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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT 03-3 Suppression of Wind-Induced Vibrations of a Seesaw-Type Oscillator by means of a Dynamic Absorber H. Lumbantobing ISSN Reports of the Department of Applied Mathematical Analysis Delft 003

2 Copyright 003 by Department of Applied Mathematical Analysis, Delft, The Netherlands. No part of the Journal may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission from Department of Applied Mathematical Analysis, Delft University of Technology, The Netherlands.

3 Suppression of Wind-Induced Vibrations of a Seesaw-Type Oscillator by means of a Dynamic Absorber H. Lumbantobing Department of Applied Mathematical Analysis, Faculty of Information Technology and Systems, Delft University of Technology, Mekelweg 4, 68 CD Delft, The Netherlands Abstract In this paper the suppression of wind-induced vibrations of a seesaw-type oscillator by means of a dynamic absorber is considered. With suppression the shift of the critical flow velocity to higher values as well as the reduction of vibration amplitudes is meant. The equations of motion are derived using Lagrange s formalism. From a linear analysis the optimal absorber tuning is obtained. That is, the optimal frequency and damping combination such that the highest critical flow velocity is obtained. A strong increase of the critical flow velocity is obtained when the absorber has a frequency close to the oscillator s frequency. From the nonlinear analysis the suppression in terms of a reduction of amplitude is shown. For a specific case, a comparison between the original and the suppressed vibration behaviour is shown. Introduction In this paper the suppression of wind-induced vibrations of a one-degree-of-freedom seesawtype oscillator, by means of a dynamic absorber, is considered. A schematic sketch of the seesaw oscillator with an absorber is given in Figure. It consists of a rigid bar, holding a cylinder at the right end. On the other end a counter weight is fixed, balancing the cylinder with respect to a hinge axis. Two springs provide for a restoring moment. It is assumed that the cylinder has a uniform cross-section along its axis. Inside of the cylinder, a dynamic absorber is placed. The absorber consists of a mass, a spring and a damper. Supervised by Dr. ir. T. I. Haaker. On leave from Department of Mathematics - FKIP, University of Cenderawasih, Papua, Indonesia

4 Suppression of Wind-Induced Vibrations The motion of the absorber mass is directed perpendicular to the oscillator s rigid bar. If the cylinder has a non-circular cross-section and is exposed to a steady wind flow, self-excited so called galloping oscillation may arise []. Several authors have studied galloping for one-degree-of-freedom structures. Notably, Parkinson & Smith [] modelled and analyzed plunge galloping for a square prism. Van der Burgh, Haaker, van Oudheusden and Lumbantobing analyzed and modelled rotational galloping of seesaw-type oscillators [3, 4, 5, 7, 8, 9, 3]. These authors applied a quasi-steady theory [] to model the aeroelastic forces acting on the cylinder. For low to moderate wind velocities, these forces may be assumed small. Then a mathematical analysis of the equations of motion can be based on an asymptotic method [4, 5]. Typically, one finds a critical flow velocity above which the equilibrium position becomes unstable and stable galloping oscillations occur. The amplitude of these oscillations grows if the flow velocity is increased. For higher flow velocities, aeroelastic stiffness forces may arise that are of the same order of magnitude as the structural stiffness force [6, 7, 8]. The resulting aeroelastic behaviour is of a potentially catastrophic nature, as a nonlinear divergence phenomenon may occur [6]. The addition of the dynamic absorber is an important means for suppressing the windinduced vibrations. With suppression the shift of the critical flow velocity to higher values as well as the reduction of vibration amplitudes is meant. Tondl [6, 7] investigated the effect of a dynamic absorber on vibration quenching of pendulum type systems. He applied the harmonic balance method to analyze the quenching effect. In [0,, ], Matsuhisa, et al. investigated the efectiveness of dynamic absorbers for ropeway gondolas, chairlifts, ropeway carrier, ships and cable suspension bridges. Here the effectiveness of a dynamic absorber for suppression of wind-induced vibrations for a seesaw-type oscillator is considered. This paper is organized as follows. In section, the equation of motion for the seesaw oscillator with dynamic absorber is derived using Lagrange s formalism. In section 3, the analysis of the model equation is presented. In section 3., a linear analysis based on the harmonic balance method is given. The optimal tuning for the absorber is derived, such that the critical velocity is shifted to the highest possible value. In section 3., a nonlinear analysis based on the two-time-scales method [5, 8] is given. In particular the reduction of vibration amplitudes is considered. For a specific example, a comparison between the original and the suppressed vibration behaviour is given. This paper is ended with some conclusions in section 4. Derivation of the model equation To derive the equation of motion the Lagrange s formalism [9] is used. The kinetic and potential energies are given by the following expressions E k = a ψ + mrẋ ψ + mẋ + mx ψ,.) E p = κψ + k x ) + mgx cosψ),.)

