A simple method for the vertical stabilization of two tethered nanosatellites in low Earth orbit is described. Without active attitude control or passive damping, the satellite system would freely oscillate like a pendulum due to the gravity gradient moment acting dierentially on the displaced satellites. It would also vibrate axially like a bouncing ball as the tether cycles from tautness to slackness. What ambient friction exists within the satellite system might eventually damp out these librations and vibrations, but with small space structures such dissipative mechanisms are meager: undesirable oscillations could persist for months. The performance benefit of engineering a passive spring-damper mechanism along the length of the tether connecting the two nanosatellites is investigated. Two-dimensional orbital-plane analysis only is given in this paper. The axial spring-damper is tuned by frequency matching the axial and in-plane librational modes so that they are coupled as strongly as possible. The coupling encourages the two modes to xchange energies. Over time, net energy is removed from the librations and dissipated by the axial spring-damper mechanism. The system is driven towards stable equilibrium. It is shown that a sti-spring is inadequate; there must be signicant play in the tether when it is in tension for the scheme to work. The tuned parameters are shown to perform well over a broad range of initial conditions, from deployment to near stable equilibrium. Except when initialized near the unstable equilibrium the method reduces the amplitude of in-plane librations by a factor of ve in about one day for most initial conditions. The axial vibrations are seen to attenuate in unison with the librations.