# Reaction Forces and Apparent Thrust in Dual Oscillating Control Moment Gyroscopes
## 2 Materials and Methods
### Impulse of Reaction Forces  
#### 2.8.1 Approximate Model  
A close look at Equations (18) and (19)--in the approximate model--reveals that the sum of the two reaction forces is not affected by the gyroscopic terms because they cancel one another; practically, they compose a couple of equal and opposite forces which cancel out \(\tau_{\rm int}\). Therefore, the total vertical inertial force \(F_{\rm tot}=F_{\rm front}+F_{\rm rear}\) is influenced only by the measure of total centripetal force \(2m\omega^{2}r\) and the instantaneous polar angle \(\theta\).  
Equations (18) and (19) are based on the assumption of a constant angular velocity \(\omega\); otherwise, its temporal velocity \(\dot{\omega}\) must be considered.  
The total impulse of the reaction forces is given by  
\[I_{\rm tot}=\int_{0}^{T}[F_{\rm front}(t)+F_{\rm rear}(t)]dt\,, \tag{26}\]  
where \(T=T_{\rm rise}+T_{\rm reset}\) is the period. In more detail, for each phase of motion, by integrating Equations (18) and (19), we have the following:  
_I. Rise phase:_\(\theta_{\rm min}\to\theta_{\rm max}\)  
\[I_{\rm front}^{\rm rise}=\left[\frac{2l_{2}m\tau\omega_{\rm rise}}{l_{1}+l_{2 }}-\frac{2L_{\rm rotor}}{l_{1}+l_{2}}\right](\cos\theta_{\rm max}-\cos\theta_{ \rm min})\,, \tag{27}\]  
\[I_{\rm rear}^{\rm rise}=\left[\frac{2l_{1}m\tau\omega_{\rm rise}}{l_{1}+l_{2}} +\frac{2L_{\rm rotor}}{l_{1}+l_{2}}\right](\cos\theta_{\rm max}-\cos\theta_{ \rm min})\,. \tag{28}\]  
_II. Reset phase:_\(\theta_{\rm max}\to\theta_{\rm min}\)  
\[I_{\rm front}^{\rm reset}=\left[\frac{2l_{2}m\tau\omega_{\rm reset}}{l_{1}+l_{ 2}}-\frac{2L_{\rm rotor}}{l_{1}+l_{2}}\right](\cos\theta_{\rm min}-\cos\theta_{ \rm max})\,, \tag{29}\]  
\[I_{\rm rear}^{\rm reset}=\left[\frac{2l_{1}m\tau\omega_{\rm reset}}{l_{1}+l_{2 }}+\frac{2L_{\rm rotor}}{l_{1}+l_{2}}\right](\cos\theta_{\rm min}-\cos\theta_{ \rm max})\,. \tag{30}\]  
In general, when comparing Equation (27) with Equation (29), we obtain  
\[I_{\rm front}^{\rm rise}=-I_{\rm front}^{\rm reset}\,. \tag{31}\]  
Also, when comparing Equation (28) with Equation (30), we obtain  
\[I_{\rm rear}^{\rm rise}=-I_{\rm rear}^{\rm reset}\,. \tag{32}\]  
By adding Equations (31) and (32) in parts, we obtain  
\[\boxed{I_{\rm front}^{\rm rise}+I_{\rm rear}^{\rm rise}=-\big{(}I_{\rm front} ^{\rm reset}+I_{\rm rear}^{\rm reset}\big{)}}\,, \tag{33}\]  
which means that the impulse of both reaction forces in the rise phase is equal and opposite to the impulse in the reset phase. Therefore, as a general rule, whatever is probably gained in the rise phase is then lost in the reset phase.  
Obviously, if \(\theta_{\rm min}=-45^{\circ}\) and \(\theta_{\rm max}=45^{\circ}\), then each term in Equations (27)-(30) vanishes, and therefore the total impulse of the inertial forces within a period is _zero_. In contrast, if, for example, \(\theta_{\rm min}=0^{\circ}\), and \(\theta_{\rm max}=45^{\circ}\), then the impulse in each phase is different than zero, but again the total impulse per period is zero.  
It is noted that the above discussion suffers from the fact that the assumed constant angular velocities do not satisfy the initial conditions of zero values at the initial (\(t=0\)), transitional (\(t=T_{\rm rise}\)), and final positions (\(t=T=T_{\rm rise}+T_{\rm reset}\)) in each phase.