Downloae from orbit.tu.k on: Jan 17, 19 Placement an tuning of resonance ampers on footbriges Krenk, Steen; Brønen, Aners; Kristensen, Aners Publishe in: Footbrige 5 Publication ate: 5 Document Version Early version, also known as pre-print Link back to DTU Orbit Citation (APA): Krenk, S., Brønen, A., & Kristensen, A. (5). Placement an tuning of resonance ampers on footbriges. In Footbrige 5: n International Conference (Vol. Abstract Book an CD ROM). Venice: University IUAV, Italy. General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights. Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research. You may not further istribute the material or use it for any profit-making activity or commercial gain You may freely istribute the URL ientifying the publication in the public portal If you believe that this ocument breaches copyright please contact us proviing etails, an we will remove access to the work immeiately an investigate your claim.
PLACEMENT AND TUNING OF RESONANCE DAMPERS ON FOOTBRIDGES Steen KRENK Professor Dept. of Mechanical Engineering Technical University of Denmark Lyngby, Denmark Aners BRØNDEN Engineer Buro Happol Glasgow, Scotlan Aners KRISTENSEN Engineer RAMBØLL Virum, Denmark Summary The placement an tuning of multiple tune mass ampers is iscusse with reference to the newly esigne Langelinie Footbrige, presently uner construction in Copenhagen. First the principles of resonance ampers are briefly reviewe, an a novel result for the calibration is presente. It has been shown that classical frequency tuning combine with minimization of the relative motion of the amper mass leas to an increase of the amping relative to the classical proceure with about 1 pct. The two esign approaches: i) minimum resonance for frequency epenent loa, an ii) robustness against resonance by sufficient amping ratio, are iscusse. A simple two-step proceure is presente for esign of multiple ampers on footbriges: first the properties of tune mass ampers for the relevant free vibration moes are estimate by generalizing the simple basic equations to moal vibrations, an then the parameters are fine tune by vibration analyses incluing the suspene masses from the resonant ampers for the other moes. Usually a single correction is sufficient to obtain equal amping in the combine system. Keywors: footbrige ynamics; vibrations; amping; tune mass ampers; multiple ampers. 1. Introuction Slener structures with variable loas are potentially susceptible to vibration problems [1]. Many moern footbriges belong to this category slener structures with long spans an peestrian loas from walking, umping an running. The loas will typically have main frequency components below about 5 Hz [,3], an footbriges crossing maor roas, rivers or railway terrains often have one or more vibration moes within this range. It is ifficult to increase the lowest frequencies beyon this range, because increase stiffness will also require increase mass. Thus, amping becomes an important issue for these briges. The lateral vibration problems of the Lonon Millennium Brige have le to great attention to this kin of vibration problem. However, most footbriges of more conventional esign have consierably larger lateral than vertical bening stiffness, leaving the irect loaing from the footfall as the main source of peestrian loa. Lateral vibrations are of the self inuce kin, where the motion of the brige eck leas to changes in the walking pattern, generating an unfavourable loa. Essentially, this is an instability that can be cure by introucing sufficient amping, whereby the loaing is eliminate. Conversely, the vertical peestrian loa is only little influence by the brige eck motion. In this case the purpose of the amping is to reuce the magnitue of the ynamic response, while the loaing is left virtually unchange. Tune mass ampers are efficiently use for this purpose. The iea of the tune mass amper is to suspen a mass in such a way that the relative motion of the mass is governe by resonance. This prouces a large relative motion which is then ampe by a conventional amper, connecting the mass an the brige eck. The present paper escribes an accurate an systematic proceure for esign of a system of tune mass ampers for amping of several vibration moes of a footbrige. Due to uncertainty regaring the precise loaing conitions, the proceure is base on an initial estimate of the necessary amping ratios of the concerne moes. The ampers are then esigne by a two step proceure: i) first an initial estimate base on the iniviual uncouple moes of vibration, an then ii) an austment base on moes incluing the amper masses of the other moes. The proceure makes use of newly foun properties of moal amping ratios for tune mass amper systems, an leas to a simple, yet accurate, istribution of the amping over the moes. The proceure is illustrate for the Langelinie Footbrige, presently uner construction in Copenhagen. First the esign proceure is presente an emonstrate, an then some particular loa cases representing umping an running are illustrate.
