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American Institute of Aeronautics and Astronautics

STABILITY ANALYSIS OF A TETHERED AEROSTAT

Casey Lambert1 Meyer Nahon2

Department of Mechanical Engineering Department of Mechanical EngineeringUniversity of Victoria McGill University

P.O. Box 3055 817 Sherbrooke St. W.Victoria, BC, Canada, V8W 3P6 Montreal, QC, Canada, H3A 2K6

AbstractThis paper presents a dynamics analysis of a streamlined aerostat tethered to the ground by a single tether. A nonlineardynamics model of this system is first assembled. The tether is modeled using a lumped mass approach, and the viscoelasticproperties of the tether are included. The aerostat is modeled using a component breakdown approach. The dynamicsequations of the cable and aerostat are then assembled into a single system of nonlinear differential equations. Alinearization of this system is then performed using a finite difference approach. The resulting linear equations of motionsare decoupled into longitudinal and lateral subsets. The stability properties of each subset is then studied as a function ofwindspeed.

1 AIAA Student Member2 AIAA Senior Member

IntroductionTethered aerostats are known to be useful inapplications where a payload must be deployed ataltitude for long durations. In these applications, theenergy consumed (and the resulting refuelingrequirement) by a powered heavier-than-air craft,renders that platform less competitive in relation to anaerostat which consumes no energy. However, tetheredaerostats can be difficult to deploy and operate due totheir large size, sensitivity to environmental conditions,and particular dynamics characteristics.

A group of radio astronomers at the NationalResearch Council in Canada is interested in using atethered aerostat system to support the receiver of alarge scale radio telescope1. As part of the proof-of-concept experiment for this system, they are presentlydeploying a tethered aerostat in Penticton, BritishColumbia, to study its performance in this application.A parallel analytical/computational study is beingconducted to study this system’s dynamiccharacteristics.

Previous dynamics investigations of tetheredaerostats can be separated into nonlinear and linearstudies. DeLaurier2 was first to study the non-lineardynamics of an aerostat attached to a comprehensivecable model. This initial work considered only 2-Dmotion and steady state wind conditions. DeLaurieralso later addressed the effect of turbulence3. Thestability of the system was analyzed by showing thatthe motion decoupled into separate lateral andlongitudinal motions. Lateral instabilities at low windspeeds were predicted (though later conversations withthe author indicate these may have been due to spuriousresults). Progress with the dynamics modeling of a

tethered aerostat was made by Jones and Krausman4

when a 3-D nonlinear dynamics model with a lumpedmass discretized tether was established. Jones andDelaurier5 built on this basic model by introducing asegmented panel method for modeling the aerostat. Thisentailed dividing the aerostat into vertical slices toaccount for the effects of turbulence variations alongthe length of the hull. Another 3-D nonlinear dynamicsmodel of a tethered aerostat was developed byHumphreys6. This model used a single partialdifferential equation to relate the motion and forcesalong the tether. Experimental validation of thisdynamics model was achieved by performing tests witha scaled model in a tow tank.

In 1973, a linear model of a tethered aerostat wasproposed by Redd et al7. Experimental data was used tovalidate their model in a steady wind. A study of thestability of the aerostat was performed, but theformulation of the linear model neglected the dynamiccross-coupling of the tether and the aerostat. Thenonlinear model developed by Jones and Delaurier5 wasused by Badesha and Jones8 to perform a linear stabilityanalysis of a large commercial aerostat by linearizingthe equations of motion of the nonlinear model. Thisanalysis included only pendulum modes and neglectedother modes of motion. The dynamics model developedby Badesha and Jones showed good agreement withexperimental data as presented by Jones and Shroeder9.In 1998, Etkin10 used a linear analysis to study thestability of a towed body. The stability of severaldifferent modes was studied as function of wind speed.Although the generalized formulation of this approachprovides relevance to a range of bodies constrained by acable including that of a buoyant tethered aerostat in a

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flow field, a detailed study of the dynamics of a specifictethered aerostat is necessary to establish confidence inthe prediction of the behavior of the system.

The present work focuses on an investigation of thedynamics of a streamlined aerostat on a single tether usingnonlinear modeling techniques and a linear stabilityanalysis. The linearization of the system was performedusing a finite difference approach rather than aconventional analytical method. The stability analysisincludes several longitudinal and lateral modes and alsoconsiders the cross-coupling between the tether andaerostat. Section 2 discusses the development of thenonlinear dynamics model of the cable and aerostat. InSection 3, the nonlinear model is linearized numericallyand the effects on stability of changing various systemparameters such as tether length and steady state pitchangle are investigated.

Dynamics ModelA 2-D schematic of the system model, which consistsof the tether and the aerostat, is shown in Figure 1. Theaerostat is modeled as a single body at the upper nodeof the leash, subject to buoyancy, aerodynamic drag andgravity.

aerostat

node 1

element 1 node 2

element 2

node n

element n node n+1

Wind

Figure 1: Discrete implementation of tethered aerostatdynamics model.

