## 2 First Order Theory of Propulsion Generation by Means of Rotation Energy  
Assume a stiff rotor disc (radius \(R\), thickness D \(<\) R, homogenate material distribution, total mass \(M\)) equipped with fixed unbalanced mass \(m_{u}\) at its periphery (Figure 1). The disc rotation axis A should be perpendicularly centered in a planar system with orthogonal axes (\(x\); y), fixed in an inertial reference frame. We apply in the following discussion the equivalent circuit mass point diagram of Figure 1 (right). Due to disc rotation, the generated centrifugal force vector written in polar coordinates is  
\[ \begin{array}{c}\overset{\rightarrow}{F}_{u}=\overset{\rightarrow}{R}m_ {u}\omega^{2}\\ \overset{\rightarrow}{R}=Re^{i(\omega t+\varphi_{0})}\end{array}  \tag{1}\]  
with magnitudes  
\(\overset{\rightarrow}{R}\) radius vector (\(i\) = imaginary unit) pointing from axis A to unbalance mass \(m_{u}\)  
\(\varphi_{0}\) rotor fixed azimuth of the unbalanced mass point \(m_{u}\)  
\(R\) radius of the rotor disc  
\(\omega=2\pi f\) (angular) revolution frequency  
\(T=\frac{1}{f}\) cycle time of rotor disc  
Equation (1) says centrifugal force vector \(\overset{\rightarrow}{F}_{u}\) is always pointing in the direction of instantaneous \(\overset{\rightarrow}{R}\). Its magnitude \(F_{u}\) increases proportional with \(m_{u}\) and proportional with rotor radius \(R\), furthermore, \(F_{u}\) is increasing quadratically with applied rotor rotation frequency \(f\). It is useful to introduce an unbalance vector \(\overset{\rightarrow}{U}\) that is vector \(\overset{\rightarrow}{F}_{u}\) divided by the square of angular rotation frequency \(\omega\):  
\[\overset{\rightarrow}{U}=U_{0}e^{i(\omega t+\varphi_{0})}=m_{u}Re^{i(\omega t +\varphi_{0})} \tag{2}\]  
Figure 1: (**Left**): Unbalanced rotor due to asymmetric mass distribution. (**Right**): equivalent circuit mass point diagram [18] with A rotation axis, perpendicular to paper plane (\(\omega\); y), \(m_{u}\) point of unbalance mass, \(M\) point of rotor disc mass, \(R\) rotor radius, \(r\) translation of \(M\) due to \(m_{u}\). Note, unbalance magnitude \(U_{0}=m_{u}R\) is independent of \(\omega\). Due to peripheric unbalance mass \(m_{u}\) there is a small displacement \(r\) of disc gravity center \(M\) arises, away from rotor rotation axis A. Note that this displacement is caused due to static balancing of the rigged disc. Additionally, the elasticity and strain of real rotor materials may contribute to real displacement, but we will not take this effect into account here. In the case of very small unbalance mass \(m_{u}\) (i.e., \(M=M_{0}+m_{u},\ m_{u}\ll M_{0}\cong M\)), the following vector equation results:  
\[M\overset{\rightarrow}{r}=m_{u}\overset{\rightarrow}{R} \tag{3}\]  
respectively, for the magnitudes  
\[Mr=m_{u}R\]  
According to Equation (3) the displacement \(r\) points always towards peripheric unbalance mass \(m_{u}\). Thus, rotation axis A, disc mass \(M\), and unbalanced mass point \(m_{u}\) are located on the same radius vector \(R\) (azimuth \(\varphi_{0}\)). Considering this we draw the conclusion: Vectorial displacement \(r\) of gravity center \(M\) can be controlled in principle via choosing magnitude and azimuth of the peripheric unbalance mass \(m_{u}\)! Now the question arises: Is this mass displacement effect insignificantly small or can it be used to the best advantage in propulsion technology?