Optimum design of linear multiple tuned mass dampers subjected to white-noise base acceleration considering practical configurations

Engineering Structures 171 (2018) 516–528

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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct

Optimum design of linear multiple tuned mass dampers subjected to whitenoise base acceleration considering practical configurations

T

⁎

Sung-Yong Kim, Cheol-Ho Lee

Department of Architecture and Architectural Engineering, Seoul National University, Seoul, Republic of Korea

A R T I C LE I N FO

A B S T R A C T

Keywords: Tuned mass damper Multiple tuned mass damper Optimization Passive control Robustness

This study investigates the application of multiple tuned mass dampers (MTMDs) for situations wherein multiple small-sized tuned mass dampers (TMDs) are required to be installed owing to practical reasons such as a space limitation, for transportation, and for ease of handling. This paper presents an optimum design and analysis of linear MTMDs with various practical configurations on a single-degree-of-freedom (SDOF) primary structure subjected to a zero-mean white-noise base-acceleration excitation. Six practical configurations are developed and comparatively analyzed, each of which is constrained with linearly distributed frequency ratios, uniformly distributed damping coefficients, linearly distributed mass ratios, and combinations thereof. Two different optimization techniques were adopted in this study. The first minimizes the nominal performance objective function while the second minimizes the mean value of the objective function while assuming that the associated structural parameters are perturbative. All the optimized parameters and plots are presented in a non-dimensional form in order to provide results that are useful for practical design. Among the six cases investigated, a three-parameter optimum solution based on frequency and damping ratios showed sufficiently satisfactory control performance and is recommended for practical design. Furthermore, this study provides the contour maps that enable designers to accommodate the moving TMD units within a permissible stroke limit.

1. Introduction A tuned mass damper (TMD) is a fascinating vibration control device that dissipates the vibration energy of a structure through the use of a damping element. Since Frahm [12] proposed its concept in his patent, numerous studies have been conducted on the optimum solution of TMDs in order to determine its parameters such as optimum frequency and optimum damping ratio. The classical solutions proposed by Den Hartog [10] and Warburton [34] are still widely used in both academic and practical applications. Various researchers, including Adam and Furtmüller [2], Bekdaş and Nigdeli [4], Farshidianfar and Soheili [11], Salvi and Rizzi [31], and Salvi and Rizzi [32], also contributed to the recent development in the optimum TMD parameters for diverse loading conditions and objective functions. However, the use of a large single TMD could cause practical problems. For example, if a heavy mass is concentrated at one point, it may result in unacceptable overloading on the primary structure. Other practical issues include transportation, space requirements, installation, and tuning in the field. For suppressing the excessive vibration of the three Spring Mountain Road pedestrian bridges located in Las Vegas, the use of a single TMD was not considered owing to its low aesthetic

⁎

Corresponding author. E-mail address: [email protected] (C.-H. Lee).

https://doi.org/10.1016/j.engstruct.2018.06.002 Received 6 October 2017; Received in revised form 1 May 2018; Accepted 1 June 2018 0141-0296/ © 2018 Published by Elsevier Ltd.

appeal and the insufficient clearance between the TMD and the roadways underneath [33]. The multiple TMD (MTMD), which refers to a system comprising multiple units of TMDs in which each TMD may have different dynamic characteristics, is one of the viable solutions to the aforementioned difficulties. In the early stages of research, the MTMD configurations with simplified and limited conditions were mainly addressed in order to reduce the number of associated design variables. For instance, the MTMD of large numbers of units with equally spaced natural frequencies and equal damping constants was studied by Xu and Igusa [35] based on an asymptotic analysis, and it was shown that such an MTMD is effective in reducing the response of the main structure. For a finite number of MTMDs with similar constraints, Joshi and Jangid [18] and Jangid [17] presented optimum parameters of MTMDs for an undamped and a damped primary structure, respectively. MTMDs with equal damping ratios and equally spaced natural frequencies were also investigated by former researchers such as Yamaguchi and Harn pornchai [36], Abé and Fujino [1], and Kareem and Kline [19]. Continued studies have been conducted on MTMDs with more relaxed constraints; for example, Igusa and Xu [16], Li [20], Hoang, Fujino, and Warnitchai [13], Zuo and Nayfeh [39], Li and Ni [22], Yang

