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Analysis and optimization of multiple tuned mass dampers with coulomb dry friction
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Analysis and optimization of multiple tuned mass dampers with coulomb dry friction Sung-Yong Kima, Cheol-Ho Leeb, a b
⁎
School of Architecture, Changwon National University, Changwon, Republic of Korea 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 Frictional tuned mass damper Statistical linearization Passive control
The optimum design and analysis of frictional multiple tuned mass dampers (FMTMDs) are presented in this study, where Coulomb dry friction is incorporated as an energy dissipation mechanism. To deal with the nonlinearity of the Coulomb dry friction force, which is a source of difficulty in the optimization process, this study adopted a statistical linearization that replaces the original nonlinear system with an equivalent linear system, and found an optimum design that minimized the root mean square (RMS) displacement of the primary structure. The equivalent damping ratio of a tuned mass damper (TMD) unit significantly decreased with the increase in number of TMDs. Four practical configurations were formulated and comparatively analyzed, by constraining the frequency ratios and the coefficient of friction (COF). Among the four investigated cases, a particular solution described using only three parameters is recommended for use in design, because it is simple and sufficiently accurate. We also analyzed the input sensitivity of nonlinear frequency responses of FMTMDs in order to provide general design guidance.
1. Introduction A tuned mass damper (TMD) is a passive control device that consists of mass, spring, and energy dissipation elements mounted to a structure to dampen its dynamic response. Traditionally, viscous dampers were used as energy dissipation devices, and TMDs with viscous dampers, or linear TMDs, have been widely studied by various researchers over the past decades. The proposed solutions have been widely applied in both research and field applications [1–3]. Linear TMDs have been utilized in a wide variety of applications to resolve vibration problems in lightweight floors, observatory towers, high-rise buildings, and other structures. Recently, new types of TMDs have been actively investigated by various researchers [4–8]. Despite their simplicity and effectiveness, linear TMDs have several drawbacks. For example, a viscous damper is vulnerable to performance degradation caused by aging. With repetitive operations over a long lifetime, the dashpot could malfunction because of temperature rise due to repetitive operation, or there may be reduced performance caused by leakage. To facilitate robust control performance, some researchers have attempted to incorporate a Coulomb-type friction mechanism into the TMD as an energy dissipator. Ricciardelli and Vickery [9] studied optimum nonlinear TMDs equipped with dry friction damping, and proposed a design formula for the optimum design. In addition, they ⁎
showed that the optimum parameters of nonlinear TMDs depended on the level of input excitation as well as the structural response. Inaudi and Kelly [10] proposed a friction-dissipating TMD in which the dissipative force was exhibited by friction dampers that acted in a direction perpendicular to the direction of motion. They demonstrated that with an appropriate design, the nonlinear system can exhibit the same level of control performance compared to its counterpart linear TMD. Gewei and Basu [11] investigated the effectiveness of a single TMD with dry friction using a harmonic and statistical linearization solution. Their results showed that a friction TMD is as effective as linear TMDs, and demonstrated that the optimal friction coefficient for the TMD was dependent on the intensity of the excitation. Rüdinger [12] investigated TMDs with power-law type nonlinear viscous damping elements, and showed that the optimum parameters for nonlinear TMDs depended on the level of displacement amplitude and input intensity, which is in contrast with linear TMDs. Wang [13] and Carpineto et al. [14] also developed TMDs that could dissipate input energy via a frictional hysteretic mechanism. In addition, multiple tuned mass dampers (MTMDs) have been investigated by various researchers. Linear MTMDs (LMTMDs) with equally spaced natural frequencies and identical viscous damping constants were studied by Xu and Igusa [15]. They demonstrated that under these conditions, the LMTMD was efficient in attenuating the
Corresponding author. E-mail address: [email protected] (C.-H. Lee).
https://doi.org/10.1016/j.engstruct.2019.110011 Received 12 July 2019; Received in revised form 24 October 2019; Accepted 26 November 2019 0141-0296/ © 2019 Published by Elsevier Ltd.
Please cite this article as: Sung-Yong Kim and Cheol-Ho Lee, Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.110011
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Unfortunately, the actual friction mechanism is not as simple and the Coulomb theory is only an approximation in practice. Although the kinetic friction coefficient τk in Eq. (1) is often assumed to be a constant, it is not exactly constant for a given relative contacting surface velocity. However, it should be noted that the approximation is a fundamentally empirical construction used for its simplicity and versatility. Furthermore, the friction force does not exhibit an instantaneous change in magnitude when the velocity is zero. Due to the discontinuity of the model at zero velocity, the Coulomb friction model results in stiff equations of motion, yielding numerical challenges such as computational burdens [35]. To minimize the computational difficulty associated with the discontinuity at zero velocity, the kinetic friction coefficient is approximately treated as being equal to the static friction coefficient [36]. Although the relationship assumed between the normal force and frictional force is not exact, the Coulomb friction model is still an acceptable representation of friction for the analysis of many physical systems, particularly those used in the design of friction TMDs [9–11,37–39].
response of the primary structure. MTMDs with equally spaced natural frequencies and equal damping ratios have also been investigated by various researchers [16–22]. Notable research works have also been conducted to investigate LMTMDs with more relaxed constraints [23–34]. These studies can be distinguished from each other in the following points: 1) the input condition considered, 2) the optimization criteria, and 3) the optimization strategies. The aforementioned studies also discussed the advantages of conventional MTMDs with viscous dampers. The results of these studies confirm that MTMDs exhibit better performance than a single TMD. In addition, they demonstrated a more robust performance when subjected to changes in the natural frequency of the primary structures under wide-band excitation. Meanwhile, frictional MTMDs (FMTMDs) can be a very effective control device, because they take advantage of LMTMDs over a linear single TMD while eliminating the disadvantages of viscous TMD. FMTMDs have the advantage of easy transportation or installation owing to the light weight of the TMD units. Additionally, they do not require a viscous damping device for each TMD units. However, to the best of our knowledge, there are no reported studies on the implementation of MTMDs with a friction mechanism and the optimal design for various configurations has not been examined either. In this study we propose using MTMDs with a Coulomb dry friction force which can exhibit more robust performance by combining the advantages of MTMDs and frictional mechanism. In this investigation, constrained optimization was performed for some configurations that are considered to be practical in real applications, such as springs of equal stiffness, constant damping elements, or equally spaced frequencies. In this study, the optimum TMD design was investigated, where the natural frequency and coefficient of friction (COF) values were constrained. To determine the optimal solution of a system with nonlinear frictional behavior, equivalent linear systems are constructed based on statistical linearization. Sensitivity analyses of the input strength were also conducted using a linearization technique to overcome the limitation of the solution.
3. Problem statement 3.1. Governing equations of motion Suppose that a system consists of a primary structure and N units of FMTMDs, where each unit of mass is connected via a spring and a friction element in parallel (see Fig. 2). The equations of motion of the system can be written as follows: N
(ms + mT ) x¨s +
mi x¨i + cs xṡ + ks xs = fs
i=1
mi (x¨s + x¨i ) + ki x i + gi = 0
i = 1, ⋯, N
(2a) (2b)
where ms , cs and ks are the mass, damping coefficient and spring constant of the primary structure; mi and ki are the mass and spring constant of the i-th TMD; gi is the frictional force that arises from the relative motion during contact with the surface between the primary structure and the i-th TMD; N is the total number of the TMDs; mT is the N total mass of TMDs, which is equal to ∑i = 1 mi;xs is the displacement of the primary structure, and x i is the relative displacement between the ith TMD and the primary structure. A dot notation denotes a derivative with respect to time, t. The external force exerted on the primary structure is denoted by fs . The Coulomb-type force term gi is modeled as shown in Fig. 3. The force term can then be modeled as follows:
2. Brief review of coulomb dry friction Coulomb model is the most commonly used model for modelling dry friction, which presumes that the static friction force must be overcome to initiate contact motion between two solid surfaces in a state of rest, and the kinetic friction is independent of the relative velocity of the contacting surfaces. According to these presumptions, the Coulomb model is analytically expressed as follows (see Fig. 1):
Fs ⩽ τs Fn v=0 F=⎧ F τ F sgn( v ) v = ≠ 0 ⎨ k n ⎩ d
∑
(1)
gi = gi (x i̇ ) = gio sgn(x i̇ ) = τi mi g sgn(x i̇ )
where v is the relative velocity between the two surfaces, τs is the static friction coefficient, τk is the kinetic friction coefficient, and Fn is the normal force.
(3)
where gio is the characteristic friction force of the i-th TMD, τi is COF, g is the gravitational acceleration, and sgn(·) denotes a signum function. In this study, it is assumed that each force term fi is neglected, and the entire system is subjected to a white-noise base acceleration. In this case, the external force term, fs , becomes − (ms + mT ) u¨ g , where u¨ g is the ground acceleration with a spectral intensity Su¨g given as:
[u¨ g (t ) u¨ g (t + Δt )] = 2πSu¨g δ (Δt ),
Fig. 1. The Coulomb friction model.
