Performance and constraints of an active vibration control system with electrodynamic inertial mass actuators

Transcription

1 Performance and constraints of an active vibration control system with electrodynamic inertial mass actuators G. Lapiccirella 1, J. Rohlfing 1,2, D. Mayer 2 1 Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany giovanni.lapiccirella@lbf.fraunhofer.de 2 Technical University Darmstadt, Research Group of System Reliability and Machine Acoustics SzM, gdarmstadt, Germany Abstract The aim of this work is to investigate the effects of different control strategies for an active vibration control system regarding the control performance, power consumption and constraints of the active vibration control system with electrodynamic inertial mass actuators. As a first step, a control system with a single inertial mass actuator mounted on a single degree of freedom has been considered. The analysis is expanded to an active vibration control system applied to a six cells laboratory truss structure. In the simulations the truss structure is modelled using a reduced state space formulation derived from a finite element model. The parameters for the models of the truss structure and the control actuators used in the simulations are derived from experimental characterizations in the laboratory. Numerical and experimental studies are shown. First results show that different control strategies can give different performance for the same power consumption. Moreover, the achievable performances are limited by nonlinear effects as well as stroke limits of the control actuators. 1 Introduction Lightweight design solutions are commonly used in aerospace, automotive, naval and civil engineering applications. Truss structures represent a good example of generic lightweight schemes and often they are used as base design for more complex constructions (e.g. bridges, buildings, vehicle frames). However, these structures are susceptible to vibration problems. Therefore, vibration control is of key importance in several practical applications. Passive and semi-active solutions have been proposed for the control of large truss structures, like for example tuned vibration absorbers [1], adaptive vibration neutralizers [2] or active struts directly embedded into the beams [3]. However, the high requirements of some use-cases require the application of Active Vibration Control (AVC) approaches. AVC represents a promising approach for the control of the vibration whenever the passive systems are inadequate. AVC on large lightweight structures has already been investigated in the past, using different control strategies and transducers. For instance, Herold et al. [4] proposed a novel dielectric stack actuator that combines active and passive approach. This actuator can efficiently absorb vibrations at some target frequency ranges and provides additional active damping at different frequency ranges. Kauba et al. [5] implemented a feedforward control approach on an AVC network consisting of four inertial mass actuators (IMAs). Rohlfing et al. [6] implemented an AVC system composed of two IMA on a Warren truss laboratory structure for vibrations reduction and structural health monitoring. However, these studies did not consider the possible constraints and performance limits that can arise due to the restricting properties of the transducers or application requirements. The performances of the AVC are limited by physical constraints (e.g. stroke limit) of the control actuators. Moreover, power consumption of active control systems may be a problem in cases where the electrical power resources are limited. Therefore, limitations on the power provided to the AVC could represent a performance constraint. In aerospace and automotive applications, 1233

2 1234 PROCEEDINGS OF ISMA216 INCLUDING USD216 the power consumption of subsystems is often an issue that needs to be tackled from the early design stages. Power consumption strongly depends by the components of the AVC. The chosen electromechanical transducer has a strong effect on the power consumption and the power flow. For example, piezoelectric transducers mainly act as capacitive electrical loads. This suggests that a great amount of the power provided to the transducer by the energy source (apparent power) oscillates between the power amplifier and the actuator without producing any effective work (reactive power). In contrast, electrodynamic voice coil motors exhibit resistive behavior at low frequencies and inductive behavior above the electrical cut-off frequency. Furthermore, electrical loads can also be affected by the mechanical behavior of the connected structure. First studies on AVC systems power consumption focused the attention on the electro -mechanical coupling between the transducer and the controlled structure. Liang et al. [7] formulated a coupled electro-mechanical analysis for piezoelectric actuators integrated in mechanical structures. They observed that the electrical impedance of the transducer is influenced by the mechanical behavior of the controlled structure. Zhou et al. [8] presented a method to predict the power flow in structures with integrated piezoelectric actuators. Similar effects are observed also in the case of different actuators, like for example electrodynamic inertial mass actuators [9]. Thus, the resulting effective electrical impedance hence influences the power consumption and power flows into the AVC. Power amplifier concepts may also influence the total power required by the active control [1], [11]. The positioning of the actuators on the controlled structure may also influence the power consumption of the AVC as well [12]. Finally, the control strategies applied for active control can also affect the amount of power required and the power flows within the AVC system. In the following sections, the effects of different control strategies on the control performance, power consumption and constraints of the active vibration control system with electrodynamic IMAs are discussed. First, results for an AVC system with a single electrodynamic IMA applied on variable stiffness single degree of freedom (SDOF) primary structure are presented. The performances and the power consumption of the AVC are discussed for different stiffness values of the primary structure and for four different control strategies. Successively, results from a study on the performance and the constraints of an AVC system comprising two IMAs applied to a laboratory six cells truss structure are presented. A brief introduction of the test laboratory structure, the inertial mass actuators and the FEM model of the truss structure are given. A velocity feedback control with and without feedback specifically designed compensation has been applied and compared. The cases in which the actuators work singularly and the case in which the two actuators work simultaneously in a decentralized network are considered. Numerical and experimental results on the performance, the constraints and power flows inside the AVC system are presented. Finally, conclusive results show that different performance levels are achieved for the same given amount of power and that the performance are limited by nonlinear effects in some specific cases. 2 Preliminary studies on a two DOF system c) d) c 2 c 1 rr 2 F a k 2 rr 1 xx 2 xx 1 k 1 xx p U ll c 2 c 1 rr 2 F a k 2 rr 1 xx 2 xx 1 k 1 xx p U ll CC c 2 c 1 rr 2 F a k 2 rr 1 xx 2 xx 1 k 1 xx p U H LMS SS c 2 c 1 rr 2 F a k 2 rr 1 xx 2 xx 1 k 1 xx p U SS H RLS SS - + Figure 1: Control strategies considered for the SDOF system a) VFB b) CVFB c) FFC and d) IMC A simple system is considered in order to get a first understanding of the effects of the control strategies on the performance and constraints of AVC systems. As preliminary study, the performance and the components of the power flow inside an AVC with an IMA and a SDOF primary structure with variable stiffness are examined. For simulations the system can be considered as a lumped parameter model of simple two DOF system, and the electrical properties of the coil are modeled as electrical impedance comprising a resistive and inductive load. However, due to the back electromotive force (EMF), the

