4. Equations of motion

IT ¼ 2ρ

MT=<sup>m</sup> <sup>¼</sup> <sup>2</sup> ffiffiffiffi

IT=mL<sup>2</sup> <sup>¼</sup> ffiffiffi

<sup>M</sup>=<sup>m</sup> <sup>¼</sup> <sup>2</sup> ffiffiffi

tether material are likely to fall within these bounds.

(kg/m<sup>3</sup> )

Tether mass properties for various materials (from Eqs. (4), (7) and (8)).

Material Density

smaller.

Table 1.

56

<sup>I</sup>=mL<sup>2</sup> <sup>¼</sup> ffiffiffi

Evaluating the integrals and simplifying:

Planetology - Future Explorations

ðL 0 a rð Þr 2

The limit χ ! 0 represents a material of infinite strength, in which case the tether mass and moment of inertia vanish. Adding the contributions of the two end

These expressions for the skyhook mass properties indicate the dependence on tether material properties, and provide key parameters for dynamical modeling. An important feature of a tether is the taper factor, the ratio of maximum to minimum cross-section area. A tether constructed from low strength material has a large taper factor, indicating its impracticality. The nominal skyhook described above with a carbon fiber tether has a taper factor of 237, in which case the diameter at the centroid is about 15 times that the end points. If the tether had the properties of carbon nanotubes the taper factor reduces to 3.3. The properties of any future

Table 1 indicates the mass properties of the nominal skyhook for several tether materials. Notionally high strength materials like steel and diamond are excluded by the very large taper factor. Aramid fibers like Kevlar are possible but the total mass is large. The strongest carbon fiber offers a solution with a skyhook mass about 4600 times the endpoint mass. If materials with still greater tensile strength become available, such as by incorporating carbon nanotubes or colossal carbon tubes into the tether material, the taper factor and skyhook mass can be much

For the skyhook configuration described here the endpoint mass is regarded as the maximum payload capability. This assumes the endpoint mass may be replaced by a docking mechanism of negligible mass to capture the payload. Engineering margins have not been included in this analysis, but the nominal configuration is a

> Strength (MPa)

Steel 2800 8000 2693 67.7 1.2 � 1025 2.9 � <sup>10</sup><sup>27</sup> 3.1 � <sup>10</sup><sup>23</sup> Diamond 3500 2800 26.7 4.1 � <sup>10</sup><sup>11</sup> 3.9 � 1013 2.7 � 1010 Aramid fiber 1440 3757 8.2 3629 1.05 � 105 783.4 Zylon (PBO) 1560 5800 5.75 315 6421 96.1 Carbon fiber (T1100S) 1790 7000 5.47 237 4596 75.8 Carbon nanotube 1340 63,000 0.45 1.58 5.46 5.06 Colossal carbon tube 116 7000 0.35 1.43 1.70 5.93

χ <sup>2</sup> Taper Factor

Mass (MT/m)

Moment (IT/mL<sup>2</sup> )

masses leads to expressions for the mass properties of the entire skyhook:

dr (6)

<sup>π</sup> <sup>p</sup> <sup>χ</sup> exp <sup>χ</sup><sup>2</sup> � �erfð Þ <sup>χ</sup> (7)

<sup>π</sup> <sup>p</sup> <sup>χ</sup>�<sup>1</sup> exp <sup>χ</sup><sup>2</sup> � �erfð Þ� <sup>χ</sup> <sup>2</sup> (8)

<sup>π</sup> <sup>p</sup> <sup>χ</sup> exp <sup>χ</sup><sup>2</sup> � � erfð Þþ <sup>χ</sup> <sup>2</sup> (9)

<sup>π</sup> <sup>p</sup> <sup>χ</sup>�<sup>1</sup> exp <sup>χ</sup><sup>2</sup> � �erfð Þ <sup>χ</sup> (10)

Skyhook length is a significant factor in the dynamics because field strength is not uniform across the structure. This differs from most problems in astrodynamics where the object of interest is small compared to the field gradient length scale, or the system can be simplified by assuming spherical symmetry.

