CONTROL OF PEDESTRIAN-INDUCED VIBRATIONS OF FOOTBRIDGES USING TUNED LIQUID COLUMN DAMPERS

Transcription

1 CONTROL OF PEDESTRIN-INDUCED VIBRTIONS OF FOOTBRIDGES USING TUNED LIQUID COLUMN DMPERS Michael REITERER & Franz ZIEGLER Institute of Rational Mechanics, Vienna University of Technology, ustria Keywords: tuned liquid column damper, tuned mass damper, footbridges, pedestrian induced vibrations bstract: In recently constructed footbridges unexpected lateral vibrations occurred due to a large number of crossing pedestrians. To control and suppress these undesired vibrations secondary passive damping devices, such as the conventional tuned mass damper (TMD), were installed to the bridge. In the present investigation it is proposed to apply a more efficient and more economic type of damping device: the tuned liquid column damper (TLCD). mechanical model of a coupled bridge/tlcd system is presented and analyzed numerically. The bridge is assumed to be a three degree-of-freedom (DOF) system. Furthermore, a small scale testing facility has been constructed in the laboratory of the TU- Institute. The experimental results are in good agreement with the theoretical predictions and indicate that TLCDs are effective damping devices for the control of pedestrian induced vibrations of footbridges. INTRODUCTION To mark the Millennium a new slender footbridge was built in London across the river Thames. When the bridge was opened in June 000, a large number of pedestrians crossed the bridge, causing unexpected excessive lateral vibrations. Two days later the vibrations gave reason to close the bridge for the following 8 months. The extensive retrofit involved the use of fluid viscous dampers and conventional TMDs to increase the structural damping with a total cost of 0 million Euro, Dallard []. Similar problems emerged at the Toda Park pedestrian cable-stayed bridge in Japan. When a sufficiently dense population of pedestrians crossed the bridge, unexpected lateral vibrations occurred and made modifications necessary. To suppress the vibrations tuned liquid damper (TLD) were installed inside the box girder, as discussed in a paper by Fujino []. In the present investigation it is proposed to apply the more efficient and more economic TLCD for suppressing pedestrian-induced vibrations of footbridges. The TLCD relies on the motion of a liquid mass in a sealed tube to counteract the external motion, while a built-in orifice plate induces turbulent damping forces that dissipate kinetic energy. For optimal tuning of TLCDs the natural frequency and equivalent linear damping coefficient have to be chosen appropriately, likewise to the conventional TMDs, as given by Den Hartog [3]. The advantages of TLCDs over other types of damping devices are: simple tuning of natural frequency and damping, where the damping can be controlled by the opening angle of the built-in orifice plate, low cost of design and maintenance and a simple construction. Furthermore, the over-linear turbulent damping of the fluid motion increases the energy dissipation rapidly in case of large vibration amplitudes of the liquid surface and contributes one more feature of superiority over conventional TMDs. Dipl.-Ing. Michael REITERER, Research ssistant, Wiedner Hauptstr. 8 / E 0, -040 Vienna, ustria. mr@allmech9.tuwien.ac.at Dr. Dr. h.c. Franz ZIEGLER, Professor, Head of Institute, Chairman of the ustrian ssociation of Earthquake Engineering (OGE), ustria. franz.ziegler@tuwien.ac.at

2 In order to study the damping effectiveness of optimally tuned TLCDs a detailed theoretical and experimental investigation is carried out. Thereby, the nonlinear turbulent damping of the fluid motion and a general plane motion of the TLCD is considered. The bridge is assumed to be a three DOF system, where the interaction force and moment from the TLCD dynamics are assigned. It will be shown that optimally tuning of the TLCD to the fundamental frequency of the bridge, whose corresponding mode has a dominant horizontal response, similar to London s Millenium Bridge, reduces the resonant peak of the first mode by 65%. Hence, the proposed TLCD turns out to be an effective damping device for the undesired pedestrian induced footbridge vibration. MECHNICL MODEL Basically, the response of footbridges due to pedestrian excitation is a combination of horizontal, vertical and rotational motions. The equations of motion for the complex hybrid bridge/tlcd system are derived by a substructure synthesis method, which splits the problem into two parts.. Substructure Separated TLCD In a first step the TLCD is separated from the bridge and considered under combined and assigned horizontal w t, vertical v t and rotational ϕ t excitations, as illustrated in Fig.. Fig. Separated TLCD under combined horizontal, vertical and rotational excitations The TLCD parameters are the horizontal length of the liquid column B, the length of the liquid column in the inclined pipe section at rest H, i.e. the total length is B+ H, the inclined and horizontal cross-sectional areas H and B, respectively, and the opening angle of the inclined pipe section π / 6< β < π /. The relative and incompressible flow of the liquid inside the pipe is described by the liquid surface displacement u. Furthermore, the reference point is chosen symmetrically, whereby the absolute acceleration a v is separated into the moving reference system, v a ax, = = az, w cosϕ v sinϕ t t t t w sinϕ + v cosϕ t t t t. () In Fig. a closed piping system is assumed, thus the air inside the pipe is compressed or released, depending on the liquid surface displacement. Hence, a non negligible pressure

