Twin Rotor Damper for Control of Wind-Induced Bridge Deck Vibrations Jörn SCHELLER * and Uwe STAROSSEK ** *Institute of Steel and Timber Construction, Faculty of Civil Engineering, Technische Universität Dresden, D-01062 Dresden, Germany E-mail: Joern.Scheller@tu-dresden.de **Structural Analysis and Steel Structures Institute, Hamburg University of Technology, D-21071 Hamburg, Germany E-mail: starossek@tuhh.de Abstract In this paper, the feasibility of the application of the new twin rotor damper for the control of wind-induced bridge deck vibrations is investigated. In the context of this problem, it is shown that the proposed active mass damper requires only little actuator power for the control of wind-induced bridge deck vibrations. The analysis is performed in the form of numerical simulations in time domain. The numerical example models an actively damped long-span suspension bridge and considers self-excited as well as buffeting forces. The evaluated results verify the effectivity and efficiency of full-scale applications of the new device. In addition, results are compared to those obtained from an alternative implementation of a conventional active mass damper. The comparison of the actuator power and energy demands reveals the superiority of the twin rotor damper. Key words: Active Mass Damper, Active Control, Wind-induced Bridge Deck Vibrations 1. Introduction Active structural control may be an effective means for the mitigation of wind-induced bridge deck vibrations. For this purpose, active mass dampers may be appropriate active control devices as applications in buildings and bridge towers suggest (Spencer and Sain (1) ). However, no such device has been implemented yet in bridge decks. The difficulty to generate sufficiently large control forces with a low power demand of the actuators can be identified as a reason for the lack of applications. Starossek and Scheller propose a new active mass damper, for which the term twin rotor damper has been coined (2). This damper uses a force generating mechanism differing from that of conventional active mass dampers. In this way, a device is designed that can efficiently generate large control forces suitable for structural control. 2. Layout and Versatility of Twin Rotor Damper for Bridges The idea is to utilize centrifugal forces generated by pairs of eccentrically rotating masses in such a way that undesired force components cancel. The basic configuration is shown in Figure 1. In this configuration, two rotors rotate with the same angular speed ω in opposite directions around horizontal axes (Fig. 1). Each rotor consists of an actuator-driven rotating rod with length r c and a mass m c attached to its free end. The rotors perform complete revolutions and both point in the upward direction at the same instant of time and in the downward direction at another point in time. The generated centrifugal forces F R are 347 1
used for the control of the supporting structure. Since the horizontal force components F h cancel at all times, solely a resulting vertical force F c = 2 F v is generated which can be used for damping vertical vibrations. F v is the vertical force component of the centrifugal forces generated by one rotor. With a constant speed of rotation ω, the generated force in Fig. 1 Basic unit of twin rotor damper. the motionless system is harmonic with circular frequency ω. Moreover, no actuator power is required to generate the dynamic force. The twin rotor damper may be used to control bridge deck vibrations. Many long-span bridges have box girder decks. In such cases, two identical basic units shown in Figure 1 may conveniently be installed in the same vertical plane within the girder as shown in Figures 2 and 3. Using identical rotor speeds, each unit generates a vertical force with identical amplitude and frequency. By purposefully adjusting the phases of the individually generated vertical forces, it is possible to generate a resultant vertical control force only, a resultant control moment only, or certain combinations of a simultaneously acting vertical control force and a control moment with a single active mass damper. Figure 2 shows both rotor units jointly generating a vertical control force. In this case, the phases of both units are identical. By simply changing the phases such that the phase of one force is opposite to the other one, as shown in the configuration of Figure 3, a moment proportional to the distance between both centers of the units is exclusively generated. Therefore, it is easily possible to switch from the control of vertical vibrations, which may be due to traffic, to the control of torsional vibrations, which may be due to wind. Moreover, by properly selecting the phase difference between the forces of both units, a dynamic vertical force and a moment with the same frequency can be generated (2). 3. Advantageous Configuration for Wind Applications Generalized two-dimensional mechanical systems may be used when modeling wind-induced bridge deck vibrations (Simiu and Scanlan (3) ). Frequently, it is sufficient to consider the vertical and the rotational degrees of freedom of such a reduced bridge deck model. Moreover, the degrees of freedom of such a model are coupled by the action of the Fig. 2 Using two basic units of Fig. 1 to generate resultant vertical force. Fig. 3 Using two basic units of Fig. 1 to generate resultant moment. 348 2