Here the skyhook is assumed to behave as a rigid body, kept in tension by the rotation and experiencing no stretching or bending. The validity of these assumptions depends on the tether material properties, but they are sufficient for the present purpose. The structure is expected to remain linear due to the large centripetal restoring force that counters any bending.

The equations of motion of a rigid body are typically obtained by a Lagrangian method using the mass properties. This formulation ignores the field gradient effect, which is important for skyhook dynamics. To see this note that the skyhook structure experiences a moment due to the two arms being subject to different field strengths according to their proximity to Earth. If the skyhook were treated as a single compact object this behavior would not be represented.

The skyhook system is modeled here as three objects connected by tethers of fixed length L as illustrated in Figure 1. The central object has the mass properties of the tethers as calculated above. This formulation represents the physical extent of the skyhook in a non-uniform field. It is also a good approximation for the mass distribution of the tether if it has a significant taper factor, in which case much of the mass is concentrated near the centroid. Based on these considerations a Newtonian formulation is used for the analysis.

The system state is described by a six element vector comprising the skyhook centroid location r ¼ ð Þ r; θ and orientation angle φ and their derivatives. The endpoint locations are specified by the vectors r1 and r2 which are functions of the state

#### Figure 1.

Skyhook geometry with the tether mass and moment of inertia represented by a compact object at the mass centroid.

vector and may be written as follows where ^<sup>t</sup> <sup>¼</sup> ð Þ cos <sup>φ</sup>; sin <sup>φ</sup> is the skyhook orientation unit vector:

$$r\_{1,2} = r \mp L\pounds \tag{11}$$

The gravitational force on each mass is projected through the centroid to obtain the net radial and azimuthal forces, and onto the normal for the torque:

$$F\_r = -\left(\frac{GM\_E}{r\_1^2}m\right)\hat{r}\_1.\hat{r} - \left(\frac{GM\_E}{r\_2^2}m\right)\hat{r}\_2.\hat{r} - \left(\frac{GM\_E}{r^2}M\_T\right) \tag{12}$$

$$F\_{\theta} = -\left(\frac{\text{GM}\_{E}}{r\_{1}^{2}}m\right)\hat{r}\_{1}\,\hat{\theta} - \left(\frac{\text{GM}\_{E}}{r\_{2}^{2}}m\right)\hat{r}\_{2}\,\hat{\theta} \tag{13}$$

$$\boldsymbol{\tau} = -\left(\frac{\mathbf{G}\mathbf{M}\_E}{r\_1^2}\boldsymbol{m}\right)\boldsymbol{L}\,\hat{\mathbf{r}}\_1\,\hat{\mathbf{t}}' + \left(\frac{\mathbf{G}\mathbf{M}\_E}{r\_2^2}\boldsymbol{m}\right)\boldsymbol{L}\hat{\mathbf{r}}\_2\,\hat{\mathbf{t}}'\tag{14}$$

Here ^t 0 ¼ ð Þ sin φ; � cos φ is a unit vector normal to the skyhook. In circular polar co-ordinates the acceleration is:

$$
\ddot{\mathbf{r}} = (\ddot{r} - r\dot{\theta}^2)\hat{\mathbf{r}} + \left(r\ddot{\theta} + 2\dot{r}\dot{\theta}\right)\hat{\boldsymbol{\theta}}\tag{15}
$$

The skyhook equations of motion are then:

$$
\ddot{r} - r\dot{\theta}^2 = F\_r/(2m + M\_T) \tag{16}
$$

$$r\ddot{\theta} + 2\dot{r}\dot{\theta} = F\_{\theta}/(2m + M\_T) \tag{17}$$

5. Energy replenishment

Altitude (dark) and ground track speed (light) of a skyhook endpoint.

Space Access for Future Planetary Science Missions DOI: http://dx.doi.org/10.5772/intechopen.88530

Figure 2.

59

Al�tude (1000 Km) / Speed (Km/s)

to achieve energy replenishment.

