A tuned liquid inerter system for vibration control

International Journal of Mechanical Sciences 164 (2019) 105171

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

A tuned liquid inerter system for vibration control Zhipeng Zhao a, Ruifu Zhang a,∗, Yiyao Jiang a,b, Chao Pan c a

Department of Disaster Mitigation for Structures, Tongji University, Shanghai 200092, China Shanghai Key Lab of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai 200433, China c College of Civil Engineering, Yantai University, Yantai 264005, China b

a r t i c l e

i n f o

Keywords: Analytical stochastic response Demand based design Inerter Tuned liquid damper Tuned liquid inerter system

a b s t r a c t This study proposes a novel tuned liquid inerter system (TLIS) that employs a tuned liquid element and an inerter-based subsystem, including damping, stiffness, and inerter elements, to completely utilize their synergy benefits. The stochastic analyses for both damped and undamped structure-TLIS systems are conducted, and the analytical solution to the displacement response is derived for an undamped structure. Using the stochastic response results, the vibration control effect of the proposed TLIS is characterized through an extensive analysis in comparison with classic tuned liquid dampers. A parametric analysis is then performed to investigate the response variation pattern of the structure-TLIS with respect to the changes in TLIS parameters. Inspired by the parametric analysis, an optimal design framework for the most economical condition is developed for the TLIS by simultaneously considering the structural performance and control force. In particular, for a TLIS in the presence of an existing tuned liquid element, the optimal damping and stiffness ratios of the inerter subsystem in the TLIS are derived in an analytical form as the functions of tuned liquid mass and inertance to ensure structural responses are minimized. Moreover, the corresponding design procedure and examples are provided to validate the effectiveness of the proposed TLIS and design methods. It is concluded that the employment of the inerterbased subsystem in the TLIS can substantially improve the structural performance and reduce the displacement response of the tuned liquid in comparison with the classic tuned liquid damper for a given liquid mass. The optimal design of the TLIS can satisfy the target structural performance demands despite optimizing control cost. Simultaneously, the function of the tuned liquid element is partly substituted by the added inerter element, which implicitly facilitates the minimization of the physical mass of the TLIS to achieve lightweight-based control.

1. Introduction Structural vibration control strategies that add additional devices to a dynamically actuated structural system have been shown to effectively protect the structure from undesired vibrations [1,2]. In recent years, the idea of using an inerter-based system has received particular attention because it is potentially beneficial for vibration control through its synthetical effects of tuning frequency, mass enhancement, and damping enhancement [3,4]. The inerter is a two-terminal mechanical element that can adjust the structural inertia characteristic without introducing considerable physical mass. Ideally, its inertia force is proportional to the relative acceleration of the two terminals and to a constant inertial mass called inertance [5]. In the 1970s, Kawamata [6] presented a liquid mass pump to make full use of the inertial resistance of flowing liquid, which is the bud of a two-terminal inertial element. Subsequently, Ikago and his co-workers proposed a two-terminal inertial system (tuned viscous mass damper, TVMD), which explicitly used mass enhancement and damping enhancement for the first time [4,7–10]. From the per-

∗

spective of the cost of mass, the inerter can easily generate an inertance thousands of times its gravitational mass [4] so that a large inertance can be introduced into the controlled system with an infinitesimal control cost of the gravitational mass (known as the mass enhancement effect). Given this benefit of the enhancement effect, employing an inerter is a potentially effective solution to realize a “lightweight” and highly efficient tuned-mass-type system [11–13]. Garrido et al. [13] proposed a rational improvement in the tuned mass damper by replacing its damping element with a TVMD. The developed device is found to be more effective than a tuned mass damper at the same mass ratio as it is simultaneously characterized by a broader suppression band. Marian and Giaralis [4] analyzed the performance of tuned mass damper with an inerter and discovered that the incorporation of the inerter can replace part of the tuned mass damper vibrating mass to achieve lightweight passive vibration control solutions. Zhang et al. [11] proposed a lightweightbased optimal design method for a wind turbine tower by utilizing an inerter-based tuned system. The desired structural performance target can be met with optimized mass. Inspired by this efficient solution,

Corresponding author. E-mail address: [email protected] (R. Zhang).

https://doi.org/10.1016/j.ijmecsci.2019.105171 Received 11 June 2019; Received in revised form 16 September 2019; Accepted 16 September 2019 Available online 17 September 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

Z. Zhao, R. Zhang and Y. Jiang et al.

various inerter-based tuned mass systems have been applied for the vibration control of different types of structures [14–18]. Nonetheless, the analyzed inerter-based tuned mass systems in the literature are all based on a solid mass that inevitably needs to be attached to the main structure and cannot provide any practical function other than energy storage. In addition to the above-mentioned tuned-mass-type system using additional solid mass, a tuned liquid damper (TLD) has been well studied to perform as a vibration control device for structural response mitigation by using the liquid element [19–23]. A TLD is usually a rigid liquid container with shallow liquid inside it, and it is commonly fixed directly to the controlled primary structure. To increase the insufficient damping ratio of this device and increase its vibration control effect, various ideas, such as employing different screens [24–26] or porous media [27], have been proposed. However, these solutions inevitably destroy the integrity and structural form of the TLD and may also present difficulties in its installation and increase subsequent maintenance costs. The optimization design of the TLD in a typical tunedlike damper is manipulated, allowing it to be expressed as an equivalent tuned mass damper [28,29] so that typical optimization equations developed for a linear tuned mass damper can be applied directly for a linearized TLD [30]. Copious supplemental liquid is required to achieve a high level of structural performance, correspondingly leading to high control cost of the mass quantity and increased installation space. Dealing with the above-mentioned drawbacks of existing systems, we propose a novel tuned liquid inerter system (TLIS) for vibration control by utilizing the synthetic advantages of the inerter-based system and tuned liquid element to pursue the lightweight-based control and improved control performance. The optimal design framework is also established for the TLIS, where the analytical design formulae are derived to determine the parameters of the TLIS. In the TLIS, an inerter-based subsystem is externally combined with a liquid container to maintain its original function and form. The liquid element is characterized as dual function by its practical function of liquid storage and its vibration control function that includes tuning effect and energy dissipation through the liquid sloshing. Stochastic analyses for both damped and undamped structures with a TLIS are conducted. Using the results, the vibration control effect of the proposed TLIS is characterized through an extensive analysis in comparison with the classic tuned liquid damper. The response variation pattern of the controlled structure with respect to the changes in the TLIS parameters is assessed in a parametric analysis. Inspired by the results of the parametric analysis, an optimal design framework is developed for the TLIS in the presence and absence of an existing tuned liquid element. Finally, the corresponding design procedure and examples are provided to validate the effectiveness of the proposed TLIS and design methods.