3 difference p= p p can built up on either side of the inclined pipe sections, whose influence on the undamped circular natural frequency of the TLCD is defined in Eq. (4). pplying the modified Bernoulli equation along the relative non-stationary streamline in the moving frame and in an instant configuration, Ziegler [4, p. 497], yields the nonlinear parametric excited equation of motion of the TLCD, a z, Ω B u + δ uu cos u a sin L + ω ϕt + κ = κ ( x, g ϕt)+ κ, g ω Ω () where the geometry coefficients κ, κ, κ and the effective length L eff of the liquid column are defined by B+ Hcosβ H sin β Bcosβ + H κ = κ = κ = L = H + H,,, eff L L L B. (3) eff eff eff The undamped circular natural frequency of the TLCD includes the air spring effect due to sealed tubes and is given by B ω g np ρ g 0 / = β+ L sin eff h eff, (4) where n denotes the polytropic index, which is determined by the type of state change of the gas. For an adiabatic process of any two atomic gas n =4. whereas for a isothermal (slow) process n =0.. ny other process is in between those two extreme situations. Furthermore, p0, ρ, g and h 5 eff denote the initial atmospheric pressure p0 = 0 Pa, the liquid density, e.g. 3 water ρ =000kg / m, the gravity constant and the effective height of the air spring, an important design variable as it will directly influence the TLCD undamped circular frequency. In Eq. () the head loss coefficient δ L due to turbulent losses along the relative streamline can be controlled by the opening angle of the built-in orifice plate. The head loss coefficients of stationary flowing fluids in relevant pipe elements are given e.g. in Idelchick [5]. The time variant stiffness parameter in Eq. () leads to parametric excited oscillations and is caused by rotation and vertical excitation of the TLCD. Consequently, under special conditions of the lightly damped TLCD, the instability phenomenon of parametric resonance occurs. Parametric resonance increases the TLCD vibration amplitude in an undesired manner and thus, the damping effectiveness of the optimally tuned TLCD might get worse. However, in a recent paper by Reiterer and Ziegler [6] it is indicated that damping of the fluid motion strongly influences the occurrence of parametric resonance. Sufficiently high damping prevents parametric resonance. In the present investigation optimal tuning of the TLCD (tuned by frequency and equivalent linear damping) leads to a sufficiently high fluid damping for the range of vertical excitation amplitudes considered and hence, no undesired worsening-effects are observed. It has to be mentioned that parametric resonance may also exist for the conventional pendulum type TMD, whose point of suspension moves vertically. In addition, the over-linear turbulent damping stabilizes the unstable motion and thus, limits the vibration amplitude of the TLCD. For the optimal design of the TLCD, which is shown in Section 4, the nonlinear turbulent damping term δ L has to be

