This study applies the extended transfer matrix method and Newton-Raphson technique with complex numbers for torsional vibration analysis of damped systems. The relationships of the vibratory amplitude, the vibratory torque, the derivatives of the vibratory angular displacement and the vibratory torque between components at the left end and the right end of the torsional vibration system are derived. The derivatives of the vibratory angular displacement and the vibratory torque are used directly in the Newton-Raphson technique to determine the eigensolutions of systems that are compared and show good agreement with the available data.

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