= Solution
A <normal mode> is a motion of a linear oscillatory system in which every coordinate has the same <angular frequency> and a fixed relative amplitude and phase. For a conservative small-displacement system it has the form
$$
\boldsymbol x(t)=q\cos(\omega t+\phi),\qquad Kq=\omega^2Mq,
$$
where $q$ is the <mode shape>, $K$ the <stiffness matrix>, and $M$ the <mass matrix>. Superposing <normal modes> gives the general small-oscillation motion when their <eigenvectors> form a complete <basis>.
Two common examples are standing transverse oscillations of a string fixed at both ends, and oscillations of coupled masses and springs. The string's mode shapes are sinusoidal along its length; the coupled system's mode shapes are fixed patterns of particle displacements. The following parts determine those patterns for the three-mass system in the source figure.
Back to article page
