A novel active mass damper for vibration control of bridges

Transcription

1 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea A novel ative mass damper for vibration ontrol of bridges U. Starossek & J. Sheller Strutural Analysis and Steel Strutures Institute, Hamburg University of Tehnology, Germany ABSTACT: The main issue onerning the ative ontrol of strutures and the reason for its rare appliation is the diffiulty in reliably generating adequate ontrol fores with a low power demand. Furthermore, the additional weight resulting from the ative devie needs to be kept small. In aordane with these requirements, a novel ative mass damper for vibration ontrol of strutures was developed. In this study, the new damper and the possibilities of fore generation are presented. The appliation of the damper for bridge dek flutter ontrol is demonstrated using a numerial example. INTODUCTION Slender strutures suh as high-rise buildings and long-span bridges are suseptible to dynami loads. In addition to earthquakes, wind is a major soure of suh loads. The effets of the resulting vibrations may be disastrous. An important example of suh a phenomenon is bridge dek flutter. It is ruial to take into aount this aeroelasti instability during the design stage, and the installation of stiffer girders may be required. As an alternative to hanging the strutural system, passive or ative ontrol has proven to effetively suppress strutural vibrations. Although both are implemented in some high-rise buildings, to date only passive ontrol has been used in bridge strutures. Ative mass dampers for bridge vibration ontrol desribed in the literature generate the ontrol fores by means of aelerating auxiliary masses. Miyata et al. (996) proposed the generation of a twisting moment by variable eentri weight and by hanging the rotational speed of a mass. Both dampers are extensively investigated in Körlin & Starossek (00). Another possibility is to adapt gyrosopi anti-roll devies used in ships (e.g. Ferry 93) for use in bridges. In this ase, the spin axis of a gyrosopi wheel is tilted and aelerated by an atuator; due to the gyrosopi effet, ontrol moments are indued. A novel ative mass damper was developed whih, in a preferred onfiguration, uses only entrifugal fores indued by rotating unbalaned masses. In ontrast to the above-mentioned systems, the ontrol fores are generated with a onstant motion, and no aeleration is needed for their generation. Thus, a very low power demand is ahieved as demonstrated using a numerial example for bridge dek flutter ontrol. DAMPE CONCEPT AND FOCE GENEATION. Basi unit The basi omponent of the new damper onsists of an atuator-driven rotating rod with a mass attahed to its free end. Two suh rotors with lumped masses m and m, lengths r and r, and

2 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea angular veloities ω and ω are arranged side by side on the same plane (Fig. a). Both rotors form a rotor pair the basi unit of the ative damper. The two parallel axes of the rotors have a distane. At time t 0 = 0, the angles referring to the global downward diretion are ϕ 0 and ϕ 0, respetively. By turning the rotors, the masses indue radial and tangential fores whih an be ombined to a resulting vertial and horizontal fore and a moment. With an appropriate hoie of parameters, manifold time histories of fores an be reated. Periodi fores develop if both harateristis of the angular veloities repeat after a onstant time interval. Approximate triangular or retangular shapes of the generated fore time histories are possible.. Some possible onfigurations If both damper masses rotate with the same onstant angular veloity in opposite diretions, i.e. ω = ω = ω, fores F y and F z are generated whih refer to a loal oordinate system (Fig. b). The diretion of the resultant fore F z is determined by ϕ0 + ϕ γ = 0 () with referene to the global downward diretion. Following the basi priniples of mehanis, the generated fores ating on the ontrolled struture are where F F y z = = ( mr mr ) ω sin ϕ + ( m + m ) g sin γ ( m r + m r ) ω os ϕ + ( m + m ) g os γ ϕ0 ϕ ϕ = ω t 0 (3) and g is the aeleration of gravity. The dynami omponents of the fores are harmoni. Note that no strutural motions are onsidered and the rotor lengths remain onstant. The (in general non-harmoni) moment M is M = ( mr + mr ) ω sin ϕsin γ ( mr mr ) ω os ϕos γ ( m m ) ( m r m r ) g sin ϕos γ ( m r + m r ) g os ϕsin γ g + K For the ontrol of lateral motions only, it is preferable to plae both units with their axes in a vertial diretion. As a result, the terms with g in Equations and drop. If the produts of mass and length for eah rotor is hosen to be the same (m r = m r = mr) and the stati omponents are omitted, a single harmoni fore is generated F z = mr ω osϕ (5) whih ats in the diretion γ (Eq. ). The dynami fore omponent in the perpendiular diretion is always zero. The ating dynami moment is ( mr ω sin ϕ mr g os ϕ) γ M = sin (6) () () X Z m Y ù ù m r r ö 0 ö0 X Z Y ö m ù r ö 0 F z M ã F y ö 0 ö ù r m (a) ö ö (b) / / Figure. otor damper: (a) basi layout; (b) diretions of the generated fores.