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S.-Y. Kim, C.-H. Lee

motion of the primary structure-LMTMD system can be written as

et al. [37], and Bozer and Özsariyildiz [5]. The main differences among these studies involve (1) considered excitation; harmonic forcing function versus ground acceleration, (2) the objective function; the RMS (root mean square) response of the primary structure versus the maximum of the frequency response and (3) optimization strategies employed. Existing studies, however, only included a comparison with other configurations in a limited manner. Li and Ni [22], for instance, only compared the performance between their non-uniformly distributed MTMD and the one with equal frequency spacing and equal damping ratio. Li [20] investigated various MTMD configurations and provided optimum parameters for the case wherein the natural frequencies are equally spaced. Hoang et al. [13] and Li and Liu [21] also conducted similar studies. While designing optimum TMDs, it is also necessary to consider the possible performance deterioration caused by the detuning effect. Several researchers proposed methods for the robust design of MTMDs including Lucchini et al. [25], Hoang and Warnitchai [14], De et al. [9], and Lin et al. [24]. However, the studies on the robust design of MTMDs are still limited. The primary objective of this study was to provide a general design guide for linear MTMDs (LMTMDs) with various practical configurations for a single-degree-of-freedom (SDOF) primary structure subjected to a zero-mean white-noise base-acceleration excitation. This study first investigates the optimum parameters of various LMTMD configurations under diverse constraints related to the frequency ratio, damping ratio, mass distribution, and combination thereof. Second, two different optimization schemes are employed for a more comprehensive investigation: nominal performance optimization (NPO) and robust performance optimization (RPO). NPO seeks a solution that minimizes the “deterministic” objective function while RPO minimizes the statistically disturbed objective function while assuming that the associated parameters are uncertain rather than deterministic. Particularly for the RPO problem, this study adopted the point estimation method (PEM), which is one of the simple and reasonable methods for evaluating the statistical properties of a complicated function. Third, a three-parameter optimum solution based on the frequency ratio and damping ratio is proposed for practical design. Finally, in order make it possible to consider the performance evaluation and stroke limitations simultaneously, this study also provides contour maps for the RMS displacement of the main structure and the largest RMS displacement of LMTMDs.

N

∑

(ms + mT ) x¨s +

mi x¨i + cs x ̇ + ks x = −(ms + mT ) u¨ g

i=1

mi (x¨s + x¨i ) + ci x i̇ + ki x i = 0

i = 1, ⋯, N

(1a) (1b)

where ms , cs , and ks are the mass, damping coefficient, and spring constant of the primary structure, respectively; mi , ci , and ki are the mass, damping coefficient, and spring constant of the i-th TMD, respectively; N is the number of TMDs; mT is the total mass of TMDs equal N to ∑i = 1 mi;xs is the displacement of the primary structure; and x i is the relative displacement between the i-th TMD and the primary structure. A dot notation signifies a differentiation with respect to time (t); u¨ g is the ground acceleration with a constant spectral intensity Su¨g given by

E [u¨ g (t ) u¨ g (t + Δt )] = 2πSu¨g δ (Δt )

(2)

where E [·] is an expectation operator, and δ (·) is the Dirac-delta function. In order to standardize the subsequent treatment, the following nondimensionalized terms were introduced:

μi =

mi , ms

γi =

ωi = ωs

ζi =

ci , 2γi mi ωs

(3a)

ki mi

ms = ks

ki −1/2 μ , ks i

(3b)

(3c) N ∑i = 1

and let μT be the total mass ratio defined by μi . Using these terms, the equations of motion given in Eqs. (1a) and (1b) become N

(1 + μT ) x¨s +

∑

μi x¨i + 2ζs ωs x ̇ + ωs2 x = −(1 + μT ) u¨ g

i=1

x¨s + x¨i + 2ζi γi ωs x i̇ + γi2 ωs2 x i = 0

i = 1, ⋯, N .

(4a) (4b)

A suitable substitution of non-dimensional variables enables us to simplify and parameterize the equations of motion. First, xs and x i can be non-dimensionalized by normalizing them with the RMS displacement of the uncontrolled structure ( x ref ). With the help of the theoretical results for the stochastic response of an SDOF system excited by a white-noise stationary process [26], the RMS displacement of the uncontrolled system can be expressed as

2. Model formulation

x ref =

2.1. Governing equations of motion

πSu¨g 2ζs ωs3

. (5)

Furthermore, on introducing non-dimensional displacements ys = xs / x ref and yi = x i / x ref , and a time scale to = ωs t , the equations of motion given in Eqs. (4a) and (4b) can be non-dimensionalized as follows:

Consider a system comprising an SDOF primary structure and N units of linear TMDs, in which a zero-mean white-noise base acceleration is exerted on the primary structure (Fig. 1). The equations of

Fig. 1. Primary structure-LMTMD system. 517

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S.-Y. Kim, C.-H. Lee N

(1 + μT ) ys″ +

∑

μi yi′ + 2ζs ys′ + ys = −(1 + μT ) wg″

Table 1 Constraints for considered LMTMD configurations.

(6a)

i=1

Configuration

ys″ + yi″ + 2γi ζi yi′ + γi2 yi = 0

i = 1, ⋯, N

(6b)

Masses

where a prime notation indicates a differentiation with respect to the non-dimensional time to , and wg″ is the non-dimensionalized ground acceleration exerted on the primary structure with its spectral intensity S wg″ given by (7)

Rearranging Eq. (6) into the matrix form yields the following expression:

