A simple hybridization of active and passive mass dampers

Transcription

1 A simple hybridization of active and passive mass dampers S. Chesné 1, C. Collette 2 1 Université de Lyon, CNRS INSA-Lyon, LaMCoS UMR 5259, F Villeurbanne, France simon.chesne@insa-lyon.fr 2 Université Libre de Bruxelles, BEAMS department, 50,F.D.Roosevelt av., 1050 Brussels, Belgium. Abstract Hybridization of active and passive systems is an interesting and promising concept whatsoever for isolation and/or damping structures. Hybridization aims at combining the best of active and passive approaches. Improvement of performance with low power consumption are combined with fail-safe aspect, which is very important in the context of embedded applications (Transport/aerospace industries). However, for large values of the control gain, some difficulties arise when attempting to couple active and passive control within the same mechanical system particularly in terms of stability for usual controller. In this paper, we present a novel and simple control law, dedicated to drastically improve the performance of classical hybrid dampers based on decentralized velocity feedback techniques. The proposed hybrid system controller is fail-safe but also unconditionally stable. Besides these assets, the control approach proposed requires a low consumption and is extremely efficient under harmonic excitation, and is extremely simple compared to usual controller. 1 Introduction Tuned Mass Dampers (TMD) or Dynamic Vibration Absorber (DVA) are small oscillators, appended to a structure to dissipate the vibrational energy a specific resonance. Since their first invention in 1909 [1], these practical and robust devices have found numerous applications ranging from civil structures to precision engineering. The beauty is that the same design rules can be used to damp nanometer vibrations or meter vibrations. The most popular is the so-called equal peak procedure [2,3]; other interesting rules can be found in [4, 5]. Only the technology changes. The major drawback of TMD is that they are tuned to damp only one specific resonance. To some extent, broadband dampers (e.g. mass mounted on a layer of elastomer) may damp several resonance modes, with a limited efficiency. A more efficient method to treat several resonances is to introduce an actuator which controls the force that the reaction mass applies on the structure. The resulting device is often called an Active Mass Damper (AMD). In order to add viscous damping to the structure, the natural way is to drive the actuator with a signal which is proportional to the velocity. However, for large values of the control gain, the AMD poles tend to be destabilized, even when the structure velocity is measured near the AMD. It is believed that this is the reason why many elaborated control strategies have been introduced to control AMDs (although it is rarely openly confessed): classical tuning [6], fuzzy controllers [7 9], pole placement [10] Lyapunov s method [11], optimal control [12 15], µ-synthesis [16], H-infinity [17] or sliding mode control [18]. On the other hand, a novel class of AMD has appeared which are trying to combine several objectives at the same time. These devices are gathered under the common name of Hybrid Mass Damper (HMD), or Hybrid Vibration Absorber (HVA) even though the objective pursued may significantly differ from one HMD to the other. For example, in [15], a HMD is presented, where an optimal control is used to combine structural damping with a restricted stroke of the actuator. In [19] a H-infinity optimal control is used to minimise both 137

