Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 3, pp. 791-806, Warsaw 2011 IMPLEMENTATION OF AN ACTIVE MASS DRIVER FOR INCREASING DAMPING RATIOS OF THE LABORATORIAL MODEL OF A BUILDING Carlos Moutinho Álvaro Cunha Elsa Caetano University of Porto, Faculty of Engineering (FEUP), Porto, Portugal e-mail: moutinho@fe.up.pt This paper describes the experimental work involving a laboratorial im- plementation of an active system to increase the damping ratios of a plane frame physical model of a building structure of three storeys. For this purpose, the use of an Active Mass Driver commanded by the Di- rect Velocity Feedback control law is suggested. The study developed to define the maximum control gain based on the classical Root-Locus technique, as well as the analysis of system stability is presented. The efficiency of the proposed control system to achieve pre-defined damping ratios is verified experimentally by observing free decay responses of the system at several natural frequencies before and after control. Key words: vibration control, active systems,ActiveMassDriver,Direct Velocity Feedback, Root-Locus design 1. Introduction Many Civil Engineering structures have vibration problems in terms of servi- ceability limit states due to several transient or periodic dynamic loads, e.g., footbridges subjected to pedestrians actions, road and railway bridges excited by traffic loads and tall buildings exposed to wind forces. In these situations, the implementation of control systems can improve the structural performance by reducing the vibration levels to acceptable values. To achieve this, several passive, active, semi-active or hybrid control devices can be used (Chu et al., 2005).This study is addressed topractical caseswhere the specificuseof active systems can be an appropriate solution. 792 C. Moutinho et al. In fact, it is generally accepted that active systems, despite constituting a powerful control scheme, are not an interesting solution to many structural problems, especially when dealing with large structures (Kobori, 2002). This stems fromthe fact that active control demands sophisticated technology, high costs and highmaintenance, and is less reliable than passive systems (Spencer andNagarajaiah, 2003). However, given the potential of active systems, active control can still be an attractive solution, especially for very flexible systems whose dynamics is dominated by the contribution of several vibration modes (Fujino, 2002). The interest in active control grows when dealing with nonli- near systems and structures that exhibit significant variability in its dynamic parameters (Soong, 1990). On the other hand, if the problemunder analysis involves harmonic vibra- tions the use of control strategies that conducts to the increasing of damping ratios of the structure is particularly adequate. This is because the amplitude of harmonic responses is strongly influenced by the respective damping ratios, meaning that an appropriate control strategy should be able to increase these damping ratios to predefined values capable to keep the maximum structural response below certain limits. The Direct Velocity Feedback (DVF) control law associated with Root-Locus techniques also constitutes a good strategy that can be used for this purpose since it has the ability to add damping to the structure while providing necessary robustness to the control system (Preumont, 1997). In fact, when some control schemes using collocated pairs of actuators and sensors are used, this strategy leads to unconditionally stable control systems and avoids spillover errors due to unmodeled higher frequency modes (Preumont and Seto, 2008). However, if anActiveMassDriver is used, the control system is no longer unconditionally stable and it may destabilize itself, particularly for high gains (Moutinho, 2008). In this context, the main objective of this work is to demonstrate how active control can be used to reduce harmonic vibrations in structures using the strategy just described. Although based in simple concepts, this article systematizes the steps and themethodologyneeded to its real implementation. 2. Review of theoretical background 2.1. Collocated versus non-collocated control It is well established in theory of system dynamics that in presence of a SISO system, if the actuator and sensor are collocated, i.e., positioned at the same point and measuring in the same direction, the transfer function that relates the systemInputandOutputhasalternating imaginarypolesandzeros. Implementation of an active mass driver... 