Dynamics of Electrodynamic

Tether Systems

Matthew S. Bitzer, Joshua R. Ellis and

Christopher D. Hall

Aerospace and Ocean Engineering

Virginia Tech

Matthew S. Bitzer, Joshua R. Ellis and

Christopher D. Hall

Aerospace and Ocean Engineering

Virginia Tech

Presentation at Presentation at

Where We Are

27,000 students27,000 students27,000 students27,000 students

2600260026002600----acre main campusacre main campusacre main campusacre main campus

>100 buildings>100 buildings>100 buildings>100 buildings

Space-Related Research @

Space-Related Research @

On Research Project Selection

He had been Eight Years upon a Project for extracting Sun-Beams out of Cucumbers, which were to be put into Vials hermetically sealed, and let out to warm the Air in raw inclement Summers. He told me he did not doubt in Eight Years more he should be able to supply the Governors Gardens with Sun-shine at a reasonable Rate.

-- Gulliver, Gulliver’s Travels

Rigid Body Spin Stability

X

Y

Z • Ixx > Iyy > Izz

• Major axisMajor axisMajor axisMajor axis spin is stablestablestablestable

• Minor axisMinor axisMinor axisMinor axis spin is stablestablestablestable

• Intermediate axisIntermediate axisIntermediate axisIntermediate axis spin is

unstableunstableunstableunstable

• Energy dissipationEnergy dissipationEnergy dissipationEnergy dissipation

changes these results

⇒ Minor axisMinor axisMinor axisMinor axis spin

becomes unstableunstableunstableunstable

• This is called the MajorThis is called the MajorThis is called the MajorThis is called the Major----

Axis RuleAxis RuleAxis RuleAxis Rule

MAJOR

INT

ER

ME

DIA

TE

MINOR

Sputnik & Explorer I

• Sputnik was launched 50 years ago!

• Professor Ronald Bracewell, a radio

astronomer at Stanford, deduced that

Sputnik was spinning about a symmetry axis,

and that it must be the major axis

• He called JPL to make sure that the Explorer I design was taking this into

account, but security prevented him from getting through

• Explorer I was designed as a minor axis spinner, launched in 1958

Spin-Stabilized Satellites

Explorer I (1958) was supposed

to be spin-stabilized about its

minor axis. It went into a flat

spin due to energy dissipation.Telstar I (1962) was spin-

stabilized about its major axis,

spinning at about 200 RPM.

Electrodynamic Tethers

Current inCurrent inCurrent inCurrent in tether tether tether tether

interacts with interacts with interacts with interacts with

earthearthearthearth’’’’s magnetic s magnetic s magnetic s magnetic

field to create a field to create a field to create a field to create a

Lorentz ForceLorentz ForceLorentz ForceLorentz Force Electron collectors Electron collectors Electron collectors Electron collectors

and emitters at and emitters at and emitters at and emitters at

each end of the each end of the each end of the each end of the

tether exchange tether exchange tether exchange tether exchange

free electrons in free electrons in free electrons in free electrons in

the ambient the ambient the ambient the ambient

plasma complete plasma complete plasma complete plasma complete

the tether circuit.the tether circuit.the tether circuit.the tether circuit.

Velocity

Curr

ent

Force

B

Force

BLiF

×=

[1]

Electrodynamic Tethers

Electrodynamic Tethers

F l Bid= ×∫

B

i

EDT System Model

• Two gyrostat end bodies with arbitrary number of momentum wheels and thrusters

• Flexible tether modeled as elastic string, Linear Kelvin-Voigt Law of Viscoelasticity

• Earth-centered, Earth-fixed, tilted dipole model for magnetic field

• Two gyrostat end bodies with arbitrary number of momentum wheels and thrusters

• Flexible tether modeled as elastic string, Linear Kelvin-Voigt Law of Viscoelasticity

• Earth-centered, Earth-fixed, tilted dipole model for magnetic field

n1

n2

n3

o3

o1o

2

a3

a1 a2

e3

b3

b1

b2

Body A

Body B

Ω

ω ν+I

Ar

u

r

Coordinate Frames

FN: inertial frame (similar to perifocal frame)

FO: orbital frame

FA: Body A body frame

FB: Body B body frame

FE: tether-fixed frame

FN: inertial frame (similar to perifocal frame)

