Solutions Manual for Simulations of Oscillatory Systems (1st Edition, 2015) by Butikov – Covers All 7 Chapters

All 7 Chapters Covered
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SOLUTIONS

, Contents

1 Free Oscillations of a Linear Oscillator
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1.2 Review of the Principal Formulas ....................................................... 5
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1.3 Questions and Problems with Answers and Solutions ........................ 6
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1.3.1 Free Undamped Oscillations ..................................................... 6
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1.3.2 Damped Free Oscillations ....................................................... 11
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1.3.3 Non-oscillatory Motion of the System .................................. 15
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2 Torsion Spring Oscillator with Dry Friction
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2.2 Review of the Principal Formulas ..................................................... 21
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2.3 Questions and Problems with Answers and Solutions ...................... 22
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2.3.1 Damping Caused by Dry Friction ....................................... 22
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2.3.2 Influence of Viscous Friction ................................................ 26
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3 Forced Oscillations in a Linear System
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3.4 Review of the Principal Formulas ..................................................... 31
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3.5 Questions and Problems with Answers and Solutions ...................... 32
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3.5.1 Steady-state Forced Oscillations.............................................. 32
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3.5.2 Transient Processes ................................................................. 44
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4 Square-wave Excitation of a Linear Oscillator
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4.8 Review of the Principal Formulas ..................................................... 59
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4.9 Questions and Problems with Answers and Solutions ...................... 60
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4.9.1 Swinging of the Oscillator at Resonance ............................ 60
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4.9.2 Non-resonant Forced Oscillations ........................................... 69
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5 Parametric Excitation of Oscillations
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5.4 Questions and Problems with Answers and Solutions ...................... 75
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5.4.1 Principal Parametric Resonance ............................................. 75
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5.4.2 Manual Control of the Parameter ......................................... 90
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5.4.3 Parametric Resonances of High Orders................................. 91
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, 4 CONTENTS

6 Sinusoidal Modulation of the Parameter
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6.4 Questions and Problems with Answers and Solutions..................... 103
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6.4.1 Principal Parametric Resonance............................................ 103
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6.4.2 The Principal Interval of Parametric Resonance................ 109
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6.4.3 The Second Parametric Resonance ...................................... 113
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7 Free Oscillations of the Rigid Pendulum
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7.5 Review of the Principal Formulas ................................................... 115
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7.6 Questions and Problems with Answers and Solutions..................... 116
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7.6.1 Small Oscillations of the Pendulum .................................. 116
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7.6.2 Oscillations with Large Amplitudes ................................... 121
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7.6.3 The Rotating Pendulum ....................................................... 133
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, Chapter 1 v g




Free Oscillations of a Linear vg vg vg vg




Oscillator
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1.2 Review of the Principal Formulas
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The differential equation of a free linear torsion oscillator:
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ϕ¨ + 2γϕ˙ +0 ω 2ϕ = 0.
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vg The frequency and the period of free oscillations without friction (at γ
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vg ≪ ω0):vg √ v
g

D 2π vg
ω0 = vg vg , T0 = vg vg . (1.2)
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An oscillatory solution (valid at γ < ω0):
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γt
ϕ(t) = A0e − vg vg vg cos(ω 1t + δ0), vg vg (1.3)

where the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
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The frequency ω1 of damped oscillations
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√
ω1 = ω 2 − γ2. (1.4)
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An equivalent form of the general solution:
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− γt
ϕ(t) v g= e (C cos ω 1t + S sin ω 1t),
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where the constants C and S are determined by the initial conditions.
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They are related to A0 and δ0:
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√
A0 =
v g C 2 + S 2,
v g
v g
vg tan δ 0 = −S/C. (1.6) vg vg vg




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