793 This property is very important in the context of control systems because it ensures that the closed-loop poles of the controlled system remain entirely within the left-half complex plane even if in the presence of uncertainty or variability of the system parameters (Preumont and Seto, 2008). In this case, it is said that the system is robust with respect to stability and the general shape of the Root-Locus plot of such a system is characterized by having several nice stable loops (Ogata, 2009). On the other hand, in the case of non- collocated systems, the pole-zero alternating property is no longer guaranteed, and so there is an effective possibility of achieving system instability in using high control gains. Figure 1 shows a typical view of the Root-Locus diagram of both situations when using a pure derivative controller (same as Direct Velocity Feedback) which has the effect of adding a zero in the open-loop transfer function (represented at the origin of the real/imaginary axis). Fig. 1. Root-Locus diagram for (a) collocated and (b) non-collocated control system 2.2. Case of a control system using an Active Mass Driver One of the most widely known actuation systems used to apply control actions in civil structures is the Active Mass Driver (AMD) which enables generation of inertial forces between the structure and the mass of the devi- ce. The main advantage of this control system consists in avoiding external connections from the structure to the ground which, in most of the cases, are undesirable from the architectural point of view. Although AMDs constitute an interesting solution for many practical situations, some design issues must be carefully observed. In fact, AMDs do not correspond to a collocated con- trol system because, although the actuator is positioned in the same location as the measurement point, the force generated with this device is applied at two different points. This effect can be clearly seen in Fig.2 which represents themodel of anAMD integrated with a single degree-of-freedom structure. In this case, the control force is calculated according to the structure response 794 C. Moutinho et al. in correspondence with x1, being applied as a pair of forces at the degrees-of- freedom related to x1 and x2. Fig. 2. SDOF structure with an ActiveMass Driver The problem with this scheme is due to the reaction force that goes to the inertial mass which is contrary to the direction of control action. As a consequence, at least one close-loop polemoves in the direction of the unstable region of the complex plane, causing system instability above a certain gain level (Moutinho, 2008). In order to analyze this situation, the Root–Locus diagram of general case of a multi-degree-of-freedom structure controlled by an AMD using Direct Velocity Feedback is represented in Fig.3a. As in the previous case, it is clear that the system is potentially unstable for a certain level of gain (gmax). To minimize this problem, a large gainmargin is desirable, which can be achieved by increasing the damping ratio of theAMD, i.e., moving the respective open- loop pole to the left side. As a result, it is possible to mobilize extra gain and extra damping in the vibration modes of the structure. Another important issue about the use of AMDs is related to its natural frequency. In the case ofFig.3a, theopen-looppoleassociatedwith theAMDis the lowest frequencypole of the system.This iswhy thevibrationmodes of the structure describe increasing damping loops, being the instability conditioned by dynamics of the AMD. On the other hand, if the natural frequency of the AMD is greater than the first natural frequency of the original structure, a zero-pole flipping is observed in the intermediate open-loop poles of the structure, as shown in Fig.3b. This is disastrous because the damping ratios of the vibration modes with frequencies below the frequency of the AMD decrease as the gain increases. As a consequence, in the Root-Locus diagram, the close-loop poles associated with these modes have loops with departure angles in the direction of the unstable region, and the AMD loop describes a Implementation of an active mass driver... 795 stable trajectory. Moreover, if the structure is lightly damped, it is necessary just a small gain to get the system unstable by one of these modes. Fig. 3. Root-Locus diagram of a controlled structure using an AMD with (a) low and (b) high frequency Asa summary , it shouldbe emphasized that an idealAMDused to control a structure shouldhaveahighdamping inorder toget a larger gainmarginand mobilize extra damping into the system, and should have a natural frequency below the first natural frequency of the original structure to avoid instability of the lower vibration modes. 