Hindawi Publishing Corporation Shock and Vibration Volume 2015, Article ID 760394, 15 pages http://dx.doi.org/10.1155/2015/760394

Research Article Seismic Response Reduction of Structures Equipped with a Voided Biaxial Slab-Based Tuned Rolling Mass Damper

Shujin Li,1 Liming Fu,1,2 and Fan Kong1

1 School of Civil Engineering and Architecture, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, China 2Hubei Synthetic Space Building Technology Co. Ltd., Wuhan 430070, China

Correspondence should be addressed to Fan Kong; [email protected]

Received 25 February 2015; Accepted 14 May 2015

Academic Editor: Kumar V. Singh

Copyright © 2015 Shujin Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper proposes a novel tuned mass damper (TMD) embedded in hollow slabs of civil structures. The hollow slabs inthis context, also referred to as “voided biaxial reinforced slabs,” feature a large interior space of prefabricated voided modules that are necessary in the construction of this special structural system. In this regard, a tuned rolling mass damper system (“TRoMaDaS”) is newly proposed, in combination with hollow slabs, to act as an ensemble passive damping device mitigating structural responses. The main advantage of this TMD configuration lies in its capacity to maintain architectural integrity. To further investigate the potential application of the proposed TRoMaDaS in seismic response mitigation, theoretical and numerical studies, including deterministic and stochastic analyses, were performed. They were achieved by deterministic dynamic modeling using Lagrange’s equation and the statistical linearization method. Finally, the promising control efficacy obtained from the deterministic/stochastic analysis confirmed the potential application of this newly proposed control device.

1. Introduction papers and several important textbooks have been published in this area, such as [2, 3]. Among these structural control Civil structures exposed to ambient dynamic excitation often methods, the most widely used and also most mature is exhibit excessive responses that need to be mitigated, from passive control, whereas active control and semiactive control considerations of safety and/or serviceability issues of the suffer from various practical or theoretical challenges that structures. Traditionally, relying on the enhancement of need further investigation. material strength and cross-sectional area, structure/element Tuned mass dampers (TMDs), attached to the primary responses can be reduced to some level. However, this structure, consist of a block of mass, a stiffness-restoring treatment of suppressing structure/element responses is not element, and an energy dissipation element. These are now economically feasible, especially for structures located in the one of the most widely used passive control devices, espe- disastrous earthquake- or gust-prone areas. An alternative cially against wind-induced vibration in high-rise buildings, and smarter measure to suppress structural response is so- because of their advantages in terms of operation, mainte- called structural control. nance, and fabrication (see [4] for details). Begun by Yao [1], structural control techniques are However, the traditional configuration of the TMDs intended to mitigate structure responses to random ambient requires dedicated large space to accommodate the huge scale excitation. They have been developed for decades and some of the mass and the corresponding functional components of them have matured with large advances in both theoretical providing damping and stiffness. As an example, the Chifley and practical aspects. The central concept of structural Tower in Sydney, a 52-storey 209-m tall steel structure, is control theory, in fact, relies on the changing of structural fitted with a single pendulum-type TMD suspended by steel parameters or external excitations appropriately, passively, cables at the 44th floor4 [ ]. The 400 t block of steel with actively, semiactively, or with combinations of them, leading dimensions of 4 × 4 × 4mandthestroke(about±910 mm) to mitigated structure responses. To date, many research of the mass indicates the size of the dedicated space needed. 2 Shock and Vibration

Moreover, the suspension cables need to stretch across two response can, in the context of earthquake excitations. In floors (from the 44th to the 46th floors) to tune the natural this study, we also show that different vertical distributions frequency of the damper to the frequency of the tower. Other and parameter configurations of the TRoMaDaS lead to examples include the platform-type TMD system, consisting different control efficiencies, necessitating a more complete 3 of a concrete mass of 373 t (about 150 m )withastrokeof investigation of the optimum design of TRoMaDaS in future ±1.14m,employedatthe63rdfloorofthe278mtallCiticorp studies. Finally, to investigate the control efficiency from a Center in New York City [4]. Even for the more compact probabilistic perspective, a statistical linearization method, configuration of the inverted pendulum-type TMD installed along with a pertinent Monte Carlo simulation, was used to at the 100 m level of the 134 m high Sky Tower in Nagoya [5], calculate the standard deviation of the response displacement Japan, the dimensions of the mass block are 2.5 m square × of a structure with and without the proposed TRoMaDaS. 3 m, with a stroke of 150 mm. Furthermore, the installation of The stochastic analysis of the structure revealed that not only a TMD as an additional component leads to, in some sense, was the displacement standard deviation reduced to a large an inconsistency in architectural styles. degreebutalsothestochasticresponseofthestructurereadily Recently, a structure system with a voided biaxial slab became stationary because of the TRoMaDaS. has been used widely in practical engineering, especially in Europe [6, 7]andChina[8]. Originating from one- spanning hollow-core slabs, recent developments on voided 2. Voided Biaxial Slab-Based Tuned Rolling slab technology have been aimed at reducing the dead Mass Damper weight and thus enhancing the span of the structure floors. 2.1. Structural System with Voided Biaxial Slabs. The voided Specifically, this is achieved by laying less heavy material, biaxial reinforced , constructed using prefabri- such as polystyrene or polypropylene made hollow modules cated voided modules and cast in situ , has between the reinforcement of beams/ribbed beams, and thus gained wide attention since its emergence as a new structural displacing concrete with less structural benefit. A typical system [6–10]. These ecofriendly voided biaxial slabs have technology used in Europe is the so-called “BubbleDeck” many advantages over conventional solid concrete slabs, such biaxial voided slab [9, 10], in which spherical hollow modules as lower total cost, reduced material use, enhanced structural composed of recycled industrial plastic are used, allowing the efficiency, and decreased construction time, that have con- hollow slab to act as a normal monolithic two-way spanning tributed to their wide application in civil construction. concrete slab. For another form of hollow module, see the “U- Taking one kind of voided biaxial slab used widely in boot” [7] technology, as an example. One of the main features China as an example, the construction procedures for this of these hollow modules is the large voided interior space. structural system are as follows. First, the prefabricated hol- In this context, we propose a novel form of distributed low box-like modules are located between the reinforcement TMDthatwehavenamedthetunedrollingmassdamper grids of the main beams and the ribbed beams. Next, box- system (“TRoMaDaS”) for this specific structure, taking like voided modules are used as the side when advantageofthelargespaceinthevoidedmodulethathas site-casting the concrete beams. In this way, only the bottom not before been considered for structural control. Specifically, of the concrete slab are needed, thus saving the rolling ball TMD, first suggested by Pirner [11]and considerable construction costs. To guarantee the integrity Naprstek´ and colleagues [12] is, for the first time, proposed between the prefabricated hollow modules and the circum- to be distributed in the hollow floors of the structures. This jacent reinforced concrete, nonstructural measures, such as arrangement of the TMDs is expected to not only suppress extending the reinforcement bar into the main beams/ribbed the structure response but also maintain the consistency of beams, may be implemented. The vertical dimension of a thearchitecturalstyleinthesenseofarchitecturalesthetics. To validate the control efficacy of the proposed passive prefabricated hollow module, according to the spans of main control system, a physical and mathematical model of the beams, ranges from 200 to 900 mm. Figure 1 shows the con- controlled chain-like MDOF structure was developed. In this struction procedures of a kind of voided biaxial slab in detail. regard, Lagrange’s equation was used to derive the coupled equationsofmotionofthecontrolledsystem,inwhichthe 2.2. Motivation of Voided Biaxial Slab-Based Tuned Rolling stroke of the rolling mass is considered a small quantity Mass Damper. We noted the unique large space in the hollow and nonlinearity is exhibited in the damping force. The modules and thus proposed to locate certain structural or derivation of the dynamic equation from the physical model architectural elements within these modules. Specifically, is quite general, in which any TRoMaDaS with arbitrary these additional elements can be designed as a part of heating, parameters can be considered. The Runge-Kutta method of ventilation and air conditioning, water supply and sewerage, the fourth order was used to solve the dynamic equations and structural control systems, without altering the architec- numerically for cases with different TRoMaDaS parameters tural appearance excessively. Among these architectural and or distributions. We conclude that, for all cases, with the structural requirements, structural safety and serviceability equipment of the proposed TRoMaDaS, the input energy are the most important issues. In this regard, we proposed of an ambient excitation is absorbed to a large degree. to install passive control devices in the hollow modules to Furthermore,ithasbeenfoundthatbecauseofthe“stick- mitigate the dynamic responses of the structure. slip” behavior of friction-type TMDs [13], the peak value of Among the passive control devices, the passive vibra- response cannot be efficiently reduced, whereas the postpeak tion absorber or the tuned mass damper (TMD) has been Shock and Vibration 3

