Section 7.5

1. The natural frequencies are $ω_{0} = 0$ $ω_{0} = 0$ and $ω_{1} = 2$ $ω_{1} = 2$ . In the degenerate natural mode with “frequency” $ω_{0} = 0$ $ω_{0} = 0$ the two masses move linearly with $x_{1} (t) = x_{2} (t) = a_{0} + b_{0} t$ $x_{1} (t) = x_{2} (t) = a_{0} + b_{0} t$ . At frequency $ω_{1} = 2$ $ω_{1} = 2$ they oscillate in opposite directions with equal amplitudes.
2. The natural frequencies are $ω_{1} = 1$ $ω_{1} = 1$ and $ω_{2} = 3$ $ω_{2} = 3$ . In the natural mode with frequency $ω_{1}$ $ω_{1}$ , the two masses $m_{1}$ $m_{1}$ and $m_{2}$ $m_{2}$ move in the same direction with equal amplitudes of oscillation. At frequency $ω_{2}$ $ω_{2}$ they move in opposite directions with equal amplitudes.
3. The natural frequencies are $ω_{1} = 1$ $ω_{1} = 1$ and $ω_{2} = 2$ $ω_{2} = 2$ . In the natural mode with frequency $ω_{1}$ $ω_{1}$ , the two masses $m_{1}$ $m_{1}$ and $m_{2}$ $m_{2}$ move in the same direction with equal amplitudes of oscillation. In the natural mode with frequency $ω_{2}$ $ω_{2}$ they move in opposite directions with the amplitude of oscillation of $m_{1}$ $m_{1}$ twice that of $m_{2}$ $m_{2}$ .
4. The natural frequencies are $ω_{1} = 1$ $ω_{1} = 1$ and $ω_{2} = \sqrt{5}$ $ω_{2} = \sqrt{5}$ . In the natural mode with frequency $ω_{1}$ $ω_{1}$ , the two masses $m_{1}$ $m_{1}$ and $m_{2}$ $m_{2}$ move in the same direction with equal amplitudes of oscillation. At frequency $ω_{2}$ $ω_{2}$ they move in opposite directions with equal amplitudes.
5. The natural frequencies are $ω_{1} = \sqrt{2}$ $ω_{1} = \sqrt{2}$ and $ω_{2} = 2$ $ω_{2} = 2$ . In the natural mode with frequency $ω_{1}$ $ω_{1}$ , the two masses $m_{1}$ $m_{1}$ and $m_{2}$ $m_{2}$ move in the same direction with equal amplitudes of oscillation. At frequency $ω_{2}$ $ω_{2}$ they move in opposite directions with equal amplitudes.
6. The natural frequencies are $ω_{1} = \sqrt{2}$ $ω_{1} = \sqrt{2}$ and $ω_{2} = \sqrt{8}$ $ω_{2} = \sqrt{8}$ . In the natural mode with frequency $ω_{1}$ $ω_{1}$ , the two masses $m_{1}$ $m_{1}$ and $m_{2}$ $m_{2}$ move in the same direction with equal amplitudes of oscillation. In the natural mode with frequency $ω_{2}$ $ω_{2}$ they move in opposite directions with the amplitude of oscillation of $m_{1}$ $m_{1}$ twice that of $m_{2}$ $m_{2}$ .
7. The natural frequencies are $ω_{1} = 2$ $ω_{1} = 2$ and $ω_{2} = 4$ $ω_{2} = 4$ . In the natural mode with frequency $ω_{1}$ $ω_{1}$ , the two masses $m_{1}$ $m_{1}$ and $m_{2}$ $m_{2}$ move in the same direction with equal amplitudes of oscillation. At frequency $ω_{2}$ $ω_{2}$ they move in opposite directions with equal amplitudes.
8. $x_{1} (t) = 2 \cos t + 3 \cos 3 t - 5 \cos 5 t, x_{2} (t) = 2 \cos t - 3 \cos 3 t + \cos 5 t$ $x_{1} (t) = 2 \cos t + 3 \cos 3 t - 5 \cos 5 t, x_{2} (t) = 2 \cos t - 3 \cos 3 t + \cos 5 t$ . We have a superposition of three oscillations, in which the two masses move (1) in the same direction with frequency $ω_{1} = 1$ $ω_{1} = 1$ and equal amplitudes; (2) in opposite directions with frequency $ω_{2} = 3$ $ω_{2} = 3$ and equal amplitudes; (3) in opposite directions with frequency $ω_{3} = 5$ $ω_{3} = 5$ and with the amplitude of motion of $m_{1}$ $m_{1}$ being 5 times that of $m_{2}$ $m_{2}$ .