3 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES 1235 effective electrical impedance is affected by the mechanical resonances of the primary structure and the control strategies applied to the AVC. For this reason, four different control strategies have been applied and analyzed in terms of performance and power consumption (Figure 1) to different primary structure stiffness settings: velocity feedback (VFB) that adds active damping to the primary structure and compensated velocity feedback (CVFB) that improves the stability and the performance of the classical VFB (as described in [13]), a feedforward control (FFC) with a digital finite impulse response (FIR) filter adapted through the use of F-x-LMS algorithm and an internal model control (IMC) where for the adaptation of the FIR filter a recursive least squares (RLS) algorithm is used. The results of this study show that different control strategies and the specific stiffness settings of the main primary structure influence the achievable control performance, the power consumption and the proportional part of reactive power. Independently from the control strategy applied, simulations show that higher values of apparent power are needed for higher stiffness settings of the primary structure. Moreover, lower values of reactive power are observed in case VFB and CVFB compared to IMC and FFC. Experimental results generally agree with the simulations. It can be stated that different control performances are achieved for the same amount of power consumption for the four control strategies in simulations and experiments. However, the experiments have shown that the performances of the AVC systems are limited by nonlinear effects well before the stroke and current limit of the IMA. This suggests that simulation models should be improved by including elements that can influence the control performance of the AVC (e.g. A/D and D/A conversion delays, amplifier concepts, stroke limitations and nonlinear characteristics) in order to be able to evaluate the performance constraints of the AVC. More detailed results are presented in reference [14]. The methodologies for the evaluation of performances and constraints of this simple system are extended to a more complex system. As first step, the applications of velocity feedback without and with compensation are considered. 3 The laboratory truss structure and the control strategies The test rig considered in this study is composed by a laboratory six cells truss structure, an AVC system with two IMAs and a primary shaker for the primary disturbance force (Figure 2). The six cells space truss structure is composed of 28 nodes, connecting 52 short and 24 long struts. The structure is mounted on rubber mounts at the four principal corners. The low stiffness of the mounts guarantees that all the rigid body modes of the structure lie below 16 Hz. Six global modes occur within the frequency range from 5 Hz to 19 Hz. Three rhombic modes occur at 52.7 Hz, 54.2 Hz and 13.2 Hz. The first two bending modes occur at Hz and Hz and finally the first torsional mode occurs at Hz. All modes above Hz are local modes of the struts. A more detailed description of the construction of the truss structure can be found in Ref. [15] and the experimental modal analysis on the six cells truss structure are given in Ref. [3]. Figure 2 : Experimental test rig comprising a six cells truss structure Conventionally, the nodes are numbered progressively from 1 to 28 starting from the low front angle on the left of the structure in Figure 2. According to this convention, the primary shaker (Data Physics V4) is

