This paper describes an algorithm which predicts steady-state torsional vibration amplitudes without resorting to more general, complicated, expensive, and time consuming advanced structural analysis programs. It applies equally well near or far from resonances. The method is a generalization of the classical Holzer analysis including damping, multiple arbitrary forcing functions, and frequency dependent stiffnesses and inertias. The algorithm requires a description of the rotating system as a series of inertias, elastic springs, and dampers. The system description may include branches, gear reductions, and frequency dependent characteristics, such as accessory drives, transmissions, and pendulum absorbers respectively. The system may also include multiple excitation locations and relative phases characteristic of multi-cylinder engines and other machines. An engine model simulates the excitation caused at each cylinder location for an arbitrary number of cylinders, an arbitrary crankshaft arrangement, and an arbitrary rotational speed and brake mean effective pressure. The driveline description readily accepts the frequency dependent damping of elastomeric couplings and other dissipative elements. The output of the algorithm is the torque and angular displacement of all points in the system as functions of time for one complete cycle of the vibration. This paper thoroughly describes the algorithm and compares the calculated results with engine test stand data.