My″ + Cy′ + Ky = fw g″

f = [−(1 + μT ), where y = [ys , y1, …, yN sponding matrices are given by

⎡1 M=⎢ ⎢ ⎢ ⎣

and

+ μT μ1 ⋯ μN ⎤ 1 1 ⋯ 0 ⎥, ⋮ ⋮ ⋱ ⋮⎥ 1 0 ⋯ 1⎥ ⎦

⎡ 2ζs 0 ⎢ 0 2γ ζ 1 1 C=⎢ ⋮ ⎢⋮ ⎢0 0 ⎣

⎡1 ⎢0 K=⎢ ⎢⋮ ⎢0 ⎣

0, ⋯, 0]T ,

0 γ12 ⋮ 0

⋯ ⋯ ⋱ ⋯

⋯

the

–

–

LMTMDγ

C† – C L L

–

U† U

U U – U

U U U U

0⎤ 0⎥ . ⋮⎥ ⎥ 2 γN ⎥ ⎦

Frequency ratios – L† – L C C

2.2.1. Spring constants uniformly distributed Given that the spring constants are uniformly distributed, the mass ratio of the i-th TMD, μi , is expressed in terms of the total mass ratio μT and the frequency ratios γi . Let us suppose that the spring constant of each TMD is identical to ko . Then, according to its definition in Eq. (3b), the mass ratio becomes

corre-

(9a)

μi =

0

⎤ 0 ⎥ ⋯ ⎥, ⋱ ⋮ ⎥ ⋯ 2γN ζN ⎥ ⎦

Spring constants

† : U = Uniformly distributed; C = Constrained automatically; and L = Linearly distributed.

(8)

]T ,

Damping coefficients

LMTMDo LMTMDζ LMTMDγζ LMTMDμ LMTMDμζ

Su¨g

2ζ S wg″ = 2 3 = s . π x ref ωs

Constraints

ko −2 γ ks i

i = 1, ⋯, N

(13)

where ks is the stiffness of the primary structure. On summing all the mass ratios of Eq. (13), we obtain the following expression:

μ ko = N T . ks ∑i = 1 γi−2

(9b)

(14)

On substituting Eq. (14) into Eq. (13), we obtain the expression of μi written in terms of the frequency ratios γi and a predetermined total mass ratio μT as follows:

(9c)

γi−2 μT N ∑i = 1 γi−2

With the introduction of a non-dimensional state vector z = [yT , y′ T ]T , a first-order state-space model can be formulated as follows:

μi =

z ′ = Az + Bwg″

Eq. (15) clearly shows that the mass ratios of each TMD can be completely expressed using the frequency ratios. The associated design vector γd is given by

(10)

where the corresponding matrices A and B are given by

(15)

γd = [γ1, ⋯, γN ]T .

O I ⎤, A=⎡ −1 −1 ⎣− M K − M C ⎦

(11a)

O B = ⎡ ⎤. − ⎣ f⎦

(11b)

(16)

2.2.2. Linearly distributed frequency ratios When the frequency ratios are linearly distributed (or the frequencies are equally spaced), only two frequency ratios are sufficient for determining all the frequency ratios. It can be easily shown that the frequency ratio of the i-th TMD is expressed as follows:

As the external loading wg″ is a steady-state stationary white noise of spectral strength S wg″ as previously assumed, the covariance matrix Q = E [zzT] can be obtained by solving the following Lyapunov equation [26]:

AQ + QAT + 2πSwg″ BB T = O.

i = 1, …, N .

γi = γ1 +

(12)

i−1 (γ −γ ) N −1 N 1

i = 1, ⋯, N .

(17)

It should be noted that the frequency ratios can be determined by the first and last frequency ratios. Thus, the associated design vector for the frequency ratios γd is reduced to

2.2. LMTMD configurations

γd = [γ1, γN ]T , This study considers a total of six practical LMTMD configurations: LMTMDo is a configuration in which no constraints are imposed on the frequency ratios or damping coefficients; in the case of LMTMDγ, the frequency ratios are linearly distributed; in the case of LMTMDζ, the damping constants are uniformly distributed; in the case of LMTMDγζ, the frequency ratios and damping ratios are, respectively, assumed to be distributed linearly and uniformly; in the case of LMTMDμ, the masses are linearly distributed; and in the case of LMTMDμζ, an equal damping constraint is additionally added to LMTMDμ. For all six configurations, the stiffness of each TMD is assumed to be identical. The constraints for the six LMTMD configurations are summarized in Table 1.

(18)

and the remaining frequency ratios can be determined using Eq. (17). Under this constraint, the distribution of masses is automatically determined. Fig. 2 graphically represents the spatial aspects of the constraint for the linearly constrained frequency ratios. 2.2.3. Uniformly distributed damping coefficients Given that the viscous damping coefficients of MTMDs are uniformly distributed, the associated damping ratios become proportional to the frequency ratios. Let us suppose that the viscous damping coefficient is identical to co . Then the corresponding constraint is given by

c1 = c2 = ⋯=cN = co. 518

(19)

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Fig. 2. Spatial aspects of constraint for linearly distributed frequency ratios.

Fig. 4. Spatial aspects of constraint for linearly distributed masses.

or equivalently, Eq. (15) can be rewritten in terms of normalized variables as follows:

space. In the geometrical sense, the fictitious quantity ζo can be read from the ζ value that coincides with the unity frequency ratio and can also be interpreted as a slope of the straight line.

2μ1 γ1 ωs ζ1 = 2μ 2 γ2 ωs ζ2 = ⋯=2μN γN ωs ζN = co/ ms.

