# Physics of Gyroscope’s “Antigravity Effect”

Authors: Ryspek Usubamatov
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

The physics of gyroscopic effects are more complex than presented in existing mathematical models. The effects presented by these models do not match the real forces acting on gyroscopic devices. New research in this area has demonstrated that a system of inertial torques, which are generated by the rotating mass of spinning objects, acts upon a gyroscope. The actions of the system of inertial forces are validated by practical tests of the motions of a gyroscope with one side support. The action of external load torque on a gyroscope with one side support demonstrates that the gyroscope’s upward motion is wrongly called an “antigravity” effect. The upward motion of a gyroscope is the result of precession torque around its horizontal axis. The novelty of the present work is related to the mathematical models for the upward and downward motions of gyroscopes influenced by external torque around the vertical axis. This analytical research describes the physics of gyroscopes’ upward motion and validates that gyroscopes do not possess an antigravity property.


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## Body

## 1. Introduction

The applied theory of gyroscopes emerged mainly during the twentieth century due to the vast application and intensification of the rotation of numerous spinning objects in engineering [1–4]. Gyroscope properties are used in many engineering calculations related to rotating parts in aerospace, shipbuilding, and other industries, and numerous publications have been dedicated to gyroscopic effects [5, 6]. Fundamental textbooks and publications about classical mechanics describe gyroscopic effects in Euler’s term of the change in angular momentum [7–9]. Nevertheless, previous analytical approaches are based on several assumptions and simplifications that lead to theoretical uncertainty about gyroscopic effects [10, 11]. Mathematical models for gyroscope properties in publications do not match the practical applications of gyroscopic devices [12–15]. All rotating objects of movable mechanisms manifest gyroscopic effects that should be computed using engineering methods. From this, researchers have coined artificial terms such as gyroscopic effects and gyroscope couples, and they have established noninertial, nongravitational properties that contradict the principles of physics.The physics of gyroscopic effects are more complex than presented in the literature. The external torque applied to the gyroscope produces a system of eight inertial torques generated by centrifugal, common inertial, and Coriolis forces, as well as by the change in the angular momentum of the spinning rotor. The actions of the system of inertial torques around the axes of the gyroscope are interrelated and manifest all gyroscopic properties that were previously unexplainable. Today, the physics and mathematical models of gyroscopes’ inertial torques are well described and have been validated [16–20]. However, the new analytical approach still has some mathematical errors. Some publications point out mathematical models’ inaccuracy when dealing with the inertial torques acting on a gyroscope [20]. In engineering, several load torques of gyroscopic devices can act in different directions around their axes of rotation. The interrelated actions of the internal and external load torques on the gyroscope pose a solvable scientific and engineering problem. The novelty of the present work describes an accurate mathematical model for gyroscope upward motion under the action of external torque and explains the physics of the gyroscopic effect that has wrongly been called an “antigravity” property.


## 2. Methodology

Recent investigations into the physical principles of gyroscopic effects have led to mathematical models that can be used to assess a system of inertial forces acting on a gyroscope. New mathematical models of internal torques have described new gyroscope properties and behaviours of gyroscopic devices. The action of external load on a gyroscope generates resistance and precession torques originated by the rotation of the mass-elements of the spinning rotor. The mathematical models for the inertial torques acting on a gyroscope are presented in several publications [16–20]. One publication contains incorrect mathematical models for the interrelation of external and internal torques acting on a gyroscope [20]. The action of the system of interrelated inertial torques around the axes of the gyroscope should be considered carefully. Analyses of the interdependent sequence action of the external and inertial torques enable the physical principles of the gyroscope motions to be formulated. The actions of the system of torques and their effects on the gyroscopic stand have been considered, and the related technical data have been published [20]. The mathematical models for gyroscope motions are presented via two examples of the actions of external and internal torques on the gyroscope with one side support for its horizontal location. For simplicity, the actions of frictional forces on the supports and pivot are not considered. Two mathematical models of gyroscope motions provide a clear picture of the physics of the acting external and internal torques. The first mathematical model considers the action of the external load torques, described as the torque Ty acting on the gyroscope around axis oyand the torque T of the gyroscope weight W acting around axis ox in a counterclockwise direction. The second mathematical model considers the external loads described as the action of the torque Ty in the clockwise direction around axis oy and the action of the torque T of the gyroscope weight in a counterclockwise direction around axis ox. This constructional peculiarity of the gyroscope stand generates a unique combination of acting inertial torques and different motions.