4 transformed to its equivalent linear expression ζ. Basically, the method of harmonic balance [4, p. 69] is used. pplying this method yields the equivalent linear viscous damping term ζ U δl = 4 30 π, (5) where U 0 denotes the relative vibration amplitude of the liquid surface. Thereby, the value of U 0 is determined from numerical simulations of the linear system and commonly chosen as U0 = Umax. Subsequently, optimal tuning of the TLCD adjusts only the circular natural frequency ω and the equivalent linear viscous damping coefficient ζ. In a second step, the interaction force and the interaction moment caused by the TLCD dynamics have to be determined. Conservation of moment renders the nonlinear interaction force v F in x' and z' direction of the moving reference system (acting on the fluid body), v v F = m a + m f f ( ) u u H uu H u κ( Ω )+ κ 4 Ω+ ( + ) Ω ( u + u κ Ω Ω)+ κ H u H + u Ω uu ( ( ) + ), (6) where Ω = ϕ t and Ω = ϕ t denotes the angular velocity and angular acceleration. The geometry dependent coefficients of Eq. (6) are given by B+ Hcosβ H sinβ κ = κ = L = H + B,, L L B. (7) In Eq. (6) mf = ρ H L defines the total fluid mass, and L is a length dependent from the cross section, which becomes equal to L eff in case of H = B. Conservation of angular momentum with respect to the accelerated reference point, see Fig., yields the nonlinear resultant interaction moments M y (acting on the fluid body), M m u a g H H u y = f κ (, z + cosϕt)+ κ + ( a, x g sinϕt) B u + ( u + uu ) + κ κ κ H Ω Ω 3 Ω, (8) H ( ) where the geometry coefficient κ 3 due to rotation is given by κ 3 3 H 3B 3B B B = + + cos β+ 3. 3L 4H H 8 H H (9) Several nonlinearities show up in Eq. (6) and (8), which are considered in the overall numerical simulation. Furthermore, the interaction moment M y includes the additional

5 undesired moment from gravity, acting on the displaced center of fluid mass C F, see Fig., which is always neglected in the theory of conventional TLCDs and TMDs.. Substructure Separated Bridge with ssigned Interaction Force and Moment In this Section the equations of motion of the separated bridge with assigned interaction force and moment from the TLCD dynamics are derived. Thereby, the bridge is assumed to be a three DOF system with its deformation, given by the horizontal displacement v, vertical displacement w and rotation ϑ, as indicated in Fig.. Fig. Separated three DOF bridge with assigned interaction forces and moment from the TLCD dynamics: Fy, Fz, Mx; time periodic excitation forces: F v, F w ; center of stiffness C S ; center of mass C M ; horizontal k v, vertical k w and torsional k ϑ stiffness The bridge is excited through the combined horizontal and vertical time periodic excitation forces F v and F w, respectively, which are acting in the center of stiffness C S. In the selected mechanical model of the bridge, the reference point C S does not coincide with the center of mass C M and thus, coupled bending and torsional motions are expected. Thereby, the components of the distance between C M and C S are denoted by c and d, as shown in Fig.. pplying conservation of momentum and moment of momentum with respect to C S yields the following nonlinear coupled equations of motion, where a small angle of rotation ϑ is assumed. In Eq. (0) the mass of the TLCD (without fluid) mt = m mf is included in the bridge mass M.

6 F F v d v v v y Fz ϑ + ζvωv + Ωv = + ϑ, M M M w c w w F F w y Fz + ϑ + ζwωw + Ωw = + ϑ +, M M M (0) M, I cw M I dv M F x y ϑ + + ζϑωϑϑ + Ωϑϑ = I I z I = I C M + c M + d M. ζ v, ζ w, ζ ϑ are the linear viscous damping coefficients of the bridge, and Ωv, Ωw, Ωϑ are the corresponding uncoupled and undamped circular natural frequencies, kv kϑ kϑ Ωv =, Ωw =, Ωϑ =. () M M I Furthermore, I denotes the mass moment of inertia with respect to C S, and z is the vertical distance between and C S. The nonlinear interaction forces and moment, given in Eq. (6) and (8), are transformed with respect to the new reference point C S to main coordinates of the bridge, ( ) F m a y f y z u u = H uu H u, + + ( ), + ϑ κ ϑ κ ϑ ( + 4 ) ϑ ( ) F m a z f z z u u =, + H u ϑ + κ( ϑ + ϑ)+ κ ( H + u ) ϑ + uu, Mx = mf u a z g H H u κ ( z a, + ϑ + ) + κ ( + )(, y + ϑ z + g ϑ) B u u κ κ + uu H ( ϑ ϑ) κ3 ϑ. (4) where the acceleration a of the previous reference point in main coordinates and in case of small angles of rotation ϑ is given by, a ay, = = az, v + w ϑ v ϑ + w. (5) Finally, the equation of motion of the separated TLCD is transformed to the main coordinates of the bridge, w v z u + δ uu L + ω + ϑ + ϑ κ ϑ g g g ω B u = κ[ v wϑ + ϑ z g + ϑ] κ ϑ. (7) Substituting the transformed interaction forces and moment, Eq. (4) in Eq. (0), yields together with Eq. (7) a set of four equations of motion for the unknown displacements v, w,

