114
A. K. Misra and S. Cohen
Clearly, the required counterweight is proportional to the maximum area of cross
section of the ribbon, and is also dependent on other material and design parameters. In Fig. 7, the ribbon, counterweight, and total mass per maximum unit area
of cross section of ribbon are plotted against nominal ribbon length, assuming a
taper ratio of 6 (ribbon material parameters are as assumed above). A ribbon with
L 0 = 100,000 km and A m = 10mm
2 would have a mass of about 1000 tons. The
corresponding counterweight mass would be about 300 tons.
4 Elastic Oscillations of the Ribbon
Both longitudinal and transverse displacements of the ribbon, u and v, shown in
Fig. 4, are expanded in series form as products of a set of generalized coordinates
and spatial basis functions. The longitudinal extension is expressed as
u(s, t) = ε 0 s +
N
i=1
a i (t)ψ i (s)
(22)
where N generalized coordinates, a i (t), and appropriate basis functions, ψ i (s), are
used to describe the extension of the ribbon from its nominal amount u 0 (s) = ε 0 s.
Since an energy method is used to derive the equations of motion, ψ i (s) need to be
only admissible functions, not comparison functions [i.e., they need to satisfy only
the geometric boundary condition, ψ i (0) = 0].
The transverse displacement is represented by
v(s, t) =
M
i=1
b i (t)ψ i (s)
(23)
where M generalized coordinates b i (t) are used to describe this displacement. Again,
ψ i (s) are appropriate admissible basis functions satisfying the geometric boundary
conditions ψ i (0) = ψ i (L 0 ) = 0. Generalized coordinates α, a i (1 = 1, 2, . . . , N ) and
b i (1 = 1, 2, . . . , M) fully define this N + M + 1 degree-of-freedom system. Here,
α is the pitch rotation (or libration) of the ribbon in the equatorial plane.
The position vectors of a ribbon element and the counterweight with respect to
the center of the Earth, respectively, are given by
r R (s) = (R E cos α + s + u)i + (v − R E sin α)j
(24)
and
r C (s) = (R E cos α + L 0 + u L )i − R E sin αj
(25)
A. K. Misra and S. Cohen
Clearly, the required counterweight is proportional to the maximum area of cross
section of the ribbon, and is also dependent on other material and design parameters. In Fig. 7, the ribbon, counterweight, and total mass per maximum unit area
of cross section of ribbon are plotted against nominal ribbon length, assuming a
taper ratio of 6 (ribbon material parameters are as assumed above). A ribbon with
L 0 = 100,000 km and A m = 10mm
2 would have a mass of about 1000 tons. The
corresponding counterweight mass would be about 300 tons.
4 Elastic Oscillations of the Ribbon
Both longitudinal and transverse displacements of the ribbon, u and v, shown in
Fig. 4, are expanded in series form as products of a set of generalized coordinates
and spatial basis functions. The longitudinal extension is expressed as
u(s, t) = ε 0 s +
N
i=1
a i (t)ψ i (s)
(22)
where N generalized coordinates, a i (t), and appropriate basis functions, ψ i (s), are
used to describe the extension of the ribbon from its nominal amount u 0 (s) = ε 0 s.
Since an energy method is used to derive the equations of motion, ψ i (s) need to be
only admissible functions, not comparison functions [i.e., they need to satisfy only
the geometric boundary condition, ψ i (0) = 0].
The transverse displacement is represented by
v(s, t) =
M
i=1
b i (t)ψ i (s)
(23)
where M generalized coordinates b i (t) are used to describe this displacement. Again,
ψ i (s) are appropriate admissible basis functions satisfying the geometric boundary
conditions ψ i (0) = ψ i (L 0 ) = 0. Generalized coordinates α, a i (1 = 1, 2, . . . , N ) and
b i (1 = 1, 2, . . . , M) fully define this N + M + 1 degree-of-freedom system. Here,
α is the pitch rotation (or libration) of the ribbon in the equatorial plane.
The position vectors of a ribbon element and the counterweight with respect to
the center of the Earth, respectively, are given by
r R (s) = (R E cos α + s + u)i + (v − R E sin α)j
(24)
and
r C (s) = (R E cos α + L 0 + u L )i − R E sin αj
(25)