Frequency band-wise passive control of linear time invariant structural systems with H∞ optimization

Journal of Sound and Vibration 332 (2013) 6044–6062

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Frequency band-wise passive control of linear time invariant structural systems with H1 optimization N. Debnath, S.K. Deb, A. Dutta n Department of Civil Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

a r t i c l e in f o

abstract

Article history: Received 30 June 2012 Received in revised form 25 May 2013 Accepted 16 June 2013 Handling Editor: J. Lam Available online 22 July 2013

A frequency band specific passive control strategy is presented based on H1 optimization for multi-degree of freedom (MDOF) linear time invariant (LTI) structural systems. Effective control can be achieved if passive control devices are designed by considering frequency bands of excitation. Minimization of maximum spectral norm or worst-case gain in the excitation frequency range is taken into account for the design of passive control devices for effective performance. A multi-storey shear planer frame coupled with a tuned mass damper (TMD) system as the passive control device is considered in the numerical simulation for controlling both displacement and acceleration subjected to base excitation. The band-specific H1 optimization problem for design of passive control devices has been transformed into GA-friendly form for the TMD system as control devices. Such a design strategy of passive control devices based on minimizing worst-case gain associated to finite frequency band is observed to provide efficient design of a TMD system with better performance than that designed based on conventional H1 optimization associated to entire frequency range. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction Many techniques are found in literature to design passive control devices for structural systems. However, search for better control strategies still exists, which can be applicable for structures subjected to wide range of excitations. In this regard, H1 optimization appears to be a good strategy in view of its applicability in frequency domain. H1 optimization for the design of passive control devices as vibration absorbers has been under investigation of many researchers for quite a long time. H1 optimization employs the minimization of H1 norm i.e. the maximum spectral norm or worst-case gain of the primary system. Initial works on optimum design using H1 minimization considered a single degree of freedom (SDOF) primary system connected to a dynamic vibration absorber (DVA). Those methods are found to be based on the fixed-point theory as detailed in Den Hartog [1]. This theory states that all frequency–response curves pass through two invariant points independent of the absorber damping. The DVA is similar to classical TMD which consists of single mass movable with SDOF and connected to the primary system with a spring and a damper. Nishihara and Asami [2] developed exact optimal solution with minimization of the maximum amplitude magnification factor in the form of H1 norm of an SDOF primary system using DVA. Asami et al. [3] proposed a series solution for analytical evaluation of the H1 norm for an SDOF primary system using DVA as the control device. Numerical evaluation of H1 norm for the design of vibration absorber was carried out by Randall et al. [4] and Thompson [5] as well as Soom and Lee [6]. The first application of minimization of H1 norm for the design of vibration absorber was reported by Ormondroyd and Den Hartog [7]. Cheung and Wong [8] proposed an n

Corresponding author. Tel.: +91 3612582401; fax: +91 3612582440. E-mail address: [email protected] (A. Dutta).

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.06.018

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alternative procedure for the H1 optimization of the DVA problem. A new set of optimum tuning frequency and damping of the absorber is derived resulting in lower maximum amplitude responses than those reported in the literature. Ozer and Royston [9] extended the classical Den Hartog approach of fixed points to MDOF undamped primary systems. Similarly, Dayou [10] extended the fixed point concept for the continuous structure with well separated natural frequencies. Cheung and Wong [11] implemented H1 optimization for vibration reduction using a simply supported rectangular plate. Mohtat and Dehghan-Niri [12] developed a novel framework for generalized robust design of a TMD system as the passive vibration controller for an uncertain linear multi-input multi-output (MIMO) structural system based on H1 optimization. They implemented genetic algorithm (GA) for H1 optimization in their study. Design of controller with specific frequency-band is observed as an emerging trend in different practical problems as well as in the problem of vibration control. A significant contribution in this regard is observed in terms of development of the generalized Kalman–Yakubovich–Popov (KYP) lemma, which was developed primarily through the works of Iwasaki et al. [13] and Iwasaki and Hara [14]. The generalized KYP lemma provides a well defined framework for designing the controller with consideration of the finite frequency band of interest. Standard/conventional H1 norm, thus, can be associated to a finite frequency band to have band-specific H1 norm based on this framework. The generalized KYP lemma helps to convert a certain frequency domain inequality (FDI) associated to finite frequency range to linear matrix inequality (LMI) condition which is suitable for numerical computations. Thus the system design specifications expressed in terms of FDIs may be reduced to the problem of LMI optimization. It may be mentioned that entire frequency range based FDIs can be interpreted by equivalent time domain inequalities (TDIs) complementing one another. Iwasaki et al. [15] presented necessary and sufficient conditions for a general FDI to hold within a specific frequency band in terms of TDIs and thus facilitated increased flexibility in the system analysis and synthesis. Iwasaki and Hara [16] further considered a control synthesis problem to meet design specifications in terms of multiple FDIs based on generalized KYP lemma. Zhang and Yang [17] presented a new technique for designing dynamic output feedback controllers through solutions of a set of LMIs, where the resulting closed-loop systems are asymptotically stable and meet the requirements of small gain specifications in finite frequency ranges. Zhang and Yang [18] further studied the problem of control synthesis via state feedback for linear time-delay systems through solutions of a set of LMIs with design specifications in finite-frequency ranges. Gao and Li [19] studied the problem of finite frequency filtering for linear discrete-time state-delayed systems on the basis of generalized KYP lemma. Li and Gao [20] further investigated the problem of robust finite frequency H1 filtering for two-dimensional (2-D) Roesser models based on the 2-D generalized KYP lemma. In the areas of active vibration control with finite frequency specification, Chen et al. [21] presented a new approach for control of building vibration over finite frequency range based on H1 control theory, generalized KYP lemma and projection lemma. Passive vibration control with finite frequency specification, however, is observed to be in early stage of development. Yang et al. [22] analysed performance of vibration absorbers associated to a frequency band in terms of energy reduction index. Vibration control of simply supported plates was considered using DVA as the control device. Mohtat and Dehghan-Niri [12] used an input weights matrix relating to TF matrix as an attempt to facilitate the scope for consideration of excitation frequency content. From the literature review, the following features are observed: (i) Application of H1 optimization using analytical approaches for design of passive control devices such as DVA or TMD is found mainly for SDOF systems. (ii) Application of H1 optimization based on analytical derivations for passive control of general structural systems is observed to be very limited. Moreover, very few works are observed for the design of passive vibration control devices for MDOF systems using numerical H1 optimization. (iii) Limited attempts with consideration of frequency band in the passive control strategy are found. (iv) Active control strategies considering finite frequency band are found to be carried out based on the generalized KYP lemma. However, a gap is observed in terms of any comprehensive study regarding the finite frequency band specific design of passive vibration control devices as well as performance assessment of such designs. The present work is an effort to address issues pertaining to frequency band specific design using H1 optimization. H1 optimization can be considered to have some comparative advantages over other passive control devices design strategies: (i) it provides an unified approach for design of all types of passive control devices; (ii) it is applicable to MDOF structural systems with freedom of input–output degree of freedom (DOF) selection; (iii) it is theoretically well established; and (iv) it has wide application in active-control design. In the present work, a finite frequency band specific design strategy of passive control devices is considered for MIMO LTI [23] structural systems using band-specific H1 optimization which retains many of the advantages of H1 optimization. However, the combined system of any primary structural system and the adopted passive control devices is required to behave as an LTI system. H1 norm i.e. worst-case gain of the coupled primary system and passive control devices is minimized in a specified frequency band matching the frequency content of excitation. A sample multi-storeyed shear frame coupled with a TMD system as the passive control device is considered for numerical simulation using the finite frequency band-specific design strategy. 2. System formulation 2.1. Equation of motion The equation of motion for n DOF dynamic systems with mass matrix M, damping matrix D, stiffness matrix K and input force vector u is represented as Mx€ þ Dx_ þ Kx ¼ u

