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Experimental Characterization and Simulation of a Tethered Spherical Helium Balloon in an Outdoor Environment
P. Coulombe-Pontbriand* and M. Nahon† McGill University, Montreal, Quebec H3A 2K6, Canada
This paper focuses on an investigation of the dynamic characteristics of a spherical aerostat on a single tether. A portable test facility was constructed to gather experimental data required to characterize the system. All experiments were in the supercritical range, at Reynolds numbers greater than 3.7 × 105. The balloon’s drag coefficient was determined based on position measurements. The balloon’s large oscillations and surface roughness, combined with the wind turbulence, resulted in a substantial increase in the drag coefficient. To further study the system, a numerical simulation was developed. The aerostat is modeled as a rigid body attached to a tether, and is subject to buoyancy, aerodynamic drag and gravity. The tether is modeled using a lumped-mass approach, while includes stiffness and damping. The dynamics simulation of the system is obtained by formulating the equations of motion of the aerostat and the cable nodes in 3D space and integrating them numerically. The simulation is then validated by comparing its results with experimental data. Finally, a modal analysis of the natural modes of the system is performed.
I. Introduction lthough tethered helium balloons are presently used in many applications, only a limited amount of research into their dynamic behaviour has been performed, and most of that focus has been directed at streamlined
aerostats as shown in Fig.1. The advent of new technologies and the desire to use tethered aerostats in increasingly sophisticated applications such as surveillance, where reliability is critical, have created a need for better understanding of the behaviour of these systems. As well, alternative balloon shapes might be more suitable for these applications, but there exists little research in the open literature. A natural candidate to study is the spherical shape tethered helium aerostat shown in Fig. 2 which has the advantages of having no preferred orientation to the wind, being relatively easy and fast to build and having the most efficient volume to free lift ratio (due to its low surface area). Surprisingly, although a spherical object attached to the ground by a single tether in a flow is one of the simplest tethered balloon systems possible, there exists little data about it.
The study of the dynamics of a spherical aerostat in a wind field and its simulation is still an almost untouched subject. Lambert1 performed a preliminary simulation of the dynamics of a spherical tethered aerostat in fluid flow but this work was not validated experimentally. No experimental data on tethered spherical aerostat motion in wind fields appears available in the open literature.
Tethered buoyant spheres in a fluid flow have been investigated in laboratory studies. To the authors’ knowledge the first study of oscillations of a tethered sphere in a fluid flow was performed at Moscow university by Kruchinin2. Scoggins3 characterized lift and drag forces of rising spherical weather balloon in 1967.Other studies in the literature are concerned with the action of * Masters Student, Dept. of Mechanical Engineering, 817 Sherbrooke St. West. † Associate Professor, Dept. of Mechanical Engineering, 817 Sherbrooke St. West, Senior Member AIAA.
A
Figure 1: Streamlined aerostat
AIAA 5th Aviation, Technology, Integration, and Operations Conference (ATIO) <br>26 - 28 September 2005, Arlington, Virginia
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surface waves on tethered buoyant spherical structure. These include the work by Harleman and Shapiro4 in 1961, Shi-Igai & Kono5 1969 and Ogihara6 in 1980. The first group to give systematic attention to the transverse oscillations of a tethered sphere in a fluid flow is Govardhan and Williamson7,8. In 2001, Jauvtis et al.9 explained the sphere oscillations by a ‘lock in’ phenomenon of the principal vortex shedding as observed on a fixed sphere and the body motion. They also discovered the existence of a mode of oscillation at much higher flow speeds that could not be explained by the classical ‘lock in’ theory since the vortex shedding of the fixed sphere in that flow regime would have no frequency content close to the natural pendulum frequency of the tethered sphere. Govardhan and Williamson10 provided the explanation of this unexpected phenomenon in 2005. They attributed the oscillations of the tethered sphere to ‘movement induced vibration’ as categorized by Naudasher & Rockwell11 in 1994 where the sphere motion generates self-sustaining vortex forces. Other research in this field includes the work of Bearman12 in 1984 and Anagnostopoulos13 in 2002.
A number of simulation studies have been performed, focusing on streamlined aerostats. DeLaurier14 in 1972, was the first to study the dynamics of a tethered streamlined aerostat with a comprehensive cable model. In 1973, Redd et al15 used experimental data to validate their linear simulation. Jones and Krausman16 in 1982 completed the first 3-D nonlinear dynamics model with a lumped mass discretized tether. Jones and DeLaurier17 further developed this concept to come up with a model based on semi-empirical values. In 2003, Lambert and Nahon18 presented a nonlinear model of a tethered streamlined aerostat using a lumped mass cable model and suggested a method to assess the stability of a single tethered aerostat by linearization of the equations of motion. The response of the streamlined tethered aerostat to extreme turbulence was studied by Stanney and Rahn 19 who used a sophisticated wind model. Lambert20 in 2005 used the results from experiments performed on a fully instrumented 18 m tethered streamlined aerostat in the scope of the LAR21 project to perform a validation of its nonlinear model.
The focus of the present work is the analysis and simulation of the dynamics of a tethered spherical aerostat in a wind field. This includes the design and construction of a portable experimental facility capable of accurately measuring the tether forces and the motion of the aerostat; the analysis of the motion data; and a computer simulation and modal analysis of the aerostat system.