5 Suppression of Wind-Induced Vibrations 3 hinge axis ϕ absorber m. l 0 x U R Figure : Schematic sketch of the structure of the seesaw oscillator with absorber. where the dot denotes the differentiation with respect to time t, ψ and x denote the angle of rotation of the seesaw structure around the hinge axis and the deflection of the absorber mass from its equilibrium position, respectively. Furthermore, a, m, κ, k and g are the moment of inertia of the seesaw structure including the absorber with respect to the hinge axis, mass of absorber, restoring force coefficient due to the springs attached to the seesaw bar, restoring force coefficient due to the spring of the absorber mass and acceleration of gravity, respectively. Here, R = R + l 0, where R is the distance from the cylinder s center to the hinge axis and l 0 is the distance from the equilibrium position of the small mass to the center of the cylinder, respectively. From.,.) one gets the Lagrangian as follows L = E k E p,.3) = a ψ + mrẋ ψ + mẋ + mx ψ κψ + k x ) mgx cosψ)..4) Then one finds that d L dt ψ ψ = a ψ + mrẍ + κψ + mx ψ + mxẋ ψ mgx sinψ),.5) d L dt ẋ L x = mẍ + mr ψ + k x mxψ + mg cosψ)..6) So one gets an unperturbed and undamped system as follows a ψ + mrẍ + κψ + mx ψ + mxẋ ψ mgx sinψ) = 0,.7) mẍ + mr ψ + k x mxψ + mg cosψ) = 0..8)

6 Suppression of Wind-Induced Vibrations 4 Assuming that b and b are damping coefficients of the main system and the dynamic absorber, respectively, and assuming that a quasi-steady aeroelastic force F = ρdlu C N α) is exerted on the cylinder, then one gets a ψ + mrẍ + κψ + b ψ + mx ψ + mxẋ ψ mgx sinψ) = ρdlr U C N α),.9) mẍ + mr ψ + k x + b ẋ mx ψ + mg cosψ) = 0,.0) where ρ, d, l, U and C N are air density, characteristic diameter of the cylinder cross section, length of the cylinder, wind velocity and aerodynamic coefficient curve, respectively. Furthermore, the dynamic angle of attack α is approximated by ψ R ψ. U Using the approximation for sinψ) and cosψ) up to quadratic terms and transforming x w mg k then one gets where a 0 = a + m3 g k a 0 ψ + mrẅ + a ψ + b ψ + mw m ψ g w k ψ +mwẇ ψ m g ẇ k ψ mgwψ = ρdlr U C N α),.) mẅ + mr ψ + k w + b ẇ mw ψ + m g k ψ mgψ = 0,.) and a = κ + m g k. Introducing Ω = a a 0, Ω = k m and Q = Ω Ω + m wẇ a ψ m g 0 a 0 k then after transforming time τ Ω t one gets ψ + mr a 0 ẅ + ψ + b a 0 Ω ψ + m a 0 w ψ m g a 0 k w ψ ẇ ψ mg wψ = ρdlr a 0 Ω a 0 Ω U C N α),.3) ẅ + R ψ + Q w + b ẇ w mω ψ + mg ψ g ψ = 0,.4) k Ω where α = ψ R Ω ψ and the dot now denotes differentiation with respect to τ. One U can assume that the wind force is small as the air density, ρ, is O0 3 ). Introducing a small parameter ɛ = a 0 ρdlr 3 and a reduced velocity µ = U Ω R and assuming m a 0 = Oɛ) and b a 0 = Oɛ), then one gets where α = ψ ψ µ, ɛβ = b λ 5 = mg k, λ 6 = g Ω ψ + ψ = ɛ λ w ψ + λ w ψ λ 3 ẅ β ψ λ wẇ ψ. +λ ẇ ψ + λ 4 wψ + µ C N α) ),.5) ẅ + R ψ + Q w + β ẇ = w ψ λ 5 ψ + λ 6 ψ,.6) a 0 Ω, β = b mω, ɛλ = m a 0, ɛλ = m g a 0 k, ɛλ 3 = mr a 0, ɛλ 4 = mg a 0, Ω

