# Inertial Propulsion of a Mobile Platform Driven by Two Eccentric Bodies
## 3 Analytical Investigation of the Inertial Propulsion System  
For the deduction of the motion equations for the IPS, we considered two eccentric bodies, which are each materialized by a mass \(m\) connected to a rigid rod of length \(r\). The two rotating masses are attached to the mobile platform of mass \(M\) and are rotating in a synchronized manner, in opposite directions. As the masses are contra-rotating perfectly synchronously, the potential movement will take place only along one axis (\(x\) in Figure 2), while along the other axis (\(y\)), the potential movements nullify each other. In relation to the fixed coordinate system \(xOy\), at the initial moment (\(t=0\)), the platform's center of mass (COM) is located at the distance \(x_{0}\), while the rods form an angle \(\varphi_{0}\) with the \(y\)-axis. At the arbitrary time \(t>0\), the system will travel a distance along the \(x\) axis, so that the momentary coordinate of the platform COM will be \(x_{M}\). The abscissa of the masses \(m\) is \(x_{m}\) and the angle described by the rotating masses is denoted by \(\varphi(t)\). The inertial force generated by the so-called \"driving masses\" \(m\) will act only along the \(x\) axis, directed opposite to the movement of the platform, and will be equal to \(-2m\ddot{x}_{m}\). Thus, [PERSON]'s second law can be written for the platform \(M\) as follows:  
Figure 1: Inertial propulsion drive with counter-rotating eccentrics. (**a**) Front view; (**b**) upper view. Figure 2: Initial and arbitrary positions of the IPS. \[M\ddot{x}_{M}=-2m\ddot{x}_{m}. \tag{1}\]  
Given that:  
\[x_{m}=\ x_{M}+rsin[\varphi(t)+\varphi_{0}], \tag{2}\]  
and \(\varphi(t)\)= \(\omega\cdot t\), where \(\omega\) is the angular velocity of the eccentric bodies, the first derivative of \(x_{m}\) may be expressed as:  
\[\dot{x}_{m}=\dot{x}_{M}+\omega rcos(\omega t+\varphi_{0}), \tag{3}\]  
or:  
\[v_{m}(t)= v_{M}(t)+\omega rcos(\omega t+\varphi_{0}), \tag{4}\]  
which means that the velocity of each driving mass is a sum of an absolute motion (motion of the platform \(v_{M}\)) and a relative motion. Further, the second derivative of \(x_{m}\) is:  
\[\ddot{x}_{m}=\ddot{x}_{M}-\omega^{2}rsin(\omega t+\varphi_{0}). \tag{5}\]  
Therefore, Equation (1) becomes:  
\[M\ddot{x}_{M}=-2m\ddot{x}_{M}+2m\omega^{2}rsin(\omega t+\varphi_{0}), \tag{6}\]  
or:  
\[\ddot{x}_{M}(M+2m)=2m\omega^{2}rsin(\omega t+\varphi_{0}), \tag{7}\]  
and the acceleration of the platform is:  
\[\ddot{x}_{M}=\frac{2m\omega^{2}r}{M+2m}sin(\omega t+\varphi_{0}). \tag{8}\]  
The platform's velocity is obtained by time integration of Equation (8):  
Figure 2:0  
where \(C\) is an integration constant. Regrouping the terms:  
Figure 2:1  
At start (\(t=0\)) the velocity of the platform is:  
Figure 2:2  
Further, considering the whole system of masses (\(M+2m\)) with the center of mass located in \(CM\), the law of impulse conservation can be written as:  
Figure 2:3  
or:  
Figure 2:4  
Applying Equation (4), Equation (12) becomes:  
Figure 2:5  
or:  
Figure 2:6  
and:Figure 2:7  
For the initial condition (\(t=0\)), Equation (15) becomes:  
Figure 2:8  
Evaluating the Equations (10) and (16), it can be concluded that, at start (\(t=0\)), the initial velocity of the whole system's CM is:  
Figure 2:9  
meaning that the systems \(CM\) will move forward only if the constant \(C\
eq 0\). Furthermore, analyzing Equation (18), one can conclude that the initial speed of the vehicles center of mass (\(v_{CM,0}\)) is the higher, the more:  
* The platform's mass \(M\) is smaller, compared to the driving masses \(m\);
* The driving masses are bigger;
* The rotational speed of the driving masses is higher;
* The length of the rods (\(r\)) on which the masses are attached is higher. Based on Equation (18), the platforms velocity expressed in Equation (10) becomes:  
\[M\ddot{x}_{M}=-2m\ddot{x}_{m}. \tag{1}\]0  
which means that the velocity (\(v_{M}\)) consist of a linear component (\(v_{CM,0}\)) and an oscillating one.