II. Design of test facility In this section, the experimental set-up for the characterization of the dynamics of a tethered aerostat is
described. The objective was to develop a compact and portable facility that would allow accurate measurement of the time history of the balloon’s motion and of the tension in the cable. The experimental set-up is divided into four subsystems: the physical system, the sensor system, the data acquisition system and the instrument platform.
A. Physical System The physical system includes the aerostat, the tether and the winch, as shown in Fig. 2. In terms of compactness,
a spherical aerostat is desirable as it has the most efficient volume to free lift ratio (due to its low surface area). A 3.5 m spherical aerostat was selected, made out of urethane coated nylon to keep the system as small as possible. The free lift FL of the balloon was 172.7 N, giving a net lift of 136.5 N after the instrument platform weight was accounted for.
The tether, made by Cortland Cable, is made of Plasma® a UHMW polyethelene fiber characterized by a very high elastic modulus and strength to weight ratio. The required cable diameter was determined assuming an operational wind gust of up to 15 m/s. To distribute the load on the aerostat, the single 1.5 mm tether splits into four 1 mm secondary lines (shown on Fig.2) at the confluence point.
The CSW-1 winch model made by A.G.O. Environmental Electronics Ltd shown in Fig. 2, was selected for the experiment. The main advantages of the CSW-1 winch are that it is compact and is battery powered. These features allow flexibility in the choice of launch site.
Figure 2: Picture of the experimental facility showing a zoom of the winch and the tether.
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Figure 4: The left picture shows a technical drawing of the instrument platform. The picture on the right is a bottom view of the aerostat showing the platform attachment.
B. Sensors and Data Acquisition System The sensors, shown in Fig. 3, are selected to
measure the motion of the aerostat, the tension in the tether, and the environmental conditions, without altering the natural behaviour of the aerostat. An accuracy of 5 cm on the position was required in order to allow characterization of the aerostat motion. One commercially available technology that can achieve this accuracy over long distances is differential GPS or DGPS22. The DGPS hardware was chosen based on compactness and accuracy, and consisted of two GPS receivers and two antennas from NovAtel. The base receiver is the DL-4 plus which uses an OEM4-G2L card. The roving receiver, mounted on the aerostat, had to be low power, compact and very light. NovAtel’s FlexPak receiver fulfilled these requirements. The position data collected by the two receivers was post-processed using the GrafNav DGPS software from Waypoint Consulting Inc. GrafNav can post-process kinematic baseline to cm level accuracy and static baseline to sub-millimetre accuracy. With this software, a position accuracy of about 5 cm was achieved at a rate of 10 Hz.
An MLP75 load cell from Transducer Techniques was selected to measure the tension in the tether. This lightweight and compact unit can measure loads up to 75 pounds with a safe overload of 150%. The load cell was mounted at the confluence point where the main tether splits into four secondary lines. Its analog signal was digitized to RS-232 at a rate of 5 Hz using a SY016 digital conditioner and amplifier from Synectic Design.
Wind monitoring was performed using three Young 05103-10 anemometers from Campbell Scientific located on a tower at 3, 5 and 10m above ground. Each sensor recorded wind speed and wind direction relative to the true north. The raw voltage signals from the sensors were sampled and stored at 300 hertz using a PMD-1208FS digitizer from Measurement Computing. They were then decoded into values of wind speed at a rate of 5 Hz, and wind direction at a rate of 0.5 Hz.
A WLAN was assembled to transmit the roving GPS and the tension data to the ground station. It consisted of a DataHunter dual RS-232 SeriaLan and a D-Link DI-614 802.11g wireless router used as an access point (Fig. 3). The SeriaLan ports were set to 115200 baud rate to comply with the GPS transmission requirement.
The airborne system was powered using a set of 8 alkaline D-cell 1.5 Volt batteries in series, which gave an autonomy of about 6.5 hours. The power consumption of the complete system was calculated to be about 4.75 W. A multithreaded software called DATAS (Dynamics Acquisition of Tethered Aerostat System) was developed in the Visual C++ environment, to acquire, store and synchronize the data coming from the different sensors. Time stamps for the different sensors were all synchronized on the GPS time (GPST).
C. Instrument Platform In order to carry the instruments aloft, a
platform, made of 3/8 inch thick construction grade clear acrylic, was designed and constructed as shown in Fig 4. A flexible vinyl membrane with a VelcroTM patch is attached to the platform using eight aluminum rods. This patch can in turn be attached solidly to the aerostat. The roving
Figure 3: The sensors/communication system
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GPS antenna sat between the patch and the aerostat skin. To further reduce the relative motion of the platform with respect to the aerostat, the acrylic base of the platform was attached to the aerostat through a set of four 1 mm stabilization Plasma® lines under tension as shown on the right of Fig. 4.
While designing the instrument platform, particular care was devoted to minimize its effect on the aerostat’s natural behaviour. A Pro/EngineeringTM technical drawing of the system was assembled and used to estimate the effect of the instrument platform on the aerostat center of gravity and moments of inertia.
