Vibration mode shape

A vibration mode shape is the spatial pattern of an eigenvibration: the relative displacement amplitude at each point when a structure vibrates freely at one natural frequency. Each mode pairs a shape (the eigenvector) with a frequency (the eigenvalue) from the free-vibration problem ([K] - omega^2 [M]){phi} = 0. For a ship, the 2-node vertical bending shape has maximum deflection near bow and stern with a node amidships; local panel modes show the stiffened plate flexing between supports. Mode shapes matter because resonance amplifies the response only where the excitation aligns with an antinode, so a propeller exciting near a stern antinode drives that mode hard. Finite element modal analysis extracts both shapes and frequencies.