Low-energy structures embedded with smart dampers

Abstract Building structures, subject to dynamic loadings or external disturbances, may undergo destructive vibrations and encounter different degrees of deformation. Modeling and control techniques can be applied to effectively damp out these vibrations and maintain structural health with a low energy cost. Smart structures embedded with semi-active control devices, offer a promising solution to the problem. The smart damping concept has been proven to be an effective approach for input energy shaping and suppressing unwanted vibrations in structural control for buildings embedded with magnetorheological fluid dampers (MRDs). In this paper, the dissipation energy in MRD is studied by using results from induced hysteretic effect of structural vibrations while the fluid is placed under a controlled magnetic field. Then, a frequency-shaped second-order sliding mode controller (FS2SMC) is designed along with a low-pass filter to implement the desired dynamic sliding surface, wherein the frequency responses of the hysteretic MRD is represented by its magnitude and phase describing functions. The proposed controller can thus shape the frequency characteristics of the equivalent dynamics for the MRD-embedded structure against induced vibrations, and hence, dissipate the energy flow within the smart devices to prevent structural damage. Simulation results for a 10-floor building model equipped with current-controlled MRDs, subject to horizontal seismic excitations validate the proposed technique for low-energy structures with smart devices. The closed-loop performance and comparison in terms of energy signals indicate that the proposed method allows not only to reduce induced vibrations and input energy, but also its spectrum can be adjusted to prevent natural modes of the structure under external excitations.


Introduction
Analysis of life cycle cost for energy-efficient buildings is evaluated based on energy consumption, assessment of environmental impact or natural hazards, and prediction of structural or non-structural damage [1]- [2]. Various elements equipped with energy-efficient features of the engineering structures likely ex- 5 perience different levels of damage subject to external dynamic loadings such as seismic events or gusty winds, depending on the specific geographic region where the structures are situated [3]. Thereby, it may increase future costs associated with post-event repair or replacement to maintain structural health or reinstate an acceptable level. Studies have shown that cumulative damage cost 10 can be higher than energy-efficient features and accordingly payback time for building energy efficiency investment will be prolonged [1]- [3].
Modern structures involve not only energy management [4, 5] but also condition assessment and safety management, whereby the integration of modeling, control and health monitoring is of crucial importance [6]. In quake-prone areas, 15 building structures often undergo vibrations in response to the ground motion caused by the seismic energy and fail to dissipate inelastic energy due to excessive lateral motion, resulting in structural deformation [7], [8]. Moreover, taller, slimmer and lighter structures using high-strength materials with the same modulus of elasticity, i.e. less stiff structures, may make them more prone to dynamic 20 loading sources, which cause discomfort and eventually, structural deterioration [9]. Thus, adequate strength and energy dissipation capacity should be rendered in the structure to limit the overall structural motion and shift away its natural frequency from the resonance region under the disturbance excitation to maintain the structural health at a controllable embodied energy level. For 25 example, the structural stiffness and damping can be adjusted whilst keep the amount of material utilized to a minimum.
It is possible to increase the stiffness of a building through selecting an appropriate structural configuration. Damping can be increased through the installation of auxiliary robust damping devices, since the damping characteristics, 30 such as inherent damping, of the core structural system is relatively ambiguous until the building is completed [10]. Alternatively, damping from external devices can be promising thanks to the extensive research conducted in the last decades, which make them a competent solution for mitigating the structural vibration problems in any dynamic application. However, active devices require 35 a large supply of energy, for example in active mass dampers.
Energy-dissipative semi-active devices, such as the magnetorheological (MR) fluid damper (MRD) [11], [12], MR elastomer base isolator [13], MR pin joint [13] provide supplementary robust damping for the attenuation of vibrations induced by excitation sources into the structure. The semi-active control systems can 40 dissipate vibration energy into heat through the adjustment of damping and stiffness characteristics of the system under a low-power control signal and failsafe operations. The controlled damping forces always oppose the motion of the structure, hence, promoting stability, as well as reducing the consequence of system uncertainties. [9,10].

