On the Stability of the Track of the Space Elevator

Pugno, Nicola; Tröger, H.; Steindl, Alois; Schwarzbart, Michael

doi:10.2514/6.iac-06-d4.2.05

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…The characteristic of the curves in Fig. 3 are the same that we gained from the continuous string model [2]. Although the current model cannot evaluate different materials, the fact that for a given stiffness a limited reachable altitude for the inner mass exists is very interesting and is also true for the continuous model.…”

Section: Elastic Dumb-bellmentioning

confidence: 78%

“…3 are shifted to the right. In a more sophisticated model [2] the mass of the rods is also taken into account and is therefore adequate to decide which material can be used in a Space Elevator. The characteristic of the curves in Fig.…”

Section: Elastic Dumb-bellmentioning

confidence: 99%

“…(2) Figure 5 shows the hyperbolic form of the gravitational field and the linear term of the spring that appears in Eqn. (2). Points of intersection correspond to equilibrium positions.…”

Section: Elastic Dumb-bellmentioning

confidence: 99%

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On stability problems of the space elevator

Schwarzbart

Steindl

Tröger

2007

Proc Appl Math and Mech

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A long dumb-bell satellite in the radial configuration represents a simple model of a Space Elevator. The dumb-bell moves with the angular velocity of the planet, but not relative to it. To calculate the stability of such a relative equilibrium we used the reduced energy-momentum method. We considered dumb-bell satellite models with rigid and elastic, but massless, rods. Figure 1 shows the model with the three DOFs r, ϑ and ϕ. A rigid dumb-bell satellite has a radial (ϕ = 0) and a tangential (ϕ = π/2) relative equilibrium position in the orbit. It is well known that the tangential one is unstable and the radial one is stable below a certain critical length that is for β > β crit = r/a ≈ 3.146 [1]. Integration of the equations of motion for a long satellite (β < β crit = 3.125), with a small deviation from the radial equilibrium position towards the planet, gives the unstable motion of the dumb-bell which is shown in Fig. 2a. When the disturbed radius is larger than the equilibrium position, the dumb-bell moves away from the planet in a helical path (Fig. 2b). We need an additional mass at the geostationary orbit to reduce the altitude of the inner mass beyond the critical value β crit , in order to stabilize the radial equilibrium position. When the inner mass of the dumb-bell touches the surface of the earth, we end up with a simple Space Elevator. For a terrestrial dumb-bell satellite a mass at the geostationary height (r gs ≈ 42164 km) of approximately 6200 times the endmass m = 1 kg is necessary to stabilize the radial configuration. Rigid dumb-bell Elastic dumb-bellNext we assumed two endmasses m, each 1 kg, connected with a mass at the geostationary orbit m s by means of massless, linear elastic springs. This model has just the interesting radial equilibrium position (ϕ = 0). With the definition of the spring stiffness c = EA/l 0 and a realistic choice of the cross-section A = 2.10 −9 m 2 we investigated the stability of the elastic dumb-bell satellite for different materials given in Tab. 1. Because of the non-linear gravitation field we have different unloaded spring lengths for the inner l 0i and outer l 0o spring. With the cyclic variable ϑ and the DOFs r i , r o and r gs (=radii of the three masses) we obtained the amended potentialwith the momentum μ 0 for the motion along the group orbit and the mass ratio α = m s /m. The first and second derivatives of Eqn. 1, with respect to the non-cyclic variables, lead to a system of equations for the equilibrium positions and their stability. Evaluation of these equations was done with a homotopy procedure with the

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Section: Elastic Dumb-bellmentioning

confidence: 78%

Section: Elastic Dumb-bellmentioning

confidence: 99%

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On stability problems of the space elevator

Schwarzbart

Steindl

Tröger

2007

Proc Appl Math and Mech

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“…Pugno [187] studied the strength of carbon nanotube-based space elevator cable and concluded that the actual strength of the cable would be reduced by ~70% of theoretical estimation due to presence of defects. Pugno et al [188] studied the stability of the track of space elevator and concluded that the track in itself is orbitally unstable, but attaching an adequately heavy satellite can stabilize it. CNTs are also used in aircraft wiring [189], propulsion [190], lightening protection [191] and stealth aircraft designs [192].…”

Section: Applicationsmentioning

confidence: 99%

An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

Khan¹

2021

Preprint

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Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

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Strength of Nanotubes and Megacables

Pugno

2010

Physical Properties of Ceramic and Carbon Nanoscale Structures

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On the Stability of the Track of the Space Elevator

Cited by 6 publications

References 15 publications

On stability problems of the space elevator

On stability problems of the space elevator

An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

Strength of Nanotubes and Megacables

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