International Journal of Mechanical Sciences 164 (2019) 105171

to the relative acceleration between the two terminals and inertance [33–38]: ( ) 𝐹𝑖𝑛 = 𝑚𝑖𝑛 𝑢̈ 2 − 𝑢̈ 1 , (1) where u1 and u2 denote the displacements of the two terminals, and the dots signify the derivative with respect to time. The intrinsic advantage of the mass enhancement effect of the inerter means that even though a large amount of inertance is purposely designed, the physical mass is almost negligible. Thus, no additional external force dependent on excitation acceleration is introduced. 2.1.2. Tuned liquid element model Consider a rigid rectangular tank (see Fig. 2) with length L, width b, and a quiescent liquid depth h, in which the liquid is considered incompressible and inviscid without rotational flow. Assuming that the liquid’s sloshing response is small compared to the fluid depth, the potential flow model is adopted in this study for its concise expression and accuracy for engineering purposes [39]. Subject to a horizontal excitation, the dynamic behavior of the tuned liquid element is simulated in two distinct patterns, namely the motion of tuned liquid mass mt moving rigidly along the tank and an equivalent sloshing mass meq representing the surface movement. The corresponding expressions are determined in terms of the entire liquid mass ml = 𝜌bhL as in Refs. [24,40]. ( ) ( ) 8𝜌𝑏𝐿2 πℎ 𝑚𝑒𝑞 = 𝛽𝑒𝑞 𝑚𝑙 ,𝑚𝑡 = 1 − 𝛽𝑒𝑞 𝑚𝑙 , 𝛽𝑒𝑞 = tanh , 𝐿 π3

(2)

where 𝛽 eq and 𝜌 refer to the equivalent sloshing mass ratio and liquid density, respectively. Referring to the linear sloshing theory, the fundamental natural frequency of liquid sloshing 𝜔l can be predicted as in Refs. [24,40]. √ ( ) πg πℎ 𝜔𝑙 = tanh (3) , 𝐿 𝐿 where g is the gravity-induced acceleration. The interaction behavior between meq and the tank wall are modeled by equivalent spring 𝑘𝑒𝑞 = 𝑚𝑒𝑞 𝜔2𝑙 and damping elements 𝑐𝑒𝑞 = 2𝜉𝑙 𝑚𝑒𝑞 𝜔𝑙 arranged in parallel. Adopting the recently developed inerter element, the proposed TLIS (see Fig. 3) is composed of a grounded inerter element min , a damping element cd , a spring element kd , and a tuned liquid element (containing mt , meq , ceq , and keq ). Consider a linear damped single-degree-offreedom structure (SDOF) structure characterized by concentrated mass m, linear spring k, and viscous damping c. The proposed TLIS is purposely settled to suppress the structural vibration response. Compared with the conventional control technology using a TLD, additional viscous damping element cd and tuning spring element kd are inserted as an interconnection between the tuned liquid element and primary structure m. The inerter element connects the tank of the tuned liquid element to the supporting ground.

2. Proposed tuned liquid inerter system (TLIS) 2.2. Stochastic analysis of the SDOF-TLIS system 2.1. Mechanical model of TLIS 2.1.1. Inerter element model In Fig. 1, the depicted inerter is a mechanical element that potentially produces a constant property (namely inertia mass [9], apparent mass [4], or inertance [5,31,32]) thousands of times its physical mass. In an ideal scenario, the inertia force Fin of the inerter is proportional

2.2.1. Solution for damped structure Subjecting the SDOF-damped structure with the TLIS (see Fig. 3) to a horizontal dynamic excitation, the governing motion equation is established according to dynamic equilibrium conditions: ⎧(𝑢̈ + 𝑢̈ + 𝑢̈ ) + 2𝜉 𝛾𝜔 𝑢̇ + 𝛾 2 𝜔2 𝑢 = −𝑢̈ 𝑠 𝑙 )𝑙 (𝑠 𝑙 )( ) 𝑔 ( ) ⎪ 𝑙 (𝑡 , 𝛽 𝑢̈ + 𝑢̈ + 𝜇 𝑢̈ 𝑡 + 𝑢̈ + 2𝜉𝜔𝑠 𝑢̇ 𝑡 + 𝜅𝜔2𝑠 𝑢𝑡 = −𝜂 𝑢̈ 𝑔 ⎨𝜂𝛽𝑒𝑞 𝑢̈ 𝑙 (+ 𝑢̈ 𝑡 + 𝑢̈ + 𝜂) 1 − ( 𝑒𝑞 )(𝑡 ) ( ) ⎪𝑢̈ + 𝜂𝛽𝑒𝑞 𝑢̈ 𝑙 + 𝑢̈ 𝑡 + 𝑢̈ + 𝜂 1 − 𝛽𝑒𝑞 𝑢̈ 𝑡 + 𝑢̈ + 𝜇 𝑢̈ 𝑡 + 𝑢̈ + 2𝜁 𝜔 𝑢̇ + 𝜔2 𝑢 = −(1 + 𝜂)𝑢̈ 𝑔 𝑠 𝑠 ⎩ (4)

Fig. 1. Mechanical model of a two-terminal inertia element (inerter).

where u, ut , and ul are the relative displacements of the primary SDOF structure to the ground, the tank to the primary structure, and the equivalent sloshing mass to the tank. 𝑢̈ 𝑔 is the acceleration of the ground motion. The other system parameters (frequency, damping ratio, and TLIS dimensionless parameters) expressed in Eq. (4) are summarized in

Z. Zhao, R. Zhang and Y. Jiang et al.

International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 2. Mechanical model of the tuned liquid element considered for simplification.

where 𝑁1 = 𝑠2 + 2𝑠𝜉𝑙 𝜔𝑠 + 𝛾 2 𝜔2𝑠 , 𝑁2 = 𝑠2 (𝜂 + 𝜇) + 2𝑠𝜉𝜔𝑠 + 𝜅𝜔2𝑠 , and 𝐷1 = 𝑠2 + 2𝑠𝜁 𝜔𝑠 + 𝜔2𝑠 . For the SDOF-TLIS system, the response quantities of interest are the displacement and absolute acceleration of the primary mass, the control force transmitted to the primary structure, and the relative displacement of the equivalent sloshing mass to the tank. The corresponding transfer functions HU , HAcc , HF , and 𝐻𝑈𝑙 are given as 𝑠2 𝑈 + 𝑈̈ 𝑔 𝑈 𝐻𝑈 (𝑠)||𝑠=i𝜔 = , 𝐻𝐴𝑐𝑐 (𝑠)||𝑠=i𝜔 = , 𝐻𝐹 (𝑠)||𝑠=i𝜔 𝑈̈ 𝑔 𝑈̈ 𝑔 =

𝜎𝑈2 =

Table 1 Parameters defined in this study for the SDOF-TLIS system. Description

Structural parameters TLISTLD

Definition √ 𝜔𝑠 = 𝑘∕ 𝑚 𝜁 = 𝑐∕2𝑚𝜔𝑠 𝜂 = 𝑚𝑙 ∕𝑚 𝛽𝑒𝑞 = 𝑚𝑒𝑞 ∕𝑚𝑙 𝛾 = 𝜔𝑙 ∕ 𝜔𝑠 𝜉𝑙 = 𝑐𝑒𝑞 ∕2𝑚𝑒𝑞 𝜔𝑙 Supplements 𝜇 = 𝑚𝑖𝑛 ∕𝑚 𝜅 = 𝑘𝑑 ∕𝑘 √ 𝜉 = 𝑐𝑑 ∕2 𝑘𝑚

Circular frequency Inherent damping ratio Liquid mass ratio Convective mass ratio Liquid frequency ratio Liquid damping ratio Inertance-mass ratio Stiffness ratio Damping ratio

𝜎𝐹2 =

(

∞

|𝐻 (𝑠)|2 𝑆 𝑑 𝑠, 𝜎 2 = |𝐻 (𝑠)|2 𝑆 𝑑 𝑠, 𝐴𝑐𝑐 ∫−∞ | 𝑈 | 0 ∫−∞ | 𝐴𝑐𝑐 | 0 ∞

∞

| |2 |𝐻 (𝑠)|2 𝑆 𝑑 𝑠, 𝜎 2 = |𝐻 (𝑠)| 𝑆 𝑑 𝑠. 𝑈𝑙 ∫−∞ | 𝐹 | 0 ∫−∞ | 𝑈𝑙 | 0

(8)

𝑈0 1 =− , 𝐻𝐹 ,0 (𝑠)||𝑠=i𝜔 = 𝐻𝐴𝑐 𝑐 ,0 (𝑠)||𝑠=i𝜔 𝑠2 + 2𝑠𝜁 𝜔𝑠 + 𝜔2𝑠 𝑈̈ 𝑔 ( ) 𝑠2 𝑈0 + 𝑈̈ 𝑔 𝜔𝑠 2𝑠𝜁 + 𝜔𝑠 = =− , (9) 𝑠2 + 2𝑠𝜁 𝜔𝑠 + 𝜔2 𝑈̈ 𝑔