1.1 The Langelinie Footbrige The esign of the Langelinie Footbrige in Copenhagen was submitte in 4, an the brige is now uner construction. The brige is a steel box girer with four spans, crossing the railway terrain behin Østerport Station an connecting the Østerbro part of the city with a park containing part of the ol city fortifications. A general view an a cross-section are shown in Fig. 1. The following analyses are base on the initial esign of the brige. In the final esign some corrections have been introuce, an etails of the results may therefore not be fully representative for the complete brige. The brige is a continuous four-cell box girer in Corten steel with moulus of elasticity 3 GPa an Poisson s ratio of ν =.3. The main imensions of the brige are shown in Fig.1. The total length is 169.8 m, an the box section is 6.85 m wie an 1. m high. The flanges vary from 3 mm in thickness over the full length of the brige with more material in compressive zones. Longituinal stiffeners an transverse bulkheas have a thickness of 1.5mm an 1 mm, respectively. The istribute mass is approximately 3 kg/m corresponing to a total mass incluing 5 transverse bulkheas of about 5.6 1 kg/m. The requirement for vibration comfort were those given by the Danish Roa Directorate [4] consisting of a reference loa scenario with two walking persons, represente by an equivalent harmonic loa an the maximum acceleration a.78 max.5 f (1) where f is the frequency of moe No. in Hz, an a is the maximum acceleration in m/s. max a) 4.8m 49.4m 49.m 3.4m b) 6.85m 1.m Fig. 1 Langelinie Footbrige, a) general view, b) cross-section. 1. Damping of vibrations from peestrian loas The initial esign of ampers for footbriges can follow either of two approaches: esign against ynamic amplification for a harmonic loa, or a esign base on robustness against ynamic amplification by introucing a sufficient amping ratio for each of the relevant moes. The present case is concerne with vertical isplacements only, an the T isplacements of the noes of the brige can be written in vector format as u = [ u1, u, ]. The equations of motion are conveniently expresse in the form of the following matrix equation, usually obtaine by a finite element moel, Mu + Cu + Ku = Q() t () where a ot enotes time erivative. M, C an K are the mass, amping an stiffness matrix, respectively, while T Q() t = [ Q(), t Q (), t ] enotes the time epenent vector of noal loas. 1 An estimate of the ynamic amplification of moe No. can be obtaine from the equations of motion () by consiering the iniviual moes one at a time. The response of an iniviual moe is given as the prouct of the moe shape vector
u an the moal amplitue r () t in the form u() t = r () t u (3) Substitution of this expression into (), followe by multiplication from the left by the transpose of the moe shape vector leas to the equation for the moal amplitue, 1 r t r t r t q t () + ω () + ω () = () m In this equation ω is the angular frequency an is the amping ratio of moe No.. The moal loa q () t is given as the prouct of the original noal loa vector Q () t an the moe shape vector u, (4) q () t = u T Q() t (5) an m is the moal mass, efine by m = umu T (6) The moal mass gives an impression of how much of the structural mass participates in the motion of this particular moe as iscusse for the Langelinie Footbrige in Section 3. The relation (6) can be use to normalize the moe shape vectors. In particular, if the moal mass m is selecte to be the mass of the brige eck, the moe shape vector reflects a typical unit isplacement. For a perioic loa the most severe situation is most often resonance. Resonance is characterize by a harmonic response with angular frequency ω. At resonance the inertial term represente by r () t precisely cancels the stiffness term ω r () t. In terms of magnitues the equation can therefore be written as ω r q = m This relation implies that the contribution to the acceleration of the vertical motion of the brige for a moe in resonance is given by q ru u m u = ω = Thus, the amplitue of the acceleration is given as the moal loa ivie by twice the amping ratio times the moal mass, q / m. It is note that this formula for the acceleration oes not contain the factor ω sometimes seen in han calculation type estimates. This factor woul be introuce, if the moal mass m were replace with the moal stiffness k. However, while the moal mass an the corresponing moe shape vectors are easily normalize in an intuitively clear way, the moal stiffness increases rapily with the