Dynamics of a Streamlined AerostatA model of the dynamics of a streamlined aerostat wasdeveloped to study its behavior in various windconditions. The aerostat model is based on astreamlined aerostat manufactured by Aeros Flightcamof Canoga Park, California, shown in Fig. 2. Thecentral goal of this investigation is the stability of theaerostat on a single tether.

The parameters of interest of this study are themotion of the aerostat and the forces generated duringits motion. The aerostat is considered to be rigid and iscapable of full 6 degrees of freedom motion in 3Dspace. The output of the simulation is the translationaland rotational position and velocity of the aerostat andthe input is a set of initial conditions and a wind field.

The methodology for the model development willbe presented in two parts. The first is the derivation ofthe mathematical equations that govern the motion ofthe aerostat. The second part describes the process ofdetermining the aerodynamic parameters of theaerostat.

Figure 2: Aeros Flightcam aerostat.

Equations of MotionThe dynamic simulation of the aerostat is obtained bysetting up and solving the equations of motion in 3-Dspace. The motion of the aerostat is described as therelative position and velocity of a body-fixedcoordinate frame to an inertial coordinate frame. Thebody frame is attached to the aerostat’s centre ofgravity and the inertial frame is fixed to an arbitrarypoint on the ground. A diagram illustrating thesereference frames in relationship to the aerostat is shownin Figure 2. The translational motion is governed byNewton’s Second Law and can be written as:

I Im�F awhere FI is the net force applied to the aerostat, m is itsmass and aI is the acceleration of the mass centre withrespect to an inertial frame. Because the aerodynamicforces will be calculated as components in the bodyframe, it becomes more convenient to solve for themotion variables also expressed in this frame. For thiscase, the acceleration is found by differentiating the

velocity with respect to the inertial frame, BddtV . This is

related to B

t�

�

V , the rate of change of the velocity as

seen in the body frame, as follows:B B B

Bd

m dt t�

��

F V V ω V

yB

xB

zB

zI

xI

yI

3


where VB = [u v w]T and � = [p q r]T is the angularvelocity of the aerostat. The rotational motion of theaerostat must satisfy Euler’s equation:

wherexx xy xz

cm cm cm cm yx yy yz

zx zy zz

I I II I II I I

� ��

� � � � � ��

M I ω ω I ω I�

where Icm is the inertia tensor and Mcm is the netexternal moment acting on the aerostat, both takenabout the mass centre. Due to the symmetry of theaerostat in the x – z plane, the components Ixy and Iyz ofthe inertia tensor are zero.

The forces and moments which influence theaerostat are due to one of the following sources:gravity, buoyancy, aerodynamic and tether tension. Thetranslational and rotational equations of motion of theaerostat can now be written as:

( ) sin( )

( )sin cos

( )( )cos cos

( )

b Hx Px Sx Ux Lx

b Hy Py Sy Uy Ly

b Hz Pz Sz Uz Lz

mg F F F F F Fm u qw rv

mg F F F F F F

m v ru pwmg F F F F F F

m w pv qu

�

� �

� �

� � � � � �

� � �

� � � � � � �

� � �

� � � � � � �

� � �

�

�

�

2 2

sin cos( ) ( )

cos cos sin

( ) ( )sin cos

( ) ( )

b b Hx Px Sx Ux Lx

xx yy zz xz

b b b b Hy Py Sy Uy Ly

yy zz xx xz

b b Hz Pz Sz Uz Lz

zz xx yy xz

z F M M M M MI p I I qr I r pq

x F z F M M M M M

I q I I pr I r px F M M M M M

I r I I pq I p qr

� �

� � �

� �

� � � � � �

� � � � �

� � � � � � �

� � � � �

� � � � � �

� � � � �

� �

�

� �

where FH, FP, FS, FU are the aerodynamic forcecontributions from the hull, port fin, starboard fin,upper fin, and FL is the force exerted by the leash on theaerostat while Fb is the buoyancy force applied in thepositive z-direction of the inertial frame and mg is theweight of the aerostat applied in the negative z-direction of the inertial frame. The angles �, �, �,represent the aerostat’s roll, pitch and yaw. The left-hand side of the equations represents the external forcesand moments acting on the aerostat while the right-handside represents the aerostat’s motion.

Component Breakdown MethodThe aerodynamic forces and moments on the aerostatare calculated by breaking down the aerostat intocomponents with known aerodynamic characteristics.This approach is based on the method developed byNahon for modeling of underwater vehicles11. Theindividual components are the hull and the three aft

fins. The method for calculating the effects of eachcomponent is as follows:� calculate the motion at a reference point on each

component,� calculate the local angle of attack and sideslip

angle,� calculate the lift and drag forces and moments,� transform these forces to the body frame,� sum up the forces and moments.

The details of this procedure are now discussed.