4 1236 PROCEEDINGS OF ISMA216 INCLUDING USD216 mounted on node 16, the AVC IMA1 on node 1 and the AVC IMA2 on node 19. The primary shaker is suspended from an aluminum frame construction using elastic wires. This configuration adds only little additional mass to the truss structure. The two AVC IMAs are directly screwed to the structure nodes with M8 stainless steel screws. The IMAs are Fraunhofer LBF in house designs. They are constructed from a linear electrodynamic voice coil motors, where the magnets are suspended via two membrane springs and they act as the inertial masses. The electromechanical characteristics of the two IMAs have been experimentally evaluated. The key parameters of the IMAs are listed in Table 1. Electro mechanical characteristics m [kg] ω n [Hz] c [Ns/m] k [N/m] L [mh] R [Ω] Ψ [N/A] Actuator 1 node Actuator 2 node Table 1: Mechanical characteristics of the actuators 3.1 Finite element model of the truss structure and experimental validation A Finite Element Model (FEM) of the test rig is realized in ANSYS. The model closely resembles the real laboratory structure. The model consists of beam elements of different lengths. The complete struts are modularly built consisting of two connection bolts, two coupling nuts and the beam. The connections between the beams, the connection bolts and the two connecting nuts are defined by elastic connections. Their stiffness and the damping values are set in order to the fit the real torque values. The model has been experimentally validated in previous works [3]. The model consists of more than 33 degrees of freedom (DOF). However, the structure connections are highly sensible to room conditions (e.g. temperature) or small mechanical changes. Therefore, the structure mechanical behavior is sensitive to environmental conditions. For the purpose of this study, the numerical model is fitted to experimental results in order to meet frequency and amplitudes of the bending mode in the x-direction (Figure 3). Table 2 shows a list comparing the measured and simulation frequencies of the modes. x z y Figure 3: Finite element model - Bending mode in x direction Modes Rigid body Rhombic Bending Torsional Frequency [Hz] Simulations Measurements Table 2: Overview of structural modes and their natural frequencies in simulation and measurements

5 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES 1237 Due to the high order of DOFs of the FEM model, for the numerical dynamic analysis the FEM is reduced and then converted to a state space formulation with 12 states and imported in Matlab Simulink. Figure 4 shows a comparison between the numerical simulations (left) and measurements (right) results obtained considering the velocity at the node position 1 relative to the input force introduced at node 16. The results show good general agreement in both the occurrence of natural modes and response magnitudes. Amplitude [db] Figure 4: Transfer functions relative to velocity at node 1 and the disturbance force at node 16 a) simulations b) measurements Amplitude [db] Control strategies The control performance and power consumption of an AVC system can depend on the applied control strategy. In this paper, a VFB and CVFB applied to an AVC system comprising two electrodynamic IMAs are investigated. VFB is a linear time invariant control approach that adds active damping to the primary structure under control. Normally, due to the 18 phase shift below the natural frequency of the IMA blocked force, the feedback loop is only conditionally stable. With increasing feedback control gain, this causes spillover effects that compromise the control performances and stability. It is possible to detune the mechanical and electro-dynamical responses of the IMA blocked force by designing an open loop compensator. This allows the implementation of higher feedback gains and therefore higher values of additional damping to the structure [13]. Figure 6 shows the open loop frequency response function (OL- FRF) of the VFB of the IMA1. The phase is close to at the IMA natural frequency (around 26 Hz). This underlines that stability problems do not occur at the natural frequency of the actuators. Hence, from preliminary results, the open loop compensator presented has proven to be ineffective. The OL-FRF crosses 18 around 1 Hz. This frequency corresponds to the displacement rigid body mode in the IMAs application direction (conventionally in the FEM indicated as x-direction). Moreover, the phase of the OL- FRF drops above the natural frequency of the actuators due to the electrical cut-off frequency of the voice coil of the IMAs. The additional phase delay causes undesired effects above the IMAs natural frequency, affecting the velocity feedback performances. In order to improve the performances and stability of the VFB loop, a phase lag-lead compensator is designed. A third order phase lag compensator delays the voltage signal anticipating the 18 crossing point to lower frequencies. The gain margin increases due to the lower amplitudes occurring at these frequencies. A fourth order lead compensator lifts the phase of the feedback signal above the actuators natural frequencies, compensating the phase drop caused by the electro-mechanical coupling. The following equations show the frequency response functions of the compensators: CC = CC llllll CC llllllll = jjjj + zz 3 llllll jjjj + pp llllll jjjj + zz 4 llllllll jjjj + pp llllllll where the z lag and p lag are the zeros and the poles of the lag compensator respectively and the z lead and p lead are the zeros and the poles of the lead compensator. The values for the zeros and the poles are set to z lag /(2π) = 1 Hz, p lag /(2π) = 1, Hz, z lead /(2π) = 8 Hz and p lead /(2π) = 16 Hz respectively. In Figure 5 the FRF of the phase lag, phase lead and the final compensator resulting by the two lead-lag compensation are shown. (1)