Fig. 2. Structure-FMTMD system. 2
(4)
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ηi =
gio / μi
=
ms ωs2 x ref
1 τi. ωs2 x ref / g
(11)
Eq. (8) can be rearranged into a matrix form as follows:
My ″ + Cy ′ + Ky + ψ = fwg″
(12)
y = [ys , y1, …, yN ]T , ψ = [0, ψ1, …, ψN ]T , f = [−(1 + μT ), 0, …, 0]T
where and
⎡1 M=⎢ ⎢ ⎢ ⎣
+ μT 1 ⋮ 1
⋯ 1 ⋮ 0
μN ⎤ ⎡ 2ζs ⋯ 0 ⎥, C = ⎢ 0 ⎢⋮ ⋱ ⋮⎥ ⎢ ⋯ 1⎥ ⎦ ⎣0
0 0 ⋮ 0
⋯ ⋯ ⋱ ⋯
⎡ 1 02 0⎤ 0 γ1 0 ⎥, K = ⎢ ⎢⋮ ⋮ ⋮⎥ ⎢ ⎥ 0⎦ ⎢0 0 ⎣
⋯ ⋯ ⋱ ⋯
0⎤ 0⎥ . ⋮⎥ ⎥ γN2 ⎥ ⎦ (13)
Fig. 3. Idealized Coulomb-type frictional force.
3.2. FMTMD configurations
where δ (·) is the Dirac-delta function, and [·] is an expectation operator. The non-dimensionalized terms were introduced to standardize the subsequent treatment as follows:
μi =
mi , ms
γi =
ωi = ωs
Four FMTMD configurations are considered in this study: FMTMDo is a configuration in which no constraints are imposed on either the frequency ratios or the COFs; FMTMDγ represents the case whereby the frequency ratios are identical; FMTMDτ denotes that the COFs are identical; FMTMDγτ is the case in which the frequency ratios and the COFs are equally spaced and identical. For all four configurations, the spring constants for each TMD are assumed to be identical. The FMTMD configurations considered in this study are tabulated in Table 1.
(5a)
ki mi
ms = ks
ki −1/2 μ ks i
(5b) N by ∑i = 1
μi and Eqs. and μT is the ratio of the total TMD mass determined (2a), , and ωi = ki/ mi and ωs = ks / ms are the natural frequencies of the i-th TMD and the main structure, respectively, and (2b) can be expressed as follows:
3.2.1. Identical spring constants As the spring constants of the MTMD are assumed to be identical, the mass ratio of the i-th TMD, μi , can be defined by the total mass ratio μT and the frequency ratios γi . If the spring constants for all TMDs are identical to ko , the mass ratio μi is determined as follows:
N
(1 + μT ) x¨s +
∑
μi x¨i + 2ζs ωs xṡ + ωs2 xs = −(1 + μT ) u¨ g
i=1
x¨s + x¨i + gi / mi + γi2 ωs2 x i = 0
i = 1, ⋯, N .
(6a)
μi = =
(6b)
To reformulate the governing equations in terms of the non-dimensional form, xs and x i were non-dimensionalized by normalizing them with the RMS displacement of the uncontrolled structure x ref . With the help of the random vibration theory, the RMS displacement of the uncontrolled system is determined using [40]:
x ref =
πSu¨g 2ζs ωs3
. (7)
γj = γ1 +
∑
μi yi″ + 2ζs ys′ + ys = −(1 + μT ) wg″
i=1
ys″ + yi″ + ψi + γi2 yi = 0
i = 1, …, N
Su¨g
=
2ζs . π
gi (x i̇ ) mi ωs2 x ref
(14)
j−1 (γ − γ1) for j = 2, …, N . N−1 N
(15)
(8b)
ψi = ψi (yi′) =
τ sgn(yi′) i = 1, …, N ωs2 x ref / g
(16)
where τ represents the identical COF. Table 1 Constraints for considered FMTMD configurations. Configuration
Constraints
(9)
The non-dimensional friction force term ψi is defined as follows:
ψi = ψi (yi′) =
i = 1, …, N .
(8a)
where the prime denotes the differentiation with respect to the nondimensional time to , and wg″ is the non-dimensional ground acceleration subjected to the primary structure, of which spectral intensity is determined as: 2 x ref ωs3
μT
3.2.3. Identical coefficients of friction (COFs) Given that the nonlinear friction force arises from the TMD and the main structure with the identical COF, the non-dimensional frictional force can be determined as follows:
N
Swg″ =
Σγi−2
3.2.2. Equally spaced frequency ratios When the frequency ratios are equally spaced, only two frequency ratios are required to determine the entire frequency ratio. In this case, any of the frequency ratios can be expressed in terms of the first and Nth frequency ratios γ1 and γN as follows:
Furthermore, by introducing the non-dimensional displacements ys = xs / x ref and yi = x i / x ref , and a time scale to = ωs t , Eqs. (2) can be rewritten as follows:
(1 + μT ) ys″ +
γi−2
= ηi sgn(yi′)
FMTMDo FMTMDγ FMTMDτ FMTMDγτ
(10)
where the normalized characteristic friction force term is determined by substituting Eq. (3) by Eq. (10) as follows:
†
3
Frictional coefficients
Spring constants
Frequency ratios
– –
I† I
– E†
I I
I I
– E
I = Identical and E = Equally spaced.
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Fig. 4. Non-dimensional friction force ψi and its derivative ∂ yi′ ψi .
Fig. 5. Optimum parameters of FMTMDs (ζs = 1% and μT = 5 %).
Fig. 4. Assuming that the responses of the equivalent system are stationary zero-mean Gaussian processes, the non-dimensional velocity of the i-th TMD, yi′, becomes Gaussian with its variance σ yi′. Consequently, the equivalent damping coefficient cieq + 1 can be evaluated using the sifting property of the Dirac delta function as follows:
3.3. Statistical linearization Statistical linearization is an approximation technique, which replaces a nonlinear system by equivalent linear equations in a statistical sense [41–43]. Using the statistical linearization technique, the nonlinear force term ψ in Eq. (21) can be replaced with an equivalent linear term that minimizes the mean square of the error [ε Tε ], where the error ε is defined by
ε = ψ − Ceqy ′
(17)
(19)
Rewriting Eq. (19) in terms of gio and τi gives:
where Ceq is a parametric matrix, which can be determined by applying the first-order necessary condition for optimality as follows:
⎡ ∂ψi ⎤ i = 1, …, N cieq + 1 = [∂ yi′ψi] = ⎢ ⎥ ⎣ ∂yi′ ⎦
2 1 η . π i σ yi′
eq cieq + 1 = 2γi ζi =
ηi =
gio / μi ms ωs2 x ref
=
τi = ωs2 x ref / g
2π γi ζieq σ yi′.
(20)
Hence, the matrix equation for the equivalent linear system can be rewritten as follows:
(18)
where ψi is the nonlinear force of the i-th TMD. The normalized Coulomb-type frictional force ψi and its derivative ∂ yi′ ψi are depicted in
My ″ + (C + Ceq) y ′ + Ky = fwg″ 4
(21)
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10% Mass ratio ȝT, %
Mass ratio ȝT, %
10% Design Curve 5% for STMD ȝT = 2%
5% ȝT = 2%
γ
10% Mass ratio ȝT, %
Mass ratio ȝT, %
10%
5% ȝT = 2%
5% ȝT = 2%
Fig. 6. Optimum frequency and equivalent damping ratios of FMTMDs depending on mass ratio (ζs = 1% ).
10% Mass ratio ȝT, %
Mass ratio ȝT, %
10%
5% ȝT = 2%
5% ȝT = 2%
γ
o
10% Mass ratio ȝT, %
Mass ratio ȝT, %
10%
5% ȝT = 2%
5% ȝT = 2%
Fig. 7. Optimum frequency ratios and COFs of FMTMDs depending on mass ratio (ζs = 1%).
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Fig. 9. Normalized frequency response functions depending upon input intensity (ζs = 1%, μT = 5% and N = 10 ).
O B = ⎡ ⎤. ⎣f ⎦
(24b)
Since the external loading wg″ is assumed as stationary white noise with spectral strength of Swg″, the covariance Q = [zzT] can be evaluated by solving the following Lyapunov equation [40]:
AQ + QAT + 2πSwg″ BBT = O.
Fig. 8. Comparison of FRFs when subjected to the targeted input level (ζs = 1%, μT = 5% and N = 10 ).
However, it is not straightforward to calculate the responses of the equivalent linear system, because the damping matrix Ceq consists of the non-dimensional velocities of TMDs. Hence, it is first required to assume the initial values of the equivalent damping matrix and to then iterate the response by solving Eq. (23).
where the matrices M, C , and K are already defined in Eqs. (13) and the equivalent damping matrix Ceq is defined as follows:
Ceq
⎡ 0 0eq ⎢ 0 c2 =⎢ ⋮ ⋮ ⎢ ⎣0 0
⋯ 0 ⎤ ⋯ 0 ⎥ ⋱ ⋮ ⎥ ⎥ ⋯ cNeq+ 1⎦
By introducting a non-dimensional state vector z = space model is formulated as follows:
z′ = Az + Bwg″
4. Optimization strategies (22)
[yT ,
In this study, the RMS displacement of the primary structure is minimized, which is expressed as follows:
T T
y ′ ] , a state-
σy2s = [ys2 ] = [(sTz)TsTz] = tr[SQ] (23)
(26)
where tr[·] refers to a trace operator, s = [1, 0, ⋯, 0]T is a weighting vector which sifts the displacement of the primary structure, and S is the weighting matrix defined as S = ssT . In the optimization process, this study set the classical solution for a single TMD (STMD) proposed by Warburton [3] as the initial value,
where
O I ⎤ A=⎡ −1 −1 eq , ⎣− M K − M (C + C ) ⎦
(25)
(24a) 6
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Fig. 10. Frequency response functions depending upon input intensity (ζs = 1%, μT = 5% and N = 10 ).
which appeared to converge stably to the optimum point for all considered cases. The Warburton’s formula used for the frequency ratios and damping ratios are as follows:
γSTMD =
1 + μT /2 1 + μT
eq , ζSTMD =
μT (1 + 3μT /4) . 4(1 + μT )(1 + μT /2)
(27)
Fig. 11. Maximum of frequency responses depending upon input intensity (ζs = 1% and μT = 5%).
