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-oscillatory Motion of the System .................................... 15
2 Torsion Spring Oscillator with Dry 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 by Dry Friction ............................................ 22
2.3.2 Influence of Viscous Friction................................................... 26
3 Forced Oscillations in a Linear System 31
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 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
3
,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
, Chapter 1 k 0
Free Oscillations of a Linear k0 k0 k0 k0
Oscillator
k0
1.2 Review of the Principal Formulas k0 k0 k0 k0
The differential equation of a free linear torsion oscillator:
k0 k0 k0 k0 k0 k0 k0 k0
ϕ¨ + 2γϕ˙ +0 ω2ϕ = 0.
k0 k0 k0 k0 k0 k0 (1.1)
The frequency and the period of free oscillations without friction (at γ ≪
k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0
k0 ω0): √ 0
k
D 2π k0
ω0 = k0 k0 , T0 = k0 k0 . (1.2)
J ω0
An oscillatory solution (valid at γ < ω0):
k0 k0 k0 k0 k0 k 0 k0
ϕ(t) = A0e−γt cos(ω1t + δ0),
k0 k0 k0 k0 k0 (1.3)
where the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0
The frequency ω1 of damped oscillations
k0 k0 k0 k0 k0
√
ω1 = ω0 2 − γ2. (1.4) k0 k0 k0 k0
An equivalent form of the general solution:
k0 k0 k0 k0 k0 k0
−
ϕ(t) k0= e γt
(C cos ω1t + S sin ω1t),
k0 k0 k0 k0 k0 k0 (1.5)
where the constants C and S are determined by the initial conditions. They
k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k 0
are related to A0 and δ0:
k0 k0 k0 k0 k0 k0
√
A0 = C2 + S2, tan δ0 = −S/C. (1.6) k 0 k 0
k 0
k0 k0 k0 k0
5
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-oscillatory Motion of the System .................................... 15
2 Torsion Spring Oscillator with Dry 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 by Dry Friction ............................................ 22
2.3.2 Influence of Viscous Friction................................................... 26
3 Forced Oscillations in a Linear System 31
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 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
3
,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
, Chapter 1 k 0
Free Oscillations of a Linear k0 k0 k0 k0
Oscillator
k0
1.2 Review of the Principal Formulas k0 k0 k0 k0
The differential equation of a free linear torsion oscillator:
k0 k0 k0 k0 k0 k0 k0 k0
ϕ¨ + 2γϕ˙ +0 ω2ϕ = 0.
k0 k0 k0 k0 k0 k0 (1.1)
The frequency and the period of free oscillations without friction (at γ ≪
k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0
k0 ω0): √ 0
k
D 2π k0
ω0 = k0 k0 , T0 = k0 k0 . (1.2)
J ω0
An oscillatory solution (valid at γ < ω0):
k0 k0 k0 k0 k0 k 0 k0
ϕ(t) = A0e−γt cos(ω1t + δ0),
k0 k0 k0 k0 k0 (1.3)
where the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0
The frequency ω1 of damped oscillations
k0 k0 k0 k0 k0
√
ω1 = ω0 2 − γ2. (1.4) k0 k0 k0 k0
An equivalent form of the general solution:
k0 k0 k0 k0 k0 k0
−
ϕ(t) k0= e γt
(C cos ω1t + S sin ω1t),
k0 k0 k0 k0 k0 k0 (1.5)
where the constants C and S are determined by the initial conditions. They
k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k0 k 0
are related to A0 and δ0:
k0 k0 k0 k0 k0 k0
√
A0 = C2 + S2, tan δ0 = −S/C. (1.6) k 0 k 0
k 0
k0 k0 k0 k0
5
