Normalmoden

Normalmoden, or normal modes in English, are the distinct patterns of motion in a system with multiple degrees of freedom in which all parts oscillate at a common frequency with fixed phase relationships. In linear, time-invariant systems, the equations of motion can be decomposed into independent single-degree-of-freedom equations by expressing the motion as a sum of these normal modes. Each normal mode has an associated natural frequency and a mode shape that describes the relative amplitudes of the system’s components.

In mechanics, the undamped free-vibration problem leads to a generalized eigenvalue problem Kφ = ω^2 M φ, where K is the stiffness matrix, M the mass matrix, φ the mode shape, and ω the natural frequency. The mode shapes are orthogonal with respect to the mass matrix, and together they form a basis for the system’s displacement space, enabling modal or normal-coordinate analysis and modal superposition.

Normal modes are observed in simple systems such as a vibrating string, two coupled pendulums, or a flexible plate. Damping and nonlinearity modify the idealized picture: eigenvalues become complex, and each mode acquires a damping ratio that governs its decay. When an external force excites the system, the response can be expressed as a sum of modal responses, each evolving at its own natural frequency.

Applications of normal modes include structural and mechanical vibration analysis, acoustics, earthquake engineering, and various fields of physics where resonant behavior and energy transfer between components are important.