(20)

2.2.4. Linearly distributed TMD masses Fig. 4 depicts the spatial distributions of the linearly distributed masses. Under this constraint, the characteristic frequency ratios become closely spaced as γi becomes smaller, because the distributions of the mass ratios of the TMDs are inversely proportional to the squared frequency ratio. The frequency ratio of the i-th TMD for the constraint is expressed as follows:

On eliminating the mass ratio μi by substituting Eq. (15) into Eq. (20), we obtain the following relationship:

γ1−1 ζ1 = γ2−1 ζ2 = ⋯=γN−1 ζN = ζo

(21)

or

ζi = γi ζo

i = 1, ⋯, N

(22)

where ζo is a fictitious damping ratio given by

ζo =

N ∑i = 1 γi−2 co

2μT ms ωs

.

μi = μ 1 +

i = 1, …, N .

(25)

On substituting Eq. (3b) into Eq. (25), we obtain

(23)

It can be observed from Eq. (22) that the damping ratio is proportional to the frequency ratio, and the only independent design variable is the fictitious damping ratio given as

ζd = ζo.

i−1 (μ −μ ) N −1 N 1

μi =

μT N ∑i = 1 γi−2

i−1 −2 −2 ⎡γ1−2 + (γ −γ1 )⎤ N −1 N ⎣ ⎦

i = 1, ⋯, N . (26)

N 1/ ∑i = 1 γi−2

into Eq. (26) and eliminating μi on On substituting μT = both sides, we obtain the following equation:

(24)

Fig. 3 shows the spatial characteristics of the constraint for equal damping coefficients. Under this constraint, the feasible points (γi, ζi ) must be located on a straight line that passes through the origin of the

γi−2 = γ1−2 +

i−1 −2 −2 (γ −γ1 ) N −1 N

i = 1, ⋯, N .

(27)

The equation implies that the mass ratios of the TMDs can be completely replaced by two frequency ratios γ1 and γN . Accordingly, the associated design vectors γd are given as follows:

γd = [γ1, γN ]T

(28)

The remaining frequency ratios can be determined using Eq. (27). 3. Optimization strategies This section introduces two different optimization strategies called NPO and RPO. In NPO, it is assumed that the only source of randomness is the loading, which can be modeled as a stochastic process while all other system parameters are treated as deterministic. In contrast, in RPO, not only is the randomness of the loading considered but also the uncertainty involved in the structural system parameters. 3.1. Nominal performance optimization The response quantities of interest in this study include the RMS displacement of the controlled main structure normalized by that of the uncontrolled one σys . Based on this definition, the non-dimensional displacement of the main structure σys should be in the range of zero to unity. It can be interpreted as a quantity for control efficiency

Fig. 3. Spatial aspects of constraint for equal damping coefficients. 519

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Furthermore, Eqs. (6a) and (6b) are non-dimensionalized in order to simplify and standardize the problem. On introducing a non-dimensionalized counterpart time scale to = ωs, p t and non-dimensional displacements ys = xs / x ref and yi = x i / x ref , the equations of motion given in Eqs. (6a) and (6b) can be reformulated as follows:

considering that σys is zero if the TMD completely suppresses the vibration of the main structure and is unity when the TMD has no effect at all. The mathematical description of the response of the quantity can be expressed as follows:

σy2s = E [ys2 ] = E [(s Tz )Ts Tz ] = tr[SQ]

(29)

N

∑

μi yi′ + 2ζs κys′ + κ 2ys = −(1 + μT ) wg″

where tr[·] is a trace operator, s = [1, is the weighting vector corresponding to sifting the structural displacement, and S is the weighting matrix that can be calculated using S = ssT . The optimization problem is formulated as follows:

(1 + μT ) ys″ +

minimize J = σys

where κ = ωs / ωs, p is a factor that quantifies the extent of natural frequency perturbation, and wg″ is the nondimensionalized ground acceleration exerted on the primary structure with a spectral intensity of

0, ⋯, 0]T

i=1

ys″ + yi″ + 2γi ζi κyi′ + γi2 κ 2yi = 0

γd, ζd

subject to γd ∈ Ωγ , ζd ∈ Ωζ

(30)

where γd and ζd are the design variable vectors defined in the previous section that correspond to appropriate constraints, and Ωγ and Ωζ are the feasible regions for γd and ζd , which are defined as positive orthants for the associated variables, respectively. In the optimization process, a feasible starting point of the design variables affects the number of function evaluations for determining the solution. One can provide the initial point based on the solutions for the aforementioned single TMD. In this study, the optimum frequency ratio and damping ratio proposed by Warburton [34], γSTMD and ζSTMD , were considered as the initial condition:

γSTMD =

ζSTMD =

1 + μT /2 1 + μT

,

μT (1 + 3μT /4) 4(1 + μT )(1 + μT /2)

S wg″ =

2ζs . π

(34)

γd, ζd

subject to γd ∈ Ωγ , ζd ∈ Ωζ

(35)

As compared to the formulated NPO problem, the RPO problem utilizes an expectation quantity to construct the objective function. Various techniques for evaluating the objective function can be adopted such as a Monte Carlo simulation [38], direct perturbation method [27,25], and response surface method [29]. From among the techniques that can be used, this study employed the point estimation method (PEM), which is a simple and efficient method for meeting the objective of this study.