7 Suppression of Wind-Induced Vibrations 5 Assuming the cylinder cross section to be symmetric, then one can approximate the aerodynamic coefficient curve up to the cubic term as follows C N α) = c α + c 3 α 3, see [0]. After substituting the aerodynamic coefficient curve into equation.5) and defining U = c µ β then one gets ψ + ψ = ɛ λ 3 ẅ β U) ψ + β c U ψ λ w ψ +λ w ψ λ wẇ ψ + λ ẇ ψ + λ 4 wψ β +c U 3 ψ 3 + 3β U ψ ψ c c +3ψ ψ + c )) β U ψ 3,.7) ẅ + R ψ + Q w + β ẇ = w ψ λ 5 ψ + λ 6 ψ..8) Introducing a final scaling to equations.7 -.8) through w = w and ψ = Rψ then d d after neglecting the bar one gets ψ + ψ = ɛ η 3 ẅ β U) ψ + β U ψ η w ψ c +η w ψ η wẇ ψ + η ẇ ψ + η 4 wψ β + c U 3 ψ 3 + 3β U ψ ψ c c +3ψ ψ + c )) β U ψ 3,.9) ẅ + ψ + Q w + β ẇ = η 5 ψ + η 6 ψ + η 7 w ψ,.0) where η = d λ, η = dλ, η 3 = Rλ 3, η 4 = dλ 4, η 5 = d λ R 5, η 6 = d λ R 6, η 7 = d and R c 3 = c 3d. Note that U = corresponds with the critical flow velocity for the seesaw R oscillator without absorber. 3 Analysis of the equation In this section the equations.9 -.0) are analyzed. Firstly, the harmonic balance method is applied for the linear analysis and then the two time scales method is applied for the nonlinear analysis. 3. Linear analysis In this section a linear analysis for equations.9 -.0) around the equilibrium position is presented. One gets the following linearized equations from.9 -.0) ) ψ + ɛβ U ψ + ɛβ U) c ψ + ɛη 3 ẅ = 0, 3.)

8 Suppression of Wind-Induced Vibrations 6 ẅ + ψ + Q w + β ẇ = 0. 3.) To apply the harmonic balance method one sets ψ = cosωτ) and w = AcosΩτ) + BsinΩτ) and substituting into equations ) then one finds Ω ) ɛβ U ɛη 3 Ω A c = 0, 3.3) β U) + η 3 ΩB = 0, 3.4) Q Ω )A + β ΩB = Ω, 3.5) β ΩA + Q Ω )B = ) Note that in this analysis no assumptions have to be made with respect to the size of ɛ. From equations ) one finds A = Ω Q Ω ) β and B = Ω 3. Q Ω ) +β Ω Q Ω ) +β Ω After substituting B into equation 3.4) one gets U = + After substituting A and U into equation 3.3) then one gets β η 3 Ω 4 β Q Ω ) + β Ω ). 3.7) Ω ) ɛβ β η 3 Ω 4 ) + c β Q Ω ) + βω ) Q Ω ) + β Ω) ɛη 3 Ω 4 Q Ω ) = ) Without a dynamical absorber the critical wind velocity for the onset of galloping is U =. From 3.7) one finds that after applying the dynamical absorber the critical value of the velocity is shifted to a higher value. A numerical result is presented in Figure showing the minimum values for the wind velocity U in dependence on the tuning coefficient Q for different values of the coefficient β of the absorber damping. The function UQ) is arranged by solving equation 3.8) for Ω then substituting the obtained Ω into equation 3.7)). For the calculation the dynamic absorber is located in the center of the cylinder, i.e. l 0 = 0. The values of some parameters are chosen from wind tunnel measurements on an actual seesaw oscillator in [4]. Figure a) shows the critical wind velocity, U, depending on tuning Q for different values of the absorber damping coefficient β with η 3 = 0.33, β =.85, ɛ = and c = 3. For smaller values of β, UQ) consists of two branches reaching its optimal value near Q =. The lower branch for Q < ) corresponds to higher frequencies Ω, the lower branch for Q > ) to the lower frequencies Ω. Figure b) shows the projection of Figure a) on the β, U)-plane. One finds that U reaches its optimal value at U = 8.375, near β = 0.08 and tuning Q =.0, with resulting oscillation frequency Ω =.09. Figure 3a) shows the oscillation frequency of the obtained periodic solution as a function of tuning Q for β = From this figure, one can see that there are some values of Q having three corresponding values of Ω. This means that three periodic solutions may exist for some β, Q) combinations,