𝐻𝑈 ,0 (𝑠)||𝑠=i𝜔 =

)

𝑈𝑡 + 𝑈 + 2𝑠𝜉𝑙 𝛾𝜔𝑠 𝑈𝑙 + 𝛾 2 𝜔2𝑠 𝑈𝑙 = −𝑈̈ 𝑔 ⎧ 𝑈𝑙 + ⎪𝜂𝛽 𝑠2 (𝑈 + 𝑈 + 𝑈 ) + 𝜂 (1 − 𝛽 )𝑠2 (𝑈 + 𝑈 ) + 𝜇𝑠2 (𝑈 + 𝑈 ) 𝑙 𝑡 𝑒𝑞 𝑡 𝑡 ⎪ 𝑒𝑞 , 𝜅𝜔2𝑠 𝑈𝑡 = −𝜂 𝑈̈ 𝑔 ⎨ +2𝑠𝜉𝜔𝑠 𝑈𝑡 + ( ) ( )( ) ( ) ⎪𝑠2 𝑈 + 𝜂𝛽𝑒𝑞 𝑠2 𝑈𝑡 + 𝑈 + 𝜂𝑠2 1 − 𝛽𝑒𝑞 𝑈𝑡 + 𝑈 + 𝜇𝑠2 𝑈𝑡 + 𝑈 ⎪ ⎩ +2𝑠𝜁 𝜔𝑠 𝑈 + 𝜔2𝑠 𝑈 = −(1 + 𝜂)𝑈̈ 𝑔

∞

For the damped structure, the mean-square responses need to be calculated numerically due to the difficulty posed by analytical derivation. For a damped SDOF structure without any control device, the displacement HU,0 , absolute acceleration HAcc,0 , and control force (provided by structural inherent stiffness and damping) HF,0 transfer functions are determined as

Table 1. Let us rewrite the governing equation in the Laplace domain and denote the notation s = i𝜔. 𝑠2

(7)

Under a white noise excitation whose input power spectra amplitude is S0 , the mean-square of the displacement 𝜎𝑈2 , absolute acceleration of 2 , control force transmitted to the primary structure the primary mass 𝜎𝐴𝑐𝑐 2 𝜎𝐹 , and relative displacement of the equivalent sloshing mass to tank 𝜎𝑈2 𝑙 can be determined as

Fig. 3. Mechanical model of the SDOF-TLIS system.

Parameters

𝜅𝜔2𝑠 𝑈𝑡 + 2𝑠𝜉𝜔𝑠 𝑈𝑡 𝑈 | , 𝐻𝑈𝑙 (𝑠)| = 𝑙. |𝑠=i𝜔 𝑈̈ 𝑔 𝑈̈ 𝑔

𝑠

where the subscript “0” denotes the case of an uncontrolled 2 SDOF structure. The corresponding 𝜎𝑈2 ,0 = 𝜋∕(2𝜁 𝜔3𝑠 ) and𝜎𝐴𝑐 = 𝜎𝐹2 ,0 = 𝑐 ,0 ∞

(5)

where U, Ut , Ul , and 𝑈̈ 𝑔 are the Laplace transforms of u, ut , ul , and 𝑢̈ 𝑔 , respectively. Correspondingly, the dynamic results in the Laplace domain can be derived as

∫−∞ |𝐻𝐴𝑐 𝑐 ,0 (𝑠)|2 𝑆0 𝑑𝑠 =

𝜋(1+4𝜁 2 )𝜔𝑠 2𝜁

are derived in a closed form.

2.2.2. Analytical solution for undamped structure Using Eq. (6), an undamped SDOF structure is considered in this section to derive the analytical form of structural displacement responses 𝜎𝑈2 . Considering that the damping ratio of the liquid is typically small [24], the assumption of the no damping effect of the liquid in the TLIS

( )( ) ⎧ −𝑠4 𝜂𝛽𝑒𝑞 + 𝜂𝜔𝑠 2𝑠𝜉 + 𝜅𝜔𝑠 𝑁1 − 𝑠2 𝛽𝑒𝑞 + 𝑁1 𝑁2 ⎪𝑈 = ( )( ) ( ) 𝑈̈ 𝑔 ⎪ 4 2 𝜔𝑠 2𝑠𝜉 + 𝜅𝜔 𝐷 𝑠4 𝜂𝛽𝑒𝑞 − 𝑁1 𝑁2 𝑠 𝑠 𝜂𝛽𝑒𝑞 − 𝑠 (𝜂 +)𝜇)𝑁1 + ( ( ( ) ) ⎪ −𝑠4 𝜇 − 2𝑠3 −𝜁 𝜂 + 𝜁 𝜂𝛽𝑒𝑞 + 𝜇𝜉𝑙 𝜔𝑠 + 𝑠2 𝜂 − 𝛾 2 𝜇 − 𝜂𝛽𝑒𝑞 + 4𝜁 𝜂𝜉𝑙 𝜔2𝑠 + 2𝑠𝜂 𝛾 2 𝜁 + 𝜉𝑙 𝜔3𝑠 + 𝛾 2 𝜂𝜔4𝑠 ⎪ 𝑈̈ 𝑔 , ( )( ) ( ) ⎨𝑈 𝑡 = 𝜔𝑠 2𝑠𝜉 + 𝜅𝜔𝑠 𝑠4 𝜂𝛽𝑒𝑞 − 𝑠2 (𝜂 + 𝜇)𝑁1 + 𝐷 𝑠4 𝜂𝛽𝑒𝑞 − 𝑁1 𝑁2 ⎪ ⎪ 𝑠4 𝜇 + 2𝑠3 𝜇(𝜁 + 𝜉)𝜔𝑠 + 𝑠2 (𝜇 + 𝜅𝜇 + 4𝜁 𝜉)𝜔2𝑠 + 2𝑠(𝜁 𝜅 + 𝜉)𝜔3𝑠 + 𝜅𝜔4𝑠 ⎪𝑈 𝑙 = ( )( ) ( ) 𝑈̈ 𝑔 𝜔𝑠 2𝑠𝜉 + 𝜅𝜔𝑠 𝑠4 𝜂𝛽𝑒𝑞 − 𝑠2 (𝜂 + 𝜇)𝑁1 + 𝐷 𝑠4 𝜂𝛽𝑒𝑞 − 𝑁1 𝑁2 ⎪ ⎩

(6)

Z. Zhao, R. Zhang and Y. Jiang et al.

International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 4. Contour plots of the acceleration comparison ratio 𝛽𝐴𝑐𝑐 by comparing the acceleration response of the SDOF-TLIS system with that of the SDOF-TLD system for inertance-mass ratio 𝜇 ∈ [0.01, 3.0] and damping ratio𝜉 ∈ [0.01, 1.0], values of the liquid mass ratio: (a) 𝜂 = 0.01, (b) 𝜂 = 0.025, and (c) 𝜂 = 0.05.

