(2019) Soil Dynamics and Earthquake Engineering DOI: 10.1016/j.soildyn.2019.03.016
Experimental and numerical assessment of the seismic response of steel structures with clutched inerters
C. M´alaga-Chuquitaypea, C. Menendez-Vicenteb, R. Thiers-Moggiaa
aDepartment of Civil and Environmental Engineering, Imperial College London, UK bDepartment of Civil Engineering, RWTH Aachen University, Aachen, Germany
Abstract Supplemental rotational inertia devices provide an e cient means of suppressing ground-induced vibrations over a large range of structural periods. The beneficial e↵ects of the inerter can be further enhanced by coupling it with a clutch system that prevents it from driving the structural response and ensures that its supplemental rotational inertia is only employed to resist the motion. In this paper, we examine the behaviour of single-degree-of-freedom and multi-degree-of-freedom structures equipped with twin inerter- clutch devices subjected to strong ground-motion. The influence of the clutch sti↵ness, gears play, viscous damping and dry friction, on the non-linear dynamics of the system are explored first, by analysing the stable periodic solutions of a structure with inerters under harmonic-sweeps. We demonstrate that, for the range of parameters typically expected in earthquake engineering practice, the influence of dry-friction and clutch damping are limited, although the clutch sti↵ness and gear play may need to be accounted for when large inertances or defective clutches are considered. Based on these findings, we propose a simplified numerical modelling strategy suitable for implementation in conventional Finite Element simulations. Small scale experiments on bare elastic structures as well as structures equipped with 3D-printed inerter and inerter-clutch twins are presented and employed for concept demonstration and for the validation of the numerical model proposed. Finally, a series of studies on detailed numerical models of multi-storey steel frames under idealized and real pulse-like ground-motions are used to demonstrate the vibration absorbing capabilities brought about by the twin inerter-clutch system and to highlight practical aspects related to their structural implementation. Keywords: Displacement reduction, inerter, dry friction, clutch, structural control, ground-motion
1. Introduction
Inerters are widely used in the automotive industry to improve the dynamic performance of vehicle suspension systems [1–3]. An inerter is a mechanical element which output force is proportional to the relative acceleration between its terminals [1]. As such, it is the mechanical equivalent of the capacitor in
5 mechanical-electrical analogous circuits [2]. Its constant of proportionality is called inertance and has units of mass. Of particular interest to earthquake engineers is the fact that an inerter can generate a significant inertial mass while keeping the associated gravitational mass minimum [1].
Email address: [email protected] (C. M´alaga-Chuquitaype)
PreprintsubmittedtoSoilDynamicsandEarthquakeEngineering February14,2019 Recent studies [4–8] have highlighted the potential advantages of employing inerters to control the seismic
10 response of civil structures. Hwang et al. [4] proposed a damper formed by a cylindrical mass rotating inside a fluid chamber. A similar proposal was made by Ikago et al. [5] with the addition of a flywheel to increase the inertial e↵ects of the device. The amplification e↵ects of multiple flywheels in series was also highlighted by Makris and Kampas [6], who evaluated the use of a rack-pinion-flywheel inerter system. They demon- strated that supplemental rotational inertia is particularly e↵ective in reducing peak displacements of long
15 period structures, but at the expense of generating appreciable forces on the support system. Importantly, their study also explored the use of a clutch to ensure the inerter is only employed to resist the structural motion. A simplified discontinuous step function was assumed to represent the e↵ects of the clutch and the idle rotation of the disengaged flywheel was disregarded. On the other hand, Takewaki et al [7] investigated the response of multi-degree-of-freedom (MDOF) structures incorporating a ball-screw-type inerter. Aspects
20 of the energy stored in the flywheel and the subsequent forces transferred to the supporting system were also ignored in this study. More rencetly, Thiers-Moggia and M´alaga-Chuquitaype [8] demonstrated that the addition of inerters to rocking structures results in lower rotation seismic demands and reduced probabilities of overturning in comparison with uncontrolled cases. The potential benefits of employing clutches was also examined in [8].
25 Other seismic applications of the inerter have sought to take advantage of the force amplification e↵ects generated by the supplemental inertial when used in conjunction with tuned mass dampers (e.g. [9], [10]). For example, Garrido et al. [11] proposed a double inerter-tuned mass damper which advantages include an improved reduction of peak deformations and a wider suppression band. Alternatively, De Domenico
30 et al. [12] studied the seismic response of structures where base isolation is combined with a tuned inerter damper. This configuration was able to reduce the displacement demand of base-isolated structures while controlling the response of the superstructure at the same time. Importantly, Makris [13, 14]derivedthe basic frequency-response and time-response functions of inerto-viscoelastic solid and fluid mechanical net- works. None of these studies employed clutches.
35 It follows from the previous discussion, that the vast majority of seismic applications of inerters to date have not considered the advantages created by coupling inerters with one-way clutches. Moreover, the two studies that incorporate clutches in their analyses employ simple acceleration-based discontinuous mathematical models on SDOF systems and ignore the idle motion of the disengaged flywheel. The validity
40 of these assumptions remains to be assessed and the potential influence of inerter-clutch non-linearities, especially when subjected to strong ground-motion, requires further study. In this paper, we build and analyse a mathematical model of a structure incorporating a pair of clutched inerters. The clutch is modelled as a non-linear spring with continuous step-wise sti↵ness and the response of the system is first studied with reference to steady-state frequency response functions under harmonic-sweep base-motion. We assess the
45 influence of gear play, flywheel damping, clutch sti↵ness and dry friction on the overall displacment response of the structure. Based on these results, a simplified Finite Element model of the rack-pinion-flywheel inerter and clutch are developed. For validation of the twin interter-clutch concept and of our modelling strategy, we perform a series of dynamic tests on reduced-scale specimens. Finally, a set of numerical studies on multi-storey frames subjected to ideal harmonic pulses as well as a large dataset of real pulse-like ground-
2 (a) Structure with rack-pinion-flywheel inerter.
(b) Structure with twin inerter-clutch system.
Figure 1: Single-degree-of-freedom structure with a single inerter (a) and with a twin inerter-clutch device (b).
50 motion records is used to highlight practical aspects related to the implementation of clutch-inerter systems in steel buildings.