### 2.1. Case Study 1

The mathematical model for the gyroscope motions considers the action of the load torqueTy around axis oy in a counterclockwise direction and the load torque T=Wgl generated by the weight W of the gyroscope around axis ox. Under these conditions, the action of the external load torque Ty generates a system of inertial torques that act around the axes of the gyroscope (Figure 1). The action of the load torque Ty generates the following inertial torques:
(a)
The resistance torqueTr.y=Tct.y+Tcr.y generated by the action of centrifugal Tct.y and Coriolis forces Tcr.y around axis oy in a clockwise direction (i.e., in the opposite direction of the action of load torque Ty)
(b)
The procession torqueTp.y=Tin.y+Tam.y generated by the common inertial forces Tin.y and the change in the angular momentum of the spinning rotor Tam.y originated at axis oy but acting around axis ox in a clockwise direction (i.e., in the opposite direction of the action of the gyroscope weight T). The value of the torque produced by the gyroscope weight T is lower than the value of the resulting inertial torques Trstx=Tp.y−Tr.x−T acting around axis ox. The latter (Trstx) causes the upward motion of the gyroscope
(c)
The procession torqueTp.y around axis ox, in turn, generates the resistance torque Tr.x=Tct.x+Tcr.x of centrifugal Tct.x and Coriolis forces Tcr.x around axis ox. These torques move in a counterclockwise direction and augment the action of the gyroscope weight TFigure 1
The torques and motions acting on the gyroscope with one side support.The action of the resulting torqueTrstx=Tp.y−Tr.x−T around axis ox produces the precession torque Tp.x=Tin.x+Tam.x generated by the common inertial forces Tin.x and the change in the angular momentum Tam.x originated on axis ox but acting around axis oy and added to the resistance torque Tr.y around axis oy, which results in the resistance torque Trst.y=Tr.y+Tp.x.The actions of loadsTy and T change the values of interrelated inertial torques acting around two axes. The action of the external torque Ty generates precession torque Tp.y, which acts around axis ox, which is opposite to the action of the gyroscope weight T. This situation leads to decreases in the value of the resulting inertial torque Trst.x and the precession torque Tp.x acting around axis oy. The changes in the values of the inertial torques are equal because they express the internal kinetic energies. This change is expressed by the coefficient of the change in the inertial torques around one axis and is explained by the following expression:
(1)η=Tp.y−TTp.y=1−TTp.y=1−Wgl2π2+9/9Jωωy,where expressions of the inertial torques [16] and the torque produced by the gyroscope weight are substituted into Equation (1). All components are as specified above.The coefficientη represents the decrease in the value of precession torque acting around axis oy when the value of inertial torque around axis ox decreases. This dependency reflects the interrelation of the inertial torques acting around two axes. An analysis of Equation (1) results in the following values of the coefficient η:
(i)
The absence of the load torqueT acting around axis ox means that the coefficient is η=1.0 (i.e., there is no change in the value of the inertial torques acting around two axes)
(ii)
The action of the load torqueTmeans that the coefficient is η<1.0 (i.e., the values of the inertial acting torques around two axes decrease)
(iii)
When the load torque is equal to the precession torque (T=Tp.y), the coefficient is η=0 (i.e., the gyroscope does not turn around axis oy but instead turns around axis ox under only the action of the gyroscope weight).This peculiarity should be represented in equations of the gyroscope motions around two axes. A mathematical model has been developed to assess a gyroscope’s motions that are caused by the action of the external and internal torques. This model involves corrections based on the interrelated action of the inertial torques and is presented by the following Euler’s differential equations:(2)Jydωydt=Ty−Tct.y−Tcr.y−Tin.x+Tam.xη,(3)−Jxdωxdt=T−Tin.y−Tam.y+Tct.x+Tcr.x,where ωx and ωy are the angular velocities of the gyroscope around axes ox and oy, respectively; Tct.x, Tct.y, Tcr.x, Tcr.y, Tin.x, Tin.y, Tam.x, and Tam.y are the internal torques generated by the centrifugal, Coriolis, and common inertial forces and the change in the angular momentum acting around axes ox and oy, respectively [16]. The sign (-) of Equation (3) means that the motion occurs in a clockwise direction. All other components are as specified above.Equations of inertial torques [16] and Equation (1) are substituted into Equations (2) and (3). The components of the torques generated by centrifugal Tct.y and inertial Tin.x forces that have the same expression are removed from Equations (2) and (3). The simplification and modification of these equations are similar to the solution presented in the manuscripts [17, 18], wherein this process is justified in detail. Modified equations are presented by the following system:
(4)Jydωydt=Ty−2π2+89Jωωy−Jωωx×1−Wgl2π2+9/9Jωωy,(5)−Jxdωxdt=T−2π2+99Jωωy−89Jωωx,(6)ωx=4π2+17ωy,where Equation (6) represents the dependency of the angular velocities of the gyroscope around axes ox and oy that were added to Equations (4) and (5). This solution is presented in manuscripts [17, 18] that use similar analytical approaches. All other parameters are as specified above.Substituting Equation (6) into the first Equation (3) and transformation gives the following equation:
(7)Jydωydt=Ty−2π2+89Jωωy−4π2+17Jωωy×1−Wgl2π2+9/9Jωωy,where all components are as specified above.


