedure can be used.For a given mass ratio and a structural system approximated by the model shown in Figure 1b, this basically involves: 1) For a selected set of values of r f and C, carry out dynamic analysis to evaluate RY m , R m θ ɺ and RY m ɺɺ as outlined previously; 2) Repeat Step 1) for a suitable range of r f and C values, and identify the combination of r f and C, denoted by (r fopt , C opt ), that leads to the lowest RY m , R m θ ɺ and/or RY m ɺɺ .
The identified (r fopt , C opt ) provides the optimum design for the TMD system.

Structural Characteristics and Peak Responses
For the analysis, an actual structure is approximated by a two-degree-of-freedom (2DOF) system whose characteristics are defined by the generalized properties of the actual structure.For the numerical example, the mass, M s , and mass moment of inertia, I s , of the generalized system, representing a 60 story tower, are considered to be equal to 2,519,208 kg and 2,472,771 kg-m2, respectively; the natural frequencies of the structure, defined as ω Y and ω θ , are equal to 0.753 and 0.959 rad/s.The damping ratios ζ Y and ζ θ for the sway and torsional mode are set equal to 1%.The stiffness and damping coefficient of the generalized system can be calculated from By considering that the structure is subjected to the wind loads defined in Eqs. ( 5) to ( 7) with α = 1.5, σ F = 27340, e σ = 0.033 and The considered values of pO θ ɺ and prO Y ɺɺ are associated with the response levels that could affect the comfort, health and/or disrupt the daily activities of the building occupants (Mendis et al., 2007;Isyumov, 1995), while the value of Y pO is selected considering a drift ratio of about 1/667.For the analysis the simulation cycle, n, is considered equal to 30, and T d is considered to be equal to 10 minutes.To see whether this selected simulation cycle n = 30 is adequate to evaluate the peak mean responses, the analysis is repeated with n = 60, and the obtained mean of Y pO , pO θ ɺ and prO Y ɺɺ are 0.289 m, 4.83 rad/s and 0.27 m/s 2 , respectively.Comparison of these results to the previous ones suggests that the use of n =30 for evaluating the mean value is adequate.Therefore, unless otherwise indicated, n equal to 30 is adopted for the remaining numerical analyses.To assess the impact of the correlation between the along wind load and the torsional moment on the responses, Y F M ρ is varied from 0 to 1.0 with an increment of 0.25 while α, σ F , and e σ are maintained to be the same.The obtained results by repeating the above analysis are also listed in Table 1.The results show that the mean values of the responses of the structure without dampers are sufficiently uniform.This is due to that the values σ F and e σ (see Eq. ( 6)) are not affected by the selected Y F M ρ value.

Responses with TMDs
To investigate the effectiveness of using the TMDs to suppress the peak responses of the considered structure, three basic TMD-structure systems are studied in the following.In the first system, the structure is fitted with three identical TMDs (see Figure 1b) (i.e., the dampers have the same mass, stiffness and damping coefficient).The use of identical dampers placed at different locations has been used by Pansare and Jangid (2002) to mitigate the response of structures under sinusoidal force.This is consistent with the decisions often made in practice in reducing cost of manufacturing dampers.Use of this TMD-structure system is aimed at reducing both the lateral and torsional responses.The second system considers two dampers, each located at each side of the center of rotation of the main structure.The properties of these two dampers are again considered to be the same; use of this system is focused mainly on the mitigation of the torsional response of the main structure, although it is likely to reduce the amplitude of the translational motion as well.
Finally, for the third system, only one damper is considered.The damper is to be located such that its center of mass coincides with the center of rotation of the main structure.
For each considered system, numerical analyses are carried out by following the procedure given above to identify the optimum TMD system.For the analysis, the wind forces are simulated according to Eqs. ( 5) to ( 7) for the values of α = 1.5, σ F = 27340 and e σ = 0.033.Further, the analysis is first carried out by considering viscous damping mechanisms for the dampers, and then repeated by considering the power law damping mechanism for the dampers.Also, the impact of the amount of damping of the dampers on the response ratios is assessed.In particular, Figure 2a shows that RY m is significantly reduced for rf close to 1 and that for certain range of r f and some values of

