Solutions Manual for Simulations of Oscillatory Systems (1st Edition by Butikov) – Complete Worked Solutions

All 7 Chapters Covered
c c c




SOLUTIONS

, Contents

1 Free Oscillations of a Linear Oscillator
c c c c 5 c




1.2 Review of the Principal Formulas ........................................................... 5
c c c c




1.3 Questions and Problems with Answers and Solutions............................... 6
c c c c c c




1.3.1 Free Undamped Oscillations ....................................................... 6
c c




1.3.2 Damped Free Oscillations ........................................................ 11
c c




1.3.3 Non-oscillatory Motion of the System ...................................... 15 c c c c




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




2.2 Review of the Principal Formulas ......................................................... 21
c c c c




2.3 Questions and Problems with Answers and Solutions............................. 22
c c c c c c




2.3.1 Damping Caused by Dry Friction ............................................ 22
c c c c




2.3.2 Influence of Viscous Friction ................................................... 26
c c c




3 Forced Oscillations in a Linear System
c c c 31 c c




3.4 Review of the Principal Formulas ......................................................... 31
c c c c




3.5 Questions and Problems with Answers and Solutions............................. 32
c c c c c c




3.5.1 Steady-state Forced Oscillations ............................................... 32 c c




3.5.2 Transient Processes ................................................................. 44
c




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




4.8 Review of the Principal Formulas ......................................................... 59
c c c c




4.9 Questions and Problems with Answers and Solutions ............................ 60
c c c c c c




4.9.1 Swinging of the Oscillator at Resonance .................................. 60
c c c c c




4.9.2 Non-resonant Forced Oscillations ............................................ 69 c c




5 Parametric Excitation of Oscillations
c c 75 c




5.4 Questions and Problems with Answers and Solutions ............................ 75
c c c c c c




5.4.1 Principal Parametric Resonance............................................... 75
c c




5.4.2 Manual Control of the Parameter ............................................. 90
c c c c




5.4.3 Parametric Resonances of High Orders..................................... 91
c c c c




3




@@SSeeisismmiciicsisoolalatitoionn


K25515_SM_Cover.indd c 5 16/12/
PM
c

, 4 CONTENTS

6 Sinusoidal Modulation of the Parameter
c c c c 103
6.4 Questions and Problems with Answers and Solutions ........................... 103
c c c c c c




6.4.1 Principal Parametric Resonance ............................................. 103
c c




6.4.2 The Principal Interval of Parametric Resonance ...................... 109
c c c c c




6.4.3 The Second Parametric Resonance ......................................... 113
c c c




7 Free Oscillations of the Rigid Pendulum
c c c c c 115
7.5 Review of the Principal Formulas ........................................................ 115
c c c c




7.6 Questions and Problems with Answers and Solutions ........................... 116
c c c c c c




7.6.1 Small Oscillations of the Pendulum ....................................... 116
c c c c




7.6.2 Oscillations with Large Amplitudes ....................................... 121
c c c




7.6.3 The Rotating Pendulum ......................................................... 133
c c




@@SSeeisismmiciicsisoolalatitoionn


K25515_SM_Cover.indd c 6 16/12/
PM
c

, Chapter 1 c




FreeOscillations ofaLinear c c c c




Oscillator
c




1.2 Review of the Principal Formulas c c c c




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




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

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




D 2π c


ω0 = , T0 = . (1.2) c c c c



J ω0
An oscillatory solution (valid at γ < ω0):
c c c c c c c




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

where the constants A0 and δ0 aredetermined by the initial conditions ϕ(0), ϕ˙(0).
c c c c c c c c c c c c c




The frequency ω1 of damped oscillations
c c c c c




√
ω1 = ω20− γ2. (1.4) c c c c




Anequivalent formofthe generalsolution:
c c c c c c




−
ϕ(t) c = e γt
(C cosω1t + S sinω1t),
c c c c c c (1.5)

where the constants C and S are determined by the initial conditions. They are
c c c c c c c c c c c c c




related to A0 and δ0:
c c c c c




√
A0 = C2 + S2, tanδ0 = −S/C. (1.6) c c
c
c c c c




5




@@SSeeisismmiciicsisoolalatitoionn



K25515_SM_Cover.indd c 7 16/12/
PM
c