(a) (b)

Figure 1: Construction procedures for a hollow-ribbed floor. (a) Locate prefabricated hollow box-like modules between the reinforcement grids. (b) Cast reinforced concrete beams between the box-like hollow modules.

researched widely and used extensively in newly constructed buildings and in the retrofitting of existing structures, because of its simplicity, relatively low construction and maintenance costs, and safety. In this regard, TMDs of the pendulum type [4]arethemostwidelyused,especially for slender towers and high-rise buildings. However, this configuration requires dedicated large vertical and horizontal Figure 2: Voided biaxial slab-based tuned rolling mass damper. dimensions to accommodate the extra weight, as mentioned in Section 1, especially for high-rise buildings with a low natural fundamental frequency. Moreover, safety measures, such as mechanical stops or other braking systems, are also lies in its architectural integrity. By locating the passive required to limit the excessive travel of the mass block. control device in an existing space (shown in construction An alternative TMD configuration is the type mounted site artwork; Figure 1), installation of the voided biaxial slab- on a large complex mechanical platform [14, 15], which based tuned rolling mass damper system (“VBS-TRoMaDaS” provides damping and stiffness using hydrostatic bearings, or“TRoMaDaS”)doesnotalterthearchitecturalinterior hydraulic cylinders, pneumatic springs, or coil springs. It can appearance excessively. A diagram of a TRoMaDaS with a be argued that, compared with TMDs of the pendulum type, single rectangular hollow module is shown in Figure 2. the platform-type has increased complexity and thus costs more in terms of daily operations and maintenance. Thus, 2.3. Characteristics of TRoMaDaS. By appropriately selecting we concluded that neither of these TMD configurations was parameters, including the radius of the arc path and the suitable for a vibration mitigation device to be installed in the oscillator (the natural frequency of the TRoMaDaS depends modules of the voided biaxial slab. on these parameters as shown in Section 3), the friction Another more compact and also simple configuration is coefficient between the arc path and the oscillator (the the one with an object rolling on a tridimensional surface. The damping capacity of the TRoMaDaS depends on this), and dampingmechanismandrestoringforceofthisconfiguration the material/density of the oscillator, an optimum con- are provided by the rolling friction and the resulting force trolled structure can be obtained. From the review above, of the gravity of the rolling mass, respectively. Specifically, itcanbeobservedthat,comparedwithpendulum-type the so-called tuned rolling mass damper (TRMD) or the ball and platform-type TMDs, the energy dissipation mechanism vibration absorber (BVA) consists of a ball rolling along an of the TRoMaDaS is more compact and simpler. These arch path located in hollow modules to absorb the structural promising features, naturally, lead to a more durable passive kinetic and potential energy as the result of the ambient control system and reduce considerably the cost of fabrica- excitations. In fact, the TRMD has been applied in several tion, maintenance, and operation. Moreover, the proposed other engineering structures, such as long-span bridges [16], TRoMaDaS is expected to suppress the coupled lateral and TV towers [11], and wind turbines [17, 18]. However, to our torsional motions of structures, due to accidental or intended knowledge, very few applications of TRMDs or BVAs in eccentricities between their mass and stiffness centers. In building structures to suppress seismic vibrations have been this regard, Singh et al. [21] investigated the efficiency of reported, except those proposed recently by Fisher and Pirner four TMDs of the dashpot-spring type, placed along two [19]andMattaetal.[20]. Moreover, compared with the orthogonal directions in pairs, to mitigate the torsionally cou- “traditional” configuration of TMDs in high-rise buildings, pled responses of irregular multistorey buildings. Jangid and where the TMDs are always installed as an additional energy- Datta [22] conducted a parametric study of the effectiveness absorbing element, the most remarkable advantage of the of multi-TMDs along the width of an eccentric structure installation of TRMDs in hollow modules embedded in slabs in reducing torsionally coupled responses. However, both 4 Shock and Vibration

m F mn,1 n,n𝑛 n ··· Mn 𝜃 TRoMaDaS Voided i,j Ri,j kn cn biaxial slab F mn−1,1 n−1 ··· Mn−1 y fi,j 𝜔i,j A . . . . 𝜓 i,j 𝜃i,j x B m2,1 m2,n2 ri,j F2 ··· M2 mi,jg N Initial state A i,j k2 c2

m1,1 m1,n1 Figure 4: Free-body diagram of a TRoMaDaS. F1 ··· M1

k1 c1 simplicity, the TRoMaDaS in each floor contains several (𝑚 ,𝑚 ,...,𝑚 ) Figure 3: Lumped-mass model of a shear-type structure equipped oscillators with different masses 𝑖,1 𝑖,2 𝑖,𝑛𝑖 rolling with a TRoMaDaS. along the axial-symmetrical sphere surfaces, where 𝑚𝑖,𝑗 represents the 𝑗thoscillatorlocatedonthe𝑖th floor. From Figure 3, it can be seen that the total degrees of freedom are 𝑛+(𝑛1 +𝑛2 + ⋅⋅⋅ + 𝑛𝑛),with𝑛 degrees of freedom of investigations focused on a theoretical analysis of mitigating translational motion of the lumped mass and (𝑛1+𝑛2+⋅⋅⋅+𝑛𝑛) torsionally coupled responses using ideal physical models, degrees of freedom of the oscillators’ rotation with respect whereas the practical feasible multi-TMDs configuration has to the centers of the arch paths, respectively. A free-body not been considered. diagram of the TRoMaDaS is shown in Figure 4,where𝑅𝑖,𝑗, In this aspect, the proposed voided biaxial slab-based 𝑟𝑖,𝑗, 𝑖=1, 2,...,𝑛𝑖, 𝑗=1, 2,...,𝑛𝑖, are the radii of the (𝑖, 𝑗)th TRoMaDaS can be used as a platform to realize opti- arc path and oscillator, respectively, 𝜃𝑖,𝑗, 𝜓𝑖,𝑗 are the rotation mally controlled structures with various damper distri- angles of the oscillator with respect to the center of the path butions/arrangement and parameters. Specifically, because and of the oscillator ball itself, and 𝜔𝑖,𝑗 = 𝜓̇𝑖,𝑗 denotes the there is no restriction on the direction of the rolling motion angular velocity of the oscillator. or the location/distribution of the oscillator, the proposed TRoMaDaS can, in fact, be fabricated as a type of spatial multi-TMDssystemthatcanbeusedtocontrolstructure 3.2. Governing Equation of the TRoMaDaS-Controlled Struc- motion in every direction and even torsional motion caused ture. Thus, the kinetic and potential energy of the controlled by eccentric structures or bidirectional base excitations. structure can be written as Moreover, to fully use the spatial movement of the oscillator, a three-dimensional TRoMaDaS with a nonaxially symmet- 𝑛 𝑛 rical three-dimensional path surface can be used to control 𝑖 2 1 ̇ ̇ motions of spatial structures with different natural mode 𝑇=∑∑ { 𝑚𝑖,𝑗 [𝑥𝑖 +𝜌𝑖,𝑗𝜃𝑖,𝑗 cos 𝜃𝑖,𝑗] 2 frequencies in two orthogonal directions simultaneously. This 𝑖=1𝑗=1 concept was initially proposed by Matta et al. [20]inthe 1 ̇ 2 1 2 1 2 (1) context of rolling-pendulum TMDs and can be used to extend + 𝑚𝑖,𝑗 [𝜌𝑖,𝑗𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗] + 𝐽𝑖,𝑗𝜔𝑖,𝑗 + 𝑀𝑖𝑥𝑖̇ }, 2 2 2 the possible application of the TRoMaDaS in structures with different frequencies in two orthogonal directions. 𝑖=1, 2,...,𝑛; 𝑗=1, 2,...,𝑛𝑖,

