Simple Harmonic Motion with Damping and Driving

[TimeInquirer]

Hello, I was asked by my professor today to graph the motion, as well as the energies, of a spring that undergoes driven and/or damped oscillation; however, I was unable to because I do not have a very good idea of how they work. Can someone explain to me, qualitatively, what it means to have a damped or driven oscillation? Also, anything note worthy of knowing to answer concept based questions. Thank you!

 

[Coffee_]

Do you know how to solve basic differential equations? Then you can derive the motion graphs yourself. By a damped oscillator in general one means an oscillator where the damping force goes linearly with the velocity.

 

[TimeInquirer]

I do know how to solve the basic differential equations. However, I do not know how to get to the motion graphs. Can you please explain?

 

[Coffee_]

First you have to set up the equations for a damped harmonic oscillator then:

##mx''(t)+bx'(t)+kx(t)=F(t)## where ##F(t)## is the driving force that one can choose. A simply damped harmonic oscillator will have ##F(t)=0##, a driven damped usually has ##F(t)=F_0 cos(wt)## but in principle can be any function of time.

Based on the solution of the differential equation you can find the function ##x(t)## and thus graph the position in function of time. A tip I will give is that in ##F(t)=0## you will have three different kind of cases you need to seperately consider and for ##F(t)=F_0cos(wt)## you can find the solution after stabilization by assuming that the solution ##x(t)## is a harmonic oscillation with the same frequency as the driving force but a different phase.

 

[houlahound]

"Can someone explain to me, qualitatively, what it means to have a damped or driven oscillation?"

damped would be a mass on a spring oscillating in a jar of honey, it amplitude would diminish with an exponential envelope.

ie; amplitude(t) = (decaying exponential)X(sine wave)

amplitude(t) = e^-at*cos(t)

the damping is held up there in the factor "a"

sorry not good at Latex or typing in pretty math functions and you only asked for qualitative.

all real motion is damped to some extent as without energy input the oscillation would die to zero given enough time.

a driven oscillator is one connected to a powered oscillator that replaces the energy lost to damping forces.

the powered oscillator will impart the most energy to the driven oscillator at a specific frequency, not unlike tuning to a radio station.

that is my newbish take on it.

a simple search will find free simulations of this stuff on the intertubes