Fig. 5. Contour plots of the displacement comparison ratio 𝛽𝑈 by comparing the displacement response of the SDOF-TLIS system with that of the SDOF-TLD system for inertance-mass ratio 𝜇 ∈ [0.01, 3.0] and damping ratio 𝜉 ∈ [0.01, 1.0], values of the liquid mass ratio: (a) 𝜂 = 0.01, (b) 𝜂 = 0.025, and (c) 𝜂 = 0.05.

is plausible. Substituting 𝜁 = 𝜉 l = 0 into Eqs. (6) to (8), the displacement response is derived in an analytical form as in Ref. [41]. 𝑁𝑈 𝜋 𝜎𝑈2 = (10) (( ) )2 , 2𝛾 2 𝜉 −1 + 𝛾 2 (𝜂 + 𝜇) + 𝜂𝛽𝑒𝑞 𝜔3𝑠 Additional details can be found in Appendix A1. The other two re2 sponse measures, that is, 𝜎𝐴𝑐𝑐 and 𝜎𝐹2 , can be obtained numerically. The derived analytical form for 𝜎𝑈2 is also used in the parametric analysis (Section 3.1) and the simplification of the optimal design in Section 3.3.1. 2.3. TLIS performance compared with classic TLD This section conducts an extensive parametric analysis to investigate the performance of the proposed TLIS compared with that of a conventional TLD. To facilitate this, the TLD is designed using the following well-known classic formulae [30]. √ ( ) √ 𝜇𝑒𝑞 1 − 𝜇𝑒𝑞 ∕4 1 − 𝜇𝑒𝑞 ∕2 𝛾= , 𝜉𝑙 = √ ( (11) )( ), 1 + 𝜇𝑒𝑞 4 1 + 𝜇𝑒𝑞 1 − 𝜇𝑒𝑞 ∕2 where 𝜇𝑒𝑞 = 𝑚𝑒𝑞 ∕(𝑚𝑡 + 𝑚) = (𝛽𝑒𝑞 𝜂)∕((1 − 𝛽𝑒𝑞 )𝜂 + 1) is defined as the equivalent liquid mass ratio for the SDOF-TLD system because the tuned liquid mass mt is directly attached to the primary mass when employing the TLD. The tuned liquid element in the TLIS is purposely designed to be the same as that in the TLD for a fair comparison. The stochastic comparison ratios are defined here by comparing the dynamic responses of the SDOF-TLIS with those of the SDOF-TLD system. 𝜎𝑈 𝜎𝑈 𝜎𝐴𝑐𝑐 𝑙 𝛽𝑈 = ,𝛽 = ,𝛽 = , (12) 𝜎𝑈 ,TLD 𝐴𝑐𝑐 𝜎𝐴𝑐 𝑐 ,TLD 𝑈𝑙 𝜎𝑈 ,TLD 𝑙

which can be calculated by consulting Section 2.2.1. Here, the subscript “TLD” refers to the SDOF-TLD system.

As an example, an SDOF structure (𝜔s = 𝜋, 𝜁 = 0.02) [42] is considered to be controlled by a TLD and a TLIS (with the same tuned liquid element as the TLD), and both variations are tested for different liquid mass ratios 𝜂 of 0.01, 0.025, and 0.05, respectively. For the supplements in the TLIS, supposing that the value of 𝜅 is constant at 0.25, the variation trends of 𝛽𝐴𝑐𝑐 , 𝛽𝑈 , and 𝛽𝑈 are calculated with continuously chang𝑙 ing 𝜇 and 𝜉. Figs. 4–6 show the results in form of surf plots, where 𝜇 and 𝜉 are represented in logarithmic scales. The variation principle derived in this section holds true for the other parameter sets. For 𝛽𝐴𝑐𝑐 (Fig. 4) and 𝛽𝑈 (Fig. 5) of structures with different typical liquid mass ratios (i.e., 𝜂 = 0.01, 0.025, and 0.05), the supplements—an inerter element, a damping element, and a tuning spring element undoubtedly improve the structural performance in terms of reducing the absolute acceleration and displacement responses, corresponding to the area 𝛽𝐴𝑐𝑐 < 1 and 𝛽𝑈 < 1. Notably, the much lower responses of 𝛽𝐴𝑐𝑐 and 𝛽𝑈 (upper-right corners in Figs. 4 and 5) are provided by the TLIS with the parameter set comprising large values of 𝜇 (> 0.50) and 𝜉 (> 0.30) by utilizing the potential ability for energy storage and dissipation. These variation trends suggest that it is a prior way to incorporate a large inertance for structural vibration control. It is also notable that this improvement results in almost no cost in terms of additional physical mass. Comparing 𝛽𝐴𝑐𝑐 and 𝛽𝑈 for different values of 𝜂 shows that this improvement provided by inerter-based system (𝜇, 𝜅, and 𝜉) in the proposed TLIS is prominent for all values of 𝜂 (𝜂 ∈ [0, 0.05]) that are commonly adopted by the tuned liquid element. Regarding the dynamic performance 𝛽𝑈 (Fig. 6) of the secondary sys𝑙 tem TLIS, the incorporation of a damping element and a tuning spring element substantially isolates the tuned liquid element, resulting in a significant reduction in the liquid dynamic response. To suppress the displacement of the liquid mass as much as possible, a large damping ratio is preferred and 𝜉 plays a dominant role compared with 𝜇. More importantly, a comprehensive examination of 𝛽𝐴𝑐𝑐 , 𝛽𝑈 , and 𝛽𝑈 reveals that 𝑙

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International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 6. Contour plots of the sloshing displacement comparison ratio 𝛽𝑈 by comparing the sloshing displacement response of the SDOF-TLIS system with that of the 𝑙 SDOF-TLD system for inertance mass ratio 𝜇 ∈ [0.01, 3.0] and damping ratio 𝜉 ∈ [0.01, 1.0], values of liquid mass ratio: (a) 𝜂 = 0.01, (b) 𝜂 = 0.025, and (c) 𝜂 = 0.05.

Fig. 7. Surf plots of acceleration response mitigation ratio 𝛼𝐴𝑐𝑐 of structure with TLIS for stiffness ratio 𝜅 ∈ [0.001, 1.0] and damping ratio 𝜉 ∈ [0.01, 1.0] (the location of the minimum 𝛼𝐴𝑐𝑐 is marked by the yellow triangle). The values of inertance mass ratio: (a) 𝜇 = 0.10, (b) 𝜇 = 0.25, (c)𝜇 = 0.50.

when 𝛽𝐴𝑐𝑐 and 𝛽𝑈 of the primary structure are minimized, the dynamic response of the secondary system, that is, the TLIS, is also controlled to a very low level (𝛽𝑈 < 0.20). The proposed TLIS is a viable solution for 𝑙 multiple target control in the controlled structure, potentially providing high performance levels for both the primary structure and the control device. 3. Optimization of TLIS 3.1. Parametric analysis A comprehensive parametric study is conducted to explore the variation pattern of the structural performance with changes to the key parameters of the TLIS (𝜇, 𝜅, 𝜉, 𝜂, and 𝛾), the purpose being to discover rational principles for TLIS parameter design. A series of stochastic response mitigation ratios are given by comparing the case of the SDOFTLIS system with that of the uncontrolled SDOF structure in a comprehensive evaluation. The acceleration response mitigation ratio 𝛼𝐴𝑐𝑐 , displacement response mitigation ratio 𝛼𝑈 , and control force response ratio 𝛼𝐹 are given as 𝛼𝐴𝑐𝑐 =

𝜎𝐴𝑐𝑐

𝜎𝐴𝑐 𝑐 ,0

, 𝛼𝑈 =

𝜎𝑈

𝜎𝑈 , 0

= , 𝛼𝐹 =

𝜎𝐹

𝜎𝐹 , 0

,

(13)

where the vibration control effectiveness (𝛼𝐴𝑐𝑐 and 𝛼𝑈 ) and the control cost (𝛼𝐹 ) are evaluated simultaneously. The SDOF structure (𝜔s = 𝜋) in Ref. [42] is adopted as an example in this section. Typically, the liquid used in the TLIS is water so that the damping effect of the liquid can be ignored to avoid overestimation of TLIS performance; using high viscosity liquid for vibration control is not considered for this reason. Supposing that 𝛾 = 1.0 and ℎ∕𝐿 = 0.50 (which are commonly used values, and the corresponding 𝛽 eq can be calculated using Eq. (2)), and