9 Suppression of Wind-Induced Vibrations u 5 4 u Q a) β β b) Figure : Critical wind velocity U in dependence on Q for different values of damping coefficient β. each corresponding with a distinct critical flow velocity. Now considering the stability of the trivial solution, one obtains a diagram as given in Figure 3b). The trivial solution is stable for 0 < U < U and for U < U < U 3 and is unstable for U < U < U and for U > U 3. So, after a first instability at U = U, re-stabilization occurs for U = U. The final instability occurs for U = U 3. For example, for the optimal damping, i.e. β = 0.08 at Q =.0 one gets three periodic solutions with frequencies Ω =.005, Ω =.08 and Ω 3 =.06, respectively. For each frequency, one obtains the critical flow velocities U = 4.048, U = and U 3 = 8.507, respectively. The diagram of the stability of the trivial solution is that of Figure 3b). 3. Nonlinear analysis In this section the equations.9 -.0) are considered again. In the analysis the nonlinear terms in the second equation are neglected because the strong linear coupling term is the most important. Note that the coefficients of the nonlinear terms contain d. From R equations.9 -.0) one gets ψ + ψ = ɛ η 3 ẅ β U) ψ + β U ψ η w ψ c +η w ψ η wẇ ψ + η ẇ ψ + η 4 wψ β + c U 3 ψ 3 + 3β U ψ ψ c c +3ψ ψ + c )) β U ψ 3, 3.9) ẅ + ψ + Q w + β ẇ = ) The two time scales method is applied by introducing new time variables ξ = τ and η = ɛτ [] and assuming ψ = ψξ, η) and w = wξ, η) then one gets

10 Suppression of Wind-Induced Vibrations 8 Ω Q a) 0 u. u u u3... Figure 3: a): Quadratic frequency of periodic solution as a function of tuning Q for β = b): Stability diagram of the trivial solution depending on the wind velocity for the optimal absorber tuning, i.e. β, Q) = 0.08,.0). b) dψξ,η) dτ = ψξ,η) ξ + ɛ ψξ,η) η, d ψξ,η) dτ = ψξ,η) ξ + ɛ ψξ,η) ξ η + ɛ ψξ,η) η, dwξ,η) dτ = wξ,η) ξ + ɛ wξ,η) η, d wξ,η) = wξ,η) + ɛ wξ,η) + ɛ wξ,η). dτ ξ ξ η η Expanding ψξ, η) = ψ 0 ξ, η) + ɛψ ξ, η) +, and wξ, η) = w 0 ξ, η) + ɛw ξ, η) + and substituting these into equations ) then one gets: A system for O) as follows A system for Oɛ) as follows ψ 0ξξ + ψ 0 = 0, 3.3) w 0ξξ + ψ 0ξξ + Q w 0 + β w 0ξ = ) ψ ξξ + ψ = ψ 0ξη η 3 w 0ξξ β U)ψ 0ξ + β U c ψ 0