2 138 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 the response and the control effort. Among the HMD, several of them have the property of being fail safe, which means that they still behave as TMD even when the feedback control is turned off. In [20] the control relies on the pole placement technique. In [21], a dual loop approach is preferred to increase the stability margins. In [22], a two degree of freedom system is studied, which can behave as a active mass damper to suppress the vibrations induced by small earthquakes, and as a tuned mass damper to suppress the vibrations of a targeted mode excited by a big earthquake. In the active configuration, a linear quadratic regulator is used with a large gain on the structural velocity. The control is switched off above a threshold value of the structure displacement, i.e. when the actuator cannot deliver the requested force anymore. The controller proposed in this paper is also fail safe and unconditionally stable. The strategy consists of placing a pair of zeros adequately in the controller in order to obtain interacting poles and zeros in the open loop transfer function as suggested in [23]. Besides these assets, the control approach proposed requires a low consumption and is extremely efficient under harmonic excitation, and is extremely simple compared to controller found in the literature. The paper is organized as follows. Section two presents the simplified flexible structure which is used to illustrate the controller efficiency. Section three contains an analyze of the stability of active/hybrid mass damper. Section four and five present the concept of the proposed controller and its design laws. Section six discusses the performances and limitations of the new controller and compares it to others. Section seven presents the experimental validation before drawing the conclusions. 2 Description of the discrete structure Consider the system shown in Fig. 1. It represents a flexible structure, excited from the base x 0. The mass suspended has been divided in three parts in order to include three resonances. In the remaining of the paper, the following numerical values have been used: m 1 =m 2 =m 3 = 100kg, k 1 = k 2 = k 3 = N/m; leading to the following eigen-frequencies f 1 = 14.1Hz,f 2 = 39.7Hz,f 3 = 57.3Hz. Dashpots constants c 1, c 2, c 3 are tuned in order to provide a modal damping of ξ = 0.01 to all modes. A HMD is appended on m 3, as shown in Fig. 1. The passive part of the HMD (in blue) consists in a mass m a a spring k a and a damper c a. The values of these parameters depends on the operating mode (see section 3). The active part (in red) consists in an absolute velocity sensors mounted on m 3, a controller H(s) and an actuator F a between m 3 and m a. x 0 x 1 x 2 x 3 x a k a k 1 k 2 k 3 m 1 m 2 m 3 c a m a c 1 c 2 c 3 x 3 F a H(s) Figure 1: Simplified model of a flexible structure with an HMD. (Blue: Passive part, Red: Active part).

3 ACTIVE NOISE AND VIBRATION CONTROL Stability analysis of active/hybrid mass damper Consider the Active Mass Damper (AMD) as shown in Fig. 1. The resonance frequency of the device (m a, c a, k a ) is tuned significantly lower than the first mode to be damped. In order to damp the structural resonances, the actuator is driven by a force proportional to the velocity of the structure measured on mass m 3 where H(s) is a simple gain. F a (s) = H(s)ẋ 3 (s) (1) The corresponding root locus is shown in Fig 2(a). The root locus of the AMD (red dotted line) shows three loops associated to the flexible modes of the structure and a small loop at low frequency linked to the dynamic of the actuator. Even if the kind of active device is very efficient, in case of failure, the first mode of the passive resulting system is uncontrolled. This is due to the fact that the actuator is not tuned on this mode. Moreover it can be seen that for high gains, the root locus passes through the imaginary axis and can creates instabilities. In order to obtain a fail-safe system, let us further choose the parameters of the uncontrolled AMD according to the Equal peak procedure to damp the first resonance of he structure, and keep the same control force (1). The resulting root locus, also shown in Fig. 2 for comparison (continuous black line) shows clearly the limitation of this approach. On the zoom close to the origin shown in Fig. 2(b), one can see that the initial pole has been duplicated by the TMD. One sees also that the lower frequency pole goes rapidly in the right half plane, leading to instability. The closed loop system will always be near this stability limit even with low gains. This is due to the absence of zero between the pole of the inertial damper and the first pole of the structure. As explained in the introduction, several elaborations of the controller have been proposed to bypass this limitation. In the next section, we will present a simple alternative controller which address the root cause of the problem by adequately placing a pair of zeros at the right location, which will allow recovering a guaranteed stability of the closed loop system. 4 A robust Hybrid Mass Damper (α-hmd) As in the previous section, consider a HMD where the mechanical parameters have been tuned in such a way that the uncontrolled device behave like a TMD tuned on the first more. The controller is still using the velocity of m 3, however, the filter is now defined as (s + α)2 H α (s) = g s 2 (2) Compared to Equ.(1), one sees that a pair of poles and zeros have been introduced. The parameter α is tuned to make the controller hyperstable. Its value will be discussed in section 5). The tuning of gain g depends on the objectives of the controller and also on the capability of the transducer (see 6 ) 1. Taking α = 2πf 1 = ω 1 the resulting rootlocus of the α-hmd is compared in Fig. 2 (continuous green line) 2. This controller has several advantages. Firstly, the whole root locus plot is in the left half plane, meaning 1 In the remaining of the paper, we will call it α-hmd. 2 Notice that, in order to avoid a constant component in the feedback loop, a high pass filter H HP (s) at very low frequency has been added. It is a first order filter with a cut off frequency equals to α/50. The resulting control force is: F a(s) = H α(s)h HP (s)ẋ 3(s) (3)