#### 2.1.1. Working Example

The components of a gyroscope with one side support move under the action of the load torquesTy and T, which are generated by the gyroscope weight and act around axes oy and ox, respectively. The value of the first torque Ty is half the value of torque T (i.e., Ty=0.5Wgl). The actions of the torques and motions are presented in Figure 1. The technical data related to the gyroscope is presented in the manuscript ([20], Table 1). Equation (7) is the equation of the gyroscope motion around axis oy. The ratio of the angular velocities around two axes is calculated using Equation (6). The angular velocity around axis oy is defined by the following solution. Substituting the initial data of the gyroscope [20] and the data presented above into Equation (7) yields the following equation:
(8)3.38437×10−4dωydt=0.5×0.146×9.81×0.0355−2π2+89×0.5543873×10−4×ωωy−4π2+17×0.5543873×10−4×ωωy1−0.146×9.81×0.03552π2+9/9×0.5543873×10−4×ωωy.Equation (8) can be simplified as follows:
(9)0.102495748dωydt=280.049918886−ωωy.Equation (10) arises by separating the variables to simplify and transform Equation (9):
(10)dωy280.049918886/ω−ωy=9.756502262ωdt.Equation (11) is the integral form of Equation (10):
(11)∫0ωydωy280.049918886/ω−ωy=9.756502262ω∫0tdt.Table 1
Technical data of the test stand with Super Precision Gyroscope, “Brightfusion Ltd.”.
Parameters
Mass (kg)
Mass moment of inertia around axesox and oy (J kg m2)
Spinning rotor with shaft (mr)
0.1159
J=mrR2/2=0.5543873×10−4 (around axis oz)
Gyroscope (M)
0.146
JiM=2/3mfrf2+mfl2+mrR2c/4+mrl2=2.284736∗10−4
Centre beam with journals and screw (b)
0.028
Jbx=mbr2b/2=0.00224×10−4 (around axis ox)Jby=mbl2b/12=0.2105833×10−4 (around axis oy)
MassE=M+b
0.174
JEx=JxM+Jbx=2.286976×10−4 (around axis ox)
Bar with screws (bs)
0.067
Jbs=mbsl2bs/12=0.51456×10−4 (around axis oy)
Arm (a)
2×0.009=0.018
Ja=mar2a/2+mala2=0.41553×10−4 (around axis oy)
TotalA=M+b+bs+a
0.259
Jy=JyM+Jb+Jbs+Ja=3.38437×10−4 (around axis oy)The left integral of Equation (11) is tabulated and presented as the integral ∫dx/a−x=−lnx+C. The right integral is simple. Solving the integrals yields the following equation:
(12)−ln280.049918886ω−ωy0ωy=9.756502262ωt0t,thus giving rise to the following:
(13)ln280.049918886ω−ωy−ln280.049918886ω=−9.756502262ωt.Equation (13) can be transformed into the equation of angular velocity for the gyroscope around axis oy:
(14)1−ωy280.049918886/ω=e−9.756502262ωt.The right component of Equation (14) contains the expression e−9.756502262ωt, which has a small value of the high order by which the angular velocity ω of the spinning rotor is around n=10000⋯30000rpm. Hence, this component of the equation can be neglected. Solving Equation (10) yields the following equation:
(15)ωy=280.049918886ω,where all components are as specified above.The angular velocityω of the gyroscope around axis oy is defined according to the rotor’s speed of n=10000.0rpm. Substituting n into Equations (15) and (6) yields the angular velocities of the gyroscope around axes oy and ox:
(16)ωy=280.049918886ω=280.04991888610000×2π/60=0.267427973rad/s=15.322494°/s,(17)ωx=−4π2+17ωy=−4π2+17×0.267427973=−15.103908738rad/s=−865.390224°/s,where the sign (-) of Equation (17) means the gyroscope precession around axis ox is in the clockwise direction, i.e., the gyroscope moves upward.The torqueTy acting around axis oy leads to slow counterclockwise rotation around axis oyand causes the intensive precessed clockwise rotation of the gyroscope around axis ox. The gyroscope moves upward from its horizontal location. This effect serves as practical evidence of the absence of the antigravity property.


