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Simulating a conical tuned liquid damper
Simulation Modelling Practice and Theory 11 (2003) 353–370 www.elsevier.com/locate/simpat
Simulating a conical tuned liquid damper Fabio Casciati a
a,*
, Alessandro De Stefano b, Emiliano Matta
b
Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy b Department of Structural Engineering and Geotechnics, Polytechnic of Turin, Corso Duca degli Abruzzi 34, 10100 Torino, Italy Received 14 May 2002; received in revised form 7 April 2003; accepted 12 April 2003
Abstract In recent years, tuned liquid dampers (TLD) have proved a successful control strategy for reducing structural vibrations. The present study focuses on the frustum-conical TLD as an alternative to the traditional cylindrical tank. If compared to the cylindrical reservoir, the cone-shaped TLD allows calibrating its natural frequency through varying liquid depth, which makes it suitable for a semi-active implementation, and attains the same level of performance with a fewer mass, at least for small fluid oscillations. A linear model is presented which can interpret TLDÕs behaviour for small excitations. For larger amplitudes, strong nonlinearities occur and the linear model is no longer predictive. Consequently, for a frustum-cone TLD subjected to harmonic excitations, a tuned mass damper (TMD) analogy is established where TMD parameters vary with the excitation amplitude. 2003 Elsevier B.V. All rights reserved. Keywords: Conical tank; Energy dissipation; Nonlinearities; Passive control; Tuned liquid damper
1. Introduction TLDs have been mainly applied in Japan. The TLD devices installed in the 77.6 m high structure of the Tokyo Airport Tower (1993), for instance, consist of 1400 tanks filled with water and floating particles. The containers, low cylinders of diameters 0.6 and 0.125 m, are stored in six layers on steel consoles. The total mass of the TLDs is approximately 3.5% of the first modal mass of the tower and its frequency is optimised to 0.74 Hz. Floating polyethylene particles were added in order to enhance *
Corresponding author. E-mail addresses: [email protected] (F. Casciati), [email protected] (A. De Stefano), [email protected] (E. Matta). 1569-190X/03/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1569-190X(03)00051-0
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energy dissipation by their collision. The behaviour of the TLD has been observed under various wind phenomena. In one of these cases with a maximum wind speed of 25 m/s, the recorded data showed that TLDs reduced the acceleration to approximately 60% of its value without control. Dampers using liquid motion have firstly been used in space satellites and marine vessels. Subsequently, some researchers promoted their application to structures on the ground, including buildings and towers. After that, TLDs have proven a successful passive vibration mitigation system [7]. In the meantime, a number of theoretical investigations have flourished aiming at modelling up TLDsÕ physical behaviour. Modi and Welt [10] conducted an experimental and analytical study on a nutation damper (annular tank), which is conceptually the same as TLD. Their theoretical investigation concerned the energy-dissipation mechanism assuming a potential flow with a nonlinear free-surface condition in conjunction with boundary-layer correction [10]. Fujino et al. developed a two-dimensional model of liquid motion in a rectangular TLD using the shallow-water wave theory, i.e. deriving the nonlinear solving system from continuity and equilibrium equations written for an incompressible and irrotational fluid [5]. The damping effects due to solid boundary friction and free-surface contamination were included by adopting the linear theory of the boundary layer (Lamb [8]) with the corrections suggested by Vandorn (1966) and Miles (1967) [8]. Wave breaking was not intended to be modelled. The whole of the experiments revealed strong nonlinearities, the effects of higher harmonics of liquid sloshing around the resonance, a hardening-spring type behaviour and a sudden jump-down in the frequency response soon after resonance. In case of wave breaking (i.e. for amplitude excitations greater than 0.5 cm), the simulation overestimated both the reaction force and the surface elevation near the end wall, while underestimating the energy dissipation (since the wave breaking contribution was not modelled). Good agreements between simulation and experimental results were observed provided that no wave breaking occurred in the experiments. Sun et al. used the same procedure to address the case of a rectangular TLD subjected to pitching vibration [12]. The nonlinear analytical model, based on shallow water wave theory while including the linear damping of liquid sloshing, agreed well with the experimental results. The overall conclusions confirmed those obtained for the mere horizontal motion. A different perspective was adopted in an other study by Sun et al. [13]. Rectangular, circular and annular tanks, subjected to harmonic base excitation, were measured experimentally. Using a SDOF TMD analogy, equivalent mass, stiffness and damping of the TLD were calibrated from the experimental results as functions of the tank base amplitude. In particular, it was found that, as the excitation amplitude increases, the natural frequency increases, the effective mass of the TLD increases too and approaches the total mass of the fluid, the damping ratio increases and exceeds 10% when wave breaking occurs. Reed et al. used the random-choice numerical method to solve the nonlinear shallow-water equations for a rectangular TLD [11]. The model captures the underlying
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physical phenomenon adequately, including wave breaking, for most of the frequency range of interest and over a wide range of amplitude excitation. It was found that the response frequency of TLD increases as excitation amplitude increases and the system is of the hardening-spring type. The aforesaid nonlinear models often show good agreement with the experimental results, in some cases even strong nonlinearities and wave breaking can be predicted. However, more predictive the simulation provided by complex nonlinear theories, more difficult to look for the optimal location of a multiple set of TLD devices on a real building. Moreover, apart from the TMD analogy, where nonlinearities conventionally reduce to changing the parameters of the oscillator with the amplitude of the excitation, nonlinear models based on shallow-water wave theory were conceived only for rectangular tanks, and the whole of different shapes (including cylindrical and conical TLDs, for instance) are still lacking in a proper theoretical model. In what follows, a cone-shaped TLD is proposed, owing to a number of expected advantages it may provide with respect to the cylindrical container. Among the most relevant ones: • a bigger effective mass (at least for deep waters or for small excitations) and a larger free-surface area; • a stronger variability of the fundamental frequency with the water depth, beyond a certain value of the liquid height; • suitability to innovative solutions for tank building technologies, such as deformable membrane-made conical containers which could be extended along the vertical direction providing a semi-actively changing geometry. Furthermore, conical reservoirs are already present in many civil buildings as a water reserve, and it would be interesting to investigate their suitability to turn into passive dynamic absorbers, though TLDs consist traditionally of shallow water tanks. A nonlinear shallow water wave theory would be hardly applicable to the case, because of the complex geometry of the tank. Consequently, a nonlinear (both frequency- and amplitude-dependent) SDOF model is proposed to simulate the dynamic behaviour of the system. As will be illustrated in the following pages, for limited excitations, a simplified linear theoretical approach for the conical and frustum-conical TLD, similar in principle to the classical HousnerÕs approach for rectangular and cylindrical tanks, has proved successful in predicting the frequency of an equivalent linear TMD. For greater base amplitudes, instead, the linear model is proven unsatisfactory, since the parameters of the oscillator are no longer constant, and the effective mass, the natural frequency and the damping coefficient of the equivalent TMD increase as the amplitude increases [3]. This makes it necessary to calibrate the oscillator properties on the experimental results, and eventually turns the TLD into a TMD whose parameters depend not only on the geometry in an analytical way (as it would be as far as the linear theory were concerned) but also on the amplitude of excitation in an experimental way (which is far less general). Yet, it will be demonstrated that the advantages of the linear theory may be partially retained even for large amplitudes if
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the effects of nonlinearity are conventionally condensed in the damping coefficient alone. This is quite confirmed by experimentation, since for large excitations the effective mass approaches the total mass and the natural frequency may also be slightly misunderstood without diminishing the control performance, thanks to a robust behaviour of the TLD in a large range of frequencies around the resonance. The investigated modelling strategy is specifically intended to provide a simple description of a peculiar structural control device, the frustum-conical tuned liquid damper, such that its basic properties could be understood for the sake of an optimal control design. Nevertheless, the physical phenomenon under scrutiny, i.e. liquid sloshing in a reservoir, is a much more general problem, and it is expected that the proposed method could be exported to other applications, for instance to the modelling of the response of any liquid containers subjected to a dynamic excitation. 2. Typology of tuned liquid damper TLD systems can be divided into two categories. The first category is represented by the sloshing damper (sloshing is the phenomenon of wave breaking on the surface of the liquid against the walls of the tank), or said, TLD in the proper sense. The dimensions of the tank or the depth of the liquid regulates the period of vibration. Damping ability is increased introducing nets or rods into the liquid. The second type is represented by the tuned liquid column damper (TLCD). Here the shape of the columns or the pressure of the air inside them regulate the period of oscillation. The damping ability is increased by regulating the opening of an orifice in the column, which generates high turbulence. Our attention, here, is devoted only to sloshing TLD. TLD, compared to tuned mass damper (TMD), has the following favourable properties: • the absence of mechanical friction provides a negligible trigger level; • the absence of mobile mechanical components reduces maintenance costs and increases reliability; • TLD can operate in all the possible directions in the horizontal plane; • the liquid mass can be shared between several tanks, which are then made small and easily transportable; • if the tank has the horizontal section varying along the vertical axis, as it happens for conical reservoirs, it is easy to tune the frequency by simply changing the contained liquid volume; • for structures having different vibration frequencies in the two principal directions, it is possible to use containers with double axial symmetry, so that a double tuning is attained. On the other hand, an unfavourable property of the TLD is the small mass density of water, which requires larger devices compared with a metallic TMD. However such disadvantage is compensated by the fact that it is easy, technically effective and low cost demanding to locate in any available space several TLD devices, working in parallel.
