Solutions Manual for Simulations of Oscillatory Systems 2nd Edition by Eugene Butikov | Complete Chapters 1–10 | 2025/2026 100% Verified Correct Revised Edition

All7ChaptersCovered
lI Il lI




SOLUTIONS

, Contents

1 Free Oscillations of a Linear Oscillator
I l I l I l I l 5 I l




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-oscillatory Motion of the System................................... 15

2 Torsion Spring Oscillator with Dry Friction
Il Il 21 Il Il Il




2.2 Review of the Principal Formulas ..................................................... 21
2.3 Questions and Problems with Answers and Solutions ........................ 22
2.3.1 Damping Caused by Dry Friction .......................................... 22
2.3.2 Influence of Viscous Friction ................................................ 26

3 Forced Oscillations in a Linear System
I l I l I l 31 I l I l




3.4 Review of the Principal Formulas ..................................................... 31
3.5 Questions and Problems with Answers and Solutions ........................ 32
3.5.1 Steady-state Forced Oscillations ............................................ 32
3.5.2 Transient Processes .............................................................. 44

4 Square-wave Excitation of a Linear Oscillator
Il Il 59 Il Il Il




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
I l I l 75 I l




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

3




@@SSeeisismmiciicsisoolalatitoionn


K25515_SM_Cover.indd 16/12/
I 10l PM
Il

, 4 CONTENTS

6 Sinusoidal Modulation of the Parameter
I l I l I l I l 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
I l I l I l I l I l 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




@@SSeeisismmiciicsisoolalatitoionn


K25515_SM_Cover.indd 16/12/
I 10l PM
Il

, Chapter 1 I l




FreeOscillationsofaLinear lI lI Il lI




Oscillator
Il




1.2 Review of the Principal Formulas Il Il Il Il




The differential equation of a free linear torsion oscillator:
Il Il Il Il Il Il Il Il




ϕ¨ + 2γϕ˙ + ω02ϕ = 0.
Il Il Il Il Il Il (1.1)

Il Il free oscillations without friction (at γ ≪ ω0):
The frequency and the period of√ Il Il lI lI lI
Il
lI Il lI Il Il Il Il




D 2π lI


ω0 = , T0 = . (1.2) Il lI Il Il



J ω0
An oscillatory solution (valid at γ < ω0):
Il Il Il Il Il I l I l




ϕ(t) = A0e−γt cos(ω1t + δ0), Il Il Il Il Il (1.3)

wheretheconstantsA0 andδ0 aredeterminedbytheinitialconditionsϕ(0), ϕ˙(0).
lI lI lI Il lI Il lI lI lI lI lI lI lI




Thefrequencyω1 ofdampedoscillations
lI lI Il lI lI




√
ω1 = ω20 − γ2. (1.4) I l Il Il Il




Anequivalentformofthegeneralsolution:
lI lI lI lI lI lI




−
ϕ(t) Il = e γt
(Ccosω1t+S sinω1t),
lI lI lI lI lI lI (1.5)

where the constants C and S are determined by the initial conditions. They are
Il Il Il Il Il Il Il Il Il Il Il I l Il




related to A0 and δ0:
Il Il Il Il Il




√
A0 = C2 + S2, tanδ0 = −S/C. (1.6) I l I l
I l
Il lI Il Il




5




@@SSeeisismmiciicsisoolalatitoionn



K25515_SM_Cover.indd 16/12/
I 10l PM
Il