(1)

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€ x_ and x are acceleration, velocity and displacement vectors respectively. The present work focuses on the secondwhere x, order structural system since passive control is commonly implemented for structural systems. However, this strategy may be applicable for passive control of an LTI system of any order. 2.2. Computation of transfer function (TF) matrix The second-order equation of motion in Eq. (1) is converted to state-space form. Further, there exists straight forward relation in terms of state-space matrices and TF matrix. The state space form of equation for state vector z, input vector u and output vector y is represented as z_ ¼ Az þ Bu

(2)

y ¼ Cz þ Eu

(3)

In the present work, the state vector is considered as z ¼ fx1 x2 …xn x_ 1 x_ 2 …x_ n gT . With this representation of state vector the state-space matrices A and B become

 0 I 0 A¼ and B ¼ M1 K M1 D M1 where I and 0 are identity matrix and matrix of zeros respectively with size n  n for each of these matrices. The remaining state space matrices C and E are dependent on the type of response measurement. For displacement measurement C ¼ ½I 0

and

E¼0

C ¼ ½0 I

and

E¼0

for velocity measurement

and for acceleration measurement

h i C ¼ M1 K M1 D

and

E ¼ M1

With the Laplace and Fourier transformation of the state-space Eqs. (2) and (3), TF matrix can be easily established. The TF matrix is given in Eqs. (4) and (5) for Laplace and Fourier domain respectively: GðsÞ ¼ CðsIAÞ1 B þ E

(4)

GðωÞ ¼ CðiωIAÞ1 B þ E (5) pffiffiffiffiffiffiffi where i is defined as i ¼ 1. In the present work, only Fourier domain is considered for the subsequent analysis. The frequency domain counterparts of the input vector u(t) and output vector y(t) in terms of U(ω) and Y(ω) respectively are well known to be related as YðωÞ ¼ GðωÞUðωÞ

(6)

3. Frequency band-specific H1 norm minimization This section contains the definition of H1 norm as well as band-specific H1 norm, input–output selection necessary for computing TF matrix and use of GA in band-specific H1 optimization. 3.1. H1 norm H1 norm [24–27] for single input single output (SISO) systems is defined as the maximum magnitude of the frequency response of the system in entire frequency range and commonly expressed as jjGjj1 ¼ max jGðωÞj ω

(7)

For MIMO systems, H1 norm is defined as maximum gain i.e. worst-case gain in entire frequency range. Gain of the system is usually evaluated (Eq. (8)) as spectral norm i.e. Euclidean vector-norm (||.||) based induced norm of TF matrix and becomes equal to the largest singular value [28] ðsð:ÞÞ of the TF matrix:   jjYðωÞjj : UðωÞ≠0 ¼ sðGðωÞÞ βðωÞ ¼ max (8) jjUðωÞjj H1 norm for MIMO systems is usually expressed as jjGjj1 ¼ sup sðGðωÞÞ ω

(9)

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Considering a frequency range from ω1 to ω2 or in short (ω1:ω2), the band-specific H1 norm can be expressed as ^ jjGjj 1¼

sup sðGðωÞÞ

ω∈ðω1 :ω2 Þ

(10)

Standard relations among H1 norm and L2 norm of input as well as output already exist and encourage the application of H1 optimization in practical problems e.g. vibration control etc. Representing the general class of structural system in the form of causal LTI system having real-valued impulse response function, it is quite straightforward to extend these relations in band specific form. Assuming the frequency contents of the input and output signals lying within the frequency-band ^ and Y^ respectively, relations between H1 ^ y, ^ U (ω1:ω2) and denoting the band specific representation of u, y, U and Y as u, norm and L2 norms (||.||2) in frequency as well as time domain can be expressed in band-specific form in terms of the following inequalities: ^ jjYjj 2

^ ≤kGk 1

(11a)

^ 2 jjyjj ^ ≤kGk 1 ^ 2 jjujj

(11b)

^ jjUjj 2

A generalized study with regard to the consideration of finite frequency-band specification as well as relevant time domain interpretation can be found in literatures e.g. [14,15]. 3.2. Input–output selection for H1 optimization The structural system without control devices is commonly referred to as the primary system (PS). A general MDOF system is presently considered as the primary system in this study. Each DOF usually behaves as both input and output locations for MDOF structural systems and this makes an MDOF system as a MIMO system. On the other hand, passive control devices are secondary systems (SS) attached to primary systems to fulfil the objective of control. A passive control system may increase the system size. Some passive control devices e.g. TMD increase the system size, while devices like added viscous dampers do not increase the system size. The second-order system matrices, as in Eq. (1), for the primary system are transformed to state space form to finally evaluate the TF matrix. In a similar way, it is always possible to form the TF matrix for the system coupled with any type of passive control devices. Finally, the goal of using passive control devices is to reduce the frequency response of input–output channels of the primary system. All the required input–output channels are influenced in an overall way by the spectral norm of TF matrix. The input–output locations from the controlled system are selected generally to match with the required input–output locations of the primary system. Thus, it is possible to have reasonable comparison of spectral norm between the primary system and the controlled system. Further, sizes of TF matrices for both the primary and controlled systems become similar. 3.3. Design of passive control devices with band-wise H1 optimization It can be observed from Eq. (6) that the frequency response of a structural system within an excitation frequency band contributes to the output. Thus, for a known frequency range of excitation, effective design of passive control devices can be achieved by controlling the frequency response behaviour within that frequency range alone. In the conventional H1 optimization, minimization of maximum spectral norm is carried out for entire frequency range. A controlled system based on conventional H1 optimization may not perform well if the frequency range with better minimization of gain as outcome of the control strategy does not match well to the frequency range of excitation. If the worst-case gain is minimized only in the excitation frequency range, then it is expected to have better minimization of spectral norm as well as control performance in that range. In the present work, band-specific H1 optimization is taken into account for efficient design of passive control devices. H1 norm for complete frequency range is commonly evaluated based on the algorithm proposed by Bruinsma and Steinbuch [29]. H1 norm for a specific frequency band can be evaluated by choosing the maximum gain out of the discrete gains computed at discrete frequencies inside that frequency band. The gain for a particular frequency ω can be computed as the maximum singular value of TF matrix G(ω) which is evaluated using Eq. (5). Smaller frequency-increment (Δω) between two consecutive discrete frequencies is desired to capture the sharp peaks available in the frequency response associated to a specific frequency band. GA [30], a heuristic optimization technique, is commonly known for its effective performance. Some of the useful advantages of GA are (a) higher tendency to find global optimal solution, (b) robust performance with function having noisy/ textured feature, constant-valued/flat feature over a large domain or having any discontinuity, (c) ability to handle large as well as poorly understood search space, (d) ease in handling arbitrary kinds of constraints, and (e) effectiveness for multiobjective optimization. Associated with these advantages, GA appears to be a good choice for the band-specific H1 optimization. In the present study, the band-specific H1 optimization is carried out with GA utilizing the numerically simulated problems.

