All 7 Chapters Covered
SOLUTIONS
,Contents
1 Free Oscillations of a Linear Oscillator 5
1.2 Revie of the Principal Formulas .......................................................... 5
1.3 Questions and Problems ith Ans ers 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 ith Dry Friction 21
2.2 Revie of the Principal Formulas ........................................................ 21
2.3 Questions and Problems ith Ans ers 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 Revie of the Principal Formulas ........................................................ 31
3.5 Questions and Problems ith Ans ers and Solutions ............................. 32
3.5.1 Steady-state Forced Oscillations ............................................ 32
3.5.2 Transient Processes .............................................................. 44
4 Square- ave Excitation of a Linear Oscillator 59
4.8 Revie of the Principal Formulas ........................................................ 59
4.9 Questions and Problems ith Ans ers and Solutions ............................. 60
4.9.1 S inging 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 ith Ans ers 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 ith Ans ers 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 Revie of the Principal Formulas....................................................... 115
7.6 Questions and Problems ith Ans ers and Solutions............................ 116
7.6.1 Small Oscillations of the Pendulum..................................... 116
7.6.2 Oscillations ith Large Amplitudes ....................................... 121
7.6.3 The Rotating Pendulum ...................................................... 133
, Chapter 1
Free Oscillations of a Linear
Oscillator
1.2 Revie of the Principal Formulas
The differential equation of a free linear torsion oscillator:
ϕ¨ + 2γϕ˙ + ω 2
0 ϕ = 0. (1.1)
The frequency and the period of√free oscillations ithout friction (at γ ≪ ω0):
D 2π
ω0 = , T0 = . (1.2)
J ω0
An oscillatory solution (valid at γ < ω0):
ϕ(t) = A0e−γt cos(ω1t + δ0), (1.3)
here the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
The frequency ω1 of damped oscillations
√
ω1 = ω20 − γ2. (1.4)
An equivalent form of the general solution:
− γt
ϕ(t) = e (C cos ω1t + S sin ω1t), (1.5)
here the constants C and S are determined by the initial conditions. They are
related to A0 and δ0:
√
A0 = C2 + S2, tan δ0 = −S/C. (1.6)
5
SOLUTIONS
,Contents
1 Free Oscillations of a Linear Oscillator 5
1.2 Revie of the Principal Formulas .......................................................... 5
1.3 Questions and Problems ith Ans ers 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 ith Dry Friction 21
2.2 Revie of the Principal Formulas ........................................................ 21
2.3 Questions and Problems ith Ans ers 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 Revie of the Principal Formulas ........................................................ 31
3.5 Questions and Problems ith Ans ers and Solutions ............................. 32
3.5.1 Steady-state Forced Oscillations ............................................ 32
3.5.2 Transient Processes .............................................................. 44
4 Square- ave Excitation of a Linear Oscillator 59
4.8 Revie of the Principal Formulas ........................................................ 59
4.9 Questions and Problems ith Ans ers and Solutions ............................. 60
4.9.1 S inging 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 ith Ans ers 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 ith Ans ers 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 Revie of the Principal Formulas....................................................... 115
7.6 Questions and Problems ith Ans ers and Solutions............................ 116
7.6.1 Small Oscillations of the Pendulum..................................... 116
7.6.2 Oscillations ith Large Amplitudes ....................................... 121
7.6.3 The Rotating Pendulum ...................................................... 133
, Chapter 1
Free Oscillations of a Linear
Oscillator
1.2 Revie of the Principal Formulas
The differential equation of a free linear torsion oscillator:
ϕ¨ + 2γϕ˙ + ω 2
0 ϕ = 0. (1.1)
The frequency and the period of√free oscillations ithout friction (at γ ≪ ω0):
D 2π
ω0 = , T0 = . (1.2)
J ω0
An oscillatory solution (valid at γ < ω0):
ϕ(t) = A0e−γt cos(ω1t + δ0), (1.3)
here the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
The frequency ω1 of damped oscillations
√
ω1 = ω20 − γ2. (1.4)
An equivalent form of the general solution:
− γt
ϕ(t) = e (C cos ω1t + S sin ω1t), (1.5)
here the constants C and S are determined by the initial conditions. They are
related to A0 and δ0:
√
A0 = C2 + S2, tan δ0 = −S/C. (1.6)
5