III. Data Analysis Two essential questions were to be answered by analysis of the experimental data. First, how does the balloon
move in turbulent wind and second, what is the nature of the forces acting on the system? The answer to the second question will provide information relevant to a simulation of the tethered aerostat system. The answer to the first question will give useful insight in our understanding of the behaviour of a spherical object in a turbulent flow and will be used in Section IV to validate simulation outputs.
A. Experimental Data The goal of the experiments was to collect position and tension data for a broad range of wind conditions. There
were a total of 9 successful flights, summarized in Table 1, where, for each flight are shown: the date of the flight;
refU the mean horizontal wind speed at 10 m height and its dispersion σU; the mean wind direction wθ and its
dispersion σθ. For Flight 1-3, the wind direction sensing was not operational.
Each flight usually lasted about 30 minutes during which the aerostat was flown at 15, 30 and 45 m tether length.
After post-processing with GrafNav, the position of the aerostat is obtained in a local inertial coordinate frame. In this frame, the coordinates of the aerostat are given by the relative position of the roving GPS antenna (on the instrument platform) with respect to the base GPS antenna. In order calculate variables such as the tether angle, the aerostat position was redefined relative to the winch (rWA) as shown in Fig. 5. Table 2 shows the characteristics of the position data at the different cable lengths for Flight 9. The mean and rms position were calculated from the instantaneous positions obtained from GrafNav over for each constant tether length segment. The components of the instantaneous position were also used to calculate the tether angle α, which was then also averaged over each constant tether length segment.
As noted earlier, the wind data was measured at 3, 5 and 10 m heights. However the real interest is at the aerostat height and therefore it is necessary to extrapolate the wind to higher altitudes. The mean wind speed with height was found to follow a power law boundary layer model23 reasonably well, expressed as
( )m
ref ref
U z zU z
⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠ (1)
Table 1: Wind condition for the different days of experimentation
Figure 5: Schematic of the relative positions of the components
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where z is the height of interest, zref the reference height which is usually taken as 10 m by meteorologists23, U(z) is the wind speed at the height of interest and Uref is the wind speed at the reference height. The exponent m was determined by fitting the power law profile to the experimental mean wind speed. For Flight 9, m was calculated to be 0.11 which is consistent with a flat field of low grass. In order to determine the wind speed at the balloon’s altitude, the wind time history at 10 m height (zref) was smoothed using a 10th order polynomial as shown in Fig. 6. This was considered a low enough order polynomial to filter out the turbulent gusts. The power law, with the proper value of exponent m for that flight, was then applied to the smoothed wind speed at the reference height of 10 m to extrapolate the wind speed at the average aerostat height corresponding to cable lengths L = 15, 30 and 45 meters, as shown in Fig. 6. These profiles were then used to get the mean wind U at the aerostat height for each flight segment.
B. Drag Force and Drag Coefficient Since the aerostat flies in an unsteady fluid flow,
calculation of the instantaneous drag force would require accurate knowledge of its accelerations. Instead, the mean drag force was calculated based upon 2 assumptions: (a) over a long period of time, the mean acceleration is zero, and (b) the mean z-component of the drag is negligible.
The first assumption is bound to be true for a tethered aerostat system since a non-zero mean acceleration over a long time period would mean that the balloon is moving away from its attachment point. For the second assumption, it is generally considered acceptable to neglect the vertical mean wind in the lower planetary boundary layer24. With these assumptions, the analysis can be reduced to a simple static system where the tether angle α is the average angle between the vertical and the tether line, the drag force is the average drag force and the lift the average lift as shown in Fig. 7. From static equilibrium, the average drag force is simply given by
( )tanD LF F α= (2)
where DF is the average drag force, LF is the average free lift (136.5 N) and α is the average angle obtained from the position measurements. The average drag force for each flight segment was obtained using eq. (2).
One of the most important physical parameters for the characterization of an aerostat is its drag coefficient. Mean values of drag coefficient DC were extracted from the mean drag force using
221 rel
DD UA
FC
ρ= (3)
where relU is the average velocity of the air relative to the
balloon and is simply equal to the average wind speed U since the mean aerostat velocity is zero. According to published data, the drag coefficient of a smooth fixed sphere, ranges from 0.40 at
Figure 6: Plot of the fitted wind speed for Flight 9
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Reynolds numbers less than 2 × 105 (known as subcritical), decreasing to 0.15 at Reynolds numbers greater than 3.5 × 105 (known as supercritical). Our experiments were all in the supercritical range. The mean drag coefficient of the tethered balloon was calculated to be 0.56 and was found relatively constant over the range of Reynolds numbers studied. This drag coefficient is about 4 times higher than for a fixed sphere in supercritical flow. This can be explained by the high level of upstream turbulence, the roughness of the aerostat and by the sphere’s oscillations. For comparison, Williamson et al. found that the oscillation of a tethered sphere in subcritical flow was about 0.75--
roughly twice the drag coefficient of a fixed sphere25. Thus it appears that the decrease in drag coefficient experienced by fixed sphere in an ideal flow beyond the critical Reynolds number may be much less in the case of a tethered aerostat in turbulent wind. This behaviour is consistent with Golsdtein’s26 results who observed a nearly constant sphere drag coefficient when the flow turbulence was increased. Similar results were found by increasing the surface roughness26. Fig. 8 shows a plot of the drag coefficients obtained compared to values from Wieselsberger27 for a fixed sphere in a wind tunnel and Scoggins3 for free floating smooth and rough spheres. It is interesting to note that our drag coefficients values are between those of the rough sphere and smooth sphere of Scoggins.
