I have some coupled nonlinear ordinary differential equations. Three equations have three 2nd order derivative coupled together. Can someone tell me how I can get the response

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I have some coupled nonlinear ordinary differential equations. Three equations have three 2nd order derivative coupled together. Can someone tell me how I can get the response, numerically? i.e t vs x , t vs theta dot plots.
Thanks in advance

Accepted Answer

Torsten
Torsten on 21 Sep 2022
Edited: Torsten on 21 Sep 2022
Solve the third equation for theta_dotdot and insert this expression in equations (1) and (2).
Then use the usual substitutions
z1 = x
z2 = xdot
z3 = y
z4 = ydot
z5 = theta
z6 = thetadot
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More Answers (1)

James Tursa
James Tursa on 21 Sep 2022
Edited: James Tursa on 21 Sep 2022
Another related approach is to isolate all the 2nd derivatives on the LHS and put everything else on the RHS. Then pick off the coefficients of the 2nd order derivatives to form a matrix problem that you solve for the 2nd derivatives. E.g., the matrix problem would look something like this based on your comment edits:
A*z = b
where
A = [ 1 0 -f*sin(theta);
0 1 f*sin(theta);
-m*e*sin(theta) m*e*cos(theta) (m*e^2+Ip)/B^2]
and
z = [x_dotdot; y_dotdot; theta_dotdot]
and
b = 3x1 vector with all the other stuff from RHS
Then solve for z to get the 2nd derivatives.
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