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Traditional design optimization of the Voigt-type dynamic vibration absorber often solved for the vertical or lateral vibration problems. However, for the damped primary system under torsional excitation, to the best of our knowledge, there is no study to solve this problem by algebraic approaches. This paper presents the analytical solutions to the optimization of dynamic vibration absorber, which is used to suppress torsional vibration of multi-degrees-of-freedom damped linear systems. The parameters considered in optimizing are dimensionless natural frequency of dynamic vibration absorber and viscous damping of absorber. First, the system equations of motion for shaft–dynamic vibration absorber system subjected to time-varying torsional moment were established. Then, closed-form formulae of optimized parameters were derived using the fixed-point theory. The obtained formulae provide exact solution for the proposed problem. To confirm the effectiveness of the obtained formulae, parametric studies on torsional vibration were performed for several sample multi-degrees-of-freedom systems with and without optimal dynamic vibration absorber. Numerical results showed that torsional vibrations of the primary system attached with optimal dynamic vibration absorber are effectively suppressed, even in the resonant conditions