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Exercise:
How can the damping of the following oscillations with linear damping i.e. exponentially decreasing envelope be characterised? abcliste abc For a mass mbO on a spring kkbO with a damping coefficient debO. abc For an oscillator with period TaO whose amplitude decreases from AaO to BaO in taO. abc For an oscillator whose amplitude decreases to one half of the initial value during one period. abc For an LC oscillator with inductance LdO capacitance CdO and resistance RdO. abcliste

Solution:
abcliste abc The damping ratio is zeta fracdeltaomega_ zebF deb times sqrtfracmbkb zebP Since zeta the oscillator is overdamped. abc The damping coefficient is given by At A_ e^-delta t Longrightarrow -delta t lnfracAtA_ Longrightarrow delta deaF The damping ratio is zeta fracdeltaomega_ approx zeaF fraclnAa/Ba times Tapitimes ta zeaP Because zeta ll the oscillation is only weakly damped. abc The half-life for the exponential decay corresponds to T_/ fracln delta Since the half-life is equal to the period it follows for the damping coefficient delta fracln T_/ fracln T The damping ratio is zeta fracdeltaomega_ approx fracln Tpi T zecF zecP This is significantly smaller than so the oscillation is weakly damped. abc The damping ratio is zeta fracdeltaomega_ fracfracRLfracsqrtLC zedF fracRdtimes sqrtfracCdLd zedP This is clearly overdamped. abcliste
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Exercise:
How can the damping of the following oscillations with linear damping i.e. exponentially decreasing envelope be characterised? abcliste abc For a mass mbO on a spring kkbO with a damping coefficient debO. abc For an oscillator with period TaO whose amplitude decreases from AaO to BaO in taO. abc For an oscillator whose amplitude decreases to one half of the initial value during one period. abc For an LC oscillator with inductance LdO capacitance CdO and resistance RdO. abcliste

Solution:
abcliste abc The damping ratio is zeta fracdeltaomega_ zebF deb times sqrtfracmbkb zebP Since zeta the oscillator is overdamped. abc The damping coefficient is given by At A_ e^-delta t Longrightarrow -delta t lnfracAtA_ Longrightarrow delta deaF The damping ratio is zeta fracdeltaomega_ approx zeaF fraclnAa/Ba times Tapitimes ta zeaP Because zeta ll the oscillation is only weakly damped. abc The half-life for the exponential decay corresponds to T_/ fracln delta Since the half-life is equal to the period it follows for the damping coefficient delta fracln T_/ fracln T The damping ratio is zeta fracdeltaomega_ approx fracln Tpi T zecF zecP This is significantly smaller than so the oscillation is weakly damped. abc The damping ratio is zeta fracdeltaomega_ fracfracRLfracsqrtLC zedF fracRdtimes sqrtfracCdLd zedP This is clearly overdamped. abcliste
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Attributes & Decorations
Branches
Damped Oscillations
Tags
damping, exponential, frequency, oscillation, period
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Difficulty
(1, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
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Decoration