9. $x_{1} (t) = 5 \cos t - 8 \cos 2 t + 3 \cos 3 t, x_{2} (t) = 5 \cos t + 4 \cos 2 t - 9 \cos 3 t$ $x_{1} (t) = 5 \cos t - 8 \cos 2 t + 3 \cos 3 t, x_{2} (t) = 5 \cos t + 4 \cos 2 t - 9 \cos 3 t$ . We have a superposition of three oscillations, in which the two masses move (1) in the same direction with frequency $ω_{1} = 1$ $ω_{1} = 1$ and equal amplitudes; (2) in opposite directions with frequency $ω_{2} = 2$ $ω_{2} = 2$ and with the amplitude of motion of $m_{1}$ $m_{1}$ being twice that of $m_{2}$ $m_{2}$ ; (3) in opposite directions with frequency $ω_{3} = 3$ $ω_{3} = 3$ and with the amplitude of motion of $m_{2}$ $m_{2}$ being 3 times that of $m_{1}$ $m_{1}$ .
10. $x_{1} (t) = - 15 \cos 2 t + \cos 4 t + 14 \cos t, x_{2} (t) = - 15 \cos 2 t - \cos 4 t + 16 \cos t$ $x_{1} (t) = - 15 \cos 2 t + \cos 4 t + 14 \cos t, x_{2} (t) = - 15 \cos 2 t - \cos 4 t + 16 \cos t$ . We have a superposition of three oscillations, in which the two masses move (1) in the same direction with frequency $ω_{1} = 1$ $ω_{1} = 1$ and with the amplitude of motion of $m_{2}$ $m_{2}$ being $8 / 7$ $8 / 7$ times that of $m_{1}$ $m_{1}$ ; (2) in the same direction with frequency $ω_{2} = 2$ $ω_{2} = 2$ and equal amplitudes; (3) in opposite directions with frequency $ω_{3} = 4$ $ω_{3} = 4$ and equal amplitudes.
11. (a) The natural frequencies are $ω_{1} = 6$ $ω_{1} = 6$ and $ω_{2} = 8$ $ω_{2} = 8$ . In mode 1 the two masses oscillate in the same direction with frequency $ω_{1} = 6$ $ω_{1} = 6$ and with the amplitude of motion of $m_{1}$ $m_{1}$ being twice that of $m_{2}$ $m_{2}$ . In mode 2 the two masses oscillate in opposite directions with frequency $ω_{2} = 8$ $ω_{2} = 8$ and with the amplitude of motion of $m_{2}$ $m_{2}$ being 3 times that of $m_{1}$ $m_{1}$ . (b) $x (t) = 2 \sin 6 t + 19 \cos 7 t, y (t) = \sin 6 t + 3 \cos 7 t$ $x (t) = 2 \sin 6 t + 19 \cos 7 t, y (t) = \sin 6 t + 3 \cos 7 t$ We have a superposition of (only two) oscillations, in which the two masses move (1) in the same direction with frequency $ω_{1} = 6$ $ω_{1} = 6$ and with the amplitude of motion of $m_{1}$ $m_{1}$ being twice that of $m_{2}$ $m_{2}$ ; (2) in the same direction with frequency $ω_{3} = 7$ $ω_{3} = 7$ and with the amplitude of motion of $m_{1}$ $m_{1}$ being $19 / 3$ $19 / 3$ times that of $m_{2}$ $m_{2}$ .
12. The system’s three natural modes of oscillation have (1) natural frequency $ω_{1} = \sqrt{2}$ $ω_{1} = \sqrt{2}$ with amplitude ratios $1 : 0 : - 1$ $1 : 0 : - 1$ ; (2) natural frequency $ω_{2} = \sqrt{2 + \sqrt{2}}$ $ω_{2} = \sqrt{2 + \sqrt{2}}$ with amplitude ratios 1 : $- \sqrt{2}$ $- \sqrt{2}$ : 1; (3) natural frequency $ω_{3} = \sqrt{2 - \sqrt{2}}$ $ω_{3} = \sqrt{2 - \sqrt{2}}$ with amplitude ratios 1 : $\sqrt{2}$ $\sqrt{2}$ : 1.
13. The system’s three natural modes of oscillation have (1) natural frequency $ω_{1} = 2$ $ω_{1} = 2$ with amplitude ratios 1 : 0 : $- 1$ $- 1$ ; (2) natural frequency $ω_{2} = \sqrt{4 + 2 \sqrt{2}}$ $ω_{2} = \sqrt{4 + 2 \sqrt{2}}$ with amplitude ratios 1 : $- \sqrt{2}$ $- \sqrt{2}$ : 1; (3) natural frequency $ω_{3} = \sqrt{4 - 2 \sqrt{2}}$ $ω_{3} = \sqrt{4 - 2 \sqrt{2}}$ with amplitude ratios 1 : $\sqrt{2}$ $\sqrt{2}$ : 1.