𝑛 𝑛 𝑛𝑖 1 2 3. Equation of Motion of a Controlled 𝑉=∑ 𝑘𝑖 (𝑥𝑖 −𝑥𝑖−1) + ∑∑𝑚𝑖,𝑗𝑔𝜌𝑖,𝑗 (1 − cos 𝜃𝑖,𝑗), 2 Structure with TRoMaDaS 𝑖=1 𝑖=1𝑗=1 (2) 𝑖= , ,...,𝑛; 𝑗= , ,...,𝑛, 3.1. Simplified Model of a TRoMaDaS-Controlled MDOF 1 2 1 2 𝑖 Structure. Consider a mass-lumped model of a shear-type structure equipped with a TRoMaDaS in each hollow floor respectively, where 𝑥𝑖, 𝑥𝑖̇ are the translational displacement (Figure 3), where (𝑀1,𝑀2,...,𝑀𝑛), (𝑘1,𝑘2,...,𝑘𝑛),and and velocity of the lumped mass, 𝜌𝑖,𝑗 =𝑅𝑖,𝑗 −𝑟𝑖,𝑗 is the radius (𝑐1,𝑐2,...,𝑐𝑛) are the mass, stiffness, and the damping coef- ̇ ficient of the main structure, respectively, and 𝐹𝑗, 𝑗= difference between the arc path and the oscillator, 𝜃𝑖,𝑗 is the 1, 2,...,𝑛, are the external ambient excitations, including angular velocity of the oscillator with respect to the center of 2 the equivalent force of base excitation or wind loading. For the arc path, and 𝐽𝑖,𝑗 = 2𝑚𝑖,𝑗𝑟𝑖,𝑗/5 is the rotational inertia of Shock and Vibration 5 the 𝑗th oscillator on the 𝑖th floor. Furthermore, (1)-(2) can be coefficient of the rolling friction (in meters) between the 𝑗th simplified to oscillator and the sphere surface on the 𝑖th floor. Combining (4)-(5), the nonconservative force can be written as 𝑛 𝑛𝑖 2 2 2 1 ̇ ̇ ̇ ̇ ,𝑠 𝑇=∑∑ 𝑚𝑖,𝑗 (𝑥𝑖 + 2𝑥𝑖𝜌𝑖,𝑗𝜃𝑖,𝑗 +𝜌𝑖,𝑗𝜃𝑖,𝑗) 𝑄nc =𝐹+[−(𝑥̇ − 𝑥̇ )𝑐 +(𝑥̇ − 𝑥̇ )𝑐 ], 2 𝑖 𝑖 𝑖 𝑖−1 𝑖 𝑖+1 𝑖 𝑖+1 𝑖=1𝑗=1 (6) 𝑖=1, 2,...,𝑛, 1 2 2̇ 1 2 + 𝑚𝑖,𝑗𝜌𝑖,𝑗𝜃𝑖,𝑗 + 𝑀𝑖𝑥𝑖̇ , 5 2 𝜃̇ 𝑅 ,o 𝑖,𝑗 𝑖,𝑗 𝑄nc =−󵄨 󵄨 𝜇 𝑚 𝑔, 𝑖= , ,...,𝑛; 𝑗= , ,...,𝑛, (3) 𝑖,𝑗 󵄨 ̇ 󵄨 𝑖,𝑗 𝑖,𝑗 1 2 1 2 𝑖 󵄨𝜃𝑖,𝑗󵄨 𝑟𝑖,𝑗 (7) 𝑛 𝑛 𝑖 2 1 1 2 𝑖=1, 2,...,𝑛; 𝑗=1, 2,...,𝑛𝑖. 𝑉=∑∑ [ 𝑘𝑗 (𝑥𝑗 −𝑥𝑗−1) + 𝑚𝑖,𝑗𝑔𝜌𝑖,𝑗𝜃𝑖,𝑗] , 𝑖=1𝑗=1 2 2 In fact, (6) is the expression for the viscous damping force 𝑖=1, 2,...,𝑛; 𝑗=1, 2,...,𝑛𝑖, and the external excitation of main structures, whereas (7) denotes the rolling friction force between the oscillator and when the angular motion of the oscillator (𝜃𝑗) is considered the sphere surface. to be small. Note in (3) that the compatibility condition of The governing equation of the controlled structure can be displacement 𝑅𝑖,𝑗𝜃𝑖,𝑗 =(𝜃𝑖,𝑗 +𝜓𝑖,𝑗)𝑟𝑖,𝑗 is used. It may be derived by invoking the Lagrange equation: argued that the small-quantity assumption for the oscillator 𝜕𝑇 𝜕𝑇 𝜕𝑉 response and (3) tend to be reasonable in cases of lower-level d ( )− + =𝑄nc, 𝑡 𝜕𝑞̇ 𝜕𝑞 𝜕𝑞 𝑖 external excitations. However, it should be noted that not d 𝑖 𝑖 𝑖 (8) only may the structure undergo nonlinear behavior but also 𝑖= , ,..., 𝑛+(𝑛 +𝑛 +⋅⋅⋅+𝑛 ), the oscillator response may surpass a certain small-quantity 1 2 2 1 2 𝑛 level in the case of intense excitations. Although the small- 𝑞 𝑞̇ quantity assumption is important and often used in the sim- where 𝑖, 𝑖 arethegeneralizeddisplacementandvelocityof 𝑖 plification of governing equations of pendulum- or rolling- the th coordinate. Combining (3) and (8) and considering (𝜃 , 𝜃̇ ) pendulum-type TMDs, further examination of the applica- the angular displacement and velocity 𝑗 𝑗 to be small bility of this assumption during intense external excitation is quantities, one can obtain necessary. 𝑛 𝑛 𝑄nc 𝑖 𝑖 The nonconservative force 𝑗 , including the external ̈ (𝑀𝑖 + ∑𝑚𝑖,𝑗) 𝑥𝑖̈ + ∑𝑚𝑖,𝑗𝜌𝑖,𝑗𝜃𝑖,𝑗 +(𝑥𝑖 −𝑥𝑖−1)𝑘𝑖 force, the damping force of the structure, and the rolling 𝑗=1 𝑗=1 friction force between the sphere surface and the oscillator, (9a) can be derived by the virtual work principle; that is, −(𝑥𝑖+1 −𝑥𝑖)𝑘𝑖+1 +(𝑥𝑖̇ − 𝑥𝑖−̇ 1)𝑐𝑖 −(𝑥𝑖+̇ 1 − 𝑥𝑖̇ )𝑐𝑖+1 𝑛 =𝐹𝑖, 𝛿𝑊 = ∑ {𝐹𝑖𝛿𝑥𝑖 𝑖=1 5𝜇𝑖,𝑗𝑔𝑅𝑖,𝑗 5𝑔𝜃𝑖,𝑗 𝜃̈ + (𝜃̇ )+ =− 5 𝑥̈ , +[−(𝑥̇ − 𝑥̇ )𝑐 +(𝑥̇ − 𝑥̇ )𝑐 ]𝛿𝑥} 𝑖,𝑗 𝜌2 𝑟 sgn 𝑖,𝑗 𝜌 𝜌 𝑖,𝑗 𝑖 𝑖−1 𝑖 𝑖+1 𝑖 𝑖+1 𝑖 (4) 7 𝑖,𝑗 𝑖,𝑗 7 𝑖,𝑗 7 𝑖,𝑗 (9b) 𝑛 𝑛 𝑖 𝛿𝑠 𝑖,𝑗 𝑖=1, 2,...,𝑛; 𝑗=1, 2,...,𝑛𝑖, − ∑∑𝑓𝑖,𝑗 ,𝑖=1, 2,...,𝑛; 𝑗=1, 2,...,𝑛𝑖, 𝑖=1𝑗=1 𝑟𝑖,𝑗 where sgn(⋅) denotes the sign function. Equations (9a) and where 𝛿𝑊 denotes the total virtual work of the nonconserva- (9b) are a set of coupled nonlinear ordinary differential equations, even with the small-quantity assumption of the tive force, 𝑠𝑖,𝑗 =𝑅𝑖,𝑗𝜃𝑖,𝑗 =𝑟𝑖,𝑗(𝜓𝑖,𝑗 +𝜃𝑖,𝑗) is the arc length of the 𝑛 𝑥̈ ∑ 𝑖 𝑚 + oscillator movements between the initial state and state being rolling motion. On the left side of (9a),theterms 𝑖 𝑗=1 𝑖,𝑗 𝑛𝑖 ̈ considered, and 𝑓𝑖,𝑗 is the moment of the rolling friction force ∑𝑗=1 𝑚𝑖,𝑗𝜌𝑖,𝑗𝜃𝑖,𝑗 can be regarded as the control force produced with respect to center of the oscillator and can be determined by the translational and rotational motion of the oscillator, by counteracting part of the external force, 𝐹𝑖.Moreover,the right side of (9b) shows that the motion of the oscillators 𝜃̇ 𝑓 =− 𝑖,𝑗 𝜇 𝑁 is caused by the supported base acceleration, which is the 𝑖,𝑗 󵄨 󵄨 𝑖,𝑗 𝑖,𝑗 󵄨𝜃̇ 󵄨 accelerationofthemainstructure.Theleftsideof(9b) shows 󵄨 𝑖,𝑗󵄨 𝑗 (5) the undamped natural frequency of the th oscillator on the 𝜃̇ 𝑖th floor depends only on the radius difference between the 𝑖,𝑗 2̇ =−󵄨 󵄨𝜇𝑖,𝑗𝑚𝑖,𝑗 (𝑔 cos 𝜃𝑖,𝑗 +𝜌𝑖,𝑗𝜃 ). arch path and the oscillator and can be written as 󵄨 ̇ 󵄨 𝑖,𝑗 󵄨𝜃𝑖,𝑗󵄨 5𝑔 In (5), 𝑁𝑖,𝑗 is the 𝑗th reaction force perpendicular to the 𝜔𝑖,𝑗 = √ . (10) 7𝜌𝑖,𝑗 tangent of the contact point on the 𝑖th floor and 𝜇𝑖,𝑗 is the 6 Shock and Vibration

