A steady prediction method of dynamic stability derivatives is presented in the unified framework of the unsteady Euler equations. New approach does not require any modification of the governing equations except addition of non-inertial force terms. The present methods are applied to compute the pitch-damping coefficients using the lunar coning and the lunar helical motions in the Cartesian coordinate frame. The results for the ANSR and the Basic Finner are in good agreement with the PNS data, range data, and the results using the unsteady prediction method. The results show that the steady approach using the unified governing equations in the Cartesian coordinate frame can be successfully applied to predict the pitch-damping coefficients.