TMD-structure System with Dampers with Viscous Damping Mechanism
ρ , RY m exceeds 1, indicating that the use of dampers increases the response of the main structure.Figure 2b shows that the reduction in the peak rotational response can be achieved for r f approximately equal to 1.27 (i.e., ω θ /ω Y ).In other words, the dampers are tuned to the torsional frequency of the structure.However, the effectiveness of the dampers in reducing the acceleration response prO Y ɺɺ (Figure 2c) is somewhat insensitive to the value of r f .This can be explained by noting that prO Y ɺɺ depends on both the translational and rotational accelerations.In all cases, the use of the TMD system effectively reduces the peak responses of the main structure.
Similar conclusions as above could be drawn based on the results shown in Figure 3 for the second TMD-structure system, and the reduction in responses for both TMD systems are similar as well.Therefore, use of two or three TMDs in reducing the peak responses for the considered structure is likely to be governed by the feasibility of the construction and installation of the TMDs.
The results for the third considered TMD-structure system are shown in Figure 4, it can be observed that the reduction in the peak translational displacement and prO Y ɺɺ is achieved for r f close to 1.Note that for this case, since there is no reduction of the torsional responses, R m θ ɺ equals one for all values of r f and is not shown.

TMD-structure System with Dampers with Power Law Damping Mechanism
As mentioned earlier, the analysis carried out for TMD-structure systems with dampers with viscous damping mechanism is repeated but considering the power law damping mechanism for the dampers.Although is not the objective of the present study, a comparison of the effectiveness of the TMDstructure system with viscous and power law damping mechanisms indicates that the use of either one or the other is effective to reduce the structural response.

Effect of the Damping Coefficient of the Damper
The results presented in Figures 2 to 7 are obtained by considering the optimum damping coefficient of the damper(s).To illustrate the effect of the damping coefficient on the effectiveness of the TMDs in mitigating the structural response, an analysis is carried out in this section considering the above mentioned three TMD-structure systems but varying the damper's damping coefficient.For the analysis, the frequency ratio r f is selected equal to those identified optimum frequency ratios r fopt based on the results presented in Figures 2 to 7       It is interest to note that at r fopt , the dampers are effective for a considerable range of C L or C PL values.

Identified Optimum TMD System
The optimum parameters of the TMD systems for the considered structure are shown in Table 2.
The results suggest that r fopt can be considered to be independent of Y F M ρ for the dampers with viscous damping mechanism and that the maximum reduction of the structural response is achieved when the dampers are tuned to the vibration frequency of the mode whose response is to be reduced.Table 2 also shows the results of the optimum parameters of TMD systems with power law damping mechanism.Similar to the case with viscous damping, r fopt for the dampers with power law damping mechanism could be considered to be independent of Y F M ρ .
Further parametric study considering different dynamic characteristics of the main structure as shown in Table 2 was also carried out and the identified optimum TMD systems for the TMDs with viscous and power law damping mechanism are summarized in Table 3.It shows that, in general, the optimum tuning frequency of the dampers is equal to the vibration frequency of the mode of the structure whose response is to be reduced (i.e., r fopt = 1).This is the case for TMDs with viscous or power law damping mechanism.It is noted that values of r fopt equal to ω θ /ω Y indicate that the frequency of the TMDs is to be tuned to the frequency of the torsional mode of the structure (i.e., ω θ ).Table 3 also shows that in most of the cases, the ratio C Lopt /C s is smaller than C PLopt /C s , this is because the TMDs with power law damping mechanism help the structure to attain the maximum reduction at higher values of C PL as indicated previously.Moreover, inspection of Table 3  5) In general the optimum tuning frequency of the dampers is equal to the vibration frequency of the mode of the structure whose response is to be reduced.6) The ratios of the optimum coefficient of viscous damping to the coefficient of damping of the main structure and the optimum coefficient of power law damping to the coefficient of damping of the main structure are more dependent on the type of response considered compare to the optimum frequency ratio.

Figure 1 :
Figure 1: Structure without and with TMDs: a) Structure without TMDs, b) Structure with TMDs.
. R is the measure of the distance from the torsional center to the line of the eccentric force 2 F Y (t) and M(t), and that the ratio of the standard deviation of the M(t), σ M , to that of F Y (t), σ F , for the wind loads defined in Eq. (6) is equal to standard-deviation-based load eccentricity e σ (i.e., σ M /σ F )to model the wind loads is advantageous since it can be employed to match prescribed values of σ F and e σ by simply setting c equal to σ F and R equal to / Y F M e σ ρ .Note that in all cases, M(t) can be expressed as 2 (

4)
Repeat Steps 1) to 3) n times to generate samples of R Y , R θ ɺ and Y R ɺɺ to evaluate their mean values denoted by RY m , R m θ ɺ and RY m ɺɺ .
By carrying out the analysis for the three considered TMD-structure systems, the obtained mean of the response ratios ( RY m , R m θ ɺ and RY m ɺɺ ) are shown in Figures 2 to 4 for values of the correlation coefficient between F Y (t) and M(t), Y F M ρ , and of frequency ratio r f .The results shown in Figure2for the first TMD-structure system suggest that the mean response ratios are affected by Y F M ρ and that maximum reduction of the peak responses (i.e., minimum value of mean response ratios) occurs at different r f values.However, the selection of the optimum TMD system is not significantly affected by the

Figure 2 :
Figure 2: Expected value of the response ratios for the first TMD system as a function of the frequency ratio for TMDs with viscous damping: a) RY m ; b) R m θ ɺ and c) RY m ɺɺ .