15. $x_{1} (t) = \frac{2}{3} \cos 5 t - 2 \cos 5 \sqrt{3} t + \frac{4}{3} \cos 10 t, x_{2} (t) = \frac{4}{3} \cos 5 t + 4 \cos 5 \sqrt{3} t + \frac{16}{3} \cos 10 t$ $x_{1} (t) = \frac{2}{3} \cos 5 t - 2 \cos 5 \sqrt{3} t + \frac{4}{3} \cos 10 t, x_{2} (t) = \frac{4}{3} \cos 5 t + 4 \cos 5 \sqrt{3} t + \frac{16}{3} \cos 10 t$ .

We have a superposition of two oscillations with the natural frequencies $ω_{1} = 5$ $ω_{1} = 5$ and $ω_{2} = 5 \sqrt{3}$ $ω_{2} = 5 \sqrt{3}$ and a forced oscillation with frequency $ω = 10$ $ω = 10$ . In each of the two natural oscillations the amplitude of motion of $m_{2}$ $m_{2}$ is twice that of $m_{1}$ $m_{1}$ , while in the forced oscillation the amplitude of motion of $m_{2}$ $m_{2}$ is four times that of $m_{1}$ $m_{1}$ .
20. $x_{1}^{′} (t) = - v_{0}, x_{2}^{′} (t) = 0, x_{1}^{′} (t) = v_{0}$ $x_{1}^{′} (t) = - v_{0}, x_{2}^{′} (t) = 0, x_{1}^{′} (t) = v_{0}$ for $t > π / 2$ $t > π / 2$
21. $x_{1}^{′} (t) = - v_{0}, x_{2}^{′} (t) = 0, x_{1}^{′} (t) = 2 v_{0}$ $x_{1}^{′} (t) = - v_{0}, x_{2}^{′} (t) = 0, x_{1}^{′} (t) = 2 v_{0}$ for $t > π / 2$ $t > π / 2$
22. $x_{1}^{′} (t) = - 2 v_{0}, x_{2}^{′} (t) = v_{0}, x_{1}^{′} (t) = v_{0}$ $x_{1}^{′} (t) = - 2 v_{0}, x_{2}^{′} (t) = v_{0}, x_{1}^{′} (t) = v_{0}$ for $t > π / 2$ $t > π / 2$
23. $x_{1}^{′} (t) = 2 v_{0}, x_{2}^{′} (t) = 2 v_{0}, x_{1}^{′} (t) = 3 v_{0}$ $x_{1}^{′} (t) = 2 v_{0}, x_{2}^{′} (t) = 2 v_{0}, x_{1}^{′} (t) = 3 v_{0}$ for $t > π / 2$ $t > π / 2$
24. (a) $ω_{1} \approx 1.0293$ $ω_{1} \approx 1.0293$ Hz; $ω_{2} \approx 1.7971$ $ω_{2} \approx 1.7971$ Hz. (b) $v_{1} \approx 28 mi/h$ $v_{1} \approx 28 mi/h$ ; $v_{2} \approx 49 mi/h$ $v_{2} \approx 49 mi/h$
27. $ω_{1} = 2 \sqrt{10}, v_{1} \approx 40.26$ $ω_{1} = 2 \sqrt{10}, v_{1} \approx 40.26$ (ft/s (about 27 mi/h), $ω_{2} = 5 \sqrt{5}, v_{2} \approx 71.18 ft/s$ $ω_{2} = 5 \sqrt{5}, v_{2} \approx 71.18 ft/s$ (about 49 mi/h)
28. $ω_{1} \approx 6.1311, v_{1} \approx 39.03 ft/s$ $ω_{1} \approx 6.1311, v_{1} \approx 39.03 ft/s$ (about 27 mi/h) $ω_{2} \approx 10.3155, v_{2} \approx 65.67 ft/s$ $ω_{2} \approx 10.3155, v_{2} \approx 65.67 ft/s$ (about 45 mi/h)
29. $ω_{1} \approx 5.0424, v_{1} \approx 32.10 ft/s$ $ω_{1} \approx 5.0424, v_{1} \approx 32.10 ft/s$ (about 22 mi/h), $ω_{2} \approx 9.9158, v_{2} \approx 63.13 ft/s$ $ω_{2} \approx 9.9158, v_{2} \approx 63.13 ft/s$ (about 43 mi/h)

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Table of Contents for Section 7.5

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Table of Contents for
Section 7.5