3 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea With the assembly of two or more rotor pairs, the magnitudes of the generated fores and moments as well as the diretion of the fores an be ontrolled. The fores and moments generated by eah rotor pair an be ombined again to resultant fores and a resultant moment. Favorable fore omponents an be amplified, unfavorable fore omponents aneled out. In partiular, large moments are generated with a large distane perpendiular to the diretion of the resultant fores of two rotor pairs. A large differene between these fores leads to a larger moment. If the fores are equal and at in opposite diretions, the resultant fore is zero and the resultant moment is at a maximum. Provided the frequeny of the generated fores remains onstant, the amplitude of the resultant fore of a rotor pair an only be altered by a variation of the lengths of the rotors beause a variation of the masses during operation is not pratial. However, two rotor pairs with the same angular veloity an be employed. With a variation of the relative phase angle between both pairs, the resultant fore amplitude an be modified. A relative phase angle of zero leads to the maximum fore, whereas a gradual shift in phase redues the generated total fore. A temporary redution of the total fore to zero is possible if two rotor pairs with idential masses and lengths are used and the pairs are in opposite phase. 3 APPLICATION FO BIDGE DECK FLUTTE CONTOL 3. General As desribed in the preeding setion, fores with various time harateristis an be purposefully generated with the rotor pairs or with ombinations of rotor pairs. Harmoni fores an be generated by onstant rotor veloities in a simple manner. Appliations for devies based on the desribed priniples an be found in the field of onstrution, e.g. vibrating plate ompators and vibration generators for determining natural frequenies of strutures. However, the onept of fore generation has not yet been applied for vibration suppression and ontrol of strutures. Suh ative mass dampers ould be used to ontrol earthquake or wind-indued vibrations in high-rise buildings and bridges. By virtue of the fore generation mehanism, vibrations with onstant or nearly onstant amplitudes an be damped partiularly effetively. Therefore, in the following, the newly developed ative mass damper is presented for the ontrol of bridge dek flutter. 3. Equations of motion The damper onsists of two rotor pairs in a bridge setion and an be loated inside of a box girder as depited in Figure. The rotors of both rotor pairs rotate in opposite diretions. In this paper, a generalized two-dimensional system with the two bridge degrees of freedom h for heaving and α for rotation is investigated (Fig. 3). As a prerequisite for the utilization of this system, no horizontal fores must be generated by the ative devie. The equations of dynami motion per unit span are given by mh&& + hh& + khh = Lse + F I α & (7) + α + k α = M + M α & α se where m is the mass; I the mass moment of inertia; h and α the strutural damping oeffi- Figure. Proposed damper onfiguration. 3