self-excited aeroelastic forces. By this action, energy is transferred between both degrees of freedom. For this reason, it is possible to control the entire motion of the bridge deck by adding damping to either of the degrees of freedom. However, assuming identical damper dimensions and masses, numerical studies show that the twin rotor damper is more effective in controlling the resulting motion if it generates damping moments only. Because of this observation, the configuration shown in Figure 3 is preferred in this study. Since in this configuration the rotors of the units rotating in the same direction are operated with identical angular velocities, the corresponding rotors may be coupled mechanically by a gear. The coupled rotors are in indifferent equilibrium because the lever arms of the weights of the associated damper masses with respect to the associated rotor axis are always identical and the resulting static moments have opposite directions. In this way, no actuator power is needed for lifting the damper masses. Moreover, if the rotors exhibit constant local angular velocities, power input is only needed to compensate for losses, which may be due to friction. Consequently, operating the rotors with rotor velocities as steady as possible requires only a very low power demand of the actuators, which becomes clear when comparing the power demand of the twin rotor damper with that of a conventional active mass damper, as described below. 4. Equations of Motion Figure 4 shows the generalized two-dimensional system with the two bridge degrees of freedom h for heaving and α for rotation. The corresponding equations of motion per unit span are given by m h chh khh Lse Lb Fres, (1) I α cαα kαα M se M b M res, where m is the mass, I the mass moment of inertia, c h and c α the structural damping coefficients, and k h and k α the structural stiffnesses. The dot indicates differentiation with respect to time t. The self-excited aeroelastic forces L se and M se due to the oncoming flow with mean wind velocity v are modeled using rational function approximation (e.g., Chen et al. (4) ). In extension to the system studied by Starossek and Scheller (2), buffeting forces L b and M b due to longitudinal wind fluctuations are included using a version given by Körlin (5) of the quasi-steady formulation presented by Simiu and Scanlan (3). The generated control forces are combined to a resultant force F res acting at midspan in the global downward direction and a resultant moment M res positive clockwise. These control forces depend on the rotor lengths and masses as well as on the rotor motions and are given by Starossek and Scheller for general parameters of the rotor damper (2). For the particular configuration of interest in this study i.e., generating control moments only for which all rotor lengths and masses are chosen to be identical and the coupled rotors point in opposite directions implying that in Figure 4 the rotor angles 2 referring to a fixed Fig. 4 Two-degree-of-freedom structural system with two units of twin rotor damper. coordinate system are given by 2 = 1 + π the following rotor forces can be specified: 349 3
Fres 4m h c, 2 2 c (2) 2 2 M res 4m a c α 4mcrc a φ 1 cos α sin φ1 α sin α φ 1 α cos α cos φ1, 4 in which a and c are the lengths as indicated in Figure 4. Inserting the given rotor forces into the equation of motion (Eq. 1) shows that the twin rotor damper generates control moments only because the terms depending on the accelerations of the bridge deck can be transferred to the left-hand side of that equation resulting in an increase of the inertia forces. 5. Application of Twin Rotor Damper in Example Bridge A construction stage of the Great Belt Bridge in Denmark is chosen for numerical simulations in time domain. The parameters of the bridge structure are given by Walther (6). The mass and the mass moment of inertia are m = 17.8 10 3 kg/m and I = 2.173 10 6 kgm, respectively. The stiffnesses k h and k α can be computed from the natural frequencies of vibration f h = 0.099 Hz and f α = 0.186 Hz. The structural damping is set to zero, i.e., c h = c α = 0. The half-width of the bridge deck is b = 15.5 m. Expressions for the aerodynamic forces are given by Körlin (5) for a cross section that has a shape similar to that of the real bridge deck. The aerodynamic derivatives of the used trapezoidal cross section are specified by Bergmann (7). The damper parameters are a = 2/3 b, c = 0, 4 m c = 0.005 m, and r c = 1.0 m. The idea of the control algorithm used in the simulations is based on adjusting the phase of the control moment to that of the resulting motion of the bridge deck such that the energy dissipated by the twin rotor damper is maximum. The control is activated when the angular displacement α exceeds 0.01 rad and deactivated when the maximum rotation does not exceed 0.005 rad within a vibration cycle. The maximum magnitude of the angular acceleration of the rotors is limited to 2 rad/s 2. Further details concerning the applied control algorithm are omitted here due to limited space and are presented by Scheller (8). The bridge deck model is subjected to longitudinal wind fluctuations that are generated using an algorithm given by Spanos and Schultz (9). The power spectrum of the wind fluctuations is described by a von Karman spectrum (Simiu and Scanlan (3) ) with an integral length scale of 80 m and a turbulence intensity of 5 %. The generated time series for a mean wind velocity of v = 39.7 m/s is shown in Figure 5a in which the mean wind velocity is added. Figure 5b presents the associated bridge responses without and with active damping in the form of time histories of the angular displacement α. Since the critical wind speed is 38.5 m/s for the system without and 39.7 m/s for the system with active damping, the response without active damping is unbounded indicating a potential instability, whereas a Fig. 5 Numerical simulations in time domain (v = 39.7 m/s): total wind velocity with fluctuations (a) and resulting angular displacements α without and with active damping (b). response with reasonably limited values is observed when the twin rotor damper is active. 350 4
