MEC702_Wk_7_Tutorial - MEC702 Wk 7 Tutorial Modeling 1 If a...

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Unformatted text preview: MEC702 Wk 7 Tutorial Modeling 1. If a massspring system with an iron ball of weight 98 nt (about 22 lb) can be regarded as undamped, and the spring is such that the ball stretches it 1.09 m (about 43 in.), how many cycles per minute will the system execute? What will its motion be if we pull the ball down from rest by 16 cm (about 6 in.) and let it start with zero initial velocity? (Solution: y(t) = 0.16 cos 3t) 2. How does the the motion in Question 1 above change if we change the damping constant c from one to another of the following three values, with y(0) = 0.16 and y 0 (0) = 0 as before? (a) Case I: c = 100 kg/sec (b) Case II: c = 60 kg/sec (c) Case III: c = 10 kg/sec 3. If a weight of 20 nt (about 4.5 lb) stretches a certain spring by 2 cm, what will the frequency of the corresponding harmonic oscillation be? The period? 4. Show that in the overdamped case, the body can pass through y = 0 at most once. 5. Show that the maxima of an underdamped motion occur at equidistant t-values and nd the distance. Higher Order Linear Homoegeneous ODEs 1. Solve the given ODEs. (a) y 000 + 25y 0 = 0. (b) y iv + 2y 00 + y = 0. (c) y iv − 4y 00 = 0. (d) (D3 − D2 − D + 1)y = 0. 2. Solve the given IVP. (a) y iv − 9y 00 − 400y = 0, y(0) = 0, y 0 (0) = 0, y 00 (0) = 41, y 000 (0) = 0. (b) 4y 000 + 8y 00 + 41y 0 + 37y = 0, y(0) = 9, y 0 (0) = −6.5, y 00 (0) = −39.75. 1 ...
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