M = I, This expression is consistent with that derived by Zhang and 4 colleagues [17]andChenandGeorgakis[18]. Equations (9a) K = (K1, K2,...,K𝑛), and (9b) canberewritteninacompactform: 2 diag 2 2 2 𝑔 1 Mz̈ + Cż + Kz + f (z, ż) = w (𝑡) , (11) K𝑖 = 5 ⋅ ( , 1 ,..., 1 ), 2 diag 𝜌 𝜌 𝜌 7 𝑖,1 𝑖,2 𝑖,𝑛𝑖 where 𝑘1 +𝑘2 −𝑘2 M1 M2 −𝑘 𝑘 +𝑘 −𝑘 M =[ ], 2 2 3 3 K1 =( ), M3 M4 d

C1 0 −𝑘𝑛 𝑘𝑛 C =[ ], (12) 00 𝑐1 +𝑐2 −𝑐2

K1 0 −𝑐2 𝑐2 +𝑐3 −𝑐3 K =[ ] C1 =( ), 0 K2 d

−𝑐𝑛 𝑐𝑛 are the mass, damping, and stiffness coefficients; the aug- 𝑇 𝑇 𝑇 𝑇 𝑇 (14) mented displacement is z =[x , 𝜃1 , 𝜃2 ,...,𝜃𝑛 ] with x = 𝑇 𝑇 (𝑥1,𝑥2,...,𝑥𝑛) , 𝜃1 =(𝜃1,1,𝜃1,2,...,𝜃1,𝑛 ) , 𝜃2 =(𝜃2,1,𝜃2,2, 1 where diag()denotes the diagonal matrix. The nonlinear dif- ...,𝜃 )𝑇 𝜃 =(𝜃,𝜃 ,...,𝜃 )𝑇 2,𝑛2 ,and 𝑛 𝑛,1 𝑛,2 𝑛,𝑛𝑛 ; the external ferential equation (11) canbesolvedbynumericalalgorithms, excitation on the structure at the right side of (11) is w = 𝑇 𝑇 𝑇 for example, a fourth-order Runge-Kutta method. [w𝑠 , 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟,0,...,0 ] with w𝑠 =(𝐹1,𝐹2,...,𝐹𝑛) ,andthenonlin- 𝑛1+𝑛2+⋅⋅⋅+𝑛𝑛 ear damping force can be written as 3.3. Numerical Examples. Consider a six-floor structure with the lumped-mass model (Figure 1)subjecttoearthquake excitations. In the present numerical example, the external 5𝜇1,1𝑔𝑅1,1 𝐹 =−(𝑀+ 𝑓 (z, ż) = [⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, ,..., ; excitation in this case can be written as 𝑖 𝑖 0 0 0 2 𝑛𝑖 𝑛 𝜌 𝑟1,1 ∑ 𝑚𝑖,𝑖)𝑥𝑔̈ (𝑡),where𝑥𝑔̈ (𝑡) is the earthquake acceleration, [ 7 1,1 𝑗=1 eachlumpedmassisassumedtobethesame,at16.32t,the 5𝜇1,2𝑔𝑅1,2 5𝜇1,𝑛 𝑔𝑅1,𝑛 linear stiffness coefficients of each storey are 13.51, 12.86, 11.58, ⋅ (𝜃̇ ), (𝜃̇ ),..., 1 1 3 sgn 1,1 𝜌2 𝑟 sgn 1,2 𝜌2 𝑟 9.65, 7.07, and 3.86 × 10 kN/m, and the damping coefficients 7 1,2 1,2 7 1,𝑛1 1,𝑛1 (13) of each storey are selected as 27.9, 23.76, 21.39, 17.87, 13.07, and 𝜇 𝑔𝑅 𝜇 𝑔𝑅 . ⋅ ̇ 5 𝑛,1 𝑛,1 ̇ 5 𝑛,2 𝑛,2 10 04 kN s/m. To reduce the dynamic response of a MDOF ⋅ sgn (𝜃1,𝑛 );...; sgn (𝜃𝑛,1), 1 2 2 structure efficiently, the natural frequency of the damper 7𝜌𝑛,1𝑟𝑛,1 7𝜌𝑛,2𝑟𝑛,2 needs to be resonant with the structure’s natural frequencies (6.28, 15.38, 24.32, 33.23, 42.13, and 51.02 rad/s). In this 5𝜇𝑛,𝑛 𝑔𝑅𝑛,𝑛 ⋅ (𝜃̇ ),..., 𝑛 𝑛 (𝜃̇ )] . respect, the natural frequency of the TRoMaDaS can be tuned sgn 𝑛,2 2 sgn 𝑛,𝑛𝑛 7𝜌𝑛,𝑛 𝑟𝑛,𝑛 𝑛 𝑛 ] to the first-mode frequency, by which the structure response is dominated. In fact, the optimum physical parameters of Furthermore, submatrices in (12) are the tuned mass damper including the frequency and the damping coefficient, in this case relative to the rolling friction 𝑛1 𝑛2 coefficient of the oscillator, are functions of the mass ratio M1 = diag (𝑀1 + ∑ 𝑚1,𝑗,𝑀2 + ∑ 𝑚2,𝑗,...,𝑀𝑛 between the main structure mass to the damper mass and 𝑗=1 𝑗=1 other structural parameters. It can be argued that, by only

𝑛 tuning the frequency of the control device to the first-mode 𝑛 frequency of the structure, preferable control efficiency can + ∑ 𝑚𝑛,𝑗), 𝑗=1 be expected to result in the case of a low mass ratio [23]. Research on control parameter optimization—in this case, 1 2 𝑛 the selection of the optimum oscillator mass, rolling friction M2 = diag (M2, M2,...,M2), coefficient, and the radii of the arc path and the oscillator—is M𝑖 =(𝑚 𝜌 ,𝑚 𝜌 ,...,𝑚 𝜌 ) , beyond the scope of the present paper. Invoking (10) and the 2 𝑖,1 𝑖,1 𝑖,2 𝑖,2 𝑖,𝑛𝑖 𝑖,𝑛𝑖 1×𝑛 𝑖 first-model frequency of the structure, the radius difference M = (M1, M2,...,M𝑛), between the arch paths and the oscillator can be obtained. 3 diag 3 3 3 The oscillators located in certain floors of the structure are 𝑇 assumed to be composed of iron with a mass density of 𝑖 5 1 1 1 3 M3 = ( , ,..., ) , 7800 kg/m ; the rolling friction coefficients between the arch 7 𝜌𝑖,1 𝜌𝑖,2 𝜌𝑖,𝑛 𝑖 𝑛𝑖×1 paths and the oscillators are chosen to be identical, 0.01 m. Shock and Vibration 7

×10−3 ×10−3 8 8

6 6

4 4

2 2

0 0

−2 −2 Displacement (m) Displacement Displacement (m) Displacement −4 −4

−6 −6

−8 −8 01020304050 0 1020304050 Time (s) Time (s)

Without TRoMaDaS Without TRoMaDaS With TRoMaDaS With TRoMaDaS (a) (b)

Figure 5: Comparison between interstorey drift ((a) first storey, (b) sixth storey) of structures with (solid line) and without (dashed line) 2 TRoMaDaS. Structure subjected to scaled El Centro acceleration (peak acceleration = 0.513 m/s ); TRoMaDaS oscillators with a mass ratio of 5% are distributed equally on every floor.