When the initial conditions were used, the convergence was stable to a point for all the considered cases. The objective function can be evaluated by solving the Lyapunov equation using well-established algorithms (for instance, see [44]). For the optimization process, the Broyden-Fletcher-Goldfarb-Shannon algorithm was utilized because of its effectiveness and good performance [45]. Followed by the optimization process, the nonlinear parameters inversely correspond to the identified optimum equivalent damping ratios. It should be noted that the expression for the nonlinear term is ∗ explicit because σ yi,L ′ is independent of the nonlinear force ηi , but is determined using the parameters of the equivalent linear system, γi∗ and ζieq ∗. Hence, seeking the optimal nonlinear parameter ηi implies finding the optimal parameters of the linearized system, which can be accomplished using the optimization procedure for linear systems. In the cases of FMTMDo and FMTMDγ , the optimal level of frictional force and the associated COF can be directly determined from the optimum solution of the equivalent linear system. As for FMTMDτ and FMTMDγτ , it is tedious to determine the optimum solution because all of the friction coefficients are restricted to be identical. Therefore, a constrained optimization scheme should be adopted.
out in this section. The damping ratio of the primary structure system was assumed to be 1%, and the total mass ratio of the FMTMDs varied in the range of 1% to 10% at an interval of 1%. Note that the distribution of the optimal solution may change depending on the assumed damping ratio of the primary structure [46]. However, in this study, the optimal solution for a primary structure with 1% damping was presented, considering that primary structures experience vibration problems usually exhibited at low damping ratios. The number of TMDs considered was in the range of 1–10.
5.1. Optimum parameters Fig. 5 depicts the spatial distributions of the optimum frequency ratio γi and the optimum equivalent damping ratio ζieq for the considered constraints. It can be observed that, with an increasing number of TMDs, the frequency bandwidth became wider, and the equivalent damping ratios decreased. In the case of FMTMDo (see Fig. 5a), the optimum tuning condition was achieved when the frequency ratios were more closely spaced in the vicinity of the natural frequency of the primary structure. As for FMTMDγ , it was observed that the optimum parameters gradually deviated from those of FMTMDo with an increase in the number of TMDs (see Fig. 5b), and the TMDs located at the ends
5. Results and discussions A numerical study for the optimum values of FMTMD was carried 7
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Fig. 14. Normalized frequency response functions of FMTMDo depending upon input intensity and frequency variations (ζs = 1% and μT = 5%).
Fig. 12. Comparison of the normalized displacement time histories of FMTMDo and LMTMD depending upon the input intensity and the number of TMDs (ζs = 1% and μT = 5%).
Fig. 15. Spatial representation of design parameters m1, mN and τ for FMTMDγτ .
increase. The optimum frequency ratios and optimum COFs are presented in Fig. 7. It is evident that as the mass ratio increases, the optimum COF of FMTMDs becomes significantly lower. It was also found that the optimum COF becomes asymptotically smaller as the number of TMDs increases. It should be noted that the optimum COFs were normalized to Xref , which means that the optimal solution depends on the input level. Note that the spatial distribution of optimum parameters for a given mass ratio lies on the straight lines for FMTMDτ and FMTMDγτ .
Fig. 13. Comparison of normalized RMS displacements of the main structure depending upon input intensity (ζs = 1% and μT = 5%).
of the bandwidth required higher equivalent damping compared to the unconstrained condition. No appreciable differences were found between FMTMDτ and FMTMDγτ (see Fig. 5c and 5d). However, compared to FMTMDo , the optimum equivalent damping ratios were distributed such that the TMDs with low frequency ratios required higher damping. The distributions of the optimum parameters for the mass ratio between 1% and 10% are presented in Fig. 6. It is evident from the figure that in comparison to the single TMD, the MTMD requires a significantly low level of equivalent damping ratio with the increase in the mass ratio. It is also evident that the optimum frequency ratios become wider as the number of TMDs and the total mass ratio μT
5.2. Frequency response with optimum parameters Fig. 8 compares the frequency response functions (FRFs) for the displacement of the primary structure. It can be observed that all considered FMTMD configurations can effectively reduce vibrations when N + 1 well-separated local modes are exhibited. Fig. 8a indicates 8
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5.3. Input intensity sensitivity analysis Given that the nonlinear nature of the FMTMD-structured system used in this study cannot be described by an impulse response (or by the law of superposition), the possible variation of the response depending on the input intensities should also be investigated. The frequency responses described in the preceding subsection are valid only when the input intensity corresponded to the target or design level. Although the design is usually performed to meet a target performance against a design load level, the actual loadings (which can be higher or lower than those assumed in the design), may also excite the structure. The effect of this loading cannot be linearly interpolated or extrapolated because of the nonlinearity of the problem. Next, the sensitivity analyses of the input intensity are presented. A primary structure with a damping ratio of 1% equipped with ten units, whose total mass ratio was 5%, was selected for this sensitivity study. The analysis was performed at input intensities ranging from 0.5 to 1.5 times the design intensity at 0.1 intervals. Two configurations FMTMDo and FMTMDγ τ were considered. Fig. 9 summarized the FRFs of the considered FMTMDs when subjected to varying input intensities. The frequency response curve became flat when a low-level input was applied, and as the input level increased beyond the design input level, the peaks of the frequency responses became sharper. Considering that the lower peak of the magnitude of the frequency response implies the reduction of the worstcase response, the results suggest that TMD units move in a “sticky” manner if the input intensity is smaller than the level considered in the design. The change in the frequency response curve depending on the input level can be inferred from Eq. (19): When the RMS acceleration of the TMDs (σ yi′) is lower in comparison to the target input, the value of the equivalent damping ratio (ζieq ) increases. Comparing Figs. 9a and 9b for FMTMDτ , a configuration where the COF was restricted to be identical, it exhibited an inferior performance as compared to FMTMDo . Figs. 10a and 10b depict the maximum of FRFs for FMTMDo and FMTMDγτ , respectively. It is evident that for both FMTMD configurations, the shape of the curves was more oscillatory if the loading level exceeded the design level. As implied in Eq. (19), the equivalent damping ratio is inversely proportional to the normalized RMS velocity of the TMD response. Thus, the increased TMD unit response (due to the increased input beyond the design level) eventually leads to a reduction in the control performance. Fig. 11 shows the maximum frequency responses as a function of the input level for the FMTMDo and FMTMDγτ configurations. In both FMTMDs, the maximum frequency response decreased as the number of TMDs increased for any input intensity. It was also observed that the increase in the maximum frequency response is slightly larger in FMTMDγτ than in FMTMDo . In addition, when the input level is higher than the target value, the maximum frequency response increases slightly more than the increase in the input level. Therefore, it is recommended to set the target input intensity at a level that is higher than the nominal design level for the robust control performance of the FMTMDs. A time domain analysis was also conducted to demonstrate the reduction in the vibration of FMTMDo over time. Fig. 12 presents the time histories of the normalized displacements of the primary structure subjected to white-noise excitation for three input intensities: lower forcing amplitude (0.5 times the design input intensity), intended forcing amplitude (the design intensity), and higher forcing amplitude (1.5 times the design input intensity). The effect of the number of TMD units is also included in this figure. Overall, the equivalent linear model (LMTMD) provides satisfactory displacement predictions compared to FMTMDo , although the discrepancy is increased for lower input intensities (Swg″/ Swg″, D = 0.5). It can also be observed that the use of multiple TMDs generally reduced the prediction error when the equivalent linear model was used. The relation between the input intensity and the RMS displacement of the main structure with 1, 5, and 10 TMD units is plotted in Fig. 13. It is evident that the FMTMDs exhibited more robust
Fig. 16. Shape parameters for optimum parameters of FMTMDγτ (ζs = 1%). Table 2 Design parameters and regression coefficients (for μT varying from 1 to 10%). Parameter
m1 [Eq. (28a)] mN [Eq. (28b)] τ /(ωs2 x ref / g ) [Eq. (28c)]
Coefficient
R2
p0
p1
p2
p3
38.98 −5547 1.423
7.40 13.78 −4.337
0.10 0.07 0.047
– – 0.049
0.9872 0.9883 0.9631
that the FRFs for FMTMDo and FMTMDγ demonstrate little difference. Again, there was no significant difference between FMTMDτ and FMTMDγτ , indicating that there will be no appreciable difference in performance if the frequency ratios are constrained. As for FMTMDτ and FMTMDγτ , as the frequency ratio increased, the blunt peak appeared at a low frequency and subsequent peaks became sharper (see Figs. 8b and 8c), and such a characteristic shape is due to the constraint condition of identical COF.
