Mode shapes in physics are specific patterns of vibration that a structure or system can exhibit when it oscillates at its natural frequencies. These patterns describe the relative displacement of different parts of the system during vibration.
In applied mathematics, mode shapes are a manifestation of
eigenvectors
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
which describe the relative displacement of two or more elements in a mechanical system or wave front.
A mode shape is a deflection pattern related to a particular natural frequency and represents the relative displacement of all parts of a structure for that particular mode.
Mathematical derivation
Mode shapes have a mathematical meaning as 'eigenvectors' or 'eigenfunctions' of the
eigenvalue problem
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...
which arises, studying particular solutions of the
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
of a system.
See also
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Normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies ...
*
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
References
Linear algebra
Vectors (mathematics and physics)
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