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An additional drawback is represented by TLDÕs nonlinear behaviour, which makes it difficult to work out a predictive simulation approach which could be easily handled in practical design.
3. Linearised analysis of the tuned liquid damper The hydrodynamic theory of the TLD is somewhat complex and difficult to handle, above all because of the intrinsic nonlinearity in the physical model. However, the nonlinearity can be neglected in the dominion of small excitations, and even for large excitations a linear model may prove useful as far as it is able to approximate such a paramount parameter as the natural frequency. The linear model Housner proposed in the first Ô60s aims at simulating––essentially for cylindrical and rectangular tanks––water response on tank walls with two lumped masses: a fixed one (representing the liquid that is not moved by the acting acceleration), rigidly connected to the tank, and one moving (determining the sloshing effect which is responsible for the energy absorption), connected to the tank by a spring and a damper [6]. More recently the simplified model of Housner has been adapted to conical tanks with the original purpose, shared in fact with HousnerÕs approach, of simulating the response of large elevated water tanks to seismic action. As for their use as a control strategy, conical or frustum-conical TLDs appear particularly effective as a passive control system, owing to the little requirements of mass, and very suitable for transformation into an adaptive control system, owing to the variability of frequency with water height.
y A
λ λ’
G H
A’
um xP’
h0
α
h2
B’ P’ B
r r
∆h
φ 250
θ
G’
yH
ϕ
α
y
yP x0
OC C’
k/2
k/2 m1
b’b
m0
h m1
A
A’
O≡G
h m0
β
D
(a)
G’
r r
B
B’
P’
D’
(b)
Fig. 1. Conical tanks: scheme of the liquid in fixed mass and moving mass (a) and scheme of the liquid motion (b).
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Let us now summarise the linearised formulation for a conical tank, in order to calculate the fundamental frequency and to estimate the parameters of fixed and moving masses (Fig. 1). Such parameters are the amount of the masses, their position and the stiffness of springs connecting the mobile mass to the tank. The formulas shown in this paragraph allow the direct calculation of the two masses (indicated as m0 and m1 ), of their heights (hm0 and hm1 respectively), of the stiffness k of the spring associated to the moving mass and consequently of the fundamental frequency x. It has already been stressed out that the mobile and fixed masses have a specific physical meaning. In particular, the mobile mass simulates the sloshing effect. Its evaluation is obtained by setting the continuity and equilibrium equations with the simplified assumption that the fluid in rest is divided in layers by plane parallel surfaces, remaining plane, even if no more parallel, during motion. Governing dynamic equilibrium condition is deduced by applying the Hamilton principle. In the modified approach for conical tanks, the centroid of plane surfaces containing fluid layers do not remain on the same vertical axis during motion, as in HousnerÕs model, but displaces laterally. 16 p 2 4 m1 ¼ qIzh2 x2 Að4Þ ; Izh2 ¼ ðh2 tgaÞ ; 81g 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 8 ; k ¼ m1 x2 ; x¼ 2 h2 tg a 27 ðF1 C1 þ F2 C2 Þ þ 1 AðiÞ ¼
9 þ 2F1 9 þ 2F2 1 lðF1 þiÞ lðF1 F2 Þ 1 lðF2 þiÞ ; i þ F1 i þ F2
F1=2 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 1 1 þ 2:16k1 ; 2
C1 ¼ 1
1 ðF1 F2 Þ ; h1 h2
C2 ¼
k1 ¼
h1 h2
ð1Þ
l¼
h1 ; h2
ð2Þ
1 tg2 a ; 4tg2 a
ðF1 F2 Þ
1
1 ðF1 F2 Þ
ð3Þ
h1 h2
The foregoing linearised theory has already experimentally proved successful in evaluating the free-decay natural frequency for a variety of deep water heights in conical tanks but there is still need to define its applicability to shallow water conical TLD.
4. Laboratory analyses The equipment used for the experimental research is the one installed in the Department of Structural Mechanics of the University of Pavia. It includes a shaking table vibrating with a single motion axis, controlled in closed loop by means of a software, with which it is possible to create an adequate drive signal for the table in order to produce the acceleration in the desired form. A hydraulic device, which
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is set into action by a servo-valve, produces the movement. The actuator receives as input an electrical signal and transforms it into a displacement. The experimental research followed two main routes: first, a comparison between the performances of cylindrical and conical TLDs as passive control systems applied on a single-degree-of-freedom test frame, subjected to harmonic, white noise and seismic base excitations; then, the response to harmonic excitations of the frustum TLD alone, on a load cell. In the first part of the experiment, TLDs were located on a SDOF steel frame, mounted on the shaking table. Both cylindrical and conical tanks have been tested. Cylindrical tanks had diameters of 10, 20 and 30 cm. Their fundamental frequency was not so much influenced by the water height (respectively 3, 2.1 Hz and 1.6–1.7 Hz for water height between 10 and 30 cm). The conical tank was shaped as a frustum of a cone with base diameter 20 cm and an inclined lateral wall with a slope of 60 to the horizontal, whose frequency––with the same range of variation of water height––varies from 1.1 to 1.4 Hz. Some of the tests were carried on with a single tank (which controls a single frequency) and others with several tanks in order to see the effect of concomitant tanks. As it was expected, the tests showed that a tank which is not tuned to the main structure works well only for signals that excite its natural frequency, not bringing any benefit to the resonance frequency of the whole system. For tuned tanks, instead, only a small amount of water proved necessary to obtain satisfactory results in terms of response attenuation; reductions of the transfer function up to 30–40% have been found and reductions of more than 50% of time domain outputs under white noise excitations have been obtained. The controlling liquid mass needed to reach such good results has been indeed small (0.64% of the total mass of the structure for the frustum TLD, corresponding to a water height of 3 cm, in tuning conditions). Experiments using sinusoidal and white noise input signals showed the achievement of decisive improvements of the resonant response of the frame (Figs. 2
Fig. 2. Comparison between control and no control case using a conical tank exciting the structure with a white noise.
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Fig. 3. Comparison between control and no control case exciting the structure with a sine wave (f ¼ 1:27 Hz; A ¼ 0:7 mm).
and 3). Large reductions of accelerations have been observed for earthquake excitations with tuned tanks, demonstrating the applicability of this control method also for such cases, except for the first seconds of sharply impulsive earthquake excitations, with a really steep gradient of acceleration. In this case, as it has been seen for the earthquake of Bagnoli–Irpinia (Fig. 4), for some seconds (of the order of 2–3 s) no difference was observed between controlled and uncontrolled structure. After these few seconds, however, the dynamic damped response amplitude fell down to a 50% (in case of the truncated cone with a water height of 3 cm) of the value observed in the uncontrolled case. For earthquakes with less impulsive character, such
Fig. 4. Comparison between control and no control case using a conical tank exciting the structure with Bagnoli Earthquake record.