After launch it is necessary to replenish the skyhook energy and circularize the orbit. If the orbital eccentricity is small there is no interaction between the skyhook and the atmosphere, so this may occur over many orbits. Electric thrusters are proposed as a suitable technology for maintaining the skyhook orbit. They produce thrust with a high specific impulse, and therefore utilize propellant very efficiently. The preferred location to apply thrust is the skyhook centroid. A force at this point maximizes energy transfer, the rate of work being the product of the thrust and orbital velocity V0. The skyhook is also very robust at the centroid, and with a local acceleration near zero it is the optimal location for solar arrays to power the thrusters. Note that mass at the centroid does not affect the skyhook structure or energy transfer rate. This means the propulsion system mass and efficiency is of no concern. The key thruster performance characteristics are the efflux velocity and mass flow rate, which together determine the propellant quantity and time needed

0 5 10 15 20

Endpoint Trajectory

Ground Track Distance (1000 Km)

Electric propulsion has been developed for tasks that require a small thrust with high specific impulse. Examples include orbital transfer and deep space missions, for which ion thrusters are the preferred technology. Energy replenishment requires a high specific impulse and sufficient thrust to limit the replenishment time. A magnetoplasmadynamic (MPD) motor is best suited for this purpose. MPD thruster technology is developmental,

An MPD thruster creates an electric current in plasma in the presence of a magnetic field. The field may be generated externally by coils or intrinsically by the current itself. In either case Lorentz force acts on the plasma and expels it at high velocity. Laboratory MPD thrusters have demonstrated 5 N of thrust with a mass

but their performance can be inferred from experimental demonstrators.

$$\ddot{\boldsymbol{\rho}} = \boldsymbol{\tau}/\mathbf{I} = \boldsymbol{\tau}/m\boldsymbol{L}^2 \left\{ \sqrt{\pi} \,\boldsymbol{\chi}^{-1} \exp\left[\boldsymbol{\chi}^2\right] \text{erf}(\boldsymbol{\chi}) \right\} \tag{18}$$

Evaluating the vector dot products and re-arranging:

$$\ddot{r} = r\dot{\theta}^2 - \frac{GM\_Em}{2m + M\_T} \frac{1}{2r} \left\{ \left(\frac{1}{r\_1} + \frac{1}{r\_2}\right) + \left(\frac{r\_1^3 + r\_2^3}{r\_1^2 r\_2^2}\right) \cos\left(\theta\_1 - \theta\_2\right) + \frac{2M\_T}{r} \right\} \tag{19}$$

$$\ddot{\theta} = -\frac{2\dot{r}\dot{\theta}}{r} + \frac{GM\_Em}{2m + M\_T} \frac{1}{2r^2} \left(\frac{r\_1^3 - r\_2^3}{r\_1^2 r\_2^2}\right) \sin\left(\theta\_1 - \theta\_2\right) \tag{20}$$

$$\ddot{\rho} = -\frac{GM\_Em}{2I} \left(\frac{r\_1^3 - r\_2^3}{r\_1^2 r\_2^2}\right) \sin\left(\theta\_1 - \theta\_2\right) \tag{21}$$

The skyhook trajectory was obtained by numerical solution of these equations of motion for the nominal parameters. The endpoint altitude and ground track speed are shown in Figure 2. Note that the minimum energy point occurs at zero ground track speed at an altitude of 532 km. The specific energy of a stationary object at this altitude is about 5% of one in orbit. The configuration could be altered to allow access at a lower altitude, but it may then incur an unacceptable risk of collision with satellites in low Earth orbit.

During launch momentum transfers from the skyhook to the payload, perturbing the skyhook orbit into an ellipse. This perturbation is small if the skyhook mass is much greater than the payload mass, as is true for most tether materials. If the tether material is sufficiently strong the skyhook mass can be small enough for the orbital perturbation to be significant. This can be overcome by placing ballast mass at the centroid.

