Детальная информация

This study guide introduces the basic principles of oscillation theory and the stability of mechanical systems. It covers methods for determining equilibrium positions and analyzing stability using Lyapunov’s criterion, as well as key aspects of linear and nonlinear systems. The guide discusses free and forced oscillations for systems with one and two degrees of freedom, the effects of resistance, and resonance phenomena. Practical examples and recommendations for solving dynamic problems are provided.

• CONTENTS
• INTRODUCTION TO THE THEORY OF OSCILLATIONS
• Determination of the equilibrium position of the system
• Stability of the equilibrium position according to Lyapunov
• Linear and nonlinear systems. Linearization
• Lagrange–Dirichlet theorem (without proof). Sylvester's criterion
• SYSTEM WITH ONE DEGREE OF FREEDOM
• Free oscillations without resistance
• Relay dissipative function
• Ф and total mechanical energy
• Influence of viscous resistance on the motion of the system
• Forced oscillations without resistance
• Beats and resonance in the absence of resistance
• Dependencies of the coefficient of dynamism and phase shift
• Forced oscillations with viscous resistance. The law of motion
• Characteristics 𝜆(𝑧) and 𝜀 (𝑧)
• Example of solving the problem on oscillations of a system with one degree of freedom.
• SYSTEM WITH TWO DEGREES OF FREEDOM
• Quadratic form of potential energy. Condition for the stability of the equilibrium position
• Quadratic form of kinetic energy
• Differential equations of motion of a system. Main forms
• Oscillations of a double mathematical pendulum
• Forced oscillations without resistance
• Dynamic vibration damper
• ELEMENTARY GYROSCOPE THEORY
• Gyroscope on a hinge
• Role of transport inertial forces
• Angular acceleration of the gyro
• Free gyroscope on the plane
• Columbus Egg