2. Non-linear dynamics of structures with clutch-inerter devices
Figure 1a presents a SDOF frame equipped with supplemental rotational inertia in the form of a rack- pinion and flywheel system adapted from [6]. In the analyses that follow, the chevron frame supporting the
55 device is assumed to be rigid while the frame supporting the structural mass m has a sway sti↵ness of k. A flywheel is connected to the vibrating mass (m) by means of a rack and pinion system where the rack is
attached to m and the pinion is concentric with the flywheel. Additionally, the flywheel has a radius Ri and
a mass mi, while the pinion has a radius ⇢.
60 The motion of such a frame (Figure 1a) when subjected to a ground accelerationu ¨g(t) is governed by:
2⇠ ! !2 1 u¨ + 1 1 u˙ + 1 u = u¨ (1) 1 1+ 1 1+ 1 1+ g
where u1 is the horizontal displacement of the mass m, ⇠1 the damping ratio (⇠1 = c/2!1m), !1 the natural circular frequency of the swaying frame, and the suppression coe cient given by [6]:
1 m R2 = i i (2) 2 m ⇢2
3 65 The solution of Eq. 1 for a suppression coe cient of = 1 and a sinusoidal ground-motion of angular fre-
quency !g =6.28 rad/s (period Tg =1.0 s) and amplitude ag =0.5 g is presented in Figure 2. Since part of the ground-motion energy is transferred to the flywheel, the angular momentum accumulated in it can drive the mass m, even at intervals during which the relative velocity of the frame would otherwise tend to slow down. This phenomena, already described by Makris and Kampas [6], can cause undesirable deformations
70 in the controlled structure. In order to ensure that the inerter is only e↵ective in resisting the motion, and not driving it, a clutch can be inserted in the load-path between the pinion and the flywheel. This clutch will disengage the flywheel from the structure when the motion of m tends to reverse after a local maximum displacement is attained. For the system to be e↵ective in both directions, a parallel inerter is needed to oppose the motion in the alternative direction, while the first inerter rotates idly, as depicted in Figure 1b.
75 Figure 2 also shows the displacement history for such arrangement (parallel inerter-clutch). It is evident from this figure that substantial reductions in peak displacements are achieved when the twin inerter-clutch
system is employed. In contrast, peak forces tranferred from the device to the structure (F (t)max) are not significantly a↵ected by the inclusion of the clutch system.
0.2 0 -0.2 -0.4 0 0.5 1 1.5 2 2.5 3 3.5 4
0.5
0
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
0.2
0
-0.2
0 0.5 1 1.5 2 2.5 3 3.5 4
Figure 2: Displacement response history of a frame with supplemental rotational inertia device mounted on an infinitely sti↵ support subjected to a sinusoidal signal of period, Tg =1.0s,andamplitudeag =0.5g.Thestructurehasanaturalperiodof 1s,asuppressioncoe cientof =1.0, and a damping ratio of ⇠ =0.02. When a parallel inerter-clutch system is employed, the force from the device only acts in opposition to the motion.
4 80 2.1. Mathematical model of clutch-inerter devices
One-way clutches engage or disengage based on their relative angular speed or relative displacement. There are several designs of one-way clutches available, but the most commonly used fall within one of three categories: i) ratchet and pawl, ii) roller and spring, and iii) sprang. Figure 3 presents a schematic com- parison of their envelope sizes according to [15]. The ratchet type is obviously the most limited in capacity,
85 furthermore, in order to comply with maximum stresses on the ratchet and the tooth, the ratchet clutch must be very large in comparison with other clutch types, which can impose some limits for building appli- cations. Ratchet-pawl clutches are also associated with non-negligible levels of rotation before engagement. By contrast, sprang clutches o↵er a good load carrying capacity and are not significantly a↵ected by friction [16]. However, the complexity of their fabrication and assembly contribute to higher costs of manufacture
90 and maintenance. The roller clutch, on the other hand, needs smaller envelopes than the ratchet-pawl type to transfer the same amount of torque, is reliable and relatively inexpensive. Also, the simple design of roller clutches made them ideal for 3D printing during the experimental phase described later in this paper. Although the focus of this paper will be on this clutch type, the sensitivity analysis presented herein cover the ranges of friction and play of sprang configurations as well, and therefore are of wider relevance.
95
Formsprang overrunning clutch Ratchet and pawl clutch
Roller clutch
Figure 3: Schematic comparison of di↵erent clutch types. Adapted from [15].
Given the physical configuration of overrunning one-way clutches, when the input driver (mass m in Figure 1) moves in the direction of engagement, the ball-spring system inside the clutch tightens and the
friction increases locking the structural mass, m, and the flywheel, mi, together in motion. During this state, the clutch is engaged and delivers a torque equal to:
T = f(k✓,i, ✓) (3)
100 where k✓,i is the clutch sti↵ness and ✓ its rotation increment. Alternatively, when m moves in the other direction, the ball-spring system loosens and no torque is transmitted to the frame. Mathematically, the torque transmitted between the flywheel and the frame can be approximated by [17]:
1 T = k [1 + tanh (✏ (u /⇢ ✓ ))] (u /⇢ ✓ ) (4) 2 ✓,i · 1 i 1 i 5 where ✏ is a parameter that controls the transition between engagement and disengagement. Besides defining the engagement-disengagement condition in clear terms, Eq. 4 has also the advantage of defining a contin-
105 uous function which facilitates the numerical continuation on which the results of this section are based. Figure 4a compares the clutch torque described by Eq. 4 for di↵erent values of ✏ and for a clutch sti↵ness
of k✓,i = 10000 kN.m/rad. It can be appreciated from Figure 4a that the di↵erence is indistinguishable for values of ✏ larger than 1000. Therefore, a value of ✏ = 1000 has been employed throughout this study.
Similarly, Figure 4b presents the influence of the clutch sti↵ness, k✓,i, on the transmitted torque, T .
110
104 5 = 10 = 1000 4 = 10000
3
2
Torque, T [kN.m] 1
0
-2 -1 0 1 2 ( / - ) [rad] 1 i
(a) Influence of the parameter ✏, k✓,i =1000kNm/rad.
104 5 k = 100 kNm/rad ,i k = 1000 kNm/rad 4 ,i k = 10000 kNm/rad ,i 3
2
Torque, T [kN.m] 1
0
-2 -1 0 1 2 ( / - ) [rad] 1 i
(b) Influence of the clutch sti↵ness k✓,i, ✏ =1000.
