Horizontal Vibrations of a Framework

Suppose a two story framework consists of columns of negligible mass, and two very stiff horizontal girders, of mass  
\[m_1 , \: m_2\]
  respectively. The columns are stiff, of stiffnesses  
\[k_1, \: k_2\]
  respectively. The joints between the columns and the girders are rigid, and vibrations are undamped. When vibrations occur the girders remain horizontal.

framework vibrating horizontally

If the horizontal displacements of the girders are  
\[x_1 , \: x_2\]
  respectively, then applyting Newton's Second Law to each girder gives
\[m_1 \ddot{x}_1=-k_1 x_1+k_2(x_2-x_1) \rightarrow \ddot{x}_1=(- \frac{k_1}{m_1}+ \frac{k_2}{m_1}) x_1+ \frac{k_2}{m_1}x_2\]

\[m_2 \ddot{x}_2=-k_2(x_2-x_1) \rightarrow \ddot{x}_2=\frac{k_2}{m_2}x_1- \frac{k_2}{m_2}x_2 \]
We can write this in matrix form as
\[ \begin{pmatrix} \ddot{x}_1 \\ \ddot{x}_2 \end{pmatrix} = \left( \begin{array}{cc} - \frac{k_1}{m_1}+ \frac{k_2}{m_1} & \frac{k_2}{m_1} \\ \frac{k_2}{m_2} & - \frac{k_2}{m_2} \end{array} \right) \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\]
The frequencies of vibration of the system are the square roots of the negatives of the eigenvales of the matrix.
\[\begin{equation} \begin{aligned} \left| \begin{array}{cc} - \frac{k_1}{m_1}+ \frac{k_2}{m_1} - \lambda & \frac{k_2}{m_1} \\ \frac{k_2}{m_2} & - \frac{k_2}{m_2} - \lambda \end{array} \right| &= \frac{\lambda^2 m_1m_2+ \lambda(m_1-k_1m_2-k_2m_2)+k_1k_2}{m_1m_2} \\ &= \frac{\lambda^2 m_1m_2+ \lambda(m_1-k_1m_2-k_2m_2)+k_1k_2}{m_1m_2}=0 \end{aligned} \end{equation}\]
If the eigenvalues - the solutions to this quadratic are  
\[\lambda_1 \: \lambda_2\]
  the the frequencies of vibrtation of the structures are  
\[\omega_1= \sqrt{-\lambda}_1 , \: \omega_2= \sqrt{-\lambda}_2\]

Add comment

Security code