### 2.2. Case Study 2

The mathematical model for the gyroscope’s motions is considered for the same gyroscope stand and technical parameters presented in Section2.1, Case Study 1. The difference is the clockwise action of the load torque Ty around axis oy. Under this condition, the actions of the gyroscope internal torques and motions are as follows:
(i)
The load torqueTy generates the resistance torque Try, which acts in a counterclockwise direction around axis oy and the precession torque Tpy, which acts around axis oxin a counterclockwise direction. This coincides with the action of the load torque T
(ii)
The action of the torque produced by the gyroscope weightT generates the resistance Trx and precession Tpx torques. The resistance torque Trx acts in a clockwise direction around axis ox. The precession torque Tpx acts in a counterclockwise direction around axis oy as the torque TAll acting torques and motions of the gyroscope with one side support are illustrated in Figure2.Figure 2
Internal torques and motions when the load torqueTy acts in the clockwise direction around axis oy.The equations of the gyroscope torques and motions around axesox and oy are represented by the following expressions:
(18)−Jydωydt=−Ty+Tct.y+Tcr.y+Tin.x+Tam.xη,(19)Jxdωxdt=T+Tin.y+Tam.y−Tct.x−Tcr.x,where the sign (-) of Equation (18) indicates a clockwise movement. All other components are as specified above.The coefficientη represents the proportional increase of the precession torque Tp.x around axis oy. The coefficient η is expressed as the ratio of the sum of the precession Tp.y and load T torques to the precession Tp.y torque, which acts around axis ox. The expression η is obtained in a similar way as in Equation (1) and is represented by the following equation:
(20)η=Tp.y+TTp.y=1+TTp.y=1+Wgl2π2+9/9Jωωy,where η is the coefficient of the proportional increase of the value of the precession torque Tp.x acting around axis oy. All other parameters are as specified above.The coefficientη denotes the increase in precession torque Tp.x around axis oy when the value of the resulting torque Trst.x around axis ox also increases. Equation (20) demonstrates the following values of the coefficient η:
(i)
The absence of the load torqueT acting around axis ox means that the coefficient η=1.0 (i.e., there is no change in the values of the inertial torques acting around two axes)
(ii)
The action of the load torqueT means that the coefficient is η>1.0 (i.e., the values of the inertial torques acting around two axes increase)
(iii)
When the load torque of the gyroscope weight is equal to the precession torque (T=Tp.y), the coefficient η=2 (i.e., there is an increase in the value of the resulting resistance torque Trtst.y acting around axis oy)Equations for inertial torques—that is, Equations (17) and (20)—are substituted into Equations (18) and (19). Equations (21)–(23) are simplified and transformed in the same manner as Equations (4)–(6):
(21)−Jydωydt=−Ty+29π2+4Jωωy+Jωωx×1+Wgl2/9π2+1Jωωy,(22)Jxdωxdt=T+29π2+1Jωωy−89Jωωx,(23)ωx=4π2+17ωy,where all parameters are as specified above.Regarding the motion around axisox, the acting torques are represented by the combined action of the gyroscope weight T with the precession torques Tp.y and of the resistance torques Tr.x in the opposite direction. The torques acting around axis oy are represented by the action of the load torque Ty and the combined resistance torques Tr.y with precession torques Tp.x acting in the opposite direction. The following solution for Equations (21)–(23) is the same as presented in Section 2.1, Case Study 1. Equation (21) can be solved by substituting the defined equation and utilising transformations to yield the following equation:
(24)−Jydωydt=−Ty+29π2+4Jωωy+4π2+17Jωωy×1+Wgl2/9π2+1Jωωy,where all components are as specified above.


#### 2.2.1. Working Example

The motions of the gyroscope with one side support are conducted under the clockwise action of the load torqueTy around axis oy. The sketch of the action of the torques and motions is presented in Figure 2. Equation (24) is the equation of the gyroscope motion around axis oy. The angular velocities of precessions around the two axes should be defined. Substituting the initial data of the gyroscope ([20], Table 1) into Equation (24) and making transformations yield the following equation:
(25)‐3.38437×10−4dωydt=−0.5×0.146×9.81×0.0355+29π2+4×0.5543873×10−4×ωωy+4π2+17×0.5543873×10−4×ωωy×1+0.146×9.81×0.03552/9π2+1×0.5543873×10−4×ωωy.Equation (25) is simplified. Then, the steps of solutions similar to the steps presented in Section 2.1, Case Study 1, are carried out. All comments related to the solutions of the equations are omitted.
(26)−0.102495748dωydt=264.651454335+ωωy,dωyωy+264.651454335/ω=−9.756502262ωdt,∫0ωydωyωy+264.651454335/ω=−9.756502262ω∫0tdt,lnωy+264.651454335ω0ωy=−9.756502262ωt0t,lnωy+264.651454335ω−ln264.651454335ω=−9.756502262ωt,1+ωy264.651454335/ω=e−9.756502262ωt,ωy=−264.651454335ω,where the sign (-) indicates a clockwise movement.
(27)ωy=−264.65145433510000×2π/60=−0.252723522rad/s=−14.479991°/s,ωx=4π2+17×0.252723522=14.273424613rad/s=817.806989°/s.The torqueTy acting around axis oy causes the slow clockwise rotation around axis oy and the intensive counterclockwise rotation of the gyroscope around axis ox. These rotations result in a counterclockwise torque around axis ox.


