In this paper, we propose and investigate a new adaptive control algorithm for mechanical systems with time-varying parameters and/or time-varying disturbances. The proposed method does not assume any structure to the time-varying parameter or disturbance. The idea is based on the expansion of the time-varying parameter/disturbance using the Taylor series expansion; this facilitates expanding a time-varying function as a finite length polynomial and a bounded residue; the coefficients of the finite length polynomial are estimated in a time interval small enough so that they can be assumed to be constant within that interval. A novel experiment is designed using a two-link mechanical manipulator to investigate the proposed algorithm experimentally. Simulation and experimental results validate the proposed new adaptive control algorithm; we discuss these results and also give some future research directions.