FO: orbital frame

FA: Body A body frame

FB: Body B body frame

FE: tether-fixed frame

3e

1e

2e

GA

GB

ˆ1a

ˆ2a

ˆ3a

ˆ1

bˆ2b

ˆ3

b

( )s,tr

Ap

Bp

3e

ds

θ

φ

rABody A

Body B

GA

PA

GB

PB

O

ˆ1

n

ˆ2

n

ˆ3

n

ˆ1o

ˆ2

o

ˆ3o

2-1 rotation sequence through α,β from FO to FE

2-1 rotation sequence through α,β from FO to FE

Solution Methods

• Two different solution methods used for

the tether partial differential equations

– Assumed Modes Assumed Modes Assumed Modes Assumed Modes – tether displacments are

expanded using assumed mode functions

– Finite Element Method Finite Element Method Finite Element Method Finite Element Method – nodal displacements

and slopes interpolated using cubic Hermite

polynomials

• Two different solution methods used for

the tether partial differential equations

– Assumed Modes Assumed Modes Assumed Modes Assumed Modes – tether displacments are

expanded using assumed mode functions

– Finite Element Method Finite Element Method Finite Element Method Finite Element Method – nodal displacements

and slopes interpolated using cubic Hermite

polynomials

Assumed Modes

Expand tether displacements using assumed mode

functions

Expand tether displacements using assumed mode

functions

Express tether position vector relative to E frameExpress tether position vector relative to E frame

Use Galerkin Method to obtain set of 2NT+NLODEs for generalized coordinates

Use Galerkin Method to obtain set of 2NT+NLODEs for generalized coordinates

Finite Elements

Express tether position vector relative to E frameExpress tether position vector relative to E frame

Use interpolation functions to express tether

displacements

Use interpolation functions to express tether

displacements

Shape functions are cubic Hermite polynomials –

ensures continuity of displacement and slope

Shape functions are cubic Hermite polynomials –

ensures continuity of displacement and slope

• Borrowing a technique from

Computational Fluid Dynamics (CFD)…

• Introduce additional terms into a set of

PDEs to “manufacture” an exact solution

• Use the manufactured exact solution to

determine discretization errors and

observed orders of accuracy

• Check that observed order of accuracy

matches theoretical order

• Borrowing a technique from

Computational Fluid Dynamics (CFD)…

• Introduce additional terms into a set of

PDEs to “manufacture” an exact solution

• Use the manufactured exact solution to

determine discretization errors and

observed orders of accuracy

• Check that observed order of accuracy

matches theoretical order

Manufactured Exact Solutions

MMES – A Simple Example

Let the desired manufactured solution beLet the desired manufactured solution be

We can then solve for the required f asWe can then solve for the required f as

Solving the PDE with this f should yield the

desired manufactured solution

Solving the PDE with this f should yield the

desired manufactured solution

Manufactured Exact Solutions

Given numerical solution and manufactured exact

solution, can determine discretization error

Given numerical solution and manufactured exact

solution, can determine discretization error

Discretization errors are then used to determine

observed order of accuracy

Discretization errors are then used to determine

observed order of accuracy

Observed order of accuracy should converge to

theoretical order as discretization is refined

Observed order of accuracy should converge to

theoretical order as discretization is refined

MMES applied to the EDT system using FEM with

cubic interpolation polynomials

MMES applied to the EDT system using FEM with

cubic interpolation polynomials

Observed order of accuracy of 4 matches theoretical order for cubic

interpolation polynomials – discretized equations accurately solve the

governing PDEs

Observed order of accuracy of 4 matches theoretical order for cubic

interpolation polynomials – discretized equations accurately solve the

governing PDEs

Manufactured Exact Solutions

Take advantage of form when selecting

manufactured solution

Take advantage of form when selecting

manufactured solution

Manufactured Exact Solutions

Slightly different approach to MMES used for

assumed modes

Slightly different approach to MMES used for

assumed modes

Manufactured Exact Solutions

Using enough assumed modes should yield “exact”

solution – and it does

Using enough assumed modes should yield “exact”

solution – and it does

Examples of Tether Motion

Three movies illustrating tether motion

Black – FEM, Blue – assumed modes

Motion equations solved using 5 finite elements

and 5 assumed modes (both transverse and

longitudinal) – note differences in solutions

Three movies illustrating tether motion

Black – FEM, Blue – assumed modes

Motion equations solved using 5 finite elements

and 5 assumed modes (both transverse and

longitudinal) – note differences in solutions

Switch to “Simple” Model

• EDT is treated as a point mass

– Equations of motion are simple

• Magnetic field is assumed to be a

tilted dipole

– Set up and solve minimum

time optimal control problem

with tether direction as the

control

Coordinate Systems

2

2n

1n

3n

3o

1o

2o

1om1

2o

1o

TLi

ψ)ˆˆ(ˆˆ131

nmoo ××=

132ˆˆˆ ooo ×=

||ˆ

3B

Bo

=

We assume the tether is

always in the plane

so that the EDT can

generate the maximum

amount of force.