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Fig. 5. Comparison between control and no control case using a conical tank exciting the structure with El Centro Earthquake record.
as El Centro (Fig. 5), these negative effects were substantially lower. These results were confirmed by a genetic algorithms optimisation study carried out, in parallel to the experimental investigations, with the purpose of searching for the best configuration of multiple dynamic absorbers as a control strategy for seismic induced vibration mitigation. The numerical procedure too demonstrates that the seismic response to strongly impulsive signals can not be significantly reduced through passive oscillators, so that the ineffectiveness of passive TLDs to address impulsive seismic actions can be regarded as a general result [9]. Comparable results have been obtained with a tuned cylindrical tank, with relevant reductions for all the categories of signals, but in that case a liquid mass of about 1.6% of the total mass of the structure was needed, instead of the only 0.64% used inside the conical tank. In the second part of the experimental investigation, the same frustum-cone was located on a load cell, which had been designed and built up for the purpose, and the load cell on the shaking table. The dynamic uncoupling between the TLD and the load cell made it possible to isolate the TLD behaviour, which instead had been previously indirectly analysed through the effects it induced on a tuned test frame. The load cell is meant to reproduce a unidirectional SDOF shear-type system, in fact consisting of two horizontal steel plates connected by four thin rectangular-cross-section columns. When a force is applied on the upper plate, the lower one being rigidly restrained, the relative displacement of the two plates induces a proportional deformation on the four columns, which can be picked up by four Wheatstone half bridges. Through a una tantum calibration procedure, the mean value from the four measurements therefore provides the applied force. For different liquid volumes (liquid depth h equal to 3, 4.5 and 6 cm) and for various amplitudes of harmonic excitations (1.25, 2.5, 5, 10 and 15 mm), the frequency
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response functions were experimentally obtained for the normalised force F 0 ¼ F =ðmAx2 Þ, where F is the force read by the load cell (therefore comprised the inertial terms due to the tank and load cell masses), m is the liquid mass, Ax2 is the amplitude of acceleration (Figs. 6–8). Whatever the liquid depth, the fundamental result was the strong nonlinear response of the frustum TLD near resonance, which is in perfect agreement with the behaviour of traditionally shaped containers.
Fig. 6. Conical TLD with 3 cm of water: experimental transfer functions for the normalised force measured at the load cell.
Fig. 7. Conical TLD with 4.5 cm of water: experimental transfer functions for the normalised force measured at the load cell.
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FORCE TRANSFER FUNCTION FOR h = 6 cm 30
A = 1.25 mm A = 2.5 mm A = 5 mm
25
2
F /(mA ω )
20
15
10
5
0 0.5
0.7
0.9
1.1
1.3
1.5
ω exc / ω 0
Fig. 8. Conical TLD with 6 cm of water: experimental transfer functions for the normalised force measured at the load cell.
The normalised frequency response function grows flat as the base excitation increases, thus showing a favourable robustness, and at the same time its peak value decreases. Some local peaks can be interpreted as the effect of higher harmonics. As the amplitude increases, the resonance frequency of the TLD increases accordingly, performing a hardening-spring behaviour. The lower bound for the fundamental frequency appears to be well defined by the value identified through the simplified linear theory. What cannot be rigorously described by a frequency response function is, however, the instability occurring near resonance for the largest excitation amplitudes, as more evidently as water depth increases. In these conditions, namely, it was found that the measured reaction force versus time can not stabilise, not even after the transient interval, into a harmonic signal (the steady-state solution of a linear system): for h ¼ 3 cm the signal is somewhat modulated as though multiplied by a periodic function of smaller frequency, whilst for h ¼ 4:5 cm and for h ¼ 6 cm an even more chaotic response is derived. For this instability, the same concept of frequency response function, strictly speaking, grows deficient, and it must be specified that the FRF herein considered conventionally consisted of the mean values of the signal as long as the instability took place. The whole of the aforementioned results agreed well with the frequency response function for the surface elevation near the end wall (Fig. 9). In particular, the nonlinearity reveals through a multiplicity of effects, progressively intensifying as the excitation amplitude and the water depth increase and the resonance frequency approaches: asymmetry between the elevation and the descent of the surface near the end wall (elevation is always much greater than descent, nearly up to one order of magnitude), concavity of water surface, three-stages oscillations, wave breaking,
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Fig. 9. Conical TLD with 3 cm of water: experimental transfer functions for the normalised surface level at the end wall.
sprays and noisy gushing, oscillation in a plane rotated with respect to the excitation plane, rotating motion superimposed on the sloshing oscillation, and finally instability, which mixes up several effects together, typically generating an alternation of clockwise and anticlockwise rotations and sloshing oscillations. All these phenomena are responsible for a significant increase in the damping of the TLD and therefore in the reduction of the peaks of the frequency response function for the normalised reaction force. From a different viewpoint, the force reduction can be explained as the physical impossibility of the liquid oscillation to increase proportionally to the excitation beyond a certain maximum (which obviously is the basic aspect of a nonlinearity), so that sloshing turns gradually into rotation and a number of dissipative effects come into existence. Some tests were additionally conducted for the same tank filled with 15 cm of water, upon an amplitude excitation equal to 2.5 mm. At resonance, instability becomes even stronger and no apparent regularity may be found in the response, which seems to be an accidental sequence of rotations in both senses and oscillations in any vertical planes (Fig. 10). At some instant, as shown by the reaction force (once subtracted the inertial terms obtained from an identical test carried on in absence of water) plotted versus the base displacement (Fig. 11), the instability even undergoes a negative damping: the contour trajectory becomes clockwise, which means the damper behaves adversely (the area enclosed by the contour represents the energy dissipation during a cycle of motion). This phenomenon should be accurately avoided, not so much because of the difficulty in predicting it but because of its intrinsic disadvantage, and great care should be generally taken in properly modifying the geometry of the reservoir in order to prevent any sort of instability. It would be practically impossible to simulate these nonlinear and unstable effects through the shallow water wave theory model (it is so much easier to simulate wave
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TIME HISTORY FOR THE REACTION 50 40 30
FORCE (N)
20 10 0 -10 -20 -30 -40 -50
0
10
20
30
40
50
60
70
80
90
100
TIME (s)
Fig. 10. Conical TLD response for an harmonic excitation amplitude of 2.5 mm and 15 cm of water height.
FORCE VERSUS DISPLACEMENT 50 40 30
FORCE (N)
20 10 0 -3
-2
-1
0
1
2
3
-10 -20 -30
3s
6s
-50
DISPLACEMENT (mm)
Fig. 11. Conical TLD energy dissipation curve for an harmonic excitation amplitude of 2.5 mm and 15 cm of water height.
breaking in a rectangular tank!) and quite uninteresting too. In fact, the only possible approach is a macroscopic phenomenological one, intended to catch the global behaviour through some few equivalent parameters. The agreement with experimental results cannot be general, but if some robustness is granted and the agreement
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is assured near resonance, there is good hope that such a strategy could be satisfactorily used in a design procedure.