Figure 4: Clutch torque from Equation 4.
By assuming modal damping, the equations of motion governing the response of a SDOF frame equipped
6 with twin inerter-clutches (Figure 1b) can be obtained as follows:
u¨ +2⇠ ! u˙ + !2u + I¯ ( ✓)+I¯ ( ✓)= u¨ (5) 1 1 1 1 1 1 a b g
✓¨ +¯c ✓˙ I ( ✓) = 0 (6) a a a a
✓¨ +¯c ✓˙ I ( ✓) = 0 (7) b b b b
wherec ¯i is the inertial mass-normalized damping coe cient, I¯a and I¯b are the mass-normalized forces
transferred from flywheels a and b to the frame, and ✓a and ✓b their corresponding flywheel rotations. Ii
115 and I¯i are related via:
u I¯ = I ⇢,✓ = i (8) i i i ⇢ Consequently,
1 k✓,a Ia( ✓)= [1 + tanh (✏ (u1/⇢ ✓a))] (u1/⇢ ✓a) (9) 2 Ja ·
1 k✓,b Ib( ✓)= [1 tanh (✏ (u1/⇢ ✓b))] (u1/⇢ ✓b) (10) 2 Jb ·
where Ja and Jb are the rotational inertias of flywheels a and b, respectively. In this formulation, each inerter will be associated with an independent suppression coe cient equal to:
J J = a , = b (11) a m⇢2 b m⇢2 Although, for practical purposes, we have assumed no mass imbalance between parallel inerters and the
120 same inertance is considered for flywheels a and b throughout our analyses (i.e. a = b = ).
2.2. Frequency response under sweep harmonic ground-motion
In this section, we examine the dynamic response of the SDOF structure with twin inerter-clutches depicted in Figure 1b and described by Eqs. 5 to 10 subjected to continuous sweep harmonic ground-
motions. To this end, the acceleration of the ground is taken asu ¨g = ag cos(!gt), where ag is the peak
125 ground acceleration amplitude and !g its circular frequency. With this in mind, we can easily recast the non-autonomous system of equations 5 to 10 as an autonomous system, such as Eq. 12.Thisrequiresthe introduction of the desired periodic forcing as a secondary oscillator close to the supercritical Andronov-Hopf bifurcation [18] such that:
7 2 2 u0 = u (1 u v ) w v ⇤ ⇤ 2 2 v0 = v (1 u v )+w u ⇤ ⇤ x10 = x0 2 x0 = A u w x 2 ⇠ w x 0 ⇤ 0 ⇤ 1 ⇤ ⇤ 0 ⇤ 0 k (1 + tanh(✏ (x x ))) (x x ) ⇤ ⇤ 1 3 ⇤ 1 3 k (1 tanh(✏ (x x ))) (x x ) ⇤ ⇤ 1 5 ⇤ 1 5 x30 = x2 2 x0 = c/m /(R ) x + k (1 + tanh(✏ (x x ))) (x x ) 2 a a ⇤ 2 ⇤ ⇤ 1 3 ⇤ 1 3 ⇤ x50 = x4 2 x0 = c/m /(R ) x + k (1 tanh(✏ (x x ))) (x x ) (12) 4 b b ⇤ 4 ⇤ ⇤ 1 5 ⇤ 1 5 ⇤
The system of autonomous di↵erential equations presented in Eq. 12 simplifies the numerical integration
130 and enables the use of standard continuation techniques for the mapping of periodic steady-state solutions under constant amplitude ground-motions. They have been used to analyse the response of a structural system under the influence of various inerter-clutch non-linearities. The factors under consideration, and
presented below, are the clutch rotational sti↵ness (k✓), viscous damping in the device (⇠a and ⇠b), play in
the rack and pinion mechanism or, if a series of gears is employed, total gear play (up), and dry friction.
135 During our investigations, we have assumed a fundamental structural frequency of ! =6.5 rad/s (consis- tent with a period of T 1 s), a mass of m = 600000 kg, and a damping coe cient c consistent with 1 ⇡ 1 a damping ratio ⇠ = 5% of the critical damping. All results of this section correspond to a peak ground
acceleration of ag =0.15 g. However, since the system is homogeneous (Eqs. 5 to 10), load scaling is re- tained and hence the tendencies observed herein are expected to be valid over di↵erent excitation intensities.
140
Influence of clutch rotational sti↵ness
Figure 5 presents the frequency response curves of the SDOF structure considered under harmonic-sweep base excitation, together with cases where a single inerter and a twin inerter-clutch system are employed. Two levels of supplemental inertial mass are presented, corresponding to suppression coe cients of =1
145 and = 5. The period lengthening e↵ect associated with the supplemental inertia is clearly observed in Figure 5. Likewise, the relative improvements generated by the clutching are also appreciable.
We can evaluate the influence of di↵erent levels of clutch sti↵ness, k✓ in Eq. 4, with reference to Figure 5
where the curves corresponding to inerter-clutches of k✓ = 1 MN.m/rad and k✓ = 100 MN.m/rad are shown.
150 When = 1, the influence of clutch sti↵ness is negligible and both response curves overlap. However, when
a larger inertance is employed (e.g. equivalent to = 5 in Figure 5), the sti↵er system (k✓ = 100 MN.m/rad) gives rise to some instabilities in the solution and a slightly jagged response. These small kinks were still evident after several modifications of the integration and search algorithm and are likely to originate from
8 inherent dynamic non-linearities of the system. By contrast, the more flexible clutch (k✓ = 1 MN.m/rad)
155 presents a smoother response even when large supplemental inertia is employed. These levels of clutch sti↵ness are closer to typical values found in mechanical systems and is therefore expected that the practical implications of strongly sti↵non-linearities will be limited.
10-1
-2 10 =1 =5
Displacement [m] Bare frame -3 10 Frame with inerter Twin inerter-clutch system, k = 100MN.m/rad Twin inerter-clutch system, k = 1MN.m/rad
10-4 1 2 3 4 5 6 7 8 9 10 [rad/s]
Figure 5: Resonance response curves under sweep harmonic excitation for a SDOF structure with di↵erent values of clutch sti↵ness, k✓ and suppression coe cient, .
