How can I simplify complex symbolic equations? Simplify not working

I'm having issues simplifying a large symbollic expression that Matlab outpiuts. I've tried using the simplify() command, but it doesn't seem to make any differences. Here's an example of the expression that I want to simplify:

syms R_s R_w r_w r_cm I_s I_r m_r m_s g tau

syms theta_s theta_b dtheta_s dtheta_b ddtheta_s ddtheta_b

eqn = (R_s*(I_r*R_s*r_w*tau - I_r*R_s*R_w*tau - I_r*R_s^2*tau + dtheta_b^2*m_r^2*r_cm^3*r_w*sin(theta_b + theta_s)^3 + dtheta_s^2*m_r^2*r_cm^3*r_w*sin(theta_b + theta_s)^3 - R_w*m_r*r_cm*tau*cos(theta_b + theta_s) - R_s^2*m_r*r_cm^2*tau*cos(theta_b + theta_s)^2 + m_r*r_cm*r_w*tau*cos(theta_b + theta_s) - R_s^2*m_r*r_cm^2*tau*sin(theta_b + theta_s)^2 + g*m_r^2*r_cm^2*r_w*cos(theta_b + theta_s)*sin(theta_b + theta_s) + 2*dtheta_b*dtheta_s*m_r^2*r_cm^3*r_w*sin(theta_b + theta_s)^3 + I_r*dtheta_b^2*m_r*r_cm*r_w*sin(theta_b + theta_s) + I_r*dtheta_s^2*m_r*r_cm*r_w*sin(theta_b + theta_s) - R_s*R_w*m_r*r_cm^2*tau*cos(theta_b + theta_s)^2 - R_s*R_w*m_r*r_cm^2*tau*sin(theta_b + theta_s)^2 + R_s*m_r*r_cm^2*r_w*tau*cos(theta_b + theta_s)^2 + R_s*m_r*r_cm^2*r_w*tau*sin(theta_b + theta_s)^2 + dtheta_b^2*m_r^2*r_cm^3*r_w*cos(theta_b + theta_s)^2*sin(theta_b + theta_s) + dtheta_s^2*m_r^2*r_cm^3*r_w*cos(theta_b + theta_s)^2*sin(theta_b + theta_s) + 2*dtheta_b*dtheta_s*m_r^2*r_cm^3*r_w*cos(theta_b + theta_s)^2*sin(theta_b + theta_s) + 2*I_r*dtheta_b*dtheta_s*m_r*r_cm*r_w*sin(theta_b + theta_s)))/(r_w*(I_s*R_s^2*m_r*r_cm^2*cos(theta_b + theta_s)^2 + I_s*R_s^2*m_r*r_cm^2*sin(theta_b + theta_s)^2 + I_r*I_s*R_s^2 + m_r^2*r_cm^2*sin(theta_b + theta_s)^2 + m_s*m_r*r_cm^2*cos(theta_b + theta_s)^2 + m_s*m_r*r_cm^2*sin(theta_b + theta_s)^2 + I_r*m_r + I_r*m_s));

simplify(eqn)

Just looking at this expression, I can see that there are sin^2(x)+cos^2(x) = 1 simplications that could be made, as well as terms that could be collected. Is there any way to make MATLAB do this for me, as opposed to me trying to do this by hand (and likely making errors)?

Answer 1

[Star Strider]

Tell the simplify function to keep working on ‘eqn’ as long as necessary:

eqns = simplify(eqn, 'Steps',150)

producing:

eqns =

(sin(theta_b + theta_s)*(R_s*(r_w*dtheta_b^2*m_r^2*r_cm^3 + I_r*r_w*dtheta_b^2*m_r*r_cm + 2*r_w*dtheta_b*dtheta_s*m_r^2*r_cm^3 + 2*I_r*r_w*dtheta_b*dtheta_s*m_r*r_cm + r_w*dtheta_s^2*m_r^2*r_cm^3 + I_r*r_w*dtheta_s^2*m_r*r_cm) - R_s*g*m_r^2*r_cm^2*r_w*(2*sin(theta_b/2 + theta_s/2)^2 - 1)) - R_s*(I_r*R_s^2*tau + I_r*R_s*R_w*tau + R_s^2*m_r*r_cm^2*tau - I_r*R_s*r_w*tau + R_s*R_w*m_r*r_cm^2*tau - R_s*m_r*r_cm^2*r_w*tau) + R_s*(R_w*m_r*r_cm*tau - m_r*r_cm*r_w*tau)*(2*sin(theta_b/2 + theta_s/2)^2 - 1))/(r_w*(I_r*m_r + I_r*m_s + m_r^2*r_cm^2 + m_r*m_s*r_cm^2 + m_r^2*r_cm^2*(sin(theta_b + theta_s)^2 - 1) + I_r*I_s*R_s^2 + I_s*R_s^2*m_r*r_cm^2))

[Anthony Stuart]

That solved it. Thanks!

[Star Strider]

As always, my pleasure!