Space Access for Future Planetary Science Missions DOI: http://dx.doi.org/10.5772/intechopen.88530

vector and may be written as follows where ^<sup>t</sup> <sup>¼</sup> ð Þ cos <sup>φ</sup>; sin <sup>φ</sup> is the skyhook orien-

The gravitational force on each mass is projected through the centroid to obtain

^r1:^<sup>θ</sup> � GME r2 2 m � �

¼ ð Þ sin φ; � cos φ is a unit vector normal to the skyhook. In circular polar

GME r2 2 m � �

€ <sup>þ</sup> <sup>2</sup>r\_ \_ θ � �θ

the net radial and azimuthal forces, and onto the normal for the torque:

^r1:^<sup>r</sup> � GME r2 2 m � �

> L ^r1:^t 0 þ

<sup>θ</sup><sup>2</sup> � �^<sup>r</sup> <sup>þ</sup> <sup>r</sup><sup>θ</sup>

þ

GMEm 2m þ MT

r3 <sup>1</sup> <sup>þ</sup> <sup>r</sup><sup>3</sup> 2 r2 1r2 2 � �

> r3 <sup>1</sup> � <sup>r</sup><sup>3</sup> 2 r2 1r2 2 � �

1 2r<sup>2</sup>

r3 <sup>1</sup> � <sup>r</sup><sup>3</sup> 2 r2 1r2 2 � �

During launch momentum transfers from the skyhook to the payload, perturbing the skyhook orbit into an ellipse. This perturbation is small if the skyhook mass is much greater than the payload mass, as is true for most tether materials. If the tether material is sufficiently strong the skyhook mass can be small enough for the orbital perturbation to be significant. This can be overcome by

The skyhook trajectory was obtained by numerical solution of these equations of motion for the nominal parameters. The endpoint altitude and ground track speed are shown in Figure 2. Note that the minimum energy point occurs at zero ground track speed at an altitude of 532 km. The specific energy of a stationary object at this altitude is about 5% of one in orbit. The configuration could be altered to allow access at a lower altitude, but it may then incur an unacceptable risk of collision

� �

Fr ¼ � GME

r2 1 m � �

<sup>F</sup><sup>θ</sup> ¼ � GME

<sup>τ</sup> ¼ � GME r2 1 m � �

The skyhook equations of motion are then:

1 2r

r2 1 m � �

<sup>r</sup>€ <sup>¼</sup> <sup>r</sup>€� <sup>r</sup>\_

<sup>r</sup>€� <sup>r</sup>\_

<sup>φ</sup>€ <sup>¼</sup> <sup>τ</sup>=<sup>I</sup> <sup>¼</sup> <sup>τ</sup>=mL<sup>2</sup> ffiffiffi

rθ € <sup>þ</sup> <sup>2</sup>r\_ \_

Evaluating the vector dot products and re-arranging:

1 r1 þ 1 r2 � �

<sup>φ</sup>€ ¼ � GMEm 2I

<sup>r</sup>1,<sup>2</sup> <sup>¼</sup> <sup>r</sup> <sup>∓</sup> <sup>L</sup>^<sup>t</sup> (11)

^r2:^<sup>r</sup> � GME

L^r2:^t

<sup>θ</sup><sup>2</sup> <sup>¼</sup> Fr=ð Þ <sup>2</sup><sup>m</sup> <sup>þ</sup> MT (16)

θ ¼ Fθ=ð Þ 2m þ MT (17)

cosð Þþ θ<sup>1</sup> � θ<sup>2</sup>

<sup>π</sup> <sup>p</sup> <sup>χ</sup>�<sup>1</sup> exp <sup>χ</sup><sup>2</sup> � �erfð Þ<sup>χ</sup> � �<sup>g</sup> (18)

<sup>r</sup><sup>2</sup> MT � �

^r2:^θ (13)

^ (15)

2 r MT m

sin ð Þ θ<sup>1</sup> � θ<sup>2</sup> (20)

sin ð Þ θ<sup>1</sup> � θ<sup>2</sup> (21)

<sup>0</sup> (14)

(12)

(19)

tation unit vector:

Planetology - Future Explorations

Here ^t 0

<sup>r</sup>€ <sup>¼</sup> <sup>r</sup>\_

co-ordinates the acceleration is:

<sup>θ</sup><sup>2</sup> � GMEm 2m þ MT

> θ € ¼ � <sup>2</sup>r\_ \_ θ r þ

with satellites in low Earth orbit.

placing ballast mass at the centroid.

58

Figure 2. Altitude (dark) and ground track speed (light) of a skyhook endpoint.