(31a)

(31b)

3.2.2. Point estimation method PEM is a class of numerical methods used for evaluating the statistical moments of a given function that consists of random input variables. A typical work for this method involves determining specific points of input variables and the associated weighting factors followed by evaluating the statistical moments of the given function at the discrete points and combining all the evaluated statistical moments with the associated weighting factors for the final calculation. The numerical answer can be treated as an approximate value of the statistical moments of the given function. The PEM is effective and powerful as compared to several relevant techniques, such as the direct integration method, Monte Carlo simulation, and response surface method, especially when the associated random variables are large in number. Some of the details of the determination of ‘points’ vary depending on the number of specific points per input variable. This study deals with the 2N + 1 scheme, which requires 2N + 1 specific points per input variable. The theoretical basis of this method can be found in the studies conducted by Rosenblueth [30] and Hong [15], and its applications can also be found in Morales and Perez-Ruiz [28] and Caramia et al. [6]. Consider the objective function of the RPO problem defined using Eq. (35) and the perturbation variable of natural frequency. In this case, the perturbation factor κ is an uncertain variable with its standard derivation σκ . The two points and associated weighting factors are deand termined as κμ = 1, κ1 = 1 + 3 σκ, and κ2 = 1− 3 σκ wμ = 2/3 and w1 = w2 = 1/3, and respectively. Hence, the mean of the objective function can be evaluated using the following formula:

RPO may be viewed as an extension of NPO in that it considers the associated structural parameters as uncertain. This uncertainty may arise from various sources such as modeling error in identifying the structural properties or the random deterioration of material or structural properties over time. However, modeling the associated parameters as random yields random response quantities such as the RMS displacement as well. In order to accommodate the random variables in the optimization, the uncertain system should be distinguished from a nominal system. 3.2.1. Frequency-perturbed system Here ωs, p is defined as the perturbed natural frequency of the primary system, which is distinguished from the nominal value ωs . As done earlier, we normalize the equations of motion with respect to the displacement of the uncontrolled primary structure. By using the wellknown theoretical results for the stochastic response of an SDOF system excited by a white-noise stationary process, the RMS displacement, or the reference displacement, of the uncontrolled system can be expressed as

2ζs ωs3, p

=

minimize J = E [σys ]

3.2. Robust performance optimization

πSu¨g

xo2 ωs3, p

(33b)

In contrast to Eq. (6), the perturbation factor (κ ) in Eq. (33) allows for the consideration of the uncertainty of the natural frequency of the primary structure. Hence, the optimization problem defined in Eq. (30) for NPO should be modified as follows:

The objective function is evaluated by solving the Lyapunov equation, which can be efficiently solved using the well-established algorithm proposed by Bartels and Stewart [3], which is implemented in a commercial program such as MATLAB®. In the optimization procedure, an iterative method was adapted for solving a sequence of quadratic programming sub-problems for its superior rate of convergence. At each iteration, in order to obtain an approximation of the Hessian matrix, the Broyden—Fletcher—Goldfarb–Shanno algorithm was adopted for its effectiveness and good performance even in the case of non-smooth optimization problems [8].

x ref =

Su¨g

i = 1, ⋯, N

(33a)

2

E [σys] =

∑ k=1

.

wk σys (κk ) + wμ σys (κμ )

(36)

where σys (κk ) denotes the RMS displacement of the main structure when the perturbation factor is κk .

(32) 520

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Fig. 5. Spatial distributions of optimum variables of LMTMDs ( μT = 5 %).

4.1. LMTMDs designed using NPO

Thus, using the PEM, the expectation of the objective function can be simply estimated by computing a linear combination of the evaluated functions for the deterministic points. Each of the functions for the deterministic points can be evaluated by following the procedure described in the NPO part. In the optimization procedure, the sequence of the quadratic programming sub-problems was used, which is the same as that adopted in the NPO part.

4.1.1. Optimum parameters Fig. 5 shows the spatial distribution of the optimum frequency ratios γi and the optimum damping ratio ζi . It can be easily observed that the optimum point for the STMD corresponds to the well-known solution of Warburton [34] (γSTMD = 0.97 and ζSTMD = 0.11), because those two points are obtained by minimizing the same objective function or RMS displacement of the main structure. It can be observed from Fig. 5 that, as compared to the optimum solution for the single TMD, the bandwidth of the optimum frequency ratios becomes wider and the optimum damping ratios tend to decrease with the increase in the number of TMDs. In the case of LMTMDo with no constraints imposed, the optimum tuning condition is achieved when the frequency ratios are more densely spaced around the natural frequency of the primary structure. In addition, it is also observed that the TMD units having natural frequencies that are closer to that of the primary structure require a slightly lower damping (see Fig. 5a).

4. Results and discussions This section discusses the optimum solutions obtained on using the NPO and RPO described above. The main system is characterized by a damping ratio of 1%, and the total mass ratio of the MTMDs is varied in the range of 1–10% at intervals of 1%, although in some cases, this value is held to be 5% for discussion purposes. The number of TMDs is in the range of one to ten.