moe number. The limit on vertical ynamic response from peestrian loas is usually consiere as a comfort criterion base on acceleration. It is seen from (8) that if the moal mass is chosen equal to the mass of e.g. the eck the response epens on the prouct of the loa with the moe shape vector, ivie by the amping ratio. Thus local moes can lea to severe response, if the loa is also local. On the other han istribute loas lea to smaller moal loa q thereby limiting the response. For istribute loaing this suggests the alternative esign principle, in which all the relevant moes, e.g. moes with natural frequencies below 5 Hz, are given a sufficient amping ratio. If this principle is aopte, the final esign shoul be checke for local loas, e.g. from a small group eliberately trying to excite the brige. (7) (8). Tune Mass Dampers Traitionally the principle of the tune mass amper is evelope for a single amper mass m attache to a moving structural mass m by a spring with stiffness k an a viscous amper with parameter c. The unampe frequency of the structure is characterize via the structural stiffness k. The iealize tune mass amper is illustrate in Fig. a. Figure b illustrates a case in which two ampers are use in parallel. This is typically use for flexible structures to
istribute the action of the ampers over the structure. A three step esign proceure for tune mass ampers on flexible structures can consist of: i) selection of optimal parameters base on the iealize systems shown in Fig., ii) evaluation of representative structural parameters for the moes to be ampe, an finally iii) austment of the parameters to account for the presence of multiple ampers. First the optimal parameters for the iealize system are ientifie, an then the reinterpretation for flexible structures is ealt with in Section 3. a) b) m m m k c x k c k c x m m k x k x Fig. Tune mass amper(s), a) single resonant mass, b) ouble masses..1 Optimal calibration The masses of the iealize system in Fig.a are characterize by the mass ratio μ = m / m, while the stiffness of structure an amper are characterize by their angular frequencies, ω = k / m, ω = k / m (9) The natural angular frequency of the structure ω is given, an the frequency of the amper is selecte to provie optimal properties of the combine system. The classical metho is to consier a frequency iagram of the ynamic amplification [5]. Two points are inepenent of the magnitue of the amping, an the frequency tuning of the amper is etermine to ω ω = 1 + μ The amper is characterize by its amping ratio. When an aitional mass from the amper is introuce, the original structural vibration moe splits into two moes with closely space frequencies. It has recently been emonstrate that for moerate mass ratio the frequency tuning (1) leas to ientical amping ratio for both these moes. The literature has various efinitions, epening on normalizing frequency, but here the amping ratio of the amper is efine by the free vibrations of the amper if the structure remains stationary, i.e. = c/mω. The magnitue of the amping ratio must be selecte large enough to prouce amping, but not so large that it limits the relative motion of the amper. It can be shown that the largest amping ratio that can be use without increasing the relative motion of the amper is given by, [6], 1 μ = (11) 1+ μ This is ifferent from the classic result [5], which has the factor 1 as 3 8. The ynamic amplification of the structural mass is illustrate in Fig. 3a, an the relative motion of the amper mass in Fig. 3b for a mass ratio of μ =.5. It is seen that the relation (11) leas to a flat maximum of the relative motion of the amper mass, an also that it essentially removes the ouble peaks in the motion of the structural mass. It has been be shown that the amping of the combine structure-amper moes is approximately equal to half the amping ratio of the amper 1 struc. Thus, the aitional 1 pct. amping introuce by (11) relative to the classical formula also leas to an increase of 1 pct. in the amping of the structural moes. (1)
