Linear Systems

oscillating systems whose properties do not change when their state changes—that is, the parameters of a linear system that characterize its properties (the elasticity, mass, and coefficient of friction of a mechanical system; the capacitance, inductance, and active resistance of an electrical system) are independent of the quantities that characterize its state (displacements and velocities in the case of a mechanical system; voltages and currents in the case of an electrical system).

The parameters of real systems always depend to some extent on state. For example, the coefficient of elasticity of a spring depends on the magnitude of the deformation (deviations from the Hooke law in the case of large deformations), and the active resistance of a conductor depends on its temperature, which in turn depends on the strength of the current passing through the conductor. Therefore, real systems may be considered linear only within certain limits of changes in their state for which changes in their parameters may be disregarded. For a very large number of real systems these limits prove to be extremely broad, and therefore most problems can be solved by regarding real systems as linear. Examples of linear systems are a pendulum (for oscillations of small amplitude), an electrical oscillatory circuit, a bridge measuring circuit, and automatic control systems. When changes in the parameters of a real system appear within the limits of possible changes in its state, the nonlinearity of the system must be taken into account.

Linear systems have properties that greatly simplify analysis of processes that transpire within them. Processes in linear systems are described by linear differential equations (hence their name). In physically different linear systems processes are described by structurally identical equations. This is the basis for physical and, in particular, electrical simulation of linear systems and computer simulation. Linear systems play a major role in physics and engineering, since they reproduce without distortion of form external influences that have the character of harmonic oscillations and because the superposition principle is valid in them.