5. TMD analogy of nonlinear TLD The most practical way of addressing the nonlinearity of the physical model is by substituting the real TLD system with an equivalent ideal tuned mass damper with parameters (mass, frequency and damping) varying with the excitation amplitude (besides depending on the geometry of the tank). In fact, this very concept of a TMD analogy is at the base of the simplified linear theory as well, except that the formulas are in that case derived from a theoretical approach. The procedure followed for the nonlinear TLD merely differs for its empirical justification and can be considered as the necessary expansion of the linear theory for the case of great amplitudes. There exist manifold criteria for the choice of the optimally fitted TMD, some of them based on time integration of reaction force signal multiplied by tank base displacement or velocity. In this study, a more elementary method is applied: for each TLD configuration (i.e. for each value of h), and for each excitation amplitude, mass, frequency and damping of the equivalent TMD are directly chosen as those able to best reproduce the experimental curve of force versus frequency (Fig. 13). In this search for the best fitted nonlinear TMD, a further simplification seems to be legitimate, which enables to consider the effective mass as equal to the total mass of the liquid. Actually, as far as a linear behaviour is concerned, one of the most favourable aspects of conical TLDs, if compared with cylindrical ones, is the greater percentage of the mobile mass to the total mass of liquid, which allows smaller total masses for identical level of performance. This aspect, together with the observation that the effective mass for a cylindrical TLD approaches the total mass when the amplitude increases, helps explaining why, for a conical TLD, the best value for the effective mass corresponds quite well to the total mass of liquid, even for not too large amplitudes. This physical benefit has the additional advantage of reducing the un-
Fig. 12. Base amplitude-dependent equivalent natural frequency (a) and damping ratio (b).
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Fig. 13. Force transfer function: fitting curves by nonlinear TMD analogy (abscissa: xexc =x0 ; ordinate: F 0 ).
known parameters to frequency and damping alone. In each case, in fact, the mass has been assumed identical to the total mass of liquid in the tank. The curve fitting can be definitely summarised in the parameters of Fig. 12a and b. The TMD frequency is there normalised to the value provided by the linear theory (respectively f0 ¼ 1:22, 1.33, 1.38 Hz for h ¼ 3, 4.5, 6 cm), which turns to be a good estimator of TLD natural frequency for small oscillations. As the excitation increases, the fundamental frequency increases too (hardening-spring type behaviour) but accordingly a robust behaviour is appreciable near resonance (the FRFs grow flatter and wider), which makes the analogy conservative and some prediction errors acceptable. Once that the mass is proved to coincide with the total mass of the liquid and the frequency is shown to slightly increase with respect to the value derived from the linear theory, the only a priori unknown and hardly predictable parameter is the damping of the equivalent TMD, which is easily obtainable from a fitting procedure. It turns out to significantly increase with the amplitude of the excitation. This fact is responsible for the decrement of the frequency response for the normalised force, but should not be looked at as an unfavourable property. As it is well known since FrahmÕs undamped oscillator (1909), a small damping for a tuned TMD (which corresponds to a high peak of the FRF for the reaction force), minimises the structural response at its original resonance but creates two new resonance frequencies at certain
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distance. If the response is to be minimised on a large bandwidth, the optimum damping 1opt (in case the condition of optimum tuning holds on) depends on the mass ratio l in a manner which is approximately given by the classical Den HartogÕs formula [4]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3l 1opt ¼ ð4Þ 8ð1 þ lÞ which suggest, for a usual mass ratio of 1%, an optimal damping of 6%, which is approximately the value experimentally obtained for a harmonic excitation amplitude of 10 mm. 6. Conclusions and future developments The first result of the present work is that the frustum conical TLD proves a valid alternative to the more classical cylindrical TLD. Though the main physical aspects of liquid motion are generally the same (nonlinearities, hardening-spring type behaviour, swirling around resonance for large excitations, jump-down after resonance), the conical shape seems to provide a greater effective mass ratio (except for the particular case of great amplitudes of harmonic excitation, for which also the cylindrical tank involves nearly all the liquid mass) and a more relevant adaptability of frequency through water height changes. In fact, the frustum-conical TLD considered in this study is only slightly different from a cylindrical TLD (because of the small water depth in comparison with the base diameter) and does not show but a few of the potentialities expected from the new geometry (for instance, for depth varying between 3 and 6 cm, the frequency only slightly varies between 1.22 and 1.38 Hz) so that further investigations are needed on really conical tanks and/or deeper water heights. At any rate, as far as small oscillations are concerned, the present study shows that the linear theory is capable of predicting the natural frequency for shallow waters, and previous researches demonstrated it reproduces well the frequencies for deep waters as well. Consequently, it seems plausible that the linear model may still stand for any conical or frustum-conical geometry, thus providing an easy tool for TLD design. Moreover, according to the linear theory, TLD fundamental frequency may be modified by changing the cone opening angle or the liquid level, thus enhancing a semi-active control of the tuning condition, particularly useful to respond to damage-induced structural properties variations or to changing harmonic excitation frequency. Damping too, for which in this series of experiments only a rough search has been made, can be properly modified by inserting stiff barriers, inside the oscillating liquid volume. In some papers, small spheres or similar materials are proposed, in order to improve the performance of the TLD. An unstable behaviour is observed at resonance for deep water heights which suggests to look for some corrections in tank geometry (e.g., pyramidal shape replacing conical shape, floodgates to canalise fluid motion, . . .) and also for lower water
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heights the rotational motion must be cared about. It is expected that a more regular behaviour could be gained through the use of more viscous fluids. The second fundamental result is that the proposed SDOF model, through an integration of the simplified linear theory with a phenomenological interpretation of nonlinearities occurring for large harmonic excitations, is able to match the main dynamic properties of the physical system. The approach is not avoid of drawbacks: • the SDOF approximation worsens with increasing excitation amplitude; • the instability cannot be reproduced, and must be accounted for only in mean terms; • the analogy is built in terms of force and not in terms of cyclic energy dissipation, therefore it cannot rigorously be concluded to be conservative; • further investigation is necessary for modelling up the response to white noise and seismic excitations. On the other hand, the greatest advantage of the approach is its simplicity, since the nonlinearity is almost entirely translated into variations of damping coefficient without affecting the mass (which can be assumed as coinciding with the total liquid mass) and affecting the frequency almost negligibly (thanks to the robust behaviour shown for large excitations) and at any rate predictably. It should be stressed out that a simple model is paramount for addressing the optimisation of a set of TLDs on a structure [1,2,14]. The proposed model is very effective in this sense: once the order of magnitude for the expected motion of the structure is established, the model makes the three equivalent dynamic parameters of a TLD directly depending on the geometry of the tank and on the water height, thus the search for the optimal control can be directly made in terms of the geometry of each container. Some more insight seems to be necessary especially for an adaptive implementation of the TLD: especially in case of a semi-active control of TLD damping through the regulation of floodgates or similar devices, a more precise model may prove needed for predict the damping coefficient, which seems to be the more controversial parameter for a tuned liquid damper. Presently, the model can be said to be rigorously validated only for the case of harmonic excitations. Future work shall be devoted to investigate more general conditions, such as nonharmonic excitations, different container shapes or different liquid properties. It is believed, however, that the proposed method is basically extendable to almost any liquid sloshing problems, at least within given restraints in the intensity of dynamic excitation.
Acknowledgements This research was supported by a grant to the first author from the Italian Ministry of Education, University and Research (MIUR), within the frame of the
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COFIN01 project with Prof. DaviÕ, the University of Ancona, as national coordinator.
