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4e Conférence spécialisée en génie des structures de la Société canadienne de génie civil 4th Strutural Specialty Conference of the Canadian Society for Civil Engineering Montréal, Québec, Canada 5-8 juin 2002 / June 5-8, 2002 THE DYNAMIC PROPERTIES OF A TUNED LIQUID DAMPER USING AN EQUIVALENT AMPLITUDE DEPENDENT TUNED MASS DAMPER M.J. TaitA, A.A. El DamattyB and N. IsyumovC A Graduate Student, Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada B Assistant Professor, Department of Civil and Environmental Engineering, The University of Western Ontario, London, Ontario, Canada C Research Director and Professor, The Boundary Layer Wind Tunnel Laboratory, The University of Western Ontario, London, Ontario, Canada ABSTRACT: In this paper, shaking table tests are conducted to determine the dynamic characteristics of a TLD at various excitation amplitudes. The results are used to describe the TLD as an equivalent amplitude dependent tuned mass damper (TMD). This model is obtained by maintaining energy dissipation equivalence. The amplitude dependent TMD model allows the use of existing mathematical tools to describe the response of a structure-TLD system. As the effects of a TMD on a structure are well understood, the developed model also provides insight into the performance of a TLD. Forced vibration tests are conducted on a structure-TLD system to verify the numerical model. Results indicate that an amplitude dependent TMD model can correctly simulate the response of a structure-TLD system. 1. INTRODUCTION An economical solution to reduce building motions to suitable levels, under dynamic loading, is to provide additional damping. A passive damping device shown to be effective is the tuned liquid damper (TLD). A TLD consists of a rigid tank, partially filled with a liquid, usually water. When a structure, fitted with a properly tuned TLD begins to sway during a dynamic loading event, it causes fluid sloshing motion inside the tank. The fluid sloshing motion imparts inertia forces approximately anti-phase to the dynamic forces exciting the structure, thereby reducing structural motion. The inherent damping mechanism of the TLD dissipates the energy of the fluid sloshing motion. A number of tall structures have been successfully fitted with TLD devices, resulting in a substantial reduction in structural motion (Noji et al. 1988, Fujii et al. 1988 and Wakahara et al. 1992). The TLD has many advantages over other passive damping devices, which include: operating under both small (wind) and large (earthquake) vibrations, extremely low probability of failure, easy to tune and inexpensive to install and maintain. However, there are some drawbacks: not all the water participates in reducing the structural motion, and the low density of water requires somewhat more of the valuable building space to be given up. A TLD operates analogously to that of the well-known tuned mass damper (TMD), however, its dynamic characteristics are non-linear. Although the construction of a TLD is simple i.e., a partially fluid filled tank, the fluid sloshing motion inside the tank is complex, especially with the introduction of baffles or screens. Therefore, its influence on the response of a structure subjected to a 1 dynamic loading is not as well understood as a TMD. Replacing the TLD with an equivalent mechanical model is an attractive proposition as it allows the use of existing TMD design concepts to be used in the initial design of a TLD. This paper focuses on replacing the TLD with an equivalent, amplitude dependent, mechanical model having the same effect on the primary structure as the TLD. This allows the use of simple iterative schemes or well-known time stepping routines to be used in the analysis of a TLD-structure system. Graham and Rodriguez (1959) developed a mechanical model, using an equivalent spring and mass, that simulates the forces that develop by sloshing fluid motion inside a tank. The model is based on the assumption of potential fluid flow, with linearized boundary conditions. This limits the model to small excitation accelerations and ignores the energy dissipated by the sloshing fluid, which is in fact an important parameter. This equivalent model has been used to represent a TLD attached to a structure (Kareem 1987), however this linear model, with an assumed inherent damping value, is unable to capture the non-linear, amplitude dependent, properties of the TLD. More recent semi-empirical models of non- linear TLD behaviour were suggested by Chaiseri (1990) and Sun et al. (1995). The concept of virtual mass and damping were used to determine an equivalent TMD, by matching both of these properties simultaneously with experimentally estimated values, which were obtained by performing shaking table tests, at several amplitudes, on a number of different TLDs. Yu et al. (1999) estimated equivalent TMD properties for a TLD, subjected to large excitation amplitudes, based on the concept of equivalent energy dissipation. Again the energy dissipated by various TLDs were determined by shaking table tests. However, this model assumes full participation of the fluid mass, which is valid under large excitation amplitudes or extremely shallow water depths. In all the above