Multidegree freedom system, free vibration, no damping
\begin {equation} \left [ m\right ] \left \{ \ddot {v}\left ( t\right ) \right \} +\left [ k\right ] \left \{ v\left ( t\right ) \right \} =\left \{ 0\right \} \label {eq:1} \end {equation}
Assume \[ \left \{ v\left ( t\right ) \right \} =\left \{ \hat {v}\right \} \sin \left ( \omega t+\theta \right ) \]
where \(\left \{ \hat {v}\right \} \) is an amplitude vector of constants. Acts like shape function. Hence \(\left \{ \ddot {v}\left ( t\right ) \right \} =-\omega ^{2}\left \{ \hat {v}\right \} \sin \left ( \omega t+\theta \right ) \). Substituting into Eq ??
\begin {align*} \left [ m\right ] \left ( -\omega ^{2}\left \{ \hat {v}\right \} \sin \left ( \omega t+\theta \right ) \right ) +\left [ k\right ] \left \{ \hat {v}\right \} \sin \left ( \omega t+\theta \right ) & =\left \{ 0\right \} \\ \left ( -\left [ m\right ] \omega ^{2}+\left [ k\right ] \right ) \left \{ \hat {v}\right \} & =\left \{ 0\right \} \end {align*}
This is an eigenvalue problem. Hence \[ \det \left ( \left [ k\right ] -\left [ m\right ] \omega ^{2}\right ) =0 \]
We obtain \(n\) unique eigenvalues \(\omega _{i}\) and corresponding \(n\) independent mode shapes \(\left \{ \hat {v}\right \} _{i}\)
\[ \left ( -\left [ m\right ] \omega _{i}^{2}+\left [ k\right ] \right ) \left \{ \hat {v}\right \} _{i}=\left \{ 0\right \} \]
Where \(\left \{ \hat {v}\right \} _{i}=\begin {Bmatrix} 1\\ v_{2}\\ \vdots \\ v_{n}\end {Bmatrix} _{i}\) or \(\left \{ \hat {v}\right \} _{i}=\begin {Bmatrix} \varphi _{1}\\ \varphi _{2}\\ \vdots \\ \varphi _{n}\end {Bmatrix} _{i}\). Hence for each \(\omega _{i}\) we get diļ¬erent shape function vector \(\left \{ \hat {v}\right \} _{i}.\) Let the mode shape matrix \(\left [ \Phi \right ] \) be
\begin {align*} \left [ \Phi \right ] & =\left [ \left \{ \hat {v}\right \} _{1},\left \{ \hat {v}\right \} _{2},\cdots ,\left \{ \hat {v}\right \} _{n}\right ] \\ & =\begin {Bmatrix} \varphi _{11} & \varphi _{12} & \cdots & \varphi _{1n}\\ \varphi _{21} & \varphi _{22} & \cdots & \varphi _{2n}\\ \vdots & \vdots & \ddots & \vdots \\ \varphi _{n1} & \varphi _{n2} & \cdots & \varphi _{nn}\end {Bmatrix} \end {align*}