Aerodynamic Forces on the FinsFor the fins, the location of the centre of pressure istaken to be at their ¼ chord line midway from the baseto the tip. The velocity of the origin of the local framefor a particular fin, VL can be found using therelationship:

L B L� � �V V ω rwhere VB is the velocity at the origin of the body frame,while rL is the position vector from the origin of thebody frame to the origin of the local frame or,specifically, from the centre of mass of the body to thecentre of pressure of the component. Once the velocityof the centre of pressure of each component is found, itmust be expressed with respect to its local frame. Thelocal angle of attack, �i and sideslip angle, �i arecalculated for each component using the relationships:

1 1tan ( ) tan ( )i ii i

i i

w vu u

� ��

� �

For the vertical tail fin, definitions for the sideslipangle, � and the angle of attack, � are reversed sincethe fin is vertical rather than horizontal. A 2-Drepresentation of an airfoil representing the tail fins isshown in Figure 3.

xi

zi

� i

Li

Di

Vi

ui

vi

1/4chord

Figure 3: Lift and drag forces relative to fin motion.

The fins are approximated as NACA 0018 airfoilsand the forces on each fin are characterized as lift, Land drag, D. The lift and drag forces for the fins aredimensionalized using the following equations:

2 21 12 2i f i L i f i DL A V C D A V C� ��

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where � is the density of air, Af is the planform area ofthe fin, Vi is the local velocity of each fin,

2 2 2i i i iV u v w� � � and CL and CD are the lift and drag

coefficients. CL can be estimated using the empiricalrelationship between the 2-D lift coefficient and theangle of attack. A linear fit to the curve for the region0<�<�stall, where �stall is the stall angle, will be used tocalculate the 2-D lift coefficient. The region of the liftcurve past the stall angle, �>�stall, is taken to be flatwith the lift coefficient retaining its maximum value.This is considered reasonable in the case of low aspectratio airfoils12. The 3-D lift coefficient can becalculated for a particular aspect ratio using:

,422

L L L lAC C C CAAA

� � ��

��

��

where Cl� is the slope of the 2-D lift curve, A is theaspect ratio and CLα is the slope of a 3-D lift curve.

The drag coefficient, CD includes both parasite andinduced drag according to the equation:

2L

D DoCC CAe�

� �

where CDo is the parasite drag coefficient and e is theOswald’s efficiency factor. It is important that if theairfoil has exceeded its stall angle, the CL used in thisequation is calculated using the actual attack angle, � toreflect the additional drag beyond the stall point.

Using these equations, the lift and drag forces areoriented according to the relative velocity of the fin.The lift force, Li is directed normal to the direction ofmotion while the drag force, Di acts in the oppositedirection of the velocity vector as shown in Figure 3.The reason the direction of the lift force is directeddownwards which is opposite to the common depictionof lift is because of our convention for assigning the z-direction in an upward direction. For use with theequations of motion outlined earlier, these forces mustbe resolved into the aerostat’s body frame. This can beachieved by first performing a simple rotation about thefin’s local y-axis.

For inclusion with the rotational equations ofmotion, the resulting moment, Mi is calculated using across-product of the fin force vector with a positionvector from the centre of mass of the aerostat to thecentre of pressure of each fin as follows:

( )i L i B� �M r F

Aerodynamic Forces on the HullUsing the method from Jones and Delaurier5 theaerodynamic influence of the hull was estimated by alift and drag force and a pitch moment. The lift force,Lh and the drag force, Dh are considered to be applied at

the nose of the aerostat. The pitch moment about thenose, Mnose accounts for the pitching tendencies of theaerostat. These quantities are calculated using thefollowing equations:

3 1 1

1

: [( ) sin(2 ) cos( )2

( ) sin sin | |]

h o k

c h

L q k k I

Cd J

��

� �

� �

�

lift

2

3 1 1

: [( ) cos

( ) sin(2 ) cos( )]2

h o c o h

k

D q Cd S

k k I

�

��

�

� �

drag

3 1 3

2

: [( ) sin(2 ) cos( )2

( ) sin sin | |]

nose o k

c h

M q k k I

Cd J

��

� �

� � �

�

moment

where qo = �VB2/2 is the dynamic pressure; � is the

density of the atmosphere, VB = 2 2 2u v w� � is thetotal velocity of the aerostat; � is the hull angle ofattack; k1 and k3 are the axial and lateral added-masscoefficients respectively; �k is the hull efficiency factoraccounting for the effect of the fins on the hull; (Cdc)his the hull cross-flow drag coefficient, referenced to J1;(Cdh)o is the hull zero-angle axial drag coefficient,referenced to the hull reference area Sh = (hullvolume)2/3. Also,

3 0

hl dAI dd

� ��

� � and 1 0

hl

hdAI d Ad

��

� ��

2 02hlJ r d� �� and 1 0

2hlJ rd�� where A is the cross sectional area of the hull, � is theaxial distance along the hull from the nose and r is thehull radius. The values for I1, I3, J1 and J3 are based onthe geometry of the aerostat. In order to utilize theseequations it is required to split the aerostat into tworegions; the hull region which extends from the nose tothe starting point of the fins, and the fin region fromthis point to the tail. Figure 4 shows a 2-D diagram ofthe aerostat and the various aerodynamic parameters.The moment calculated about the nose, Mnose of theaerostat is not appropriate for our simulation since theequations of motion sum the moments at the centre ofgravity of the aerostat.