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4. Example and numerical simulation A multi-storeyed shear frame coupled with a TMD as the passive control device is considered for numerical simulation to have observations on the band specific designs as well as their performance evaluation. It may be mentioned that similar strategy is applicable for design of any passive control device subjected to the condition that the controlled system also behaves as an LTI system like the primary system. 4.1. Shear frame system A numerically simulated multi-storeyed shear frame building system coupled with a TMD at roof top is considered for demonstration of the strategy. The behaviour of shear frame model is well understood and has been considered as the primary system, while the TMD system is considered as the secondary system. A typical n storied shear frame coupled with the single-TMD (STMD) system and the multiple-TMD (MTMD) system is shown in Fig. 1(a) and (b) respectively. The mass matrix M is considered as diag(m1 m2 … mi …mn) where mi represents the mass of ith storey. Referring to the stiffness of ith storey as ki the stiffness matrix of the primary system K is expressed as 2 3 k2 0 k1 þ k2 7 6 k k2 þ k3 k3 7 6 2 6 7 6 7 ⋱ K¼6 7 6 kn1 kn1 þ kn kn 7 5 4 0 kn kn The number of DOF of the primary system is considered as 10 for the numerical simulation. Hence, DOF vector for the primary system becomes [1 2 …. 10]T according to the DOF numbering pattern as shown in Fig. 1(a) or (b). Mass and stiffness at each of the storey are taken as mi ¼10,000 kg and ki ¼ 18,000 kN m  1 respectively. Damping is assumed to be proportional. The natural frequencies (ω) in rad/s along with the assumed damping ratios (ξ) in percentage are presented in Table 1 for all the modes. 4.2. Optimization problem using a TMD system as the passive control device Both of the STMD and MTMD systems are considered in the numerical simulations. The total mass of the TMD system is taken as 2.5 percent of total shear frame mass whether it is an STMD or an MTMD system. Each TMD component-device adds one DOF. If it is assumed that k numbers of TMD components with DOF-vector as {(n+1) … (n+j) … (n+k)}T are attached to a shear frame system with DOF-vector as {1 2 … r … n}T, then the interaction stiffness and damping matrices n+1

n+k

n+1 -o-o-o n

n

n-1

n-1

2

2

1

1

w

w

Fig. 1. A general shear frame primary system with n numbers of DOF is coupled with (a) an STMD system and (b) an MTMD system.

Table 1 Natural frequencies and considered damping ratios of the shear frame system. Mode

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

ω (rad/s) ξ (%)

6.3 1.0

18.9 1.1

31.0 1.2

42.4 1.3

52.9 1.4

62.2 1.5

70.1 1.6

76.4 1.7

81.1 1.8

83.9 1.9

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between the rth DOF associated floor-mass and (n+j)th DOF associated TMD-mass are given, respectively, by the following equations: " # kTMD kTMD KSS ¼ (12a) kTMD kTMD " DSS ¼

cTMD

cTMD

cTMD

cTMD

# (12b)

These interaction stiffness and damping matrices are associated to the DOF vector, {r (n+j)}T. Finally, these matrices are assembled to the stiffness and damping matrices of the primary system to form the equation of motion of the controlled system. The TMD system is assumed to be positioned at roof top in all the simulations carried out in the present study. Further, the optimization variables are considered as natural frequency (ωTMD) and damping ratio (ξTMD) of the TMD system. In all the cases, optimization is carried out using a single TMD device. Using these two parameters as variables, stiffness (kTMD) and damping coefficient (cTMD) of a TMD device can be written as kTMD ¼ mTMD  ωTMD 2

(13)

cTMD ¼ 2  mTMD  ωTMD  ξTMD

(14)

where mTMD is the mass of that TMD device and it is known. In the numerical study, control of displacement and acceleration is considered. Velocity is generally not controlled, though control of velocity is equally applicable like displacement and acceleration. The multi-storey shear frame is considered to be subjected to base excitation. A TMD system adds extra DOF to the primary system increasing the sizes of dynamic matrices as in Eq. (1) and finally increases the size of the system. In case of base excitation all the DOF are excited and act as input locations. All the DOF locations belonging to the primary system are considered for achieving the control of response. The input and output locations belonging to the shear frame system (PS) are selected for the computation of TF matrices for both the cases of primary system as well as controlled system. It helps to have a reasonable comparison of spectral norm. 4.3. Band-specific H1 optimization Three numbers of frequency bands of excitation are considered here as 1–25 rad/s, 25–50 rad/s and 50–75 rad/s. Overall range of considered frequencies i.e. 1–75 rad/s well represents the frequency range of the shear frame system with first and last natural frequencies as 6.3 and 83.9 rad/s respectively. Moreover three numbers of frequency bands appears reasonable choice for observation on the band-specific design strategy using the shear frame with 10 well separated modes. The objective functions along with the considered ranges of the variables are presented in Table 2 for different single-objective optimization cases. Each of the optimization cases is carried out for both displacement and acceleration. 4.3.1. Implementation of GA GA solver of MATLAB [31] with required customizations is used to carry out the optimization. The associated GA parameters are detailed as below: (a) (b) (c) (d)

Real coded GA is employed as real-coded population based GA shows many advantages over binary-coded GA [32]. Population size is taken as 100, which can be considered reasonable for the present optimization problem. A penalty function approach [33] is employed in GA for constraint handling. Fitness scaling is observed to influence the performance of GA [34]. Raw objective function scores based scaling as well as their ranking based scaling are commonly used. The effect of spread of the raw objective function scores is removed with the rank based fitness scaling strategy. A rank based power law scaling strategy has been used where scaled fitness

Table 2 Objective functions (either displacement or acceleration) and associated variables for different single objective optimization cases. Optimization case no. 1 2 3 4