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6. Conclusions
performance under varying input intensities as the number of TMD units increased, which corroborates the observations made based on Fig. 11. Sensitivity analysis of the structural frequency variation was also performed. A case study was performed with the assumption of a natural frequency variation of 10%. Fig. 14 compares the normalized FRFs of FMTMDo as a function of the input intensity and frequency variations for the case of ζs = 1% and μT = 5%. It can be seen that the maximum normalized frequency response increased during frequency variation. Moreover, when the frequency deviated from perfect tuning, the shape of the FRFs showed a flattened and undulated frequency response as the loading level became lower than the designed level. Thus, for the robust control efficiency of TMDs, it is advantageous to set the input strength Swg″ to be slightly higher than what was nominally expected. Furthermore, increasing the number of TMDs also provided robustness of the control efficiency. In the region of the frequency bandwidth, multiple units of FMTMD provided higher and more robust control efficiency compared to a single FTMD. When the natural frequency was lower than the nominal frequency, a slightly higher frequency response than that of a single TMD was obtained. However, it should be noted that frequency amplification did not reach a problematic level, because the frequency bandwidth for the response amplification was narrower than for a single TMD.
This study investigated the optimum design and analysis of frictional multiple tuned mass dampers (FMTMDs), in which a Coulombtype frictional force is incorporated as an energy dissipation mechanism. Four practical configurations of FMTMDs were considered by constraining the frequency ratio and/or the coefficient of friction. This study first applied statistical linearization that replaces the nonlinear Coulomb-type frictional force with an equivalent linear viscous damping force to calculate the RMS response of the structure, and then found the optimal solution by conducting constrained optimization on the linearized system. In addition, this study analyzed the response of the structure-FMTMD systems in the frequency domain to demonstrate the performance of the optimal FMTMDs, as well as the robustness of the control performance on the variations in the dynamic properties of the system and input intensity. The results of this study can be summarized as follows: 1. This study determined the optimum design parameters for four FMTMD configurations for mass ratios varying from 1 to 10%. It was found that with an increasing number of TMDs, the bandwidth of the frequency ratios becomes wider and the equivalent damping ratios decreased. Moreover, as the mass ratio increases, the optimum COF becomes significantly lower. 2. Based on the input-sensitivity analysis, it was found that increasing the input excitation beyond the design level can decrease the equivalent damping ratio, thereby causing a higher and more frivolous response. Therefore, it is recommended to set the target input intensity at a higher level than the nominal design level for the robust control performance of the FMTMDs. 3. For FMTMDγτ , in which the frequency ratios and COFs are equally spaced and the COFs are identical, this study proposed an approximate formula that can be determined using only three design parameters (the first and the last frequency ratios γ1 and γN , and the COF τ ).
5.4. Approximate solution for FMTMDγτ Until now, we have discussed optimum solutions for various practically probable FMTMD configurations. Generally, however, optimum solutions cannot be explicitly formulated because too many design variables are involved. Moreover, the optimum parameters for FMTMDγτ can be described with only three design variables: γ1, γN , and τ. The geometric descriptions for the design parameters of FMTMDγτ are presented in Fig. 15. For a given μT , a plane normal to the backbone curve, parallel to the (γ − τ ) plane is formed, on which the optimum frequency ratio and the COF of STMD can be determined (point A in Fig. 15). On the plane, three lines l1, l2 and l3 with slopes m1, mN and τ respectively, determine two intersection points B1 and BN . All of the optimum points can, therefore, be defined by internally dividing the line l3 into N equal internals. Fig. 16 displays the parameters m1, mN and τ as a function of different mass ratios and the number of TMDs needed to obtain the design formula for FMTMDγτ for mass ratios μT varying 1 to 10%. Based on the figure, it can be inferred that 1) m1 increases, and both mN and COF τ decrease with an increase in the mass ratio and 2) COF τ decreases exponentially with an increase in the number of TMDs. Multiple regressions were performed to derive expressions for m1, mN and τ , and the regression coefficients are tabulated in Table 2. p
m1 = p0 exp(−p1 μT 2 )
mN =
p p0 exp(−p1 μT 2 )
τ /(ωs2 x ref / g )
=
p p0 exp[p1 N p2 μT 3]
Declaration of Competing Interest The authors declared that there is no conflict of interest. References [1] Den Hartog J. Mechanical vibrations. McGraw-Hill; 1956. [2] Luft R. Optimal tuned mass dampers for buildings. J Struct Div 1979;105(12):2766–72. [3] Warburton G. Optimum absorber parameters for various combinations of response and excitation parameters. Earthq Eng Struct Dynam 1982;10(3):381–401. [4] De Domenico D, Ricciardi G. An enhanced base isolation system equipped with optimal tuned mass damper inerter (TMDI). Earthq Eng Struct Dynam 2018;47(5):1169–92. [5] Giaralis A, Taflanidis A. Optimal tuned mass-damper-inerter (TMDI) design for seismically excited MDOF structures with model uncertainties based on reliability criteria. Struct Control Health Monit 2018;25(2):e2082. [6] Pietrosanti D, De Angelis M, Basili M. Optimal design and performance evaluation of systems with Tuned Mass Damper Inerter (TMDI). Earthq Eng Struct Dynam 2017;46(8):1367–88. [7] De Angelis M, Giaralis A, Petrini F, Pietrosanti D. Optimal tuning and assessment of inertial dampers with grounded inerter for vibration control of seismically excited base-isolated systems. Eng Struct 2019;196:109250. [8] Cao L, Li C. Tuned tandem mass dampers-inerters with broadband high effectiveness for structures under white noise base excitations. Struct Control Health Monit 2019;26(4):e2319. [9] Ricciardelli F, Vickery B. Tuned vibration absorbers with dry friction damping. Earthq Eng Struct Dynam 1999;28(7):707–23. [10] Inaudi J, Kelly J. Mass damper using friction-dissipating devices. J Eng Mech 1995;121(1):142–9. [11] Gewei Z, Basu B. A study on friction-tuned mass damper: harmonic solution and statistical linearization. J Vib Control 2010. 1077546309354967. [12] Rüdinger F. Tuned mass damper with nonlinear viscous damping. J Sound Vib 2007;300(3–5):932–48. [13] Wang M. Feasibility study of nonlinear tuned mass damper for machining chatter suppression. J Sound Vib 2011;330(9):1917–30. [14] Carpineto N, Lacarbonara W, Vestroni F. Hysteretic tuned mass dampers for
(28a) (28b) (28c)
Using the slope-intercept form of the equation of a line, two optimum frequency ratios (γ1 and γN ) can be calculated using the following equations (see points B1 and BN in Fig. 15):
γ1 =
γN =
τSTMD − m1 γSTMD τ − m1
(29a)
τSTMD − mN γSTMD τ − mN
(29b)
where γSTMD is the optimum frequency ratio for STMD proposed by [3] [see Eq. (27)] and τSTMD is the τ value for an STMD obtained by substituting N = 1 into Eq. (28c). The optimum frequency ratios of each TMD can be determined by internally dividing the values γ1 and γN . 10
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[31] Wen B, Moustafa MA, Junwu D. Seismic response of potential transformers and mitigation using innovative multiple tuned mass dampers. Eng Struct 2018;174:67–80. [32] Bozer A, Özsarıyıldız ŞS. Free parameter search of multiple tuned mass dampers by using artificial bee colony algorithm. Struct Control Health Monit 2018;25(2):e2066. [33] Elias S, Matsagar V, Datta T. Distributed tuned mass dampers for multi-mode control of benchmark building under seismic excitations. J Earthquake Eng 2019;23(7):1137–72. [34] Elias S, Matsagar V, Datta TK. Along-wind response control of chimneys with distributed multiple tuned mass dampers. Struct Control Health Monit 2019;26(1):e2275. [35] Pennestrı ̀ E, Rossi V, Salvini P, Valentini PP. Review and comparison of dry friction force models. Nonlinear Dyn 2016;83(4):1785–801. [36] Popov VL. Contact mechanics and friction- physical principles and applications. Springer; 2010. [37] Love J, Tait M. The peak response distributions of structure–DVA systems with nonlinear damping. J Sound Vib 2015;348:329–43. [38] Sinha A, Trikutam KT. Optimal Vibration absorber with a friction damper. J Vib Acoust 2017;140(2):021015. [39] Kim S-Y, Lee C-H. Peak response of frictional tuned mass dampers optimally designed to white noise base acceleration. Mech Syst Signal Process 2019;117:319–32. ISSN 10961216. [40] Lutes L, Sarkani S. Random vibrations: analysis of structural and mechanical systems. Butterworth-Heinemann; 2004. [41] Roberts J, Spanos P. Random vibration and statistical linearization. Courier Corporation; 2003. [42] Socha L. Linearization in analysis of nonlinear stochastic systems: recent results–Part I: Theory. Appl Mech Rev 2005;58(3):178–205. ISSN 0003-6900. [43] Socha L. Linearization in analysis of nonlinear stochastic systems, recent results–part II: Applications. Appl Mech Rev 2005;58(5):303–15. ISSN 0003-6900. [44] Bartels R, Stewart G. Solution of the matrix equation AX+XB=C. Commun ACM 1972;15(9):820–6. [45] Coleman T, Branch M, Grace A. Optimization toolbox for use with MATLAB: user’s guide. Mathworks Incorporated; 1999. [46] Ghosh A, Basu B. A closed-form optimal tuning criterion for TMD in damped structures. Struct Control Health Monit 2007;14:681–92.