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The level of possible damage of individual structural members, e.g., beams, columns, and roof/floor slabs can be determined by the transmitted external dynamic loading into structural vibrations. The induced energy can then be decomposed into different forms, i.e. kinetic, damping, recoverable elastic strain and irrecoverable hysteretic dissipation in the structure during a loading event 50 [14], [15], [7]. Semi-active control with MR fluid devices provides energy-efficient protection of engineering structures [11]-[18] by dissipating excess energy into heat through the fluid. This heat is then transferred to the environment by convection and/or conduction [17]- [20]. The frequency domain approach to structural control allows for a roll-off of the control action at high frequencies and specify the disturbance attenuation over desired bands. The frequency-shaping (FS) technique to the linear-quadratic 60 (LQ) design was first proposed in [21] with the cost functional expressed via the frequency variable ω. A discrete time approach to the frequency-shaping LQ control using the Parseval's theorem was reported in [22] for active suspension system. A challenging requirement for these structural control systems remains strong robustness in face of system uncertainties and large disturbances. For 65 this, sliding mode control (SMC) is known as a discontinuous robust control [23]- [25], which forcibly confines the system's states to a user-chosen sliding surface by varying the control structure in the state space. To extend the SMC design to the frequency domain, frequency-shaped SMC (FSSMC) has been developed and applied to various mechanical systems including flexible robot manipulators 70 [26], [27], active vibration control [28], and hard disk drives [29].
In FSSMC, the sliding surface is obtained by applying a desired linear operator to the original sliding function for shaping the system equivalence dynamics in the frequency domain [30,28]. An output feedback FSSMC was studied in [31] for damping out structural vibrations of a smart flexible cantilever beam, 75 where the system states are implicitly obtained by measuring the output at a faster rate than the control input. The works mentioned above have not clearly explained on the dissipation of vibration-induced energy in the controlled smart devices. To date the analysis of the energy flow in the structures has not been directly addressed for control and monitoring to achieve energy-efficient em-80 bedded structures under vibrations induced from external loadings. Here, the frequency domain advantage is taken into account within a modelling and control framework to analyze the energy relationships of the smart devices in the structures, and to design a robust controller to achieve low-energy and resilient structures against dynamic loadings such as earthquakes or gusty winds. are compared between the uncontrolled case, the Lyapynov-based control and the proposed FS2SMC in terms of kinetic, damping, strain, and input energy signals to illustrate the capability of a low-energy smart structure in suppressing quake-induced vibrations. A conclusion is finally drawn in Section 6.

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The smart device considered in this section is the magnetorheological damper (MRD). To experimentally study the dissipation and energy-related aspects in the device for analysis of low-energy resilient structures embedded with MRDs, the RD-8041-1 damper manufactured by LORD Corporation is characterized by using a thermal camera.
where f MR = c MRẋ , the equivalent damping coefficient c MR = k MR ζ MR /(πf ), x = E sin(ωt) is the sinusoidal displacement of amplitude E and angular frequency ω = 2πf . The derivation of (1) is given in Appendix. To characterize this hysteretic force-velocity relationship, the MRD is mounted on a Schenck 110 machine, the Instron ElectroPuls TM E10000, as shown in Fig. 1(a).
To record the dissipation of the damper RD-8041-1 in response to the magnetic field strength at a magnetization current i(t), we used a Testo 875-2i thermal camera. The mechanical energy dissipated in the magnetorheological fluid inside the damper housing is converted into heat, depending on the magnetic field strength. Thus, an increase in the magnetization current i will result in a temperature rise in the fluid inside the MRD housing, as depicted in Fig. 1 Taking into account also the Joule effect of the coil resistance R, the power P of the system can be given by where R = 5 Ω (7 Ω) at the ambient temperature (at 71 • C) and i ∈ [0 2]A for the MRD RD-8041-1 used in experiments.   (a) One energy cycle, 2π/ω = 0.5 s.

