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Question 16.P.5: The vibrations of the bar in Problem 16.4 are resisted by da...

The vibrations of the bar in Problem 16.4 are resisted by damping forces that are viscous in nature. Assuming that the distributed damping coefficient is proportional to the mass per unit length and that damping in the first mode is 5 % of critical, derive the damping ratios for the second and the third modes. Then using the first three modes obtain an expression for the damped axial vibrations of the bar.

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\begin{aligned}&\xi_{1}=0.05 \quad \xi_{2}=0.0122 \quad \xi_{3}=0.0082 \\&\omega_{d 1}=0.6155 \sqrt{\frac{E A_{1}}{m_{1} L^{2}}} \\&\omega_{d 2}=2.5261 \sqrt{\frac{E A_{1}}{m_{1} L^{2}}} \\&\omega_{d 3}=3.7571 \sqrt{\frac{E A_{1}}{m_{1} L^{2}}}\end{aligned}

Modal responses

\begin{aligned}&y_{1}=1.0776 \frac{P_{0} L}{E A_{1}}\left[1-e^{-\xi_{1} \omega_{1} t}\left(\cos \omega_{d 1} t+\frac{\xi_{1} \omega_{1}}{\omega_{d 1}} \sin w_{d 1} t\right)\right] \\&y_{2}=-0.06397 \frac{P_{0} L}{E A_{1}}\left[1-e^{-\xi_{2} \omega_{2} t}\left(\cos \omega_{d 2} t+\frac{\xi_{2} \omega_{2}}{\omega_{d 2}} \sin \omega_{d 2} t\right)\right] \\&y_{3}=-0.02892 \frac{P_{0} L}{E A_{1}}\left[1-e^{-\xi_{3} \omega_{3} t}\left(\cos \omega_{d 3} t+\frac{\xi_{3} \omega_{3}}{\omega_{d 3}} \sin \omega_{d 3} t\right)\right] \\&u(x, t)=\sum_{i=1}^{3} f^{(i)}(x) y_{i}(t)\end{aligned}

where f^{(i)}(x) are as in Example 15.2.

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