4 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea ients; and k h and k α the strutural stiffnesses. The bridge dek width is b and the dot indiates differentiation with respet to time. The self-exited aeroelasti fores L se and M se due to the onoming flow with wind speed v are inluded, whereas buffeting fores are not onsidered in this study, but will be inluded in further researh. Desriptions of the motion-indued aerodynami fores in time domain are well established (e.g. Chen et al. 000). The generated ontrol fores are ombined to a resultant fore F ating at midspan in the global downward diretion and a resultant moment M positive lokwise. 3.3 Fore and moment generated by damper Numbering the rotors - from left to right in Figure 3, the equations of motion for rotor i referring to the global oordinate system are mi [ di ( α&& sin α + α& os α) ri ( ϕ&& i os ϕi ϕ& i sin ϕi )] = Fyi mi [ h&& g + di ( α&& os α α& sin α) ri ( ϕ&& i sin ϕi + ϕ& i os ϕi )] = Fzi ( Ii + miri ) ϕ&& i miri [( h&& g) sin ϕi + di ( α&& sin( ϕi α) α& os( ϕi α) )] = M Mi with the rotor mass m i = 0.5 m ; the rotor inertia I i = 0.5 I ; the onstant rotor lengths r = r = r l and r 3 = r = r r ; d = d = (a ) and d = d 3 = (a 0.5 ), with a as indiated in Figure 3. M Mi are moments whih have to be applied by the atuators. The angles of the rotors - are onstrained to that of rotor ϕ 3 ; = ϕ ; ϕ = ϕ + θ ϕ = ϕ θ (9) where θ is a onstant angle defining a phase shift between the rotor pairs. In this way, only one degree of freedom, namely ϕ, is additionally introdued to those of the bridge. The interrelation between this angle whih refers to the global oordinate system and ϕ, the angle overed by the rotor during time t, is ϕ (0) = ϕ + ϕ0 + α α0 with the angles of the rotor ϕ 0 in the global oordinate system and the bridge dek α 0, both at time t 0 = 0. In the following, this point in time is hosen to oinide with a maximum in the rotational degree of freedom of the bridge dek, i.e. α 0 = α max. The orresponding equations for the other rotor angles an be determined from Equations 9 and 0. The resultant ontrol fores are obtained by the use of Equations 8 and 9 F M m = m h&& + rl = m a with the abbreviations X Z v Y r l ö / / r l ( ϕ&& p + ϕ& p ) m + α && + r a l m rl ö k á á k h M se [ ϕ&& os α p + ( α&& sin α ( ϕ& α& ) os α) p ] a a [ ϕ&& sin α p + ( α&& os α + ( ϕ& α& ) sin α) p ] m / r r L ö I 3 / m / se m,i r r ö I / m / I / m / I / Figure 3. Two degree of freedom system with rotor damper. h h á 3 / / + K (8) ()

5 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea p p 3 ( ϕ + θ) ; p = os ϕ + V os( ϕ + θ) ( ϕ + θ) ; p = os ϕ V os( ϕ + θ = sin ϕ + V sin = sin ϕ V sin ) where V = r r / r l and onstant terms are omitted. The first term of eah Equation reveal an inrease in the mass and mass moment of inertia in Equations 7. The other terms in Equation represent rotor motion dependent fores. The moment omponents thereof are influened by the rotary motion of the bridge dek. It an be easily verified that, with the onstraints in Equation 9, no resultant horizontal fore is indued by the rotors. 3. Generation of fores or moments only If the length of all rotors is hosen to be idential (V = ), two interesting ases an be speified. The motion of the rotors indues only fores and no moments if both rotors of eah rotor pair are oaxially loated ( = 0) and the phase angle between the left and right rotor pairs vanishes, i.e. θ = 0. Equation with beomes F M = m = m h&& + m r a α&& l ( ϕ&& ϕ + ϕ& sin ϕ ) os On the other hand, if the rotor pairs are in opposite phase, i.e. θ = 80, it follows from Equations and that: F = && mh M = + m a α && + m r a l [ ϕ&& os α sin ϕ + ( α&& sin α ( ϕ& α& ) os α) os ϕ ] In this ase, only moments and no fores are generated by the rotor motion. Thus, with the same devie, it is possible to generate fores or moments only by simply hanging the phase angle θ. 3.5 Parameters to be set in ase of harmoni fores and moments In the pure flutter state, the vibrations of the bridge dek exhibit harmoni motions of frequeny ω with onstant amplitudes in both degrees of freedom h and α (Starossek 99). For the generation of harmoni fores, ϕ& & = 0. Sine the rotors have to omplete one revolution in a vibration period of the bridge dek, ϕ& = ω. This also implies ϕ & >> α& and ϕ& >> α& for small angles α. Inorporating this and approximations for small angles, the following is obtained from Equations and : m F = + mh&& rl ϕ& = + m M α + m a && r ϕ l a & ( os ϕ + V os( ϕ + θ) ) ( os ϕ + V os( ϕ + θ) ) On inspetion of Equation 5 and as suggested in the previous setion, the magnitude of the indued fores F and M an be ontrolled by speifying θ and V. These parameters also in- Im () (3) () (5) V M F è V è ø ö Figure. epresentation of generated fore amplitudes in the omplex plane. e 5