11 Suppression of Wind-Induced Vibrations 9 η w 0 ψ 0 ξξ + η w 0 ψ 0ξξ η ψ 0ξ w 0 w 0ξ + η w 0ξ ψ 0ξ + η 4 ψ 0 w 0 + c 3 β U c ψ β U ψ0 ψ 0 ξ c ) +3ψ 0 ψ0 ξ + c β U ψ3 0 ξ, 3.33) w ξξ + ψ ξξ + Q w + β w ξ = ) w 0ξη + ψ 0ξη From equation 3.3) one gets a general solution as follows β w 0η. 3.34) ψ 0 ξ, η) = Aη) cosξ + φη)). 3.35) After substituting the solution into equation 3.3) then the equation becomes The general solution of equation 3.36) is with ) r, = β ± β 4Q, f η) = Q )Aη), f η) = Q ) +β β Aη) Q ) +β w 0ξξ + β w 0ξ + Q w 0 = Aη) cosξ + φη)). 3.36) w 0 ξ, η) = c η)e r ξ + c η)e r ξ + f η) cosξ + φη)), + f η) sinξ + φη)), 3.37) where r and r are either real negative or complex with negative real part, c η), c η) and φη) can be determined from initial conditions. From equations 3.35) and 3.37) one gets ψ 0ξ = Aη) sinξ + φη)), ψ 0ξη = daη) sinξ + φη)) Aη) cosξ + φη)) d φη), ψ 0ξξ = Aη) cosξ + φη)), w 0ξ = r c η)e r ξ + r c η)e r ξ f η) sinξ + φη)) + f η) cosξ + φη)), w 0η = e r ξ d c η) + e r ξ d c η) + sinξ + φη)) d f η) + cosξ + φη)) d f η)

12 Suppression of Wind-Induced Vibrations 0 + ) d φη) f η) sinξ + φη)) + f η) cosξ + φη)), w 0ξξ = r c η)e r ξ + r c η)e r ξ f η) cosξ + φη)) f η) sinξ + φη)), w 0ξη = r e r ξ d c η) + r e r ξ d c η) sinξ + φη)) d f η) + cosξ + φη)) d f η) ) d φη) f η) cosξ + φη)) + f η) sinξ + φη)). After substituting these into the equations of order Oɛ), one gets where with ψ ξξ + ψ = g ξ, η), 3.38) w ξξ + Q w + β w ξ + ψ ξξ = g ξ, η), 3.39) g ξ, η) = a 0 + a c η)e r ξ + a c η)e r ξ + η Aη) c η)e r ξ + c η)e r ξ ) cosξ + φη)) + η 4 η ) Aη) c η)e r ξ + c η)e r ξ ) cosξ + φη)) + η Aη) r c η) e r ξ + r c η) e r ξ +r + r )c η)c η)e r +r )ξ η r c η)e r ξ + r c η)e r ξ )) sin ξ + φη)) + a 3 c η)e r ξ + a 4 c η)e r ξ ) cos ξ + φη))) + a 5 c η)e r ξ + a 6 c η)e r ξ ) sin ξ + φη))) + a 7 cosξ + φη)) + a 8 sinξ + φη)) + a 9 cosξ + φη))) + a 0 sinξ + φη))) + a cos3ξ + φη))) + a sin3ξ + φη))), g ξ, η) = b r e r ξ + b r e r ξ + b 3 cosξ + φη)) + b 4 sinξ + φη)) a 0 = Q )η 4 Aη) D0, a = r η β Aη) D0 η 3 r, a = r η β Aη) D0 η 3 r,

13 Suppression of Wind-Induced Vibrations a 3 = η Aη) Q ) β r ), a 4 = η Aη) Q ) β r ), a 5 = η Aη) r Q + β r ), a 6 = η Aη) r Q + β r ), a 7 = β U c + dφη) + 3 c 3 β 4 + U c + η 3 ) + η 4D0 Q ) ) Aη) Q ) )) + 3β Aη) 3, a 8 = daη) + ) β U) + β η 3 Aη) 3 c 3 c + ) β U 4 β U c + β η Q ) ) Aη) 3, a 9 = Q ) η4 η ) Aη), a 0 = ) η 4 η β Aη), ) ) a = 3 c 3 c 4 U β U η β Q c + ) Aη) 3, D0 a = c 34 c ) 3β U β U c + 3β η Q ) ) Aη). D0 b = d c η) + β c η) ), b = d c η) + β c η) ) b 3 = df η) b 4 = d f η) f η) dφη) f η) d φη) = Q ) + β. Aη) d φη) d Aη) d φη) β f η), + β f η), From equation 3.38) one knows that the secular terms are the terms containing cosξ + φη)) and sinξ + φη)). After equating to zero the coefficient of the secular terms, one gets + η 4D 0 β U c + dφη) + η 3Q ) ) Aη) ) Q ) + 3β) c 3 + β U ) Aη) 3 = 0, 3.40) c daη) + β β U + β ) η 3 Aη) 3 c3 c + β U ) + β ) η Q ) Aη) 3 = ) 4β c U D0