The mass ratio of the TRoMaDaS mass to the total mass TRoMaDaS. This phenomenon of response reduction delay of main structure, according to engineering considerations, canbealsoobservedfromthedisplacementdataofashaking couldbeselectedtobe5%.Fourcaseswithdifferentoscillator table experiment of a wind turbine structure, controlled by distributions or different levels of earthquake excitations were a similar ball vibration absorber (BVA) rolling at the top considered: of the nacelle [17]. The failure in peak response reduction may be attributable to several reasons; the most significant (a) Oscillators are equally distributed on six floors of is the stick-slip [13] behavior of ball oscillators. The steel ball structures subject to El Centro record NS component oscillators, initially resting at the bottom of the arch path, only of the 1940 Imperial Valley earthquake (El Centro begintomovealongthepathsuntiltheaccelerationofthe excitation for short) and JMA record NS component main structure exceeds the maximum acceleration of the ball of the 1995 Kobe earthquake (Kobe excitation for 2 oscillator caused by the friction force/moment. That is, the short), with a peak value of 0.513 m/s . TRoMaDaS can be regarded as being “activated” only when (b) The damper and structure parameters are the same as the oscillator starts rolling along the arch path after strong in case (a), whereas the peak values of El Centro and 2 structure acceleration has already occurred, thus leading to JMAKobeexcitationincreaseto1.026m/s . a delay in response reduction and the failure of peak-value (c) The structure and excitation parameters are the same mitigation. The maximum absolute values of the interstorey as in case (b) except that oscillators with mass ratios drift of the controlled and uncontrolled structure and the of3%and2%areinstalledonthethirdandthefourth corresponding response reduction rates, shown in Table 1, floors, respectively. further confirm the response reduction delay effect. Also, the rotation angles of the oscillators, shown in Table 1,donot (d) All the parameters are the same as in case (c), except exceed the limitation of the small quantity assumption, which that the oscillator on the fourth floor is split into 10 ∘ is often regarded as 0.3 rad (or about ca. 20 ). smaller oscillators with equal mass. The effect of the TRoMaDaS on mitigating the response Figures 5 and 6 show the comparison of inter-storey drift during the postpeak interval may benefit the damage behav- in case (a) with and without the TRoMaDaS of structures ior of structures because of material fatigue. One of the subject to El Centro and Kobe excitations. From these plots, widely used approaches to investigate damage behavior is itcanbeseenthatthepeakresponses(fortheElCentro based on the amount of energy absorbed by the materials. case, at about 3-4 s, and for the Kobe case, at about 7–9 s) To further investigate the effectiveness of TRoMaDaS, the were not reduced significantly, whereas responses during energy dissipation efficiency of the controlled structure was thetimeintervalafterthepeakvaluedecreasedtoalarge investigated. Figure 7 shows the amount of total energy input degree because of the energy dissipation mechanism of the by the earthquake excitation and dissipated by the structural 8 Shock and Vibration

Table 1: Peak response of inner-storey drifts of structures with and without TRoMaDaS with an earthquake acceleration peak value of 0.513 m/s2. Excitation Storey 1 2 3 4 5 6 Max. uncontr. disp. (mm) 6.29 6.52 6.81 7.06 7.13 7.69 Max. contr. disp. (mm) 6.02 6.30 6.77 6.98 6.92 6.73 El Centro Disp. red. rate (%) 4.29 3.37 0.58 1.13 2.94 12.48 Max. ang. disp. (rad) 0.0073 0.0077 0.0087 0.0268 0.0714 0.1870 Ma x . u n c ont r. ( m m ) 7. 3 5 7. 3 9 7. 3 5 7. 8 5 8 . 78 1 0 . 02 Max. contr. disp. (mm) 6.17 6.24 6.45 6.91 7.83 8.64 Kobe Disp. red. rate (%) 16.05 15.56 12.24 11.97 10.82 13.77 Max. ang. disp. (rad) 0.0063 0.0069 0.0099 0.0180 0.0621 0.1865

×10−3 8 0.01

6

4 0.005

2

0 0

−2 Displacement (m) Displacement Displacement (m) Displacement −4 −0.005

−6

−8 −0.01 0 1020304050607080 0 20406080 Time (s) Time (s)

Without TRoMaDaS Without TRoMaDaS With TRoMaDaS With TRoMaDaS (a) (b)

Figure 6: Comparison between the interstorey drift ((a) first storey, (b) sixth storey) of structures with (solid line) and without (dashed line) 2 TRoMaDaS. Structure subjected to scaled Kobe acceleration (peak acceleration = 0.513 m/s ); TRoMaDaS oscillators with a mass ratio of 5% are distributed equally on every floor.

damping and by the TRoMaDaS. It can be seen that, for the important factor that should be considered. In this situation, El Centro and Kobe excitation, respectively, about 37.5% and the stroke of the oscillator, say, the rotation angle of the ball, 40.1% of the input energy were dissipated by the TRoMaDaS. needs more investigation, not only from the point of view of To further investigate the response reduction efficiency the small quantity assumption for rotation angle, but more of the TRoMaDaS in case (b) of intensive earthquake exci- importantly also from the aspect of practical considerations. 2 tations, scaled base acceleration with a peak value 1.026 m/s According to numerical investigations, the rotation angle of was considered. The maximum absolute interstorey drifts of the oscillators can be reduced by properly locating these the controlled and uncontrolled structures to the El Centro oscillators on certain floors. For example, in case (c), the andKobeexcitationareshowninTable2.However,the TRoMaDaS is installed in the third and the fourth floor with maximumabsoluterotationangleoftheoscillatorinthe 3%and2%ofthestructuremass,respectively.Comparedwith topfloorreached1.05radfortheElCentroexcitationand case (b), the results of case (c) show that although the control 1.01 rad for Kobe excitation. These values, obviously, exceed efficiency in mitigating the peak response deteriorated, the the limitation of the small quantity assumption made in the rotation angle was reduced greatly, by about 50%. The rota- derivation of the governing equations, and thus the data tion angle of oscillators can be further reduced by splitting obtained in this situation cannot be used to assess the per- the oscillator with excess angular displacement. For example, formance of the TRoMaDaS in reducing structure responses. in case (d), the oscillator in the fourth floor is split into 10 In the parameter optimization procedure of a “tradi- smaller ball oscillators with equal mass. In this situation, tional” tuned mass damper, the stroke of the damper is an with a slight expense in terms of a deteriorated response Shock and Vibration 9

Table 2: Peak response of inner-storey drifts of structures with and without TRoMaDaS with an earthquake acceleration peak value of 1.026 m/s2. Excitation Storey 1 2 3 4 5 6 Max. uncontr. disp. (mm) 12.59 13.04 13.62 14.11 14.27 15.38 (b) 10.36 10.79 11.46 11.81 11.56 13.14 Max. contr. disp. (mm) (c) 11.50 12.23 12.74 12.75 12.33 13.01 (d) 12.06 12.79 13.28 13.24 12.68 13.00 (b) 17.71 17.25 15.86 16.30 18.99 14.56 El Centro Disp. red. rate (%) (c) 8.66 6.21 6.46 9.36 13.60 15.41 (d) 4.21 1.92 2.50 6.17 11.14 15.47 (b) 0.0075 0.0270 0.0993 0.3191 0.6343 1.0499 Max. ang. disp. (rad) (c) — — 0.2409 0.5213 — — (d) — — 0.2730 0.2816 — — Max. uncontr. disp. (mm) 14.70 14.78 14.71 15.70 17.56 20.04 (b) 12.03 12.25 12.71 13.59 15.32 16.36 Max. contr. disp. (mm) (c) 12.40 12.55 13.03 13.87 15.68 13.01 (d) 12.45 12.63 13.11 13.92 15.67 13.01 (b) 18.16 17.12 13.60 13.44 12.76 18.36 Kobe Disp. red. rate (%) (c) 15.65 15.09 11.42 11.66 10.71 35.08 (d) 15.31 14.55 10.88 11.34 10.76 35.08 (b) 0.0061 0.0309 0.1199 0.3082 0.6185 1.0127 Max. ang. disp. (rad) (c) — — 0.2216 0.5352 — — (d) — — 0.2618 0.2593 — —

1500 2000 Absorbed energy

1500 1000 Energy (J) Energy (J) 1000 Absorbed energy 500 500

0 0 01020304050 0 1020304050607080 Time (s) Time (s)

Input Input Structure damped Structure damped Structure damped + kinetic Structure damped + kinetic Structure damped + strain + kinetic Structure damped + strain + kinetic (a) (b)

Figure 7: Cumulative energy time history for TRoMaDaS-damped MDOF structure. Structure subjected to scaled El Centro (shown in (a) 2 2 with peak acceleration = 0.513 m/s ) and Kobe (shown in (b) with peak acceleration = 0.513 m/s ) acceleration; TRoMaDaS oscillators with a mass ratio of 5% are distributed equally on every floor.