6 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea fluene the diretion of the generated fores, whereas ϕ 0 determines the relative phase between the bridge motion and the generated fores. Thus, ϕ 0 influenes the degree of damping or exitation of the bridge motion. The degrees of freedom of the bridge are eah optimally damped if the indued fore F and moment M at in the opposite diretion of the vertial bridge dek veloity h & and rotational veloity α&, respetively. In the following, relationships for the parameters in dependeny of V are given whih omply with this ondition. The rotor motion dependent parts of Equations 5 an be desribed in the omplex plane (Fig. ). Multiplying the first of Equations 5 with b, the rotor motion dependent parts are m ~ m ~ Fϕ b = rl b ϕ& F ; M ϕ = rl a ϕ& M (6) where ~ ~ F = os ϕ + V os( ϕ + θ) ; M = os ϕ + V os( ϕ + θ) (7) an be interpreted as real parts of the omplex numbers F and M in Figure. Defining ψ as the phase angle between α and h and aordingly between M and F the phase angle θ follows from Figure V sin θ = tanψ (8) V It an be seen that two solutions exist for θ, i.e. two ombinations of generated fore amplitudes are possible. Furthermore, by means of Figure, these fore amplitudes are F, = V + ± M, = V + m ( ) ( V ) V + os ψ ( ) ( V ) V + os ψ where the plus sign in the first equation and the minus sign in the seond equation apply for 90 < θ 90 and the other signs for 90 < θ 70. For the existene of solution of Equations 8 and 9, ψ 90 and ψ 70. Additionally for Equation 9 < V (0) ψ tan must be satisfied for 90 < ψ < 0 and 0 < ψ < 90, whereas for 90 < ψ < 80 and 80 < ψ < 70 ψ tan (9) V < () applies. In Equations 0 and, the restritions V > and V < are inorporated, whih ensure that both fore omponents at in the opposite diretion of h & and α&, respetively. It an be shown that for ψ = 90 and ψ = 70, the length of all rotors must be the same, i.e. V =. The ratio of the generated fores n = F / M an now be hosen arbitrarily and the phase angle θ is determined from n osθ = () n + where 0 < θ < 80 and 80 < θ < 0 for ψ = 90 and ψ = 70, respetively. The equivalent of Equation 9 is F = + osθ; M = osθ (3) 6

7 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea As mentioned above, the angle ϕ 0 ontrols the relative phase between the bridge dek motion and the generated fores. It an be shown that the generated fores at in the opposite diretions of the veloities h & and α & if ϕ 0 fulfills the following ondition: V θ ϕ = os () V sin θ tan 0 where 90 < ϕ 0 < 90 for 80 < θ < 360 and 90 < ϕ 0 < 70 for 0 < θ < 80. As with all ative systems, the seletion of a phase angle ϕ would lead to an exitation of the struture. Therefore, the solution range must be seleted arefully. For the interesting ase of the generation of fores only, i.e. V =, θ = 0, F =, M = 0, the relative phase angle whih ensures the fore ats in the opposite diretion of the veloity is ϕ 0 = 70 ψ. Similarly, if moments are exlusively generated, i.e. V =, θ = 80, F = 0, M =, the phase angle is ϕ 0 = 90 for all phase angles ψ of the bridge motion. For ompleteness, if ψ = 0, then either θ = 0 with F = V +, M = V and ϕ 0 = 70 or θ = 80 with F = V, M = V + and ϕ 0 = 90, where in both ases V. Similarly, if ψ = 80, then either θ = 0 with F = V + and M = V + or θ = 80 with F = V + and M = V +, where in both ases V and ϕ 0 = 90. EXAMPLE In this setion, an example for the appliation of the novel ative mass damper for flutter ontrol is given. It is intended to demonstrate the apability of the rotor damper. Details onerning the performane and the applied ontrol algorithm are omitted here due to limited spae and will be presented elsewhere. A onstrution stage of the Great Belt Bridge in Denmark is hosen for numerial simulations in time domain. The strutural parameters, taken from Walther (99) are m = kg/m, I = kgm, b = 5.5 m, f h = Hz and f α = 0.86 Hz. The strutural damping is set to zero, i.e. h = α = 0. The self-exited aeroelasti fores L se and M se are modeled using rational funtion approximation (e.g. Chen et al. 000) and expressions are taken from Körlin (006). The damper parameters are a = /3 b, = 0, m = m, I = 0, r l = r r =.0 m. Sine V = and with θ = 80 moments only are generated for ontrol. Sine the angular veloities of rotor and 3 are always idential, they are assumed to be mehanially oupled by a gear. The same applies for rotor and. Beause of θ = 80, the oupled rotors are in indifferent equilibrium and no energy is needed for lifting the damper masses. Furthermore, to ahieve low power onsumption, the maximum magnitude of the angular aeleration of the rotors is limited to 0. rad/s during the ontrol proess. During the start-up of the damper, the angular aeleration of the rotors is rad/s. Although a small ontrol mass and rotor length are hosen, the ritial flutter speed is signifiantly enhaned from 38.5 m/s to 3.9 m/s, i.e. by %. It should be noted that the ahieved ritial wind speed depends on the Power [Nm/m/s] Power [Nm/m/s] (a) Time [s] 3000 (b) Time [s] Energy [Nm/m] Time [s] Figure 5. Total mehanial power and energy of atuators (v = 3.9 m/s): (a) mehanial power of rotor damper atuators; (b) power of omparison system; () mehanial energy of rotor damper atuators. () 7

