We have three equations as follows:
$A_p=F \cos(\alpha+ \phi) - \mu N^{'}_{S_1} - \mu N_{S_1} - W \sin \theta = m ( \ddot x - r \ddot \theta (\sin (\gamma+ \phi)) )$
$B_p=F \cos(\alpha+ \phi) - \mu N^{'}_{S_1} + \mu N_{S_1} - W \cos \theta = m ( -r \ddot \theta (\sin (\gamma + \phi)) )$
$C_p=F e_G \cos \alpha + F \sin \alpha(x_G-l_1)+ N^{'}_{S_1} (x_G-l_1-a ) \cos \phi - N_{s_1}(x_G-l_1-a-s_1)\cos \phi +\mu N^{'}_{S_1} (h_1+ \frac{d}{2}-e_g) \cos \phi - \mu N_{S_1} (h_1+ \frac{d}{2}+e_g) \cos \phi = I_G \ddot \phi $
the unknown is $N_{S_1}, N^{'}_{S_1},\ddot \phi(t), \phi(t) $. I think (not sure) we can replace $\phi(t)= \frac{1}{2} \ddot > \phi(t)\cdot t^2$.
Infact I want to solve the following:
$A_p=(1.051250801 \times 10^6t-9.885179829 \times 10^6 t^2+ 4.744120386 \times 10^7 t^3 - 1.282599777 \times 10^8 t^4 + 1.970136609 \times 10^8t^5 - 1.604249861 \times 10^8 t^6 + 5.377070775 \times 10^7 t^7) \Bigg( \cos \Bigg( \frac {1}{1800} \pi \Bigg ) - \sin \Bigg( \frac {1}{1800} \pi \Bigg) \phi(t) \Bigg ) - 0.16 N^{'}_{s_1} - 0.16 N_{s_1} - 3534.543 \sin \Big( \frac{1}{18} \pi \Big)=42628.97791 - 157.3659448 \Bigg ( \frac {d^2}{dt^2} \phi (t) \Bigg) (0.3068020853 + 0.9517733346 \phi(t)) $
$B_p=(1.051250801 \times 10^6t-9.885179829 \times 10^6 t^2+ 4.744120386 \times 10^7 t^3 - 1.282599777 \times 10^8 t^4 + 1.970136609 \times 10^8t^5 - 1.604249861 \times 10^8 t^6 + 5.377070775 \times 10^7 t^7) \Bigg( \sin \Bigg( \frac {1}{1800} \pi \Bigg ) +\cos \Bigg( \frac {1}{1800} \pi \Bigg) \phi(t) \Bigg ) - 0.16 N^{'}_{s_1} + 0.16 N_{s_1} - 3534.543 \cos \Big( \frac{1}{18} \pi \Big)=-157.3659448 \Bigg ( \Bigg ( \frac{d^2}{dt^2} \phi(t) \Bigg ) (0.3068020853 + 0.9517733346 \phi(t)) \Bigg) $
$C_p=0.006(1.051250801 \times 10^6t-9.885179829 \times 10^6 t^2+ 4.744120386 \times 10^7 t^3 - 1.282599777 \times 10^8 t^4 + 1.970136609 \times 10^8t^5 - 1.604249861 \times 10^8 t^6 + 5.377070775 \times 10^7 t^7) \cos \Bigg( \frac {1}{1800} \pi \Bigg)+ 1.5485 (1.051250801 \times 10^6t-9.885179829 \times 10^6t^2+4.744120386 \times 10^7 t^3 - 1.282599777 \times 10^8 t^4 + 1.97013660910 \times 10^8 t^5 + 5.377070775 \times 10^7 t^7) \sin \Bigg( \frac {1}{1800} \pi \Bigg ) + 0.4157 N^{'}_{s_1} - 0.0127 N_{s_1} + 0.16 N^{'}_{s_1} (0.3000000000) - 0.16 N_{s_1} (0.3120000000) = 229.38 \Bigg ( \frac{d^2}{dt^2} \phi(t) \Bigg ) $
I want to solve this problem, I ran into this question from my two month research on my works, anyone can help me? is it possible to solve it via MAPLE or other things to get solutions for this complex problem? I means how can calculate the $\phi$? or at least very good approximation?