8 3 x / x s 6 4 x / x s 1.6.8 1 1. 1.4 ω /ω.6.8 1 1. 1.4 ω /ω Fig. 3 Dynamic amplification for harmonic loa, a) motion of structural mass, b) relative motion of amper mass. The full line correspons to amping given by (11), while the ashe line is the classical result [5].. Double ampers When mounting ampers on flexible structures like footbriges it may be esirable to use two or more ampers for the same moe in orer to limit the size of the iniviual amper an to istribute the effect of the ampers. The iealize moel with two ampers is shown in Fig. b. This is a system with three egrees of freeom, an therefore three moes of vibration. If the two ampers are ientical there are two moes, where the ampers exhibit ientical motion. These moes correspon to those escribe above. When the ampers are correctly tune these moes will have a amping ratio equal to half the amping ratio of the ampers, 1 struc. In the thir moe the two ampers oscillate in opposite motion, leaving the structural mass at rest. This moe has the amping ratio of the ampers, struc. For flexible structures with ouble ampers, these will typically be balance to provie equal participation, an the aitional moe will therefore typically involve even larger motion of the ampers relative to the motion of the structure, an accoringly the amping is also larger than the orinary moes typically about ouble. This aitional moe is therefore not a esign problem. 3. Multiple Dampers on Flexible Structures When the simple results for frequency tuning (1) an calibration of amping (11) are to be use for multiple ampers on flexible structures three questions have to be aresse: i) efinition of suitable generalize parameters to be use in the esign formulae, ii) interaction effects between the ampers, an iii) possible changes in the moe shapes ue to the aitional mass of the ampers. Equivalent parameters can be efine in terms of the unampe moe shapes of the structure an use for preliminary esign of the amper system as escribe in Section 3.1. A subsequent etaile esign can then be obtaine from a free vibration analysis making use of complex moe shapes an frequencies, incluing the effect of the ampers. This proceure is briefly inicate in Section 3. an illustrate for the Langelinie brige in Section 4. 3.1 Equivalent moal parameters The preliminary esign analysis is base on the moes of the structure without any ampers mounte. For each of these moes the isplacement of the brige eck is given by the representation (3) in terms of the moal amplitue r () t an the corresponing moe shape vector u. One or more ampers are installe for moe. They are all tune to the same frequency ω,, an the esign is base on the situation, in which they move in phase. The motion of this group of ampers can then be represente by the moe shape vector u in a form similar to (3) for the brige eck, u () t = r () t, u (1) As seen from this representation the motion of each amper in the group is proportional to the local motion of the brige eck. The effective mass of the group of ampers can therefore be calculate by an expression similar to (6) for the moal mass of the brige,
m = um u T,, (13) where M, is a iagonal matrix containing the masses of the ampers for moe in the appropriate iagonal positions. The equivalent mass ratio for moe is then given by μ = m, / m,. The frequency tuning then follows from (1), an the calibration of the ampers from (11). When mounting a tune mass amper, the original moe of vibration splits into two moes with closely space frequencies. In the optimal amper configuration each pair of moes has ientical amper ratio. Typically the preliminary esign leas to somewhat ifferent amping ratios, when ampers are introuce for several moes. This is ue to sensitivity to the aitional mass of the other ampers. The importance of these effects epens on the number of moes to be ampe, an thereby the amper mass ae to the structure, an on the stiffness of the brige eck. The interaction effects can be estimate an the preliminary esign improve by recalculating the natural frequencies for each moe with all the amper masses for the other moes inclue with spring stiffness, but without ampers. This gives a goo estimate of the interaction effect, an in practice constitutes a suitable basis for the esign of the amper system. 3. Analysis by complex moes an frequencies The esign proceure outline above provies an estimate of the necessary masses, the stiffness of their support, an the necessary amping, incluing an estimate of the interaction effect of multiple ampers, but neglecting any change in moe shapes introuce by the amping. The behavior of the brige with ampers can be investigate by performing a free vibration analysis of the brige, incluing the tune mass ampers in the moel. In the free vibrations the motion of the ampers is not in phase with the motion of the brige, an the vibrations can therefore not be escribe by a conventional real-value moal analysis. In orer to escribe vibrations that are not in phase a complex-value moal analysis is