References [1] M. Abe, Y. Fujino, Dynamic characterization of multiple tuned mass dampers and some design formulas, Earthquake Engineering and Structural Dynamics 23 (1994) 813–835. [2] M. Abe, T. Igusa, Tuned mass dampers for structures with closely spaced natural frequencies, Earthquake Engineering and Structural Dynamics 24 (1995) 247–261. [3] F. Casciati, A. De Stefano, E. Matta, A proposal of a frustum-conical tuned liquid damper for suppressing large amplitude motions of structures, in: Proceedings of the Third World Conference on Structural Control, Como, 2002. [4] J.P. Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, 1956. [5] Y. Fujino, L. Sun, B.M. Pacheco, P. Chaiseri, Tuned liquid damper (TLD) for suppressing horizontal motion of structures, Journal of Engineering Mechanics 118 (1992) 2017–2030. [6] G.W. Housner, The dynamic behaviour of water tanks, Bulletin of the Seismological Society of America 53 (1963) 381–387. [7] A. Kareem, The next generation of tuned liquid dampers, in: the First World Conference on Structural Control, Los Angeles, 1994. [8] H. Lamb, Hydrodynamics, New York, Dover, 1945. [9] E. Matta, Sistemi di attenuazione della risposta dinamica ad oscillatori a massa solida e fluida, Thesis (in Italian), Department of Structural and Geotechnical Engineering, Politecnico di Torino, 2002. [10] V.I. Modi, F. Welt, Vibration control using nutation dampers, Proceedings of the Royal Society of London Series A 297 (1967) 459–475. [11] D. Reed, J. Yu, H. Yeh, S. Gardsson, Investigation of tuned liquid dampers under large amplitude excitation, Journal of Engineering Mechanics 124 (1998) 405–413. [12] L.M. Sun, Y. Fuyino, K. Koga, A model of tuned liquid damper for suppressing pitching motions of structures, Earthquake Engineering and Structural Dynamics 24 (1995) 625–636. [13] L.M. Sun, Y. Fujino, P. Chaiseri, B.M. Pacheco, The properties of tuned liquid dampers using a TMD analogy, Earthquake Engineering and Structural Dynamics 24 (1995) 967–976. [14] G.B. Warburton, Optimum absorber parameters for various combinations of response and excitation parameters, Earthquake Engineering and Structural Dynamics 10 (1982) 381–401.
Simulating a conical tuned liquid damper Fabio Casciati a
a,*
, Alessandro De Stefano b, Emiliano Matta
b
Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy b Department of Structural Engineering and Geotechnics, Polytechnic of Turin, Corso Duca degli Abruzzi 34, 10100 Torino, Italy Received 14 May 2002; received in revised form 7 April 2003; accepted 12 April 2003
Abstract In recent years, tuned liquid dampers (TLD) have proved a successful control strategy for reducing structural vibrations. The present study focuses on the frustum-conical TLD as an alternative to the traditional cylindrical tank. If compared to the cylindrical reservoir, the cone-shaped TLD allows calibrating its natural frequency through varying liquid depth, which makes it suitable for a semi-active implementation, and attains the same level of performance with a fewer mass, at least for small fluid oscillations. A linear model is presented which can interpret TLDÕs behaviour for small excitations. For larger amplitudes, strong nonlinearities occur and the linear model is no longer predictive. Consequently, for a frustum-cone TLD subjected to harmonic excitations, a tuned mass damper (TMD) analogy is established where TMD parameters vary with the excitation amplitude. 2003 Elsevier B.V. All rights reserved. Keywords: Conical tank; Energy dissipation; Nonlinearities; Passive control; Tuned liquid damper
1. Introduction TLDs have been mainly applied in Japan. The TLD devices installed in the 77.6 m high structure of the Tokyo Airport Tower (1993), for instance, consist of 1400 tanks filled with water and floating particles. The containers, low cylinders of diameters 0.6 and 0.125 m, are stored in six layers on steel consoles. The total mass of the TLDs is approximately 3.5% of the first modal mass of the tower and its frequency is optimised to 0.74 Hz. Floating polyethylene particles were added in order to enhance *
Corresponding author. E-mail addresses: [email protected] (F. Casciati), [email protected] (A. De Stefano), [email protected] (E. Matta). 1569-190X/03/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1569-190X(03)00051-0
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energy dissipation by their collision. The behaviour of the TLD has been observed under various wind phenomena. In one of these cases with a maximum wind speed of 25 m/s, the recorded data showed that TLDs reduced the acceleration to approximately 60% of its value without control. Dampers using liquid motion have firstly been used in space satellites and marine vessels. Subsequently, some researchers promoted their application to structures on the ground, including buildings and towers. After that, TLDs have proven a successful passive vibration mitigation system [7]. In the meantime, a number of theoretical investigations have flourished aiming at modelling up TLDsÕ physical behaviour. Modi and Welt [10] conducted an experimental and analytical study on a nutation damper (annular tank), which is conceptually the same as TLD. Their theoretical investigation concerned the energy-dissipation mechanism assuming a potential flow with a nonlinear free-surface condition in conjunction with boundary-layer correction [10]. Fujino et al. developed a two-dimensional model of liquid motion in a rectangular TLD using the shallow-water wave theory, i.e. deriving the nonlinear solving system from continuity and equilibrium equations written for an incompressible and irrotational fluid [5]. The damping effects due to solid boundary friction and free-surface contamination were included by adopting the linear theory of the boundary layer (Lamb [8]) with the corrections suggested by Vandorn (1966) and Miles (1967) [8]. Wave breaking was not intended to be modelled. The whole of the experiments revealed strong nonlinearities, the effects of higher harmonics of liquid sloshing around the resonance, a hardening-spring type behaviour and a sudden jump-down in the frequency response soon after resonance. In case of wave breaking (i.e. for amplitude excitations greater than 0.5 cm), the simulation overestimated both the reaction force and the surface elevation near the end wall, while underestimating the energy dissipation (since the wave breaking contribution was not modelled). Good agreements between simulation and experimental results were observed provided that no wave breaking occurred in the experiments. Sun et al. used the same procedure to address the case of a rectangular TLD subjected to pitching vibration [12]. The nonlinear analytical model, based on shallow water wave theory while including the linear damping of liquid sloshing, agreed well with the experimental results. The overall conclusions confirmed those obtained for the mere horizontal motion. A different perspective was adopted in an other study by Sun et al. [13]. Rectangular, circular and annular tanks, subjected to harmonic base excitation, were measured experimentally. Using a SDOF TMD analogy, equivalent mass, stiffness and damping of the TLD were calibrated from the experimental results as functions of the tank base amplitude. In particular, it was found that, as the excitation amplitude increases, the natural frequency increases, the effective mass of the TLD increases too and approaches the total mass of the fluid, the damping ratio increases and exceeds 10% when wave breaking occurs. Reed et al. used the random-choice numerical method to solve the nonlinear shallow-water equations for a rectangular TLD [11]. The model captures the underlying
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physical phenomenon adequately, including wave breaking, for most of the frequency range of interest and over a wide range of amplitude excitation. It was found that the response frequency of TLD increases as excitation amplitude increases and the system is of the hardening-spring type. The aforesaid nonlinear models often show good agreement with the experimental results, in some cases even strong nonlinearities and wave breaking can be predicted. However, more predictive the simulation provided by complex nonlinear theories, more difficult to look for the optimal location of a multiple set of TLD devices on a real building. Moreover, apart from the TMD analogy, where nonlinearities conventionally reduce to changing the parameters of the oscillator with the amplitude of the excitation, nonlinear models based on shallow-water wave theory were conceived only for rectangular tanks, and the whole of different shapes (including cylindrical and conical TLDs, for instance) are still lacking in a proper theoretical model. In what follows, a cone-shaped TLD is proposed, owing to a number of expected advantages it may provide with respect to the cylindrical container. Among the most relevant ones: • a bigger effective mass (at least for deep waters or for small excitations) and a larger free-surface area; • a stronger variability of the fundamental frequency with the water depth, beyond a certain value of the liquid height; • suitability to innovative solutions for tank building technologies, such as deformable membrane-made conical containers which could be extended along the vertical direction providing a semi-actively changing geometry. Furthermore, conical reservoirs are already present in many civil buildings as a water reserve, and it would be interesting to investigate their suitability to turn into passive dynamic absorbers, though TLDs consist traditionally of shallow water tanks. A nonlinear shallow water wave theory would be hardly applicable to the case, because of the complex geometry of the tank. Consequently, a nonlinear (both frequency- and amplitude-dependent) SDOF model is proposed to simulate the dynamic behaviour of the system. As will be illustrated in the following pages, for limited excitations, a simplified linear theoretical approach for the conical and frustum-conical TLD, similar in principle to the classical HousnerÕs approach for rectangular and cylindrical tanks, has proved successful in predicting the frequency of an equivalent linear TMD. For greater base amplitudes, instead, the linear model is proven unsatisfactory, since the parameters of the oscillator are no longer constant, and the effective mass, the natural frequency and the damping coefficient of the equivalent TMD increase as the amplitude increases [3]. This makes it necessary to calibrate the oscillator properties on the experimental results, and eventually turns the TLD into a TMD whose parameters depend not only on the geometry in an analytical way (as it would be as far as the linear theory were concerned) but also on the amplitude of excitation in an experimental way (which is far less general). Yet, it will be demonstrated that the advantages of the linear theory may be partially retained even for large amplitudes if
