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Damped Harmonic Oscillator Derivation

Welcome to Chafia Physic. In this post we will derrive the equation of motion of Damped Harmonic Oscillator. To derrive the equation of motion first we must draw the diagram of this motion in 1 Dimensional system.

In this system as we can see. If we give the box a Force to the right, at instant spring force and friction will comes up in opposite direction with the force we gift. With Newton second Law we can write the total of Force acting on the box to be.

Suppose we set x is equal to e^lamda t, we can subtitute it to differential equation. Subtitution will give us.

Simplify the equation we get.

Now we looking for lamda.

From earlier we set x = e^lambda*t. Here we get two lambda, so x will have two solution. Because there are two solution of x, complete equation of Damped Harmonic Oscillator  is sumation or superpotition of this two equation.
 With this equation the are three posibilities related with how big the magnitude of gamma and omega.

 First We will Solve the case when the box in overdamped situation.

Overdamped Oscillation

In this case gamma i larger beter than omega zero. So the equation that we will get is.

As we can see from this equation, there is no cosine or sinus form represent int this equation. This equation told us, if A box or something in overdamped situation. The motion of box motion will drastically stop exponentially.

Critically Damped

In his case if we use omega is equal to zero, solution will just depend on e^-gamma*t times (c1+c2) that we know it can be replaced with just one constant C. One thing that we must rememmber in secon order differential equation, if we derrive a function twice, we must have at least two constant that related with it.
So in Critically Damped situation we can't use omega equal to zero, but we can use omega very very close to zero. Because omega is close to zero, we can rewrite the x Damped Harmonic Oscillator to be.


Underdamped Oscillator

So we now in the last case, Underdamped oscillator. Same like before let's write the case problem (omega form) and subtitute it in x Damped Harmonic Oscillator equation.

From the subtitution we now we get imaginary form in our equation. In Classical Mecanics we dont working with imaginary part, so we will assume that.

From euler equation we can change x equation form.
From the derrivation now we get in Underdamped Oscillation the box will oscillate before the box stop because lambda decrease over time.


OK this All sumarry of all cases we derrive:

I think that is i can share with you in this post, I hope it can be usefull for you...
Thank you for coming...

Visit Wiki hookes law to get know more about spring and 1dimensional oscillatio

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