14 Suppression of Wind-Induced Vibrations From equation 3.4) one obtains daη) from which the critical flow velocity follows as = β β U + β ) η 3 Aη) + 3 c3 c + β U ) + β ) η Q ) Aη) 3, 3.4) 4β c U D0 β η 3 U = + ). β Q ) + β Note that the highest flow velocity is obtained for Q =. The nontrivial critical point of equation 3.4) is A = ) 4β c U β U) + β η 3 ). 3.43) 3 c 3 D0c + βu ) + β β c η UQ ) In absence of a dynamic absorber one gets the following A 0 = ) 4β c U U). 3.44) 3 c 3 c + βu ) Comparing A from 3.43) and with A 0 from 3.44) one obtains A A 0 = + β η 3 β U) + β β c η UQ ) 3 c 3 D 0 c +β U ). 3.45) From equation 3.43) it follows readily that, for case Q and U > + β η 3 β one gets A = ) 4β c U {β U) + β η 3 }, 3 c 3 D0c + βu ) + β β c η UQ ) < ) 4β c U U), 3 c 3 c + βu ) = A ) From here one concludes that the dynamic absorber is useful to suppress the amplitude of the vibrations. A similar result holds for Q >. From equation 3.45), it can be shown that the optimal amplitude suppression, i.e. the lowest ratio of A over A 0, is obtained for Q =. In Figure 4, the amplitude ratio A A 0 as a function of wind velocity U for various values of tuning parameter Q is shown. The same specific parameter values are used as for the linear example, i.e. one sets η = 0.04, η 3 = 0.33, β =.85, c = 3, c 3 = and

15 Suppression of Wind-Induced Vibrations 3 Q = 0.7 Q =. 0.8 A A0 0.6 Q = 0.95 Q = Q = u Figure 4: Comparison of amplitude ratios A A 0 various values of Q versus wind velocity U for β = 0.08 and β = The figure shows clearly that the optimal suppression result is obtained for Q =. In that case the critical flow velocity is the highest and the vibration amplitude is the lowest. Figure 5 shows the comparison of amplitude and critical wind velocity for the seesaw oscillator with and without a dynamic absorber. One can see that the critical velocity is shifted from U = to U = 0.7. Also the amplitude of the vibrations is suppressed to a lower value for U >. Note that this critical flow velocity compares with the highest critical flow velocity, U, as obtained from the linear example. The asymptotic analysis does not reproduce the re-stabilization phenomenon as obtained from the linear analysis. This may be understood from the fact that the linear analysis does not assume ɛ to be small. The asymptotic analysis under-estimates the actual critical flow velocity. However, the amplitude occurring after the first critical flow velocity are rather small. The asymptotic analysis adequately calculate the final instability after which oscillations with larger amplitudes are found. 4 Conclusions The suppression of wind-induced vibrations of a seesaw-type oscillator by means of a dynamic absorber has been considered in this paper. The absorber is placed inside the cylinder of the seesaw oscillator. The linear analysis shows that an optimal combination of absorber tuning Q and damping coefficient β exists, such that the critical flow velocity is shifted to the highest possible value. This analysis also shows that a re-stabilization phenomenon may occur for a flow velocity above the critical flow velocity. Subsequently, a second critical flow velocity is