ignoring the inherent damping of the SDOF-TLIS system, the variation trends of 𝛼𝑈 (analytical Eq. (10)), 𝛼𝐴𝑐𝑐 , and 𝛼 F (see the numerical results in Section 2.2.2) are calculated with continuously changing 𝜅 and 𝜉 using Eq. (13). Figs. 7 and 8 show the results in the form of surf plots, where 𝜅 and 𝜉 are represented in logarithmic scales. As expected for a tuned-like system, the performance of the TLIS improves (i.e., 𝛼𝐴𝑐𝑐 and 𝛼𝑈 reduce correspondingly) with increasing inertance-mass ratio 𝜇, similar to a TLD system. Obviously, this improvement in structural performance occurs at the expense of increased control costs (evaluated by 𝛼𝐹 ). Comparing the vibration pattern of 𝛼𝐴𝑐𝑐 (Fig. 7) and 𝛼𝑈 (Fig. 8) for given constant 𝜇, both vibration shapes are characterized as a valley in the geomorphology, whose bottoms (𝛼𝐴𝑐𝑐 and 𝛼𝑈 with low values) are highlighted by the blue-colored areas. A more detailed comparison reveals that the locations of the parameter set (𝜅 and 𝜉) with the minimum responses of 𝛼𝐴𝑐𝑐 and 𝛼𝑈 are depicted by the yellow and red triangles inside the curves, respectively. Plotting the two locations of the yellow and red triangles in Fig. 8 shows that they are close to each other, implying that the minimizations of 𝛼𝐴𝑐𝑐 and 𝛼𝑈 potentially refer to almost the same optimal design result. More importantly, the minimization of 𝛼𝑈 can be expressed in a mathematical form and solved using the analytical solutions in Section 2.2.2, which will be explained in the next section. If the minimization of 𝛼𝐴𝑐𝑐 occurs prior to that of 𝛼𝑈 , the optimal results can be preliminarily substituted by those obtained from the minimization of 𝛼𝑈 , because these results are derived in closed form for easy to use and provide the simultaneous efficacious reduction of 𝛼𝐴𝑐𝑐 and 𝛼𝑈 . Inspecting the results of 𝛼𝐹 in Fig. 9, the surf plots have the appearance of slopes whose bottoms are located at the lower right corners, unlike those of 𝛼𝐴𝑐𝑐 and 𝛼𝑈 . Inspired by this phenomenon, it is reasonable to balance the vibration control effect and control cost (evaluated

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International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 8. Surf plots of displacement response mitigation ratio 𝛼𝑈 of structure with TLIS for stiffness ratio𝜅 ∈ [0.001, 1.0] and damping ratio 𝜉 ∈ [0.01, 1.0] (the locations of the minimum 𝛼𝐴𝑐𝑐 and 𝛼𝑈 are marked by the yellow and red triangles, respectively). The values of inertance mass ratio: (a) 𝜇 = 0.10, (b) 𝜇 = 0.25, (c) 𝜇 = 0.50.

Fig. 9. Surf plots of control force response ratio 𝛼𝐹 of structure with TLIS for stiffness ratio 𝜅 ∈ [0.001, 1.0] and damping ratio 𝜉 ∈ [0.01, 1.0]. The values of inertance mass ratio: (a) 𝜇 = 0.10, (b) 𝜇 = 0.25, (c) 𝜇 = 0.50.

by 𝛼 F ) to seek the optimization result with the minimum control cost while meeting specific performance demands. This solution is defined as the most economical solution in this study.

3.2. Overview of optimal design criteria The analysis results obtained thus far indicate that the proposed TLIS is an effective and economical approach for structural vibration control, because it entails almost no cost for the additional physical mass of the inerter. If the primary structure was installed with an existing water tank as a tuned liquid element, the TLIS can also be interpreted as a developed control device by introducing the inerter-based system as a supplement to the existing tuned liquid element. As summarized above, the optimal design in this study is divided over two conditions, that is, the optimization of the TLIS in the presence and absence of an existing tuned liquid element. In the first condition, the optimization design of the TLIS is limited by the given tuned liquid element with a constant 𝜂 so that the vibration control force is partially determined by this existing liquid mass, which is not an essential key index in this design. The key parameters (𝜅, 𝜉, 𝜇, and 𝛾) of the TLIS should be determined toward attaining the best control effect (i.e., minimum structural response). In the second condition, the TLIS design includes consideration of the parameters of the tuned liquid element (𝜂 and 𝛾) and supplemental inerter-based system (𝜇, 𝜅, and 𝜉). The most economical solution with simultaneous consideration of structural performance and control cost is suggested.

3.3. Parameter determination approach 3.3.1. Minimization of 𝛼 Acc Assuming a tuned liquid element with a given liquid mass, the parameter determination of the TLIS includes four key parameters, that is, 𝛾, 𝜇, 𝜅, and 𝜉, where the liquid frequency ratio 𝛾 can be adjusted by changing the length L and width b of the liquid tank without changing the fixed value of the liquid mass and the quiescent liquid depth h. As stated in the optimal design criterion, to produce a manifest improvement for the existing tuned liquid element, the parameter set of 𝜅 and 𝜉 marked by the red triangle (Fig. 8) is adopted as a rational result to minimize the acceleration and displacement responses simultaneously. Inspired by the contour plots in Fig. 8, this design condition can be described mathematically in that the partial derivatives of 𝛼𝑈 with respect to both 𝜅 and 𝜉 are set as zero. 𝜕𝛼𝑈 𝜕𝜅

=0,

𝜕𝛼𝑈 𝜕𝜉

=0 .

(14)

Substituting Eqs. (10) and (13) into Eq. (14) and solving these parameter design constraints, the values of 𝜅 and 𝜉 can be quantified as the functions of 𝜇, 𝜂, 𝛾, and 𝛽 eq , whose detailed expressions can be found in Appendix A2. These derived analytical formulae can be used for design purposes to identify the supplement stiffness and damping parameters of the TLIS with an existing tuned liquid element when 𝜇 is predetermined. When 𝜂 and 𝛽 eq (parameters of the existing tuned liquid element) are constant, the vibration mitigation effect of the TLIS is merely dependent on the undetermined parameters 𝜇 and 𝛾. To further investigate the impact of 𝜇 and 𝛾 on the performance of the TLIS, a parametric analysis is

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International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 10. Responses of structure-TLIS system with an existing tuned liquid element for inertance-mass ratio 𝜇 ∈ [0.01, 10.0] and liquid frequency ratio 𝛾 ∈ [0.80, 1.20]. (a) acceleration response mitigation ratio 𝛼 Acc and (b) displacement response mitigation ratio 𝛼 U .

Fig. 11. Contour plots of acceleration response mitigation ratio 𝛼 Acc and control force response ratio 𝛼 F of structure with TLIS for stiffness ratio 𝜅 = 0.5, inertance-mass ratio 𝜇 ∈ [0.01, 3.0] and damping ratio 𝜉 ∈ [0.01, 1.0].

conducted for the example used in Section 3.1, where 𝜅 and 𝜉 are optimized using Eq. (14). Fig. 10 shows the corresponding results of 𝛼𝐴𝑐𝑐 and 𝛼𝑈 . It is observed that a larger inertance (𝜇 > 0.50) not only significantly reduces the dynamic response of the structure (𝛼𝐴𝑐𝑐 and 𝛼𝑈 ), but also enhances the robustness of the TLIS against changes in 𝛾. As shown in the area marked by the dashed lines, for a TLIS designed with a large inertance, 𝛼𝐴𝑐𝑐 and 𝛼𝑈 are insensitive to the change in the liquid sloshing frequency thereby resisting the unwanted disturbance of this frequency and avoiding a significant reduction in vibration mitigation effect. Through an effective parameter search and optimization of 𝜇 and 𝛾, it is possible to maximize the control effect of the TLIS compared with that using a bare tuned liquid element. In this section, the essential performance index 𝛼𝐴𝑐𝑐 is the suggested aspect that needs to be minimized. It also represents the performance level of shear force (for a base-fixed structure) and the isolation effect (for an equivalent isolated structure). Until now, the parameter determination problem for the TLIS design problem is interpreted in a mathematical form in the extremum condition. ⎧minimize 𝛼 (𝜇, 𝜅, 𝜉, 𝛾) 𝐴𝑐𝑐 { ⎪ ⎨subject to 𝜕𝛼𝑈 ∕𝜕𝜅=0 . ⎪ 𝜕𝛼𝑈 ∕𝜕𝜉=0 ⎩

