Introduction:
I have a mathematic model of trebuchet(a siege engine) with 3 degrees of freedom $\gamma, \varphi, \beta$. Firstly a projectile is propelled on the ground and then hurled into the air. There is such a condition (that is a virtual work principle VWP for dynamic rigid bodies) that in any moment $t$ all generalized forces $Q_{\gamma}, Q_{\varphi}, Q_{\beta}$ must equal $0$.
I want to know the ground reaction force $N_{we}$. It is "inside" the vector:
$\overset{\rightharpoonup }{N}=(0,N_{\text{we}},0)$
I have 3 equations for these generalized forces, One of the generalized forces $Q_{\gamma }$ cancels out the ground reaction force (index "w" means that variable is a vector):
$Q_{\gamma }=\overset{\rightharpoonup }{B_2}.\frac{\partial l_{\text{OC}_2 w}}{\partial \gamma (t)} \overset{\rightharpoonup }{+B_3}.\frac{\partial l_{\text{OC}_3 w}}{\partial \gamma (t)} \overset{\rightharpoonup }{+G_1}.\frac{\partial l_{\text{OC}_1 w}}{\partial \gamma (t)}+\overset{\rightharpoonup }{B_1}.\frac{\partial l_{\text{OC}_1 w}}{\partial \gamma (t)}+\overset{\rightharpoonup }{G_2}.\frac{\partial l_{\text{OC}_2 w}}{\partial \gamma (t)}+\overset{\rightharpoonup }{G_3}.\frac{\partial l_{\text{OC}_3 w}}{\partial \gamma (t)}+\overset{\rightharpoonup }{N}.\frac{\partial \left(l_{2 w}+l_{4 w}\right)}{\partial \gamma (t)}=0$
$Q_{\varphi }=\overset{\rightharpoonup }{B_2}.\frac{\partial l_{\text{OC}_2 w}}{\partial \varphi (t)} \overset{\rightharpoonup }{+B_3}.\frac{\partial l_{\text{OC}_3 w}}{\partial \varphi (t)} \overset{\rightharpoonup }{+G_1}.\frac{\partial l_{\text{OC}_1 w}}{\partial \varphi (t)}+\overset{\rightharpoonup }{B_1}.\frac{\partial l_{\text{OC}_1 w}}{\partial \varphi (t)}+\overset{\rightharpoonup }{G_2}.\frac{\partial l_{\text{OC}_2 w}}{\partial \varphi (t)}+\overset{\rightharpoonup }{G_3}.\frac{\partial l_{\text{OC}_3 w}}{\partial \varphi (t)}+\overset{\rightharpoonup }{N}.\frac{\partial \left(l_{2 w}+l_{4 w}\right)}{\partial \varphi (t)}=0$
$Q_{\beta }=\overset{\rightharpoonup }{B_2}.\frac{\partial l_{\text{OC}_2 w}}{\partial \beta (t)} \overset{\rightharpoonup }{+B_3}.\frac{\partial l_{\text{OC}_3 w}}{\partial \beta (t)} \overset{\rightharpoonup }{+G_1}.\frac{\partial l_{\text{OC}_1 w}}{\partial \beta (t)}+\overset{\rightharpoonup }{B_1}.\frac{\partial l_{\text{OC}_1 w}}{\partial \beta (t)}+\overset{\rightharpoonup }{G_2}.\frac{\partial l_{\text{OC}_2 w}}{\partial \beta (t)}+\overset{\rightharpoonup }{G_3}.\frac{\partial l_{\text{OC}_3 w}}{\partial \beta (t)}+\overset{\rightharpoonup }{N}.\frac{\partial \left(l_{2 w}+l_{4 w}\right)}{\partial \beta (t)}=0$
To determine that $N_{we}$ and simultaneously satisfy that mentioned VWP condition I joined these 3 equations in one:
$Q_{\gamma}+ Q_{\varphi}+ Q_{\beta}=0$
you can see that it only equals $0$ when all components equal $0$. Is that equation right?
If so I can solve it and find $N_{we}$. However, when I do it I got wrong solution. But then probably I have wrong those above 3 equations for $Q_{\gamma}, Q_{\varphi}, Q_{\beta}$