A method for estimating a steady state response of a drill string in a borehole includes calculating a first displacement of the drill string in a frequency domain for a first excitation force frequency and a number of multiples of this frequency using an equation of motion of the drill string. The equation of motion has a static force component, an excitation force component, and a non-linear force component with respect to at least one of a deflection and a derivative of the deflection of the drill string. The method further includes: transforming the first displacement from the frequency domain into a time domain; calculating a non-linear force in the time domain; calculating a frequency domain coefficient derived from the calculated non-linear force in the time domain; and calculating a second displacement of the drill string in the frequency domain using the equation of motion and the frequency domain coefficient.