Figure 3 :
Figure 3: Expected value of the response ratios for the second TMD system as a function of the frequency ratio for TMDs with viscous damping: a) RY m ; b) θ ɺ R m and c) Y R m ɺ ɺ .

Figure 4 :
Figure 4: Expected value of the response ratios for the third TMD system as a function of the frequency ratio for TMDs with viscous damping: a) RY m and b) Y R m ɺ ɺ .
The obtained results are shown in Figures 5 to 7. Comparison of the results shown in these figures to those presented in Figures 2 to 4 indicates that the conclusions drawn from the latter are equally applicable to the former.

Figure 5 :
Figure 5: Expected value of the response ratios for the first TMD system as a function of the frequency ratio for TMDs with power law damping: a) RY m ; b) θ ɺ R m and c) Y R m ɺ ɺ .

Figure 6 :
Figure 6: Expected value of the response ratios for the second TMD system as a function of the frequency ratio for TMDs with power law damping: a) RY m ; b) θ ɺ R m and c) Y R m ɺ ɺ .

Figure 7 :
Figure 7: Expected value of the response ratios for the third TMD system as a function of the frequency ratio for TMDs with power law damping: a) RY m and b) Y R m ɺ ɺ .
. The results from the analyses for the systems shown in Figure2to 7 are presented in Figures 8 to 13, respectively.The results show that the mean response ratios ( RY m , R m θ ɺ and RY m ɺɺ ) vary with the ratio C L /C s or C PL /C s , and depend on the considered damping mechanism.In general for the dampers with the power law damping mechanism, the mean response ratio decreases as C PL /C s , increases.However, for the dampers with viscous damping mechanisms, the mean response ratio decreases initially as C L /C s increases, and a slight increase in the mean response ratio can be observed as C L /C s increases further (i.e., right side of the figures).This implies that a significant further increase in C L /C s is not effective in reducing the peak responses.

Figure 8 :
Figure 8: Expected value of the response ratios for the first TMD system as a function of the ratio CL/Cs for TMDs with viscous damping: a) RY m ; b) R m θ ɺ and c) RY m ɺɺ .

Figure 9 :
Figure 9: Expected value of the response ratios for the second TMD system as a function of the ratio CL/Cs for TMDs with viscous damping: a) RY m ; b) R m θ ɺ and c) RY m ɺɺ .

Figure 10 :
Figure 10: Expected value of the response ratios for the third TMD system as a function of the ratio CL/Cs for TMDs with viscous damping: a) RY m and b) RY m ɺɺ .

Figure 11 :
Figure 11: Expected value of the response ratios for the first TMD system as a function of the ratio CL/Cs for TMDs with power law damping: a) RY m ; b) R m θ ɺ and c) RY m ɺɺ .

Figure 12 :
Figure 12: Expected value of the response ratios for for the second TMD system as a function of the ratio CL/Cs for TMDs with power law damping: a) RY m ; b) R m θ ɺ and c) RY m ɺɺ .

Figure 13 :
Figure 13: Expected value of the response ratios for the third TMD system as a function of the ratio CL/Cs for TMDs with power law damping: a) RY m and b) RY m ɺɺ .
that the mean of Y pO , pO θ ɺ and prO Y ɺɺ with a simulation cycle of 30 are equal to 0.292 m, 4.83x10-3 rad/s (4.8 milli-rad/s) and 0.267 m/s2 (27.2 milli-g), respectively, for T d equal to 10 min.
the calculated mean of pO Y , pO θ ɺ and prO Y ɺɺ , by using the procedure outlined in the previous sections, are shown in Table 1.Note that σ F and e σ are cho-sen such

Table 1 :
Statistics of the peak responses of the main structure without TMDs.

Table 2 :
indicates that the ratios C Lopt /C s and C PLopt /C s are more dependent on the type of response considered than r fopt .Lopt /C s r fopt C Lopt /C s r fopt C Lopt /C s r fopt C PLopt /C s r fopt C PLopt /C s r fopt C PLopt /C s Optimum values of the TMD system for the structure considered. 1

Table 3a :
Optimum values of the TMD system to reduce RY

Table 3b :
Optimum values of the TMD system to reduce