(15)

In the process of solving this optimization problem, 𝜅 and 𝜉 are determined as the functions of undetermined parameters (𝜇, 𝛾) in the ana-

lytical formulae to meet the constraint conditions. The optimal design results can be obtained numerically until the minimum 𝛼𝐴𝑐𝑐 is reached. 3.3.2. Performance-demand-based design for TLIS Regarding the criterion for the second design condition, namely, design of a TLIS without an existing tuned liquid element, it is rational and practical to consider the structural performance (𝛼𝐴𝑐𝑐 ) and control cost (𝛼𝐹 ) simultaneously. The two evaluation indices are compared in a more detailed manner via the contour plots shown in Fig. 11 for 𝜂=0.05, 𝛾=1.0, and 𝜅 = 0.5 as an example. When drawing the two contour lines in one figure, the solid lines refer to the contour of 𝛼 Acc , while 𝛼𝐹 is drawn in a color-filled contour format. Each solid yellow line represents a specific level of 𝛼𝐴𝑐𝑐 , and any point on the contour has the same value of 𝛼𝐴𝑐𝑐 . However, under the premise of the same 𝛼 Acc , the control costs of different points (located on the same solid line) vary within a wide range because the points are located in different color-filled segments. From the perspectives of phenomenon-based inspiration and control cost minimization, the optimal design results correspond to the parameter set referring to the minimized control cost, as marked by the yellow points. Following the demand-oriented design framework, for a given target acceleration mitigation level 𝛼𝐴𝑐 𝑐 ,𝑡 , the optimization problem of TLIS can be expressed mathematically to pursue the most economical solutions. {

minimize 𝛼𝐹 (𝜇, 𝜅, 𝜉, 𝛾, 𝜂) . subject to 𝛼𝐴𝑐𝑐 (𝜇, 𝜅, 𝜉, 𝛾, 𝜂) ≤ 𝛼𝐴𝑐 𝑐 ,𝑡

(16)

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International Journal of Mechanical Sciences 164 (2019) 105171

Table 2 Optimal results in the extremum condition obtained with and without consideration of 𝜁 = 𝜉 l = 0. 𝜇

Optimal design parameters 𝜅 𝜉

0.25 0.50 1.00

0.19 (0.14) 0.27 (0.26) 0.35 (0.36)

0.04 (0.05) 0.12 (0.11) 0.24 (0.24)

∗

The values within the rounded brackets are obtained from the analytical formulae in Appendix A2. Table 3 Case information pertaining to existing tuned liquid elements of Cases I to III. Case ID

𝜂

Structural response levels 𝛼U 𝛼 Acc

Case I Case II Case III

0.01 0.025 0.05

0.78 0.68 0.60

0.79 0.71 0.64

Based on the above-mentioned discussion and design criteria, the proposed design procedure for the TLIS for the two conditions can be summarized in the following chart. 4. Examples In this section, the proposed TLIS is designed for the two conditions to show the design results of the proposed method and illustrate their respective effectiveness: (1) the optimization of TLIS in the presence of an existing tuned liquid element, and (2) the optimization of TLIS for the target 𝛼 Acc in the absence of an existing tuned liquid element.

Table 4 Results of optimized TLIS using Eq. (15) in Cases I to III. Case ID

Optimal parameters 𝛾 𝜇 𝜅

𝜉

Performance results 𝛽 Acc 𝛽U 𝛽𝑈𝑙

Case I Case II Case III

1.19 1.53 1.18

0.16 0.22 0.22

0.37 0.38 0.43

0.67 0.94 0.90

0.32 0.36 0.34

0.39 0.42 0.47

0.04 0.07 0.11

4.1. Design of TLIS with a given tuned liquid element First, we illustrate the problem associated with the application of the derived analytical design formulae, which are relevant to an undamped structure and undamped liquid, to the design of a damped structure and damped liquid element (the model in Section 2.2.1). As an example, TLISs with different inertance-mass ratios are considered in Table 2, corresponding to the primary structure in Section 2.3, and 𝜂 = 0.025, 𝛾 = 0.9, and 𝜉 l = 0.05 (i.e., typical values for a conventional optimized TLD). The optimal results obtained for the damped and undamped structures (using the analytical design presented in Appendix A2) are summarized in Table 2. It can be observed that the derived analytical formulae can optimize 𝜅 and 𝜉 with satisfactory accuracy compared with the values obtained from the numerical analysis of a damped structure. In addition, as the inertance-mass ratio increases, the differences between the two methods reduce to an acceptable range. Assuming that a large inertance is preferentially designed to achieve significant vibration control effect and increase the robustness of the TLIS (as analyzed in Section 3.3.1), the given analytical design formulae can be adopted to simplify the TLIS design and ensure accuracy. Utilizing the proposed design method in Section 3.3.1, specific improvements to the typical tuned liquid elements due to the inerterbased system are presented in this section. Three cases are summarized in Table 3, where the existing tuned liquid elements are considered with three commonly used values of 𝜂 (0.01, 0.025, and 0.05) and predesigned using Eq. (11). The corresponding control effects (𝛼 Acc and 𝛼 U ) Fig. 12. Proposed parameter determination procedures for TLIS.

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International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 13. Acceleration responses of structure with existing tuned liquid element and designed TLIS compared with the original structure (in Case III: 𝛼 Acc = 0.60 and 𝛽 Acc = 0.43) under the excitations of (a) El Centro earthquake, (b) Kobe earthquake, (c) Chi-Chi earthquake, and (d) Niigata earthquake.

Fig. 14. Displacement responses u of structure with existing tuned liquid element and designed TLIS compared with the original structure (in Case III, 𝛼 U = 0.64 and 𝛽 U = 0.47) under the excitations of (a) El Centro earthquake, (b) Kobe earthquake, (c) Chi-Chi earthquake, and (d) Niigata earthquake.

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International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 15. Displacement responses ul of structure with existing tuned liquid element and designed TLIS (in Case III:𝛽𝑈𝑙 = 0.11) under the excitations of (a) El Centro earthquake, (b) Kobe earthquake, (c) Chi-Chi earthquake, and (d) Niigata earthquake.

Fig. 16. Comparative results of Cases IV to VI in terms of (a) acceleration response mitigation ratio 𝛼 Acc and (b) liquid mass ratio 𝜂.

are also listed. With the given 𝜂, the other design parameters (𝛾, 𝜇, 𝜅, and 𝜉) of the TLIS are optimized by following the procedure seen in Fig. 12, and the results are summarized in Table 4. Inspection of the optimal parameter results shows that a large inertance is always designed to pursue the minimum responses of controlled structures, especially for the tuned liquid element with a relatively large 𝜂. In addition, as shown by the performance, the secondary inerter-based subsystem significantly improves the dynamic performance (with the minimum 𝛽 Acc and 𝛽 U being 0.37 and 0.39, respectively). Due to the added damping element cd and spring element kd , the tuned liquid element is isolated, and the corresponding displacement response is reduced to a low level (𝛽𝑈𝑙 ≤ 0.11). Time history analyses were conducted for verification by adopting the designed parameters listed in Table 4. As examples, the dynamic responses of structures under four typical excitations [43], namely, the Imperial Valley (1940) (El Centro N-S record), Kobe (1995), Chi-Chi (1999), and Niigata (2004) earthquakes, are shown in Figs. 13–15. The original structure refers to the uncontrolled SDOF structure. Simultane-

ously, the corresponding values of 𝛼 Acc , 𝛽 Acc , 𝛼 U , 𝛽 U , and 𝛽𝑈𝑙 are shown. Under the excitations with rich frequency component, the optimized TLIS is undoubtedly effective at reducing the structural accelerations and displacements, which is accompanied by a low displacement response of the tuned liquid. More importantly, the improvements due to the addition of the inerter-based subsystem to a bare tuned liquid element (a conventional tuned-like system) cause the new system (TLIS) to exhibit a more effective response mitigation in the early stage and suppress the peak responses in a timelier manner. 4.2. Design of TLIS under target 𝛼 Acc In this section, the TLIS is designed under the most economical design condition by referring to the design equations in Eq. (16). It has been shown in the previous section the supplements of the inerter-based system are undoubtedly efficacious at providing higher controlled structural performance compared with that provided by the same tuned liquid element. It is meaningful to investigate the performance of the TLIS

