Question $5.$

Figure $5($Q$5)$ shows a simple model of a small bridge type structure, which is undergoing

refurbishment. Work being carried out on the bridge has resulted in the application of dynamic

load of Fcos$(t).$ Given, $a=0\mathrm{.}6m,b=0\mathrm{.}4m$ and $c=0\mathrm{.}2m$

Figure $5($Q$5)$

Using the coordinate system shown in the diagram, derive the equations

of motion and write them in the matrix form with the first element of

the displacement vector being the vertical displacement $y.$ Then, if

$m=6000kg,l=900kgm2,k=150k\frac{N}{m},$ the node shapes are

$\{u{\}}_1=\{[1],[-8\mathrm{.}15]\}$ and $\{u{\}}_2=\{[1],[0\mathrm{.}817]\}.$ Using this information, transform the

equations of motion into principal coordinates. Lastly, using the

equation of motion in principal coordinates, find the natural

frequencies of the system.