3. Characterization of the experimental control system 3.1. Description of the physical model, equipments and instrumentation This section describes the laboratorial implementation of an AMD used to reduce harmonic vibrations in a plane frame physical model, by artificially increasing its initial damping ratios bymeans of this device. For this purpose, it was developed a model of a shear building structure with 3 storeys which was supported in a shaking table, and a small Active Mass Driver to control vibrations, as shown in Fig.4. The physical model is composed by 3 rigid iron masses connected to each other and to the base through aluminum columns. The total mass of each level including the iron mass, mass of the aluminum connections, mass of the sensor and themass of each half part of the support columns is m1 =15.16kg, m2 =15.16kgand m3 =12.76kg, correspondingto the1st, 2ndand3rdfloors, respectively. The aluminum columns have 400mmof height, 120mmof width and 7mm of thickness, and are clamped at each level and at the base. The aluminummodulus of elasticity was evaluated at about 60Gpa. 796 C. Moutinho et al. Fig. 4. General view of the experimental setup In order to excite the system with harmonic loads, the physical model was fixed on a shaking table composed of a sliding platform connected to an electromagnetic shaker powered by a current amplifier (see Fig.5a) whichwas specially developed to this test. The total mass mobilized at the base of the model including the platform mass of the shaking table, the moving mass of the electromagnetic shaker, the support mass of the model, the mass of the sensor and themass of each half part of the support columns is m0 =40.51kg. Fig. 5. (a) Detail of the shaking table and (b) detail of the AMD To control vibrations in the physical model, an Active Mass Driver was installed at the top level. This device is composed of an active 2.89kg mass, which slides with low friction through 2 circular metallic threads connected to the AMD body which has total mass of 2.65kg (see Fig.5b). The active mass is connected to a small electromagnetic shaker which is responsible for application of inertial forces between the active mass and the structure. The Implementation of an active mass driver... 797 axis of this electromagnetic shaker has a spring of stiffness k = 3840N/m causingdampedharmonicmotionof theactivemasswhen left in freevibration. Fig. 6. (a) Detail of the accelerometer (b) signal conditioners and power amplifiers The system response was continuously measured with accelerometers po- sitioned at the base of the model, at each floor and at the active mass of the AMD (see Fig.6a). The force developed between the active mass of the AMD and the top level was also measured with a small load cell. The electric cur- rent generated by the table shaker and by the AMD shaker was also observed in order to monitor the force applied with these devices. Figure 6(b) shows the power amplifiers of the shaking table and the AMD, and the signal con- ditioners of the accelerometers. All the transducers mentioned before, as well as the electromagnetic shakers, were operated by a digital computer working with LabVIEWTM package software, using an acquisition board which per- forms the signal analog/digital conversion. A Fourier analyzer was also used in the identification of the modal parameters of the structure. M=        40.51 0 0 0 0 0 15.16 0 0 0 0 0 15.16 0 0 0 0 0 15.41 0 0 0 0 0 2.89        K=        74.50 −74.50 0 0 0 −74.50 149.00 −74.50 0 0 0 −74.50 149.00 −74.50 0 0 0 −74.50 78.34 −3.84 0 0 0 −3.84 3.84        Fig. 7. Identification of degrees-of-freedom of the structure and correspondingmass and stiffness matrices 798 C. Moutinho et al. Taking into account the description of the physical model and data men- tioned previously, themass and stiffnessmatrices of the laboratorial structure were defined in correspondence with the degrees-of-freedom shown in Fig.7. Thesematrices were used in analytical calculations discussed in the next Sec- tions. 