Question 18.P.6: The rollers at joint R of the truss in Problem 18.5 are lock...

Dynamics of Structures

The rollers at joint R of the truss in Problem 18.5 are locked converting the support to a pinned support. Find the new frequencies and mode shapes. Identify the symmetric and antisymmetric mode shapes.

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\begin{aligned}\omega_{1} &=0.5107 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{2}=1.2204 \sqrt{\frac{E}{\rho L^{2}}} \\\omega_{3} &=1.5864 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{4}=1.7216 \sqrt{\frac{E}{\rho L^{2}}} \\\Phi &=\left[\begin{array}{rrrr}0.0000 & 0.1354 & -1.3292 & 0.0000 \\0.9046 & 0.0000 & 0.0000 & -0.9832 \\0.0000 & 1.0615 & 0.1828 & 0.0000 \\0.7405 & 0.0000 & 0.0000 & 0.7822\end{array}\right]\end{aligned}

Modes 1 and 4, which involve vertical motions of nodes $B$ and $D$ , but no horizontal motion, are symmetric modes. Modes 2 and 4 that involve only horizontal motions of nodes $\mathrm{B}$ and $\mathrm{D}$ are antisymmetric modes.

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