Type Single objective Single objective Single objective Single objective

Objective function (minimization type) " Js ¼

# sup

ω∈ð1:1Þ

sðGðωÞÞ

" J1 ¼ " J2 ¼

ω∈ð0:25Þ

#

sup sðGðωÞÞ

" J3 ¼

#

sup sðGðωÞÞ

ω∈ð25:50Þ

sup sðGðωÞÞ

ω∈ð50:75Þ

#

ωTMD bound (rad/s)

ξTMD bound

0–75

0.01–0.4

0–75

0.01–0.4

0–75

0.01–0.4

0–75

0.01–0.4

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value of an individual with rank nr is considered as proportional to ð1=nr β Þ with β¼0.5. (e) Regarding the selection operation, popular selection schemes are observed as fitness-proportionate selection with roulette-wheel (RWS), ranking selection, stochastic universal sampling (SUS) selection and tournament selection. Tournament (binary) selections are found to perform well for attaining convergence with reduced computation time requirement [35]. Further, tournament selection without replacement is observed to perform better than with replacement version [36]. Binary tournament selection without replacement is adopted for the present study. Tournament selection has the additional property that it is not generally affected with the choice of fitness scaling. (f) In the reproduction phase involving creation of population for the next generation, consideration of an elitist strategy commonly leads to better convergence. The elitist strategy carries over the individuals with best fitness values as elite children to the next generation unaltered and hence good individuals are not lost. Rest of the elite children are formed using either crossover or mutation based on crossover probability/fraction. Presently elite children count is considered as 2. It is well known notion in GA to consider low probability/rate of mutation and high probability/fraction of crossover. Blend crossover (BLX-α) [37], a well known effective crossover operator is used here with widely applied value of α as 0.5. A uniform scheme for real-coded mutation is considered, where decision is made on first whether a variable will be mutated or not based on the mutation probability rate and if the decision becomes affirmative then mutation is carried out with a random number chosen uniformly in the range of that variable. In order to ascertain appropriate GA parameters to be used for the optimization, multiple (10) runs using GA are carried out to find the best objective function scores for different crossover fractions as [0, 0.2, 0.4, 0.6, 0.8, 1] used with each of the commonly chosen mutation rates as [0.001, 0.005, 0.01, 0.05]. The objective function based on mid-frequency band (25–50 rad/s) specific H1 norm with consideration of acceleration output is considered for this exercise. The convergence plot is shown in Fig. 2(a–d) and it can be concluded that crossover fraction as 0.8 and mutation rate as 0.05 can be chosen for the optimization exercise. The employed GA parameters associated to the single-objective optimization are presented in Table 3.

4.3.2. Optimal solutions and their performance The values of design variables of the TMD system using conventional H1 optimization are shown in Table 4. The values of design variable using the band-specific H1 optimization for three bands are shown in Table 5. An interesting observation can be noted that frequency of TMD for a frequency band becomes close to that band. Similar observation can be seen in the x 10-3 1.54

1.52

1.52

Best objective function score

Best objective function score

x 10-3 1.54 1.5 1.48 1.46 1.44 1.42 1.4 1.38 1.36

1.5 1.48 1.46 1.44 1.42 1.4 1.38 1.36

0

0.2

0.4

0.6

0.8

1

0

0.2

Crossover fraction x 10-3

0.6

0.8

1

0.8

1

x 10-3

1.54

1.54

1.52

1.52

Best objective function score

Best objective function score

0.4

Crossover fraction

1.5 1.48 1.46 1.44 1.42 1.4 1.38 1.36

1.5 1.48 1.46 1.44 1.42 1.4 1.38 1.36

0

0.2

0.4

0.6

Crossover fraction

0.8

1

0

0.2

0.4

0.6

Crossover fraction

Fig. 2. Multiple (10) GA runs based best objective function scores along with mean-showing line for different crossover fractions evaluated for the mutation probability rate as (a) 0.001, (b) 0.005, (c) 0.01 and (d) 0.05.

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Table 3 Considered GA parameters for single-objective optimization. GA parameters

Single objective GA

1. Population type 2. Population size 3. Elite count 4. Selection type 5. Crossover type 6. Crossover fraction 7. Mutation type 8. Mutation rate 9. Stopping criteria a. Tolerance (0.000001) b. Maximum generation no (set enough long so that criteria-a occurs earlier)

Double vector (real-coded) 100 2 Binary tournament without replacement BLX-α (α ¼ 0.5) 0.8 Uniform 0.05 Average change (relative) in the fitness value less than the tolerance 3000

Table 4 Design of the TMD system using conventional H1 optimization. Optimization case no.

Response type

Natural frequency (rad/s)

Damping ratio

1 1

Displacement Acceleration

6.060 10.440

0.136 0.399

Table 5 Design of the TMD system using band-specific H1 optimization. Optimization case no.

Response type

Natural frequency (rad/s)

Damping ratio

2 2 3 3 4 4

Displacement Acceleration Displacement Acceleration Displacement Acceleration

6.059 8.778 29.797 33.660 48.456 55.328

0.133 0.399 0.185 0.201 0.156 0.142

-80

-60

Spectral norm (dB)

Spectral norm (dB)

-50

Controlled system Primary system

-90 -100 -110 -120 -130 -140

-70 -80 -90 -100 -110

-150

-120

-160

-130 0

10 20 30 40 50 60 70 80 90 100

Frequency (rad/sec)

Controlled system Primary system 0

10 20 30 40 50 60 70 80 90 100

Frequency (rad/sec)

Fig. 3. Comparison of spectral norm between the uncontrolled system and the controlled system designed with conventional H1 optimization, controlling (a) displacement and (b) acceleration.

work of Rana and Soong [38], where the control of response using TMD is achieved by tuning the frequency of TMD to the frequency of target mode to be controlled. Profiles of spectral norm of the controlled system obtained using the conventional H1 optimization for displacement and acceleration control are shown in Fig. 3(a) and (b) respectively. From Fig. 3(a) and (b), a bias is visible in the form of higher minimization of spectral norm around stronger modes and lower elsewhere. Profiles of spectral norm of the displacement controlled system obtained using the band-specific H1 optimization for three frequency bands are shown in Fig. 4(a–c). Similar plots for acceleration control are shown in Fig. 5

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Controlled system Primary system

-90 -100 -110 -120 -130

-90 -100 -110 -120 -130

-140

-140

-150

-150 0

10

20

30

40

50

60

70

80

Controlled system Primary system

-80

Spectral norm (dB)

Spectral norm (dB)

-80

90

0

10

20

30

Frequency (rad/sec)

50

60

70

80

90

Controlled system Primary system

-80

Spectral norm (dB)

40

Frequency (rad/sec)

-90 -100 -110 -120 -130 -140 -150 0

10

20

30

40

50

60

70

80

90

Frequency (rad/sec)

-40

-40

-50

-50

-60

-60

Spectral norm (dB)

Spectral norm (dB)

Fig. 4. Comparison of spectral norm between the uncontrolled system and the displacement controlled system designed with band-specific H1 optimization for the frequency band (a) 0–25 rad/s, (b) 25–50 rad/s and (c) 50–75 rad/s.