C. Aerostat Oscillations As shown by Govardhan & Williamson7, a tethered sphere in a steady fluid flow will tend to oscillate both in the streamwise and transverse directions. The present experiment shows that a tethered sphere also oscillates in an unsteady flow such as an outdoor wind. In order to study the oscillatory motion of the balloon, it is convenient to decompose the motion into directions along and transverse to the mean flow. To do this, the aerostat position was expressed in a reference frame where the x-axis is aligned in the mean direction of the wind. We found no obvious periodic behavior in the motion of the balloon in the streamwise direction, most probably due to the fluctuating wind speed. However, the transverse oscillations are quite well defined even though their amplitudes vary over time as shown in Fig. 9 for Flight 9.
The amplitude of the transverse oscillations of the aerostat can be characterized by the normalized amplitude A* defined as
* 2 yADσ ′= (4)
where σy is the rms value of the y-position and D is the aerostat diameter. This quantity is plotted in Fig. 10 as function of the reduced velocity
( )redn
UU f D= where fn is the natural pendulum frequency given by
Lm
Ff
en
L
π21
= (5)
where me is the aerostat mass, including added mass (half the displaced air mass for a sphere). Also shown on the figure are the results of Govardhan and Williamson7. Our results match reasonably well for reduced velocities ranging from 5 to 10. However, at reduced velocities above 10, our amplitude results do not exhibit the saturation found by Govardhan and Williamson for buoyant spheres. Our data for different L/D collapse onto a single curve, as
Figure 9: Oscillatory behaviour of the aerostat for Flight 9
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was the case for the results of Williamson and Govardhan, in which the normalized amplitude was found to be independent of the tether length when plotted against the reduced velocity.
Power spectra of the balloon transverse position for the different flights all exhibit a sharp peak within 30% of the system’s natural pendulum frequency given eq. (5). As expected, the dominant frequency of the power spectrum decreases as the tether length increases. There was no clear relation between the frequency of the transverse oscillation and the reduced velocity or the Reynolds number.
D. Nature of the Oscillations The characteristics of the transverse oscillations gives insight into the type of the force acting on the aerostat. It
first suggests that a periodic force acts transverse to the mean flow. Also, it must have a strong component close to the system’s natural pendulum frequency in order to excite the system at this frequency. Williamson et al. 7, 9, 10 have hypothesized that the oscillation of a tethered sphere in a fluid flow results from a periodic forcing created by vortex shedding at a frequency close to natural frequency, known as the “lock in” phenomenon. This scenario is unlikely in our experiment since they were performed at supercritical Reynolds number (Re > 2.5×105) where there exists no clear structure in the wake of the sphere28.
Other possible scenarios explaining the nature of the lateral forces, such as those conjectured by Kruchinin2 or Willmarth and Enlow29, were investigated and rejected. The most plausible explanation comes from Jauvtis et al9. who discovered in 2001 an unexpected mode of oscillation of the tethered sphere at high-reduced velocity---from 15 to 40---outside the ‘lock in’ regime. This falls almost exactly in the range of operation of our aerostat. Govardhan & Williamson10 demonstrated in 2005 that these oscillations are due to ‘movement induced excitation’. They also found that this phenomenon was independent of the Reynolds number over which they performed the experiments that is, Re = 3000-9000. If we assume that this phenomenon is fully independent of the Reynolds number, it would provide an explanation for the transverse oscillations that our aerostat exhibits.
IV. Simulation of a Tethered Spherical Aerostat We now proceed to develop a simulation of the tethered spherical aerostat and compare its results to those of our
experiment. Once validated, this simulation could be used as a tool to better understand the behaviour of the system.
A. Aerostat/Tether Model The simulation developed in this work was based on an existing numerical model created by Lambert18. It uses a
lumped-mass model of the cable in which the cable is first discretized into a series of n elements and the mass of each element is lumped into its n+1 end nodes as shown in Fig. 11 Two types of forces were taken into account in modeling the tether: external and internal forces. The external forces consist of aerodynamic drag and gravity. The internal forces include the internal damping modeled as a viscous dashpot and the axial stiffness modeled as a linear spring. The lumped mass approximation allows for the motion of each node to be calculated in each of the three degrees of freedom resulting in a set of 3n equations of motion (not including the fixed node 0).
Lambert’s model was not capable of accounting for the lowering of the aerostat CG due to the presence of the instrument platform, and the new model takes this effect into account. Once the aerodynamic center of pressure and the CG are no longer coincident, rotational moments are introduced. In the proposed model, the aerostat and the secondary lines are treated as a single rigid body with the last node located at the CG, as shown in Fig. 11.
Figure 11: Sketch of the proposed aerostat model.
Figure 10: Comparison normalized oscillation amplitude with those of Govardhan and Williamson.