require. The iea originally presente by Foss [7] - is to represent the motion as the real part of a complexvalue harmonic function, u( t) = Re[exp( iωt) u ] (14) The free vibrations can then be foun from a linear eigenvalue problem of the ouble size of the original system of equations, K C M u + iω = (15) iω M M u The matrix C contains the amping coefficients. When the system inclues local ampers, as in the present case, the angular frequency ω, etermine from the eigenvalue problem, becomes complex. The amping ratio of the corresponing moe is etermine from the imaginary part of the complex frequency, Im[ ω ] = ω Thus, the amping ratio of any moe of the brige with ampers can be etermine by solving the eigenvalue problem (15). This can be one by stanar numerical software. The solution inclues the full amper system an therefore accounts for interaction effects as well as any change of moe shapes. The full complex analysis in the following to emonstrate the accuracy of the two-step esign proceure outline in Section 3.1 consisting of a preliminary esign base on the unampe moe shapes of the structure without ampers, followe by an austment base on recalculate natural frequencies incluing the other amper masses. Thus, a full complex moal analysis will most often not be neee in the actual esign proceure. (16) 4. Dampers for Langelinie Footbrige The Langelinie Footbrige has four vibration moes with resonant frequencies below 5 Hz. The vibration moes are normalize such that the moal mass of each moe equals the physical mass of the brige (56 tons) an are presente in Fig. 4. These vibration moes are prone to human loaing an it is shown that even a small peestrian group violates the comfort criteria if only the structural amping is regare. The structural amping for steel footbriges is usually
assume to be.4% an the structural response is evaluate by (7) for the four vibration moes. This inicates that the implementation of aitional amping is essential for the serviceability of the footbrige. The ampers for the Langelinie Footbrige are esigne on the basis of a mass ratio of μ =.5 an are positione at maximum crests within each vibration moe inicate with blue boxes in Fig. 4. In the following the ampers are configure by a two step proceure, an the effectiveness of the ampers is evaluate by the resulting amping ratio in the ifferent moes foun by solving the eigenvalue problem (15). 4 4 Fig.4 The first four vibration moes of the Langelinie Footbrige with amper positions inicate. 4.1 Preliminary esign of ampers from unampe structural moes The physical mass of the ampers in the iniviual moes are etermine on the moal mass of the brige alone. Thus the weight of the other ampers is isregare. The equivalent mass properties of the ampers within each moe are etermine by (13) such that the mass ratio μ = m, / m is fulfille. The remaining physical parameters of the ampers are etermine by (1) an (11).The parameters of the ampers for the relevant moes are presente in Table. The amping in the iniviual moes are presente by ζ struc. It is notice that implementation of ouble ampers yiel an extra moe characterize by a high level of amping as mentione in section.. Equal moal amping is not achieve in any of the moes but the fourth moe. This is most severe for the first moe where only 6% of critical amping is achieve. This is somewhat smaller than the expecte level of amping aroun 7.5% of critical. Table. Damper properties an effect base on unampe moe shapes of structure without ampers. Moe 1 3 4 Damper in span(s) / 3 1 / 3 1 / 4 Damper mass (kg) 445 / 445 4 / 4 355 / 355 3 μ eff.5.499.498.5 f, (Hz) 1.67.31 3.4 4.59 f, (Hz) 1.59..9 4.37 amper.154.154.154.155 struc.6 /.11.81 /.96.95 /.81.78 /.8.141.137.149 4. Improve esign by incluing amper masses in the moes The secon step in the esign proceure inclues the mass an stiffness properties of the ampers for the other vibration moes. Thus, these properties are inclue in a real-value eigenvalue analysis of the global system matrices. In principle an iterative process must be carrie out in orer to implement the correct mass of the other ampers as they are moifie in the esign of the amper(s), but usually only one step is neee. The new esign of the full amper configuration is presente in Table 3 The total weight of the ampers is increase app. 5% corresponing to the extra mass introuce in each vibration moe. This esign approach is notice to implement a higher level of amping in the moes an is most visible in the first moe, where the amping is increase by 3%.