In order to account for this, the force/momentsystem Lh, Dh, Mnose was replaced by an equivalentsystem at the aerostat’s center of pressure, c.p. Thecenter of pressure is the point at which theforce/moment system has zero moment. It is locatedalong the centre-line of the aerostat at a distance fromthe nose which is determined at each time step of thesimulation. The distance from the nose to the c.p., xn iscalculated by equating Mnose to the moment generatedabout the nose by the lift force, Lh using the followingequation: ( )nose h nM L x� � �

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Lh

VB

Dh

Mnose�

�

lh

x

z

(a)

Lh

Dh-xn

rcp/cm

centre of pressure, c.p.

(b)Figure 4: Schematic of aerodynamic parameters of hull;(a) original parameters, (b) parameters with Mnosereplaced by forces applied at center of pressure.

Using the convention of Jones and Delaurier5, theorientation of the lift and drag forces on the hull followsa different convention from that of the fins. For the fins,the forces are directed according the fin’s motion, whilethe hull forces are directed according to its body axisindependent of the direction of motion. The lift forceacts perpendicular to the aerostat’s central axis (x-direction in the body frame) while the drag force acts inthe negative x-direction.

The configuration of the aerostat shown previouslyare for a 2-D representation. Figure 5 shows the motionvariables and hull forces for 3-D motion. The angle ofattack of the hull, � and the angle of the lift force in theyz-plane, � are found using the motion variables asfollows:

1 1cos tanB

u vandV w

� ��

� � � ��

� �� In order to represent the aerodynamic forces in the

body frame, the lift and drag forces are rotated by theangle �. For the rotational equations of motion, theresultant moment, Mcm about the centre of mass can befound using the equation: ( )� �cm cp/cm h BM r F

yB

xB

zB

�

uv

wDh

�

Lh

VB

�

Figure 5: Orientation of the hull motion and forces.

Added mass is included in the analysis for the hullby defining the effective mass of the aerostat, me as thesum of the true mass, m and the added mass, ma asfollows: e am m m� �

where ma = ki�V and ki is the added mass coefficient13

in the appropriate direction, and V is the hull volume.

Tether ModelThe tether is used to constrain the motion of the aerostatand this must be accounted for in the dynamics modelby combining the aerostat and tether models. This workuses a lumped-mass model of the tether in order tomodel its dynamic behaviour. In this type of model,which is discussed in more detail by Driscoll andNahon14, the continuous cable is first discretized intoelements. The mass of each element is lumped at itsendpoints (called nodes). The internal stiffness anddamping characteristics of the cable are modeled aslumped parameter stiffness and damping elementsconnecting those nodes. This type of model, shown inFigure 6, has been validated for a variety of underwatersystems with excellent agreement with in-fieldmeasurements14.

The position of each node is described with respectto an inertial reference frame, by a three-componentvector ri = [ri

X riY ri

Z]T. Each cable element isconsidered to be a straight elastic element, subject toforces at its end points. This method of modeling allowseach cable element to possess distinct properties, suchas density and stiffness.

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ki

ki+1

node i-1

node i

node i+1

element i

element i+1

b

b

Figure 6: Schematic of visco-elastic internal forces forthe tether model.

The orientation of each cable element isrepresented using a Z-Y-X (�, �, �) Euler angle set.Since the torsion of the cable is not included in themodel, the � rotation about the inertial Z axis isconstrained to zero. The Euler angles i

� and i� can becalculated from the coordinates of the appropriate nodalend points14.

Internal ForcesThe internal forces acting within an element are due toits viscoelastic properties. These are representedschematically in Figure 6. The tension in the cable dueto its structural stiffness is considered to act only in thetangential direction and is modeled by a linear tension-strain relationship. The friction between the braids ofthe cable tends to create a damping effect. This effectis assumed to be linear with the strain rate. The totaltension in a cable element due to these effects istherefore written as:

, i i

i uiu

l lT AE bl

� � �

�

� � ��

where il and iul are the stretched and unstretched

length of the i-th element, A is its cross sectional area, Eis the effective Young’s modulus of the cable, b is theviscous damping coefficient of the cable and is thestrain.

External ForcesThe external forces acting on the cable element arethose due to aerodynamic drag and gravity. The dragforce acting on the cable element, per unit length, canbe calculated according to Morison’s equation

21

2i i

c d c uD C d l�� v

where Dc is the aerodynamic drag force; � is the local

density of air; dc is the cable diameter; iul is the

element’s unstretched length, Cd is the normal dragcoefficient of the cable; and vi is the local velocity ofthe geometric center of the i-th cable element withrespect to the surrounding air. These velocities mustaccount not only for the motion of the cable element,but also the motion of the surrounding air. The dragcoefficient is modified by loading functions14, whichaccount for the nonlinear breakup of drag between thenormal and tangential directions. Once the drag forelements i and i+1 are calculated, half of each value isapplied to the i-th node which joins the two elements.Finally, the gravitational force acting on a cableelement is applied, based on the element’s density andvolume.

