Three particles 1, 2 and 3 are fixed to three horizontal elastic strings, each of length  {jatex options:inline}l{/jatex}, and sibject to horizontal forces . {jatex options:inline}p_1 sin (k_1t), \: p_2 sin (k_2t), \: p_3 sin (k_3t){/jatex}.

Suppose the displacements are  {jatex options:inline}x_1, \: x_2{/jatex}  and  {jatex options:inline}x_3{/jatex}  respectively. If the increase  {jatex options:inline}x_1{/jatex}  in the length of string 1 is small then we can treat  {jatex options:inline}T{/jatex}  as constant. Resolving vertically for each particle in turn: Particle 1:  {jatex options:inline}m_1 \ddot{x}_1= -Tsin \theta_1 +T sin \theta_2+ p_1 sin (k_1t){/jatex}
Particle 2:  {jatex options:inline}m_2 \ddot{x}_2= -Tsin \theta_2 -T sin \theta_3+ p_2 sin (k_2t){/jatex}
Particle 2:  {jatex options:inline}m_3 \ddot{x}_3= Tsin \theta_3 -T sin \theta_4+ p_3 sin (k_3t){/jatex}
For each  {jatex options:inline}x{/jatex}  small  {jatex options:inline}tan \theta =\frac{x}{l} \simeq \theta{/jatex}  so the above equations become
Particle 1:  {jatex options:inline}m_1 \ddot{x}_1= -\frac{Tx_1}{l} +\frac{T(x_2-x_1}{l}+ p_1 sin (k_1t){/jatex}
Particle 2:  {jatex options:inline}m_2 \ddot{x}_2= -\frac{T(x_2-x_1)}{l} -\frac{Tx_2-x_3)}+p_2 sin (k_2t){l}{/jatex}
Particle 2:  {jatex options:inline}m_3 \ddot{x}_3= T\frac{x_2-x_3}{l} - \frac{x_3}{l}+p_3 sin (k_3t){/jatex}
Dividing by  {jatex options:inline}m_1, \: m_2, \: m_3{/jatex}  respectively and writing in matrix form gives
{jatex options:inline} \begin{pmatrix}\ddot{x}_1 \\ \ddot{x}_2 \\ \ddot{x}_3 \end{pmatrix} = \left( \begin{array}{ccc} -\frac{2T}{m_1l} & \frac{T}{m_1l} & 0 \\ \frac{T}{m_2l} & -\frac{2Tx_2}{m_2l} & \frac{T}{m_2l} \\ 0 & \frac{T}{m_3l} & - \frac{T}{m_3l} \end{array} \right) \begin{pmatrix}x_1 \\ x_2 \\ x_3 \end{pmatrix} + \begin{pmatrix}\frac{p_1}{m_1} sin (k_1t) \\ \frac{p_2}{m_2} sin (k_2t) \\ \frac{p_3}{m_3} sin (k_3t) \end{pmatrix}{/jatex}
The eigenvalues of the matrix are  {jatex options:inline}\lambda{/jatex}  and natural frequencies of vibration are  {jatex options:inline}f=2 \pi \sqrt{- \lambda}{/jatex}.
If any od the forcing frequencies  {jatex options:inline}f=_i- 2 \pi k_i{/jatex}  are equal to the corresponding natural frequencies then we have resonance.