Eigenfrequencies

Eigenfrequencies, also called natural frequencies, are the frequencies at which a system tends to oscillate when displaced from equilibrium and released, in the absence of sustained external forcing. They depend on the system’s structural properties, such as mass distribution, stiffness, and geometry, and they determine how the system responds to disturbances and vibrations.

In linear discrete systems described by M ẍ + K x = 0, one can seek harmonic solutions of the form x(t) = φ e^{i ω t}. Substituting this into the equation gives (K − ω^2 M) φ = 0. Nontrivial solutions exist only when det(K − ω^2 M) = 0, yielding a set of eigenvalues ω^2. The square roots ω are the eigenfrequencies, and the corresponding vectors φ are the mode shapes, describing how the system deforms in each natural vibration pattern. In multi-degree-of-freedom systems, each eigenfrequency comes with a distinct mode shape, and the modes form a basis that diagonalizes the undamped equations of motion.

Damping and driving forces modify the response but do not change the definition of the undamped eigenfrequencies; damping shifts and broadens resonances and can produce complex eigenvalues in some formulations. In continuous media, such as beams or plates, the eigenvalue problem becomes a differential equation with boundary conditions, yielding a discrete spectrum of natural frequencies that depend on material properties and geometry.

Applications of eigenfrequencies include predicting resonance, guiding design to avoid destructive vibrations, and performing experimental modal analysis to identify a structure’s dynamic behavior. They underpin engineering analyses across mechanical, civil, and aeronautical domains.