P2214HW2Key - D A ) 2 b) Z min = (1-1 2 Q 2 ) 1 / 2 = A max...

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1 Free oscillations a) x ( t ) vs t curves for the underdamped, overdamped and critically damped cases are plotted in figs.[1-5], for different initial conditions. b) (1) 0.1 percent; ω 0 τ = 22 . 36 (2) 1 percent; ω 0 τ = 7 . 05 (3) 10 percent; ω 0 τ = 2 . 18 c) (1) Auto suspension system; Critically damped: towards underdamped side (2) Door bell: Underdamped (3) Speaker Cone: Underdamped (4) Pogo stick: Underdamped (5) Drum head: Critical/Under (6) Cymbal: Underdamped (7) Bungee cord+person: Underdamped 2 Driven oscillations a) | A ( ω D ) | = f 0 /m q ( ω 2 0 - ω 2 D ) 2 +( 2 ω D τ A ) 2 , tan φ = - 2 ω D τ A ( ω 2 0 - ω 2 D ) c) | A max | = A 0 Q d) δm m = 2 Q = 0 . 0002 e) δA max A max = 1 2 δm m = 5 × 10 - 7 f) ω max = 0, A max = f 0 2 0 3 Free and driven oscillations a) Underdamped b) ν max = 4 Hz or ω max = 25 . 2 rad/sec c) ω = 24 . 97 25 . 27, δω = 0 . 3 d) Q = 83 . 7, e) ν max increases, δω decreases and Q increases. 4 Power in driven oscillations c) φ = π 2 , the driving force and the velocity are in phase. 1
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5 Mechanical Impedence a) | Z | = m q ( ω 2 0 - ω 2 D ) 2 + ( 2 ω
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Unformatted text preview: D A ) 2 b) Z min = (1-1 2 Q 2 ) 1 / 2 = A max . 6 Travelling waves a) + x direction b) A = 2 mm, k = 2 cm-1 , = 2500 rad/sec c) 0 . 0008 , . 0016 , . 0024 sec 7 Water wave a) 2 . 09 sec, 12 . 6 cm b) 0 . 5cm-1 , 0 . 48 crests/sec c) -ve x direction d) 6 cm/sec e) 12 cm/sec Figure 1: x ( t ) vs t for an Underdamped oscillator ( = 10 , = 1) 2 Figure 2: x ( t ) vs t for a critically damped oscillator ( = 1 , = 0 . 1) Figure 3: x ( t ) vs t for a critically damped oscillator ( = 1 , = 0 . 1) 3 Figure 4: x ( t ) vs t for an over damped oscillator ( = 0 . 1 , = 0 . 1) Figure 5: x ( t ) vs t for an over damped oscillator ( = 0 . 1 , = 0 . 1) 4...
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P2214HW2Key - D A ) 2 b) Z min = (1-1 2 Q 2 ) 1 / 2 = A max...

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