0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 00
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
4 . 5
5
5 . 5
R e d u c e d V e l o c i t y U / ( fn * D )
A*
L / D = 4 . 2 7L / D = 8 . 5 5L / D = 1 2 . 8 3G o va r d h a n a n d W i l l i a m s o n ( 1 9 9 7 ) L / D = 8 . 9 3
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The dynamics simulation is obtained by setting up and simultaneously solving all the equations of motion (EOM) in 3-D space using a 4th order Runge-Kutta integration scheme. These include the 3n EOM of the tether nodes and the 6 translational and rotational EOM of the aerostat. The coupling of the tether model and the aerostat model was achieved by connecting the upper end of the last element of the tether to the confluence point of the aerostat as shown in Fig. 11. Also shown on the figure is the tension force from the nth element FT. This force is included in the equations of motion of the aerostat.
The motion of the aerostat is defined as the relative position and orientation of a body-fixed coordinate frame attached to the aerostat CG with respect to an inertial frame located at an arbitrary point on the ground as shown in Fig. 12. The equations of motion of the aerostat are obtained in the body frame as discussed by Lambert1 and are written as:
[ ][ ][ ]
sin sin
cos sin cos sin
cos cos cos cos
e B Dx Tx
e B Dy Ty
e B Dz Tz
m qw rv u F mg F F
m ru pw v F mg F F
m pv qu w F mg F F
θ θ
θ φ θ φ
θ φ θ φ
− + = − + + +
− + = − + +
− + = − + +
&
&
&
(6)
where FB is the buoyancy force, mg is the forced gravity, FDx, FDy and FDz are the components of the drag force and FTx, FTy and Ftz are the components if the tether tension. The orientation of the of the aerostat is represented using a Z-Y-X Euler angle set ( ), ,ψ θ φ as discussed by Etkin30. The component form of the rotational equations of motion is derived using Euler’s equation and is expressed in component form as:
( )( )( )
xx zz yy Bx Tx Dx
yy xx zz By Ty Dy
zz yy xx Bz Tz Dz
I p I I qr M M M
I q I I pr M M M
I r I I pq M M M
+ − = + +
+ − = + +
+ − = + +
&
&
&
(7)
where MBx, MBy and MBz are the component of = ×B B BM r F and MTx, MTy and MTz are the component of = ×T T TM r F and finaly, MDx, MDy and MDz are the component of = ×D D DM r F . The z-axis of the body frame is
directed along the vertical axis of symmetry of the balloon as shown in Fig 12. This convention results in moment arms rB, rD and rT all being aligned with the z body axis.
E. Wind Model The wind was modeled using a constant mean wind speed on which turbulent gusts are superimposed. The
mean wind varies with height according to the power law given by eq. (1) and the exponent m was found to vary between 0.11 and 0.14, depending on the day of acquisition. The turbulent gusts were generated based on desired statistical properties including the turbulence intensity, scale length and spectra31. The wind properties were calculated at each time step, as discussed by Nahon et al.31 though the turbulent intensities were set based on E.S.D.U.32 data to match our experimental conditions.
Fig. 13 shows power spectral densities of the wind along the mean wind direction, for two of the flights. The simulated turbulence shows a good match of the measured turbulence throughout the bandwidth of our sensor.
Figure 13: Comparison of experimental and simulated wind spectrum
Figure 12: Idealized sketch of the aerostat system showing the body-fixed and inertial frames
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F. Lateral Forces Govardhan & Williamson10 discussed a method to characterize the forces responsible for the transverse oscillation by treating the aerostat as a simple second order system where the transverse motion and vortex force are approximated by sine functions. Based on their results, the transverse force discussed in Section III D was implemented in our simulation as follows:
1) The simulation is started including only wind turbulence, and no additional transverse force. 2) The rms value of y is computed at each time step and used to calculate the lateral motion amplitude
2amp rmsY y= .
3) As the motion amplitude increases the magnitude of the sinusoidal lateral force ( )0 sinF tω increases
according to 20 2 amp e nF Y m ζω=
4) When maximum force amplitude Fmax is reached, the magnitude of the sinusoidal lateral force is set constant to Fmax. Fmax is calculated from Govardhan and Williamson10 results for a tethered sphere at similar reduced velocity.
G. Physical parameters In order to solve the equations of motion, we must specify the physical parameters consistent with the aerostat
system used in the experiment. Most physical parameters such the aerostat net lift and diameter were obtained by direct measurement. To estimate the remaining parameters, a three-dimensional CAD model was generated using PRO-E. The CAD model was constructed to accurately represent the actual aerostat, complete with the thin-walled shell of the hull, the contained helium, which is considered to be rigid, and the instrument platform with the sensors positioned appropriately. The mass, volume and moments of inertia of the aerostat are presented in Table 3 as well as various geometrical parameters.
H. Nonlinear Simulation Results and Comparison The ultimate validation of a model comes from its ability to represent reality. Flight 9 was selected as the basis
for comparison because the three tether length flight segments are sufficiently representative of the results for the different days of experimentation. For the comparison, the inertial x-axis is aligned along the direction of the mean wind, the y-axis is transverse to the wind and the z-axis is directed along the gravity vector.
Since the wind is responsible for the majority of the dynamics of the system, the following steps were taken to ensure that the wind field in the simulation was statistically similar to the in-field conditions. First, the mean wind at 10m Uref was determined as discussed in Section III A. Then the vertical profile of the mean wind was generated using a power-law boundary layer profile as discussed in that section. Finally, the turbulence intensity was adjusted in the simulation to match the measured turbulence for the test period.
