It is well known that transient rotordynamic analyses involve numerical integration of the equations of motion in order to study the response of the system under an applied forcing function. A common problem that arises in such simulations is the choice of step-size that needs to be used to obtain numerically stable results. Traditional numerical integration techniques such as the Runge-Kutta algorithms not only require splitting up second order differential equations as two first order equations, but also necessitate multiple integrations at each time-step, thus increasing the solution time. The Newmark-beta and Wilson-theta algorithms are some of the prevalent methods that have been used for transient simulations in rotordynamics. However, those single-step methods are only conditionally stable, and require iterations to converge to a solution at each time step, thus making it pseudosingle-step. In the more recent years, a modified form of the Rosenbrock algorithm has been proposed as a numerically stable and true single-step mathematical formulation for the integration of structural dynamics problems. In this paper, the modified Rosenbrock algorithm has been applied to a transient start-up multi-degree-of-freedom rotordynamics problem. A constant time step-size algorithm has been used for the simulations, and results of the transient analysis have been presented. The fact that a multi-degree-of-freedom system can be solved without condensation of the higher order modes makes the superior numerical damping characteristics of the algorithm become evident.