3.2. Identification of the modal parameters of the system In order to identify the modal properties of the system, in particular, the natural frequencies,mode shapes anddamping ratios, themodelwas subjected to several tests and the experimental valueswere comparedwith the analytical ones obtained from the numerical model defined in the previous Section (with the exception of damping ratios which can only be evaluated via experimental tests). The natural frequencies of the systemwere evaluated with the help of the Fourier analyzer. In this case, FRFs were obtained experimentally by relating the input force applied at the base of the model by the shaking table and the output accelerationsmeasured at several degrees-of-freedom. For this purpose, a frequency range from 0 to 25Hz was stipulated, and the average of 5 FRFs estimates was considered. Each FRFwas estimated based on time series with an acquisition time of 16s,whichmeans that the frequency resolution achieved is 0.0625Hz. Figure 8 shows the magnitude of the FRF obtained, relating the input force at the base of the model and the output acceleration at the top floor. The natural frequencies of the system are clearly identified by the peaks on the graph and their values, as well as the analytical ones, are listed inTable 1. Fig. 8. FRF relating the input force at the base of the model and the output acceleration at the top floor Implementation of an active mass driver... 799 Table 1. Identified versus analytical natural frequencies Frequency Identified Calculated [Hz] [Hz] 1 5.50 5.45 2 7.35 7.35 3 15.50 14.60 4 22.50 22.25 Themethodused to identify thevibrationmode shapeswas simply exciting the structureat resonant frequencies at thebaseof themodelusing the shaking table andmeasuring the amplitude and phase shift of the system response at several degrees-of-freedom. This method is adequate to be applied to small models and gives excellent results because when the structure is subject to a harmonic force with the excitation frequency equal to the natural frequency of the system, the contribution of other vibration modes is negligible when compared with the resonant mode. Figure 9 shows the graphical representation of the mode shapes in cor- respondence with the natural frequencies indicated in Table 1. An excellent agreement can be observed between the identified and calculated mode sha- pes in the samemanner as it was observed with the natural frequencies. This means that the analytical model accurately represents the system properties in terms of mass and stiffness. Fig. 9. Identified versus analytical mode shapes To characterize completely the system parameters, it was necessary to evaluate thedampingproperties of the structure. In this case, themethodused 800 C. Moutinho et al. consists in exciting thephysicalmodelwithaharmonic force of frequencyequal to any of those that were identified as the natural frequencies of the system. By suddenly stopping the excitation, it is easy to measure the free vibration responseandestimate the respectivedamping ratio byanalyzing the freedecay response envelope, given by the equation y=Aexp(−ζωt) (Chopra, 2006). This procedure was adopted to estimate the modal damping ratios of se- veral vibration modes of the physical model. The results obtained are sum- marized in Table 2, and the time series indicating the system response for each natural frequency are plotted in Fig.10. Notice that the study of this equation applied to different sets of time intervals allows concluding that the modal damping ratios vary slightlywith the displacements amplitude. For this reason, their values were estimated in an intermediate range of the structu- re response, corresponding approximately to the mean value of the damping ratio. Table 2. Identified modal damping ratios Mode Identified Modal damping frequency [Hz] ratio [%] 1 5.50 3.20 2 7.35 1.80 3 15.50 0.35 4 22.50 0.22 Fig. 10. Free vibration responses and the estimated decay envelopes Implementation of an active mass driver... 801 Based on the modal damping ratios indicated in Table 2, the damping matrix of the systemwas evaluated using the superposition ofmodal damping matrices [kg/s] (Chopra, 2006), resulting C=        35.30 −3.74 −11.47 −16.11 −3.99 −3.74 7.82 −0.94 −1.91 −1.23 −11.47 −0.94 10.23 2.91 −0.73 −16.11 −1.91 2.91 15.00 0.11 −3.99 −1.23 −0.73 0.11 5.85        4. Implementation of active control 4.1. Description of the control system The objective of the control system is to reduce vibration levels of the physical model, when it is excited by harmonic loads. When the frequency of excitation is equal to any of the natural frequencies of the structure, i.e., when the resonance occurs, the systemmay experience large amplitudemotion, de- pending on the damping ratio of the respective vibrationmode. Therefore, an appropriate control strategy should be able to increase the system damping, causing a significant reduction in the structural response.As it is known,when the resonance occurs the amplitude of the dynamic system response is obta- ined bymultiplying the static response by 1/2ζ (being ζ the damping ratio), which suggests that, if the static response is known, the damping ratio should be chosen in order to keep the dynamic response below certain pre-defined limits. As seen before, the modification of the characteristics of the structure in terms of the modal damping