Question 18.P.5: The plane truss PQRS, shown in Figure P18.5, is pin supporte...

Dynamics of Structures

The plane truss PQRS, shown in Figure P18.5, is pin supported at $\mathrm{P}$ and supported on rollers at R. Members PQ and QR each have a cross-sectional area of $0.7 A$ , members PS and RS have a cross sectional area $A$ , and member QS has cross-sectional area $0.5 \mathrm{~A}$ . Find the frequencies and mode shapes of the system using (a) consistent mass matrix, (b) lumped mass matrix.

p18.5

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\begin{aligned}& \text { (a) } \omega_{1}=0.3263 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{2}=0.7367 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{3}=1.4436 \sqrt{\frac{E}{\rho L^{2}}} \\& \omega_{4}=1.7156 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{5}=2.2696 \sqrt{\frac{E}{\rho L^{2}}} \\& \Phi=\left[\begin{array}{rrrrr}0.247 & 0.400 & 0.906 & 0.261 & -0.860 \\0.707 & 0.561 & 0.142 & -0.971 & -0.087 \\-0.258 & 0.519 & -0.811 & 0.038 & -0.496 \\0.654 & 0.353 & -0.045 & 0.762 & 0.160 \\-0.463 & 0.556 & 0.304 & -0.069 & 1.086\end{array}\right] \\& \text { (b) } \omega_{1}=0.2842 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{2}=0.6780 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{3}=1.1193 \sqrt{\frac{E}{\rho L^{2}}} \\& \omega_{4}=1.3509 \sqrt{\frac{E}{\rho L^{2}}} \quad \omega_{5}=1.5877 \sqrt{\frac{E}{\rho L^{2}}} \\& \Phi=\left[\begin{array}{rrrrr}-0.206 & 0.386 & 0.656 & -0.308 & -0.684 \\0.626 & 0.438 & 0.152 & 0.748 & -0.133 \\-0.213 & 0.517 & -0.633 & -0.038 & -0.234 \\0.577 & 0.242 & -0.034 & -0.583 & 0.193 \\-0.391 & 0.559 & 0.320 & 0.062 & 0.712\end{array}\right]\end{aligned}

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