Tether Attachment PointThe coupling of the tether model and the aerostat modelwas achieved by connecting the uppermost node of thetether to the base of the aerostat’s flying lines as shownin Figure 7. The aerostat and its flying lines (everythingwithin the dashed box) are treated as a single rigidbody. Hence, the two flying lines shown can beinterpreted as rigid members. In reality, for an aerostatof this type, many flying lines would be present andarranged in a harness (see Figure 2) in order to rigidlysecure the aerostat. It would be difficult to incorporatethe effects of the flexibility of a complex harness andtherefore the rigid body approximation is deemedadequate for this level of investigation.

ZB

YB

XB

xl

-zl

Fl

Fb mgxb

zbx

Figure 7: Connection point of tether and aerostat.

The force from the top node of the tether or leash, Fl isincluded in the equations of motion of the aerostat. Forthe rotational equations of motion, the moment from theleash force is calculated using:

0l

l

xwhere

z

� ��

cm l l lM r F r

The actual location of the attachment point relativeto the aerostat can have a significant effect on thebehavior of the aerostat. Aeros Flightcam offered somegeneral insights as to where to place the leash

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attachment point, based on their experience withtethered aerostats. It was recommended that, in theinterests of stability, the aerostat should have a pitchedup attitude of about 5 to 10o and that the verticaldistance of the attachment point from the aerostatcentreline, zl should be about 1½ times the aerostatdiameter.

The distance xl was calculated to yield the selectedpitch angle by summing the moment about theaerostat’s centre of mass in a zero wind equilibriumposition shown in Figure 7. The moment equation is asfollows:

cos sin cos sin 0b b b b lz l lz lF x F z F x F z� � � ��

which can be rearranged as:

tan b b lz l

lz l b b

F x F xF z F z

��

�

�

The buoyancy, Fb, the leash force, Flz and theposition of the centre of buoyancy, xb and zb are allfixed quantities for the aerostat. Therefore, once valuesfor two of the other three remaining variables, θ, xl andzl are selected, this equation can be used to solve for theremaining variable. For our purposes, we selected acertain pitch angle, � and vertical leash attachmentposition, zl and solved for the unknown horizontal leashattachment position, xl.

Physical and Aerodynamic ParametersIn order to solve the equations of motion outlinedearlier, it is necessary to define numerous physicalparameters for this particular aerostat. The aerostat hasa length of 18.3 m, a diameter of 7.7 m, and a totalvolume of 519 m3. Its detailed specifications can befound in Reference 15. As shown in Fig. 2, there is aninternal air envelope (called a ballonet) used to regulatehelium pressure. The tail fins are made of the samematerial as the hull and are inflated with helium.Multiple strands of nylon cord connect the face of eachfin to the face of the adjacent fin. The nylon cord iscinched tight and the tension in the cords keeps the finsin place. The fins are arranged in an inverted Yconfiguration. The two dihedral fins are offset by anangle of 112.5o from the vertical fin.

To obtain the physical parameters of the aerostat, a3-D CAD model was generated using PRO-E. TheCAD model was constructed to accurately represent theactual aerostat, complete with a thin-walled shell of thehull and ballonet envelopes, solid models of the gasescontained within and a plate which houses the ballonetblowers. The appropriate density was assigned to eachpart and the various physical parameters werecomputed by PRO-E. The mass and volume of therelevant parts are given in Reference 15. Otherparameters of interest obtained in this manner are thelocation of the aerostat’s centre of buoyancy, c.b., thecentre of mass, c.m., and the inertia tensor. It is noted

that the mass of the gases are a significant contributionto the overall mass of the aerostat. When modelinglighter-than-air vehicles, it is essential to account forthe mass of these internal gases when determining anyphysical parameters.

The aerodynamic parameters used in the dynamicsmodel must also be estimated for this particularaerostat. The location of the centre of pressure for eachcomponent was calculated using the explanation givenearlier and is listed in Reference 15.

As noted earlier, the aerostat was split into tworegions; the hull region which includes everything fromthe nose to the start of the fins and the fin region whichincludes the fins and the section of the hull back fromthe start of the fins.

The value of CL� takes into account losses arisingfrom the fins’ location at the rear of the vehicle. Theflow field in the vicinity of the fins will haveexperienced disruptions from passing over the hull,resulting in a reduction of the fins’ capability toproduce lift.