ratio can be obtained using an Active Mass Driver commanded by a derivate controller. The definition of control gains can be studied using the Root-Locusmethod which also allows evaluating the system stability. An important issue that must be initially considered is the location of the actuator system. Themain rule is that the actuator should not be positioned at the point where the significant vibratingmodes have reduced modal components. Using thedescribed control scheme, a real control systemwas implemented in the plane frame physical model composed of anAMDpositioned at the top floor to reduce the vibrations caused by a harmonic excitation applied by the shaking table. The system response at the top floor is continuously measured using an accelerometer connected to the signal conditioner which performs conversion of theaccelerations into velocities by integrating the signal fromthe 802 C. Moutinho et al. transducer.At each time instant, the control force is calculated bymultiplying the value of the velocity by a pre-defined gain g and applied to the structure by the AMD shaker connected to the respective power amplifier. The block diagram of this close-loop control system is represented in Fig.11.The transfer function G(s)was obtained directly from the systemcha- racteristics described in thepreviousSection, relating the system input/output at the top floor (DOF no. 3). Fig. 11. Block diagram of the closed-loop control system 4.2. Root-Locus design The Root-Locus diagram of this control system is represented in Fig.12. This plot clearly shows the advantages of this method to design AMDs, be- cause even if no previous information was available, looking at this diagram, it is possible to immediately conclude that (i) the system has four natural frequencies slightly damped because the open-loop poles are close to the ima- ginary axis; (ii) the system is unstable for high gains because in this situation there are close-loop poles in the unstable region; (iii) there is a small gainmar- gin until instability is reached because the AMD has relatively low damping; (iv) when the gain increases from zero, the damping ratios of the vibration modes increase, despite the natural frequencies remain approximately the sa- me. The choice of the control gain affects simultaneously all the close-loop poles locations, which means that it is not possible to select the characteristics of each vibration mode individually. For this reason, when Root-Locus method is used, it must be selected the dominant close-loop pole as the one that contributes more significantly to the system response. The gain is adjusted to move this pole to amore convenient position, conditioning the locations of the other poles. In the case of the present plane frame physical model, when the gain increases, all themodal damping ratios increase in correspondencewith a specific gain value, which is fixed when the target close-loop pole reaches the desired location in correspondence with the pre-defined damping ratio. Implementation of an active mass driver... 803 Fig. 12. Root-Locus diagram of the plane frame controlled by an AMD In the case of this test, the AMD developed has a relatively low damping (ζTMD ≈ 3.2%) due to construction issues, which limits the efficiency of this control system. However, even in this situation it is possible to significantly increase the damping ratios of the structure to maximum values according to the maximum achievable gain g = 82, until instability occurs. If this gain is selected, the close-loop polesmove to the locations marked with small dots in the Root-Locus diagram, which are in correspondence with the new damping ratios listed in Table 3. Table 3. Calculated modal damping ratios for control gains g = 0 and g= gmax =82 Mode ζi [%] for g=0 ζi [%] for g=82 (without control) (control with gmax) 1 3.20 – 2 1.80 7.09 3 0.35 1.44 4 0.22 0.44 4.3. Experimental results In order to experimentally verify the efficiency of the described control sys- tem, the damping ratios of the physical model were evaluated after switching on the AMDwith an intermediate gain g=60. In this circumstance, themo- del was excited with a harmonic loadwith a frequency in correspondencewith 804 C. Moutinho et al. Fig. 13. Free vibration responses and estimated decay envelopes for control gain g=60 and system response for unstable control gain g=150 the identified natural frequencies of the system and, after stopping the excita- tion, the free decay response was recorded in order to evaluate the respective modal damping ratio. The results obtained are represented in Fig.13 and are summarized inTable 4, where the experimental values are also comparedwith the analytical ones. Table 4. Identified versus calculated modal damping ratios for control gain g=60 Mode Identified Calculated [%] [%] 1 – 0.83 2 6.05 5.72 3 1.26 1.15 4 0.42 0.38 System instability was also verified when the control gain exceeds its ma- ximumvalue, by introducing a clearly excessive gain of g=150. As expected, the noise in the sensors was sufficient to excite the system, which became unstable by the uncontrolled harmonic vibration of the structure with the Implementation of an active mass driver... 