# Reaction Forces and Apparent Thrust in Dual Oscillating Control Moment Gyroscopes
## 4 Discussion
Since the 1960s, both industrial and academic research groups have reported anomalous reaction forces during gyroscopic experiments, which have been interpreted by some as evidence of thrust generation or loss of weight [20, 21, 22, 23, 24, 25, 49]. These findings appear to contradict well-established physical laws. The analysis of such systems is more intricate than that of the Dean drive, primarily because the configuration of gyroscopic set-ups evolves over time. This complexity arises from the continuously changing orientation of the gyroscopes' axes, which leads to corresponding variations in the angular momentum of the overall mechanical system. In contrast to the Dean-drive motion (proportional to \(\omega^{2}\)), the inertial force in contra-rotating gyroscopes is proportional to the product of two angular velocities (\(\omega_{\text{servo}}\omega_{\text{rotor}}\)). The former (\(\omega_{\text{servo}}\)) is due to the oscillation of gyro's axle, while the latter (\(\omega_{\text{rotor}}\)) is the spin. Since the spin does not affect the bending strength of gyro's axle even it takes an extremely high value, it is deduced that gyroscopes may develop higher inertial forces than those of the rotating masses (class of Dean drive devices). A typical control moment gyroscope (CMG) system consists of an array of four gyroscopes [26]. Nevertheless, without loss of generality, and for the sake of simplicity, the ability to generate internal torque is illustrated here using only two gyroscopes. To elaborate, consider a satellite in orbit. While at rest--for example, during sleep--an astronaut exerts no torque on the spacecraft. However, internal torque can later be applied through hand movements or, equivalently, by actuating a mechanism that alters the relative orientation (e.g., the included angle) between the axes of two spinning gyroscopes, such as those considered in the present study. In such cases, [PERSON]'s third law dictates that the internal torque generated is proportional to the angular velocity of reorientation, resulting in a change in the satellite's orientation (i.e., its attitude) without affecting its translational motion; that is, the center of mass of the satellite remains fixed along its orbital path while only its rotational state is altered. When this principle is translated to a terrestrial set-up (as is the case in the present paper), the internal torque produced--such as through forced gyroscopic precession--must be counterbalanced by a _couple_ of ground reaction forces which have a zero sum. Therefore, the nonzero reaction forces are due to two separate reasons: (1) the distribution of the dead weight \(F_{S}\) and (2) the reaction forces due to the oscillating masses in the interval \([\theta_{\text{min}},\theta_{\text{max}}]\). As a result, if one measures only the reaction forces at select points--particularly those that increase and even during a short time interval--then this may create the illusion of a net propulsive force. However, such an effect does not violate Newtonian mechanics, as it merely reflects an internal redistribution of forces within a statically constrained system and not true propulsion. A relevant interesting phenomenon of supposed weight loss was demonstrated in Professor [PERSON]'s 1974 lectures at University College London. In this experiment, a precessing gyroscope was mounted on an L-shaped aluminum stand positioned at the edge of a table. When the non-spinning gyroscope was horizontally locked, its weight produced a torque that caused the stand to topple. In contrast, when the gyroscope was spinning at precession, no such overturn occurred (see the roughly 28 min mark in the video [57], where Professor [PERSON] appears to demonstrate that a gyroscope undergoing forced precession exhibits a reduction in apparent weight).