4 140 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Root Locus HMD α-hmd AMD Imaginary Axis (seconds -1 ) Real Axis (seconds ) Root Locus HMD α-hmd AMD Imaginary Axis (seconds -1 ) Real Axis (seconds -1 ) Figure 2: (a) Root locus for HMD, AMD and α-hmd; (b) Detail near the origin of the root locus.

5 ACTIVE NOISE AND VIBRATION CONTROL 141 Poles with TMD Im(s) ω b ω i Initial pole without TMD ω a Zeros of the controler (value ) Hyperstability limits Re(s) Figure 3: Hyperstability limits on the pole map, without structural damping. an unconditional stability of the feedback system (infinite gain margin). Secondly, the two branches move directly away from the imaginary axes (this will be further discussed in section 5). Thirdly, we can observe that the damping authority on all modes is equal or larger than for the AMD. Fourthly, as it is based on a TMD device, in case of failure of the active part, the first mode is still passively controlled by the TMD. The very small loop close to the origin is due to the high-pass filter introduced by Eq. 3. This filter does not change the controller behavior for the flexible modes. 5 Designing the α-hmd Due to the presence of a TMD tuned on the mode to control, the pole of this mode ω i is replaced by two new poles. One has a frequency slightly lower than the resonance frequency of the initial system, and one has a frequency slightly higher. Let us call them respectively ω a and ω b. The resulting root locus is shown in Fig.3. As long as α ]ω a ; ω b [, the branches of the root locus go immediately inside the left half plane. Actually, using the Routh criterion of stability it can be analytically found that the controlled system recovers its hyperstability if: ω a α ω b (4) Fig. 4 further illustrates the departure angles of the root locus branches for extremal cases α = ω 1 and α = ω 2. In these cases, one of the departure angle is equal to ±π/2 and its branch is tangent to the imaginary axis. In case of lightly damped system, the limits defined by Equ.(4) can be extended due to the fact that the poles (ω a and ω b ) are no more on the imaginary axis. The α parameter can be optimized depending on the application. In the following, it has been chosen that α = ω i ensuring the condition expressed in Equ.(4) is verified. The next section studies the performance of the α-hmd. 6 Performances and limitations of the α-hmd In the following α-hmd is compared with the passive TMD and the purely active AMD. Each of these devices considered to be mounted alternatively on the system shown in Fig.1. As these devices are clearly

6 142 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 = ω a Im(s) = ω b Im(s) 2 > /2 2 = /2 ω b ω b ω i ω i ω a ω a 1 =- /2 1 <- /2 Figure 4: Departure angles on the root locus, left: α = ω a, right: α = ω b, without structural damping. different, the device parameters have been chosen arbitrarily. The gain g α and the gain of the AMD are both tuned to double the damping on the first mode of the system with TMD. To sum up, the initial uncontrolled system has a damping factor on the first mode of 1%, 3.5% for the passive control system (TMD) and around 14% for the active and hybrid systems. TMD or Hybrid TMD are usually designed for applications where the disturbance is harmonic, or narrow band (e.g. helicopters). More analysis under harmonic excitation can be found in [24]. In this paper, one analyses the system response in the frequency domain while it is subjected to a uniform broadband disturbance on x 0. The displacement of the mass x 3 where the control device is attached is considered. It represents directly the performances of the control in term of structural damping. Figure 5 shows the transmissibility function x 3 /x 0 between m 3 and a solicitation coming from the ground x 0. For the α-hmd controller one sees that the initial peak of the controlled mode is replaced by a huge antiresonance due to the introduction of the zeros at this pulsation, associated with the dynamical effect of the TMD. The resulting absorption capacity is greatly improved. The two main resulting peaks are well separated as expected when looking the root locus Fig. 2. Moreover, as for the AMD, the α-hmd damps the other flexible modes, whereas it is not the case for the purely passive TMD. In [24], one can see that the nice performance of the α-hmd controller have been obtained at the cost of a much larger relative displacement of the moving mass (m a ) over a wide frequency range. At the resonance frequency of the first mode, the α-hmd controller requires a smaller force than the classical AMD. It is due to the fact that it corresponds to the tuning frequency of the TMD. For other frequencies, the α-hmd controller requires a larger force, this is a consequence of the fact that, in this example, its mass is 5 times lighter than the mass of the AMD and that the gains have been tuned to provide an equivalent damping. Nevertheless, these drawbacks (huge relative displacements and important active force on a wide frequency range), do not appear as critical. Indeed, the envisaged applications for these kind of TMD or hybrid TMD are mainly harmonics and not clearly broadband (see [24]).