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International Journal of Mechanical Sciences 164 (2019) 105171

Table 5 Case information and design results in Section 4.2 (Cases IV to VI). Case ID

𝛼 Acc,t

Optimal parameters 𝜂 𝛾 𝜇

𝜅

𝜉

Performance results 𝛼F 𝛽 Acc 𝛽𝜂

Case IV Case V Case VI

0.60 0.45 0.30

0.0031 0.0056 0.0182

0.062 0.118 0.182

0.03 0.06 0.24

1.00 0.75 0.50

0.98 0.96 0.86

0.08 0.16 0.81

93.8% 88.8% 63.6%

0.054 0.085 0.191

Fig. 17. Transfer function curves of acceleration and control force of structure with designed TLIS in Cases IV to VI. (a) acceleration response HAcc (i𝜔) and (b) control force response HF (i𝜔).

Fig. 18. Acceleration responses of structure with typical TLD and designed TLIS (in Case IV, 𝛼 Acc = 0.60 and 𝛽 Acc = 1.0) under the excitations of (a) El Centro earthquake and (b) Chi-Chi earthquake.

when it provides the same or even higher vibration mitigation effects than an optimized TLD. Taking Case III as a measure of contrast, the target acceleration mitigation ratios of the TLIS are predetermined as 𝛼 Acc,t = 0.60, 0.45, and 0.30, which are respectively 1.0, 0.75, 0.50 times of 𝛼 Acc provided by the TLD. The corresponding design results are summarized in Table 5 and plotted in Fig. 16, where 𝛽𝜂 = 1 − 𝜂𝑇 𝐿𝐼𝑆 ∕𝜂𝑇 𝐿𝐷 is the liquid mass reduction ratio between 𝜂 of the TLIS and that of the TLD. Table 5 shows that the proposed design method under the most economical condition can effectively ensure optimized control cost. Moreover, the control cost (𝛼 F ) increases when a higher vibration mitigation effect is required. Simultaneously, the optimal design criterion implicitly optimizes the tuned liquid mass into a small value to reduce the control force of the TLIS, substantially reflecting the design conception of lightweight-based optimization. Only a much smaller tuned liquid mass (the minimum 𝛽 𝜂 being 63.6%) is needed for the TLIS to provide the same (𝛽 Acc = 1.0) or

even higher (𝛽 Acc = 0.75, 0.50) structural performance compared with that offered by the TLD. As the target performance demand increases when 𝛼 Acc,t is decreased from 0.60 to 0.30, the tuned liquid mass is gradually increased to meet the demand, which is consistent with the principle of the tuned-like system. Further, a large inertance is beneficial to reduce the physical weight of the tuned liquid mass by partially substituting its function. The frequency–domain transfer function curves of HAcc (i𝜔) and HF (i𝜔) are also provided for design Cases IV to VI. These curves show that the designed TLISs are effective for response mitigation. As 𝛼 Acc,t decreases, the resonant vibration reduces more obviously, while the frequency bandwidth of vibration mitigation increases at the expense of a larger control force (Fig. 17). To verify the dynamic performance of the TLD and TLIS structures under various excitations, time history analyses are conducted. The structural displacement and acceleration responses for Case IV are

Z. Zhao, R. Zhang and Y. Jiang et al.

International Journal of Mechanical Sciences 164 (2019) 105171

Fig. 19. Displacement responses u of structure with typical TLD and designed TLIS (in Case IV, 𝛼 U = 0.64) under the excitations of (a) El Centro earthquake and (b) Chi-Chi earthquake.

shown in Figs. 18 and 19, respectively, as examples under the excitations of the El Centro and Niigata earthquakes. The time domain response curves indicate that the designed lightweight TLIS is effective at suppressing the concerned responses at the same time, where 𝛽 Acc and 𝛽 U are both close to 1.0. Under different excitations, the TLIS optimized using the stochastic method can provide the desired structural performance (same as that of the TLD structure) at the optimized control cost and a much lower added liquid mass.

multi-degree-of-freedom structures with the TLIS will be studied in the future. Declaration of Competing Interest This manuscript has not been published or presented elsewhere in part or in entirety and is not under consideration by another journal. We have read and understood your journal’s policies, and we believe that neither the manuscript nor the study violates any of these.

5. Conclusion Acknowledgments This study proposes a novel TLIS by employing the synergy benefits of an inerter-based subsystem and a tuned liquid element to achieve the lightweight-based control and improved control performance. Correspondingly, an optimal design framework is developed for the TLIS by simultaneously considering the structural performance and control force, especially yielding analytical design formulae for the TLIS in the presence of an existing tuned liquid element to optimize the control effect. The main conclusions of this study can be summarized as follows. •

•

•

•

The supplementary inerter-based subsystem in the TLIS partly substitutes the function of the tuned liquid mass at the same cost as that of virtually negligible physical mass, which makes the TLIS an effective solution for lightweight control. While providing the same or even higher vibration mitigation effects as the optimized TLD, more than 60% of the liquid mass is saved in the designed TLIS. The proposed TLIS is substantially effective for vibration mitigation of structural absolute acceleration and displacement responses. In comparison with the classic TLD for a given liquid mass, the employment of the inerter-based subsystem in TLIS undoubtedly improves the structural performance. For the example used in this study, the acceleration and displacement responses of the SDOF-TLIS are only 40% those of the classic SDOF-TLD. Furthermore, the dynamic response of the tuned liquid is also significantly reduced. The optimal design method that considers the most economical design condition can effectively meet the target structural performance demand and optimize the control cost simultaneously. This method also implicitly allows the reduction of tuned liquid mass, reflecting the design conception of lightweight-based optimization. In particular, to optimize the TLIS in the presence of an existing tuned liquid element, the derived analytical design formulae can be used to determine the additional stiffness and damping elements of the inerter subsystem in the TLIS that constitute the most efficient parameter set to minimize the structural responses. The parametric analysis indicates that the utilization of a large inertance in the TLIS increases its resistance to liquid vibration frequency disturbance and provides substantial vibration mitigation effects.

Notably, this study was only concerned with SDOF structures with the TLIS. The structural performance and optimal design method of the

