normalmodi

Normalmodi, or normal modes, are the independent patterns of motion in which all parts of a system with several degrees of freedom oscillate at the same frequency with fixed phase relations. In a linear, conservative system, the motion can be decomposed into a sum of normal modes that do not exchange energy with each other.

Mathematically, for a system described by M q'' + K q = 0, seek solutions q(t) = φ cos(ω t). Substituting yields (K − ω^2 M) φ = 0. Nontrivial solutions exist when det(K − ω^2 M) = 0; the corresponding ω_i^2 are the squared normal frequencies, and φ_i are the mode shapes. In the time domain, q(t) = Σ_i a_i φ_i cos(ω_i t + δ_i).

Examples include a vibrating string fixed at both ends, with φ_n(x) ∝ sin(nπx/L) and ω_n = nπ c / L; a chain of coupled masses and springs yields a small set of normal modes, each representing an in-phase or out-of-phase motion.

Applications span mechanical engineering, acoustics, structural dynamics, and quantum mechanics, where normal modes describe the natural vibration patterns of molecules, crystals, or fields. In practice, damping and nonlinearities modify the ideal picture; modes may be damped, and energy slowly leaks between modes in real systems. The concept originated in 19th-century physics, with further development through linear algebra and the theory of eigenvalues; it remains a fundamental tool across sciences.