Influence of accessory damping
As with any other mechanical device, clutches and inerters dissipate energy during operation. This
160 energy dissipation can be modelled as viscous damping, as is the case of the terms ca and cb in Eqs. 6 and 7. In order to assess the influence of the damping-generated forces on the e ciency of the system, we varied
the damping ratio associated with the flywheels from ⇠a = ⇠b =0.001 to ⇠a = ⇠b =0.20. Besides, the value
of ⇠a = ⇠b =0.05, usually adopted in earthquake engineering applications is also included in our analyses.
A high damping ratio of ⇠a = ⇠b =0.20 is considered to study how the system response changes due to
165 a marked increase in the viscous coe cient associated with an ine cient mechanism design. On the other
hand, ⇠a = ⇠b =0.001 represents a relatively small value of damping well below the levels of dissipation in real systems. The results of our analyses are presented in Figure 6. It can be observed from this figure that the level of damping seems to a↵ect more the response to low excitation frequencies. Similarly, all damping values assumed lead to very similar resonant displacements. Also, a mild narrowing of the resonance curve
170 can be observed for higher damping ratios in comparison with the lightly damped case.
Influence of play
In a practical implementation of an inerter and a twin inerter-clutch system, recoil may arise at any point between the di↵erent parts of the mechanism involved. In our case, we only consider play at the interface between the pinion and the rack and assume that all the play present in the system can be transferred to
9 = = 0.001 a b = = 0.05 a b = = 0.2 a b 10-1 Displacement [m]
10-2
1 1.5 2 2.5 3 3.5 4 4.5 [rad/s]
Figure 6: Resonance response curves under sweep harmonic excitation for a SDOF structure with di↵erent values of twin inerter-clutch damping for a suppression coe cient of =1.
175 this interface. Therefore, Eqs. 9 and 10 can be modified to include the total play up as:
1 k✓,a Ia( ✓)= [1 + tanh (✏ (u1/⇢ ✓a up/⇢))] (u1/⇢ ✓a up/⇢) (13) 2 Ja ·
1 k✓,b Ib( ✓)= [1 tanh (✏ (u1/⇢ ✓b up/⇢))] (u1/⇢ ✓b up/⇢) (14) 2 Jb · Besides the idealized no-play case, two other values of play gap are assumed for our analyses: a play
gap corresponding to 10 times the standardized value in electro-mechanical systems (up =0.001 m [20]) to
represent imperfect or heavily worn devices, and a drastically large play gap of up = 2 cm that corresponds to a faulty installation. The responses associated with those three cases are compared in Figure 7.
180 It appears from Figure 7 that play diminishes the e↵ectiveness of the inerter, which is reflected in the reduction of the period lengthening e↵ect associated with it. However, the fluctuations that are associated with the motion over the play gap are not expressed into significant non-smoothness of the frequency response function. In general, large values of play-gap are required for their e↵ect to be appreciable in the steady-state response of the system.
185 Influence of dry friction
The energy dissipation in the inerter or twin inerter-clutches can also be modelled as dry friction. In order to preserve the practicalities of a continuous function for the solution of the system of equations, the dry friction force in the device is represented by:
2arctan ⌘(˙u1/⇢ ✓˙b) F = f (15) µ µ ⇣ ⇡ ⌘ Therefore, instead of using viscous damping in Eqs. 6 and 7, we introduce the corresponding friction
10 u = 0 m p u = 0.001 m p u = 0.02 m p 10-1 Displacement [m]
10-2
1 1.5 2 2.5 3 3.5 4 [rad/s]
Figure 7: Resonance response curves under sweep harmonic excitation for a SDOF structure with di↵erent values of play gap for a suppression coe cient of =1.
190 force Fµ as:
Fµ⇢ ✓¨a + Ia( ✓) = 0 (16) Ja
Fµ⇢ ✓¨b + Ib( ✓) = 0 (17) Jb
Figure 8 presents the friction force for a value of ⌘ = 1000000 and a force fµ = 1 kN, while Figure 9
shows the results of our analyses for di↵erent friction forces. A value of fµ =0.1 kN is relatively large [20], but still within the range of inherent dry friction levels present in sti↵inerter mechanisms. By contrast,
a value of fµ = 1 kN corresponds to the case of imperfect devices or to the use of supplemental friction
195 appliances. It can be appreciated from Figure 9 that for the range of typical friction levels, the resonance curves remain practically undisturbed. We can conclude that typical structures with inerter or inerter-clutch devices under seismic ground-motions will not be adversely a↵ected by normal levels of dry friction.
3. Numerical modelling of supplemental rotational inertia
200 3.1. Modelling strategy
After considering the influence of various sources of non-linearity in the response of SDOF structures equipped with inerters and clutches, we propose a numerical modelling strategy that can incorporate their main behavioural characteristics. The model is developed in OpenSees [19] but it can be implemented in any Finite Element software with non-linear material modelling capabilities. It consists of two nodes (node
205 1 and node 2) connected as depicted in Figure 10. An angular mass is assigned to the rotational degree of freedom in node 1. Node 1 represents the pivot of the flywheel, whereas node 2 corresponds to the pinion gear. We model the rack-pinion mechanism, which transforms the horizontal relative displacement between 11 1500
1000
500
0
-500
-1000
-1500 -2 -1 0 1 2
Figure 8: Dry friction force from Equation 15.
the terminals to a rotation in the flywheel, as a rigid link between nodes 1 and 2. If linear geometric transformation is considered for the link element, the horizontal relative displacement and the rotation of
210 the system are related according to: u = u (t) u (t)= ⇢✓. A rotational inertia, I , is assigned to r 2,u 1,u w1 the rotational degree of freedom in node 1 to represent the flywheel inertial e↵ect. The values of ⇢ and Iw1 can be modified to take into account a series of flywheels as far as the corresponding suppression coe cient, 2 1 mi Ri = 2 m ⇢2 ,isrespected.
215 The contact model adopted between nodes 3 and 2 can be adapted to incorporate the non-linearities explored in the previous section. At its most basic level, an equalDOF constraint can be adopted in the u direction assuming perfect transmission between the pinion and rack. Alternatively, play can be incorpo- rated by employing a contact material model or a hysteretic material idealization. Likewise, a dash-pot can be introduced between nodes 3 and 2 to model viscous damping in the device.