structural vibration mitigation. J Sound Vib 2014;333(5):1302–18. [15] Xu K, Igusa T. Dynamic characteristics of multiple substructures with closely spaced frequencies. Earthq Eng Struct Dynam 1992;21(12):1059–70. [16] Yamaguchi H, Harnpornchai N. Fundamental characteristics of multiple tuned mass dampers for suppressing harmonically forced oscillations. Earthq Eng Struct Dynam 1993;22(1):51–62. [17] Abé M, Fujino Y. Dynamic characterization of multiple tuned mass dampers and some design formulas. Earthq Eng Struct Dynam 1994;23(8):813–35. [18] Kareem A, Kline S. Performance of multiple mass dampers under random loading. J Struct Eng 1995;121(2):348–61. ISSN 0733-9445. [19] Jangid R, Datta T. Performance of multiple tuned mass dampers for torsionally coupled system. Earthq Eng Struct Dynam 1997;26(3):307–17. [20] Joshi A, Jangid R. Optimum parameters of multiple tuned mass dampers for baseexcited damped systems. J Sound Vib 1997;202(5):657–67. ISSN 0022-460X. [21] Jangid R. Optimum multiple tuned mass dampers for base-excited undamped system. Earthq Eng Struct Dynam 1999;28(9):1041–9. [22] Araz O, Kahya V. Effects of manufacturing type on control performance of multiple tuned mass dampers under harmonic excitation. J Struct Eng 2018;1(3):117–27. [23] Igusa T, Xu K. Vibration control using multiple tuned mass dampers. J Sound Vib 1994;175(4):491–503. ISSN 0022-460X. [24] Li C. Optimum multiple tuned mass dampers for structures under the ground acceleration based on DDMF and ADMF. Earthq Eng Struct Dynam 2002;31(4):897–919. ISSN 1096-9845. [25] Hoang N, Fujino Y, Warnitchai P. Optimal tuned mass damper for seismic applications and practical design formulas. Eng Struct 2008;30(3):707–15. [26] Zuo L. Effective and robust vibration control using series multiple tuned-mass dampers. J Vib Acoust 2009;131(3):031003. [27] Li H, Ni X. Optimization of non-uniformly distributed multiple tuned mass damper. J Sound Vib 2007;308(1):80–97. [28] Fu T, Johnson E. Distributed mass damper system for integrating structural and environmental controls in buildings. J Eng Mech 2010;137(3):205–13. [29] Yang F, Sedaghati R, Esmailzadeh E. Optimal design of distributed tuned mass dampers for passive vibration control of structures. Struct Control Health Monit 2015;22(2):221–36. [30] Hussan M, Rahman MS, Sharmin F, Kim D, Do J. Multiple tuned mass damper for multi-mode vibration reduction of offshore wind turbine under seismic excitation. Ocean Eng 2018;160:449–60.
11
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Analysis and optimization of multiple tuned mass dampers with coulomb dry friction Sung-Yong Kima, Cheol-Ho Leeb, a b
⁎
School of Architecture, Changwon National University, Changwon, Republic of Korea 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 Frictional tuned mass damper Statistical linearization Passive control
The optimum design and analysis of frictional multiple tuned mass dampers (FMTMDs) are presented in this study, where Coulomb dry friction is incorporated as an energy dissipation mechanism. To deal with the nonlinearity of the Coulomb dry friction force, which is a source of difficulty in the optimization process, this study adopted a statistical linearization that replaces the original nonlinear system with an equivalent linear system, and found an optimum design that minimized the root mean square (RMS) displacement of the primary structure. The equivalent damping ratio of a tuned mass damper (TMD) unit significantly decreased with the increase in number of TMDs. Four practical configurations were formulated and comparatively analyzed, by constraining the frequency ratios and the coefficient of friction (COF). Among the four investigated cases, a particular solution described using only three parameters is recommended for use in design, because it is simple and sufficiently accurate. We also analyzed the input sensitivity of nonlinear frequency responses of FMTMDs in order to provide general design guidance.
1. Introduction A tuned mass damper (TMD) is a passive control device that consists of mass, spring, and energy dissipation elements mounted to a structure to dampen its dynamic response. Traditionally, viscous dampers were used as energy dissipation devices, and TMDs with viscous dampers, or linear TMDs, have been widely studied by various researchers over the past decades. The proposed solutions have been widely applied in both research and field applications [1–3]. Linear TMDs have been utilized in a wide variety of applications to resolve vibration problems in lightweight floors, observatory towers, high-rise buildings, and other structures. Recently, new types of TMDs have been actively investigated by various researchers [4–8]. Despite their simplicity and effectiveness, linear TMDs have several drawbacks. For example, a viscous damper is vulnerable to performance degradation caused by aging. With repetitive operations over a long lifetime, the dashpot could malfunction because of temperature rise due to repetitive operation, or there may be reduced performance caused by leakage. To facilitate robust control performance, some researchers have attempted to incorporate a Coulomb-type friction mechanism into the TMD as an energy dissipator. Ricciardelli and Vickery [9] studied optimum nonlinear TMDs equipped with dry friction damping, and proposed a design formula for the optimum design. In addition, they ⁎
showed that the optimum parameters of nonlinear TMDs depended on the level of input excitation as well as the structural response. Inaudi and Kelly [10] proposed a friction-dissipating TMD in which the dissipative force was exhibited by friction dampers that acted in a direction perpendicular to the direction of motion. They demonstrated that with an appropriate design, the nonlinear system can exhibit the same level of control performance compared to its counterpart linear TMD. Gewei and Basu [11] investigated the effectiveness of a single TMD with dry friction using a harmonic and statistical linearization solution. Their results showed that a friction TMD is as effective as linear TMDs, and demonstrated that the optimal friction coefficient for the TMD was dependent on the intensity of the excitation. Rüdinger [12] investigated TMDs with power-law type nonlinear viscous damping elements, and showed that the optimum parameters for nonlinear TMDs depended on the level of displacement amplitude and input intensity, which is in contrast with linear TMDs. Wang [13] and Carpineto et al. [14] also developed TMDs that could dissipate input energy via a frictional hysteretic mechanism. In addition, multiple tuned mass dampers (MTMDs) have been investigated by various researchers. Linear MTMDs (LMTMDs) with equally spaced natural frequencies and identical viscous damping constants were studied by Xu and Igusa [15]. They demonstrated that under these conditions, the LMTMD was efficient in attenuating the
Corresponding author. E-mail address: [email protected] (C.-H. Lee).
https://doi.org/10.1016/j.engstruct.2019.110011 Received 12 July 2019; Received in revised form 24 October 2019; Accepted 26 November 2019 0141-0296/ © 2019 Published by Elsevier Ltd.
Please cite this article as: Sung-Yong Kim and Cheol-Ho Lee, Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.110011
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Unfortunately, the actual friction mechanism is not as simple and the Coulomb theory is only an approximation in practice. Although the kinetic friction coefficient τk in Eq. (1) is often assumed to be a constant, it is not exactly constant for a given relative contacting surface velocity. However, it should be noted that the approximation is a fundamentally empirical construction used for its simplicity and versatility. Furthermore, the friction force does not exhibit an instantaneous change in magnitude when the velocity is zero. Due to the discontinuity of the model at zero velocity, the Coulomb friction model results in stiff equations of motion, yielding numerical challenges such as computational burdens [35]. To minimize the computational difficulty associated with the discontinuity at zero velocity, the kinetic friction coefficient is approximately treated as being equal to the static friction coefficient [36]. Although the relationship assumed between the normal force and frictional force is not exact, the Coulomb friction model is still an acceptable representation of friction for the analysis of many physical systems, particularly those used in the design of friction TMDs [9–11,37–39].
response of the primary structure. MTMDs with equally spaced natural frequencies and equal damping ratios have also been investigated by various researchers [16–22]. Notable research works have also been conducted to investigate LMTMDs with more relaxed constraints [23–34]. These studies can be distinguished from each other in the following points: 1) the input condition considered, 2) the optimization criteria, and 3) the optimization strategies. The aforementioned studies also discussed the advantages of conventional MTMDs with viscous dampers. The results of these studies confirm that MTMDs exhibit better performance than a single TMD. In addition, they demonstrated a more robust performance when subjected to changes in the natural frequency of the primary structures under wide-band excitation. Meanwhile, frictional MTMDs (FMTMDs) can be a very effective control device, because they take advantage of LMTMDs over a linear single TMD while eliminating the disadvantages of viscous TMD. FMTMDs have the advantage of easy transportation or installation owing to the light weight of the TMD units. Additionally, they do not require a viscous damping device for each TMD units. However, to the best of our knowledge, there are no reported studies on the implementation of MTMDs with a friction mechanism and the optimal design for various configurations has not been examined either. In this study we propose using MTMDs with a Coulomb dry friction force which can exhibit more robust performance by combining the advantages of MTMDs and frictional mechanism. In this investigation, constrained optimization was performed for some configurations that are considered to be practical in real applications, such as springs of equal stiffness, constant damping elements, or equally spaced frequencies. In this study, the optimum TMD design was investigated, where the natural frequency and coefficient of friction (COF) values were constrained. To determine the optimal solution of a system with nonlinear frictional behavior, equivalent linear systems are constructed based on statistical linearization. Sensitivity analyses of the input strength were also conducted using a linearization technique to overcome the limitation of the solution.