studies, no additional energy dissipating mechanisms such as damping screens were present in the TLD. This semi-empirical model proposed in the current study uses the concept of equivalent energy dissipation, as suggested by Yu et al. (1999), but does not assume that the entire fluid mass participates. The equivalent mechanical model, which has an amplitude dependent mass, damping, and stiffness, is determined by a non-linear least squares fitting routine using data obtained from shaking table tests. Values obtained for the equivalent TMD are then compared to those obtained from ideal potential flow, assuming linearized boundary conditions. Performance of the equivalent mechanical model is then evaluated by comparing experimental test results conducted on a TLD-structure system to those predicted by the proposed model. 2. IDENTIFICATION OF ESSENTIAL TLD PARAMETERS A structure-TLD system with the TLD tuned to a particular mode of the structure is shown in Figure 1a. Here the primary structure is characterized by its generalized mass M*, generalized damping C* and generalized stiffness K*. An auxiliary amplitude dependent spring-mass-damper can be used to represent the TLD as illustrated in Figure 1b. The amplitude dependent mass, damping and stiffness of the amplitude dependent TMD are given by mtld(X1), ctld(X1) and ktld(X1), respectively. The additional mass ma, rigidly attached to the primary structure, represents the portion of the fluid inside the tank that does not participate in the sloshing motion of the water. 2 X1 Equivalent Amplitude Dependent TMD K* K* cTLD(X1) M* M* ζs ζs mo mTLD(X1) kTLD(X1) (a) (b) Figure 1. (a) Structure-TLD System (b) Amplitude Dependent TMD Analogy 2.1 TLD Properties Figure 2 shows the dimensions of the tank that was tested in the current experimental study. In this figure, the dimensions L, b, and h represent the tank length in the direction of excitation, the tank width (perpendicular to excitation) and the water depth, respectively. The fundamental sloshing frequency, fw for the water inside this tank can be estimated using linear wave theory according to the following equation: [1] 1 πg πh fw = tanh( ) 2π L L where g is the acceleration of gravity. Substituting L and h into the above equation leads to a value of fw ≈ 0.545 Hz. Direction of Excitation h = 119 L = 966 b = 360 Figure 2. TLD tank Dimensions (given in millimetres) This TLD represents a 1:10 model of one of the prototype multiple tanks projected for a real building having a fundamental frequency of approximately 0.172 Hz. According to dimensional analysis theory, the fundamental fluid sloshing frequency of the prototype tank is estimated to be equal to fw/101/2 = 0.172 Hz. This means that the sloshing frequency of the prototype tank was tuned to the fundamental frequency of the building. The tuning ratio Ω, is an important parameter which strongly influences the performance of a TLD, and is given by 3 fw [2] Ω= fs where fs is the natural frequency of the structure. A second significant parameter that affects the response of the structure-TLD system is the mass ratio, µ which is defined by m TLD [3] µ= M* M* is the generalized mass of the structure in the mode of vibration, which is to be damped and mTLD is the effective mass of the fluid given by the following relationship [4] m TLD = αm w where α represents the percentage of the fluid mass contributing to the fundamental sloshing mode and mW is the total mass of the fluid inside the tank. The value of, mw can be varied by changing either the width of the tank, b, or the number of tanks attached to the building. For this particular project a target value of µ ≈ 1.70 % was chosen A third key parameter that influences the performance of a structure-TLD system is the inherent damping ratio, ζTLD of the sloshing motion inside the tank. Warburton (1982) has established an expression for determining optimum inherent damping for a TMD as a function of the mass ratio, µ. Due to the analogy existing between both the TMD and TLD devices, Warburton’s approach was used, and an initial target value of ζTLD ≈ 6.5 % was estimated from the mass ratio used in this study. The main source of damping of the sloshing motion within the tank without any additional damping devices arises from viscous dissipation in the boundary layers at the solid boundaries of the tank, and from free surface contamination. Assuming linear wave theory, the inherent damping ratio of sloshing liquid inside a rectangular tank without additional devices has been estimated by Sun (1991) to be 1 ν h [5] ζ TLD = (1 + ) 2h π fw b where ν = liquid kinematic viscosity. The estimated inherent damping for the TLD being tested was calculated to be approximately 0.45%, which is significantly less than the estimated optimal value. Equation 5 suggests that either increasing the fluid viscosity or decreasing the fluid depth would increase the damping. Neither of these is practical. Often the TLD is used as a water storage tank, preventing the use of a higher viscosity liquid in order to increase the inherent damping. Alternatively, using extremely shallow water depths requires the use of multiple tanks in order to achieve the desired mass ratio, resulting in larger space requirements. The flow-dampening devices chosen to increase the inherent damping ratio are shown in Figure 3. Energy dissipation occurs due to the losses in the flow through the slats, which can be easily installed. Altering the solidity ratio by adjusting the space between the slats, allows the inherent damping to be changed. The slats used in the experiments were approximately 5mm thick, spaced 7mm apart, resulting in a solidity ratio of 0.42. The slats were placed to a height of 0.75 h, which exceeded the free surface level at the slat locations for all tests conducted. The screen location was chosen based on previous work by Fediw (1992). The slats were placed inside the tank at 0.4L and 0.6L, respectively. 