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In the case of LMTMDγ, it is observed that the optimum parameters gradually deviate from the trend of LMTMDo as the number of the TMDs increases (see Fig. 5b). Under this configuration, the TMDs located at the ends of the bandwidth require a relatively larger damping ratio as compared to that of the unconstrained condition. There is no considerable difference between LMTMDγ and LMTMDγζ except for the distribution of the frequency ratios (see Fig. 5c and d). However, as compared to LMTMDo, the optimum damping ratios are distributed in a manner that forms straight lines passing through the origin in the (γ −ζ ) space. This pattern is understandable with recalling the constraint defined by Eq. (22). As compared to LMTMDo, LMTMDμ and LMTMDμζ show quite different patterns. Fig. 5e shows the comparison between LMTMDo and LMTMDμ. In contrast to LMTMDo, a larger number of TMDs should be allocated in the low-frequency range because of the constraint relation shown in Eq. (27). It is also observed that the optimum damping ratios are significantly larger in the low-frequency range. LMTMDμζ also shows a clearly different pattern as compared to the optimum LMTMDo (refer to Fig. 5f) and requires much higher damping. The optimum frequency ratios and damping ratios for the mass ratios between 1% and 10% are depicted in Fig. 6. It can be observed that, irrespective of the MTMD configurations, the optimum frequency ratios

Fig. 7. Non-dimensional RMS displacement of main structure σys depending on LMTMD configuration.

become wider and the damping ratios decrease as the number of TMDs increases. The patterns in the (γ −ζ ) domain explained in the previous section are also observed in the case of all the considered mass ratios. 4.1.2. Comparison of control performance Fig. 7 presents the non-dimensional RMS displacements of the main

Fig. 6. Spatial representation of optimum frequency ratios and optimum damping ratios depending upon mass ratio. 522

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structure (σys ) with various LMTMD configurations. It should be noted that the possible values of σys should range from zero to unity. The unity value of σys corresponds to the uncontrolled case. It can be observed that, for all the configurations, the control performance improves with the increase in the total mass ratio μT . The number of TMDs, however, does not affect the control efficiency significantly. In particular, the control performance worsens in the cases of LMTMDμ and LMTMDμζ as the number of TMDs increases. Although not clearly distinguished in Fig. 7, it was found that the control performance is better in the order of LMTMDo, LMTMDγ, LMTMDζ, LMTMDγζ, LMTMDμ, and LMTMDμζ but the difference among the first four configurations was very small. Two main findings can be summarized as follows: (1) LMTMDo is optimal configuration while the others can be regarded as sub-optimal configurations; and (2) LMTMDs with mass ratio constraints (LMTMDμ and LMTMDμζ) are inefficient as compared to the other configurations. The performance of MTMDs depending upon the damping ratio of the primary structure is discussed below. By analogy with an SDOF system, the effective damping ratio can be expressed as shown in [7].

ζ seff =

πSu¨g 2ωs3 σx2s

=

ζs σy2s

. (37)

Fig. 8 shows the increase in the damping ratio obtained by subtracting the damping ratio of the primary structure ζs from the effective damping ratio calculated using Eq. (37). It can be observed that 1) the higher the total mass ratio μT , the better is the control performance; 2) the number of TMDs does not contribute to the control performance significantly if the mass ratios are the same, and 3) when the primary structure has a low damping ratio of approximately 2%, the TMDs exhibit good control performance but the performance degrades when the damping ratio becomes high (say 10%). 4.1.3. Total amount of damping required N The total amount of damping (cT = ∑i = 1 ci ) required is compared in Fig. 9 in order to provide information for a more efficient design. Fig. 9a shows the required amount of damping normalized by the total TMD mass times the natural frequency of the primary structure (cT / mT ωs ). Irrespective of the mass ratio and number of TMDs, the total damping required increases in the order of LMTMDo, LMTMDγ, LMTMDζ, LMTMDγζ, LMTMDμ, and LMTMDμζ. It can be observed that while there is no significant difference among the first four configurations, the cases wherein the mass ratios are constrained show a considerable increase of total damping required. It is also found that the total damping required increases as the mass ratio increases but decreases as the number of TMDs increases. The total amount of damping normalized by that of the STMD is also shown in Fig. 9b. As compared to the STMD, the amount of damping decreases irrespective of the mass ratio in the case of LMTMDo, but the extent of this decrease becomes

Fig. 9. Comparison of total amount of damping required.

slightly larger under sub-optimum conditions.

4.1.4. Comparison of transfer functions To illustrate the efficiency of the considered MTMDs, a more detailed study was conducted for a primary structure with a damping ratio of 1% equipped with an MTMD of 10 units with a total mass ratio of 5%. Fig. 10 compares the frequency response functions (FRFs). Although there exist some minor differences in their shapes, all the LMTMDs considered effectively reduce the response while exhibiting N + 1 well-separated local modes. Fig. 10a compares the FRFs for LMTMDo and LMTMDγ, which shows little difference. Moreover, no appreciable difference is observed between LMTMDζ and LMTMDγζ (see Fig. 10b and c). Based on these comparisons, it can be concluded that there is no considerable difference in control performance once a constraint on the frequency ratios is included. As compared to LMTMDo, both LMTMDζ and LMTMDγζ differ in that a sharp peak appears in the low frequency range and the subsequent peaks become less sharp as the frequency increases (see Fig. 10b and c). This characteristic shape is a result of the use of a damping constraint with features that were identical to those discussed in the preceding section on the damping coefficient restraint. However, those for mass-constrained models (LMTMDμ and LMTMDμζ) exhibit an irregular pattern, and the maxima of the FRFs are considerably higher than those of the other cases, which means that, even if optimized, the mass constraint approach is not effective in reducing the vibration. The control efficiency worsens when both the mass ratio and damping ratios are constrained (LMTMDμζ).