21ˆˆ oo −

Equations of Motion

z

y

x

vz

vy

vx

=

=

=

m

fz

rv

m

fy

rv

m

fx

rv

zz

y

y

xx

+−=

+−=

+−=

3

3

3

µ

µ

µ

Begin with 6 first-order Two-Body Problem

Equations of Motion:

Equations of Motion

Force is perpendicular to magnetic field and is

expressed in inertial frame.

Equations of Motion

Hamiltonian for optimal control law:

0=

+++++= zvzyvyxvxzyx vvvzyxH λλλλλλ

Equations of Motion

Six equations for the Lagrange costates:

Two-Dimensional Results

EDT Trajectory Constant Thrust

Note: All quantities are nonNote: All quantities are nonNote: All quantities are nonNote: All quantities are non----

dimensionalizeddimensionalizeddimensionalizeddimensionalized

•The EDT reaches target orbit faster than a constant thrust

spacecraft

•True for all comparisons with the same initial conditions

[2]

Two-Dimensional Results

• Near-invariance in costate initial conditions and

time of flight when thrust and phase angle are

increased by same order of magnitude

– Families of solutions can be formed

– Near-invariance breaks down for high thrust and phase

angles

Three-Dimensional Solutions

• The range of inclination angles studied is 0-7.5º.

• Difficulties stem from extreme sensitivity of initial

costates as inclination angle increases.

Three-Dimensional Solutions

• The control angles for maneuvers in different

orbit planes have the same pattern except for one

difference…

Between i = 2º and 3º, EDT rotates in opposite direction at t ≈1.4 TU, but control angles are essentially identical.

• High-fidelity model of electrodynamic tether systems– First-principles derivation of equations of motion

– Two discretization methods

– Manufactured solutions verify numerical methods

• Simplified model used to obtain time-optimal orbit transfer trajectories

• Next steps: characterize domains within which simplified models match high-fidelity model results

• Contact: Christopher Hall cdhall@vt.educdhall@vt.educdhall@vt.educdhall@vt.edu

Summary

Some Backup Slides

• From MMES results we know that codes for

assumed modes and FEM accurately solve

governing PDEs

• Is one solution method “better” than the other –

does one method converge to exact solution

faster?

• Use Richardson Extrapolation to answer this

question

• From MMES results we know that codes for

assumed modes and FEM accurately solve

governing PDEs

• Is one solution method “better” than the other –

does one method converge to exact solution

faster?

• Use Richardson Extrapolation to answer this

question

Richardson Extrapolation

Use two solutions to extrapolate exact solutionUse two solutions to extrapolate exact solution

Richardson Extrapolation

Use extrapolated solution to estimate how “close” solution

is to the exact solution

Use extrapolated solution to estimate how “close” solution

is to the exact solution

Cannot extrapolate tether displacements – must use a scalar

quantity like length or tension at the end bodies to measure

“closeness” to exact solution

Cannot extrapolate tether displacements – must use a scalar

quantity like length or tension at the end bodies to measure

“closeness” to exact solution

Richardson Extrapolation –

Preliminary ResultsFEM solution with 64 elements used as “exact” solution – lower refinement FEM

and assumed modes solutions compared to this solution

FEM solution with 64 elements used as “exact” solution – lower refinement FEM

and assumed modes solutions compared to this solution

Error convergence about the same, but FEM simulation time much shorterError convergence about the same, but FEM simulation time much shorter

Solution Algorithm

Simulated annealing narrows search field to

identify potential global minima:Simulated annealing

emulates the metallurgical

process for minimizing the

internal energy in the

metal. The “energy state”

is lowered slowly so that

generated test points can

escape local minima and

lead to possible global

minima.

T1

T2

T3

Global Minima