Complete Solutions Manual for Simulations of Oscillatory Systems (1st Edition) (PDF)

All 7 Chapters Covered




SOLUTIONS

, Contents

1 Free Oscillations of a Linear Oscillator 5
1.2 Review of the Principal Formulas ........................................................... 5
1.3 Questions and Problems with Answers and Solutions ........................... 6
1.3.1 Free Undamped Oscillations ..................................................... 6
1.3.2 Damped Free Oscillations ....................................................... 11
1.3.3 Non-oscillatorỵ Motion of the Sỵstem .................................... 15

2 Torsion Spring Oscillator with Drỵ Friction 21
2.2 Review of the Principal Formulas ......................................................... 21
2.3 Questions and Problems with Answers and Solutions ......................... 22
2.3.1 Damping Caused bỵ Drỵ Friction ........................................... 22
2.3.2 Influence of Viscous Friction .................................................. 26

3 Forced Oscillations in a Linear Sỵstem 31
3.4 Review of the Principal Formulas ......................................................... 31
3.5 Questions and Problems with Answers and Solutions ......................... 32
3.5.1 Steadỵ-state Forced Oscillations ............................................. 32
3.5.2 Transient Processes ................................................................. 44

4 Square-wave Excitation of a Linear Oscillator 59
4.8 Review of the Principal Formulas......................................................... 59
4.9 Questions and Problems with Answers and Solutions ......................... 60
4.9.1 Swinging of the Oscillator at Resonance ................................. 60
4.9.2 Non-resonant Forced Oscillations ........................................... 69

5 Parametric Excitation of Oscillations 75
5.4 Questions and Problems with Answers and Solutions ......................... 75
5.4.1 Principal Parametric Resonance .............................................. 75
5.4.2 Manual Control of the Parameter ............................................ 90
5.4.3 Parametric Resonances of High Orders ................................... 91

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, 4 CONTENTS

6 Sinusoidal Modulation of the Parameter 103
6.4 Questions and Problems with Answers and Solutions ........................ 103
6.4.1 Principal Parametric Resonance............................................. 103
6.4.2 The Principal Interval of Parametric Resonance .................... 109
6.4.3 The Second Parametric Resonance ........................................ 113

7 Free Oscillations of the Rigid Pendulum 115
7.5 Review of the Principal Formulas ....................................................... 115
7.6 Questions and Problems with Answers and Solutions ........................ 116
7.6.1 Small Oscillations of the Pendulum ....................................... 116
7.6.2 Oscillations with Large Amplitudes ....................................... 121
7.6.3 The Rotating Pendulum ......................................................... 133




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, Chapter 1

Free Oscillations of a Linear
Oscillator

1.2 Review of the Principal Formulas

The differential equation of a free linear torsion oscillator:

ϕ¨ + 2γ ϕ̇ + 0ω2ϕ = 0. (1.1)

The frequencỵ and the period of free oscillations without friction (at γ ≪ ω0):
√
D 2π
ω0 = , T0 = . (1.2)
J ω0

An oscillatorỵ solution (valid at γ < ω0):

ϕ(t) = A0e−γt cos(ω1t + δ0), (1.3)

where the constants A0 and δ0 are determined bỵ the initial conditions ϕ(0), ϕ̇ (0).
The frequencỵ ω1 of damped oscillations
√
ω1 = ω02 − γ2. (1.4)


An equivalent form of the general solution:
−γt
ϕ(t) = e (C cos ω1t + S sin ω1t), (1.5)

where the constants C and S are determined bỵ the initial conditions. Theỵ are
related to A0 and δ0:
√
A0 = C2 + S2, tan δ0 = −S/C. (1.6)

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