normalmodanalyse

Normalmodanalyse, or normal mode analysis, is a technique for studying the vibrational behavior of linear dynamical systems by expressing motion as a sum of independent modes called normal modes. It is widely used in engineering and physics to understand how structures or mechanisms respond to excitations.

In a linear system with mass matrix M, damping C and stiffness K, the equation of motion is M x'' + C x' + K x = F(t). For undamped, unforced motion (C = 0, F = 0), one seeks solutions of the form x(t) = Φ q(t). Substituting gives a set of decoupled equations in the modal coordinates q, leading to the eigenproblem (K − ω^2 M) φ = 0. The vectors φ are the mode shapes and ω the natural frequencies. With damping, the eigenvalues become complex and the problem is treated as a generalized eigenproblem. If M and K are symmetric and damping is proportional or small, the equations approximately decouple in modal coordinates.

The typical procedure is: construct the system matrices M, C, K; solve the generalized eigenproblem to obtain natural frequencies and mode shapes; project initial conditions into the modal basis to obtain modal amplitudes; solve the decoupled modal equations and reconstruct the physical motion as x(t) = Φ q(t). In practice, this method is often combined with finite element modeling for complex structures.

Applications include structural vibration analysis, aerospace and automotive engineering, civil engineering, acoustics, and molecular vibrations. Benefits are reduced computational complexity and clearer insight into resonance and energy distribution, while limitations arise from assumptions of linearity, proportional damping, and well-separated frequencies.

Related concepts include modal analysis, eigenvalue problems, Fourier methods, and the finite element method.