-70 -80 -90 -100 -110

Controlled system Primary system

-120 -130

-70 -80 -90 -100 -110

Controlled system Primary system

-120 -130

0

10

20

30

40

50

60

70

80

90

0

10

20

30

Frequency (rad/sec)

40

50

60

70

80

90

Frequency (rad/sec)

-40

Spectral norm (dB)

-50 -60 -70 -80 -90 -100 -110

Controlled system Primary system

-120 -130

0

10

20

30

40

50

60

70

80

90

Frequency (rad/sec) Fig. 5. Comparison of spectral norm between the uncontrolled system and the acceleration controlled system designed with band-specific H1 optimization for the frequency band (a) 0–25 rad/s, (b) 25–50 rad/s and (c) 50–75 rad/s.

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(a–c). Profile of spectral norm of the primary system is also included in these figures for comparisons. The spectral norm here is represented in dB scale. If the value of spectral norm is α, its value in dB is expressed as   α αðdBÞ ¼ 20log10 (15) αref where αref is the reference value and is taken as unity in the absence of any specified value. Finally, the performance of the band-specific H1 optimization is demonstrated over the conventional H1 optimization. The shear frame system coupled with a TMD device designed for a specific frequency band is excited using white noise with frequency contents matching to that specific band. The same white noise is also applied to the primary system coupled with TMD which is designed with conventional H1 optimization. Responses (displacement or acceleration) of both the cases of controlled systems are compared separately with the response from the primary system based on the same white noise excitation. The white noise is used as the base-acceleration to excite the structural system. In case of base-acceleration based excitation, the input vector u is computed as u ¼ wML

(16)

1200

1200

1000

1000

1000

800 600 400

800 600 400

200

200

0

0

0 10 20 30 40 50 60 70 80 90 100

Frequency (rad/sec)

FFT (magnitude)

1200

FFT (magnitude)

FFT (magnitude)

where w is the white noise base-acceleration, M is mass matrix of the primary or controlled structural system and L is influence vector which is a vector of ones in case of the present example. It can be noted that the response is computed as relative response with respect to the base of shear frame. The white noise in the form of Gaussian white noise is generated with the MATLAB routine “wgn”. Noise is generated for quite a long duration (200 s) with sampling time as 0.01 s. This helps the observation of control performance for a reasonably long duration. Next, band-limited filtering is carried out with Butterworth filter (fifth order) to get band-limited noise with the help of MATLAB routine “butter”. Frequency bands are selected as 0–25 rad/s, 25–50 rad/s and 50–75 rad/s. Frequency contents of the band-limited noises in the form of Fast Fourier transform (FFT) are shown in Fig. 6. Control performances based on displacement and acceleration are shown in Figs. 7–12 respectively using normalized response with respect to maximum absolute value of uncontrolled response. Response of the top storey is considered here for the observation of control performance. A reasonably longer duration (200 s) is considered for comparison of responses between the uncontrolled and the controlled system. MATLAB routine “lsim” is used to compute the responses considering zero initial condition. From Figs. 7–12, it is obvious that TMD systems designed based on a frequency band-specific H1 optimization perform significantly better against excitation involving that frequency band. It is also obvious that TMD systems designed based on conventional H1 optimization are not always efficient against excitation of any frequency content. Finally it can be noted that many white noise samples have been considered for the similar study and good control performance have been observed in a very consistent manner.

800 600 400 200

0 10 20 30 40 50 60 70 80 90 100

0

0 10 20 30 40 50 60 70 80 90 100

Frequency (rad/sec)

Frequency (rad/sec)

Fig. 6. Frequency content of band-limited Gaussian white noise for band (a) 0–25 rad/s, (b) 25–50 rad/s and (c) 50–75 rad/s.

Primary system(Gray) Controlled system(Black)

Response (displacement)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

1

Primary system(Gray) Controlled system(Black)

0.8

Response (displacement)

1 0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

Time (sec)

0

20

40

60

80 100 120 140 160 180 200

Time (sec)

Fig. 7. White noise (with 0–25 rad/s frequency content) induced displacement (at top storey) comparison between the uncontrolled system and (a) the controlled system using TMD designed with conventional H1 optimization and (b) the controlled system using TMD designed with band-specific H1 optimization for the band, 0–25 rad/s.

N. Debnath et al. / Journal of Sound and Vibration 332 (2013) 6044–6062

1

Primary system(Gray) Controlled system(Black)

Response (displacement)

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

1

Primary system(Gray) Controlled system(Black)

0.8

Response (displacement)

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0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

0

20

40

60

Time (sec)

80 100 120 140 160 180 200

Time (sec)

Fig. 8. White noise (with 25–50 rad/s frequency content) induced displacement (at top storey) comparison between the uncontrolled system and (a) the controlled system using TMD designed with conventional H1 optimization and (b) the controlled system using TMD designed with band-specific H1 optimization for the band, 25–50 rad/s.

Primary system(Gray) Controlled system(Black)

Response (displacement)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

1

Primary system(Gray) Controlled system(Black)

0.8

Response (displacement)

1 0.8

-0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

0

20

40

60

Time (sec)

80 100 120 140 160 180 200

Time (sec)

Fig. 9. White noise (with 50–75 rad/s frequency content) induced displacement (at top storey) comparison between the uncontrolled system and (a) the controlled system using TMD designed with conventional H1 optimization and (b) the controlled system using TMD designed with band-specific H1 optimization for the band, 50–75 rad/s.

1 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

Primary system(Gray) Controlled system(Black)

0.8

Response (acceleration)

Response (acceleration)

1

Primary system(Gray) Controlled system(Black)

0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

0

20

40

Time (sec)

60

80 100 120 140 160 180 200

Time (sec)

Fig. 10. White noise (with 0–25 rad/s frequency content) induced acceleration (at top storey) comparison between the uncontrolled system and (a) the controlled system using TMD designed with conventional H1 optimization and (b) the controlled system using TMD designed with band-specific H1 optimization for the band, 0–25 rad/s.

4.4. Response control against excitation associated to multiple frequency-bands In the previous section passive control systems are considered targeting only a single frequency band of excitation. There may be interest in designing passive control devices to provide effective response control subjected to excitations of frequency content associated to multiple frequency bands. Two preliminary exercises are carried out here in this regard as pointed out below: (a) Implementation of multi-objective GA [39] using a STMD system while objective functions being band-specific H1 norms encompassing different bands.

