Ordinary Differential Equations by Greenberg Michael D.;

Ordinary Differential Equations by Greenberg Michael D.;

Author:Greenberg, Michael D.;
Language: eng
Format: epub
Publisher: Wiley
Published: 2014-05-27T16:00:00+00:00


From Fig. 7b we see that resonance occurs when Ω equals either of the two natural frequencies 1 and .

Note that there is a forcing function F sin Ωt in the x1-equation (16a), but no forcing function in the x2-equation (16b), so it might be tempting to conclude that there will be a nonzero particular solution for x1(t) but not for x2(t). Yet, we see in (17a,b) that there are nonzero particular solutions (the curly-bracketed terms) for both x1 and x2. This is because of the coupling of (16a) and (16b), for if we re-express (1b) as + (k2 + k3)x2 = k2x1 + F2(t), then even if F2(t) = 0, the k2x1 coupling term, on the right, acts as a forcing function for x2(t).

Closure. For two masses we found the free vibration to be a linear combination of two oscillatory modes, at the natural frequencies. For the forced vibration we let F1(t) = F sin Ωt and F2(t) = 0. As for the free vibration, we solved the system of coupled differential equations by elimination and focused on the particular solutions — the “forced vibration.” Figure 7b revealed a resonance phenomenon when the forcing frequency equals either of the two natural frequencies.

EXERCISES 4.2

THE FREE VIBRATION

1. Consider (1), with F1(t) = F2(t) = 0. Let m1 = m2 = k1 = k3 = 2, and k2 = 3.

(a) Solve for x1(t) and x2(t) and give your solution in the sine and cosine form analogous to (9), and identify the natural frequencies.

(b) Also give your solution in the single-sine-with-phase-angle form analogous to (10).

(c) Cite any specific set of initial conditions that will lead to a purely low-mode motion.

(d) Cite any specific set of initial conditions that will lead to a purely high-mode motion.

(e) Cite any specific set of initial conditions that will lead to a mixed-mode motion.

2. (a)–(e) The same as Exercise 1, but using m1 – 18, m2 = 8, k1 = 27, k2 = 18, and k3 = 2.

3. (a)–(e) The same as Exercise 1, but using m1 = 36, m2 = 4. k1 = 81, k2 = 9, and k3 = 1.

4. (a)–(e) The same as Exercise 1, but using parameter values given by your instructor.

5. Application of Initial Conditions. Above (12) we said that it is more convenient to use the solution form (9) than the equivalent form (10) when we apply the initial conditions. We proceeded to use (9) and readily obtained the final result (13). To see why we preferred (9) over (10), here we ask you to use (10) instead. Show that you do obtain the same final result (13), although obtaining it will be trickier. You should find that G = a, H = 0, and ϕ = π/2.

6. Is (15) Periodic? Recall the solution given by (15),

(6.1a,b)

It is tempting to think that these are periodic functions of t because they are linear combinations of periodic functions. [A function f(t) is periodic with period T if its value repeats as t is incremented by T, that is, if f(t + T) = f(t) for all t.



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