MATLAB: Retrieving 2nd derivative after solving second order ODE via ode45

2nd derivativeode45second derivativesecond order odestate-space

I have solved a simple pendulum problem (a pendulum driven by a motor at the joint), using ode45. I first defined the function in terms of the state space variables in a file called function1dof.m:
function func = F(t,x)
    m = 1;
    g = 10; 
    L = 1; Lc = L/2; 
    I = 0.3333; % kg-m^2
    u = 3.8; % Torque in N-m
    func = zeros(2,1); 
    func(1) = x(2); 
    func(2) = (3/(m*(L^2)))*(u - m*g*Lc*sin(x(1)));
  end
Then, I called this function in another script called solver1dof.m:
t0 = 0; tf = 5; 
x10 = 0; x20 = 0; 
[t,x] = ode45(@function1dof,[t0,tf],[x10,x20]);
However, when I defined my states such that the system is a first order ODE (so that ode45 would solve it), I realized the output gives the 0th and 1st derivatives, and not the 2nd derivatives. Is there some way I could calculate the 2nd derivative of the angle in this pendulum problem? (Note: The independent variable in this problem is the angle of the pendulum).

Best Answer

t0 = 0; tf = 5; 
x10 = 0; x20 = 0; 
[t,x] = ode45(@function1dof,[t0,tf],[x10,x20]);
for i=1:numel(t)
  t_actual = t(i);
  x_actual = x(i,:);
  ders = function1dof(t_actual,x_actual);
  second_derivative(i) = ders(2);
end
plot(t,second_derivative)
Best wishes
Torsten.
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