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(b) A possible design of a TMD system targeting frequency-content of excitation associated to multiple bands e.g. 1st and 3rd frequency bands. Multi-objective optimization is a requirement for designing the passive control system while targeting multiple frequency-bands of excitations. Sole aim of this exercise is to have a simple observation on the multi-objective-GA while objective functions being band-specific H1 norms. Band-wise H1 optimization using a STMD system is carried out using NSGA-II [39] based controlled elitist GA [40]. Controlled elitist GA uses a pareto-ratio which limits the number of individuals on the pareto front and side by side favours individuals with lower fitness values to increase the diversity of the population. A moderate value for pareto-ratio is used as 0.5, which leads to 50 individuals in pareto front out of population size of 100. Pareto-ratio with value 0.5 is observed to perform reasonably well for different types of example problems [40]. The objective functions as defined in Table 2 are utilized for multi-objective optimization and are shown in the Table 6 along with the ranges of the variables considered. Convergence based stopping criteria are used with a suitable tolerance. The adopted GA parameters are shown in Table 7. Using the pareto-optimal [41] solutions, different objective function scores (normalized between 0 and 1) are plotted individually with respect to the most important optimization variable i.e. frequency of TMD as shown in Fig. 13. It is observed from this figure that the frequency corresponding to the minimum objective function score for a frequency-band shows agreement with the Table 5. Further, using the final variables and final objective function scores, an optimization variable set associated to the minimum score of any objective function can be chosen as the best optimization variable set for that objective function. Best optimization variable set for a frequency band is

1 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

Primary system(Gray) Controlled system(Black)

0.8

Response (acceleration)

Response (acceleration)

1

Primary system(Gray) Controlled system(Black)

0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

0

20

40

60

Time (sec)

80 100 120 140 160 180 200

Time (sec)

Fig. 11. White noise (with 25–50 rad/s frequency content) induced acceleration (at top storey) comparison between the uncontrolled system and (a) the controlled system using TMD designed with conventional H1 optimization and (b) the controlled system using TMD designed with band-specific H1 optimization for the band, 25–50 rad/s.

1 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

Primary system(Gray) Controlled system(Black)

0.8

Response (acceleration)

Response (acceleration)

1

Primary system(Gray) Controlled system(Black)

0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

0

20

40

60

Time (sec)

80 100 120 140 160 180 200

Time (sec)

Fig. 12. White noise (with 50–75 rad/s frequency content) induced acceleration (at top storey) comparison between the uncontrolled system and (a) the controlled system using TMD designed with conventional H1 optimization and (b) the controlled system using TMD designed with band-specific H1 optimization for the band, 50–75 rad/s.

Table 6 Objective functions (either displacement or acceleration), as defined in Table 2, along with the associated variables for multi-objective optimization case. Optimization cases no.

Type

Objective function (minimization type)

ωTMD bound (rad/s)

ξTMD bound

5

Multi-objective (three)

J1, J2, J3

0–75

0.01–0.4

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Table 7 Considered GA parameters for multi-objective optimization. GA parameters

Multi-objective GA

1. Population type 2. Population size 3. Selection type 4. Crossover type 5. Crossover fraction 6. Mutation type 7. Mutation rate 8. Stopping criteria a. Tolerance (0.0001) b. Maximum generation no (set enough long so that criteria-a occurs earlier)

Double vector (real-coded) 100 Binary tournament without replacement BLX-α (α ¼ 0.5) 0.8 Uniform 0.05 Average change (relative) in spread of pareto solutions less than the tolerance 3000

Objective function score

0.8 0.6 0.4 0.2

Band-1 Band-2 Band-3

1

Objective function score

Band-1 Band-2 Band-3

1

0.8 0.6 0.4 0.2 0

0 0

10

20

30

40

50

60

70

80

90 100

Frequency (rad/sec)

0

10

20

30

40

50

60

70

80

90 100

Frequency (rad/sec)

Fig. 13. Relation between individual objective function scores (normalized between 0 and 1) and frequencies of TMD based on the final pareto-optimal solutions for (a) displacement and (b) acceleration. Table 8 Best design variables of the TMD system for displacement control for different bands based on multi-objective optimization. Optimization case no. (Frequency bands in rad/s)

2 (0–25)

3 (25–50)

4 (50–75)

Frequency (rad/s) Damping ratio

6.059 0.133

29.921 0.180

48.637 0.149

Table 9 Best design variables of the TMD system for acceleration control for different bands based on multi-objective optimization. Optimization case no. (Frequency bands in rad/s)

2 (0–25)

3 (25–50)

4 (50–75)

Frequency (rad/s) Damping-ratio

8.779 0.399

33.661 0.201

55.292 0.140

actually the best design of TMD for that frequency band. The best designs of TMD for displacement and acceleration control are presented in Tables 8 and 9 respectively. Best design of TMD for a frequency band, as mentioned in Tables 8 and 9, shows good agreement with the single-objective optimization based design as presented in Table 5. Therefore multi-objective GA can be considered to perform well with band-specific norms being the objective functions. In this section, an MTMD system using two TMD components is chosen for achieving consistent control under excitation with frequency content matching to both the 1st and 3rd frequency bands. It may be difficult for an STMD system to perform for both the bands. An MTMD system can be reasonably developed following a common strategy of MTMD system design, where assemblages of individually designed TMD components [38,42] corresponding to different target frequencies are done. This strategy is followed in this study as well. The frequency and damping of the two TMD components are chosen as similar to 1st and 3rd bands based values from Table 5. For example, frequencies and damping ratios are considered as [8.778, 55.328] and [0.399, 0.142] respectively for acceleration control for these two TMD components. The total TMD mass

N. Debnath et al. / Journal of Sound and Vibration 332 (2013) 6044–6062

-80

-60

Spectral norm (dB)

Spectral norm (dB)

-50

Controlled system Primary system

-90 -100 -110 -120 -130 -140

-70 -80 -90 -100 -110

-150

-120

-160

-130 0

10

20

30

40

50

60

70

80

6057

90 100

Controlled system Primary system 0

10

20

Frequency (rad/sec)

30

40

50

60

70

80

90 100

Frequency (rad/sec)

Fig. 14. Profiles of spectral norm of the controlled system using the considered MTMD system (with two TMD components) controlling (a) displacement and (b) acceleration.

Primary system(Gray) Controlled system(Black)

Response (displacement)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

1

Primary system(Gray) Controlled system(Black)

0.8

Response (displacement)

1 0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

20

40

60

80 100 120 140 160 180 200

Time (sec)

0

20

40

60

80 100 120 140 160 180 200

Time (sec)

Fig. 15. Performance of displacement control of the controlled system using the considered MTMD system (with two TMD components) against the excitation of frequency content matching to (a) 1st band and (b) 3rd band.