16 Suppression of Wind-Induced Vibrations 4 A amplitude without a dynamic absorber amplitude with a dynamic absorber u Figure 5: Comparison of amplitude and critical wind velocity for the seesaw oscillator with and without a dynamic absorber for β = 0.08 and Q =. found. The nonlinear analysis using the two time scales method, shows that the absorber suppresses the vibration amplitudes that exist above the critical flow velocity. The single) critical flow velocity obtained from the asymptotic analysis corresponds with the highest critical flow velocity obtained from the linear analysis. The asymptotic analysis neglects the first instability and does not reproduce the re-stabilization phenomenon. One concludes that a small dynamic absorber is capable of suppressing wind-induced vibrations of a seesaw-type oscillator in the following way,. The critical flow velocity is shifted to a higher value from U = to U = + β η 3 β Q ) +β ).. The amplitudes of the oscillations occurring above the critical flow velocity are reduced. References [] Blevins, R. D., Flow induced vibration nd edition), Van Nostrand Reinhold, New York, 990. [] Parkinson, G. V. & Smith, J. D., The square prism as an aeroelastic non-linear oscillator, Quart. J. Mech. and Appl. Math, Vol XVII, Pt. 964), pp

17 Suppression of Wind-Induced Vibrations 5 [3] Van der Burgh, A. H. P., Haaker, T. I. and van Oudheusden, B. W., A new aeroelastic oscillator, theory and experiments, Proceedings of the design engineering technical conferences, Vol ), part A, pp , September 7-0, Boston, Massachusetts. [4] Haaker, T. I., Quasi-steady modelling and asymptotic analysis of aeroelastic oscillators, Ph.D. Thesis, Department of Applied Analysis, Delft University of Technology, The Netherlands, 996. [5] Haaker, T. I. & Van der Burgh, A. H. P., On the dynamics of aeroelastic oscillators with one degree of freedom, SIAM Journal on Applied Mathematics Vol ), No. 4, pp [6] Haaker, T. I. & Van Oudheusden, B. W., One-degree-of-freedom rotational galloping under strong wind conditions, International Journal Non-Linear Mechanics, Vol 3, No ), pp [7] Lumbantobing, H. & Haaker, T.I., Aeroelastic oscillations of a single seesaw oscillator under strong wind conditions, Journal of Indonesian Mathematical Society, Vol.6, No.5 000), pp [8] Lumbantobing, H. & Haaker, T.I., Aeroelastic oscillations of a seesaw-type oscillator under strong wind conditions, Journal of Sound and Vibrations, Vol. 57, Issue 3 00), pp [9] Lumbantobing, H. & Haaker, T.I., On the parametric excitation of a nonlinear aeroelastic oscillator, Proceedings of ASME International Mechanical Engineering Congress & Exposition 00 IMECE00) New Orleans - Louisiana, USA), paper number IMECE [0] Matsuhisa, H., Nishihara, O., Sato, K., Otake, Y. and Yasuda, M., Design of dynamic absorber for a gondola lift, Proceedings Asia-Pacific Vibration Conference 995, pp. 5-0, Kuala Lumpur. [] Matsuhisa, H., Rongrong, G. U., Wang, Y., Nishihara, O. and Sato, S., Vibrations control of a ropeway carrier by passive dynamic vibration absorbers, JSME international journal, Series C, Vol. 38, No ), pp [] Matsuhisa, H. and Honda, Y., Vibration reduction of ropeway carriers, ships and cable suspension bridge, preprint 003, correspondence address: matsu@prec.kyoto-u.ac.jp [3] Van Oudheusden, B. W., On the quasi-steady analysis of one-degree-of-freedom galloping with combined translational and rotational effects, Nonlinear Dynamics 8 995), pp [4] Verhulst, F., Nonlinear differential equations and dynamical systems nd edition). Springer-Verlag, Berlin Heidelberg, 996.

18 Suppression of Wind-Induced Vibrations 6 [5] Murdock, J. A., Perturbations : Theory and Methods, Society for Industrial & Applied Mathematics, 995. [6] Tondl, A., Kotek, V. and Kratochvil, C. Vibration of pendulum type systems by means of absorbers, st edition, CERM akademické nakladatelstvi, s.r.o., Brno, Czech Republic, 00. [7] Tondl, A., Quenching of self-excited vibrations, Elsevier science publishers, Amsterdam, 99. [8] Nayfeh, A. H., Mook, D. T., Nonlinear Oscillations, John Wiley & Sons, Inc., New York, 995. [9] Goldstein, H., Classical mechanics, nd edition, Addison - Wesley publishing company, Inc., 980. [0] Nigol, O. & Buchan, P. G., Conductor galloping part I - Den Hartog mechanism, IEEE Trans. Pow. App. Sys, Vol. PAS-00, No. 98), pp [] Online help.