4 Figure 3. Photo of Internal Damping Screens 2.1 Shaking Table Experiments The dynamic properties of a TLD depend on both excitation amplitude, A and excitation frequency, fe. The objective of this component of the study was to determine the energy dissipated per cycle by the sloshing fluid, under several excitation amplitudes. Wave Probes F1(t)=Fc(t)+Fmw(t)+Fsw(t) Mo F2(t)=Fc(t)+Fmw(t) Shaking Table Motion Figure 4. Schematic of Shaking Table Set-Up A schematic of the experimental set-up is shown in Figure 4. The set-up included a rigid support frame attached to a shake table. As shown in the figure, the TLD was attached to the top part of the frame using four cables providing only vertical support for the tank. The lateral support of the tank was provided by two load cells that connected the bottom of the tank to the rigid intermediate member of the frame. In the bottom section, a similar attachment was done for a ballast mass, MO equal to the total TLD mass, including the total fluid mass, mW and the tank mass. The data recorded included the shaking table displacement and acceleration. In addition the base shear forces were measured using the load cells attached to both the TLD and ballast mass. The temporal free surface was measured at six locations simultaneously along the tank using capacitance type wave probes. The base shear F1 measured by the load cells connecting the tank to the rigid frame can be separated into three components. The inertial force due to the container, Fc the inertial force due to the mass of the contained fluid, Fmw and the fluid sloshing forces Fs. Meanwhile, the base shear force F2, measured by the load cells connecting the ballast mass to the frame represents the summation of Fc and Fmw. The liquid sloshing forces, Fs(t) can be evaluated by subtracting the outputs of the two base shear forces, i.e. 5 [6] Fs ( t ) = F1 ( t ) − F2 ( t ) The frame assembly was subjected to harmonic excitations with the excitation amplitude, A ranging from 2.5 mm to 12.5 mm. The forcing frequency ratio β, is defined as the ratio of the excitation frequency, fe and the fundamental sloshing frequency fw, fe [7] β= fw The tests were conducted by varying the ratio β in the approximate range of 0.65 < β < 1.50, i.e. 0.35 Hz < fe < 0.80 Hz. The water was quiescent at the start of all tests conducted, and the data were recorded after the fluid sloshing motion reached a steady state condition. 2.2 Measured Energy Dissipation of Sloshing Fluid An estimate of the energy dissipated per cycle by the TLD, Ew, can be obtained by the area enclosed by the force-displacement loop, given by [8] E w = ∫ Fs dx T where T is the period of the shake table motion and x is the shaking table displacement. A non- dimensional parameter Ew’ is evaluated using the following relation Ew [9] E 'w = 1 m w (ωA ) 2 2 where mw is the mass of the fluid, ω is the excitation frequency (rad/sec), A is the maximum amplitude of the shake table displacement and the term ½ mw(ωA)2 represents the maximum kinetic energy of the contained fluid when treated as a solid. The variation of Ew’ with the excitation frequency ratio β is provided in Figure 5 for the 5.0 mm test. 2.3 Energy Dissipation Matching Scheme The expression for the non-dimensional energy dissipation by an equivalent SDOF, Ed' representing a TMD subjected to harmonic base excitation can be expressed as 2 ω [10] E = α H(ω) ' d ω 2π sin φ TLD where the mechanical admittance function |H(ω)| is given by 1 [11] H(ω) = 2 ω 2 2 1 − + 2ζ ω TLD ω TLD ω TLD 6 and the phase angle by φ by 2ζ ω TLD ω TLD [12] φ = a tan 2 ω 1− ω TLD Non-Dimensional Dissipated Energy 50 Equivalent Model 40 Experimental Value 30 20 10 0 0.4 0.6 0.8 1 1.2 1.4 1.6 β Figure 5. Non-dimensional energy dissipation curves from shaking table tests and equivalent model Using a weighted non-linear least squares fitting routine (Matlab 1999), Ed’ was fit to Ew’ in order to determine the equivalent TMD parameters mTLD, fTLD and ζTLD for each amplitude of excitation. Figure 5 compares the experimentally obtained energy dissipation curve with the fitted equivalent TMD curve for the 5.0 mm excitation amplitude. Good agreement is found between the two curves for the all cases, indicating that the energy dissipation characteristics can be matched with an equivalent SDOF system. This indicates that even at higher amplitudes of excitation, a SDOF still models the energy dissipation characteristics of a TLD well. 3.0 EQUIVALENT AMPLITUDE DEPENDENT TMD The effect of amplitude on the mass, damping and natural frequency parameters of the TLD are shown in Figure 6. The general trend indicates