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the effects of nonlinearity are conventionally condensed in the damping coefficient alone. This is quite confirmed by experimentation, since for large excitations the effective mass approaches the total mass and the natural frequency may also be slightly misunderstood without diminishing the control performance, thanks to a robust behaviour of the TLD in a large range of frequencies around the resonance. The investigated modelling strategy is specifically intended to provide a simple description of a peculiar structural control device, the frustum-conical tuned liquid damper, such that its basic properties could be understood for the sake of an optimal control design. Nevertheless, the physical phenomenon under scrutiny, i.e. liquid sloshing in a reservoir, is a much more general problem, and it is expected that the proposed method could be exported to other applications, for instance to the modelling of the response of any liquid containers subjected to a dynamic excitation. 2. Typology of tuned liquid damper TLD systems can be divided into two categories. The first category is represented by the sloshing damper (sloshing is the phenomenon of wave breaking on the surface of the liquid against the walls of the tank), or said, TLD in the proper sense. The dimensions of the tank or the depth of the liquid regulates the period of vibration. Damping ability is increased introducing nets or rods into the liquid. The second type is represented by the tuned liquid column damper (TLCD). Here the shape of the columns or the pressure of the air inside them regulate the period of oscillation. The damping ability is increased by regulating the opening of an orifice in the column, which generates high turbulence. Our attention, here, is devoted only to sloshing TLD. TLD, compared to tuned mass damper (TMD), has the following favourable properties: • the absence of mechanical friction provides a negligible trigger level; • the absence of mobile mechanical components reduces maintenance costs and increases reliability; • TLD can operate in all the possible directions in the horizontal plane; • the liquid mass can be shared between several tanks, which are then made small and easily transportable; • if the tank has the horizontal section varying along the vertical axis, as it happens for conical reservoirs, it is easy to tune the frequency by simply changing the contained liquid volume; • for structures having different vibration frequencies in the two principal directions, it is possible to use containers with double axial symmetry, so that a double tuning is attained. On the other hand, an unfavourable property of the TLD is the small mass density of water, which requires larger devices compared with a metallic TMD. However such disadvantage is compensated by the fact that it is easy, technically effective and low cost demanding to locate in any available space several TLD devices, working in parallel.
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An additional drawback is represented by TLDÕs nonlinear behaviour, which makes it difficult to work out a predictive simulation approach which could be easily handled in practical design.
3. Linearised analysis of the tuned liquid damper The hydrodynamic theory of the TLD is somewhat complex and difficult to handle, above all because of the intrinsic nonlinearity in the physical model. However, the nonlinearity can be neglected in the dominion of small excitations, and even for large excitations a linear model may prove useful as far as it is able to approximate such a paramount parameter as the natural frequency. The linear model Housner proposed in the first Ô60s aims at simulating––essentially for cylindrical and rectangular tanks––water response on tank walls with two lumped masses: a fixed one (representing the liquid that is not moved by the acting acceleration), rigidly connected to the tank, and one moving (determining the sloshing effect which is responsible for the energy absorption), connected to the tank by a spring and a damper [6]. More recently the simplified model of Housner has been adapted to conical tanks with the original purpose, shared in fact with HousnerÕs approach, of simulating the response of large elevated water tanks to seismic action. As for their use as a control strategy, conical or frustum-conical TLDs appear particularly effective as a passive control system, owing to the little requirements of mass, and very suitable for transformation into an adaptive control system, owing to the variability of frequency with water height.
y A
λ λ’
G H
A’
um xP’
h0
α
h2
B’ P’ B
r r
∆h
φ 250
θ
G’
yH
ϕ
α
y
yP x0
OC C’
k/2
k/2 m1
b’b
m0
h m1
A
A’
O≡G
h m0
β
D
(a)
G’
r r
B
B’
P’
D’
(b)
Fig. 1. Conical tanks: scheme of the liquid in fixed mass and moving mass (a) and scheme of the liquid motion (b).
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Let us now summarise the linearised formulation for a conical tank, in order to calculate the fundamental frequency and to estimate the parameters of fixed and moving masses (Fig. 1). Such parameters are the amount of the masses, their position and the stiffness of springs connecting the mobile mass to the tank. The formulas shown in this paragraph allow the direct calculation of the two masses (indicated as m0 and m1 ), of their heights (hm0 and hm1 respectively), of the stiffness k of the spring associated to the moving mass and consequently of the fundamental frequency x. It has already been stressed out that the mobile and fixed masses have a specific physical meaning. In particular, the mobile mass simulates the sloshing effect. Its evaluation is obtained by setting the continuity and equilibrium equations with the simplified assumption that the fluid in rest is divided in layers by plane parallel surfaces, remaining plane, even if no more parallel, during motion. Governing dynamic equilibrium condition is deduced by applying the Hamilton principle. In the modified approach for conical tanks, the centroid of plane surfaces containing fluid layers do not remain on the same vertical axis during motion, as in HousnerÕs model, but displaces laterally. 16 p 2 4 m1 ¼ qIzh2 x2 Að4Þ ; Izh2 ¼ ðh2 tgaÞ ; 81g 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g 8 ; k ¼ m1 x2 ; x¼ 2 h2 tg a 27 ðF1 C1 þ F2 C2 Þ þ 1 AðiÞ ¼
9 þ 2F1 9 þ 2F2 1 lðF1 þiÞ lðF1 F2 Þ 1 lðF2 þiÞ ; i þ F1 i þ F2
F1=2 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 1 1 þ 2:16k1 ; 2
C1 ¼ 1
1 ðF1 F2 Þ ; h1 h2
C2 ¼
k1 ¼
h1 h2
ð1Þ
l¼
h1 ; h2
ð2Þ
1 tg2 a ; 4tg2 a
ðF1 F2 Þ
1
1 ðF1 F2 Þ
ð3Þ
h1 h2
The foregoing linearised theory has already experimentally proved successful in evaluating the free-decay natural frequency for a variety of deep water heights in conical tanks but there is still need to define its applicability to shallow water conical TLD.
4. Laboratory analyses The equipment used for the experimental research is the one installed in the Department of Structural Mechanics of the University of Pavia. It includes a shaking table vibrating with a single motion axis, controlled in closed loop by means of a software, with which it is possible to create an adequate drive signal for the table in order to produce the acceleration in the desired form. A hydraulic device, which
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is set into action by a servo-valve, produces the movement. The actuator receives as input an electrical signal and transforms it into a displacement. The experimental research followed two main routes: first, a comparison between the performances of cylindrical and conical TLDs as passive control systems applied on a single-degree-of-freedom test frame, subjected to harmonic, white noise and seismic base excitations; then, the response to harmonic excitations of the frustum TLD alone, on a load cell. In the first part of the experiment, TLDs were located on a SDOF steel frame, mounted on the shaking table. Both cylindrical and conical tanks have been tested. Cylindrical tanks had diameters of 10, 20 and 30 cm. Their fundamental frequency was not so much influenced by the water height (respectively 3, 2.1 Hz and 1.6–1.7 Hz for water height between 10 and 30 cm). The conical tank was shaped as a frustum of a cone with base diameter 20 cm and an inclined lateral wall with a slope of 60 to the horizontal, whose frequency––with the same range of variation of water height––varies from 1.1 to 1.4 Hz. Some of the tests were carried on with a single tank (which controls a single frequency) and others with several tanks in order to see the effect of concomitant tanks. As it was expected, the tests showed that a tank which is not tuned to the main structure works well only for signals that excite its natural frequency, not bringing any benefit to the resonance frequency of the whole system. For tuned tanks, instead, only a small amount of water proved necessary to obtain satisfactory results in terms of response attenuation; reductions of the transfer function up to 30–40% have been found and reductions of more than 50% of time domain outputs under white noise excitations have been obtained. The controlling liquid mass needed to reach such good results has been indeed small (0.64% of the total mass of the structure for the frustum TLD, corresponding to a water height of 3 cm, in tuning conditions). Experiments using sinusoidal and white noise input signals showed the achievement of decisive improvements of the resonant response of the frame (Figs. 2
Fig. 2. Comparison between control and no control case using a conical tank exciting the structure with a white noise.