(considered as 2.5 percent of the total structure) is distributed equally between the two masses of TMD components. Two TMD components are finally assembled to form the MTMD system. The profiles of spectral norm are shown in Fig. 14(a) and (b) for displacement and acceleration control respectively. Minimization of spectral norm is observed in both the 1st and 3rd frequency bands, though the degree of minimization is expectedly lower than the single band specific designed TMD as shown in Figs. 4 and 5. Demonstrations of displacement control under excitation with frequency content corresponding to 1st and 3rd bands are shown in Fig. 15(a) and (b) respectively. Similar demonstrations are also shown in Fig. 16(a) and (b) for acceleration control. Under the excitation of both the frequency bands, reasonable control is observed. However, magnitude of control using such heuristically designed TMD is observed to be lesser than the case of single band specific designed TMD as is evident from Figs. 7(b), 9(b), 10(b) and 12(b). 4.5. Band-specific H1 optimization under earthquake excitation Finally, it may be an interesting issue to observe the performance of H1 optimization for finite frequency range of interest over entire frequency range, while the shear frame system is subjected to some significant earthquake records. In this regard, three significant earthquake records are considered: (1) 1940 El Centro earthquake recorded at El Centro site, (2) 1994 Northridge earthquake recorded at Sylmar country hospital parking lot, and (3) 1995 Kobe earthquake recorded at KJMA station. An approximate frequency band of interest is suggested as 0.3–8.8 Hz (or, 1.885–55.292 rad/s) [21] based on different ground motions including earthquakes 1 and 2. The frequency band for Kobe earthquake (Fig. 17(a)) is identified as 0.666–3.039 Hz using 0.4 as the threshold value [21] over the normalized (to unity) frequency spectrum (Fig. 17(b)). Numerical simulations are carried out considering frequency range of interest as 1.885–55.292 rad/s. It can be noticed that this frequency band is quite wide considering the modal frequencies of the shear frame system. Moreover, TMD is found effective close to the frequency zone around the frequency at which it is tuned. Thus, in view of wide range of frequency of the considered ground motions, it is clear that MTMD system should be introduced. It can be observed from Figs. 3–5 that gain gets peak values close to the modal frequencies and five such peaks exist in the range of 1.885–55.292 rad/s. The peak values of gain for displacement are nearly [1.23  10  4, 1.269  10  5, 4.336  10  6, 2.136  10  6, 1.276  10  6]; while for acceleration value are nearly [0.004958, 0.004532, 0.004167, 0.003845, 0.003571]. It is observed that only 1st peak is significant in comparison to the rest four peaks for displacement, while all the peaks are significant in case of acceleration. Two sub-bands are considered for displacement, one around the peak one and

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another retaining the rest peaks. On the other hand, five sub-bands are considered for acceleration with each sub-band around each peak. A Single TMD device is designed for each of the sub-bands and subsequently the MTMD system is formed assembling all the designed TMD components for all the sub-bands. Total assigned mass for the MTMD system is shared among the TMD components i.e. among the sub-bands heuristically. It is difficult to have a single decision regarding the mass sharing among the multiple TMD components. One particular mass distribution based designed MTMD system can show better efficiency for one earthquake record than others based on the frequency content of the earthquake records. The sub-bands and associated mass shares are mentioned in Table 10. Comparative analysis is carried out here considering three TMD system schemes as (a) an STMD system designed using entire frequency based H1 optimization (STMD-EF), (b) an MTMD system where each TMD component is designed for associated sub-band using band-specific H1 optimization

1 0.6

Primary system(Gray) Controlled system(Black)

0.8

Response (acceleration)

Response (acceleration)

1

Primary system(Gray) Controlled system(Black)

0.8 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

-1

-1 0

20

40

60

80 100 120 140 160 180 200

0

20 40 60 80 100 120 140 160 180 200

Time (sec)

Time (sec)

Fig. 16. Performance of acceleration control of the controlled system using the considered MTMD system (with two TMD components) against the excitation of frequency content matching to (a) 1st band and (b) 3rd band.

1 0.5 0.5

0 -0.5 0

5

10

15

20

25

30

35

40

45

50

0

0

5

Time (sec)

10

15

Frequency (Hz)

Fig. 17. (a) Time history and (b) frequency spectrum of 1995 Kobe earthquake recorded at KJMA station.

Table 10 Used sub-bands and associated mass shares for design of the MTMD system targeting earthquake excitations. Response type

Band

Lower limit (rad/s)

Upper limit (rad/s)

Mass share

Displacement

1 2 1 2 3 4 5

1.8850 14.1372 1.8850 14.1372 25.7611 37.0708 48.0664

14.1372 55.2920 14.1372 25.7611 37.0708 48.0664 55.2920

0.99 0.01 0.4 0.3 0.2 0.075 0.025

Acceleration

Table 11 Obtained design parameters as frequency (rad/s) and damping ratio of each of the constituent TMD components for MTMD-FF and MTMD-EF. Response type

Band

Frequency (MTMD-FF)

Frequency (MTMD-EF)

Damping ratio (MTMD-FF)

Damping ratio (MTMD-EF)

Displacement

1 2 1 2 3 4 5

6.0614 18.864 6.2983 18.7314 30.7267 42.2217 52.7945

6.0626 6.3360 9.2250 8.9215 8.3950 7.4255 6.4786

0.1325 0.0139 0.0873 0.0740 0.0586 0.0339 0.0186

0.1355 0.0158 0.4000 0.4000 0.4000 0.4000 0.2577

Acceleration

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(MTMD-FF) and (c) an MTMD system where each TMD component is designed using entire frequency based H1 optimization (MTMD-EF). The MTMD system shows efficiency in different ways, which has motivated to consider MTMDEF using similar mass-shares like MTMD-FF for more equitable comparisons. Usefulness of the STMD-EF is mainly in terms of a standard reference, since the MTMD-EF is not guaranteed to perform better than the STMD-EF in all aspects. The design is carried out using band-specific optimization employing similar GA parameters as mentioned earlier in Table 3. The ranges of optimization variables as frequency and damping ratio for a TMD device are considered as earlier i.e. 0–75 rad/s and 0.01– 0.4. The design parameters are shown in Table 11 as obtained from optimization using GA. A comparison for gain is shown in Fig. 18. From Fig. 18, it can be concluded that MTMD-FF performs fairly better in case of acceleration, though marginally better in case of displacement. Further, time history observation are obtained corresponding to all the considered ground motions for inter-storey displacement and acceleration as shown in Figs. 19–24. It can be noted that though optimization is carried out for controlling displacement, inter-storey drift is chosen to show the effectiveness of control-performance. 5th and 10th stories are chosen for inter-storey displacement, while DOF 5 and 10 are chosen for acceleration. Comparisons are carried out among the TMD system schemes considering the control performance in reduction (percent) of maximum as well as root-mean-square (RMS) inter-storey drift and acceleration. Maximum and RMS inter-storey drift and acceleration of the primary system are used as the reference for comparisons. These comparisons are presented in Tables 12 and 13 for

-40

-70

Spectral norm (dB)

-80 -90 -100 -110 -120

-50

Spectral norm (dB)

No TMD STMD-EF MTMD-EF MTMD-FF

-70 -80 -90

No TMD STMD-EF MTMD-EF MTMD-FF

-100

-130 -140

-60

0

10

20

30

40

50

-110

60

0

10

Frequency (rad/sec)

20

30

40

50

60

Frequency (rad/sec)

Fig. 18. Comparison of spectral norm among different TMD system schemes, intended for earthquake excitations, in case of (a) displacement and (b) acceleration.

