All 7 Chapters Covered
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SOLUTIONS
, Contents
1 Free Oscillations of a Linear Oscillator
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1.2 Review of the Principal Formulas ............................................................... 5
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1.3 Questions and Problems with Answers and Solutions ................................. 6
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1.3.1 Free Undamped Oscillations .......................................................... 6
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1.3.2 Damped Free Oscillations ............................................................ 11
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1.3.3 Non-oscillatory Motion of the System ......................................... 15 q q q q
2 Torsion Spring Oscillator with Dry Friction
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2.2 Review of the Principal Formulas ............................................................. 21
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2.3 Questions and Problems with Answers and Solutions ............................... 22
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2.3.1 Damping Caused by Dry Friction ................................................. 22
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2.3.2 Influence of Viscous Friction ....................................................... 26
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3 Forced Oscillations in a Linear System
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3.4 Review of the Principal Formulas ......................................................... 31
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3.5 Questions and Problems with Answers and Solutions ........................... 32
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3.5.1 Steady-state Forced Oscillations .............................................. 32
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3.5.2 Transient Processes ................................................................. 44
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4 Square-wave Excitation of a Linear Oscillator
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4.8 Review of the Principal Formulas ......................................................... 59
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4.9 Questions and Problems with Answers and Solutions ........................... 60
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4.9.1 Swinging of the Oscillator at Resonance .................................. 60
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4.9.2 Non-resonant Forced Oscillations ............................................ 69 q q
5 Parametric Excitation of Oscillations
q q q 75
5.4 Questions and Problems with Answers and Solutions ........................... 75
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5.4.1 Principal Parametric Resonance .............................................. 75
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5.4.2 Manual Control of the Parameter ............................................ 90
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5.4.3 Parametric Resonances of High Orders.................................... 91
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3
,4 CONTENTS
6 Sinusoidal Modulation of the Parameter
q q q q 103
6.4 Questions and Problems with Answers and Solutions ......................... 103
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6.4.1 Principal Parametric Resonance ............................................. 103
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6.4.2 The Principal Interval of Parametric Resonance ..................... 109
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6.4.3 The Second Parametric Resonance ........................................ 113
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7 Free Oscillations of the Rigid Pendulum
q q q q q 115
7.5 Review of the Principal Formulas........................................................ 115
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7.6 Questions and Problems with Answers and Solutions ......................... 116
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7.6.1 Small Oscillations of the Pendulum ....................................... 116
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7.6.2 Oscillations with Large Amplitudes ....................................... 121
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7.6.3 The Rotating Pendulum ......................................................... 133
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, Chapter 1 q
Free Oscillations of a Linear Osc
q q q q q
illator
1.2 Review of the Principal Formulas
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The differential equation of a free linear torsion oscillator:
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ϕ¨ + 2γϕ˙ + ω02ϕ = 0.
q q q q q q
(1.1)
q q free oscillations without friction (at γ ≪ ω0):
The frequency and the period of√ q q q q q q q q q q q q q
D 2π q
ω0 = , T0 = . (1.2) q q q q
J ω0
An oscillatory solution (valid at γ < ω0):
q q q q q q q
ϕ(t) = A0e−γt cos(ω1t + δ0),
q q q q q (1.3)
where the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
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The frequency ω1 of damped oscillations
q q q q q
√
ω1 = ω20− γ2. (1.4) q q q q
An equivalent form of the general solution:
q q q q q q
− γt
ϕ(t) q= e (C cos ω1t + S sin ω1t),
q q q q q q (1.5)
where the constants C and S are determined by the initial conditions. They are rel
q q q q q q q q q q q q q q
ated to A0 and δ0:
q q q q
√
A0 = C2 + S2, tan δ0 = −S/C. q (1.6)q
q
q q q q
5
q q q
SOLUTIONS
, Contents
1 Free Oscillations of a Linear Oscillator
q q q q 5 q
1.2 Review of the Principal Formulas ............................................................... 5
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1.3 Questions and Problems with Answers and Solutions ................................. 6
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1.3.1 Free Undamped Oscillations .......................................................... 6
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1.3.2 Damped Free Oscillations ............................................................ 11
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1.3.3 Non-oscillatory Motion of the System ......................................... 15 q q q q
2 Torsion Spring Oscillator with Dry Friction
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2.2 Review of the Principal Formulas ............................................................. 21
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2.3 Questions and Problems with Answers and Solutions ............................... 22
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2.3.1 Damping Caused by Dry Friction ................................................. 22
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2.3.2 Influence of Viscous Friction ....................................................... 26
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3 Forced Oscillations in a Linear System
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3.4 Review of the Principal Formulas ......................................................... 31
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3.5 Questions and Problems with Answers and Solutions ........................... 32
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3.5.1 Steady-state Forced Oscillations .............................................. 32
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3.5.2 Transient Processes ................................................................. 44
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4 Square-wave Excitation of a Linear Oscillator
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4.8 Review of the Principal Formulas ......................................................... 59
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4.9 Questions and Problems with Answers and Solutions ........................... 60
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4.9.1 Swinging of the Oscillator at Resonance .................................. 60
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4.9.2 Non-resonant Forced Oscillations ............................................ 69 q q
5 Parametric Excitation of Oscillations
q q q 75
5.4 Questions and Problems with Answers and Solutions ........................... 75
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5.4.1 Principal Parametric Resonance .............................................. 75
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5.4.2 Manual Control of the Parameter ............................................ 90
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5.4.3 Parametric Resonances of High Orders.................................... 91
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3
,4 CONTENTS
6 Sinusoidal Modulation of the Parameter
q q q q 103
6.4 Questions and Problems with Answers and Solutions ......................... 103
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6.4.1 Principal Parametric Resonance ............................................. 103
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6.4.2 The Principal Interval of Parametric Resonance ..................... 109
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6.4.3 The Second Parametric Resonance ........................................ 113
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7 Free Oscillations of the Rigid Pendulum
q q q q q 115
7.5 Review of the Principal Formulas........................................................ 115
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7.6 Questions and Problems with Answers and Solutions ......................... 116
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7.6.1 Small Oscillations of the Pendulum ....................................... 116
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7.6.2 Oscillations with Large Amplitudes ....................................... 121
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7.6.3 The Rotating Pendulum ......................................................... 133
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, Chapter 1 q
Free Oscillations of a Linear Osc
q q q q q
illator
1.2 Review of the Principal Formulas
q q q q
The differential equation of a free linear torsion oscillator:
q q q q q q q q
ϕ¨ + 2γϕ˙ + ω02ϕ = 0.
q q q q q q
(1.1)
q q free oscillations without friction (at γ ≪ ω0):
The frequency and the period of√ q q q q q q q q q q q q q
D 2π q
ω0 = , T0 = . (1.2) q q q q
J ω0
An oscillatory solution (valid at γ < ω0):
q q q q q q q
ϕ(t) = A0e−γt cos(ω1t + δ0),
q q q q q (1.3)
where the constants A0 and δ0 are determined by the initial conditions ϕ(0), ϕ˙(0).
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The frequency ω1 of damped oscillations
q q q q q
√
ω1 = ω20− γ2. (1.4) q q q q
An equivalent form of the general solution:
q q q q q q
− γt
ϕ(t) q= e (C cos ω1t + S sin ω1t),
q q q q q q (1.5)
where the constants C and S are determined by the initial conditions. They are rel
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ated to A0 and δ0:
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√
A0 = C2 + S2, tan δ0 = −S/C. q (1.6)q
q
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