Section 5.6

1. $x (t) = 2 \cos 2 t - 2 \cos 3 t$ $x (t) = 2 \cos 2 t - 2 \cos 3 t$
2. $x (t) = \frac{3}{2} \sin 2 t - \sin 3 t$ $x (t) = \frac{3}{2} \sin 2 t - \sin 3 t$
3. $x (t) = \sqrt{138388} \cos (10 t - α) + 5 \cos (5 t - β)$ $x (t) = \sqrt{138388} \cos (10 t - α) + 5 \cos (5 t - β)$ with $α = 2 π - \tan^{- 1} (1 / 186) \approx 6.2778$ $α = 2 π - \tan^{- 1} (1 / 186) \approx 6.2778$ and $β = \tan^{- 1} (4 / 3) \approx 0.9273$ $β = \tan^{- 1} (4 / 3) \approx 0.9273$ .
4. $x (t) = 2 \sqrt{106} \cos (5 t - α) + 10 \cos 4 t$ $x (t) = 2 \sqrt{106} \cos (5 t - α) + 10 \cos 4 t$ with $α = π - \tan^{- 1} (9 / 5) \approx 2.0779$ $α = π - \tan^{- 1} (9 / 5) \approx 2.0779$
5. $x (t) = (x_{0} - C) \cos ω_{0} t + C \cos ω t$ $x (t) = (x_{0} - C) \cos ω_{0} t + C \cos ω t$ , where $C = F_{0} / (k - m ω^{2})$ $C = F_{0} / (k - m ω^{2})$
7. $x_{sp} (t) = \frac{10}{13} \cos (3 t - α)$ $x_{sp} (t) = \frac{10}{13} \cos (3 t - α)$ with $α = π - \tan^{- 1} (12 / 5) \approx 1.9656$ $α = π - \tan^{- 1} (12 / 5) \approx 1.9656$
8. $x_{sp} (t) = \frac{4}{25} \cos (5 t - α)$ $x_{sp} (t) = \frac{4}{25} \cos (5 t - α)$ with $α = 2 π - \tan^{- 1} (3 / 4) \approx 5.6397$ $α = 2 π - \tan^{- 1} (3 / 4) \approx 5.6397$
9. $x_{sp} (t) = \frac{3}{\sqrt{40001}} \cos (10 t - α)$ $x_{sp} (t) = \frac{3}{\sqrt{40001}} \cos (10 t - α)$ with $α = π + \tan^{- 1} (199 / 20) \approx 4.6122$ $α = π + \tan^{- 1} (199 / 20) \approx 4.6122$
10. $x_{sp} (t) = \frac{10}{793} \sqrt{61} \cos (10 t - α)$ $x_{sp} (t) = \frac{10}{793} \sqrt{61} \cos (10 t - α)$ with $α = π + \tan^{- 1} (171 / 478) \approx 3.4851$ $α = π + \tan^{- 1} (171 / 478) \approx 3.4851$
11. $x_{sp} (t) = \frac{\sqrt{10}}{4} \cos (3 t - α)$ $x_{sp} (t) = \frac{\sqrt{10}}{4} \cos (3 t - α)$ with $α = π - \tan^{- 1} (3) \approx 1.8925$ $α = π - \tan^{- 1} (3) \approx 1.8925$ $x_{tr} (t) = \frac{5}{4} \sqrt{2} e^{- 2 t} \cos (t - β)$ $x_{tr} (t) = \frac{5}{4} \sqrt{2} e^{- 2 t} \cos (t - β)$ with $β = 2 π - \tan^{- 1} (7) \approx 4.8543$ $β = 2 π - \tan^{- 1} (7) \approx 4.8543$
12. $x_{sp} (t) = \frac{5}{3 \sqrt{29}} \cos (3 t - α)$ $x_{sp} (t) = \frac{5}{3 \sqrt{29}} \cos (3 t - α)$ with $α = π + \tan^{- 1} (2 / 5) \approx 3.5221$ $α = π + \tan^{- 1} (2 / 5) \approx 3.5221$ $x_{tr} (t) = \frac{25}{6 \sqrt{29}} e^{- 3 t} \cos (2 t - β)$ $x_{tr} (t) = \frac{25}{6 \sqrt{29}} e^{- 3 t} \cos (2 t - β)$ with $β = \tan^{- 1} (5 / 2) \approx 1.1903$ $β = \tan^{- 1} (5 / 2) \approx 1.1903$
13. $x_{sp} (t) = \frac{300}{\sqrt{1469}} \cos (10 t - α)$ $x_{sp} (t) = \frac{300}{\sqrt{1469}} \cos (10 t - α)$ with $α = π - \tan^{- 1} (10 / 37) \approx 2.8776$ $α = π - \tan^{- 1} (10 / 37) \approx 2.8776$ $x_{tr} (t) = 2 \sqrt{\frac{113314}{1469}} e^{- t} \cos (5 t - β)$ $x_{tr} (t) = 2 \sqrt{\frac{113314}{1469}} e^{- t} \cos (5 t - β)$ with $β = 2 π - \tan^{- 1} (421 / 12895) \approx 6.2505$ $β = 2 π - \tan^{- 1} (421 / 12895) \approx 6.2505$
14. $x_{sp} (t) = \sqrt{485} \cos (t - α)$ $x_{sp} (t) = \sqrt{485} \cos (t - α)$ with $α = \tan^{- 1} (22) \approx 1.5254$ $α = \tan^{- 1} (22) \approx 1.5254$ $x_{tr} (t) = \sqrt{3665} e^{- 4 t} \cos (3 t - β)$ $x_{tr} (t) = \sqrt{3665} e^{- 4 t} \cos (3 t - β)$ with $β = π + \tan^{- 1} (52 / 31) \approx 4.1748$ $β = π + \tan^{- 1} (52 / 31) \approx 4.1748$
15. $C (ω) = 2 / \sqrt{4 + ω^{4}}$ $C (ω) = 2 / \sqrt{4 + ω^{4}}$ ; there is no practical resonance frequency.
16. $C (ω) = 10 / \sqrt{25 + 6 ω^{2} + ω^{4}}$ $C (ω) = 10 / \sqrt{25 + 6 ω^{2} + ω^{4}}$ ; there is no practical resonance frequency.
17. $C (ω) = 50 / \sqrt{2025 - 54 ω^{2} + ω^{4}}$ $C (ω) = 50 / \sqrt{2025 - 54 ω^{2} + ω^{4}}$ ; there is practical resonance at frequency $ω = 3 \sqrt{3}$ $ω = 3 \sqrt{3}$ .
18. $C (ω) = 100 / \sqrt{422500 - 1200 ω^{2} + ω^{4}}$ $C (ω) = 100 / \sqrt{422500 - 1200 ω^{2} + ω^{4}}$ ; there is practical resonance at frequency $ω = 10 \sqrt{6}$ $ω = 10 \sqrt{6}$ .
19. $ω = \sqrt{384} rad/sec$ $ω = \sqrt{384} rad/sec$ (approximately 3.12 Hz)
20. $ω \approx 44.27 rad/sec$ $ω \approx 44.27 rad/sec$ (approximately 422.75 rpm)
21. $ω_{0} = \sqrt{(g / L) + (k / m)}$ $ω_{0} = \sqrt{(g / L) + (k / m)}$
22. $ω_{0} = \sqrt{k / (m + I / a^{2})}$ $ω_{0} = \sqrt{k / (m + I / a^{2})}$
23. (a) Natural frequency: $\sqrt{10} rad/s$ $\sqrt{10} rad/s$ (approximately 0.50 Hz); (b) amplitude: approximately 10.625 in.