The added mass coefficients, k1 and k3 wereestimated using data for ellipsoids13. The hull efficiencyfactor, �k was found by averaging values given for sixdifferent aerostats from Jones and Delaurier5. Thecross-flow drag coefficient for the hull, (Cdc)h wasestimated using the drag coefficient for a cylinder. Theestimation of the zero angle axial drag, (Cdh)0 wasbased on the drag coefficient for a streamlined shapewith a fineness ratio of 2.416, where the fineness ratio isthe length of the body divided by its maximumdiameter. The hull reference area, Sh is defined as the(hull volume)2/3. The evaluation of the integral for I1gives the cross-sectional area of the hull at the hull finregion boundary, Ah. The integral for J1 is equivalent tothe cross-sectional area of the hull region of the aerostatin the xz-plane. The value for this was found using theCAD model and the evaluation tools of PRO-E. Theintegrals for I3 and J2 are not easily solved unless aclosed form equation for the aerostat shape is availableand in our case it was not. Therefore, to estimate thesequantities, an average was used of the equivalent non-dimensional terms given for six aerostat shapes5.

Tether PropertiesThe tether that was modeled has physical andaerodynamic parameters listed in Table 1. The Plasmatether is manufactured by Puget Sound Rope. Themanufacturer provided the data for the density andstrength of the cable. The damping ratio and elasticmodulus were estimated from experimental testsperformed on a sample of Plasma tether. The dampingcoefficient, b used in the tether model is calculatedusing the following equation:

cb b��

8


where bc is the critical damping coefficient.

Table 1: Tether parameters.Parameter Valuediameter, Dt 6 mmdensity, ρt 840 kg/m3

minimum strength 35.6 kNelastic modulus, E 37.4 GPadamping ratio, ζ 0.017

Linear Stability AnalysisThe nonlinear dynamics model of the streamlinedaerostat on a single tether is useful for obtaining timehistories of the aerostat’s motion in response to aparticular wind or winch input. However, in order toacquire an assessment of the stability of the aerostat, alinear approximation of the system was derived and itseigenvalues and eigenvectors were studied.

The nonlinear dynamics simulation can be writtenas a set of functional relationships where the derivativeof each state variable is dependent on the full set ofstate variables, i.e. ( )�X F X� (1)

where X =[ 1 1 1 1 1 1, , , , , , ..., , , , , , ,n n n n n nx x y y z z x x y y z z� � � ��

, , , , ,� � � � � �� ]T is the state vector. It contains theposition and velocity of all the discretized elements. Thisincludes each of the tether nodes as well as the aerostat. Forthe aerostat node, there are an additional six state variablesrepresenting the rotational motion. Therefore, the totalnumber of state variable is 6n + 6, where n is the numberof tether nodes. Note that there are no state variables for thebottom-most tether node since it is fixed to the ground.Hence the translational variables with the subscript nrepresent the aerostat node.

Linear ModelLinearizing equation (1) leads to:

�X AX�

where the state matrix, A is defined as: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

�

�

�

� � �

� � ��

� � ��

� ��

� ��

FAX

��

��

� � � �

��

The most direct method for obtaining the statematrix is to perform a numerical differentiation byfinite difference of the nonlinear differential equations.The first step is to choose a reference equilibriumcondition about which the system will be linearized.Next, each state variable is perturbed slightly from its

equilibrium value. The ensuing response of every statevariable is observed and compared to its equilibriumvalue. The difference between the response and theequilibrium value is divided by the perturbation toapproximate each element of the state matrix. Forexample, in the case of the first element in the first row,we have:

01 11

1 1

f ffx x

��

� ��

where 1f � is the value of f1 due to a perturbation of a

particular state variable (in this case, 1x� ), while 01f is

the value of f1 at the reference equilibrium conditionand 1x�� is the perturbation value of 1x� . The value forthe perturbation is typically very small (10-5 was used).The reference equilibrium condition is one in which thetethered aerostat system is in a steady state wind field inthe absence of turbulence. The process can be repeatedfor a variety of wind speeds U in the range of interest.

Once A is formed, it is necessary to verify itsvalidity through a comparison of the linear andnonlinear responses to a specified initial condition.Matlab was used to obtain the linear response. Thenonlinear response was obtained using the nonlineardynamics model described in the preceding section. Allsix position variables for the aerostat weresimultaneously given an initial perturbation from theequilibrium condition. The perturbation was 0.1 m forthe translational variables and 0.01 rad for the rotationalvariables. The linear and nonlinear responses for theposition variables are shown in Figure 8. The goodagreement between the two responses demonstrates thesuccess of the linearization process and providesjustification for pursuing the linear stability analysis.