3. Problem statement 3.1. Governing equations of motion Suppose that a system consists of a primary structure and N units of FMTMDs, where each unit of mass is connected via a spring and a friction element in parallel (see Fig. 2). The equations of motion of the system can be written as follows: N
(ms + mT ) x¨s +
mi x¨i + cs xṡ + ks xs = fs
i=1
mi (x¨s + x¨i ) + ki x i + gi = 0
i = 1, ⋯, N
(2a) (2b)
where ms , cs and ks are the mass, damping coefficient and spring constant of the primary structure; mi and ki are the mass and spring constant of the i-th TMD; gi is the frictional force that arises from the relative motion during contact with the surface between the primary structure and the i-th TMD; N is the total number of the TMDs; mT is the N total mass of TMDs, which is equal to ∑i = 1 mi;xs is the displacement of the primary structure, and x i is the relative displacement between the ith TMD and the primary structure. A dot notation denotes a derivative with respect to time, t. The external force exerted on the primary structure is denoted by fs . The Coulomb-type force term gi is modeled as shown in Fig. 3. The force term can then be modeled as follows:
2. Brief review of coulomb dry friction Coulomb model is the most commonly used model for modelling dry friction, which presumes that the static friction force must be overcome to initiate contact motion between two solid surfaces in a state of rest, and the kinetic friction is independent of the relative velocity of the contacting surfaces. According to these presumptions, the Coulomb model is analytically expressed as follows (see Fig. 1):
Fs ⩽ τs Fn v=0 F=⎧ F τ F sgn( v ) v = ≠ 0 ⎨ k n ⎩ d
∑
(1)
gi = gi (x i̇ ) = gio sgn(x i̇ ) = τi mi g sgn(x i̇ )
where v is the relative velocity between the two surfaces, τs is the static friction coefficient, τk is the kinetic friction coefficient, and Fn is the normal force.
(3)
where gio is the characteristic friction force of the i-th TMD, τi is COF, g is the gravitational acceleration, and sgn(·) denotes a signum function. In this study, it is assumed that each force term fi is neglected, and the entire system is subjected to a white-noise base acceleration. In this case, the external force term, fs , becomes − (ms + mT ) u¨ g , where u¨ g is the ground acceleration with a spectral intensity Su¨g given as:
[u¨ g (t ) u¨ g (t + Δt )] = 2πSu¨g δ (Δt ),
Fig. 1. The Coulomb friction model.
Fig. 2. Structure-FMTMD system. 2
(4)
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ηi =
gio / μi
=
ms ωs2 x ref
1 τi. ωs2 x ref / g
(11)
Eq. (8) can be rearranged into a matrix form as follows:
My ″ + Cy ′ + Ky + ψ = fwg″
(12)
y = [ys , y1, …, yN ]T , ψ = [0, ψ1, …, ψN ]T , f = [−(1 + μT ), 0, …, 0]T
where and
⎡1 M=⎢ ⎢ ⎢ ⎣
+ μT 1 ⋮ 1
⋯ 1 ⋮ 0
μN ⎤ ⎡ 2ζs ⋯ 0 ⎥, C = ⎢ 0 ⎢⋮ ⋱ ⋮⎥ ⎢ ⋯ 1⎥ ⎦ ⎣0
0 0 ⋮ 0
⋯ ⋯ ⋱ ⋯
⎡ 1 02 0⎤ 0 γ1 0 ⎥, K = ⎢ ⎢⋮ ⋮ ⋮⎥ ⎢ ⎥ 0⎦ ⎢0 0 ⎣
⋯ ⋯ ⋱ ⋯
0⎤ 0⎥ . ⋮⎥ ⎥ γN2 ⎥ ⎦ (13)
Fig. 3. Idealized Coulomb-type frictional force.
3.2. FMTMD configurations
where δ (·) is the Dirac-delta function, and [·] is an expectation operator. The non-dimensionalized terms were introduced to standardize the subsequent treatment as follows:
μi =
mi , ms
γi =
ωi = ωs
Four FMTMD configurations are considered in this study: FMTMDo is a configuration in which no constraints are imposed on either the frequency ratios or the COFs; FMTMDγ represents the case whereby the frequency ratios are identical; FMTMDτ denotes that the COFs are identical; FMTMDγτ is the case in which the frequency ratios and the COFs are equally spaced and identical. For all four configurations, the spring constants for each TMD are assumed to be identical. The FMTMD configurations considered in this study are tabulated in Table 1.
(5a)
ki mi
ms = ks
ki −1/2 μ ks i
(5b) N by ∑i = 1
μi and Eqs. and μT is the ratio of the total TMD mass determined (2a), , and ωi = ki/ mi and ωs = ks / ms are the natural frequencies of the i-th TMD and the main structure, respectively, and (2b) can be expressed as follows:
3.2.1. Identical spring constants As the spring constants of the MTMD are assumed to be identical, the mass ratio of the i-th TMD, μi , can be defined by the total mass ratio μT and the frequency ratios γi . If the spring constants for all TMDs are identical to ko , the mass ratio μi is determined as follows:
N
(1 + μT ) x¨s +
∑
μi x¨i + 2ζs ωs xṡ + ωs2 xs = −(1 + μT ) u¨ g
i=1
x¨s + x¨i + gi / mi + γi2 ωs2 x i = 0
i = 1, ⋯, N .
(6a)
μi = =
(6b)
To reformulate the governing equations in terms of the non-dimensional form, xs and x i were non-dimensionalized by normalizing them with the RMS displacement of the uncontrolled structure x ref . With the help of the random vibration theory, the RMS displacement of the uncontrolled system is determined using [40]:
x ref =
πSu¨g 2ζs ωs3
. (7)
γj = γ1 +
∑
μi yi″ + 2ζs ys′ + ys = −(1 + μT ) wg″
i=1
ys″ + yi″ + ψi + γi2 yi = 0
i = 1, …, N
Su¨g
=
2ζs . π
gi (x i̇ ) mi ωs2 x ref
(14)
j−1 (γ − γ1) for j = 2, …, N . N−1 N
(15)
(8b)
ψi = ψi (yi′) =
τ sgn(yi′) i = 1, …, N ωs2 x ref / g
(16)
where τ represents the identical COF. Table 1 Constraints for considered FMTMD configurations. Configuration
Constraints
(9)
The non-dimensional friction force term ψi is defined as follows:
ψi = ψi (yi′) =
i = 1, …, N .
(8a)
where the prime denotes the differentiation with respect to the nondimensional time to , and wg″ is the non-dimensional ground acceleration subjected to the primary structure, of which spectral intensity is determined as: 2 x ref ωs3
μT
3.2.3. Identical coefficients of friction (COFs) Given that the nonlinear friction force arises from the TMD and the main structure with the identical COF, the non-dimensional frictional force can be determined as follows:
N
Swg″ =
Σγi−2
3.2.2. Equally spaced frequency ratios When the frequency ratios are equally spaced, only two frequency ratios are required to determine the entire frequency ratio. In this case, any of the frequency ratios can be expressed in terms of the first and Nth frequency ratios γ1 and γN as follows:
Furthermore, by introducing the non-dimensional displacements ys = xs / x ref and yi = x i / x ref , and a time scale to = ωs t , Eqs. (2) can be rewritten as follows:
(1 + μT ) ys″ +
γi−2
= ηi sgn(yi′)
FMTMDo FMTMDγ FMTMDτ FMTMDγτ
(10)
where the normalized characteristic friction force term is determined by substituting Eq. (3) by Eq. (10) as follows:
†
3
Frictional coefficients
Spring constants
Frequency ratios
– –
I† I
– E†
I I
I I
– E
I = Identical and E = Equally spaced.
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Fig. 4. Non-dimensional friction force ψi and its derivative ∂ yi′ ψi .
Fig. 5. Optimum parameters of FMTMDs (ζs = 1% and μT = 5 %).
Fig. 4. Assuming that the responses of the equivalent system are stationary zero-mean Gaussian processes, the non-dimensional velocity of the i-th TMD, yi′, becomes Gaussian with its variance σ yi′. Consequently, the equivalent damping coefficient cieq + 1 can be evaluated using the sifting property of the Dirac delta function as follows:
3.3. Statistical linearization Statistical linearization is an approximation technique, which replaces a nonlinear system by equivalent linear equations in a statistical sense [41–43]. Using the statistical linearization technique, the nonlinear force term ψ in Eq. (21) can be replaced with an equivalent linear term that minimizes the mean square of the error [ε Tε ], where the error ε is defined by
ε = ψ − Ceqy ′
(17)
(19)
Rewriting Eq. (19) in terms of gio and τi gives:
where Ceq is a parametric matrix, which can be determined by applying the first-order necessary condition for optimality as follows:
⎡ ∂ψi ⎤ i = 1, …, N cieq + 1 = [∂ yi′ψi] = ⎢ ⎥ ⎣ ∂yi′ ⎦
2 1 η . π i σ yi′
eq cieq + 1 = 2γi ζi =
ηi =
gio / μi ms ωs2 x ref
=
τi = ωs2 x ref / g
2π γi ζieq σ yi′.
(20)
Hence, the matrix equation for the equivalent linear system can be rewritten as follows:
(18)
where ψi is the nonlinear force of the i-th TMD. The normalized Coulomb-type frictional force ψi and its derivative ∂ yi′ ψi are depicted in
My ″ + (C + Ceq) y ′ + Ky = fwg″ 4
(21)
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10% Mass ratio ȝT, %
Mass ratio ȝT, %
10% Design Curve 5% for STMD ȝT = 2%
5% ȝT = 2%
γ
10% Mass ratio ȝT, %
Mass ratio ȝT, %
10%
5% ȝT = 2%
5% ȝT = 2%
Fig. 6. Optimum frequency and equivalent damping ratios of FMTMDs depending on mass ratio (ζs = 1% ).