220 The reactive force developed by the model is proportional to the horizontal relative acceleration between nodes 1 and 2. Since this is the defining characteristic of an inerter, our model can be used to represent not only rack-pinion-flywheel systems, but other types of inerter devices. Moreover, the proposed model can be combined with di↵erent configurations of springs and dashpots in order to represent other forms of tuned
225 inerter-based dampers.
On the other hand, our implementation of the clutching e↵ects takes advantage of the TCL scripting capabilities of OpenSees. To this end, the clutch engagement-disengagement criterion based on relative accelerations and velocities adopted in [6] is implemented in our numerical models through a for-loop.
230 This formulation is broadly consistent with Eq. 4 but leads to a more practical implementation than a displacement-based alternative in the context of finite element simulations. To this end, the force condi- tion adopted for the clutch is implemented in our numerical model through a for-loop. This loop reads
12 f = 0 N f = 100 N f = 1000 N 10-1 Displacement [m]
10-2
1 1.5 2 2.5 3 3.5 4 [rad/s]
Figure 9: Resonance response curves under sweep harmonic excitation for a SDOF structure with di↵erent friction levels for a suppression coe cient of =1.
translating mass node 3 contact model node 2
v radius, rigid link u node 1 rotational inertia, Iw
Figure 10: Schematic diagram of the numerical model for supplemental rotational inertia devices.
displacements, velocities (or accelerations) and calculates the oscillatory state at which the system is at any time-step. Then, an if-else condition modifies the mass of the inerter as required. This approximation allows
235 the modelling of an ideal clutch system, and is verified at any time-step during the transient analysis. The simple TCL script is presented in Listing 1.
Listing 1: TCL script for simulating the transmission in the clutch mechanism in OpenSees
for set i1 $i < [ expr $Number time steps+1] incr 1 analyze 1 $dt { }{ }{ }{ set Acc i[nodeAccel$topnode 1]
240 set Vel i[nodeVel$topnode 1] set ver [ expr $Acc i/$Veli] if $ver>=0 set auxi 1 { }{ else } { set auxi 0
245 } mass $inerters node $Mass $Mass [ expr $J $auxi ] ⇤ set i[expr $i+1]
}
13 3.2. Numerical results
250 Figure 11 presents the response of a SDOF system modelled as described above under a sinusoidal pulse
of amplitudeu ¨g,max and duration Tg while Figure 12 summarizes the response spectra for the same system. We can observe from Figure 11 that the clutched structure (green curve) reaches its first local maximum displacement before the system with the single inerter (red curve). Subsequent displacement peaks occur almost simultaneously in the clutched structure and in the structure with a single un-clutched inerter with
255 decreasing amplitudes. We can conclude that the numerical model of the inerter and clutch mechanism proposed is e↵ective in disconnecting the supplemental mass and preventing the inerter from driving the motion.
Figure 11: Displacement, acceleration and force response history of a SDOF incorporating a single inerter with and without clutch under a sinusoidal ground-motion of T1/Tp =2forasuppressioncoe cientof =1.Theuseofatwinclutch-inerter device ensures that the inertance is used only to oppose the motion.
The improved response brought about by the twin inerter-clutch system is also reflected in the spectra of Figure 12. The incorporation of a clutch mechanism o↵ers lower peak displacements, specially within
260 the period range of 0.5 265 in agreement with previous studies [6]. It is interesting to note that the twin inerter-clutch device may yield slightly larger displacement than the single inerter in the spectral regions of T1/Tg < 0.4 and, more notably, for T1/Tg > 3.5. The latter are due to juddering of the numerical model at the point of sharp transition between engagement and disengagement and require further studies. The following section o↵ers experimental validation of the proposed modelling strategy. 14 4 2 0 0123456 0.2 0.1 0 0123456 1.5 1 0.5 0 0123456 Figure 12: Relative displacement, absolute acceleration and transferred force spectra of the supplemental rotational inertia device subjected to sinusoidal pulses (¨ug,max =0.5g)forasuppressioncoe cient =1.0obtainedfromnumericalsimulations. The cases of an uncontrolled structure, a structure with a single inerter (that may drive the response) and a pair of clutched inerters (that only oppose the motion) are presented. 270 4. Experimental programme 4.1. Testing set-up and specimen details After considering the numerical results of the simplified FE model described above, we carried out a series of small scale tests in order to demonstrate the twin inerter-clutch concept and provide further validation of our modelling approach. To this end, we examined the response of a SDOF structure with supplemental 275 rotational inertia in the form of a rack-pinion-flywheel system pin-connected to it. Figure 13 illustrates the experimental setup for the cases where: (i) a single inerter, and (ii) a parallel clutch-flywheel system, are employed. An un-controlled bare SDOF configuration was also tested for comparison purposes. The column on the left sustains a mass of m = 190 g while the mass of the flywheel system is kept constant at mi = 30 g. The disengaged structure has a natural period of T1 = 1 s. Other dimensions are given in Figure 13. 280 The specimen was shaken under near-harmonic base-motion while laying horizontal on the surface of the shaking table. Therefore, a PTFE sheet was provided to minimize friction. 15 (a) Schematic view of the SDOF structure with a rack-pinion-flywheel in- (b) General view of the SDOF structure with a erter. rack-pinion-flywheel inerter. (c) Schematic view of the SDOF structure with a twin inerter-clutch system. (d) General view of the SDOF structure with a twin inerter-clutch system. Figure 13: Experimental set-up. The roller overrunning clutch implemented in the parallel system consists of an outer shaft, which rotates in one direction only, glued to the flywheel, and an inner shaft, which rotates in both directions and is at- 285 tached to the pinion. The clutch mechanism was designed and 3D printed on high quality PLA resin in order to achieve the required level of detail and smoothness. The servo-controlled small shaking table employed can apply uni-directional motions up to 5 Hz with an stroke of 35 mm. Accelerometers with a frequency ± range extending up to 1600 Hz and an amplitude range of 16 g were employed to read accelerations at the base of the structure and at the mass m. Near harmonic base-motions were applied at forcing frequencies 290 of !g = 12.57, 15.71, 18.85, 21.99 and 25.13 rad/s and for peak base displacement amplitudes of ug = 15 and 20 mm. A minimum of 7 repetitions per test were conducted. 