DecouplingThe stability of vehicle models is typically analyzed bydecoupling the motion variables into lateral andlongitudinal subsets. The longitudinal variables aredefined as translational motion in the x and z directionsand rotational motion about the y-axis(i.e. �� ,,,,, 1111

�� nnnn zzxx ). The lateral variables

are defined as translational motion in the y-directionand the rotational motion about the x and z axes(i.e. �� ,,,,, 11 ��

�� nn yy ). For a tethered aerostatsymmetrical about the xz-plane it can be shown thatlongitudinal and lateral decoupling will occur. Toensure that this holds true for our tethered aerostatsystem, the state vector and the state matrix wererearranged to separate the longitudinal and lateralsystems. The new state vector, X� is as follows:

9


1 1

1 1

1

1where andlong nlong lat

lat

n

x yx yzz y

z�

�

� �

� �

� � � ��

� � � � � ��

XX X X

X

� �

� �

��

� �

0 20 40 60 80−0.1

−0.05

0

0.05

0.1

0.15

x (m

)

0 20 40 60 80−1

−0.5

0

0.5

1

1.5

y (m

)

0 20 40 60 80−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

z (m

)

time (s)

0 20 40 60 80−0.4

−0.2

0

0.2

0.4

0.6

roll,

φ (d

eg)

0 20 40 60 80−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

pitc

h, θ

(deg

)

0 20 40 60 80−0.4

−0.2

0

0.2

0.4

0.6

yaw

, ψ (d

eg)

time (s)linearnon−linear

Figure 8: Comparison of linear and non-linearsimulation results of aerostat motion for L = 300 m andwind speed U = 10 m/s.

To be consistent with the new arrangement of X� ,the state matrix, A must be rearranged by exchangingthe appropriate row and column for each variable thatwas repositioned. Once this process is complete we candetermine whether the longitudinal and lateral variablesare, in fact, decoupled. If the longitudinal and lateralsystems are decoupled, the new state matrix, A� will bepartitioned into 4 distinct sub-matrices; a longitudinalsub-matrix, Along and a lateral sub-matrix, Alat as well astwo null sub-matrices. The dimensions of and locationof each sub-matrix is as follows:

� �

� � � �

(4 2) (2 4)(4 2) (4 2)

(2 4) (4 2) (2 4) (2 4)

long n nn n

latn n n n

� � ��

� � � � � �

� ��

A 0A

0 A

For our system, it was found that the lower leftmatrix was in fact comprised of all zeros, however the

upper right sub-matrix matrix had several elements withsmall magnitudes. Since these values were quite small(the largest being on the order of 10-4) compared to themagnitude of the elements of the other matrices (thelargest being on the order of 104) it can be concludedthat the motion is essentially decoupled. Therefore, thesystem can be analyzed as two separate systems asfollows:

long long long

lat lat lat

�

�

X A X

X A X

�

�

Eigenvalues and EigenvectorsThe aerostat’s natural motion and stability ischaracterized by the eigenvalues and eigenvectors ofthe corresponding state matrix. Each eigenvalue (or pairof eigenvalues), written as 1,2 d i� � �� , represents aparticular mode of the motion, while its correspondingeigenvector provides the relationship of each statevariable in that mode. The single-tethered aerostatsystem has 3n + 3 modes, where n is the number ofelements used to approximate the tether. For the systemto be stable, all the real parts of the eigenvalues must benegative. The damped frequency, �d is the imaginarypart of the eigenvalue and the natural frequency anddamping ratio are found from:

2 2 ,n d� � �� sinn

��

��

where �n is the natural or undamped frequency and � isthe damping ratio.

Each element of the corresponding eigenvectorrepresents the magnitude and phase of the response of aparticular state variable. It is important to note that themagnitude and phase of the state variables obtainedfrom the eigenvectors are relative to each other and arenot absolute. In the case of complex conjugateeigenvalues, the corresponding eigenvectors consist ofcomplex conjugates mirrored about the real axis.

ResultsThe cable was discretized into 10 elements, yieldinglongitudinal and lateral subsystems of order 42 and 24respectively. Matlab was used to compute theeigenvalues and eigenvectors for Along and Alat. Becausethe number of modes was large (a total of 33), we choseto only study the four lowest frequency modes in eachsubsystem, since the high frequency modes are notlikely to yield significant motion in the actual system.

The results for a baseline case are presented inFigure 9. The conditions for this case are: tether length,L = 300 m with a steady-state pitch angle, θ0 = -4o

(pitched up). All other physical parameters correspondto the values presented earlier for the Aeros Aerostat.The four longitudinal and four lateral modes considered

10


here were found to be stable over the full range of windspeeds. The remaining modes omitted from thepresentation were observed to be stable as well.

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50Longitudinal Modes

perio

d, τ

n (s) pendulum

pitchingaxial springtether fundamental

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

dam

ping

rat

io, ζ

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Lateral Modes

perio

d, τ

n (s) pendulum

rollingtether fundamentaltether 2nd harmonic

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

dam

ping

rat

io, ζ

U (m/s)

Figure 9: Longitudinal and lateral modes for baselinecase with L = 300 m, θ0 = -4o.

To gain a better appreciation of the results, theeigenvectors for each of the modes were studied inattempt to classify the motion. Details of theclassification of each lateral mode are as follows:� pendulum mode – The amplitude of the position

and velocities increase linearly from the base to thetop of the tether. The magnitude of the yaw rotation,ψ of the aerostat is appreciable which suggests thatyawing motion is coupled with the pendulumoscillation. This mode has by far the lowestfrequency which is expected for the pendulummode, considering the size of the system.