Table 3. Damper properties an effect, when incluing amper mass an stiffness in the moal analysis. Moe 1 3 4 Damper in span(s) / 3 1 / 3 1 / 4 Damper mass (kg) 4775 / 4775 475 / 475 375 / 375 95 μ eff.499.499.5.5 f, (Hz) 1.58.6 3.11 4.6 f, (Hz) 1.51.16.96 4.38 amper.154.154.155.154 struc.78 /.84.86 /.89.85 /.91.78 /.79.143.137.151 The ampers can be fine tune to equal amping ratio by use of the complex moal analysis technique. Equal moal amping can be obtaine by changing the amper stiffness (or amper mass) in small steps. In the present case fine tuning by change of the amper stiffness leas to amping ratios for the four moes of.81,.87,.88 an.78, respectively. The extra highly ampe moes, associate with the uplicate ampers for the first three moes are not affecte by the fine tuning. The following loa cases are base on the amper configuration presente in Table 3. 5. Local Loas from Jumping an Running Local loas can be prouce by either umping or running groups of people. When umping or running the contact uration t p constitutes only part of the footfall perio T. The time variation of the loaing is escribe by series of halfsine impulses with maximum value kf p, kf sin ( / ) for t t p π t t p p Ft () = (17) for tp < t T F is the equivalent loa for full contact, an the peak factor kp = π T /tp is etermine by the ratio of the contact uration an the footfall perio T. This loa pattern is now use to illustrate the effect of a group of 5 people umping at one location, an a person running across the brige. Aitional loa cases an amper configurations have been consiere in [8]. 5.1 Jumping group of people The response to a group of 5 people in fully synchronize umping at the same location with frequency f = 1/ T =.5 Hz is consiere. This frequency lies in the mile of the range suggeste by Bachmann et al. [1]. The contact uration is taken as t p =.5T =. s, yieling a peak factor of k p = π. The static weight of one person is set to.8 kn, leaing to a total static loa of the group of 4 kn. Thus, the peak loa attains the magnitue of F 13 kn. peak Table 4. Maximum response in loa cases 1 an. Loa case 1 a [m / s ].376.349 max In loa case 1 the perioic loa (17) is place in the mile of span, an in loa case the loa is place in the mile of span 4. The response is foun by irect numerical time-integration. The maximum acceleration obtaine in the two loa cases is presente in Table 4, an the corresponing response curves for the noes with maximum acceleration are presente in Fig. 5. None of the loas violates the comfort criterion (1). In loa case 1 the first harmonic component of the loa is near resonance with one of the split moes originating from the secon vibration moe an is seen from
Fig.5a to be governe by this vibration moe. Thus, a fairly large response results aroun the excitation frequency f =.5 Hz. In loa case the secon harmonic component of the loa is near resonance with one of the split moes originating from the fourth vibration moe. The corresponing acceleration recor is shown in Fig. 5b, clearly illustrating a response frequency aroun 5 Hz. A Fourier expansion of the loa (17) reveals the first an secon force amplitue to be 1.57 an.67 respectively. The importance of the local moes is illustrate by the fact that the response of loa case is of the same magnitue as that of loa case 1, even though the force is less than half the magnitue. All things equal the force amplitues can change within the group, if the assumption of total synchronization is not fulfille. This woul lea to a smaller response..5 a [m/s ].5 1 3 4 5 t [s] 1 3 4 5 t [s] Fig.5 Acceleration recors in mile of loae span ue to umping group,a) span, b) span 4. 