Fig. 8. Effective damping increase. 523

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Fig. 10. Comparison of FRFs for various LMTMD configurations ( μT = 5% and N = 10 ).

mass side should be selected.

4.2. LMTMDs designed using RPO The optimum parameters obtained using RPO are discussed in this section. Among the possible uncertain system parameters, this study is focused on the variation in the natural frequency of the primary structure. The unique feature of the optimum parameters and frequency responses obtained using RPO are discussed and compared with the NPO solutions.

4.2.2. Comparison of transfer functions The transfer functions obtained using NPO and RPO are further discussed using the case study results shown in Fig. 12. Both solutions indicate that an increase in the total mass ratio provides a more robust control. As compared to the case of μT = 2% and N = 3, a heavier LMTMD ( μT = 5% and N = 3) is more effective in suppressing the frequency response as discussed in the preceding section. Increasing the number of TMDs also affects the control performance — although not as significant as the mass ratio — especially when the extent of perturbation is large. The performance of the system obtained on increasing the number of TMDs is enhanced because each TMD unit, which is finely divided, has more freedom in dissipating vibrational energy. Under the same conditions, as compared to the NPO solutions, the RPO solutions more effectively suppress the frequency responses when the natural frequency is perturbed to higher frequency side.

4.2.1. Optimum parameters The optimum parameters obtained using RPO and NPO are compared as shown in Fig. 11, in which a 10% perturbation is assumed for the natural frequency of the primary structure. Firstly, a general trend was observed as per which, when the natural frequency perturbation of the primary structure is considered, the optimum frequency band becomes wider and the optimum damping ratio increases. Such a trend can be observed more clearly by examining the FRFs in greater detail as depicted in Fig. 12. An interesting observation is that the optimum line obtained using RPO under a light mass ratio coincides with that obtained using NPO under a heavier mass ratio (see Fig. 12b). This implies that both increasing the mass ratio and securing the robustness result in a similar mechanism in suppressing the FRF. If an attempt is made to determine the optimum parameters for a pre-selected mass ratio but with increased robustness, the optimum parameters for the heavier

4.3. Approximate solution for LMTMDγζ In general, optimum solutions cannot be simply formulated in explicit form because of the existence of numerous design variables. The optimum parameters for LMTMDγζ, however, can be determined using 524

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Fig. 11. Comparison of optimum parameters obtained using NPO and RPO ( μT = 5% and 10% perturbation of the natural frequency is assumed).

various mass ratios and number of TMDs required to derive an approximate optimum formula for LMTMDγζ. As can be observed from the figure, the slopes m1 and mN tend to decrease with an increasing number of TMDs and mass ratio, while the fictitious damping ratio ζo decreases exponentially. Based on these observations on the spatial distribution of the optimum parameters, an attempt was made to derive the expressions for m1, mN , and ζo using multiple regression. The regression coefficients are summarized in Table 2.

just three design variables: two variables for the frequency ratios γ1 and γN , and a fictitious damping ratio ζo . For practical design purposes, it is desirable to provide the approximate design equation for this case because LMTMDγζ shows a control performance that is comparable to that of the optimal LMTMDo. The geometric relationship of the design parameters for LMTMDγζ is shown schematically in Fig. 13. Firstly, the backbone curve is provided as a skeleton for determining the optimum parameters of the STMD, which is adopted from Warburton [34]. For a given μT , the plane that is normal to the backbone curve and parallel to the (γ −ζ ) plane is formed such that one can determine the optimum frequency ratio and optimum damping ratio for the STMD (point A in Fig. 13; refer to Eq. (31)). Furthermore, on that plane, three straight lines l1, l2 , and l3 of which the slopes are respectively m1, mN , and ζo determine two intersection points B1 and BN . Hence, all the optimum points can be defined by internally dividing the line with slope ζo by N equal internals. Fig. 14 shows the parameters m1, mN , and ζo depending on the

m1 = p0 + p1 μT + p2 exp[−p3 (N −1)]

(38a)

mN = p0 + p1 μT + p2 exp[−p3 (N −1)]

(38b)

p

ζo = ζSTMD exp[p0 μT 1 (N −1) p2].

(38c)

By using the slope-intercept form of the equation of a line, two optimum frequency ratios (γ1 and γN ) and the corresponding damping ratios (ζ1 and ζN ) can be calculated using Eq. (39) (see points B1 and BN 525

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Fig. 12. FRFs under perturbation of natural frequency.