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Fig. 3. Comparison between control and no control case exciting the structure with a sine wave (f ¼ 1:27 Hz; A ¼ 0:7 mm).
and 3). Large reductions of accelerations have been observed for earthquake excitations with tuned tanks, demonstrating the applicability of this control method also for such cases, except for the first seconds of sharply impulsive earthquake excitations, with a really steep gradient of acceleration. In this case, as it has been seen for the earthquake of Bagnoli–Irpinia (Fig. 4), for some seconds (of the order of 2–3 s) no difference was observed between controlled and uncontrolled structure. After these few seconds, however, the dynamic damped response amplitude fell down to a 50% (in case of the truncated cone with a water height of 3 cm) of the value observed in the uncontrolled case. For earthquakes with less impulsive character, such
Fig. 4. Comparison between control and no control case using a conical tank exciting the structure with Bagnoli Earthquake record.
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Fig. 5. Comparison between control and no control case using a conical tank exciting the structure with El Centro Earthquake record.
as El Centro (Fig. 5), these negative effects were substantially lower. These results were confirmed by a genetic algorithms optimisation study carried out, in parallel to the experimental investigations, with the purpose of searching for the best configuration of multiple dynamic absorbers as a control strategy for seismic induced vibration mitigation. The numerical procedure too demonstrates that the seismic response to strongly impulsive signals can not be significantly reduced through passive oscillators, so that the ineffectiveness of passive TLDs to address impulsive seismic actions can be regarded as a general result [9]. Comparable results have been obtained with a tuned cylindrical tank, with relevant reductions for all the categories of signals, but in that case a liquid mass of about 1.6% of the total mass of the structure was needed, instead of the only 0.64% used inside the conical tank. In the second part of the experimental investigation, the same frustum-cone was located on a load cell, which had been designed and built up for the purpose, and the load cell on the shaking table. The dynamic uncoupling between the TLD and the load cell made it possible to isolate the TLD behaviour, which instead had been previously indirectly analysed through the effects it induced on a tuned test frame. The load cell is meant to reproduce a unidirectional SDOF shear-type system, in fact consisting of two horizontal steel plates connected by four thin rectangular-cross-section columns. When a force is applied on the upper plate, the lower one being rigidly restrained, the relative displacement of the two plates induces a proportional deformation on the four columns, which can be picked up by four Wheatstone half bridges. Through a una tantum calibration procedure, the mean value from the four measurements therefore provides the applied force. For different liquid volumes (liquid depth h equal to 3, 4.5 and 6 cm) and for various amplitudes of harmonic excitations (1.25, 2.5, 5, 10 and 15 mm), the frequency
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response functions were experimentally obtained for the normalised force F 0 ¼ F =ðmAx2 Þ, where F is the force read by the load cell (therefore comprised the inertial terms due to the tank and load cell masses), m is the liquid mass, Ax2 is the amplitude of acceleration (Figs. 6–8). Whatever the liquid depth, the fundamental result was the strong nonlinear response of the frustum TLD near resonance, which is in perfect agreement with the behaviour of traditionally shaped containers.
Fig. 6. Conical TLD with 3 cm of water: experimental transfer functions for the normalised force measured at the load cell.
Fig. 7. Conical TLD with 4.5 cm of water: experimental transfer functions for the normalised force measured at the load cell.
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FORCE TRANSFER FUNCTION FOR h = 6 cm 30
A = 1.25 mm A = 2.5 mm A = 5 mm
25
2
F /(mA ω )
20
15
10
5
0 0.5
0.7
0.9
1.1
1.3
1.5
ω exc / ω 0
Fig. 8. Conical TLD with 6 cm of water: experimental transfer functions for the normalised force measured at the load cell.
The normalised frequency response function grows flat as the base excitation increases, thus showing a favourable robustness, and at the same time its peak value decreases. Some local peaks can be interpreted as the effect of higher harmonics. As the amplitude increases, the resonance frequency of the TLD increases accordingly, performing a hardening-spring behaviour. The lower bound for the fundamental frequency appears to be well defined by the value identified through the simplified linear theory. What cannot be rigorously described by a frequency response function is, however, the instability occurring near resonance for the largest excitation amplitudes, as more evidently as water depth increases. In these conditions, namely, it was found that the measured reaction force versus time can not stabilise, not even after the transient interval, into a harmonic signal (the steady-state solution of a linear system): for h ¼ 3 cm the signal is somewhat modulated as though multiplied by a periodic function of smaller frequency, whilst for h ¼ 4:5 cm and for h ¼ 6 cm an even more chaotic response is derived. For this instability, the same concept of frequency response function, strictly speaking, grows deficient, and it must be specified that the FRF herein considered conventionally consisted of the mean values of the signal as long as the instability took place. The whole of the aforementioned results agreed well with the frequency response function for the surface elevation near the end wall (Fig. 9). In particular, the nonlinearity reveals through a multiplicity of effects, progressively intensifying as the excitation amplitude and the water depth increase and the resonance frequency approaches: asymmetry between the elevation and the descent of the surface near the end wall (elevation is always much greater than descent, nearly up to one order of magnitude), concavity of water surface, three-stages oscillations, wave breaking,
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Fig. 9. Conical TLD with 3 cm of water: experimental transfer functions for the normalised surface level at the end wall.
sprays and noisy gushing, oscillation in a plane rotated with respect to the excitation plane, rotating motion superimposed on the sloshing oscillation, and finally instability, which mixes up several effects together, typically generating an alternation of clockwise and anticlockwise rotations and sloshing oscillations. All these phenomena are responsible for a significant increase in the damping of the TLD and therefore in the reduction of the peaks of the frequency response function for the normalised reaction force. From a different viewpoint, the force reduction can be explained as the physical impossibility of the liquid oscillation to increase proportionally to the excitation beyond a certain maximum (which obviously is the basic aspect of a nonlinearity), so that sloshing turns gradually into rotation and a number of dissipative effects come into existence. Some tests were additionally conducted for the same tank filled with 15 cm of water, upon an amplitude excitation equal to 2.5 mm. At resonance, instability becomes even stronger and no apparent regularity may be found in the response, which seems to be an accidental sequence of rotations in both senses and oscillations in any vertical planes (Fig. 10). At some instant, as shown by the reaction force (once subtracted the inertial terms obtained from an identical test carried on in absence of water) plotted versus the base displacement (Fig. 11), the instability even undergoes a negative damping: the contour trajectory becomes clockwise, which means the damper behaves adversely (the area enclosed by the contour represents the energy dissipation during a cycle of motion). This phenomenon should be accurately avoided, not so much because of the difficulty in predicting it but because of its intrinsic disadvantage, and great care should be generally taken in properly modifying the geometry of the reservoir in order to prevent any sort of instability. It would be practically impossible to simulate these nonlinear and unstable effects through the shallow water wave theory model (it is so much easier to simulate wave
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TIME HISTORY FOR THE REACTION 50 40 30
FORCE (N)
20 10 0 -10 -20 -30 -40 -50
0
10
20
30
40
50
60
70
80
90
100
TIME (s)
Fig. 10. Conical TLD response for an harmonic excitation amplitude of 2.5 mm and 15 cm of water height.