5. Running across the brige The Langelinie Footbrige is intene to be use for public events like the Copenhagen Marathon. Running may involve groups of ifferent sizes an ifferent running styles, involving ifferent egrees of synchronization. For full synchronization the total loa will be N times that of a single runner conitions being otherwise equal while for complete lack of synchronization the effective loa will only be N times that of a single runner. The following scenario therefore involves only a single runner, an the total loa must be obtaine by appropriate scaling. The loa of each footfall is again moele by half-sine pulses given by (17) with the same frequency f an contact urationt p. This time however the force is not stationary but moving along the length of the brige giving the loa a transient character. The strie length of the runner is set to l s = 1.3m. The loa of one person is set to.8 kn. The establishment of the global vector Q() t in () is schematically presente in Fig. 6. The length of each element correspons to the strie length of the runner such that a time ifference for which the loa is applie at aacent noes correspons to the perio T. T Q i t p Q i+1 Fig.6 Time history of loa components at noe i an i + 1 for running. t The maximum acceleration is foun to be a max =.77 m/s an occurs at the mile of the fourth span, when the runner passes this point. The response of this noe is presente in Fig. 7. The vertical otte lines inicate when the runner passes a support, thus the runner has left the brige after the last line at t = 51.1s. This correspons to a forwar spee of 3.3m/s. With reference to Fig. 4 it is evient that the response is governe by the fourth vibration moe which is activate through the secon harmonic of the footfall frequency.
.1 a [m/s ].1 1 3 4 5 6 Fig.7 Response at the mile of span 4 ue to loa case with one runner crossing the brige. t [s] 6. Conclusions A two-step esign proceure for tune mass ampers on footbriges has been presente an illustrate with application to the new Langelinie Footbrige in Copenhagen. It is base on the close relation between the moal response to a harmonic loa an the amping ratio for that moe. The esign is base on proviing the necessary moal amping ratio for vibration moes with frequency below say 5Hz. In the present analyses a fairly conservative value of struct.8 has been use. More etaile values can be estimate from the moal loas by use of the relation (7). The actual esign of the amper system has two steps: i) an initial estimate of the ampers for any particular moe is obtaine from the frequency an amper formulae (1) an (11) using the generalize mass of the ampers given by (13), ii) the amper parameters are auste by re-calculating each moe incluing the amper masses an springs from the ampers for all the other moes. In practice this leas to nearly optimal tuning of the couple vibration problem of structure an ampers. Equal amping ratio for all relevant moes correspons to a loaing scenario, where the moal loas are roughly proportional to the moal masses. Local moes are typically associate with smaller moal masses, an it is therefore avisable to check the esign for local loas. The effect of local loas has been illustrate by the response to regular umping an the passage of an iniviual runner. Acknowlegment. The present work has been supporte by the Danish Technical Research Council through the proect Damping Mechanisms in Dynamics of Structures an Materials. 7. References [1] BACHMANN, H. et al., Vibration Problems in Structures Practical Guielines, Birkhäuser, 1997, 34 pp. [] BACHMANN, H., Lively Footbriges a Real Problem, Footbrige, Paris,, pp. 18-3. [3] WILLFORD, M., Dynamic actions an reactions of peestrians, Footbrige, Paris,, pp. 66-73. [4] DANISH ROAD DIRECTORATE, Broteknik, ve- og stibroer. Belastnings og beregningsregler, Danish Roa Directorate, November.. [5] DEN HARTOG J. P., Mechanical Vibrations (4 th en), McGraw-Hill, New York, 1956. (Reprinte by Dover, New York 1985) [6] KRENK, S., Frequency analysis of the tune mass amper, Journal of Applie Mechanics, 5, Vol. 7, 5, pp. 936-94. [7] FOSS, K.A., Coorinates which uncouple the equations of motion of ampe linear ynamic systems, Journal of Applie Mechanics, Vol. 35, 1958, pp. 361-367. [8] KRISTENSEN, A. an BRØNDEN, A., Tune mass ampers on structures, M.Sc. thesis, Department of Mechanical Engineering, Technical University of Denmark, June 5, 17 pp.