4.2. Experimental results and validation Figure 14 summarizes the results of our experimental campaing. It presents the mean peak acceleration 295 ratios for all tests conducted and for the three specimens: (i) un-controlled structure, (ii) SDOF with a sin- gle inerter, and (iii) SDOF with twin inerter-clutches. At lower excitation frequencies (e.g. 2.0Hz or 12.56 16 rad/s and 2.5Hz or 15.71 rad/s) the reduction on accelerations achieved by the parallel inerter configuration is very evident. Nonetheless, for higher frequencies (e.g. 3.0Hz or 18.85 rad/s and specially 3.5Hz or 22 rad/s), the results are mixed and the single inerter outperforms the parallel clutched-inerter system in a 300 number of instances. Although exacerbated by some manufacturing and installation imperfections, these results follow the response depicted in the top frame of Figure 12 where very close predictions of acceleration demands are evident for T1/Tg ratios lower than 0.5. Unfortunately, experimental constraints prevented us from testing at higher frequencies ranges. In general, our experimental results have confirmed the significant improvements in the dynamic response generated by supplemental rotational inertia. They have also verified 305 the versatility o↵ered by small-scale 3D printed specimens for dynamic experimentation. 2 Without inerter u = 20 mm g,max u = 15 mm g,max 1.5 With single inerter 1 0.5 With twin inerter-clutch system 0 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Figure 14: Spectrum of the peak relative acceleration for the cases (i) without inerter, (ii) single inerter and (iii) parallel inerter. A second motivation to carry out the testing campaign was to provide experimental validation for the numerical model proposed in the previous section. To this end, we constructed a numerical model reproduc- ing the mechanical characteristics of the physical tests. Figure 15 shows a typical comparison between the 310 numerical predictions and the experimental results for the history of accelerations of the specimens without inerter, with a single inerter and with the twin inerter-clutch system. These comparisons are promising. Despite the high non-linearity and a lack of redundancy of the model structures tested, the numerical simula- tions agree very well with the experimental response. Not only the peak accelerations coincide in magnitude, but a good level of agreement in their phases can also be observed. The minor mismatches can be attributed 315 to inherent friction which is notoriously di cult to quantify ans simulate precisely. This results validate our modelling strategy and provide confidence in its implementation. 17 1 0.5 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 (a) SDOF without inerter. 1 0.5 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 (b) SDOF with rack-pinion-flywheel inerter. 1 0.5 0 -0.5 -1 0 0.5 1 1.5 2 2.5 3 (c) SDOF with twin inerter-clutch system. Figure 15: Comparison of experimentally observed relative acceleration histories and their numerical predictions by means of OpenSees models. 18 5. Implementation in steel frames In this section, we investigate the dynamics of multi-storey buildings with and without twin inerter- clutch devices subjected to ideal harmonic pulses as well are real ground-motion records with a wide range 320 of pulse periods and amplitudes. The structures considered are modelled in OpenSees with due account for geometric and material non-linearities and incorporate the numerical representations of inerter and clutch discussed above. Our discussion is centred around peak deformations and base shears and starts with a detailed description of the buildings and the modelling assumptions considered. It continues by describing the building response to ideal harmonic pulses before introducing the dataset of real records employed and 325 presenting statistical comparisons made on the basis of the response history analyses performed. 5.1. Structural systems and non-linear models The benchmark structures proposed by the Structural Control Committee of the American Society of Civil Engineers [21] are selected for the performance comparisons carried out in this section. In particular, the 3-storey and 9-storey building configurations presented in Figure 16 are employed. The 3-storey base 330 structure was designed by Brandow & Johnston Associates for the SAC Steel Project to meet the code requirements for Los Angeles and has a 36.58 m by 54.87 m plan. The building has a total height of 11.88 m and is representative of a low rise structure. The seismic masses for the di↵erent floors range between 957 000 kg and 1 040 000 kg. Only its 2D in-plane behaviour is studied here. The columns are fixed at the ground level and all the connections are fully-rigid except for the external frame where pinned connections 335 are provided as shown in Figure 16. The beams are made of mild steel with a yield strength of 248 MPa while steel with a yield strength of 345 MPa is employed for the columns. The floors are steel-concrete composite and are assumed to provide full rigid diaphragm action. The first three eigen-modes of this structure correspond to the eigen-frequecies of 6.22, 19.23 and 36.63 rad/s, respectively. Further details of the building can be found in [21]. 340 The 9-story steel frame being considered is also shown in Figure 16 and can be taken as representative of a medium-rise building. The inter-storey heights of this building are 5.49 m and 3.96 m for the first storey and upper storeys, respectively. The building has a 3.65 m-high basement level. The seismic mass of the entire building above ground is 9 000 000 kg and its first three natural frequencies are 2.78, 7.41 and 12.88 345 rad/s. As with the 3-storey building, the 9-storey building lateral resisting system is formed by fully-rigid moment resisting steel frames. Only the response of the planar 2D frame presented in Figure 16 is consid- ered. S345 steel is assumed for the columns and further details of the structure can be found in [21]. The steel sections used for the 3-storey and 9-storey buildings are also reported in Figure 16. Both structures (3-storey and 9-storey) are analysed in two conditions: (i) in their bare uncontrolled configuration, and (ii) 350 when they are equipped with a twin inerter-clutch device mounted on top of a very rigid frame connecting the ground and the first floor. A a suppression coe cient of =1.0 is used in all cases. The steel frames are modelled in 2 dimensions in the opensource finite element framework OpenSees [19]. Force-based non-linear beam-column elements are used for beams and columns. A Steel01 material model 355 with 0.5% strain hardening and the corresponding yield strength is assigned to the fibre sections. Each 19 W24x68 W27x84 3.96m W14x257 W30x99 3.96m W36x135 3.96m W14x283 3.96m W36x135 3.96m W36x135 W14x370 3.96m W36x135 3.96m W36x160 W14x455 W24x68 3.96m W36x160 3.96m W30x116 5.49m 3.96m W36x160 W14x500 W33x1118 W14x500 3.96m 3.65m W14x257 9.15m 9.15m 9.15m 9.15m 9.15m 9.15m 9.15m 9.15m 9.15m Figure 16: Elevation view of the 3-storey (left) and 9-storey (right) buildings proposed by the Structural Control Committee of the American Society of Civil Engineers as benchmark control problems and used in this study. Pinned connections are indicated by filled circles. section is discretized into 50 fibres and 2 force-based elements with 7 Gauss-Lobatto integration points are employed per structural element. Geometric non-linearities are accounted for by means of a corotational transformation. A sti↵ness-only dependent damping with ⇠ =0.05 is assumed in all cases. 