7 the rotation ϑ, and the displacement of the liquid surface u. The performed numerical simulations of the coupled bridge/ TLCD system are presented in the Section 4. 3 EXPERIMENTL INVESTIGTION In addition to the theoretical investigation a small scale bridge model has been constructed in the laboratory of the Institute of Rational Mechanics at the Vienna University of Technology. side and front view of the experimental model set-up is shown in Fig. 3. The three DOF model is given by the rigid bridge profile on top of a skeletal box girder. lightweight cantilevered beam, made of brazen profiles / 0 / 50, provides the spring supports of the bridge profile. To prevent undesired natural vibrations of the brazen profiles, several stiffening elements, made of Styrofoam, are mounted along the cantilevered beam. The TLCD made of a plastic tube is fixed symmetrically to the bridge profile, where orifice plates induce over-linear turbulent damping. Furthermore, a novel contactless excitation device has been developed to simulate the moving pedestrian. lthough it is widely known that people walk with a frequency of about Hz, it is not commonly known that about 0% of the vertical dynamic force acts laterally, as described by Bachmann [8]. The frequency of this lateral dynamic force is always half the walking pace, i.e. about Hz. Hence, the dynamic excitation device must induce different dynamic excitation forces in horizontal and vertical directions. Therefore, the bridge profile is equipped with two electrical coils, which are located in the environment of a homogenous magnetic field. Subsequently, feeding the electrical coils with a defined periodic current excites the bridge profile horizontal and vertical in the desired manner. Fig. 3 Side and front view of the experimental model set-up The software LabView 7.0 (installed in a Macintosh computer) provides the time periodic excitation signals. The displacements of the rigid bridge profile are measured by means of contact-less laser transducers, Type optoncdt 605. ll measured signals are recorded by means of the software BEM-DMCplusV3.7 (installed in a Macintosh computer) through the board of Digital mplifier System DMCplus (Hottinger Baldwin Messtechnik, HBM). To measure the liquid surface displacement inside the TLCD an electronic resistance measurement device is used. Therefore the piping system is equipped with two pairs of wire electrodes, whose resistance depends on the water level inside the pipe. Further details of the measurement device are given in Reiterer and Hochrainer [9]. Free vibration tests of the three DOF bridge and TLCD were performed to determine natural frequencies and equivalent linear

8 viscous damping coefficients. ll measured and selected parameters of the laboratory model, which are used to perform numerical simulations, are listed in Tab.. Tab. Parameters of the three DOF bridge model and TLCD BRIDGE STRUCTURE M =57. kg kv = N / m ζ v = 004. f = ω =0. π 959 Hz c = m kw =74. 83N / m ζ w = f ω = =. 678 π Hz d= m kϑ = 580. Nm ζ ω v = f 3 3 = =. 070 Hz π I = kgm z = m OPTIMLLY TUNED TLCD β = π /4 = = m B= 00. m H = 05. m H B Leff = 040. m mf = 0. kg ω optimal ζ optimal The optimal circular natural frequency ω and linear damping coefficient ζ are determined in the following Section 4. The experimentally obtained results due to pedestrian excitation are presented and discussed in Section 5. 4 NUMERICL NLYSES The following numerical simulations are performed according to the experimental investigation by considering the parameters listed in Tab.. Hence, the linear natural frequencies of the laboratory model are given as f = Hz, f =. 678Hz and f3 =. 070 Hz. Thereby, it is important to mention that the corresponding mode of the fundamental frequency f has a dominant horizontal response, which is similar to the vibration characteristics of London s Millenium Bridge. On the other hand the corresponding mode shapes of the second and the third natural frequency, f and f 3, respectively, have dominant vertical and rotational components. Due to recent problems in footbridge constructions the attached TLCD is tuned to the fundamental horizontal vibration mode. Basically, optimal tuning of conventional TMDs adjusts the design parameters δ * opt * ω = = * * ω + µ * 3µ, ζ =, * 8+ µ * ( ) (8) where the star indicates that these parameters are in accordance with the conventional design parameters of a conjugate TMD problem, as given in Den Hartog [3, p. 9]. Hochrainer [7, p. 98] derived the analogy between TMD and equivalent TLCD. pplying this analogy and taking into account that