6 1238 PROCEEDINGS OF ISMA216 INCLUDING USD216 Amplitude[dB] Lead Lag Lead-Lag Frequency[Hz] Figure 5: Compensator FRF a) Amplitudes b) Phase The improved CVFB can be observed in Figure 6. Phase [ ] Lead Lag Lead-Lag Frequency[Hz] a) b) Open loop IMA1 [db] CVFB VFB Open loop IMA1 [db] CVFB VFB Phase[ ] 9-9 Phase[ ] Figure 6: Open loop FRF of IMA1 at node 1 - a) Simulations b) Measurements VFB (dashed blue line), CVFB (continuous black line) Figure 7 shows the Nyquist plots for the OL-FRF of the feedback loop using IMA1 located at node 1. In the graphs the OL-FRF at the natural frequency of the IMA1 for the VFB and CVFB cases are marked with diamonds. The markers show that at this frequency the OL-FRF is close to the positive real axis. This corresponds to a phase of the degrees in the Bode plot in Figure 6. This indicates that stability problems do not occur around the IMA natural frequency in both the uncompensated and the compensated cases. However, a crossing of the negative real axis (at -18 ) occurs around the natural frequency of the first rigid body mode at about 1 Hz for the VFB control. If the feedback gain is increased, the circles grow and eventually touch the Nyquist instability point at (-1, j) causing the system to go unstable. Also for all frequencies were the OL-FRF is located inside the unit circle around the Nyquist stability point the feedback control creates spillovers instead adding active damping to the structure. For the CVFB control the compensators rotates the low frequency circles of the OL-FRF clockwise, away from the negative real axis. This improves the stability and the performance of the feedback control by allowing higher gains (improvement of the gain margin). In addition, the phase lead term of the compensation rotates the higher frequency circles of the OL-FRF slightly anti clockwise, insuring that they are located completely on the right hand side of the Nyquist plane. This guarantees that no additional spillover occur at higher frequencies.

7 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES x x Imaginary Real x 1-3 Real x 1-3 Figure 7 : Nyquist plots of the open loop FRF of IMA1 at node 1 a) Simulations b) Measurements VFB (dashed blue line), CVFB (continuous black line). The diamonds indicate the position of the Actuator natural frequency in the uncompensated case (greed) and the compensated case (red) Imaginary Simulations A simulation Matlab Simulink model has been used in the time domain analysis of the AVC performances and constraints. A state-space formulation of the reduced FEM is used to simulate the truss structure mechanical behavior. The IMAs are modelled using a lumped parameter formulation. The Analog-to- Digital elements are simulated with a sixth order butterworth antialiasing digital filter, a death time delay and a direct current decoupling. Digital-to-Analog elements have been modeled with a low pass sixth order butterworth reconstruction digital filter. The power amplifier is assumed to be ideally linear. The sampling frequency used for the time domain simulations is Hz. For this analysis, different use cases are considered: IMA1 active and IMA2 passive, IMA1 passive and IMA2 active and finally when both actuators work actively in a decentralized feedback arrangement. The feedback gains for the VFB and CVFB are gradually increased from zero up to the value that approximately achieves 6 db reductions around Hz. This frequency corresponds to the first bending mode of the truss structure in x- direction. The performances of the control system are calculated in decibel by taking the ratio between the squared velocity of the controlled structure at the nodes 1 (IMA1) and 19 (IMA2) and the squared velocity at the same points on the truss structure without the two actuators. Reductions are evaluated in the frequency range from to 2 Hz. The power consumption and the power flows into the AVC are evaluated by using the following general formula in the time domain: SS = VV rrrrrr II rrrrrr, (2) PP = II(tt) VV(tt), (3) SS 2 = PP 2 + QQ FF 2, (4) where SS denotes the apparent power, PP the active power and QQ FF the reactive power. The subspcripts rrrrrr denotes mean square values they are evaluated as in equation 5. xx rrrrrr = 1 nn xx2 (tt ii ) nn ii (5)