10% Mass ratio ȝT, %
Mass ratio ȝT, %
10%
5% ȝT = 2%
5% ȝT = 2%
γ
o
10% Mass ratio ȝT, %
Mass ratio ȝT, %
10%
5% ȝT = 2%
5% ȝT = 2%
Fig. 7. Optimum frequency ratios and COFs of FMTMDs depending on mass ratio (ζs = 1%).
5
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Fig. 9. Normalized frequency response functions depending upon input intensity (ζs = 1%, μT = 5% and N = 10 ).
O B = ⎡ ⎤. ⎣f ⎦
(24b)
Since the external loading wg″ is assumed as stationary white noise with spectral strength of Swg″, the covariance Q = [zzT] can be evaluated by solving the following Lyapunov equation [40]:
AQ + QAT + 2πSwg″ BBT = O.
Fig. 8. Comparison of FRFs when subjected to the targeted input level (ζs = 1%, μT = 5% and N = 10 ).
However, it is not straightforward to calculate the responses of the equivalent linear system, because the damping matrix Ceq consists of the non-dimensional velocities of TMDs. Hence, it is first required to assume the initial values of the equivalent damping matrix and to then iterate the response by solving Eq. (23).
where the matrices M, C , and K are already defined in Eqs. (13) and the equivalent damping matrix Ceq is defined as follows:
Ceq
⎡ 0 0eq ⎢ 0 c2 =⎢ ⋮ ⋮ ⎢ ⎣0 0
⋯ 0 ⎤ ⋯ 0 ⎥ ⋱ ⋮ ⎥ ⎥ ⋯ cNeq+ 1⎦
By introducting a non-dimensional state vector z = space model is formulated as follows:
z′ = Az + Bwg″
4. Optimization strategies (22)
[yT ,
In this study, the RMS displacement of the primary structure is minimized, which is expressed as follows:
T T
y ′ ] , a state-
σy2s = [ys2 ] = [(sTz)TsTz] = tr[SQ] (23)
(26)
where tr[·] refers to a trace operator, s = [1, 0, ⋯, 0]T is a weighting vector which sifts the displacement of the primary structure, and S is the weighting matrix defined as S = ssT . In the optimization process, this study set the classical solution for a single TMD (STMD) proposed by Warburton [3] as the initial value,
where
O I ⎤ A=⎡ −1 −1 eq , ⎣− M K − M (C + C ) ⎦
(25)
(24a) 6
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Fig. 10. Frequency response functions depending upon input intensity (ζs = 1%, μT = 5% and N = 10 ).
which appeared to converge stably to the optimum point for all considered cases. The Warburton’s formula used for the frequency ratios and damping ratios are as follows:
γSTMD =
1 + μT /2 1 + μT
eq , ζSTMD =
μT (1 + 3μT /4) . 4(1 + μT )(1 + μT /2)
(27)
Fig. 11. Maximum of frequency responses depending upon input intensity (ζs = 1% and μT = 5%).
When the initial conditions were used, the convergence was stable to a point for all the considered cases. The objective function can be evaluated by solving the Lyapunov equation using well-established algorithms (for instance, see [44]). For the optimization process, the Broyden-Fletcher-Goldfarb-Shannon algorithm was utilized because of its effectiveness and good performance [45]. Followed by the optimization process, the nonlinear parameters inversely correspond to the identified optimum equivalent damping ratios. It should be noted that the expression for the nonlinear term is ∗ explicit because σ yi,L ′ is independent of the nonlinear force ηi , but is determined using the parameters of the equivalent linear system, γi∗ and ζieq ∗. Hence, seeking the optimal nonlinear parameter ηi implies finding the optimal parameters of the linearized system, which can be accomplished using the optimization procedure for linear systems. In the cases of FMTMDo and FMTMDγ , the optimal level of frictional force and the associated COF can be directly determined from the optimum solution of the equivalent linear system. As for FMTMDτ and FMTMDγτ , it is tedious to determine the optimum solution because all of the friction coefficients are restricted to be identical. Therefore, a constrained optimization scheme should be adopted.
out in this section. The damping ratio of the primary structure system was assumed to be 1%, and the total mass ratio of the FMTMDs varied in the range of 1% to 10% at an interval of 1%. Note that the distribution of the optimal solution may change depending on the assumed damping ratio of the primary structure [46]. However, in this study, the optimal solution for a primary structure with 1% damping was presented, considering that primary structures experience vibration problems usually exhibited at low damping ratios. The number of TMDs considered was in the range of 1–10.
5.1. Optimum parameters Fig. 5 depicts the spatial distributions of the optimum frequency ratio γi and the optimum equivalent damping ratio ζieq for the considered constraints. It can be observed that, with an increasing number of TMDs, the frequency bandwidth became wider, and the equivalent damping ratios decreased. In the case of FMTMDo (see Fig. 5a), the optimum tuning condition was achieved when the frequency ratios were more closely spaced in the vicinity of the natural frequency of the primary structure. As for FMTMDγ , it was observed that the optimum parameters gradually deviated from those of FMTMDo with an increase in the number of TMDs (see Fig. 5b), and the TMDs located at the ends
5. Results and discussions A numerical study for the optimum values of FMTMD was carried 7
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Fig. 14. Normalized frequency response functions of FMTMDo depending upon input intensity and frequency variations (ζs = 1% and μT = 5%).
Fig. 12. Comparison of the normalized displacement time histories of FMTMDo and LMTMD depending upon the input intensity and the number of TMDs (ζs = 1% and μT = 5%).
Fig. 15. Spatial representation of design parameters m1, mN and τ for FMTMDγτ .
increase. The optimum frequency ratios and optimum COFs are presented in Fig. 7. It is evident that as the mass ratio increases, the optimum COF of FMTMDs becomes significantly lower. It was also found that the optimum COF becomes asymptotically smaller as the number of TMDs increases. It should be noted that the optimum COFs were normalized to Xref , which means that the optimal solution depends on the input level. Note that the spatial distribution of optimum parameters for a given mass ratio lies on the straight lines for FMTMDτ and FMTMDγτ .
Fig. 13. Comparison of normalized RMS displacements of the main structure depending upon input intensity (ζs = 1% and μT = 5%).
of the bandwidth required higher equivalent damping compared to the unconstrained condition. No appreciable differences were found between FMTMDτ and FMTMDγτ (see Fig. 5c and 5d). However, compared to FMTMDo , the optimum equivalent damping ratios were distributed such that the TMDs with low frequency ratios required higher damping. The distributions of the optimum parameters for the mass ratio between 1% and 10% are presented in Fig. 6. It is evident from the figure that in comparison to the single TMD, the MTMD requires a significantly low level of equivalent damping ratio with the increase in the mass ratio. It is also evident that the optimum frequency ratios become wider as the number of TMDs and the total mass ratio μT
5.2. Frequency response with optimum parameters Fig. 8 compares the frequency response functions (FRFs) for the displacement of the primary structure. It can be observed that all considered FMTMD configurations can effectively reduce vibrations when N + 1 well-separated local modes are exhibited. Fig. 8a indicates 8
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5.3. Input intensity sensitivity analysis Given that the nonlinear nature of the FMTMD-structured system used in this study cannot be described by an impulse response (or by the law of superposition), the possible variation of the response depending on the input intensities should also be investigated. The frequency responses described in the preceding subsection are valid only when the input intensity corresponded to the target or design level. Although the design is usually performed to meet a target performance against a design load level, the actual loadings (which can be higher or lower than those assumed in the design), may also excite the structure. The effect of this loading cannot be linearly interpolated or extrapolated because of the nonlinearity of the problem. Next, the sensitivity analyses of the input intensity are presented. A primary structure with a damping ratio of 1% equipped with ten units, whose total mass ratio was 5%, was selected for this sensitivity study. The analysis was performed at input intensities ranging from 0.5 to 1.5 times the design intensity at 0.1 intervals. Two configurations FMTMDo and FMTMDγ τ were considered. Fig. 9 summarized the FRFs of the considered FMTMDs when subjected to varying input intensities. The frequency response curve became flat when a low-level input was applied, and as the input level increased beyond the design input level, the peaks of the frequency responses became sharper. Considering that the lower peak of the magnitude of the frequency response implies the reduction of the worstcase response, the results suggest that TMD units move in a “sticky” manner if the input intensity is smaller than the level considered in the design. The change in the frequency response curve depending on the input level can be inferred from Eq. (19): When the RMS acceleration of the TMDs (σ yi′) is lower in comparison to the target input, the value of the equivalent damping ratio (ζieq ) increases. Comparing Figs. 9a and 9b for FMTMDτ , a configuration where the COF was restricted to be identical, it exhibited an inferior performance as compared to FMTMDo . Figs. 10a and 10b depict the maximum of FRFs for FMTMDo and FMTMDγτ , respectively. It is evident that for both FMTMD configurations, the shape of the curves was more oscillatory if the loading level exceeded the design level. As implied in Eq. (19), the equivalent damping ratio is inversely proportional to the normalized RMS velocity of the TMD response. Thus, the increased TMD unit response (due to the increased input beyond the design level) eventually leads to a reduction in the control performance. Fig. 11 shows the maximum frequency responses as a function of the input level for the FMTMDo and FMTMDγτ configurations. In both FMTMDs, the maximum frequency response decreased as the number of TMDs increased for any input intensity. It was also observed that the increase in the maximum frequency response is slightly larger in FMTMDγτ than in FMTMDo . In addition, when the input level is higher than the target value, the maximum frequency response increases slightly more than the increase in the input level. Therefore, it is recommended to set the target input intensity at a level that is higher than the nominal design level for the robust control performance of the FMTMDs. A time domain analysis was also conducted to demonstrate the reduction in the vibration of FMTMDo over time. Fig. 12 presents the time histories of the normalized displacements of the primary structure subjected to white-noise excitation for three input intensities: lower forcing amplitude (0.5 times the design input intensity), intended forcing amplitude (the design intensity), and higher forcing amplitude (1.5 times the design input intensity). The effect of the number of TMD units is also included in this figure. Overall, the equivalent linear model (LMTMD) provides satisfactory displacement predictions compared to FMTMDo , although the discrepancy is increased for lower input intensities (Swg″/ Swg″, D = 0.5). It can also be observed that the use of multiple TMDs generally reduced the prediction error when the equivalent linear model was used. The relation between the input intensity and the RMS displacement of the main structure with 1, 5, and 10 TMD units is plotted in Fig. 13. It is evident that the FMTMDs exhibited more robust
Fig. 16. Shape parameters for optimum parameters of FMTMDγτ (ζs = 1%). Table 2 Design parameters and regression coefficients (for μT varying from 1 to 10%). Parameter
m1 [Eq. (28a)] mN [Eq. (28b)] τ /(ωs2 x ref / g ) [Eq. (28c)]
Coefficient
R2
p0
p1
p2
p3
38.98 −5547 1.423
7.40 13.78 −4.337
0.10 0.07 0.047
– – 0.049
0.9872 0.9883 0.9631
that the FRFs for FMTMDo and FMTMDγ demonstrate little difference. Again, there was no significant difference between FMTMDτ and FMTMDγτ , indicating that there will be no appreciable difference in performance if the frequency ratios are constrained. As for FMTMDτ and FMTMDγτ , as the frequency ratio increased, the blunt peak appeared at a low frequency and subsequent peaks became sharper (see Figs. 8b and 8c), and such a characteristic shape is due to the constraint condition of identical COF.
