MATLAB problem with Kapitsa pendulum(mostly about second order differentiation)

  1. 2. Relevant equations
    Theory for Kapitsa pendulum predicts that the motion consists of 2 parts:
    x(t)=X(t)+ x˜(t)
    (1)
    With fast oscillations:
    x˜(t)=−AX(t)sin(wt) /(w^2)
    (2)
    and a slowly varying motion X(t) which satisfies the following equation,
    X''=(1−(A^2)/(2w^2))X
    (3)

    1. The problem statement, all variables and given/known data
    1. Find the analytical solution of the equation (3) and derive the criterion of stability of the
    point X = 0 in terms of the parameters A and w.For the stable case, nd the period
    of oscillations of X(t) in terms of A and w, and the shape of the trajectory in the phase
    plane. For the unstable case, nd the growth rate of deviation from the equilibrium point
    and nd the trajectory in the (X; X') phase plane.
    2. Rewrite the equation x''=x(1+Asin(wt)) as a system of two rst order ODE's so that the Matlab programme ode45 could be used. Create a function le for the right-hand-side of this system to be used by ode45 to solve it.
    3.Consider the stable trajectories, with emphasis on the values of w close to the critical value wcrit. Find approximate periods for X(t). For this, you will need to run solutions for different time intervals and find for the interval which leads to the first approximate return of the trajectory to the initial phase-space point (not necessarily an exact return, because recurrence in X does not imply exact recurrence in x). Do these periods agree with the analytical formula you obtained above (in part 1)? What happens to the period when w tends to its critical value?

    3. The attempt at a solution
    Ok,this is what i currently have:
    After solving the first question, i know that:
    x(t)=x(0)cos(sqrt(A^2/(2w^2)-1)*t)
    I know that if A^2/(2w^2)-1 is less than 0, then it is unstable, otherwise it is stable
    The period seems to be 2*pi/sqrt(A^2/(2w^2)-1)
    But i don't know how to find the shape of the trajectory in the phase plane, from what i know, i have to insert the formula into Matlab to obtain the graph .
    For the unstable case, find the growth rate of deviation(i don't know what the growth rate of deviation is and how to find it) from the equilibrium point and find the trajectory in the (X; X') phase plane.( please help me with this too)

    2. Rewrite the equation x''=x(1+Asin(wt)) as a system of two rst order ODE's so that the Matlab programme ode45 could be used.
    For this, i can split it into two part:
    x'=y
    y'=x(1+Asin(wt))
    But i don't know how to create a function file for the right-hand-side of this system to be used by ode45 to solve it.
     
    Last edited: Feb 26, 2012
  2. jcsd
  3. can anyone help me?
     
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