805 frequency of the AMD vibration mode. This situation was clearly predicted by the analysis of the Root-Locus diagram plotted in Fig.12. 5. Conclusions This paper describes a laboratorial implementation of an active damping sys- tem to reduce vibrations in a 3-DOF physical model subjected to harmonic vibrations.When a resonance occurs, the amplitude of the dynamic response is strongly influenced by the damping ratio of the respective vibration mo- de. This means that a good control strategy should be able to increase the damping in the system in order to keep its response below certain limits. For this purpose, an AMD commanded by the Direct Velocity Feedback control law, resulting in a control system that applies an inertial force propor- tional to the velocity at the control point can be used.However, the use of the AMDconstitutes a non-collocated sensor/actuator scheme, causing system in- stability particularly for high gains. To analyze this problem, the Root-Locus diagram is a powerful method that allows examination of the system stabi- lity as well as designing the control gain necessary to reach some structural properties, particularly in terms of damping ratios. In order to experimentally verify the efficiency of this control system, the physicalmodelwas subjected to resonant harmonic loads applied by a shaking table, aiming at the evaluation of several modal damping ratios by analyzing the respective free decay envelopes. Itwas observed a good agreement between the experimental and analytical results corresponding to a significant increase of the damping ratios of the systemwith a consequent important reduction of its harmonic response. Acknowledgements The authors acknowledge the support provided by the Portuguese Foundation for Science and Technology (FCT) in the context of the research project Control of vibrations in Civil Engineering structures (POCTI/ECM/55310/2004). References 1. Chopra A., 2006, Dynamics of Structures Theory and Applications to Ear- thquake Engineering, 3rd edition, Prentice-Hall 806 C. Moutinho et al. 2. Chu S., Soong T., Reinhorn A., 2005, Active, Hybrid, and Semi-active Structural Control: A Design and Implementation Handbook, John Wiley & Sons, Ltd. 3. Fujino Y., 2002, Vibration, control and monitoring of long-span bridges – Recent researchdevelopments and practice in Japan,Journal of Constructional Steel Research, 58, 71-97 4. Kobori T., 2002, Past, Present and future in seismic response control in civil rngineering structures, 3rd World Conference on Structural Control, 1, 9-14 5. Moutinho C., 2008,Vibration Control in Civil Engineering Structures, PhD thesis, FEUP 6. Ogata K., 2009,Modern Control Engineering, 5th edition, Prentice-Hall 7. PreumontA., 1997,VibrationControl of Active Structures –An Introduction, Kluwer Academic Publishers 8. Preumont A., Seto K., 2008, Active Control of Structures, John Wiley & Sons, Ltd. 9. Soong T., 1990, Active Structural Control – Theory and Practice, Longman Scientific &Technical 10. Spencer Jr.B.,NagarajaiahS., 2003,State of the art of structural control, Journal of Structural Engineering, ASCE, 129, 845-856 Zastosowanie aktywnego układu redukcji drgań do zwiększenia współczynników tłumienia laboratoryjnego modelu budynku Streszczenie W pracy opisano badania doświadczalne przeprowadzone na stanowisku labora- toryjnym, których celembyła analiza układu aktywnej redukcji drgań zwiększającego współczynniki tłumienia w płaskim, ramowymmodelu trzykondygnacyjnego budyn- ku.Cel ten zrealizowanonamodelu z inercyjnymukłademredukcji drgań sterowanym w prędkościowej pętli sprzężenia zwrotnego. Przedyskutowano problem stateczności modelu metodą linii pierwiastkowych (Root-Locus), pozwalającą na określenie mak- symalnego dopuszczalnego współczynnika wzmocnienia w układzie sterowania. Sku- teczność zaproponowanejmetody sterowania ukierunkowanej na uzyskanie żądanych właściwości tłumiących zweryfikowano eksperymentalnie poprzez obserwację spadku amplitudy drgań swobodnych pobudzanych wcześniej przy kilku częstościach wła- snychmodelu z włączonym i wyłączonym układem sterowania. Manuscript received March 3, 2011; accepted for print May 4, 2011