## 2.1. Case Study 1

The mathematical model for the gyroscope motions considers the action of the load torqueTy around axis oy in a counterclockwise direction and the load torque T=Wgl generated by the weight W of the gyroscope around axis ox. Under these conditions, the action of the external load torque Ty generates a system of inertial torques that act around the axes of the gyroscope (Figure 1). The action of the load torque Ty generates the following inertial torques:
(a)
The resistance torqueTr.y=Tct.y+Tcr.y generated by the action of centrifugal Tct.y and Coriolis forces Tcr.y around axis oy in a clockwise direction (i.e., in the opposite direction of the action of load torque Ty)
(b)
The procession torqueTp.y=Tin.y+Tam.y generated by the common inertial forces Tin.y and the change in the angular momentum of the spinning rotor Tam.y originated at axis oy but acting around axis ox in a clockwise direction (i.e., in the opposite direction of the action of the gyroscope weight T). The value of the torque produced by the gyroscope weight T is lower than the value of the resulting inertial torques Trstx=Tp.y−Tr.x−T acting around axis ox. The latter (Trstx) causes the upward motion of the gyroscope
(c)
The procession torqueTp.y around axis ox, in turn, generates the resistance torque Tr.x=Tct.x+Tcr.x of centrifugal Tct.x and Coriolis forces Tcr.x around axis ox. These torques move in a counterclockwise direction and augment the action of the gyroscope weight TFigure 1
The torques and motions acting on the gyroscope with one side support.The action of the resulting torqueTrstx=Tp.y−Tr.x−T around axis ox produces the precession torque Tp.x=Tin.x+Tam.x generated by the common inertial forces Tin.x and the change in the angular momentum Tam.x originated on axis ox but acting around axis oy and added to the resistance torque Tr.y around axis oy, which results in the resistance torque Trst.y=Tr.y+Tp.x.The actions of loadsTy and T change the values of interrelated inertial torques acting around two axes. The action of the external torque Ty generates precession torque Tp.y, which acts around axis ox, which is opposite to the action of the gyroscope weight T. This situation leads to decreases in the value of the resulting inertial torque Trst.x and the precession torque Tp.x acting around axis oy. The changes in the values of the inertial torques are equal because they express the internal kinetic energies. This change is expressed by the coefficient of the change in the inertial torques around one axis and is explained by the following expression:
(1)η=Tp.y−TTp.y=1−TTp.y=1−Wgl2π2+9/9Jωωy,where expressions of the inertial torques [16] and the torque produced by the gyroscope weight are substituted into Equation (1). All components are as specified above.The coefficientη represents the decrease in the value of precession torque acting around axis oy when the value of inertial torque around axis ox decreases. This dependency reflects the interrelation of the inertial torques acting around two axes. An analysis of Equation (1) results in the following values of the coefficient η:
(i)
The absence of the load torqueT acting around axis ox means that the coefficient is η=1.0 (i.e., there is no change in the value of the inertial torques acting around two axes)
(ii)
The action of the load torqueTmeans that the coefficient is η<1.0 (i.e., the values of the inertial acting torques around two axes decrease)
(iii)
When the load torque is equal to the precession torque (T=Tp.y), the coefficient is η=0 (i.e., the gyroscope does not turn around axis oy but instead turns around axis ox under only the action of the gyroscope weight).This peculiarity should be represented in equations of the gyroscope motions around two axes. A mathematical model has been developed to assess a gyroscope’s motions that are caused by the action of the external and internal torques. This model involves corrections based on the interrelated action of the inertial torques and is presented by the following Euler’s differential equations:(2)Jydωydt=Ty−Tct.y−Tcr.y−Tin.x+Tam.xη,(3)−Jxdωxdt=T−Tin.y−Tam.y+Tct.x+Tcr.x,where ωx and ωy are the angular velocities of the gyroscope around axes ox and oy, respectively; Tct.x, Tct.y, Tcr.x, Tcr.y, Tin.x, Tin.y, Tam.x, and Tam.y are the internal torques generated by the centrifugal, Coriolis, and common inertial forces and the change in the angular momentum acting around axes ox and oy, respectively [16]. The sign (-) of Equation (3) means that the motion occurs in a clockwise direction. All other components are as specified above.Equations of inertial torques [16] and Equation (1) are substituted into Equations (2) and (3). The components of the torques generated by centrifugal Tct.y and inertial Tin.x forces that have the same expression are removed from Equations (2) and (3). The simplification and modification of these equations are similar to the solution presented in the manuscripts [17, 18], wherein this process is justified in detail. Modified equations are presented by the following system:
(4)Jydωydt=Ty−2π2+89Jωωy−Jωωx×1−Wgl2π2+9/9Jωωy,(5)−Jxdωxdt=T−2π2+99Jωωy−89Jωωx,(6)ωx=4π2+17ωy,where Equation (6) represents the dependency of the angular velocities of the gyroscope around axes ox and oy that were added to Equations (4) and (5). This solution is presented in manuscripts [17, 18] that use similar analytical approaches. All other parameters are as specified above.Substituting Equation (6) into the first Equation (3) and transformation gives the following equation:
(7)Jydωydt=Ty−2π2+89Jωωy−4π2+17Jωωy×1−Wgl2π2+9/9Jωωy,where all components are as specified above.