Inter-storey drift (m)

No TMD STMD-EF MTMD-EF MTMD-FF

0.02 0.01 0 -0.01 -0.02 -0.03

0

5

Inter-storey drift (m)

x 10-3

0.03

10 15 20 25 30 35 40 45 50 55

6

No TMD STMD-EF MTMD-EF MTMD-FF

4 2 0 -2 -4 -6 0

5

10

15

20

Time (sec)

25

30

35

40

45

50

55

Time (sec)

0.06

No TMD STMD-EF MTMD-EF MTMD-FF

0.04 0.02 0 -0.02 -0.04 -0.06 0

5

10

15

20

25

Time (sec)

30

35

40

45

Inter-storey drift (m)

Inter-storey drift (m)

Fig. 19. Inter-storey drifts under 1940 El Centro earthquake at (a) 5th storey and (b) 10th storey.

0.02 0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02

No TMD STMD-EF MTMD-EF MTMD-FF

0

5

10

15

20

25

30

35

Time (sec)

Fig. 20. Inter-storey drifts under 1995 Kobe earthquake at (a) 5th storey and (b) 10th storey.

40

45

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Inter-storey drift (m)

No TMD STMD-EF MTMD-EF MTMD-FF

0.02 0.01 0 -0.01 -0.02 -0.03

0

10

20

30

40

50

Inter-storey drift (m)

x 10-3

0.03

8 6 4 2 0 -2 -4 -6 -8

60

No TMD STMD-EF MTMD-EF MTMD-FF

0

10

20

30

40

50

60

Time (sec)

Time (sec)

10

No TMD STMD-EF MTMD-EF MTMD-FF

5 0 -5 -10

0

5

Acceleration (m/sec/sec)

Acceleration (m/sec/sec)

Fig. 21. Inter-storey drifts under 1994 Northridge earthquake at (a) 5th storey and (b) 10th storey.

No TMD STMD-EF MTMD-EF MTMD-FF

10 5 0 -5 -10

10 15 20 25 30 35 40 45 50 55

0

5 10 15 20 25 30 35 40 45 50 55

Time (sec)

Time (sec)

No TMD STMD-EF MTMD-EF MTMD-FF

20 10 0 -10 -20 0

5

10

15

20

25

30

35

40

Acceleration (m/sec/sec)

Acceleration (m/sec/sec)

Fig. 22. Acceleration responses under 1940 El Centro earthquake at (a) 5th DOF and (b) 10th DOF.

No TMD STMD-EF MTMD-EF MTMD-FF

30 20 10 0 -10 -20 -30

45

0

5

10

15

Time (sec)

20

25

30

35

40

45

Time (sec)

10

No TMD STMD-EF MTMD-EF MTMD-FF

5 0 -5 -10 0

10

20

30

Time (sec)

40

50

60

Acceleration (m/sec/sec)

Acceleration (m/sec/sec)

Fig. 23. Acceleration responses under 1995 Kobe earthquake at (a) 5th DOF and (b) 10th DOF.

15

No TMD STMD-EF MTMD-EF MTMD-FF

10 5 0 -5 -10 -15 0

10

20

30

40

50

60

Time (sec)

Fig. 24. Acceleration responses under 1994 Northridge earthquake at (a) 5th DOF and (b) 10th DOF.

inter-storey drift and acceleration respectively. It can be concluded based on the observations in Figs. 19–24 as well as Tables 12 and 13 that band-wise H1 optimization considering the finite frequency band of excitation has provided comparatively better design. It can be observed that the control performance based on acceleration is superior as compared to those based on inter-storey drift.

5. Conclusion H1 optimization is carried out associated to the excitation frequency bands for effective design of passive control devices for structural systems. A numerical simulation is carried out using a multi-storey shear frame coupled with an STMD or an

N. Debnath et al. / Journal of Sound and Vibration 332 (2013) 6044–6062

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Table 12 Control performance of different TMD system schemes with reference to the maximum and RMS inter-storey drift value observed for the uncontrolled system. Earthquake El Centro

Storey 5 10

Kobe

5 10

Northridge

5 10

Ref-type

Ref-value (m)

Control (%) STMD-EF

Control (%) MTMD-EF

Control (%) MTMD-FF

Maximum RMS Maximum RMS

0.029061 0.0068821 0.0068 0.001409

40.4957 52.9813 26.6542 36.5189

40.4985 52.9311 26.2921 36.3505

40.5389 52.9128 27.1361 36.9245

Maximum RMS Maximum RMS

0.068195 0.022467 0.01945 0.0046664

11.1796 63.7723 19.5395 48.0583

11.1735 63.7722 19.4079 47.845

11.1804 63.8611 19.6658 49.7543

Maximum RMS Maximum RMS

0.030536 0.0069977 0.0083289 0.0015263

7.2275 46.6492 7.1166 34.4293

7.5034 46.7757 7.0513 34.4741

8.5858 46.8821 8.3635 37.4738

Table 13 Control performance of different TMD system schemes with reference to the maximum and RMS acceleration value observed for the uncontrolled system. Earthquake El Centro

DOF 5 10

Kobe

5 10

Northridge

5 10

Ref-Type

Ref-Value (m/s2)

Control (%) STMD-EF

Control (%) MTMD-EF

Control (%) MTMD-FF

Maximum RMS Maximum RMS

10.0072 1.8459 13.2975 2.6172

15.677 32.483 24.6122 32.6058

18.5139 34.8202 28.7035 35.5955

21.3792 41.1226 29.1445 41.5025

Maximum RMS Maximum RMS

26.7561 5.8752 35.7723 8.5967

25.8914 18.4912 15.0353 15.9363

29.7668 31.0417 14.9089 28.6904

36.2148 51.4068 18.5666 51.1444

Maximum RMS Maximum RMS

10.8222 2.046 15.7518 2.8739

25.7119 33.9937 15.1118 33.8182

23.644 36.3796 13.2762 36.986

25.0963 41.1457 19.8833 42.3809

MTMD system. The band-specific H1 optimization problem has been transformed into a GA-friendly form for design of the passive controllers. Following concluding remarks are considered from the overall observation of the present study.

 It is demonstrated for different cases of base-excitations using white noise as well as earthquake records that a TMD  

system designed with band-specific H1 optimization associated to the band of interest shows better performance in control than that designed with conventional H1 optimization for entire frequency range. Excitation frequency-band specific H1 optimization can be considered as a better design tool for passive control devices than the conventional H1 optimization while retaining many of the advantages of conventional H1 optimization. The band-wise H1 optimization is expected to perform well for other passive control devices as well and observation on this issue remains as future scope of work.

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