reduction, the angular displacement is greatly reduced, below from the energy perspective. From these plots, it was further the limitations of the small quantity assumption. Figures 8 confirmed that, for both the El Centro and Kobe excitation, and 9 show a comparison of interstorey drift of the first and the peak values were not mitigated significantly, whereas the top floor with and without the TRoMaDaS in case (d). over 50% of the input seismic energy was dissipated by the Figure 10 shows the efficiency of the TRoMaDaS in case (d) proposed TRoMaDaS. 10 Shock and Vibration

0.015 0.02

0.01 0.015

0.01 0.005 0.005 0 0 Displacement (m) Displacement −0.005 (m) Displacement −0.005

−0.01 −0.01

−0.015 −0.015 0 1020304050 01020304050 Time (s) Time (s)

Without TRoMaDaS Without TRoMaDaS With TRoMaDaS With TRoMaDaS (a) (b)

Figure 8: Comparison between interstorey drift ((a) first storey, (b) sixth storey) of structures with (solid line) and without (dashed line) 2 TRoMaDaS. Structure subjected to scaled El Centro acceleration (peak acceleration = 1.026 m/s ). TRoMaDaS oscillators with a mass ratio of 5% are installed on the third (one ball) and fourth floors (10 balls with equal mass) of 3% and 2% of the main structure mass, respectively.

0.015 0.02

0.015 0.01 0.01 0.005 0.005

0 0

−0.005 Displacement (m) Displacement −0.005 (m) Displacement −0.01 −0.01 −0.015

−0.015 −0.02 020406080 020406080 Time (s) Time (s)

Without TRoMaDaS Without TRoMaDaS With TRoMaDaS With TRoMaDaS (a) (b)

Figure 9: Comparison between the interstorey drift ((a) first storey, (b) sixth storey) of structures with (solid line) and without (dashed line) 2 TRoMaDaS. Structure subject to scaled Kobe acceleration (peak acceleration = 1.026 m/s ). TRoMaDaS oscillators are installed on the third (one ball) and the fourth floor (10 balls with equal mass) of 3% and 2% of the main structure mass, respectively.

4. Stochastic Dynamic Analysis control efficiency of the proposed device in a different, prob- abilistic manner. Furthermore, the optimization of control Itcanbeseenfromthenumericalanalysisabovethat,because parameters in future investigations should be implemented of the uncertain input of the earthquake excitation, the con- similarly,basedonastochasticperspective. trol efficiency of the TRoMaDaS appears to differ for different Stochastic dynamic analysis as an engineering application excitations even with the same intensity (e.g., compare case was begun in 1958 by Crandall [24] for mechanical engineer- (d) with the El Centro and Kobe excitations). Using the ing. This theory was soon used to investigate the random nonlinear random vibration theory, one can investigate the responses of civil structures to quantify the uncertainty Shock and Vibration 11

7000 8000 6000

Absorbed energy 5000 6000

4000 4000 Energy (J) Energy (J) 3000 Absorbed energy 2000 2000 1000

0 0 01020304050 0 1020304050607080 Time (s) Time (s)

Input Input Structure damped Structure damped Structure damped + kinetic Structure damped + kinetic Structure damped + strain + kinetic Structure damped + strain + kinetic (a) El Centro excitation (b) Kobe excitation

Figure 10: Cumulative energy time history for TRoMaDaS-damped MDOF structure. Structure subjected to scaled El Centro (shown in (a) 2 2 with peak acceleration = 1.026 m/s ) and Kobe (shown in (b) with peak acceleration = 1.026 m/s ) acceleration; TRoMaDaS oscillators are installed in the third (one ball) and the fourth floor (10 balls with equal mass) of 3% and 2% of the main structure mass, respectively.

𝑐 0 propagation of ambient stochastic excitations. Later, stochas- C =[ ], tic methods considering nonlinear and even hysteretic behav- 00 ior in civil structures were developed by researchers. These methods include stochastic averaging [25], statistical lin- 𝑘 0 [ ] earization/nonlinearization [26], moment closure [27], and K = [ 5𝑔] , Monte Carlo simulations [28]. Among them, the statistical 0 𝜌 linearization and Monte Carlo methods are two that are used [ 7 ] widely, although the latter often suffers from computational 𝑤 (𝑡) inefficiency problems, especially for large-scale civil struc- w ={ }, tures. In this section, to consider the uncertainty propagation 0 and to further extend the stochastic analysis of a TRoMaDaS- (16) controlled MDOF structure, statistical linearization was used to determine the standard deviation of displacement, in a closed form, of a SDOF structure with a controlling device. where 𝑤(𝑡) is the Gaussian white noise excitation with power spectrum 𝑆0. The linearized equation of motion of the 4.1. Analytical Solution for SDOF Structure with TRoMaDaS. structure with a TRoMaDaS (15) canbewrittenintheform For simplicity and for the purposes of this preliminary ̈ ̇ investigation of the stochastic control efficiency of the pro- MZ (𝑡) + (C + C𝑒) Z (𝑡) + KZ (𝑡) =−w (𝑡) , (17) posed control system, a single-degree-of-freedom (SDOF) structural system with a single oscillator ball subject to white 𝑇 noise excitation was considered. The stochastic differential where Z =(𝑋,Θ) is the augmented displacement vector equation of motion of this system can be written as and C𝑒 is the additional equivalent damping matrix that can ̈ ̇ ̇ be appropriately determined by minimizing the difference MZ + CZ + KZ + f (Z, Z)=w (𝑡) , (15) between (15) and (17) in the mean squared sense (see [26]for details). In this regard, for a chain-like controlled structural Z where is the stochastic response and systems, C𝑒 can be determined by 𝑀+𝑚 𝑚𝜌 [ ] M = [ 5 ] , 𝑒 𝜕𝑓𝑖 1 𝑐𝑖,𝑗 =𝐸{ }, 𝑖=1, 2;𝑗=1, 2, (18) [ 7𝜌 ] 𝜕𝑥𝑗̇ 12 Shock and Vibration

