(1) From Section 5.2, Eqn 5.2-1, the stiffnesses of the members AB and CD are :
\hspace{5em}k_{AB} =\frac{12EI}{L^3} N/mm;\qquad k_{CD}=\frac{12E(3I)}{(2L)^2}=4.5\frac{EI}{L^3} \ N/mm
Hence
\hspace{5em}\begin{matrix}k=k_{AB} + k_{CD} & = & 16.5\frac{EI}{L^3} N/mm \\ \\ & = & 16 500 \frac{EI}{L^3} N/m\end{matrix}
From Eqn 10.3-17,
\hspace{5em} \omega =\surd \frac{K}{M}
(where k is in newtons per metre and M in kilograms), then
\hspace{5em} \omega =\surd{ \left(\frac{16500 EI}{ML^3}\right)}= 128 \surd{\left( \frac{EI}{ML^3}\right)} radians Persecond
From Eqn 10.3-20,
\hspace{5em} T=\frac{2\pi }{\omega }= \frac{2\pi }{128 }\surd \left(\frac{ML^3}{EI} \right) seconds
\hspace{5em} ƒ =\frac{1}{T} = \frac{128}{2\pi }\surd \left(\frac{EI}{ML^3} \right) cycles per second
(2) From Eqn 10.3-25, if the initial conditions are x=x_{\tau } \ m \ and \ \dot{x}=\dot{x}_{\tau} \ m/s at t = τ seconds, then
\hspace{5em} x=\frac{\dot{x}_{\tau }}{\omega } \sin \omega \left(t- \tau \right) + x_\tau \cos \omega \left(t- \tau \right) \ m (10.3-26)
In this example, x_{\tau }=l \ mm = 0.001 \ l \ m, \dot{x}_{\tau }=0, \tau =1s, \ and \ \omega =128\surd \left[\left(EI\right)/\left(ML^3\right) \right], hence
\hspace{5em} x=0.001/\cos \left[128 \surd\left(\frac{EI}{ML^3} \right)\left(t-1\right) \right] m
The reader should pay particular attention to units in structural dynamics problems. The stiffness k should be in absolute units, e.g. N/m; the mass M should be in mass units, e.g. kg. Similarly, in using Eqn 10.3-26, x should be expressed in metres, \dot{x} in m/s, and t and τ in seconds.
\hspace{5em} \begin{matrix} m_{1} \\ m_{2} \end{matrix} =\pm i\surd{\frac{k}{M}=\pm i\omega } (10.3-17)
\hspace{5em} x=\frac{\dot{x}_{\tau }}{\omega } \sin \omega \left(t- \tau \right) + x_\tau \cos \omega \left(t- \tau \right) (10.3-25)
(for x=x_\tau \ and \ dx/dt= \dot{x}_{\tau } at t=τ).
\begin{matrix}\vphantom{} \\ \begin{matrix} 1 \phantom{\frac{-6}{L^2}}\\ 2 \phantom{\frac{-6}{L^2}}\\ 3 \phantom{\frac{-6}{L^2}}\\ 4 \phantom{\frac{-6}{L^2}}\\ 5 \phantom{\frac{-6}{L^2}}\\ 6 \phantom{\frac{-6}{L^2}}\\ 7 \phantom{\frac{-6}{L^2}}\\ 8 \phantom{\frac{-6}{L^2}}\\ 9 \phantom{\frac{-6}{L^2}}\\ 10 \phantom{\frac{-6}{L^2}}\\ 11 \phantom{\frac{-6}{L^2}}\\ 12\phantom{\frac{-6}{L^2}} \end{matrix} \begin{bmatrix} P_{x1} \phantom{\frac{-6}{L^2}}\\ P_{y1} \phantom{\frac{-6}{L^2}}\\ P_{z1} \phantom{\frac{-6}{L^2}}\\ m_{x1} \phantom{\frac{-6}{L^2}}\\ m_{y1} \phantom{\frac{-6}{L^2}}\\ m_{z1} \phantom{\frac{-6}{L^2}}\\ P_{x2} \phantom{\frac{-6}{L^2}} \\ P_{y2} \phantom{\frac{-6}{L^2}}\\ P_{z2} \phantom{\frac{-6}{L^2}}\\ m_{x2} \phantom{\frac{-6}{L^2}}\\ m_{y2} \phantom{\frac{-6}{L^2}} \\ m_{z2}\phantom{\frac{-6}{L^2}} \end{bmatrix} \end{matrix} = \begin{matrix} \vphantom{} & \hspace{0em} \begin{matrix}1&& 2 &&& 3&&& 4&&& 5&& 6&&& 7&&& 8&&& 9&& 10&&& 11&&& 12\end{matrix} \\ \begin{matrix} E \\ \vphantom{E} \\ \vphantom{} \\ \vphantom{} \\ \vphantom{} \\ \vphantom{}\\ \vphantom{} \\ \vphantom{} \\ \vphantom{} \\ \vphantom{} \\ \vphantom{} \\\vphantom{} \\\vphantom{} \end{matrix}\vphantom{} & \begin{bmatrix} {A}/{L} & 0 & 0 & 0 & 0 & 0 & -{A}/{L} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{12I_{z}}{L^3} & 0 & 0 & 0 & \frac{6I_{z}}{L^2} & 0 & \frac{-12I_{z}}{L^3}& 0 & 0 & 0 & \frac{6I_{z}}{L^2} \\ 0 & 0 &\frac{12I_{y}}{L^3}& 0 & \frac{-6I_{y}}{L^2} & 0 & 0 & 0 &\frac{-12I_{y}}{L^3} & 0 & \frac{-6I_{y}}{L^2} & 0 \\ 0 &0 & 0 & \frac{J}{2L(1+\nu )} & 0 & 0 & 0 & 0 & 0 & \frac{-J}{2L(1+\nu )} & 0 & 0 \\ 0 & 0 & \frac{-6I_{y}}{L^2} & 0 & \frac{4I_{y}}{L} & 0 & 0 & 0 & \frac{6I_{y}}{L^2}& 0 &\frac{2I_{y}}{L}& 0 \\ 0 &\frac{6I_{z}}{L^2} & 0 & 0 & 0 & \frac{4I_{z}}{L} & 0 & \frac{-6I_{z}}{L^2} & 0 & 0 & 0 &\frac{2I_{z}}{L} \\ -A/L &0 & 0 & 0 & 0 & 0 & A/L & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{-12I_{z}}{L^3} & 0 & 0 & 0 & \frac{-6I_{z}}{L^2} & 0 & \frac{12I_{z}}{L^3} & 0 & 0 & 0 & \frac{-6I_{z}}{L^2} \\ 0 & 0 &\frac{-12I_{y}}{L^3} & 0 & \frac{6I_{y}}{L^2} &0 & 0 & 0 &\frac{12I_{y}}{L^3} & 0 & \frac{6I_{y}}{L^2} &0 \\ 0 & 0 & 0 & \frac{-J}{2L(1+\nu )} & 0 & 0 & 0 & 0 & 0 & \frac{J}{2L(1+\nu )} & 0 & 0 \\ 0 & 0 &\frac{-6I_{y}}{L^2}& 0 &\frac{2I_{y}}{L} & 0 &0 & 0 & \frac{6I_{y}}{L^2} & 0 &\frac{4I_{y}}{L} & 0 \\ 0 & \frac{6I_{z}}{L^2} & 0 & 0 & 0 & \frac{2I_{z}}{L} & 0 &\frac{-6I_{z}}{L^2} & 0 & 0 & 0 &\frac{4I_{z}}{L} \end{bmatrix} \end{matrix} \begin{matrix}\vphantom{} \\ \begin{bmatrix} d_{x1} \phantom{\frac{-6}{L^2}}\\ d_{y1} \phantom{\frac{-6}{L^2}}\\ d_{z1} \phantom{\frac{-6}{L^2}}\\ \theta_{x1} \phantom{\frac{-6}{L^2}}\\ \theta_{y1} \phantom{\frac{-6}{L^2}} \\ \theta_{z1} \phantom{\frac{-6}{L^2}}\\ d_{x2} \phantom{\frac{-6}{L^2}} \\ d_{y2} \phantom{\frac{-6}{L^2}}\\ d_{z2} \phantom{\frac{-6}{L^2}}\\ \theta_{x2} \phantom{\frac{-6}{L^2}}\\ \theta_{y2} \phantom{\frac{-6}{L^2}}\\ \theta_{z2} \phantom{\frac{-6}{L^2}}\end{bmatrix}\end{matrix} (5.2-1)
