In this tutorial, the traditional vibration theory course is supplemented with basic information on the principles of building dynamic models of machine mechanisms and drives, their mathematical description and methods for calculating typical vibration modes. In a concise form, methods of vibration protection and methods of rational dynamic synthesis of mechanisms and drives of machines are described, taking into account inertial and elastic dissipative characteristics. A number of special problems of increased complexity that arise during engineering calculations and design of mechanisms that form oscillatory systems with non-stationary and non-linear connections are considered. Designed for students of technical universities and can be useful in the preparation of bachelors, masters and graduate students of relevant specialties, as well as for engineers and students of the faculty of advanced training of teachers in the theory of mechanisms and machines.