9 * κ µ µ = + µ κ ( ) m f, µ =, M (9) relates the optimal design parameters of the TMD to the corresponding values in the linearized equation of motion of the TLCD. Substituting Eq. (5) in Eq. () gives, δ opt = δ * opt ( ) + µ κ *, opt, opt = , ζ = ζ = (0) Thereby, the ratio of fluid mass m f to total bridge mass M is obtained from Tab. : µ = (i.e. about 8 %), and κ is defined in Eq. (3). The value of µ is higher than the recommended mass ratio µ = 05. 3% for real bridge structures. By means of the value δ opt = , given in Eq. (0), the optimal natural frequency of the TLCD is determined, f ω ω = = δopt = Hz. π π () The obtained optimal design parameters, Eq. (0) and (), of the attached TLCD are considered in the corresponding numerical analyses. The numerically and experimentally assigned horizontal F v and vertical F w pedestrian excitation forces are shown in Fig. 4. Fv (t) Fw (t) time [s] time [s] Fig. 4 Numerically and experimentally assigned horizontal F v and vertical F w pedestrian excitation forces, according to Bachmann [8] ccording to Bachmann [8] the excitation forces are a superposition of three harmonics, 3 F t F sin π f ϕ, F t F sin π f ϕ, () ()= ( ) ()= ( ) v vn vn vn w wn wn wn n= n= where the amplitudes and phase angles are given by 3

10 F = 04. N, F = N, F = 05. N, ϕ = π / 5, ϕ = 3π / 5, ϕ = 0, v v v3 v v v3 F = 00. N, F = 003. N, F = N, ϕ = 0, ϕ = π /, ϕ = π /. w v v3 w w w3 (3) ll numerical simulations are performed by the software Matlab 6.5. It is emphasized that the nonlinear turbulent damping term δ L uu in Eq. () is replaced by its equivalent linear one: ζ ω u, which is obtained from the previous optimization process, Eq. (0). This approximation is only valid when the criterion for preventing parametric resonance is fullfilled, given in Reiterer and Ziegler [7]. For this study the required equivalent linear fluid damping to prevent parametric resonance at the critical vertical excitation frequency fw = f is given by, v0 ζ, requ = ω = , g (4) where v0 = 00. m is selected according to the experimental result and ω is given in Eq. (). The required value ζ, requ = turns out to be much lower than the optimal one ζ = 0. 7, given in Eq. (0), and thus, no undesired worsening effects on the optimal damping characteristics of the attached TLCD are expected. The results obtained by numerical simulation are presented and discussed in the following Section. 5 COMPRISON OF EXPERIMENTL ND NUMERICL RESULTS The experimentally and numerically obtained steady state horizontal displacement v of the bridge reference point C S with and without TLCD is shown in Fig. 5. Thereby, the excitation frequencies of the combined horizontal and vertical periodic excitation forces F v and F w, respectively, are assumed to be: fv= 095. Hz, fv = fv, fv3 = 3fv and fw= 90. Hz, fw = fw, fw3 = 3fw, i.e. the excitation frequency f v is close to the fundamental natural frequency f. v [mm] Experiment 40 v [mm] 30 Numerical Simulation without TLCD with TLCD time [s] time [s] Fig. 5 Experimentally and numerically obtained results of steady state horizontal displacement v of the bridge reference point C S with and without TLCD; combined horizontal F v and vertical F w excitation forces The light line represents the steady state vibration without TLCD and the heavy line shows the reduced response due to the optimal tuned TLCD. The experimental and numerical results are in good agreement. nother important response parameter, the dynamic magnification factor (DMF) is shown in Fig. 6 for both experiment and numerical simulation. By inspection,