� rolling mode – The dominant motion of this modeis the rolling of the aerostat as large relativeamplitudes with the roll angle, � and rotational

velocity, �� are observed.� 1st tether harmonic – The magnitude of the position

of the tether nodes is a maximum at the middle node

which corresponds to a simple concave/convexshape.

� 2nd tether harmonic – The motion for the nodesunder node 5 and above node 5 are 180o out ofphase. The magnitude of the position of the tether isa maximum at nodes 2 and 7 and is a minimum atnode 5 which corresponds to an ‘S’ shaped tether.As expected the frequency of this mode is twice ashigh as the 1st tether harmonic.

Classification of the longitudinal eigenvectorsfollowed a similar process. The characteristics andclassification of the 4 lowest frequency longitudinalmodes are as follows:� pendulum mode – The dominant motion variables

for this mode are the position and velocity of thetether nodes in the x-direction, 1...nx and 1...nx� . Themotion in the z-direction is negligible. Theeigenvectors corresponding to 1...nx and 1...nx� are90o out of phase. The amplitudes of 1...nx and 1...nx�increase linearly from the base to the top of thetether.

� pitching mode – The dominant motion of this modeis the pitching of the aerostat as large relativeamplitudes of the pitch angle, � and rotationalvelocity, �� are observed.

� axial spring mode – The dominant motion variablesfor this mode are the position and velocity in z-direction, 1...nz and 1...nz� . The magnitudes of themotion variables in the x-direction are negligible.

� 1st tether harmonic – Like the pendulum mode, thedominant motion variables for this mode are theposition and velocity of the tether nodes in the x-direction and the motion in the z-direction isnegligible. The magnitude of the x-position of thetether nodes is a maximum at the middle nodewhich corresponds to a simple concave/convexshape.

Reference FrequenciesThe basic motion characteristics from the linearanalysis can be compared with the analytical solutionsfor an ideal tethered aerostat, as a means of checkingthe validity of the dynamics model. The analyticalreference frequencies for the motion of a buoyant masson a string are obtained as follows:

for the pendulum mode:

ne

B mgm L

��

�

for the axial spring mode:

ne

EAm L

� �

11


for the tether’s transverse mode for the nth harmonic:

, 1,2...( )

tn

t

n nL A B mg

��

�

where me is the effective mass of the aerostat includingadded mass. For the pendulum modes, the relevantadded mass included for the longitudinal case is in thex-direction and for the lateral case it is in the y-direction. For the axial spring mode the added mass inthe z-direction is included. Added mass is not requiredfor the gravity force used in determining the net lift ofthe aerostat given as B – mg since the added mass onlyapplies to the inertial properties of the aerostat. Thetheoretical natural frequencies are used to find thetheoretical periods of each mode using 2 /n n� � �� .The results are presented in Table 2 for a comparisonwith the periods given in Figure 9 from the linearanalysis at U = 1 m/s.

Table 2: Comparison of theoretical and linear modelresults for period, τn of certain oscillatory modes for

baseline case.Long.

pendulumLat.

pendulumaxial

spring1st tetherharmonic

Theor. 43.9 s 58.0 s 2.96 s 1.81 slinear 44.4 s 67.4 s 2.96 s 1.78 s

% diff. 1.1 16.2 ≈ 0 1.7

Close agreement is observed for all modes with theexception of the lateral pendulum mode which has adifference of 16.2% with the theoretical value. Thisdiscrepancy can be attributed to the fact that thecomparison uses the results from the linear model at U= 1 m/s while the theoretical period corresponds to asystem with U = 0. Overall, the good agreementbetween the linear model and the theoreticaloscillations provides a first level of assurance of thegeneral validity of the dynamics model.

ConclusionsA nonlinear model of a tethered aerostat wasdeveloped, based on a lumped mass approach for thetether, and a component breakdown approach for theaerostat. This model was then linearized to allow anexamination of the stability of the tethered aerostat invarious winds. The behavior observed at low windspeeds correlated well with analytical predictions. Itwas found that the system remained stable at all windspeeds, and that the stability improved with increasingwind speed.

AcknowledgementsThe authors would like to thank staff at the

National Research Council of Canada, and at AerosFlightCam for their technical support. Funding for this

work was received from NSERC, the CanadaFoundation for Innovation and the BC Science Council.

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and 3.0x106 on the Low-speed AerodynamicCharacteristics of Three Low-Aspect-RatioSymmetrical Wings With Rectangular Plan Forms,NACA Res. Memorandum RM LG2G18:1-13, 1952. 13J.N. Newman. Marine Hydrodynamics, MIT Press,1989.14R. Driscoll, M. Nahon. Mathematical Modeling andSimulation of Moored Buoy System, Proceedings ofOceans, 1:517-523, 1996.15C. Lambert, Dynamics Modeling and ConceptualDesign of a Multi-tethered Aerostat System, M.A.Sc.Thesis, Department of Mechanical Engineering,University of Victoria, 2002.16B.W. McCormick. Aerodynamics, Aeronautics andFlight Mechanics, John Wiley & Sons, 1995.

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