Table 4 presents a comparison of measured and simulated results for aerostat position, tether tension and normalized amplitude A* of the lateral motion. Fig. 14 shows time histories of the measured and simulated results for Flight 9 at a cable length of 15 m for a representative period of 200 s. The time history of the simulated and experimental results is not expected be identical since the simulated wind turbulence is only statistically similar to the experimental one; not identical. An exact comparison would only be possible if a precise measurement of the wind at the aerostat altitude were imported in the simulation.
Table 3: Physical parameters of the aerostat
Parameter Value Parameter Value
Aerostat diameter 3.5 m
Confluence point (from
CG)
rTz
-3.66 m
Aerostat volume 22.45 m3
Centre of buoyancy and
of pressure (from CG)
rBz, rDz
0.52 m
Platform mass 3.69 kg
Centre of gravity (from
bottom center ) zcm
1.23 m
Total system weight 137.3 N
Drag Coefficient
(CD) 0.56
Helium density 0.169 kg/ m3 Ixx 25.64 kg⋅m2
Air density 1.229 kg/ m3 Iyy 25.64 kg⋅m2
Net lift 136.5 N Izz 16.44 kg⋅m2
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Although the simulation and experiment exhibit similar behavior, the difference between the results is as high as 60% for certain variables. The results for the mean x-position (along-wind) are quite good, but the error in the standard deviation seems to increase with tether length, with the experimental standard deviation being higher. This could be attributed to the fact the experimental wind direction was varying. As a result, the transverse oscillations might appear in the experimental x-position as can be seen in the top graph of Fig 14. The experimental z-motion and tether tension contain clear large amplitude, high frequency spikes that are not predicted by the simulation. These lead to large differences in the standard deviation of z-position and tension. The spikes are believed to be caused by wind gusts in the z-direction which are not properly replicated in the simulation. At present, there is no way to verify this since we have no means to measure the z-component of the wind speed in the field. The simulated and experimental mean y-position, are off by up to 1 m and the oscillation amplitudes are smaller in the simulation than in the experiment. This suggests that the amplitude of the periodic lateral force discussed in Section IV F is larger for our experiment than for Williamson’s tethered sphere. A stronger forcing input in the simulation would elicit larger oscillations and possibly a lower mean y-motion. No other source of data was found on tethered spheres for the flow regime in which our experiment is performed and it is therefore difficult to determine a better forcing input than that derived from Williamson’s work.
Table 4: Comparison of experimental and simulated data
Figure 14: Comparison of simulated and experimental results
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V. Linear Model A non-linear model is interesting in the sense that it allows us to study the dynamics of our system in the time
domain. However, it does not allow a more comprehensive analysis nor can it provide clear conclusions concerning the modes of oscillation. A linear model was therefore developed to analyze the model in the frequency domain.
The dynamics model of a system can be thought as a set of functional relationships where the derivative of each state variable is dependent on the full set of state variables, namely f(X)X =& . For our aerostat system, the state variables include the positions and velocities in the three orthogonal directions (x, y, z) of each of the n-1 tether nodes, and twelve translational and rotational variables for the aerostat itself, thus
1 1 1 1 1 1[ , , , , , ,..., , , , , , , , , , , , ]Tn n n n n nx x y y z z x x y y z z φ φ θ θ ψ ψ=X & & && & & && & and there are a total 6n + 6 states where n is the
number of nodes. The linear form of the equations of motion of the aerostat system can then be expressed as =X AX& where the state matrix A has the following form:
1 1 1
1 1
2 2 2
1 1
6 6 6 6 6 6
1 1
n n n
f f fx xf f fx x
f f fx x
ψ
ψ
ψ+ + +
∂ ∂ ∂⎡ ⎤⎢ ⎥∂ ∂ ∂⎢ ⎥
∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂= ⎢ ⎥⎢ ⎥⎢ ⎥∂ ∂ ∂⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦
A
K&
K&
M M O M
K&
(8)
with fn being the elements of f(X). The elements of A are obtained by finite difference of the nonlinear differential equations, as was done by Lambert1 for a streamlined aerostat system. In order to assess the validity of the model, the linear and non-linear aerostat responses to very small perturbations of the 6 aerostat state variables were compared. The agreement between the two models was excellent33 indicating the success of the linearization process. The spherical aerostat model has a total of 3n+3 degrees of freedom and therefore as many modes of oscillations. Each mode is defined either by a complex conjugate pair of eigenvalues of the matrix A34, and its frequency and damping can be deduced from those eigenvalues. Each element of the eigenvector of dimension 6n+6 corresponding to a particular eigenvalue describes the magnitude and phase response of one of the state variables34 in that mode. As was the case in Lambert’s analysis of a tethered streamlined aerostat, it is possible to decouple the state variables into independent lateral and longitudinal independent subsets. Thus, the state vector X and state matrix A were reorganized into lateral and longitudinal subsets33 and the longitudinal and lateral subsystems were analyzed separately. The cable was discretized into n = 10 elements, yielding longitudinal and lateral subsystems of order 42 and 24 respectively. Matlab was used to find the eigenvalues and eigenvectors of the system, allowing the identification of the 33 modes of oscillation (21 longitudinal and 12 lateral). For the sake of brevity, only the four lowest longitudinal and lateral modes of oscillation are presented here graphically in Fig 15. It is worth mentioning that all the modes were stable. The results were compared to those obtained from the solution to the string equation (one dimensional wave equation)33, which in principle are only valid for zero wind speed. Table 5 shows numerical results for a case with a tether length L = 45 m and wind speed U = 1 m/s. The frequencies of oscillations obtained from the linear model show excellent agreement with the analytical solutions. These modes of oscillation were also observed experimentally, it was found that the transverse pendulum mode was dominant and its frequency was within 30% of that predicted. The axial spring mode was also observed in the spectral density of the tension and its frequency was within 50% of that predicted. The discrepancy can be attributed to uncertainties in the tether properties.