### 2.1.1. Working Example

The components of a gyroscope with one side support move under the action of the load torquesTy and T, which are generated by the gyroscope weight and act around axes oy and ox, respectively. The value of the first torque Ty is half the value of torque T (i.e., Ty=0.5Wgl). The actions of the torques and motions are presented in Figure 1. The technical data related to the gyroscope is presented in the manuscript ([20], Table 1). Equation (7) is the equation of the gyroscope motion around axis oy. The ratio of the angular velocities around two axes is calculated using Equation (6). The angular velocity around axis oy is defined by the following solution. Substituting the initial data of the gyroscope [20] and the data presented above into Equation (7) yields the following equation:
(8)3.38437×10−4dωydt=0.5×0.146×9.81×0.0355−2π2+89×0.5543873×10−4×ωωy−4π2+17×0.5543873×10−4×ωωy1−0.146×9.81×0.03552π2+9/9×0.5543873×10−4×ωωy.Equation (8) can be simplified as follows:
(9)0.102495748dωydt=280.049918886−ωωy.Equation (10) arises by separating the variables to simplify and transform Equation (9):
(10)dωy280.049918886/ω−ωy=9.756502262ωdt.Equation (11) is the integral form of Equation (10):
(11)∫0ωydωy280.049918886/ω−ωy=9.756502262ω∫0tdt.Table 1
Technical data of the test stand with Super Precision Gyroscope, “Brightfusion Ltd.”.
Parameters
Mass (kg)
Mass moment of inertia around axesox and oy (J kg m2)
Spinning rotor with shaft (mr)
0.1159
J=mrR2/2=0.5543873×10−4 (around axis oz)
Gyroscope (M)
0.146
JiM=2/3mfrf2+mfl2+mrR2c/4+mrl2=2.284736∗10−4
Centre beam with journals and screw (b)
0.028
Jbx=mbr2b/2=0.00224×10−4 (around axis ox)Jby=mbl2b/12=0.2105833×10−4 (around axis oy)
MassE=M+b
0.174
JEx=JxM+Jbx=2.286976×10−4 (around axis ox)
Bar with screws (bs)
0.067
Jbs=mbsl2bs/12=0.51456×10−4 (around axis oy)
Arm (a)
2×0.009=0.018
Ja=mar2a/2+mala2=0.41553×10−4 (around axis oy)
TotalA=M+b+bs+a
0.259
Jy=JyM+Jb+Jbs+Ja=3.38437×10−4 (around axis oy)The left integral of Equation (11) is tabulated and presented as the integral ∫dx/a−x=−lnx+C. The right integral is simple. Solving the integrals yields the following equation:
(12)−ln280.049918886ω−ωy0ωy=9.756502262ωt0t,thus giving rise to the following:
(13)ln280.049918886ω−ωy−ln280.049918886ω=−9.756502262ωt.Equation (13) can be transformed into the equation of angular velocity for the gyroscope around axis oy:
(14)1−ωy280.049918886/ω=e−9.756502262ωt.The right component of Equation (14) contains the expression e−9.756502262ωt, which has a small value of the high order by which the angular velocity ω of the spinning rotor is around n=10000⋯30000rpm. Hence, this component of the equation can be neglected. Solving Equation (10) yields the following equation:
(15)ωy=280.049918886ω,where all components are as specified above.The angular velocityω of the gyroscope around axis oy is defined according to the rotor’s speed of n=10000.0rpm. Substituting n into Equations (15) and (6) yields the angular velocities of the gyroscope around axes oy and ox:
(16)ωy=280.049918886ω=280.04991888610000×2π/60=0.267427973rad/s=15.322494°/s,(17)ωx=−4π2+17ωy=−4π2+17×0.267427973=−15.103908738rad/s=−865.390224°/s,where the sign (-) of Equation (17) means the gyroscope precession around axis ox is in the clockwise direction, i.e., the gyroscope moves upward.The torqueTy acting around axis oy leads to slow counterclockwise rotation around axis oyand causes the intensive precessed clockwise rotation of the gyroscope around axis ox. The gyroscope moves upward from its horizontal location. This effect serves as practical evidence of the absence of the antigravity property.


## 2.1.1. Working Example

The components of a gyroscope with one side support move under the action of the load torquesTy and T, which are generated by the gyroscope weight and act around axes oy and ox, respectively. The value of the first torque Ty is half the value of torque T (i.e., Ty=0.5Wgl). The actions of the torques and motions are presented in Figure 1. The technical data related to the gyroscope is presented in the manuscript ([20], Table 1). Equation (7) is the equation of the gyroscope motion around axis oy. The ratio of the angular velocities around two axes is calculated using Equation (6). The angular velocity around axis oy is defined by the following solution. Substituting the initial data of the gyroscope [20] and the data presented above into Equation (7) yields the following equation:
(8)3.38437×10−4dωydt=0.5×0.146×9.81×0.0355−2π2+89×0.5543873×10−4×ωωy−4π2+17×0.5543873×10−4×ωωy1−0.146×9.81×0.03552π2+9/9×0.5543873×10−4×ωωy.Equation (8) can be simplified as follows:
(9)0.102495748dωydt=280.049918886−ωωy.Equation (10) arises by separating the variables to simplify and transform Equation (9):
(10)dωy280.049918886/ω−ωy=9.756502262ωdt.Equation (11) is the integral form of Equation (10):
(11)∫0ωydωy280.049918886/ω−ωy=9.756502262ω∫0tdt.Table 1
Technical data of the test stand with Super Precision Gyroscope, “Brightfusion Ltd.”.
Parameters
Mass (kg)
Mass moment of inertia around axesox and oy (J kg m2)
Spinning rotor with shaft (mr)
0.1159
J=mrR2/2=0.5543873×10−4 (around axis oz)
Gyroscope (M)
0.146
JiM=2/3mfrf2+mfl2+mrR2c/4+mrl2=2.284736∗10−4
Centre beam with journals and screw (b)
0.028
Jbx=mbr2b/2=0.00224×10−4 (around axis ox)Jby=mbl2b/12=0.2105833×10−4 (around axis oy)
MassE=M+b
0.174
JEx=JxM+Jbx=2.286976×10−4 (around axis ox)
Bar with screws (bs)
0.067
Jbs=mbsl2bs/12=0.51456×10−4 (around axis oy)
Arm (a)
2×0.009=0.018
Ja=mar2a/2+mala2=0.41553×10−4 (around axis oy)
TotalA=M+b+bs+a
0.259
Jy=JyM+Jb+Jbs+Ja=3.38437×10−4 (around axis oy)The left integral of Equation (11) is tabulated and presented as the integral ∫dx/a−x=−lnx+C. The right integral is simple. Solving the integrals yields the following equation:
(12)−ln280.049918886ω−ωy0ωy=9.756502262ωt0t,thus giving rise to the following:
(13)ln280.049918886ω−ωy−ln280.049918886ω=−9.756502262ωt.Equation (13) can be transformed into the equation of angular velocity for the gyroscope around axis oy:
(14)1−ωy280.049918886/ω=e−9.756502262ωt.The right component of Equation (14) contains the expression e−9.756502262ωt, which has a small value of the high order by which the angular velocity ω of the spinning rotor is around n=10000⋯30000rpm. Hence, this component of the equation can be neglected. Solving Equation (10) yields the following equation:
(15)ωy=280.049918886ω,where all components are as specified above.The angular velocityω of the gyroscope around axis oy is defined according to the rotor’s speed of n=10000.0rpm. Substituting n into Equations (15) and (6) yields the angular velocities of the gyroscope around axes oy and ox:
(16)ωy=280.049918886ω=280.04991888610000×2π/60=0.267427973rad/s=15.322494°/s,(17)ωx=−4π2+17ωy=−4π2+17×0.267427973=−15.103908738rad/s=−865.390224°/s,where the sign (-) of Equation (17) means the gyroscope precession around axis ox is in the clockwise direction, i.e., the gyroscope moves upward.The torqueTy acting around axis oy leads to slow counterclockwise rotation around axis oyand causes the intensive precessed clockwise rotation of the gyroscope around axis ox. The gyroscope moves upward from its horizontal location. This effect serves as practical evidence of the absence of the antigravity property.


