Results 1 to 3 of 3

Math Help - Nonlinear Oscillations

  1. #1
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782

    Nonlinear Oscillations

    Hi guys! I think I posted a question similar to this before. I think i've made some progress in solving questions of this type, but i'm still getting stuck on bits of it.

    Find the general solution to \ddot{x}-2\dot{x}-3x=10 \sin t.
    The method for this is long so i'll just put the answer.

    This comes out to be x(t)=Ae^{3t}+Be^{-t}-2 \sin t + \cos t where A,B are constants.

    Write this equation in the three variables x,y=\dot{x} and \dot{s}=1 so that the equations are autonomous and first order.
    \dot{x}=y
    \dot{y}=10 \sin s +3x+2y
    \dot{s}=1

    We know that \frac{ds}{dt}=1. This gives \frac{dx}{dt}=\frac{ds}{dt}. \frac{dx}{ds}=\frac{dx}{ds}.

    Applying a similar method gives:

    \frac{dy}{dt}=\frac{ds}{dt}.\frac{dy}{ds}=\frac{dy  }{ds}

    We also have:

    <br /> <br />
\frac{d^2x}{dt^2} = \frac{d}{dt} \left(\frac{dx}{dt}\right) = \frac{d}{ds} \left( \frac{dx}{ds}\right) \cdot \frac{ds}{dt} = \frac{d^2x}{ds^2} = \frac{dy}{ds}<br />
since y=\frac{dx}{ds}.

    This gives my 3 equations as:

    \frac{dt}{ds}=1
    \frac{dx}{ds}=y
    \frac{dy}{ds}-2y-3x=10 \sin t

    (Kudos to Danny for this!)

    Write down the corresponding solution (x(t),y(t),s(t)).
    x(p) \in \mathbb{R}^3
    x(p)=(\dot{x}, \dot{y},\dot{s})=(y,10 \sin s+3x+2y,1)

    But we know the general solution.

    (x(t),y(t),s(t))=(Ae^{3t}-2 \sin t+ \cos t, 3Ae^{3t}-2 \cos t - \sin t, t+C)

    The vector field in \mathbb{R}^3 is now periodic in s with period 2 \pi.

    Write down the Poincare map in the plane s= 2 \pi for a set of initial conditions (x_0,y_0) lying in the plane s=0.
    So we're going from s=0 to s= 2 \pi

    (x(0),y(0),0)=(x_0,y_0,0)

    This gives:

    A+1=x_0
    3A-2=y_0
    C=0

    \Rightarrow 4A-1=x_0+y_0
    \Rightarrow A=\frac{1}{4}(x_0+y_0+1)

    So this gives:

    (x(t),y(t),s(t))= \left( \frac{1}{4}(x_0+y_0+1)e^{3t}-2 \sin t+ \cos t, \frac{3}{4}(x_0+y_0+1)e^{3t}-2 \cos t- \sin t, t \right)

    Find it's fixed point and discuss it's stability in terms of eigenvalues.
    This is where I get stuck. I only have one example in my notes that does not have an e^{3t} in it. As for the part on eigenvalues, I have no idea how to even get a matrix from this!!

    I'd appreciate some help on this, there are still more bits of the question that I have to try and do!!
    Last edited by Showcase_22; January 10th 2010 at 02:46 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jan 2010
    Posts
    5

    equation

    Are you solving the equation \ddot{x}-\ddot{x}-3x=10\sin t or \ddot{x}-2\ddot{x}-3x=10\sin t? I found complex eigenvalues \lambda=\frac{1}{2}\pm\frac{1}{2}\sqrt{13} instead of the \lambda=3,-1 that you did.

    Also, for getting matrices, assume that x=x and y=x'. Differentiating these gives x'=y and y'=x'', which you can then use to turn any second order equation into a system of first order equations.
    Last edited by mr fantastic; January 10th 2010 at 02:26 AM. Reason: Fixed some math tags
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    I'm so sorry! I did mean to write <br /> <br />
\ddot{x}-2\ddot{x}-3x=10\sin t<br />
.

    I also think i've tried x=x and y=\dot{x}. Haven't I just written it as <br /> <br />
\dot{y}=10 \sin s +3x+2y<br />
?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to find an oscillations formula?
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: May 6th 2010, 04:39 AM
  2. Oscillations - Model Springs
    Posted in the Advanced Applied Math Forum
    Replies: 2
    Last Post: April 13th 2008, 09:20 AM
  3. Equilibria and Oscillations question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 12th 2008, 12:05 AM
  4. Vertical Oscillations
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: February 13th 2008, 12:39 AM
  5. Damped Oscillations
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 25th 2007, 09:45 AM

Search Tags


/mathhelpforum @mathhelpforum