Normalsperiod

Normalsperiod is a term used in the study of linear dynamical systems to denote the period of a specific normal mode of vibration. It describes the time required for that mode to complete one full cycle.

In mathematical terms, consider a system described by M x'' + K x = 0, where M is the mass matrix and K is the stiffness matrix. The normal modes are found by solving K v = ω^2 M v. Each eigenpair (ω, v) defines a mode with period T = 2π/ω. In informal use, this T is referred to as the normalsperiod.

Example: For a single degree-of-freedom mass-spring, m x'' + k x = 0, giving ω = sqrt(k/m) and T = 2π sqrt(m/k). In a linear chain of masses and springs, a mode with wave number k has ω_k = 2 sqrt(k/m) sin(k a / 2), yielding T_k = 2π/ω_k.

Notes and limitations: The term is not universally standardized; many authors simply speak of the period of a mode. In damped or nonlinear systems, the oscillation is not perfectly periodic and the notion of a single normalsperiod may be ambiguous or amplitude-dependent. In such cases, the period may vary with time or be approximated for small oscillations.

See also: normal mode, period, angular frequency, eigenvalue problem, lattice dynamics.