## 2.2. Case Study 2

The mathematical model for the gyroscope’s motions is considered for the same gyroscope stand and technical parameters presented in Section2.1, Case Study 1. The difference is the clockwise action of the load torque Ty around axis oy. Under this condition, the actions of the gyroscope internal torques and motions are as follows:
(i)
The load torqueTy generates the resistance torque Try, which acts in a counterclockwise direction around axis oy and the precession torque Tpy, which acts around axis oxin a counterclockwise direction. This coincides with the action of the load torque T
(ii)
The action of the torque produced by the gyroscope weightT generates the resistance Trx and precession Tpx torques. The resistance torque Trx acts in a clockwise direction around axis ox. The precession torque Tpx acts in a counterclockwise direction around axis oy as the torque TAll acting torques and motions of the gyroscope with one side support are illustrated in Figure2.Figure 2
Internal torques and motions when the load torqueTy acts in the clockwise direction around axis oy.The equations of the gyroscope torques and motions around axesox and oy are represented by the following expressions:
(18)−Jydωydt=−Ty+Tct.y+Tcr.y+Tin.x+Tam.xη,(19)Jxdωxdt=T+Tin.y+Tam.y−Tct.x−Tcr.x,where the sign (-) of Equation (18) indicates a clockwise movement. All other components are as specified above.The coefficientη represents the proportional increase of the precession torque Tp.x around axis oy. The coefficient η is expressed as the ratio of the sum of the precession Tp.y and load T torques to the precession Tp.y torque, which acts around axis ox. The expression η is obtained in a similar way as in Equation (1) and is represented by the following equation:
(20)η=Tp.y+TTp.y=1+TTp.y=1+Wgl2π2+9/9Jωωy,where η is the coefficient of the proportional increase of the value of the precession torque Tp.x acting around axis oy. All other parameters are as specified above.The coefficientη denotes the increase in precession torque Tp.x around axis oy when the value of the resulting torque Trst.x around axis ox also increases. Equation (20) demonstrates the following values of the coefficient η:
(i)
The absence of the load torqueT acting around axis ox means that the coefficient η=1.0 (i.e., there is no change in the values of the inertial torques acting around two axes)
(ii)
The action of the load torqueT means that the coefficient is η>1.0 (i.e., the values of the inertial torques acting around two axes increase)
(iii)
When the load torque of the gyroscope weight is equal to the precession torque (T=Tp.y), the coefficient η=2 (i.e., there is an increase in the value of the resulting resistance torque Trtst.y acting around axis oy)Equations for inertial torques—that is, Equations (17) and (20)—are substituted into Equations (18) and (19). Equations (21)–(23) are simplified and transformed in the same manner as Equations (4)–(6):
(21)−Jydωydt=−Ty+29π2+4Jωωy+Jωωx×1+Wgl2/9π2+1Jωωy,(22)Jxdωxdt=T+29π2+1Jωωy−89Jωωx,(23)ωx=4π2+17ωy,where all parameters are as specified above.Regarding the motion around axisox, the acting torques are represented by the combined action of the gyroscope weight T with the precession torques Tp.y and of the resistance torques Tr.x in the opposite direction. The torques acting around axis oy are represented by the action of the load torque Ty and the combined resistance torques Tr.y with precession torques Tp.x acting in the opposite direction. The following solution for Equations (21)–(23) is the same as presented in Section 2.1, Case Study 1. Equation (21) can be solved by substituting the defined equation and utilising transformations to yield the following equation:
(24)−Jydωydt=−Ty+29π2+4Jωωy+4π2+17Jωωy×1+Wgl2/9π2+1Jωωy,where all components are as specified above.