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6. Conclusions
performance under varying input intensities as the number of TMD units increased, which corroborates the observations made based on Fig. 11. Sensitivity analysis of the structural frequency variation was also performed. A case study was performed with the assumption of a natural frequency variation of 10%. Fig. 14 compares the normalized FRFs of FMTMDo as a function of the input intensity and frequency variations for the case of ζs = 1% and μT = 5%. It can be seen that the maximum normalized frequency response increased during frequency variation. Moreover, when the frequency deviated from perfect tuning, the shape of the FRFs showed a flattened and undulated frequency response as the loading level became lower than the designed level. Thus, for the robust control efficiency of TMDs, it is advantageous to set the input strength Swg″ to be slightly higher than what was nominally expected. Furthermore, increasing the number of TMDs also provided robustness of the control efficiency. In the region of the frequency bandwidth, multiple units of FMTMD provided higher and more robust control efficiency compared to a single FTMD. When the natural frequency was lower than the nominal frequency, a slightly higher frequency response than that of a single TMD was obtained. However, it should be noted that frequency amplification did not reach a problematic level, because the frequency bandwidth for the response amplification was narrower than for a single TMD.
This study investigated the optimum design and analysis of frictional multiple tuned mass dampers (FMTMDs), in which a Coulombtype frictional force is incorporated as an energy dissipation mechanism. Four practical configurations of FMTMDs were considered by constraining the frequency ratio and/or the coefficient of friction. This study first applied statistical linearization that replaces the nonlinear Coulomb-type frictional force with an equivalent linear viscous damping force to calculate the RMS response of the structure, and then found the optimal solution by conducting constrained optimization on the linearized system. In addition, this study analyzed the response of the structure-FMTMD systems in the frequency domain to demonstrate the performance of the optimal FMTMDs, as well as the robustness of the control performance on the variations in the dynamic properties of the system and input intensity. The results of this study can be summarized as follows: 1. This study determined the optimum design parameters for four FMTMD configurations for mass ratios varying from 1 to 10%. It was found that with an increasing number of TMDs, the bandwidth of the frequency ratios becomes wider and the equivalent damping ratios decreased. Moreover, as the mass ratio increases, the optimum COF becomes significantly lower. 2. Based on the input-sensitivity analysis, it was found that increasing the input excitation beyond the design level can decrease the equivalent damping ratio, thereby causing a higher and more frivolous response. Therefore, it is recommended to set the target input intensity at a higher level than the nominal design level for the robust control performance of the FMTMDs. 3. For FMTMDγτ , in which the frequency ratios and COFs are equally spaced and the COFs are identical, this study proposed an approximate formula that can be determined using only three design parameters (the first and the last frequency ratios γ1 and γN , and the COF τ ).
5.4. Approximate solution for FMTMDγτ Until now, we have discussed optimum solutions for various practically probable FMTMD configurations. Generally, however, optimum solutions cannot be explicitly formulated because too many design variables are involved. Moreover, the optimum parameters for FMTMDγτ can be described with only three design variables: γ1, γN , and τ. The geometric descriptions for the design parameters of FMTMDγτ are presented in Fig. 15. For a given μT , a plane normal to the backbone curve, parallel to the (γ − τ ) plane is formed, on which the optimum frequency ratio and the COF of STMD can be determined (point A in Fig. 15). On the plane, three lines l1, l2 and l3 with slopes m1, mN and τ respectively, determine two intersection points B1 and BN . All of the optimum points can, therefore, be defined by internally dividing the line l3 into N equal internals. Fig. 16 displays the parameters m1, mN and τ as a function of different mass ratios and the number of TMDs needed to obtain the design formula for FMTMDγτ for mass ratios μT varying 1 to 10%. Based on the figure, it can be inferred that 1) m1 increases, and both mN and COF τ decrease with an increase in the mass ratio and 2) COF τ decreases exponentially with an increase in the number of TMDs. Multiple regressions were performed to derive expressions for m1, mN and τ , and the regression coefficients are tabulated in Table 2. p
m1 = p0 exp(−p1 μT 2 )
mN =
p p0 exp(−p1 μT 2 )
τ /(ωs2 x ref / g )
=
p p0 exp[p1 N p2 μT 3]
Declaration of Competing Interest The authors declared that there is no conflict of interest. References [1] Den Hartog J. Mechanical vibrations. McGraw-Hill; 1956. [2] Luft R. Optimal tuned mass dampers for buildings. J Struct Div 1979;105(12):2766–72. [3] Warburton G. Optimum absorber parameters for various combinations of response and excitation parameters. Earthq Eng Struct Dynam 1982;10(3):381–401. [4] De Domenico D, Ricciardi G. An enhanced base isolation system equipped with optimal tuned mass damper inerter (TMDI). Earthq Eng Struct Dynam 2018;47(5):1169–92. [5] Giaralis A, Taflanidis A. Optimal tuned mass-damper-inerter (TMDI) design for seismically excited MDOF structures with model uncertainties based on reliability criteria. Struct Control Health Monit 2018;25(2):e2082. [6] Pietrosanti D, De Angelis M, Basili M. Optimal design and performance evaluation of systems with Tuned Mass Damper Inerter (TMDI). Earthq Eng Struct Dynam 2017;46(8):1367–88. [7] De Angelis M, Giaralis A, Petrini F, Pietrosanti D. Optimal tuning and assessment of inertial dampers with grounded inerter for vibration control of seismically excited base-isolated systems. Eng Struct 2019;196:109250. [8] Cao L, Li C. Tuned tandem mass dampers-inerters with broadband high effectiveness for structures under white noise base excitations. Struct Control Health Monit 2019;26(4):e2319. [9] Ricciardelli F, Vickery B. Tuned vibration absorbers with dry friction damping. Earthq Eng Struct Dynam 1999;28(7):707–23. [10] Inaudi J, Kelly J. Mass damper using friction-dissipating devices. J Eng Mech 1995;121(1):142–9. [11] Gewei Z, Basu B. A study on friction-tuned mass damper: harmonic solution and statistical linearization. J Vib Control 2010. 1077546309354967. [12] Rüdinger F. Tuned mass damper with nonlinear viscous damping. J Sound Vib 2007;300(3–5):932–48. [13] Wang M. Feasibility study of nonlinear tuned mass damper for machining chatter suppression. J Sound Vib 2011;330(9):1917–30. [14] Carpineto N, Lacarbonara W, Vestroni F. Hysteretic tuned mass dampers for
(28a) (28b) (28c)
Using the slope-intercept form of the equation of a line, two optimum frequency ratios (γ1 and γN ) can be calculated using the following equations (see points B1 and BN in Fig. 15):
γ1 =
γN =
τSTMD − m1 γSTMD τ − m1
(29a)
τSTMD − mN γSTMD τ − mN
(29b)
where γSTMD is the optimum frequency ratio for STMD proposed by [3] [see Eq. (27)] and τSTMD is the τ value for an STMD obtained by substituting N = 1 into Eq. (28c). The optimum frequency ratios of each TMD can be determined by internally dividing the values γ1 and γN . 10
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