8 IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 3-7 July 008, Seoul, Korea amplitude of the bridge motion when the damper is ativated sine the ontrol proess is nonlinear. In this study, the ontrol is ativated when α exeeds 0.0 rad (0.57 ). To highlight the advantageous low power onsumption of the new damper, the new system is ompared with an ative mass damper suggested in Miyata et al. (996) and desribed in Körlin & Starossek (00). The ontrol moment in this omparison system stems from inertia fores due to the aeleration and deeleration of a rotating mass, similar to a flywheel. The ontrol mass of this system is m S = m, the mass moment of inertia I S = m S r l, with r l as above. Thus, the weight and the height of the damper mass are the same as those of the rotor damper. The mass performs harmoni motions with suh an amplitude that the ritial wind speed is the same as with the rotor damper. The phase angle between the generated moment and the bridge dek rotation is 90 as in the ase of the new damper. In Figure 5a and b, the total mehanial power of the atuators of the two systems is illustrated. After start-up, the total power of the system with the rotors is omparatively marginal (Fig. 5a) sine only power to sustain the relative phase angle between the rotor position and the bridge rotation is needed. Sine the total mehanial atuator energy (Fig. 5) of the new rotor damper beomes negative after some time, the new system onverts energy of the air flow into mehanial energy with a low energy input. 5 CONCLUSIONS A novel ative mass damper was presented. The mehanism of fore generation was explained and parameters for advantageously generating fores were determined. It was shown that various kinds of fores for strutural ontrol an be generated. One onfiguration was desribed in detail. For the ase of harmoni fore generation, the parameters to be set for optimal damping of the two bridge degrees of freedom were given. The apability of the new damper to ontrol flutter was proven by an example. Even with a damper mass of only 0.5 % of the generalized bridge dek mass, the ritial wind speed was substantially inreased. The damper possesses an extremely small power demand and generates energy. Further researh inludes buffeting effets. Beause of the adaptability of the fore generation, other appliations of the new damper are oneivable, e.g. the ontrol of vertial bridge dek vibrations indued by moving traffi. A patent appliation has been filed for the new damper. ACKNOWLEDGEMENTS This researh is funded by the Deutshe Forshungsgemeinshaft DFG (German esearh Foundation) whih is gratefully aknowledged. EFEENCES Chen, X., Matsumoto, M. & Kareem, A Time domain flutter and buffeting response analysis of bridges. Journal of Engineering Mehanis 6(): 7-6. Ferry, E.S. 93. Applied Gyrodynamis. New York: John Wiley & Sons. Körlin, Aktive mehanishe Kontrolle winderregter Brükenshwingungen. PhD thesis, Hamburg University of Tehnology, Hamburg, Germany. Körlin,. & Starossek, U. 00. Ative mass dampers for flutter ontrol of bridges. 8th international onferene on flow-indued vibrations; Pro. intern. onf., Paris, 6-9 July 00. Miyata, T., Yamada, H. & Dung, N.N Proposed measures for flutter ontrol in long span bridges. Strut. eng. in onsideration of eonomy, environment and energy; eport IABSE ongress, Copenhagen, Denmark, 6-0 June 996. Zurih: IABSE. Starossek, U. 99. Brükendynamik. Braunshweig/Wiesbaden: Friedr. Vieweg & Sohn. Walther, J.H. 99. Disrete vortex method for two-dimensional flow past bodies of arbitrary shape undergoing presribed rotary and translational motion. PhD thesis, Tehnial University of Denmark, Lyngby, Denmark. 8