For lower mean wind velocities, the action of the twin rotor damper improves the standard deviations of the angular displacement α by 20 to 60 %. 6. Demonstration of Low Power and Energy Demand To demonstrate the power efficiency of the twin rotor damper, the mechanical power to be provided by the actuators is compared to that of another damper that uses the acceleration of a mass to generate damping moments. Such an active mass damper was suggested by Miyata et al. (10) and is also described by Körlin and Starossek (11). The control moment in this comparison system stems from inertia forces due to the acceleration and deceleration of a rotating mass, similar to a flywheel. The comparison system is depicted in Figure 6. To obtain identical total weights and heights of the damper masses used in both systems, the damper mass of the comparison system is m rm = 4 m c and has a radius of r rm = r c. Assuming that the rotating mass has the form of a thin-walled empty cylinder, the mass moment of inertia about the center of rotation is given by I rm = m rm r rm 2. Linear state feedback is used to provide a suitable input signal for the actuator of the rotating mass damper. The controller gain is adjusted such that both types of active dampers achieve the same performance, i.e., the standard deviations of the resulting angular displacements of the bridge deck are equal with both active dampers. Figure 7a shows the resulting total mechanical powers of the actuators of both active dampers as a function of time for the wind input in Figure 5a. Power losses are neglected. As can be seen from the figure, the power demand in the system with the twin rotor damper is hardly recognizable in the figure and thus significantly smaller than that in the comparison system. Compared to that in the latter system, the maximum of the total actuator power in the system with the twin rotor damper is smaller by a factor of about 250. Figure 7b presents the associated total mechanical energies of the actuators of both active dampers, which are obtained by integrating the respective mechanical powers. At the end of the shown time segment, the energies of both active dampers are equal and negative. Hence, in both systems, energy of the air flow is converted into mechanical energy, i.e., energy is generated by the actuators. However, it can be clearly seen from the figure that in the system with the rotating mass energy is cyclically input into the system by the actuators and subsequently dissipated, whereas the system with the twin rotor damper almost steadily dissipates energy from the vibrating system. Fig. 6 Comparison system using accelerations and decelerations of a rotating mass. 7. Conclusions It can be concluded that the new twin rotor damper is an effective device to mitigate and limit wind-induced bridge deck vibrations of long-span bridges. Compared to the power demand of conventional active mass dampers, the power demand of the new device is extremely small. Moreover, the new device dissipates energy from the vibrating system almost steadily. In addition, the twin rotor damper is a simple mechanical device that can have a low self-weight if the rotor length is properly chosen. 351 5
Fig. 7 Total mechanical power of actuators (a) and mechanical energy of actuators (b) of systems with twin rotor damper (see response in Fig. 5) and with rotating mass damper using accelerations. References (1) B.F. Spencer and M.K. Sain, Controlling buildings: a new frontier in feedback, IEEE Control Systems, Vol. 17, No. 6, 1997, pp.19-35 (2) U. Starossek and J. Scheller, A novel active mass damper for vibration control of bridges, Fourth International Conference on Bridge Maintenance, Safety, and Management (IABMAS'08), Seoul, Korea, July, 2008 (3) E. Simiu and R.H. Scanlan, Wind effects on structures, John Wiley & Sons, New York, 1996 (4) X. Chen, M. Matsumoto, and A. Kareem, Time domain flutter and buffeting response analysis of bridges, ASCE Journal of Engineering Mechanics, Vol. 126, No. 1, pp.7-16, 2000 (5) R. Körlin, Aktive mechanische Kontrolle winderregter Brückenschwingungen, PhD thesis, Hamburg University of Technology, Germany, 2006 (6) J.H. Walther, Discrete vortex method for two-dimensional flow past bodies of arbitrary shape undergoing prescribed rotary and translational motion, PhD thesis, Technical University of Denmark, Lyngby, 1994 (7) D. Bergmann, Experimentelle Ermittlung der instationären aerodynamischen Beiwerte von Brückenprofilen im Wasserkanal, Technical Report, Institute of Aerodynamics and Gas Dynamics, Stuttgart University, ordered by the Structural Analysis and Steel Structures Institute of Hamburg University of Technology, Germany, 2002 (8) J. Scheller, Power-efficient active structural vibration control by twin rotor dampers, accepted PhD thesis, Hamburg University of Technology, Germany (9) P.D. Spanos and K.P. Schultz, Two-stage order-of-magnitude matching for the von Karman turbulence spectrum, Proceedings of the 4th International Conference on Structural Safety and Reliability, Kobe, Japan, May, 1985, pp.i-211-i-217 (10) T. Miyata, H. Yamada, and N.N. Dung, Proposed measures for flutter control in long span bridges, 15th IABSE Congress on Structural Engineering in Consideration of Economy, Environment and Energy, Copenhagen, Denmark, June, 1996, pp.781-786 (11) R. Körlin and U. Starossek, U. 2004. Active mass dampers for flutter control of bridges, 8th International Conference on Flow-Induced Vibrations, Paris, France, July, 2004 Acknowledgements This research is partially funded by the Deutsche Forschungsgemeinschaft DFG (German Research Foundation) which is gratefully acknowledged. 352 6