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Damped Harmonic Oscillator

The document discusses the damped harmonic oscillator. It begins by presenting the Newton's second law motion equation for a damped harmonic oscillator. It then describes how this equation can be solved by assuming a solution form, which leads to an auxiliary equation. The three cases - underdamped, critically damped, and overdamped - are determined by the roots of the auxiliary equation and the damping coefficient. For an underdamped oscillator, the solution is an exponentially decaying sinusoid where the oscillator approaches zero while oscillating.

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Md Chand Asique
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1K views3 pages

Damped Harmonic Oscillator

The document discusses the damped harmonic oscillator. It begins by presenting the Newton's second law motion equation for a damped harmonic oscillator. It then describes how this equation can be solved by assuming a solution form, which leads to an auxiliary equation. The three cases - underdamped, critically damped, and overdamped - are determined by the roots of the auxiliary equation and the damping coefficient. For an underdamped oscillator, the solution is an exponentially decaying sinusoid where the oscillator approaches zero while oscillating.

Uploaded by

Md Chand Asique
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Damped Harmonic Oscillator

The Newton's 2nd Law motion equation is

This is in the form of a homogeneous second


order differential equation and has a
solution of the form

Index

Substituting this form gives an auxiliary


Periodic
equation for λ
motion
concepts

The roots of the quadratic auxiliary equation


are

Damping coefficient

Undamped oscillator

Driven oscillator
The three resulting cases for the damped
oscillator are

 
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Damping Coefficient
Index

Periodic
When a damped oscillator is subject to a damping force which is linearly
dependent upon the velocity, such as viscous damping, the oscillation will
have exponential decay terms which depend upon a damping coefficient. If motion
the damping force is of the form concepts

then the damping coefficient is given by

This will seem logical when you note that the damping force is proportional
to c, but its influence inversely proportional to the mass of the oscillator.
 
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Underdamped Oscillator
Index

Periodic
motion
concepts

Reference

Barger &
Olsson

For any value of the damping coefficient γ less than the critical damping
factor the mass will overshoot the zero point and oscillate about x=0. The
behavior is shown for one-half and one-tenth of the critical damping factor.
Also shown is an example of the overdamped case with twice the critical
damping factor.

Note that these examples are for the same specific initial conditions, i.e., a
release from rest at a position x0. For other initial conditions, the curves
would look different, but the the behavior with time would still decay
according to the damping factor.

Damped Oscillator Further details for underdamped case

 
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Underdamped Oscillator
When a damped oscillator is underdamped, it approaches zero faster than in
the case of critical damping, but oscillates about that zero.
The equation is that of an
exponentially decaying
sinusoid.
Index

Periodic
The damping coefficient is less motion
than the undamped resonant concepts
frequency . The sinusoid
frequency is given by

but the motion is not strictly


periodic.

 
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