# Theory of oscillations

Definition of theory of oscillations. General approach to oscillation phenomena in different fields of science and technics. The basis of theory of oscillations, its development, application to different processes in nature and technics. Mathematical treatment and experimental research by Rayleigh, Poincare, Liapunov, Van-der-Pol, Mandelstam, Andronov. Kinematic and dynamic approach to theory of oscillations. The choice of models for investigation and classification of oscillating systems.

General treatment of systems with one degree of freedom. Conservative systems. Influence of the initial conditions. Kinetic and potential energy of oscillating movement. The presentation of movement with a help of phase plane. Singular points, center and saddle, types of movement and phase trajectories, separatrices. Oscillations in weakly non-linear system. Harmonic approach. Non-isochronous oscillations in non-linear system. Dissipative system. Types of singular points and phase portraits of dissipative systems. The stepwise method. The construction of phase trajectories by isocline method. Lienard method.

Force and parametric action. Force harmonic action on linear system. Superposition method. General solution. Forced oscillations. Resonance. Action of non-harmonic force. Non-linear systems under the external action. Weakly non-linear systems. An approximate study of non-linear systems with forced oscillations. Parametric action. The system with periodically varying parameters. The basic theory of parametric generation. Parametric resonance in linear and non-linear systems. An approximate study of parametric generation in systems with weak non-linearity. Parametric generators of Mandelstam and Papaleksi. Parametric regeneration. Forced oscillations in systems with parametric regeneration. Single tuned parametric amplifier.

Calculating oscillations in weakly non-linear systems with low damping. Basic equations for slowly varying amplitude calculation. Stability of steady state motions. Version of slowly varying amplitude and phase method. Application of SVA method to investigation of free and forced oscillations, parametric generation and regeneration.

General definition of self-oscillating systems and peculiarity of their energetics. Types of relaxation of self-oscillating systems. The denerated relaxation systems and substitution of fast movement by jumps. The conditions of jump. The transition form relaxation systems to resonator systems. Qualitative method of phase plane. Thompson's self-oscillating systems. Quasi-linear approach to Thompson's systems. Application of SVA method. Soft and hard modes of excitation of oscillations and their presentation on phase plane. Influence of external harmonic force on self oscillating system with one degree of freedom. Suppression of oscillations.

Number of degrees of freedom of oscillatory system. The non-uniqueness of dividing of complex system into partial. The frequencies of normalized oscillations and coefficients of amplitude distribution. Wien's graph. The coupling and degree of coupling as the energy exchange between the partial systems. The time of energy transfer and daming in real systems. Forced oscillations in systems with two degrees of freedom (conservative and weakly dissipative systems). Orthogonality of external force and free oscillation. Reciprocity and its behavior in systems with two degrees of freedom.

Double-tuned parametric amplification. Non-regenerated double-tuned parametric amplifier. Physical meaning of the maximum value of amplification coefficient. Regenerative double-tuned parametric amplifier. Relation between the amplification coefficient and bandwidth in this amplifier. Double-tuned parametrical oscillator with asynchronic and synchronic frequencies. Self-oscillating system with two degrees of freedom. Basic regimes of oscillation. Possibility of chaotic oscillations appearence. Definition of strange attractor. Phenomenon of frequency pulling. Regions of oscillations damping. Application of pulling phenomenon for frequency stabilization.

Description of oscillations in linear systems with n degrees of freedom by matrix form. Normal coordinates. Orthogonality of normal oscillations. Extremal properties of eigen frequencies. Forced oscillations in systems with n degrees of freedom. Systems with n degrees of freedom with a nonlinear reactivity. Manley-Rowe relations. Their physical meaning and application to analysis of double-tuned parametric amplifiers.

Fundamentals of the theory of distributed systems. Telegraph equations and their application to non-quasi-static systems. Definition of dispersion, phase and group velocity in distributed system. Eigen oscillations in finite-length systems. Influence of boundary conditions and lumped nonhomogeneity. Reciprocity and its behavior in distributed systems. Laser as a distributed self-oscillating system. Decomposing of laser electric field in a series of normal modes of a resonator. Condition for self-exitation of the mode. Two-mode regime. Phase plane analysis of weak and strong coupling between the modes. Self-synchronization of modes applied to the case of three modes laser. Short light pulses regime of generation.

1. V.V.Migulin, V.V.Medvedev, E.R.Mustel, and V.N.Parygin. "Fundamentals of theory of oscillations". Moscow, Nauka, 1988.

2. S.P.Strelkov. "Introduction to theory of oscillations". Moscow, Nauka, 1964.