Figure 15: Modes of oscillation of a tethered sphere
American Institute of Aeronautics and Astronautics
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Expressions for the damping ratio of the lateral (transverse to wind) and longitudinal (along wind) pendulum modes can be obtained analytically by considering the aerostat system as a simple damped harmonic oscillator where the tether provides the restoring force and the damping is created by the projection of aerodynamic drag along the direction of motion of the aerostat33. The analytical expressions for the damping ratios are given by:
ne
Dy
ne
Dx
mAUC
mAUC
ωρ
ζ
ωρ
ζ
4
2
=
= (9)
The values of the analytical and simulated damping ratio for x motion and y motion of the aerostat with L = 45m and U = 1 m/s are presented in Table 5. They show excellent agreement, thus tending to demonstrate the validity of the linear model, and the underlying nonlinear model. The expression for the y damping coefficient given by eq. (9) implies that the damping of the lateral pendulum mode (transverse to wind) increases linearly with speed. For example, a wind speed U = 5 m/s would results in a damping ratio ζy = 0.9, which is close to
critical damping. Thus, in order to generate the transverse oscillation observed experimentally, a strong lateral forcing close to the pendulum frequency is required.
VI. Conclusion
The design and construction of a portable experimental facility for the characterization of the dynamics of tethered spherical aerostat was successfully achieved. The set-up was used to record the dynamics variables of a tethered spherical aerostat. The mean drag coefficient CD = 0.56 of the tethered sphere was determined using a quasi-static approximation, a value 3-4 times higher than for a fixed sphere at supercritical Reynolds numbers. Our experiment demonstrated that a tethered sphere in a turbulent flow field will oscillate strongly in the transverse direction. The amplitude of transverse oscillations was found to increase with increasing reduced velocity but was independent of tether length. Our normalized amplitudes match reasonably well those of Govardhan and Williamson for the lower reduced velocity (5-10) range, but they do not exhibit the saturation in amplitude of A*=1 at reduced velocities higher than 13. The most plausible explanation for the strong oscillations observed experimentally is that they result from a ‘movement induced excitation’.
A simulation of the tethered spherical aerostat was assembled based on a prior model created by Lambert. An attempt to validate the non-linear dynamics simulation was made by comparing the simulation results to experimental results of three flights. It was shown that the simulation is in good agreement with the experiment although better understanding of the lateral forces and z-wind gusts is required. A numerical linearization of the system’s equations of motion was performed. The frequencies of the four lowest modes of oscillation were compared to corresponding frequencies obtained analytically and excellent agreement was observed. The damping of the pendulum mode was found to be very high, which suggests that the lateral oscillations seen in the experiment are caused by strong forcing at that frequency, as conjectured by Govardhan & Williamson.
References
1Lambert, C. M., "Dynamics Modeling and Conceptual Design of Multi-Tethered Aerostat System," M.Sc. Thesis, Dept. of Mechanical Engineering, Universtiy of Victoria, Victoria, Canada, 2002. 2Kruchinin, N. N., "Transverse Oscillations of Pendulous Sphere in a Flow," Moscow Mechanics Bulletin (English Translation of Vestnik Moskovskogo Universiteta, Mekhanika), Vol. 32, No. 3-4, 1977, pp. 8-13. 3Scoggins, J. R., "Sphere Behavior and the Measurement of Wind Profile," NASA TN D-3994, Vol. No. 1967, pp. 1-53.