8 124 PROCEEDINGS OF ISMA216 INCLUDING USD216 For all the simulations the same white noise primary excitation signal is used, and the rrrrrr value of the primary force is 6.87 N. Stroke analysis of the actuators is also performed by evaluating the difference of the displacement between the proof mass of the IMAs and the relative position node AVC system with one actuator In this section, simulation results of the performances and the power consumption of the AVC are evaluated in case only a single actuator is actively working. Figure 8 shows the frequency response amplitudes of the velocities at nodes 1 and 19 relative to the applied disturbance force introduced at node 16. The two figures refer to the cases where only one IMA is used and the gain is set to give reductions at Hz. For this gain setting the VFB control also produces control reductions in the frequency range from 16 Hz to 15 Hz. The control strategy is not able to produce any reduction around the torsional mode at Hz. High spillover amplitudes are produced around the rigid body mode at 1 Hz. These spillover effects result in high power consumption and in a high stroke of the proof mass that would exceed the IMAs stroke limit. The CVFB control produces higher reductions in the entire frequency range of interest without producing any considerable control spillover. Moreover, the stroke of the IMA proof mass is well below its practical limit. x 1 / F [db rel. ms -1 / N] x 19 / F [db rel. ms -1 / N] Figure 8: Simulated frequency response functions a) velocity at node 1 relative to force disturbance when only IMA1 is active, b) velocity at node 19 relative to disturbance force when only IMA2 is active Uncontrolled structure (dotted red line), VFB control with higher gain applied (dashed blue line), CVFB control with higher gain applied (continuous black line) Figure 9 shows the average performances at the two node positions over the total apparent power consumed by the AVC system actuators a) for the case that only IMA1 is used actively and b) for the case that only IMA2 is active. The apparent power is considered as reference for the power consumption since it represents the most conservative estimation. Thicker lines are used to mark operation modes where the AVC system achieves overall reductions in the frequency range of interest from Hz to 2 Hz. It should be noted that small reductions are already obtained by simply applying the passive IMAs to the primary truss structure without applying any active control. The results show that with VFB control no additional reductions are achieved in the overall response due to the high control spillover at low frequencies. In contrast, with the CVFB control higher overall reductions are achieved with increasing feedback gain, i.e. increasing power consumption. In the case that only IMA 2 is used, the average performances increase gradually. However, only little additional overall control reductions are achieved; although Figure 8 shows that IMA 1 produces control reductions at node 1, where it is mounted, it is found that for increasing feedback gains the response at node 19 increases. Hence, the overall performance that can be achieved with IMA 1 is very limited also with feedback compensation. The results in Figure 9 also show that in order to achieve 6 db reductions at Hz, the VFB control consumes significantly more power than the CVFB control. The percentages of reactive power for the two control strategies are indicated in the boxes. The results indicate that the CVFB control produces higher values of reactive power. Therefore, CVFB control systems may benefit from using electrical power amplifier that are capable of recuperating reactive electrical power.

9 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES % Reactive % Reactive V c 2 / V 2 [db] 2 V c 2 / V 2 [db] % Reactive % Reactive Apparent Power [V A] Apparent Power [V A] Figure 9: Simulations: average reductions over total apparent power (and percentage of reactive) consumed with a) IMA 1 active b) IMA2 active - VFB (dashed blue line) and CVFB (dashed black line) AVC system with two decentralized actuators This section presents simulation results of the AVC with two decentralized control units. Figure 1 shows the amplitudes of the FRFs between the nodal velocities and disturbance force when the highest gains of the two control units has been set to give -6 db at Hz. As in the single IMA case, the VFB control produces reductions above 16 Hz. However, high spillover around 1 Hz is also created. Also in this case, these high amplitudes result in high strokes outside the linear operation limits of the IMA used in the experiments. The CVFB control produces reductions at both nodes positions around all resonance peaks of the truss structure frequency range of interest, without producing considerable spillover control. x 1 / F [db rel. ms -1 / N] x 1 / F [db rel. ms -1 / N] Figure 1: Simulated frequency response functions a) velocity at node 1 relative to force disturbance b) velocity at node 19 relative to disturbance force Uncontrolled structure (dotted red line), VFB control (dashed blue line) and CVFB control (continuous black line) with highest gain applied Figure 11 shows the average overall performances achieved at the two nodes over the total apparent power consumed by the two IMAs in the case both the actuators are used in a decetralised velocity feedback AVC system configuration. As for the single actuator cases, the performance of the VFB control is limited due to poor stability and high control spillover at low frequencies. Therefore the AVC system with VFB does not produce higher reductions with increasing power consumption. Moreover, as in the single actuator case, the power consumption drastically increases as the VFB control loop gets closer to