360 5.2. Response to sinusoidal pulses The seismic response of the 3 and 9-storey structures devoid of any control mechanism as well as with a twin inerter-clutch device at their base subjected to single sinusoidal pulses of acceleration amplitudeu ¨g,max and duration Tg is considered first. To this end, the peak displacement and peak base shear forces are pre- sented in figures 17 and 18 for the 3-storey and 9-storey buildings, respectively. In both cases, maximum 2 365 storey displacements are normalized by the energetic length of the ground-motion (¨ug,maxTg )whichisa meassure of its persistance [22, 23] while maximum base shears are normalized by the pulse amplitude times the structural mass (¨ug,maxm). It is evident from figures 17 and 18 that the inerter is highly e↵ective in suppressing building deforma- 370 tions over the full spectra of excitations. Nevertheless, its beneficial e↵ects are stronger for lower frequencies or larger period ratios (e.g T1/Tg > 3). These displacement reductions are evident for both the low rise (3-storey) and mid-rise (9-storey) steel buildings. Interestingly, at longer period ratios the displacements of the controlled structures seem to increase monotonically in both cases. In the case of the 3-storey building (Figure 17), this is accompanied by notable juddering in the clutch acceleration histories that alters the 20 0.2 0.15 0.1 0.05 Clutched-inerters No inerter 0 0 2 4 6 8 10 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 Figure 17: Relative displacement and shear force spectra of the 3-storey building with and without twin inerter-clutch devices subjected to sinusoidal pulses of amplitudeu ¨g,max =0.5g and duration Tg.Theinerter,whenemployed,hasasuppression coe cient of =1.0 375 overall response influencing mildly the peak deformations at long period ratios. As mentioned before, this peculiar response, which is know to happen in other torsionally clutched mechanisms [24], requires further investigations to discern the specific conditions under which it may arise in practice. Overall, reductions in peak deformations of up to 70% are experienced by the 3-storey steel building and up to 60% in the 9-storey structure. 380 A comparison of the bottom frames of figures 17 and 18 o↵ers an equally interesting conclusion. The significant displacement response enhancements mentioned before are not associated with any noticeable increase in peak base shear forces. Moreover, a reduction of maximum base shears is observed in the 9-storey building. This is significant since one of the main benefits of a clutched inerter device relative 385 to a single inerter implementation is to reduce the amount of forces transferred to the structure. This has important implications on the design for the frame and the associated costs of implementation of the system. 5.3. Response to real earthquake records Previous sections have examined the fundamental dynamic behaviour of steel structures equipped with clutched inerter devices by subjecting them to idealized pulse excitations or continuous harmonic-sweep 390 ground-motions. However, real earthquakes possess important non-coherent components that may a↵ect the e ciency of the structural control device proposed in this paper. Therefore, in this section, we assess 21 0.2 0.15 0.1 0.05 Clutched-inerters No inerter 0 012345 0.8 0.6 0.4 0.2 0 012345 Figure 18: Relative displacement and shear force spectra of the 9-storey building with and without twin inerter-clutch devices subjected to sinusoidal pulses of amplitudeu ¨g,max =0.5g and duration Tg.Theinerter,whenemployed,hasasuppression coe cient of =1.0 the e↵ectiveness of the twin inerter-clutch system by subjecting the non-linear models described above to a set of 202 real pulse-like ground motion records. To this end, acceleration records from 21 earthquakes with magnitudes Mw ranging from 5.4 to 7.9 obtained from the Pacific Earthquake Engineering Research Center 395 (PEER) database are employed. Table 1 summarizes the catalogue of earthquakes considered. A range of methods can be employed to characterize the relationship between Engineering Demand Pa- rameters (EDPs) and Intensity Measures (IMs) based on the results of response history analyses. These include the Incremental Dynamic Analysis (IDA) [25], sometimes confused with the Multiple-Stripe Anal- 400 ysis [26], as well as the Cloud Analysis [27, 28], among others. The application of IDA relies on a limited number of records with multiple levels of linear scaling. Therefore care must be taken when selecting the ground-motion seed and scaling procedure since di↵erent ground-motion characteristics have di↵erent scaling laws and linear scaling of spectral ordinates may lead to ground motions with non-representative frequency content or duration. Alternatively, simple linear regression predictions can be made in the logarithmic 405 EDP-IM space based on the results of a reasonable number of response-history analyses [28]. This last method, commonly known as Cloud Analysis is employed here to evaluate the relative performance of the twin clutch-inerter device. It constitutes a first step towards a more detailed probabilistic response assess- ment that necessitates the definition of site-specific hazard scenarios [29] 22 Table 1: Ground motion database used in the analyses Earthquakename Year Magnitude Mw Mechanism NumberofRecords San Fernando 1971 6.61 Reverse 1 TabasIran 1978 7.35 Reverse 1 Coyote Lake 1979 5.74 Strike Slip 4 Imperial Valley-06 1979 6.53 Strike Slip 12 Irpinia Italy-01 1980 6.9 Normal 2 Westmorland 1981 5.9 StrikeSlip 1 Morgan Hill 1984 6.19 Strike Slip 2 Kalamata Greece-02 1986 5.4 Normal 1 San Salvador 1986 5.8 Strike Slip 2 Superstition Hills-02 1987 6.54 Strike Slip 2 Loma Prieta 1989 6.93 Reverse Oblique 6 Cape Mendocino 1992 7.01 Reverse 1 Landers 1992 7.28 Strike Slip 3 Northridge-01 1994 6.69 Reverse 14 Kobe 1995 6.9 Strike Slip 4 Kocaeli 1999 7.51 Strike Slip 4 Chi-Chi Taiwan 1999 7.62 Reverse Oblique 36 Chi-Chi Taiwan-04 1999 6.2 Strike Slip 1 Chi-Chi Taiwan-06 1999 6.3 Reverse 2 Duzce Turkey 1999 7.14 Strike Slip 1 Denali Alaska 2002 7.9 Strike Slip 1 Total 202 410 We conducted statistical regression analyses on the basis of the results from the non-linear response history analyses performed considering the earthquakes database described in Table 1 were employed to conduct statistical regression analyses. As before, the structural demands are described in terms of peak deformations and forces. To this end, the peak inter-storey drift along the height of the building, , was adopted as a deformation based EDP, while the spectral acceleration at the fundamental period of the 415 building, Sa(T1), is adopted as IM. The common assumption of considering a power law distribution of demands, Dm, in terms of IM is followed: b Dm = aIM (18) Accordingly, Equation 18 becomes a straight line when plotted on a ln(D ) ln(IM) plane: m ln(Dm)=ln(a)+bln(IM) (19) 420 where a and b are the linear regression coe cients. Figures 19 and 20 show the results of the cloud analyses and the corresponding fitted seismic demand models for two cases: i) no inerter, ii) pair of clutched-inerters ( =1.0). Results for peak drift and peak base shears are presented in figures 19 and 20 for the 3 and 9-storey buildings, respectively. It should be 425 noted that the drifts associated with global yielding for the 3-storey and 9-storey buildings are 1% and 1.5%, respectively, and therefore, figures 19 and 20 can be considered representative of a wide range of structural responses including those which development significant levels of yielding. 