7 ACTIVE NOISE AND VIBRATION CONTROL 143 Transmissibility x 3 /x 0 30 Without control TMD AMD α-hmd Magnitude (db) Frequency (Hz) Figure 5: Transmissibility function x 3 /x 0 for various control devices. 7 Experimental validation 7.1 Test setup A first experimental validation is led on a flexible structure. It is a cantilever steel beam (Length: 58cm, width: 10cm, thickness:1cm). The TMD is Micromega products initially designed for purely active control (ADD-5N; html). Its moving mass is 160gr, its resonant frequency is around 21Hz and its internal damping of ξ = 11.9%. Its location on the beam is L abs = 48cm. An accelerometer is fixed nearby the actuator. The targeted mode is the first one. The mass ratio comparing the moving mass of the TMD and the effective modal mass of the flexible structure is µ = With this mass ratio, the theoretical optimal damping of the TMD should be ξ opt = 17.9%(instead of ξ = 11.9%). The structure and the control device is shown in figure 6. In practice, neither the damping or the resonant frequency are perfectly tuned to correspond to optimal values as defined in [3]. Despite this, good performance is obtained showing the robustness of the approach. The frequency response of the structure without TMD can be seen in figure 7 (black dotted line). With the passive TMD (black continue line), one can see a reduction of 18dB of the amplitude of the first mode. The passive device has no effect on other modes at higher frequencies. 7.2 Hybrid control results Figure 7 shows the acceleration response of the structure to a force disturbance applied through the transducer, for various values of the gain g (equation 2). It clearly shows very good performances of the control law to damp the first modes of the passive TMD (35dB for higher gains). An interaction of the low frequency pole of the controlled system with the electronical part of the setup prevents the use of higher gains in the control loop. An unexpected pair of pole/zero appears around 10Hz when the gain increases. At the time of writing this paper is origin is still under investigation.

8 144 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Figure 6: Picture of the experimental set-up used to test the proposed controller. Cantilever beam, clamped at one end, and equipped with the AMD (Micromega Dynamics ADD-5N) at the other end. The motion of the beam is measured by a piezoelectric accelerometer, collocated with the AMD. Acceleration/Commande [db] Without TMD g=0 (Passive TMD) g=0.5 g=1 g= Frequency (Hz) Figure 7: Compliance at the transducer location without and with TMD, for various values of the gain g. Nevertheless, without particular tuning, the first mode is completely damped. One can see also see (fig. 8) that the damping factor of the second mode is multiplied by 4 (ξ passive = 0.09% to ξ g=2 = 0.35%). The vibration are reduced of more than 12dB. Effects on higher order modes is not clearly visible. 8 Conclusion A new control law for Hybrid Mass Damper has been presented in this paper, which has been referred to as α-hmd. Besides its simplicity, the proposed control law belongs to the restricted class of hyper stable control, which means that the theoretical stability of the closed loop system is guaranteed. Design guidelines have been presented to tune the control parameter adequately for an arbitrary plant. The performance of the α-hmd has been discussed numerically on a three degree-of-freedom model, and compared to a TMD, and an AMD. It shows many advantages [24]. A first experimental validation illustrates the simplicity and the efficiency of this approach.

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