11 an excellent agreement is observed, and it can be seen that the installation of the optimal tuned TLCD reduces the resonant peak of the first mode of the bridge by 65%. DMF Experiment without TLCD with TLCD 4,00 DMF,00 0,00 65 % 65 % 8,00 6,00 4,00,00 Numerical Simulation without TLCD with TLCD f v: 0,7 0,9,,5 f v :,4,8,,5 f v3:,,7 3,3 3,75 f w :,4,8,,50 f w:,8 3,6 4,4 5,0 f w3: 4, 5,4 6,6 7,75 excitation frequencies fv, fw [Hz] 0,00 f v: 0,7 0,8 0,9,0,, f v :,4,6,8,0,,4 f v3:,,4,7 3,0 3,3 3,6 f w:,4,6,8,0,,4 f w :,8 3, 3,6 4,0 4,4 4,8 f w3 : 4, 4,8 5,4 6,0 6,6 7, excitation frequencies fv, fw [Hz] Fig. 6 Experimental and numerical obtained result of the DMF; horizontal displacement v; combined horizontal F v and vertical F w excitation forces For the sake of completeness Fig. 7 shows the experimentally and numerically derived DMF of the bridge vertical displacement w. Experiment Numerical Simulation DMF DMF without TLCD with TLCD without TLCD with TLCD f w :,4,8,,5 f w:,8 3,6 4,4 5,0 f w3 : 4, 5,4 6,6 7,5 f v : 0,7 0,9,,5 f v :,4,8,,5 f v3 :,,7 3,3 3,75 excitation frequencies fv, fw [Hz] f w:,4,6,8,0,,4 f w :,8 3, 3,6 4,0 4,4 4,8 f w3 : 4, 4,8 5,4 6,0 6,6 7, f v: 0,7 0,8 0,9,0,, f v :,4,6,8,0,,4 f v3 :,,4,7 3,0 3,3 3,6 excitation frequencies fv, fw [Hz] Fig. 7 Experimental and numerical obtained result of the DMF; vertical displacement w; combined horizontal F v and vertical F w excitation force Thereby, the combined horizontal and vertical time periodic excitation forces are sweeped within the frequency range of interest. gain, the experiment and numerical simulations are in good agreement. Fig. 7 indicates that the attached TLCD has no or only a little influence on the vertical displacement w within the range of vertical excitation amplitudes considered.

12 6 CONCLUSION The aim of this investigation was to study the reduction of undesired pedestrian induced footbridge vibrations by an innovative passive damping device. It is proposed to install the more efficient and more economic tuned liquid column damper (TLCD) at discrete sections along the bridge. detailed theoretical and experimental investigation has been carried out, considering nonlinear turbulent damping and a general plane motion of the TLCD. The equation of motion of the TLCD turns out to be time variant and hence, sensitive to parametric resonance. The instability phenomenon of parametric resonance increases the liquid surface displacement in an undesired manner and hence, the performance of the attached TLCD might get worse. However, previous investigations indicate that optimal damping of the fluid motion, which is higher than the natural one, is crucial in order to prevent parametric resonance. For this study the optimal damping coefficient turns out to be ζ = 0. 7 which is much higher than the required one ζ, requ = and thus, no undesired worsening effects are observed. The bridge is assumed to be a three DOF system, whereby the nonlinear interaction forces and interaction moment from the TLCD dynamics are assigned. The TLCD is tuned to the fundamental frequency of the bridge, whose corresponding mode has a dominant horizontal response characteristic, similar to London s Millenium Bridge. The numerically and experimentally obtained results are in good agreement and indicate the large damping effect of the optimally tuned TLCD. It is shown that the attached TLCD reduces the resonant peak of the first mode by 65%. In conclusion the proposed TLCD turns out to be an effective damping device for undesired pedestrian footbridge vibrations. References [] DLLRD ET. L., The London Millenium Footbridge, The Structural Engineer, Vol. 79 / No., (p. 7-33), 00 [] NKMUR S. & FUJINO Y., Lateral Vibration on a Pedestrian Cable-Stayed Bridge, Structural Engineering International, (p ), Vol. 4, 00 [3] DEN HRTOG J. P., Mechanical Vibrations, Reprint of the 4 th ed., McGrawHill, 956 [4] ZIEGLER F., Mechanics of Solids and Fluids, nd ed., Vienna New York, Springer, 998 [5] IDELCHICK, I. E., Handbook of Hydraulic Resistance, Coefficient of Local Resistance and Friction, vailable from the U.S. Department of Commerce, Springfield, 960 [6] REITERER M. & ZIEGLER F., Combined Seismic ctivation Of SDOF-Building With Passive TLCD ttached, in: Proc. 3 th World Conference on Earthquake Engineering, Vancouver, B.C., Canada, Paper No. 6, (5 pages), 004 [7] HOCHRINER M. J., Control of vibrations in civil engineering structures with special emphasis on tall buildings, Dissertation, Institut für llgemeine Mechanik (E0), TU- Wien, -040 Wien, 00 [8] BCHMNN H. & MMNN W., Vibrations in Structures Induced by Man and Machines, Structural Engineering Documents, International ssociation for Bridge and Structural Engineering (IBSE), 987 [9] REITERER M. & HOCHRINER M.J., Parametric Resonance In Tuned Liquid Column Dampers: n Experimental Investigation, IZ, 004, in print.