10 1242 PROCEEDINGS OF ISMA216 INCLUDING USD216 instability. For CVFB control, the control performances gradually increase with increasing power consumption for relative small amounts of apparent power. By further increasing the power consumption until the 6 db reductions at Hz are achieved, the average overall performance of the AVC system reduces but does not get worse than the uncontrolled case. Comparing the results to those for the system with only a single active control IMA, it is possible to observe that the case with the single IMA2 can achieve the highest average performances consuming less power than the two actuators solution and the single IMA1 active. The results indicate that it is very important to find good positions for the control actuators. The choice of control position will depend on the controllability of the modes of interests at this position, but also on the feedback loop stability that can be achieved. In future studies, it is therefore intended to investigate more possible control positions on the truss structure and also to investigate the optimal distribution of multiple IMAs. 6 4 IMA % Reactive V c 2 / V 2 [db] 2 IMA2: 13.3% Reactive IMA1: 6.4 % Reactive -2 IMA2: 25.5 % Reactive Apparent Power [V A] Figure 11: Simulations: average reductions over total apparent power (and percentage of reactive) consumed by the two actuators working together- VFB (dashed blue line) and CVFB (dashed black line) The percentages of reactive power for the two actuators for the VFB and CVFB control cases are given in the boxes on the right hand side of Figure 11. The outline styles of the boxes refer to the lines in the graphs, blue dashed for the VFB control and continuous black for the CVFB. As for the single IMA cases, the percentage of reactive power for the CVFB control are higher than for VFB control. For the case of CVFB control, a high percentage of the apparent power involved is reactive in the IMA1. For VFB control, the percentage of reactive power is about the same for both control units. 3.4 Measurements The experimental studies are conducted on the laboratory test rig described in the beginning of section 3. As for the simulations, the control performances and the power consumption of two control strategies, VFB and CVFB, are investigated. First, the case where the AVC system is working with a single control unit and then the case where both actuators work in decentralized network arrangement are investigated. The IMA1 and IMA2 are positioned at the node 1 and node 19 as indicated in Figure 2. An electrodynamic primary shaker introduces a disturbance force at node 16. Low pass filtered white noise with a bandwidth of 5 Hz is used as input to the primary shaker. Since the power consumption of the AVC system can depend on the disturbance force amplitude, the same root mean square amplitude of 6.87 N used in the simulations is also used in the laboratory experiments. The control strategies are implemented on a dspace system. A sampling frequency of Hz is used in order to allow a continuous time control design and keeping the additional phase delays as small as possible. The procedure for the evaluation of the performance and power consumption are the same as used in the simulation studies. Also for the measurements, the gains are increased from to values that produce approximately 6 db reductions at Hz, corresponding to the first bending model. The force and the acceleration at the position 16 are measured by a PCB TLD288D1 impedance-head. The accelerations at

11 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES 1243 node 1 and 19, and the accelerations of the magnets are used four PCB type 352C33 accelerometers. The accelerations are then filtered through low pass Kemo filters and acquired through a dspace system AVC system with one actuator In first place, the performances and the power consumption of the AVC with single actuators working are evaluated. Figure 12 shows the frequency response amplitudes of the velocity at node 1 relative to the disturbance force introduced at node 16 in the case only IMA1 is active in a) and the velocity at node 19 relative to the disturbance force introduced at node 16 in the case only IMA2 is active. The amplitudes refer to the cases where the gains that achieve approximately -6 db at Hz for both VFB and CVFB control. In the range from 1 Hz to 12 Hz two resonances arise when the two actuators are mounted on top of the structure. This causes the modes above 12 Hz to slightly shift their resonances to higher frequencies. In the case of VFB, high spillover amplitudes are produced at around 1 Hz as already observed in the simulations. These high amplitudes cause also high amplitudes in the IMAs stroke, bringing them to their linear operation limit and not allowing any further gain increase. CVFB performs good reductions over the whole frequency range of interest without producing any excessive spillover effects in agreement with simulations. x 1 / F [db rel. ms -1 / N] x 19 / F [db rel. ms -1 / N] Figure 12: Measuered frequency response functions a) velocity at node 1 relative to force disturbance when only IMA1 is active, b) velocity at node 19 relative to disturbance force when only IMA2 is active Uncontrolled structure (dotted red line), VFB control with highest gain applied (dashed blue line), CVFB control with highest gain applied (continuous black line) Figure 13 refers to the average reductions over the total apparent power consumption for the case only IMA1 is active in a) and the case only IMA2 is active in b). The lines plotted with higher thickness mark the results up to the point where highest reductions are achieved. In agreement with simulations, small reductions are already obtained by simply applying the actuators to the structure without applying any control. In the case of CVFB, the performances gradually increase with the power consumption in case only IMA1 is active. In the other case, the average performances increase gradually until a maximum average reduction is achieved. Finally, they worsen until the power is increased to the point that provides 6 db reductions at Hz. The performances of VFB are strongly affected by the spillover at low frequencies. Increasing the power consumption, VFB performs average reductions for relatively small amount of apparent power. However, further increasing the power, the spillover effects drastically compromise the performances. Moreover, the power consumption is also strongly affected. In fact, power consumption increases faster as the gain is increased and therefore the spillover. As for simulations, VFB performances are constrained from the instability before the linear operation stroke limits. In Figure 13, the percentages of the involved reactive power are shown for both control strategies in both cases. In general, higher percentages of reactive power are involved for CVFB control. This behavior is in agreement with simulation results. The higher percentages of reactive power indicate that the biggest part of the power provided to the IMAs is not producing any active work. This power could be regenerated by