Mode name
Simulated frequency ω (rad/s)
Analytical frequency ω
(rad/s)
Axial Pendulum (long.) 7.31 7.28
Transverse Tether 2nd harmonic
(long. and lat.) 41.45 42.28
Transverse Tether fundamental
(long. and lat.) 21.05 21.14
Damping ratio Simulated Value
Theoretical Value
xζ 0.37 0.36
yζ 0.l8 0.18
Table 5: Theoretical and simulated frequencies and damping ratio
American Institute of Aeronautics and Astronautics
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4Harleman, D. R. F. and Shapiro, W. C., "The Dynamics of a Submerged Moor Sphere in Oscillatory Waves," Coastal Engineering, Vol. 2, No. 1961, pp. 746-765. 5Shi-Igai, H. and Kono, T., "Study on Vibration of Submerged Spheres Caused by Surface Waves," Coastal Engineering in Japan, Vol. 12, No. 1969, pp. 29-40. 6Ogihara, K., "Theoretical Analysis on the Transverse Motion of a Buoy by a Surface Wave," Applied Ocean Research, Vol. 2, No. 1980, pp. 51-56. 7Govardhan, R. and Williamson, C. H. K., "Vortex-Induced Motions of a Tethered Sphere," Journal of Wind Engineering and Industrial Aerodynamics, Vol. 69-71, No. 1997, pp. 375. 8Williamson, C. H. K. and Govardhan, R., "Dynamics and Forcing of a Tethered Sphere in a Fluid Flow," Journal of Fluids and Structures, Vol. 11, No. 3, 1997, pp. 293. 9Jauvtis, N., Govardhan, R. and Williamson, C. H. K., "Multiple Modes of Vortex-Induced Vibration of a Sphere," Journal of Fluid and Structures, Vol. 15, No. 2001, pp. 555-563. 10Govardhan, R. N. and Williamson, C. H. K., "Vortex-Induced Vibrations of a Sphere," Journal of Fluid Mechanics, Vol. 531, No. 2005, pp. 11. 11Naudasher, E. and Rockwell, D., Flow Induced Vibrations: An Engineering Guide, Balkema, 1994 12Bearman, P. W., "Vortex Shedding from Oscillating Bluff Bodies," Annual Review of Fluid Mechanics, Vol. 16, No. 1984, pp. 195. 13Anagnostopoulos, P., Flow-Induced Vibrations in Engineering Practice, WIT Press, Boston, 2002 14Jones, S. P., "A Stability Analysis for Tethered Aerodynamically Shaped Balloons," AIAA Journal of Aircraft, Vol. 9, No. 9, 1972, pp. 646-651. 15Redd, T., Bland, R. and Bennet, R. M., "Stability Analysis and Trend Study of a Balloon Tethered in a Wind, with Experimental Comparisons," NASA TN D-7272, Vol. No. 1973, pp. 1-109. 16Jones, S. P. and Krausman, J. A., "Nonlinear Dynamic Simulation of a Tethered Aerostat," Vol. 19, No. 8, 1982, pp. 679. 17Jones, S. P. and Delaurier, J. D., "Aerodynamic Estimation Techniques for Aerostat and Airships," AIAA Journal of Aircraft, Vol. 20, No. 2, 1983, pp. 120-126. 18Lambert, C. and Nahon, M., "Stability Analysis of a Tethered Aerostat," AIAA Journal of Aircraft, Vol. 40, No. 4, 2003, pp. 705. 19Stanney, K. A. and Rahn, C. D., "Response of a Tethered Aerostat to Simulated Turbulence," American Society of Mechanical Engineers, Design Engineering Division , Vol. 117, Anaheim, United States, 2004, pp. 321 20Lambert, C. M. and Nahon, M., "Study of a Multi-Tethered Aerostat System - Experimental Observations and Model Validation," submitted to AIAA Journal of Aircraft, Vol. No. 2005, pp. 21Dewdney, P., Nahon, M. and Veidt, B., "The Large Adaptive Reflector: A Giant Radio Telescope with an Aero Twist," Canadian Aeronautics and Space Journal, Vol. 48, No. 4, 2002, pp. 239. 22Hofmann-Wellenhof, B., Lichtenegger, H. and Collins, J., Global Positioning System: Theory and Practice, Springer-Verlag, Wien; New York, 2001 23Guyot, G., Physics of the Environment and Climate, Wiley and Sons, Toronto, 1998 24Stull, R. B., An Introduction to Boundary Layer Meteorology, Kluwer Academic Press, 1988 25Schlichting, H., Boundary Layer Theory, McGraw-Hill, New York, 1979 26Goldstein, S. and Aeronautical Research Council (Great Britain), Modern Developments in Fluid Dynamics: An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes, Dover Publications, New York, 1965 27Wieselsberger, C. V., "Weitere Festllungen Uber Dei Geset Des Flussigkeits Un Luftwiderstandes," Physikalische Zeitschrift, Vol. 23, 1922, pp. 219-224. 28Taneda, S., "Visual Observations of the Flow Past a Sphere at Reynolds Numbers between 10^4 and 10^6," Vol. 85, No. 1, 1978, pp. 192. 29Willmarth, W. W. and Enlow, R. L., "Aerodynamic Lift and Moment Fluctuations of a Sphere," Journal of Fluid Mechanics, Vol. 36, No. 3, 1969, pp. 417-432. 30Etkin, B., "Stability of a Towed Body," AIAA Journal of Aircraft, Vol. 35, No. 2, 1998, pp. 197-205. 31Nahon, M., Gilardi, G. and Lambert, C., "Dynamics/Control of a Radio Telescope Receiver Supported by a Tethered Aerostat," Journal of Guidance, Control, and Dynamics, Vol. 25, No. 6, 2002, pp. 1107. 32Enginneering-Sciences-Data-Unit, "Characteristics of Atmospheric Turbulence near the Ground," Item Number 74031, No. 1974, pp. 33Coulombe-Pontbriand, P., "Simulation and Experimental Validation of Tethered Spherical Aerostat Model," M.Sc. Thesis, Dept. of Mechanical Engineering, McGill University, Montreal, Canada, 2005. 34Inman, D. J., Enginneering Vibration, Pearson/Prentice Hall, Upper Saddle River; NJ, 2001