that the mass ratio, damping ratio and frequency ratio all increase with increased amplitude, which is in agreement with findings by Sun et al. (1995). The equivalent amplitude dependent parameters were estimated by fitting functions to the experimental results shown in Figure 6. The equivalent viscous damping ratio was found to increase linearly on the excitation amplitude, which indicates it is a function of velocity squared, i.e., ζTLD ∝ A/L. This is expected as the pressure loss across the screens and hence the energy dissipated is related non-linearly to the water velocity. The increase in frequency ratio with excitation amplitude was found to be significantly less 7 than in studies conducted on TLDs without damping screens. This suggests that the screens reduce the “hardening-spring type” behaviour of sloshing fluid. This is beneficial as the tuning ratio parameter, Ω will be remain closer to the optimal value over a larger excitation amplitude range. 1.25 15 1.060 1 12.5 1.045 0.75 10 1.030 (%) /m /f 0.5 7.5 1.015 f m 0.25 Experimental Data 5 1.000 Equivalent Model 0 2.5 0.985 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 A/L A/L A/L (a) (b) (c) Figure 6. Equivalent TMD properties (a) mass ratio; (b) damping ratio (c) frequency ratio The mass ratio was found to increase from an initial value of 0.75 mw to approximately 0.82 mw. An estimate of the effective mass, mTLD, can be obtained from potential flow theory using the following equation (Graham and Rodriquez 1952) h 8 tanh(( 2n − 1)π ) mn = L m [13] w 3 3 h π (2n − 1) ( ) L Substituting the values of the tank length, L and water depth, h into equation 13 an effective mass value of 0.77 was calculated. This is in good agreement with the value of 0.75 obtained experimentally for the smallest base excitation amplitude of 2.5 mm shown in Figure 6(a). This implies that under small excitation amplitudes the effective mass from potential flow provides a good estimate for the participating mass. 4.0 MODEL VERIFICATION In order to verify the above amplitude dependent TMD structure-TLD, forced vibration experiments were conducted. Figure 7 shows a schematic of the experimental set-up. The properties of the structure were set in order to obtain a mass ratio µ ≈ 1.70 % and a tuning ratio Ω ≈ 1.0, matching that of a real structure- TLD system. The system was excited using a stepped-sine frequency sweep, ranging from approximately 0.65 fs < fe < 1.25 fs. The excitation amplitude was also varied in order to evaluated the performance of the equivalent amplitude dependent TMD model. 8 5.30 m Sinusoidal Excitation Generalized Stiffness Load of Structure - K * Cell Load Cell TLD Generalized Mass of Structure - M * Figure 7. Schematic of Two Degree of Freedom Structure-TLD System Results from the equivalent amplitude dependent model are compared with experimentally obtained frequency response values in Figure 8. It can be seen that good agreement was found between the model and the experimental results. The model accurately predicts the excitation frequency corresponding to the maximum response value and also provides a good estimate of the maximum response value. If a linear TMD model had been used to represent the TLD, the model response curve would be amplitude independent. However, from the experimental results it can be seen that ignoring the non-linear TLD properties would lead to erroneous response predictions of the structure-TLD system subjected to various excitation amplitudes. It is evident that the non-linear TLD properties can not be ignored. By using an equivalent amplitude dependent TMD an accurate estimate of the structure-TLD system can be made. 40 Po = 17.5 N Experimental 35 Po = 17.5 N Model Po = 10.0 N Experimental 30 Po = 10.0 N Model 25 /U 20 15 U 10 5 0 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Excitation Frequency (Hz) Figure 8. Comparison of frequency response curves obtained from experiments and the proposed amplitude dependent TMD model. 9 5.0 CONCLUSIONS An equivalent amplitude dependent TMD model has been used to represent a TLD. The proposed model is based on equivalent energy dissipation. The model provides both qualitative and quantitative information regarding the dynamic properties of a TLD. The equivalent amplitude dependent model reveals the following: • The mass ratio at low excitation amplitudes is approximately equal to that found using potential flow theory. An increase in excitation amplitude is associated with an increase in the fluid mass ratio participating in the sloshing motion • Both the sloshing frequency and the inherent damping of the TLD increase as the excitation amplitude is increased. • An equivalent amplitude dependent TMD can be used to accurately model a TLD by matching the energy dissipated by the device. • Modeling the amplitude dependent parameters is essentially in order to obtain the correct response of a structure-TLD system. The experimental cases presented in this study are limited and further experimental data and theoretical studies are needed to generalize the above-observed trends. 6.0 REFERENCES Chaiseri P. (1990), ‘Characteristics, Modelling and Application of the Tuned Liquid Damper’, PhD Thesis, University of Tokyo, Tokyo, Japan. 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