23 The results of the regression analyses presented in figures 19 and 20 show a strong correlation between the 430 selected intensity measure, Sa(T1), and the seismic demands, validating the estimation model proposed in Eq. 18. Moreover, the standard deviation of the residuals ranged between 0.15 and 0.29, which is indicative of the robust regression results. Overall, the steel buildings equipped with twin clutch-inerter devices show significantly smaller seismic demands for the whole range of IMs considered. In terms of maximum drifts, reductions of around 35% are observed in the 3-storey frame at low demand levels while 30% reductions can 435 be achieved in the 9-storey frame. Importantly, the inerter ability to mitigate peak drifts decreases with increasing earthquake intensity or as the non-linear behaviour of the structure dominates the response. A tendency that is more pronounced in the 9-storey frame (b =0.95) in comparison with the 3-storey structure (b =0.77). It is worth noting, however, that the suppression coe cient ( ) which defines the inerter’s ability to control the structural response can be further increased with the inclusion of gear systems [6]. In general, 440 the results of figures 19 and 20 are consistent with previous findings of this paper. (a) Regression for peak inter-storey drift, . (b) Regression for maximum base shear, Vi Figure 19: Seismic demand analysis for the 3-storey building with and without twin inerter-clutch devices subjected to the suite of records described in Table 1 Another important observation emerging from figures 19 and 20 is that the deformation reductions mentioned above are accompanied by base shear reductions and, very importantly, a corresponding reduction in the variability of the response. Overall, up to 40% and 25% reductions in base shear are observed for 445 the 3 and 9 storey steel frames, respectively. However, when the intensity of the earthquake motion is large (e.g. Sa(T1) > 1g for the 9-storey building and Sa(T1) > 0.5g for the 3-storey) the clutched inerters seem unable to operate further reductions in base shears. 24 (a) Regression for peak inter-storey drift, . (b) Regression for maximum base shear, Vi Figure 20: Seismic demand analysis for the 9-storey building with and without twin inerter-clutch devices subjected to the suite of records described in Table 1 6. Conclusions In this paper, we have analysed the response of SDOF and MDOF steel structures incorporating twin 450 inerter-clutch dampers. We started by studying the e↵ects of non-linearities on the dynamics of inerter- equipped structures under harmonic-sweep ground-motions by examining their frequency response curves. The influence of the following factors was evaluated: clutch sti↵ness, gears play, viscous damping and dry friction. 455 For clutch sti↵ness values close to those typically found in mechanical systems, the practical implications of sharp transitions of sti↵ness are limited. However, as the clutch sti↵ness increases, and for very large suppression coe cients (in the order of 5), strong non-linearities arise in the steady-state response at ⇡ particular frequencies. On the other hand, energy dissipation in the form of viscous damping in the flywheel can a↵ect the response under low frequencies of excitation. A mild narrowing of the resonance curves is 460 observed for high damping ratios corresponding to very imperfect devices. By contrast, dry friction, in quantities usually found in practice, does not seem to a↵ect the structural response. Besides, play gaps in the gearing system diminish the e↵ectiveness of the inertial damper. This is reflected in the reduction of the period lengthening e↵ect associated with the supplemental inertia. Also, generally large values of play-gap are required for their e↵ect to become apparent in the structural frequency response functions. 465 In addition, we have proposed a simplified numerical modelling strategy suitable for implementation in conventional Finite Element simulations. Our model is simple, yet versatile enough to incorporate all the sources of non-linearity explored above. For concept demonstration and for the validation of the numerical model proposed, we conducted a series of small scale experiments on bare elastic structures as well as struc- 470 tures employing 3D-printed inerter and inerter-clutch twins. The experimental and numerical results were found to be in close agreement. This shows that the proposed FE models can capture the actual response 25 of supplemental rotational inertia systems inclusive of clutching mechanisms. Finally, we have studied the dynamic response of multi-storey steel frames equipped with inerter and 475 twin inerter-clutches subjected to idealized harmonic pulses and a large dataset of real pulse-like acceleration records. We found that supplemental inertia is highly e↵ective in suppressing steel building deformations, es- pecially for lower frequencies of excitation (e.g T1/Tg > 3). Moreover, the use of twin inerters with clutches, can significantly reduce the peak base shear forces to levels below those observed in bare steel structures. Importantly, this response enhancements are accompanied by notable reductions in the variability of the 480 response. Nevertheless, juddering in the clutch, caused by the sharp changes in the dynamic mass of the sys- tem generated by the clutch, can produce very complex acceleration histories that may lead to undesirable increments in peak displacements at long periods. Besides, the development of plasticity in steel buildings may hamper the e ciency of the pair of clutched inerters. These matters are subject of current and future investigations. 485 Overall, the numerical and experimental results presented in this paper prove the significant benefits brought about by clutched supplemental rotational inertial systems. Moreover, this paper provides the necessary analytical and numerical tools for further studies. 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