2 𝑇 where f(Z, Ż)=(0, 5𝜇𝑔𝑅/(7𝜌 𝑟)sgn(Θ))̇ is the nonlinear in an explicit closed form although numerical algorithms can damping in (15). Furthermore, according to (18), be used to obtain the relationship in a discrete form. In this regard, a closed-form integration approach was developed by 00 𝐶 =[ ], Roberts and Spanos [26], providing an accurate method for eq eq (19) determining the standard deviation from the power spectrum 0 𝑐22 density(seetheAppendixforthestandarddeviationofthe 𝜇𝑔𝑅 responses in detail). 𝑐eq =𝛽 = 5 √ 2 , 22 eq 2 (20) 7𝑟𝜌 𝜎Θ̇ 𝜋 4.2. Numerical Example. Consider a SDOF structure with 𝜎 3 where Θ̇ is the standard deviation of the angular velocity of thelumpedmassbeing8× 10 kg. The stiffness and the 5 the oscillator. Furthermore, combining with (10) yields the damping ratio are 2 × 10 N/m and 0.02, respectively. The equivalent damping ratio of the nonlinear equation of motion structure is initially at rest and subject to stochastic earth- of the oscillator as quakeexcitation,modeledbyazero-meanGaussianwhite −3 −3 𝛽 noise with the power spectrum strength being 3× 10 m⋅s . eq 5𝑔 𝜇𝑅 𝜁 = = √ , (21) The natural frequency of the structure is 𝜔0 = 5 rad/s and the eq 𝜔 𝜋𝜌3 𝑟𝜎 2 0 14 Θ̇ ratio between the mass of the oscillator ball and the primary structure is selected to be 5%. To obtain preferable control 𝜔 where 0 is the natural frequency of the oscillator. efficiency, the natural frequency of the oscillator ball is tuned Next, the standard deviation of the response of an equiv- to agree with the one of the structure. In this regard, the radii alent linear system can be determined according to linear of the oscillator ball and of the arc path are 𝑟=0.337 m and random vibration theory; that is, 𝑅=𝑟+𝜌=0.617, where 𝜌=0.280 m. According to (10),the ∞ ∞ radius difference between these two radii is a factor on which 2 𝜎Ẋ = ∫ SẊ Ẋ (𝜔) d𝜔=∫ 𝜔 SXX (𝜔) d𝜔, (22) the natural frequency of the oscillator depends. −∞ −∞ To validate the randomness in the control efficiency, a comparison of the response displacements between uncon- where SẊ Ẋ (𝜔) is the power spectrum density matrix of the system displacement that can be determined by trolled and controlled structures subjected to two different sample excitations is plotted in Figure 11.Fromtheseplots,it 𝑇 can be concluded that the structure response is not mitigated SXX (𝜔) = H (𝜔) SFF (𝜔) H (𝜔) (23) efficiently until the response surpasses a certain level. This with SFF(𝜔) being the power spectrum density matrix of the is probably, according to the analysis in Section 3.3,dueto excitation w(𝑡) = (𝑤(𝑡), 0).Clearly,becauseofthespecial the stick-slip behavior of the TRoMaDaS. Moreover, it can be form of the excitation vector, in the matrix H(𝜔),only𝐻11(𝜔) seen that the “rear” part of the response displacement of the and 𝐻21(𝜔) are needed for the autospectrum 𝑆𝑋𝑋(𝜔) and controlledstructure,showninFigures11(a) and 11(b),behaves 𝑆ΘΘ(𝜔). In this regard, these frequency transfer functions can differently, confirming the need for a probabilistic investiga- be solved by tion of the control efficiency. Figure 12 shows a comparison of the standard deviation of the response displacement between M (p;𝜔) h (𝜔) = N, (24) the structures with and without TRoMaDaS. It can be seen from this figure that not only is the displacement standard where deviation reduced to a large extent but also that the stochastic M (p;𝜔) response of the structure readily becomes stationary because of the TRoMaDaS. −𝜔2 (𝑀+𝑚) + 𝑖𝜔𝑐 2+ 𝑘−𝜔 𝑚𝑝 [ ] = [ 𝜔2 𝑔] , 5. Concluding Remarks −5 −𝜔2 + 𝑖𝜔𝛽 + 5 𝜌 eq 𝜌 [ 7 7 ] A novel tuned mass damper system for civil structures named (25) 𝐻 (𝜔) tuned rolling mass damper system or TRoMaDaS has been h (𝜔) ={ 11 }, proposed in this paper. Compared with the traditional instal- 𝐻21 (𝜔) lation of tuned mass dampers, the most promising feature of the proposed passive control system is that it does not require 1 N ={ }, a dedicated space to accommodate the oscillator and thus 0 maintains the integrity of civil buildings. To investigate the seismic response reduction of the proposed passive control 𝑝 = (𝑀,𝑚,𝑐,𝑘,𝜌,𝛽 ) where eq is parameters of equivalent system, the equation of motion of the controlled system sub- linear systems. ject to external dynamic excitation was derived. In this regard, At this point, it can be argued that (20) and (22)–(24) a physical model of the passively controlled structure was constitute a mutually dependent relationship between 𝜎Θ̇ modeled as a chain-like MDOF system with arbitrary number and equivalent damping coefficient that can be solved inan of oscillators rolling within the hollow floors. Lagrange’s iterative manner. However, (22) is always difficult to obtain equation was used to derive the equation of motion of Shock and Vibration 13

0.15 0.15

0.1 0.1

0.05 0.05

0 0

−0.05 −0.05

−0.1 −0.1 Structure displacement Structure Structure displacement Structure

−0.15 −0.15

−0.2 −0.2 0 5 10 15 20 25 0 5 10 15 20 25 Time (s) Time (s)

Displacement response with TRoMaDas Displacement response with TRoMaDas Displacement response without TRoMaDas Displacement response without TRoMaDas (a) (b)

Figure 11: Comparison of displacement responses between controlled (dashed line) and uncontrolled (solid line) structures subject to −3 −3 different samples ((a) first sample, and (b) second sample) of Gaussian white noise with power spectrum strength 𝑆0 = 3 × 10 m⋅s . The frequency of the oscillator balls with a 5% mass ratio is tuned to the frequency of the main structure.

0.05 It can be concluded that, because of the so-called “stick- slip” behavior of friction-type TMDs, although the peak response cannot be mitigated efficiently, a large amount of 0.04 seismic energy input is dissipated by the TRoMaDaS. This promising feature of TRoMaDaS allows its installation in structures where fatigue is the leading cause of damage. 0.03 Furthermore, it was found that the location/distribution of the TRoMaDaS influences not only the response reduction of 0.02 the controlled structure, but also the angular displacement of the oscillators. Moreover, an increased number of oscillators locatedonacertainfloorhavetheeffectofdecreasingtheir 0.01 angular displacement, providing an alternative means of st. d. of structure displacement structure of d. st. limiting the stroke of the oscillator. Finally, the statistical linearization method was used to investigate the control 0 0 5 10 15 20 25 efficiency of the proposed control system in a probabilistic Time (s) manner. It was found that not only was the displacement standard deviation reduced by a large extent but also the Displacement response with TRoMaDas stochastic response of the structure readily became stationary Displacement response without TRoMaDas because of the TRoMaDaS. Stationary st. d. of displacement The investigations detailed in this paper on the potential Figure 12: Comparison of the standard deviation (st. d.) of the structural control capability of the proposed TRoMaDaS are response displacements between the uncontrolled structure, calcu- preliminary. Further studies should focus on investigations lated by a Monte Carlo simulation (solid line), and the controlled regarding the three-dimensional motion of the oscillator, structure, calculated by a Monte Carlo simulation (dashed line), and torsional control capacity of irregular structures, releasing of a statistical linearization method (dash-dot line). the small-quantity assumption for oscillators, and optimum control parameter selection for the proposed TRoMaDaS.

Appendix the complex coupled nonlinear dynamic system. Numerical methods such as the fourth-order Runge-Kutta method were From (22)-(23), it can be seen that the standard deviation of used to solve the dynamic equation to investigate the control thestationaryresponsecanbewritteninintegralformas efficiency of the TRoMaDaS. Several cases with different ∞ TRoMaDaS distributions and different numbers of oscillators Ξ𝑚 (𝜔) 𝐼𝑚 =∫ d𝜔, (A.1) were studied. −∞ Λ 𝑚 (−𝑖𝜔) Λ 𝑚 (𝑖𝜔) 14 Shock and Vibration where Conflict of Interests 2𝑚−2 2𝑚−4 Ξ𝑚 (𝜔) =𝜉𝑚−1𝜔 +𝜉𝑚−2𝜔 +⋅⋅⋅+𝜉0, The authors declare that there is no conflict of interests (A.2) 𝑚 𝑚−1 regarding the publication of this paper. Λ 𝑚 (𝜔) =𝜆𝑚 (𝑖𝜔) +𝜆𝑚−1 (𝑖𝜔) +⋅⋅⋅+𝜆0.

