This paper deals with the tracking control problem of a general class of nonholonomic dynamic systems with reference signals that may exponentially decay. By introducing a time-varying coordinate transformation and using the cascaded-design approach, smooth time-varying controllers are constructed, which render the tracking error dynamics globally Kappa-exponentially stable. The result shows that the persistent-excitation condition or not coverging to zero for reference signals is not necessary even for the globally Kappa-exponential tracking of nonholonomic systems. The proposed method is applied to the tracking control of an underactuated surface vessel.