MRD describing function model
To obtain the energy spectrum of the smart structure, the describing function (DF) technique, or the harmonic balance method, is used to dervive the frequency response of the MRD embedded. DF is a mathematical approach for the design and analysi s of systems containing nonlinearity [23]. Here, by using a computational implementation of the DF [32], the magnitude and phase DFs for RD-8041-1 are plotted in Fig. 4 from its experimentally obtained characteri-145 zation data. In general, the gain N (E, f, i) decreases as the amplitude E and/or frequency f increases. On the other hand, the DF gain of MRD increases with the magnetization i within its operational range.
Since the DF is a complex quantity, i.e. N e jφ , DF of MRD exhibits a phase shift φ as shown in Fig. 4(b). A rational approximation technique yields the following expressions [13]:

Energy balance equation of buildings with smart devices
The problem of energy balance in a building structure embedded with smart dampers is considered in this section. Since internal forces within an engineering structure can be derived using relative displacements and velocities, we herein compute the input energy in terms of the relative motion.

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Given an n degree-of-freedom (dof) shear structure of mass M , stiffness K, and viscous damping C, embedded with n MRDs subject to dynamic loading sources with acceleration vectorẍ g , the governing equation can be described by where Γ ∈ R n×n is a factor matrix taking into account the location and number of MRDs; f MR (x, i) is the controllable damping force vector; x = r − x g denotes the relative displacement between the ground and each mass as shown in Fig.   5(a). Here, r and x g are vectors of the absolute displacements of the floors and ground with respect to a reference frame xx − yy and an inertial frame xx − yy , By integrating both sides of (4) over the structural response path from t 0 (a) (b) Figure 5: Smart building integrated with energy-dissipative devices; (when the ground motion excitation starts) to t, we obtain: where E k , E ζ , E s , and E i represent the relative kinetic, damping, strain, and input energies, respectively. The absolute kinetic and input energies can also be defined respectively by 1 2ṙ T Mṙ and t t0r T Mẋ g dt. Consequently, the residual absolute energy terms can be derived in the same way.

FS2SMC design
It can be shown that the nonlinear non-affine dynamic system (4) of a singleinput embedded smart structure can be rendered to an n-th order nonlinear system of the form [11]:ż where z ∈ R n is the state, u ∈ R is the control, a(.) and b(., .) are some smooth nonlinear functions. Our goal is to design a robustly stabilizing control u = U (z) for example, to get a steeper roll-off of |L(jω)|, ω ∈ [0, ∞) for large values of frequency ω. Herein, L(s) = b0 s 2 +a1s+a0 gives a |L(jω)| with an asymptotic slope of −40 dB/decade above cut-off frequency. The algebraic manipulation of (6) gives where H = ∂h ∂xż + ∂g ∂xż u and v =u is denoted as the new control variable [33]. Then, we can derive the best approximation of the continuous control law that whereĝ andĤ are the nominal models of g and H, respectively. A reaching control input v R is added tov to ensure that the plant dynamics reach the sliding surface in finite time [25]. Substituting v =v + v R into (7) where δ(z) . Suppose the perturbation term δ(z) satisfies the inequality for some known positive definite function (z). With V = 1 2σ 2 chosen as a Lyapunov function candidate for (7), the time derivative of V can be computed To achieve the control objective, the robust signal v R is selected as where κ > 0, η > 0, so that the term g(z)σv R is negative and dominates over the residual term g(z)|σ| (z) whenσ = 0, giving the net results to force |σ| to reach zero. Finally, we have Substituting (11) into (10) yieldṡ By integrating the differential inequality over the time interval t 0 ≤ τ ≤ t, 200 we obtain V (t) ≤ V (t 0 )e −2κg0(t−t0) . Thus, V (t) will tend to zero exponentially where κ is the decay rate at which the sliding surface is attained. From (10) and (13), we obtainσσ ≤ −g 0 η|σ| − g 0 κσ 2 . Since g 0 κσ 2 ≥ 0, and by neglecting the nonlinear term, we also have d dt |σ(t)| ≤ −g 0 κ|σ(t)| ⇒ |σ(t)| ≤ |σ(t 0 )|e −κg0(t−t0) to substantially reduce the amplitude of the switching term in the control and 205 hence, the commonly encountered chattering problem associated with sliding mode control.