Equation (A.1) can be further calculated by division of two Acknowledgments determinants involving coefficients in (A.2);thatis, This work was supported financially by the National Natural 󵄨 𝜉 𝜉 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ 𝜉 󵄨 󵄨 𝑚−1 𝑚−2 0 󵄨 Science Foundation of China (NSFC) (Grant no. 51408451), 󵄨 −𝜆 −𝜆 −𝜆 −𝜆 ⋅⋅⋅ 0 ⋅⋅⋅ 0 󵄨 󵄨 𝑚 𝑚−2 𝑚−4 𝑚−6 󵄨 the Natural Science Foundation of Hubei Province (Grant no. 󵄨 0 −𝜆𝑚−1 −𝜆𝑚−3 −𝜆𝑚−5 −𝜆𝑚−7 ⋅⋅⋅ 00󵄨 󵄨 0 𝜆 −𝜆 −𝜆 −𝜆 ⋅⋅⋅ 00󵄨 󵄨 𝑚 𝑚−2 𝑚−4 𝑚−6 󵄨 2014CFB841), and the Fundamental Research Funds for the 󵄨 . 󵄨 󵄨 . 󵄨 Central Universities of China (Grant no. WUT-2014-IV-051). 󵄨 00 ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ −𝜆2 𝜆0 󵄨 𝜋 𝐼𝑚 = ⋅ . (A.3) 󵄨 𝜆𝑚−1 −𝜆𝑚−3 𝜆𝑚−5 𝜆𝑚−7 ⋅⋅⋅ 0 ⋅⋅⋅ 0 󵄨 𝜆 󵄨 −𝜆 −𝜆 −𝜆 −𝜆 ⋅⋅⋅ 0 ⋅⋅⋅ 0 󵄨 𝑚 󵄨 𝑚 𝑚−2 𝑚−4 𝑚−6 󵄨 References 󵄨 0 −𝜆𝑚−1 −𝜆𝑚−3 −𝜆𝑚−5 −𝜆𝑚−7 ⋅⋅⋅ 00󵄨 󵄨 0 𝜆 −𝜆 −𝜆 −𝜆 ⋅⋅⋅ 00󵄨 󵄨 𝑚 𝑚−2 𝑚−4 𝑚−6 󵄨 [1]J.T.P.Yao,“Conceptofstructuralcontrol,”Journal of Structural 󵄨 . 󵄨 󵄨 . 󵄨 Division, vol. 98, no. 7, pp. 1567–1574, 1972. 󵄨 00⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ −𝜆2 𝜆0 󵄨 [2]T.T.SoongandG.F.Dargush,Passive Energy Dissipation Sys- Specifically, the standard deviation of the rotational angular tems in , John Wiley & Sons, Chichester, displacement of the oscillator with respect to the center of the UK, 1997. arc path can be calculated as [3]G.W.Housner,L.A.Bergman,T.K.Caugheyetal.,“Struc- ∞ ∞ tural control: past, present, and future,” Journal of Engineering 2 󵄨 󵄨2 Mechanics,vol.123,no.9,pp.897–971,1997. 𝜎Θ = ∫ 𝑆ΘΘ (𝜔) d𝜔=∫ 𝑆0 󵄨𝐻21 (𝜔)󵄨 d𝜔 (A.4) −∞ −∞ [4] K. C. S. Kwok and B. Samali, “Performance of tuned mass dampers under wind loads,” Engineering Structures,vol.17,no. which can be calculated with (A.3) and 9, pp. 655–667, 1995. 2𝑚 [5] Y. Tamura, “Suppression of wind-induced vibrations of build- 𝜆4 =𝑀+ , (A.5a) 7 ings,” Journal of Wind Engineering,vol.1990,no.44,pp.71–84, 1990. 𝜆 =𝑐+(𝑀+𝑚) 𝛽 , 3 eq (A.5b) [6] T. Lai, Structural behavior of BubbleDeck slabs and their appli- cation to lightweight bridge decks [M.S. thesis],Massachusetts 𝑐𝛽 𝑞 (𝑀+𝑚) eq5 Institute of Technology, Cambridge, Mass, USA, 2009. 𝜆2 =𝑘+ , (A.5c) 7𝜌 [7] A. Churakov, “Biaxial hollow slab with innovative types of 𝑐𝑔 voids,” Construction of Unique Buildings & Structures,vol.6,pp. 𝜆 =𝛽 𝑘+5 . 70–88, 2014. 1 eq (A.5d) 7𝜌 [8]B.-D.Li,J.-J.Li,andL.-M.Fu,“Researchonthebonding properties of case-in-place reinforced concrete dense rib cavity The standard deviation of the rotational angular velocity ceiling,” Journal of Wuhan University of Technology,vol.31,no. of the oscillator with respect to the center of the arc path can 10, pp. 44–47, 2009. be calculated as [9] M. Schnellenbach-Held and K. Pfeffer, “Punching behavior of ∞ ∞ 󵄨 󵄨2 biaxial hollow slabs,” and Concrete Composites,vol.24, 𝜎2 = ∫ 𝑆 (𝜔) 𝜔=∫ 𝑆 𝜔2 󵄨𝐻 (𝜔)󵄨 𝜔 Θ̇ Θ̇ Θ̇ d 0 󵄨 21 󵄨 d (A.6) no. 6, pp. 551–556, 2002. −∞ −∞ [10] W. B. Ali and G. S. Urgessa, “Structural capacities of spherically which can be also solved with (A.3) and 𝜉3 = 1, 𝜉2 =𝜉1 =𝜉0 = voided biaxial slab,”in Structures Congress,pp.785–796,Boston, 0, and 𝜆𝑖, 𝑖=1, 2, 3, 4, as shown in (A.5a), (A.5b), (A.5c),and Mass, USA. (A.5d). [11] M. Pirner, “Actual behaviour of a ball vibration absorber,” The standard deviation of the primary structure displace- Journal of Wind Engineering and Industrial Aerodynamics,vol. ment thus can be determined as 90,no.8,pp.987–1005,2002. ∞ ∞ [12] J. Naprstek,´ C. Fischer, M. Pirner, and O. Fischer, “Non-linear 2 󵄨 󵄨 𝜎𝑋 = ∫ 𝑆𝑋𝑋 (𝜔) d𝜔=∫ 𝑆0 󵄨𝐻11 (𝜔)󵄨 d𝜔 (A.7) model of a ball vibration absorber,” in Computational Methods −∞ −∞ in Earthquake Engineering,vol.30ofComputational Methods 𝜉 = 𝜉 = 𝜉 =− 𝑔/( 𝜌) + 𝛽 𝜉 = 𝑔2/( 𝜌2) in Applied Sciences, pp. 381–396, Springer, Dordrecht, The with 3 0, 2 1, 1 10 7 eq, 0 25 49 , Netherlands, 2013. and 𝜆𝑖, 𝑖=1, 2, 3, 4, as shown in (A.5a), (A.5b), (A.5c),and [13] F.Ricciardelli and B. J. Vickery, “Tuned vibration absorbers with (A.5d). dry friction damping,” Earthquake Engineering & Structural Dynamics,vol.28,no.7,pp.707–723,1999. Disclosure [14] K. Kitamura, T. Ohkuma, J. Kanda, Y. Mataki, and S. Kawabata, “Chiba Port Tower: full-scale meaurement of wind actions (part The English in this document has been checked by at least two I): organisation, measurement system and strong wind data,” professional editors, both native speakers of English. Journal of Wind Engineering, vol. 37, pp. 401–410, 1988. Shock and Vibration 15

[15]K.Ohtake,Y.Mataki,T.Ohkuma,J.Kanda,andH.Kitamura, “Full-scale measurements of wind actions on Chiba Port Tower,” Journal of Wind Engineering and Industrial Aerodynamics,vol. 43, no. 1–3, pp. 2225–2236, 1992. [16] M. Pirner, “Dissipation of kinetic energy of large-span bridges,” Acta Technica CSAV,vol.39,pp.645–655,1994. [17]Z.-L.Zhang,J.-B.Chen,andJ.Li,“Theoreticalstudyand experimental verification of vibration control of offshore wind turbines by a ball vibration absorber,” Structure and Infrastruc- ture Engineering: Maintenance, Management, Life-Cycle Design and Performance,vol.10,no.8,pp.1087–1100,2013. [18]J.L.ChenandC.T.Georgakis,“Tunedrolling-balldampers for vibration control in wind turbines,” Journal of Sound and Vibration,vol.332,no.21,pp.5271–5282,2013. [19] O. Fisher and M. Pirner, “The ball absorber—a new tool for passive energy dissipation of vibrations of high buildings,” in Proceedings of the 7th International Seminar on Seismic Isolation, Passive Energy Dissipation and Active Control of Vibrations of Structures, pp. 103–110, Assisi, Italy, 2002. [20] E. Matta, A. De Stefano, and B. F. J. Spencer, “A new pas- sive rolling-pendulum vibration absorber using a non-axial- symmetrical guide to achieve bidirectional tuning,” Earthquake Engineering & Structural Dynamics,vol.38,no.15,pp.1729– 1750, 2009. [21] M. P.Singh, S. Singh, and L. M. Moreschi, “Tunedmass dampers for response control of torsional buildings,” Earthquake Engi- neering & Structural Dynamics,vol.31,no.4,pp.749–769,2002. [22]R.S.JangidandT.K.Datta,“Performanceofmultipletuned mass dampers for torsionally coupled system,” Earthquake Engineering & Structural Dynamics,vol.26,no.3,pp.307–317, 1997. [23] F. Rudinger,¨ “Tuned mass damper with nonlinear viscous damping,” Journal of Sound and Vibration,vol.300,no.3–5,pp. 932–948, 2007. [24] S. H. Crandall, Random Vibration, MIT Press, Cambridge, Mass, USA, 1958. [25] J. B. Roberts and P. D. Spanos, “Stochastic averaging: an approximate method of solving random vibration problems,” International Journal of Non-Linear Mechanics,vol.21,no.2,pp. 111–134, 1986. [26] J. B. Roberts and P. D. Spanos, Random Vibration and Statistical Linearization,Dover,NewYork,NY,USA,2003. [27] S. H. Crandall, “Non-gaussian closure for random vibration of non-linear oscillators,” International Journal of Non-Linear Mechanics, vol. 15, no. 4-5, pp. 303–313, 1980. [28] P. D. Spanos and B. A. Zeldin, “Monte Carlo treatment of random fields: a broad perspective,” Applied Mechanics Reviews, vol. 51, no. 3, pp. 219–237, 1998. International Journal of

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