12 1244 PROCEEDINGS OF ISMA216 INCLUDING USD216 using suitable amplifier concepts. However, the percentages of reactive power seem to be higher of approximately a factor 2 compared to the simulations for both VFB and CVFB V c 2 / V 2 [db] % Reactive V c 2 / V 2 [db] % Reactive % Reactive Apparent Power [V A] % Reactive Apparent Power [V A] Figure 13: Measurements: average reductions over total apparent power (and percentage of reactive) consumed with a) IMA 1 active b) IMA2 active - VFB (dashed blue line) and CVFB (dashed black line) AVC system with two decentralized actuators Finally, experimental studies on the AVC with two control units in a distributed decentralized arrangement are investigated. The performance and the power consumption of the AVC are evaluated when VFB and CVFB are applied. In Figure 14, the frequency response functions between the input force at the node 16 and the velocity at the node 1 and 19 are respectively shown when the highest gain that approximately provides 6 db reductions at the first bending mode at Hz is applied. In the case of VFB control, high spillover amplitudes are produced around 1 Hz confirming simulation results. At this gain setting, the high amplitudes cause the stroke saturation of IMAs. Therefore, further increasing of the gains brings the two IMAs outside their linear operation constraints. CVFB control achieves reductions over the whole frequency range of interest without producing any considerable spillovers. x 1 / F [db rel. ms -1 / N] x 19 / F [db rel. ms -1 / N] Figure 14: Measured frequency response functions a) velocity at node 1 relative to force disturbance b) velocity at node 19 relative to disturbance force Uncontrolled structure (dotted red line), VFB control (dashed blue line) and CVFB control (continuous black line) with highest gain applied Figure 15 shows the total power consumption over the average performances of the AVC system when the two IMAs work in a decentralized velocity feedback network. The VFB control overall performance is again strongly limited by spillover effects, which also causes high power consumption. The overall performance of the AVC system with CVFB control increase gradually with increasing feedback gain and

13 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES 1245 power consumption until a minimum is reached. Further increasing of the gain and power consumption does not result in higher overall performances. In Figure 15, the percentages of reactive power involved into the network are shown for both IMAs and control strategies. Similarly to the single IMAs cases, CVFB control shows higher values of reactive power compared to the VFB. The percentages are close to the experimentally estimated percentages for the single actuators cases in Figure 13. The comparison among the different cases considered in the experimental study shows that the highest average performances relative to the power consumption are achieved using the single IMA1 mounted on node 1. 6 V c 2 / V 2 [db] 4 2 IMA1. 3.2% Reactive IMA2: 37.5% Reactive IMA1: 63.4 % Reactive Apparent Power [V A] IMA2: 61.1 % Reactive Figure 15 : Measured apparent power over reductions: IMA1 active: a) node 1 b) node 19; IMA2 active: c) node 1 d) node 19 VFB (dashed blue lines) and CVFB (continuous black line) 4 Summary and conclusions The work presented in this paper estimates the performance, the constraints and the power consumption of AVC system with electrodynamic IMAs for different velocity feedback strategies. In previous studies, a control system with a single IMA applied to a single degree of freedom system with variable stiffness has been investigated. The effects on the performances and the power consumption under different control strategies and for different stiffness configurations of the primary structure are presented. Results indicate that different control strategies give different control performance for given values of apparent power. Therefore, results motivated to apply similar methodologies to more complicated systems. AVC system with two IMAs applied to a six cells laboratory truss structure is considered. The performance and power consumption of VFB and CVFB control are investigated. First, the OL FRF of the VFB is analyzed. Results show that the stability of the feedback loop is strongly unstable due to the presence of the rigid body modes at low frequencies. Hence, a phase lead-lag OL compensator is designed in order to improve the stability and the gain margin of the feedback loop. Successively, the simulation and experimental results have been presented for a) the case of using a single IMA and b) the case of using two decentralized IMAs at the same time. The feedback gains have been gradually increased from to the point where the AVC reaches 6 db reductions around the natural frequency of the first bending mode of the truss structure. Simulations show that VFB control is mainly limited by poor stability properties. Due to the high control spillover, power consumption drastically increases with the increasing feedback gains. The CVFB control is found to perform significantly better. The improved stability of the control feedback loop of the CVFB reduces the spillover effects at low frequency and gives better performances over the entire frequency range of interest. Results show that slightly higher performances are achieved with lower values of apparent power in simulation and measurements in the case of AVC with a single IMA. Therefore, results indicate that it is very important to find good positions for the control actuators. The choice of control position will depend on the controllability of the modes of interests at this position, but also on the feedback loop stability that can be achieved. While the controllability of a specific mode at a specific location may be estimated in simulations or a classical experimental modal analysis, the criterion

