Reasonable analysis of hypersonic flows requires a thermodynamic non-equilibrium model to properly simulate strong shock waves or high pressure and temperature states in the flow field. The energy relaxation method (ERM) has been used to model such a non-equilibrium effect which is generally expressed as a hyperbolic system of equations with a stiff relaxation source term. Relaxation time that is multiplied with source terms is responsible for nonequilibrium in the system. In the present study, a numerical analysis has been carried out with varying values of relaxation time for several hypersonic flows with AUSM (advection upstream splitting method) as a numerical scheme. Vibration modes of thermodynamic nonequilibrium effects are considered. The results obtained showed that, as the relaxation time reduces to zero, the solution marches toward equilibrium, while it shows non-equilibrium effects, as the relaxation time increases. The present computations predicted the experiment results of hypersonic flows with good accuracy. The work carried out suggests that the present energy relaxation method can be robust for analysis of hypersonic flows.