Retrieving 2nd derivative after solving second order ODE via ode45

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).

Answer 1

[Torsten]

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.

[Chinmay Sridhar]

Thank you!

[Siddharth Jain]

I have 6 DOF, how can this be modified for that?

[Torsten]

The derivatives of your solution variables can be retrieved after integration by

[t,y] = ode45(@fcn,tspan,y0);
y_derivatives = zeros(size(y));
for i=1:numel(t)
  t_actual = t(i);
  y_actual = y(i,:);
  y_derivatives(i,:) = fcn(t_actual,y_actual).';
end
plot(t,y_derivatives)