Another Driven Harmonic Oscillator problem

In summary, you are given a solution to a driven oscillator equation, and you are only asked to verify that the solution satisfies the boundary conditions.
  • #1
Benzoate
422
0

Homework Statement



A driven oscillator satisfies the equation

x'' + omega2=F0cos(omega(1+episilon)t]

where episilon is a positive constant. Show that the solution that satisfies the iniitial conditions x=0 and x'=0 when t=0 is

x= (F0*sin(.5episilon*omega * t) sin(omega(1+.5episilon)t)/(episilon(1+.5episilon)omega^2)


Homework Equations





The Attempt at a Solution



cos(omega*t)=cos omega(1+.5episilon)t-.5omega*t)
cos(omega(1+episilon)t)=cos(omega(1+.5episilon)t + .5omega*t)

let x=ceiwpt

x'= ciwpeiwpt
x''= -c*wpe]iwpt
c= F0/(-wp2+omega^2)

Therefore
x= (F0/(-wp2+omega^2))*ceiwpt

Am I heading in the right direction

Should I derived an equation for the complimentary solution?
 
Last edited:
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  • #2
Benzoate said:

Homework Statement



A driven oscillator satisfies the equation

x'' + omega2=F0cos(omega(1+episilon)t]

where episilon is a positive constant. Show that the solution that satisfies the iniitial conditions x=0 and x'=0 when t=0 is

x= (F0*sin(.5episilon*omega * t) sin(omega(1+.5episilon)t)/(episilon(1+.5episilon)omega^2)


Homework Equations





The Attempt at a Solution



cos(omega*t)=cos omega(1+.5episilon)t-.5omega*t)
cos(omega(1+episilon)t)=cos(omega(1+.5episilon)t + .5omega*t)

let x=ceiwpt

x'= ciwpeiwpt
x''= -c*wpe]iwpt
c= F0/(-wp2+omega^2)

Therefore
x= (F0/(-wp2+omega^2))*ceiwpt

Am I heading in the right direction

Should I derived an equation for the complimentary solution?

You are given a solutions, and all you are asked to do it verify that what you are given is indeed a solution. You are not asked to solve the equation for x(t) you are only asked to check that the given solution is correct.

So... take what you are given and first check that it satisfies the boundary conditions. For very first, check that it satisfies x(0)=0. Can you verify this?
 
  • #3
olgranpappy said:
You are given a solutions, and all you are asked to do it verify that what you are given is indeed a solution. You are not asked to solve the equation for x(t) you are only asked to check that the given solution is correct.

So... take what you are given and first check that it satisfies the boundary conditions. For very first, check that it satisfies x(0)=0. Can you verify this?

my equation for my driven response looks like:

xp=(F0*eiwp*t

For complementary/transient motion should my equation be:

xc=A*cos(wp*t)+B*sin(wp*t)=0
 
  • #4
Have you even read the previous post? You are not required to solve any equations, you need to simply check that the given solution satisfies the boundary value problem.
 
  • #5
Hootenanny said:
Have you even read the previous post? You are not required to solve any equations, you need to simply check that the given solution satisfies the boundary value problem.

But don't I need to find the equation for the driven response and the equation for the transient motion?

general solution

x=xc+xp= A*cos(wp*t)+B*sin(wp*t)+(F0*eiwp*t)/(w0^2-w^2)^2 and now that I have a general solution I can plug in my initial conditions that will helped me find A and B and I now that I have A and B, I can convert my general solution to the particular solution in the book?
 
Last edited:
  • #6
Benzoate said:
But don't I need to find the equation for the driven response and the equation for the transient motion?

general solution

x=xc+xp= A*cos(wp*t)+B*sin(wp*t)+(F0*eiwp*t)/(w0^2-w^2)^2 and now that I have a general solution I can plug in my initial conditions that will helped me find A and B and I now that I have A and B, I can convert my general solution to the particular solution in the book?
No, you don't need to find the general solution, you are given the particular solution! Of course you could go through all the trouble of solving the differential equation, then imposing the boundary conditions and finally showing that your solution is equivalent to that given. However, a much more straight forward method would be to simply substitute the given solution into the differential equation and check that the equation (and all boundary conditions) are satisfied.

Do you follow?
 
  • #7
Hootenanny said:
No, you don't need to find the general solution, you are given the particular solution! Of course you could go through all the trouble of solving the differential equation, then imposing the boundary conditions and finally showing that your solution is equivalent to that given. However, a much more straight forward method would be to simply substitute the given solution into the differential equation and check that the equation (and all boundary conditions) are satisfied.

Do you follow?

yes. Thanks.
 

1. What is a driven harmonic oscillator?

A driven harmonic oscillator is a physical system that exhibits oscillatory behavior when subjected to an external driving force.

2. How does a driven harmonic oscillator differ from a simple harmonic oscillator?

A simple harmonic oscillator has a constant amplitude and frequency, while a driven harmonic oscillator has a variable amplitude and frequency due to the influence of an external force.

3. What is the equation of motion for a driven harmonic oscillator?

The equation of motion for a driven harmonic oscillator is given by:

x'' + ω0^2x = F0cos(ωt)

Where x is the displacement from equilibrium, ω0 is the natural frequency, F0 is the amplitude of the driving force, and ω is the frequency of the driving force.

4. How does the amplitude of a driven harmonic oscillator change with respect to the driving frequency?

The amplitude of a driven harmonic oscillator is largest when the driving frequency matches the natural frequency of the system. As the driving frequency deviates from the natural frequency, the amplitude decreases.

5. What are some real-life examples of a driven harmonic oscillator?

A swinging pendulum, a spring-mass system with a vibrating base, and a guitar string are all examples of driven harmonic oscillators in everyday life.

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