Smart structural control
To further implement the proposed control strategy, described in the previous section, to low-energy MRD-embedded structures, a modal transformation is first applied to the structure dynamics, e.g. of a multi-floor building. Thus, with the transform x = Φq (Φ is a nonsingular transformation matrix) for the modal coordinate vector q [28], we can obtain a set of n second-order motion equations decoupled from (4) for each mode, m ∈ [1, n], as: where ω m , ζ m , u m , q m , d m , and µ mr are respectively the m-th modal frequency, damping ratio, entry of the modal control u = Φ −1 M −1 Γf MR = Ωf MR , modal coordinate, disturbance component and the mrth modal coupling term of the damping matrix. From (6) and (7), the following frequency-shaped sliding function is designed: where v m =u m is the new control instead of the modal control u m . We derive the equivalent controlv m to achieveσ m = 0 as followŝ v m =ω 2 mqm + 2ζ mωmqm + n r=1,r =mμ whereω m ,ζ m ,d m , andμ mr are desired values chosen for the m-th modal frequency, modal damping, first derivative of the disturbance, and modal coupling from the damping matrix, respectively. By applying the control law (12) for v m =v m + v Rm , we obtain the following FS2SMC that can ensure the conditionσ mσm ≤ −η m |σ m | − κ mσ 2 m as in (13), by taking β m = m + η m sufficiently large.

Application and Simulation
For application, we now consider a 10-storey shear building model [35]

Modal decomposition and control design
Since the mass matrix is nonsingular, (4) is written for this case as where ε kr ∈ [0 γ k N k e jφ k ] denotes inter-floor damping from the MRDs mounted between the k-th and (k − 1)-th floors and γ k is a factor taking into account the placement and number of MRDs. Equation (19) can be rewritten as Since the damping capability always takes its strongest effect at the level where the MRDs are installed, DF MR can be considered as diagonally dominant, thus including only terms γ k N k e jφ k . Other the coupling terms from the residual modes and modal decomposition errors can be lumped to disturbance d k . Taking the assumption of Rayleigh damping [7], [15], the frequency response function (FRF) matrix, H(jω), of the smart structure is therefore obtained as Here, the transfer function of the DF model (3) is approximated as ω k N 0k is the MRD equivalent time constant estimated at normalized values of amplitude E = 1, current i = 1 and first modal frequency ω 1 = 1. Hence, the FRF of the smart structure is In our design, the controller parameters are chosen as κ k = 10, η k = 1.25 and L(s) is a Butterworth filter, i.e. L(s) = 1 s 2 +1.4142s+1 . Notably, for the control law (17), a boundary layer [23] may be used in lieu of the signum function to smooth the response if necessary. From the modal control u k = v k (t)dt, the damping force can be computed as f MR(k) = Ω −1 u k in which Ω −1 is the inverse mode participation matrix. The controllable force range should be constrained by the maximum capacity, i M and the residual force at zero current in the passive control case. For example, the relation between the magnetization current and damping force has the following form: where a 0 = 0.127, a 1 = −0.00094 and a 2 = 0.0000021 for the RD-8041-1 at    nite matrix, and y is the system state [12].    suppression, the FS2SMC is more energy-efficient than the LC since it requires 300 a smaller amount of control energy E c but results in more absorption of input energy with less stiffness, damping and strain energies at the output. Indeed, owing to the incorporation of the frequency-depending relationships of forcedisplacement and force-velocity of the smart devices into the system model and control design, and hence, the ability to effectively shape the frequency responses 305 of the overall smart structure, the proposed controller can adjust the embodied energy to alter its spectrum in a desired bandwidth, roll off from the resonance region to limit the peak value of the mechanical and transmitted input energy terms, resulting in a low-energy structure while avoiding natural modes of the integrated structural system in dealing with any external loading source. 310

Conclusion
We have presented a frequency domain-based method for modeling and control of low-energy structures embedded with smart devices to mitigate the struc-

Appendix
Energy dissipated per cyclic oscillation: Given periodic displacement x = E sin(ωt) and velocityẋ = Eω cos(ωt), the energy dissipated by a MRD in one vibration cycle can be determined by the area enclosed within as hysteresis loop, as shown for example in Fig. 2(a): where the damping force function is f MR = c MRẋ = ±c MR ω √ E 2 − x 2 and with 330 its conjugate variable, the displacement, lying on an ellipse ( fMR cMRωE ) 2 +( x E ) 2 = 1 depicted in Fig. 11.
The loss coefficient or damping ratio ζ MR of the MRD can be defined as the ratio of damping energy loss per radian divided by the strain energy, i.e.,