FORCE VERSUS DISPLACEMENT 50 40 30
FORCE (N)
20 10 0 -3
-2
-1
0
1
2
3
-10 -20 -30
3s
6s
-50
DISPLACEMENT (mm)
Fig. 11. Conical TLD energy dissipation curve for an harmonic excitation amplitude of 2.5 mm and 15 cm of water height.
breaking in a rectangular tank!) and quite uninteresting too. In fact, the only possible approach is a macroscopic phenomenological one, intended to catch the global behaviour through some few equivalent parameters. The agreement with experimental results cannot be general, but if some robustness is granted and the agreement
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is assured near resonance, there is good hope that such a strategy could be satisfactorily used in a design procedure.
5. TMD analogy of nonlinear TLD The most practical way of addressing the nonlinearity of the physical model is by substituting the real TLD system with an equivalent ideal tuned mass damper with parameters (mass, frequency and damping) varying with the excitation amplitude (besides depending on the geometry of the tank). In fact, this very concept of a TMD analogy is at the base of the simplified linear theory as well, except that the formulas are in that case derived from a theoretical approach. The procedure followed for the nonlinear TLD merely differs for its empirical justification and can be considered as the necessary expansion of the linear theory for the case of great amplitudes. There exist manifold criteria for the choice of the optimally fitted TMD, some of them based on time integration of reaction force signal multiplied by tank base displacement or velocity. In this study, a more elementary method is applied: for each TLD configuration (i.e. for each value of h), and for each excitation amplitude, mass, frequency and damping of the equivalent TMD are directly chosen as those able to best reproduce the experimental curve of force versus frequency (Fig. 13). In this search for the best fitted nonlinear TMD, a further simplification seems to be legitimate, which enables to consider the effective mass as equal to the total mass of the liquid. Actually, as far as a linear behaviour is concerned, one of the most favourable aspects of conical TLDs, if compared with cylindrical ones, is the greater percentage of the mobile mass to the total mass of liquid, which allows smaller total masses for identical level of performance. This aspect, together with the observation that the effective mass for a cylindrical TLD approaches the total mass when the amplitude increases, helps explaining why, for a conical TLD, the best value for the effective mass corresponds quite well to the total mass of liquid, even for not too large amplitudes. This physical benefit has the additional advantage of reducing the un-
Fig. 12. Base amplitude-dependent equivalent natural frequency (a) and damping ratio (b).
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Fig. 13. Force transfer function: fitting curves by nonlinear TMD analogy (abscissa: xexc =x0 ; ordinate: F 0 ).
known parameters to frequency and damping alone. In each case, in fact, the mass has been assumed identical to the total mass of liquid in the tank. The curve fitting can be definitely summarised in the parameters of Fig. 12a and b. The TMD frequency is there normalised to the value provided by the linear theory (respectively f0 ¼ 1:22, 1.33, 1.38 Hz for h ¼ 3, 4.5, 6 cm), which turns to be a good estimator of TLD natural frequency for small oscillations. As the excitation increases, the fundamental frequency increases too (hardening-spring type behaviour) but accordingly a robust behaviour is appreciable near resonance (the FRFs grow flatter and wider), which makes the analogy conservative and some prediction errors acceptable. Once that the mass is proved to coincide with the total mass of the liquid and the frequency is shown to slightly increase with respect to the value derived from the linear theory, the only a priori unknown and hardly predictable parameter is the damping of the equivalent TMD, which is easily obtainable from a fitting procedure. It turns out to significantly increase with the amplitude of the excitation. This fact is responsible for the decrement of the frequency response for the normalised force, but should not be looked at as an unfavourable property. As it is well known since FrahmÕs undamped oscillator (1909), a small damping for a tuned TMD (which corresponds to a high peak of the FRF for the reaction force), minimises the structural response at its original resonance but creates two new resonance frequencies at certain
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distance. If the response is to be minimised on a large bandwidth, the optimum damping 1opt (in case the condition of optimum tuning holds on) depends on the mass ratio l in a manner which is approximately given by the classical Den HartogÕs formula [4]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3l 1opt ¼ ð4Þ 8ð1 þ lÞ which suggest, for a usual mass ratio of 1%, an optimal damping of 6%, which is approximately the value experimentally obtained for a harmonic excitation amplitude of 10 mm. 6. Conclusions and future developments The first result of the present work is that the frustum conical TLD proves a valid alternative to the more classical cylindrical TLD. Though the main physical aspects of liquid motion are generally the same (nonlinearities, hardening-spring type behaviour, swirling around resonance for large excitations, jump-down after resonance), the conical shape seems to provide a greater effective mass ratio (except for the particular case of great amplitudes of harmonic excitation, for which also the cylindrical tank involves nearly all the liquid mass) and a more relevant adaptability of frequency through water height changes. In fact, the frustum-conical TLD considered in this study is only slightly different from a cylindrical TLD (because of the small water depth in comparison with the base diameter) and does not show but a few of the potentialities expected from the new geometry (for instance, for depth varying between 3 and 6 cm, the frequency only slightly varies between 1.22 and 1.38 Hz) so that further investigations are needed on really conical tanks and/or deeper water heights. At any rate, as far as small oscillations are concerned, the present study shows that the linear theory is capable of predicting the natural frequency for shallow waters, and previous researches demonstrated it reproduces well the frequencies for deep waters as well. Consequently, it seems plausible that the linear model may still stand for any conical or frustum-conical geometry, thus providing an easy tool for TLD design. Moreover, according to the linear theory, TLD fundamental frequency may be modified by changing the cone opening angle or the liquid level, thus enhancing a semi-active control of the tuning condition, particularly useful to respond to damage-induced structural properties variations or to changing harmonic excitation frequency. Damping too, for which in this series of experiments only a rough search has been made, can be properly modified by inserting stiff barriers, inside the oscillating liquid volume. In some papers, small spheres or similar materials are proposed, in order to improve the performance of the TLD. An unstable behaviour is observed at resonance for deep water heights which suggests to look for some corrections in tank geometry (e.g., pyramidal shape replacing conical shape, floodgates to canalise fluid motion, . . .) and also for lower water
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heights the rotational motion must be cared about. It is expected that a more regular behaviour could be gained through the use of more viscous fluids. The second fundamental result is that the proposed SDOF model, through an integration of the simplified linear theory with a phenomenological interpretation of nonlinearities occurring for large harmonic excitations, is able to match the main dynamic properties of the physical system. The approach is not avoid of drawbacks: • the SDOF approximation worsens with increasing excitation amplitude; • the instability cannot be reproduced, and must be accounted for only in mean terms; • the analogy is built in terms of force and not in terms of cyclic energy dissipation, therefore it cannot rigorously be concluded to be conservative; • further investigation is necessary for modelling up the response to white noise and seismic excitations. On the other hand, the greatest advantage of the approach is its simplicity, since the nonlinearity is almost entirely translated into variations of damping coefficient without affecting the mass (which can be assumed as coinciding with the total liquid mass) and affecting the frequency almost negligibly (thanks to the robust behaviour shown for large excitations) and at any rate predictably. It should be stressed out that a simple model is paramount for addressing the optimisation of a set of TLDs on a structure [1,2,14]. The proposed model is very effective in this sense: once the order of magnitude for the expected motion of the structure is established, the model makes the three equivalent dynamic parameters of a TLD directly depending on the geometry of the tank and on the water height, thus the search for the optimal control can be directly made in terms of the geometry of each container. Some more insight seems to be necessary especially for an adaptive implementation of the TLD: especially in case of a semi-active control of TLD damping through the regulation of floodgates or similar devices, a more precise model may prove needed for predict the damping coefficient, which seems to be the more controversial parameter for a tuned liquid damper. Presently, the model can be said to be rigorously validated only for the case of harmonic excitations. Future work shall be devoted to investigate more general conditions, such as nonharmonic excitations, different container shapes or different liquid properties. It is believed, however, that the proposed method is basically extendable to almost any liquid sloshing problems, at least within given restraints in the intensity of dynamic excitation.
Acknowledgements This research was supported by a grant to the first author from the Italian Ministry of Education, University and Research (MIUR), within the frame of the
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COFIN01 project with Prof. DaviÕ, the University of Ancona, as national coordinator.
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