We consider adaptive stabilization for a class of nonlinear second-order systems. Interpreting the system states as position and velocity, the system is assumed to have unknown, nonparametric position-dependent damping and stiffness coefficients. Lyapunov methods are used to prove global convergence of the adaptive controller. Furthermore, the controller is shown to be able reject constant disturbances and to asymptotically track constant commands. For illustration, the controller is used to stabilize the van der Pol limit cycle, the Duffing oscillator with multiple equilibria, and several other example systems.