14 1246 PROCEEDINGS OF ISMA216 INCLUDING USD216 on finding locations that provide high feedback loop stability and as little as possible local and remote control spillovers seems more complicated. In future studies, it is therefore intended to investigate more possible control positions on the truss structure and also to investigate the optimal distribution of multiple IMAs. Moreover, this highlights the importance of monitoring more nodes of the truss structure in order to get a good estimate of the global performance of the AVC systems. Therefore it is intended to use more accelerometer sensors on the truss structure in future studies. Finally, simulations and experimental results underline that the percentages of reactive power involved are higher in the case CVFB control rather than VFB control. A higher percentage of reactive power indicates that such AVC system may benefit from electrical power amplifiers that can recuperate the reactive part of the power in order to achieve energy efficient operation. Table 3 summarizes the main results obtained in this study. IMA1 IMA2 Control strategy Maximum reduction [db] Power Apparent [V A] Reactive [%] Sim Meas Sim Meas Sim Meas VFB CVFB VFB CVFB VFB CVFB Table 3: Main results summary The results presented in this paper are part of an ongoing research study. For this reason, at this stage is too early to draw final conclusions. Nevertheless, the results suggest the importance to consider a holistic model of the AVC to evaluate the performance constraints and power consumption estimation of vibration control systems. Acknowledgements The authors gratefully acknowledge the European Commission for its support of the Marie Sklodowska-Curie program through the ITN ANTARES project (GA 66817).

15 FP7 ANTARES: ENERGY EFFICIENT SMART STRUCTURES 1247 References [1] N. Debnath, S. Deb, A. Dutta, "Multi-modal vibration control of truss bridges with tuned mass dampers under general loading," Journal of Vibration and Control, pp. 1-2, 215. [2] D. Mayer, S. Herold, T. Pfeiffer, J. Pöllmann, T. Röglin, G. de Rue, "Development and realization of distributed and adaptive vibration neutralizers," in ISMA 212, Leuven, 212. [3] D. Mayer, H. Stoffregen, O. Heuss, J. Pöllmann, E. Abele, T. Melz, "Additive Manufacturing of Active Struts for Piezoelectric Shunt Damping," Journal of Intelligent Material Systems and Structures, pp. 1-12, 215. [4] S. Herold, W. Kaal, T. Melz, "Novel Dielectric stack actuators for dynamic applications," in ASME 212, Stone Mountain, 212. [5] M. Kauba, J. Militzer, D. Mayer, H. Hanselka, "Multi - Channel narrowband Filtered - x - Least - Mean - Square algorithm with reduced calculation complexity," in ECCOMAS conference, Saarbrücken, 211. [6] J. Rohlfing, J. Hansmann, D. Mayer, "Structural Health monitoring using active vibration contol units applied to a strut structure laboratory demonstrator," in The sixth world conference on Structureal control and Monitoring, Barcelona, 214. [7] G. Liang, F. P. Sun, C. A. Rogers, "Coupled Electro - Mechanical Analysis of Adaptive Material Systems - Determation of the Actuator power consumption and system energy transfer," Journal of Intelligent material systems and structures, pp. 12-2, [8] S.-W. Zhou, C. A. Rogers, "Power flow and consumption in piezoelectrically actuatted structures," AIAA Journal, pp , [9] J. Millitzer, C. Ranisch, J. Kloepfer, "Electrical Power - Hardware - In - The - Loop simulation for the early validation of power amplifiers used in Active Vibration Control," in 4 Smarts, Darmstadt, 216. [1] D. J. Leo, "Energy analysis of Piezoelectric actuated structures driven by linear amplifiers," Journal of intelligent material systems and structures, pp , [11] J. A. Main, D. V. Newton, L. Massengill, E. Garcia, "Efficient power amplifiers for piezoelectric applications," IOP science, pp , [12] J. Paschedag, G. Koch, B. Lohmann, "Energy saving Actuator arrangements in an actively damped engine supsension system" in 17th World congress - The international Federation of Automatic control, Seoul, Korea, 28. [13] J. Rohlfing, S. Elliott, P. Gardonio, "Feedback compensator for control units with proof-mass electrodynamic actuators," Journal of Sound and Vibration, vol. 331, no. 15, pp , 212. [14] G. Lapiccirella, J. Rohlfing, T. Jungblut, "Power consumption and performance limit estimation of Smart Actuators for Active Vibration Control," in 4Smarts, Darmstadt, 216. [15] D. Flaschenträger, J. Thiel, J. Rausch, H. Atzrodt, S. Herold, T. Melz, R. Wethschützky, "Implementation and Characterisation of the Dynamic Behaviour of a Three - Dimensional Truss Structure for Evaluatiing Smart Devices," Leuven, 21.

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