### 2.2.1. Working Example

The motions of the gyroscope with one side support are conducted under the clockwise action of the load torqueTy around axis oy. The sketch of the action of the torques and motions is presented in Figure 2. Equation (24) is the equation of the gyroscope motion around axis oy. The angular velocities of precessions around the two axes should be defined. Substituting the initial data of the gyroscope ([20], Table 1) into Equation (24) and making transformations yield the following equation:
(25)‐3.38437×10−4dωydt=−0.5×0.146×9.81×0.0355+29π2+4×0.5543873×10−4×ωωy+4π2+17×0.5543873×10−4×ωωy×1+0.146×9.81×0.03552/9π2+1×0.5543873×10−4×ωωy.Equation (25) is simplified. Then, the steps of solutions similar to the steps presented in Section 2.1, Case Study 1, are carried out. All comments related to the solutions of the equations are omitted.
(26)−0.102495748dωydt=264.651454335+ωωy,dωyωy+264.651454335/ω=−9.756502262ωdt,∫0ωydωyωy+264.651454335/ω=−9.756502262ω∫0tdt,lnωy+264.651454335ω0ωy=−9.756502262ωt0t,lnωy+264.651454335ω−ln264.651454335ω=−9.756502262ωt,1+ωy264.651454335/ω=e−9.756502262ωt,ωy=−264.651454335ω,where the sign (-) indicates a clockwise movement.
(27)ωy=−264.65145433510000×2π/60=−0.252723522rad/s=−14.479991°/s,ωx=4π2+17×0.252723522=14.273424613rad/s=817.806989°/s.The torqueTy acting around axis oy causes the slow clockwise rotation around axis oy and the intensive counterclockwise rotation of the gyroscope around axis ox. These rotations result in a counterclockwise torque around axis ox.


## 2.2.1. Working Example

The motions of the gyroscope with one side support are conducted under the clockwise action of the load torqueTy around axis oy. The sketch of the action of the torques and motions is presented in Figure 2. Equation (24) is the equation of the gyroscope motion around axis oy. The angular velocities of precessions around the two axes should be defined. Substituting the initial data of the gyroscope ([20], Table 1) into Equation (24) and making transformations yield the following equation:
(25)‐3.38437×10−4dωydt=−0.5×0.146×9.81×0.0355+29π2+4×0.5543873×10−4×ωωy+4π2+17×0.5543873×10−4×ωωy×1+0.146×9.81×0.03552/9π2+1×0.5543873×10−4×ωωy.Equation (25) is simplified. Then, the steps of solutions similar to the steps presented in Section 2.1, Case Study 1, are carried out. All comments related to the solutions of the equations are omitted.
(26)−0.102495748dωydt=264.651454335+ωωy,dωyωy+264.651454335/ω=−9.756502262ωdt,∫0ωydωyωy+264.651454335/ω=−9.756502262ω∫0tdt,lnωy+264.651454335ω0ωy=−9.756502262ωt0t,lnωy+264.651454335ω−ln264.651454335ω=−9.756502262ωt,1+ωy264.651454335/ω=e−9.756502262ωt,ωy=−264.651454335ω,where the sign (-) indicates a clockwise movement.
(27)ωy=−264.65145433510000×2π/60=−0.252723522rad/s=−14.479991°/s,ωx=4π2+17×0.252723522=14.273424613rad/s=817.806989°/s.The torqueTy acting around axis oy causes the slow clockwise rotation around axis oy and the intensive counterclockwise rotation of the gyroscope around axis ox. These rotations result in a counterclockwise torque around axis ox.


## 3. Results and Discussion

Mathematical models for the gyroscopic effects lead to equations for the motions of the gyroscope with one side support for the main external torque acting around the vertical axis. The models of the gyroscope motions around two axes are based on the actions of the external and internal torques that are generated by centrifugal, common inertial, and Coriolis forces. They are also based on the change in angular momentum. The new analytical approach to the gyroscopic problem demonstrates that the action of the external load torque around the vertical axis generates the precession torques that turn the gyroscope upward or downward around the horizontal axis. These motions depend on the direction of the rotation of the spinning rotor. Mathematical models for the motions of a gyroscope with one side support in which actions carried out by external and internal torques around two axes are confirmed by observations made during practical tests.


## 4. Conclusion

Previous mathematical models for gyroscopic effects contain many assumptions and simplifications that have not been validated in practice. This new study of gyroscopic effects examines the actions of the system of inertial torques generated by the known inertial forces of classical mechanics. The mathematical models for the motions of a gyroscope with one side support and the action of counterclockwise and clockwise load torques around the vertical axis explain the gyroscope’s upward and downward motion. The upward motion of the gyroscope is not an antigravity property, as was once thought, but is the result of the action of the precession torque generated by the load torque. The value of the precession torque is greater than the value of the torque produced by the gyroscope weight. The analytical models for the gyroscope’s upward and downward motions clearly describe the physics of such gyroscopic effects.



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*Source: 4197863-2019-12-20.xml*