This study was supported by the National Natural Science Foundation of China (Grant no. 51978525 and 51778489), the Shanghai Pujiang Program(Grant no. 17PJ1409200), and the Fundamental Research Funds for the Central Universities (Grant no. 22120180064). Appendix A1. Closed-form expressions of mean-square response for the undamped SDOF-TLIS system 𝑁𝑈 𝜋 (( ) )2 2𝛾 2 𝜉 −1 + 𝛾 2 (𝜂 + 𝜇) + 𝜂𝛽𝑒𝑞 𝜔3𝑠 ⎡𝜂 4 𝜅(1 + 𝜅) + 𝜇 2 + 𝜅 2 (1 + 𝜇)2 − 𝜅𝜇(2 + 𝜇) + 4(1 + 𝜇)𝜉 2 ⎤ ⎡ ( ) ⎥ ⎢( 3 2 ) ⎢ ⎢ 𝛾 − 𝛾 3 2 ⎢+ 2𝜂 𝜅(𝜇+𝜅(2+𝜇)) +2𝜉 +)2𝜂(𝜇+ 𝜅(−1 − 2𝜇 ⎥ 2 + 4𝜇𝜉 2 ⎢ ⎥ 𝜅( 1 + 𝜇) 2 + 𝜇) + 6 𝜉 + ( ) ⎢ ⎢+ 𝜂 2 (1 + 𝜅 (−3 + 𝜇 2 ) + 𝜅 2 (6 + 𝜇(6 + 𝜇)) + 4(3 + 𝜇)𝜉 2 ) ⎥ ⎢ ⎣ ⎦ ⎢ ⎡ ⎢ ⎛𝜂 3 𝜅(1 + 𝜅) + 𝜇+ 𝜂 2 𝜅(𝜇+ 𝜅(5 + 𝜇))⎞ ⎢ ⎢𝜅 2 (𝜂+𝜇)2 − 2𝛾 2 ⎜+ 𝜅(−1 − 2𝜇+𝜅(2+3𝜇)) ⎟ ⎢ ⎢ ⎜ ⎟ 𝑁𝑈 = ⎢ ⎝+ 𝜂(1 + 𝜅(−3 + 𝜅(6 + 4𝜇))) ⎠ ⎢ ⎢ ⎢−4𝛾 2 (2𝜂(3 + 𝜂) − (−3 + 𝜇)(1 + 𝜇))𝜉 2 ( ) ⎢ ⎢ 1 + 𝜅) + 𝜅(−2 + 5𝜅 + 4(−1 + 𝜅)𝜇) + 2 𝜇+ 8𝜉 2 ⎞ ⎛2𝜂 3 𝜅( ⎢+ 𝜂𝛽𝑒𝑞 ⎢ ( ) 4 2 2 ⎟ ⎢ ⎢+ 𝛾 ⎜+ 𝜂 𝜅(9𝜅 + 2(1 + 𝜅)𝜇) +8𝜉 ( ) ⎜ ⎟ ⎢ ⎢ 2 + 𝜇)) + 12𝜉 2 ⎝+(2𝜂 1 + 3𝜅(−1 + 𝜅( ⎠ ⎢ ⎢ ( ( ) ⎢ ⎢+ 𝜂𝛽𝑒𝑞 −2𝜅 2 (𝜂+𝜇) + 𝛾 2 1 + −3 + 𝜂 2 𝜅 + (6 + 𝜂(6 + 𝜂))𝜅 2 ) ) ⎢ ⎢ 2 2 ⎣ ⎣+ 4(3 + 𝜂 − 𝜇)𝜉 + 𝜂𝜅 𝛽𝑒𝑞 𝜎𝑈2 =

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤⎥ ⎥⎥. ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦⎦ (A1)

A2. Analytical expressions of optimal 𝜉 and 𝜅 in the presence of a tuned liquid element Substituting Eqs. (10) and (13) into Eq. (14), and solving these parameter design constraints, the values of 𝜅 and 𝜉 can be quantified as functions of 𝜇, 𝜂, 𝛾, and 𝛽 eq . ( ( ) 𝑁 ) 𝑁𝜉 𝜅 𝜇, 𝜂, 𝛾, 𝛽𝑒𝑞 = 𝜅 , 𝜉 𝜇, 𝜂, 𝛾, 𝛽𝑒𝑞 = , 𝐷𝜅 𝐷𝜉

(A2)

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International Journal of Mechanical Sciences 164 (2019) 105171

where (( ) )(( ) 𝑁𝜅 = −𝛾 2 −1 + 𝛾 2 (𝜂 + 𝜇) + 𝜂𝛽𝑒𝑞 −1 + 𝛾 2 (1 + 𝜂) ( ) ) × (−2 − 𝜇 + 𝜂(−1 + 𝜂 + 𝜇)) + 𝜂 −3 + 𝜂 2 𝛽𝑒𝑞 , ⎡( )2 𝐷𝜅 = ⎢ −1 + 𝛾 2 (1 + 𝜂)2 (1 + 𝜂 + 𝜇) ⎢ ⎣ ⎛(𝜂 + 𝜇)2 − 2𝛾 2 (1 + 𝜂)(2 + 3𝜇 + 𝜂(4 + 𝜂 + 𝜇))⎞⎤ ⎟⎥, + 𝜂𝛽𝑒𝑞 ⎜+𝛾 4 (1 + 𝜂)(5 + 4𝜇 + 𝜂(7 + 2𝜂 + 2𝜇)) ( ) ⎟⎥ ⎜ ⎝+𝜂𝛽𝑒𝑞 𝛾 2 (6 + 𝜂(6 + 𝜂)) − 2(𝜂 + 𝜇) + 𝜂𝛽𝑒𝑞 ⎠⎦

(A3)

1∕2

⎡𝜂 4 𝜅(1 + 𝜅) + 𝜇 2 + 𝜅 2 (1 + 𝜇)2 − 𝜅𝜇(2 + 𝜇) + 2𝜂 3 (𝜅(𝜇 ⎤⎤ ⎡ ⎢ ⎥⎥ ⎢( ) 𝜅( 2 + 𝜇) + ) ) 2 ( ( )⎥⎥ ⎢ 𝛾 − 𝛾3 ⎢ 2 2 ⎢+2𝜂(𝜇+𝜅(−1 − 2𝜇+ 𝜅(1 + 𝜇)(2 + 𝜇))) + 𝜂 1 + 𝜅 −3 + 𝜇 ⎥⎥ ⎢ ⎢+𝜅 2 (6 + 𝜇(6 + 𝜇))) ⎥⎥ ⎢ ⎣ ⎦⎥ ⎢ ⎢ ⎥ ⎡ ⎤ ⎛𝜂 3 𝜅(1 + 𝜅) + 𝜇+ 𝜂 2 𝜅(𝜇+ 𝜅(5 + 𝜇))⎞ ⎢ ⎥ ⎢𝜅 2 (𝜂+𝜇)2 − 2𝛾 2 ⎜+𝜅(−1 − 2𝜇+ 𝜅(2 + 3𝜇)) ⎥ ⎟ ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ 𝑁𝜉 = ⎢ ⎥ ⎝+ 𝜂(1 + 𝜅(−3 + 𝜅(6 + 4𝜇))) ⎠ ⎢ ⎥ ⎢ ⎥ ( ) ⎥ ⎢ 3 2 ⎢ ⎥ ⎛2𝜂 𝜅(1 + 𝜅) + 𝜅(−2 + 5𝜅 + 4(−1 + 𝜅)𝜇) + 2 𝜇+ 8𝜉 ⎞⎥ ⎢ ⎢+𝜂𝛽𝑒𝑞 ⎢+𝛾 4 ⎜+𝜂 2 (𝜅(9𝜅 + 2(1 + 𝜅)𝜇) +8𝜉 2 ) ⎥ ⎟⎥ ) ( ⎢ ⎥ ⎜ ⎟⎥ ⎢ 2 ⎝+2𝜂 1 + 3𝜅(−1 + 𝜅(2 + 𝜇)) + 12𝜉 ⎠⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢+𝜂𝛽 (𝛾 2 (1 + 𝜅 (−3 + 𝜂 2 + (6 + 𝜂(6 + 𝜂))𝜅 )) ⎥ ⎢ ⎥ 𝑒𝑞 ⎢ ⎥ ⎢ ⎥ ) ⎢ ⎥ ⎣ ⎦ − 2𝜅 2 (𝜂+𝜇) +𝜂𝜅 2 𝛽𝑒𝑞 ⎣ ⎦ [ ( )2 2 2 𝐷𝜉 = 2𝛾 −1 + 𝛾 (1 + 𝜂) (1 + 𝜂+ 𝜇) ( 2 )]1∕2 2𝛾 (1 + 𝜂)(2 + 𝜂) − 2𝜂(3 + 𝜂) + 𝜂𝛽𝑒𝑞 . + (−3 + 𝜇)(1 + 𝜇) +𝜂(3 + 𝜂 − 𝜇)𝛽𝑒𝑞

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