Fig. 14. Shape parameters for approximate optimum parameters for LMTMDγζ.

in Fig. 13). Finally, the optimum parameters of each TMD can be determined by internally dividing the values γ1 and γN , and ζ1 and ζN with N equal internals. Fig. 13. Spatial representation of design parameters m1, mN , and ζo for LMTMDγζ.

γ1 =

526

ζSTMD−m1 γSTMD ζo−m1

; ζ1 = γ1 ζo,

(39a)

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Table 2 Design parameters and regression coefficients. Parameter

m1 [Eq. (38a)] mN [Eq. (38b)] ζo [Eq. (38c)]

γN =

Coefficient

p0

p1

p2

p3

0.5742 −0.4306 −0.4948

2.3400 0.7147 −0.0458

0.2812 0.0469 0.4395

0.2806 0.2153 –

ζSTMD−mN γSTMD ζo−mN

R2

; ζN = γN ζo,

0.9723 0.9414 0.9919

(39b)

5. Stroke consideration and design recommendations 5.1. Stroke consideration For design purposes, it is desired to determine the maximum stroke of TMDs to encase the moving masses. Especially when designing the MTMD, because the required damping exponentially decreases on increasing the number of TMDs, sufficient space to accommodate the TMD motion should be provided in the housing. Lin et al. [23] developed a two-stage design procedure while considering both structural and TMD responses. In terms of H∞ norm, limiting the maximum stroke of the MTMDs is helpful for both the original design purpose itself and decreasing the peak point of the FRF within an appropriate weighting region. Fig. 15a depicts the RMS displacement of the main structure in its ordinate and the maximum RMS displacement of the TMDs in its abscissa for LMTMDo and LMTMDγζ. As stated in the previous section, for the performance comparison, both the control performance and the maximum RMS displacement of the two TMDs are similar. In the case of the TMD stroke, the maximum RMS displacement rapidly increases on increasing the number of TMDs. Such a trend is predictable when recalling the dependency between the number of TMDs and the optimum damping ratios. For practical design purposes, it is more convenient to transform the information in Fig. 15a into another (N −μT ) space in a contour form (Fig. 15b). Fig. 15b can be used as a design contour for evaluating the control efficiency and for limiting the maximum TMD stroke.

Fig. 15. Performance and stroke grid for design.

5.2. Design recommendations

stroke. 5. It should be checked if the stroke of the LMTMDs selected at the previous step exceeds the prescribed stroke limitation. If it is exceeded, two solutions are possible: the number of TMDs should be reduced or the chamber should be made sufficiently spacious to accommodate the stroke.

Some useful design recommendations are presented below for the LMTMDs discussed in this paper. The use of LMTMDγζ is recommended in design. 1. The dynamic properties such as the structural mass ms , structural damping coefficient cs , and structural stiffness ks should be determined. These can be either assumed appropriately or estimated using field measurements. 2. The anticipated structural response should be evaluated, for instance, the structural displacement xs , and the performance level to be attained should be determined. N 3. The total mass ratio of the LMTMDs μT = ∑i = 1 μi based on the relationship between the total mass ratio and controlled structural RMS response normalized to the uncontrolled one should be determined (see Fig. 7). 4. The optimum frequency ratios and optimum damping ratios must be selected using Eqs. (38) and (39). Do not struggle with choosing the total number of TMDs from performance perspective, since it does not significantly affect vibration mitigation level. However, it should be noted that increasing the number of TMDs reduces the space required for the installation of each unit and the total damping required for optimum performance but results in a larger TMD

6. Summary and conclusions MTMDs, which refers to a system comprising multiple units of TMDs in which each TMD may have different dynamic characteristics, is a viable solution when small-sized MTMDs should be installed owing to practical reasons such as space limitations, transportation, and ease of handling. In this study, an attempt was made to develop a general design guide for optimum linear MTMDs with various configurations on a SDOF primary structure subjected to a zero-mean white-noise baseacceleration excitation. Six constraint conditions are considered in this study including constraints on the frequency ratio, damping ratio, and mass distribution. The optimum frequency and damping ratios were first obtained using NPO or RPO, and the control efficiency was then comprehensively evaluated. The results of this study can be summarized as follows.

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1. As compared to the optimum solution of the single TMD, the bandwidth of the optimum frequency ratios for LMTMDs is wider and the optimum damping ratios decrease on increasing the number of TMD units irrespective of the constraint conditions imposed. 2. It was found that the control efficiency is better in the order of LMTMDo, LMTMDγ, LMTMDζ, LMTMDγζ, LMTMDμ, and LMTMDμζ although the difference among the first four is small. LMTMDs with mass constraints (LMTMDμ and LMTMDμζ) were shown to be less efficient and should not be used. 3. The effects of constraint conditions and number of TMD units on the total amount of damping required were also analyzed to provide information for a more efficient design. 4. The comparison of the optimum parameters obtained using NPO and RPO (while assuming the natural frequency perturbation of 10%) indicated that the optimum frequency band becomes wider and the optimum damping ratio increases when RPO is adopted. 5. On noting that LMTMDγζ exhibits a control efficiency comparable to the optimal LMTMDo with a great reduction in the number of design variables, this study proposed an approximate formula for obtaining the optimal parameters of LMTMDγζ.

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