Contemporary engineering widely uses the gyroscopes, which are main units for navigation and control systems work on a principle of maintaining the axis of a spinning rotor in a space. This gyroscope property formulated by mathematical models based on Euler’s the principle of the change in the angular momentum. Nevertheless, the actual acting forces and motions of the gyroscopes do not match theoretical approach. This circumstance commits researchers to find true mathematical solutions. New research results in the area of gyroscopic devices have demonstrated that the gyroscope effects have an origin that is more complex than presented in publications. Investigations manifested that rotating mass of the gyroscopes generates several inertial torques based on the action of the centrifugal, common inertial and Coriolis forces as well as the change in the angular momentum. These torques are interrelated and acted at one time around two gyroscope axes, and represented the internal resistance and precession torques. Practically, the gyroscopic devices run with the action of the frictional forces on the supports and pivots that have the effect upon on gyroscope motions. This work represents an analytic solution for motions of the gyroscope with one side support based on the action of the load, internal and frictional torques. A mathematical model for the gyroscope motions under the action of the external and internal torques is validated by practical tests.

Gyroscope, Theory, Prorpety, Test, Torque.

Contemporary engineering widely uses the gyroscopes, which are main units for navigation and control systems work on a principle of maintaining the axis of a spinning rotor in a space. This gyroscope property formulated by mathematical models based on Euler’s the principle of the change in the angular momentum. Nevertheless, the actual acting forces and motions of the gyroscopes do not match theoretical approach. This circumstance commits researchers to find true mathematical solutions. New research results in the area of gyroscopic devices have demonstrated that the gyroscope effects have an origin that is more complex than presented in publications. Investigations manifested that rotating mass of the gyroscopes generates several inertial torques based on the action of the centrifugal, common inertial and Coriolis forces as well as the change in the angular momentum. These torques are interrelated and acted at one time around two gyroscope axes, and represented the internal resistance and precession torques. Practically, the gyroscopic devices run with the action of the frictional forces on the supports and pivots that have the effect upon on gyroscope motions. This work represents an analytic solution for motions of the gyroscope with one side support based on the action of the load, internal and frictional torques. A mathematical model for the gyroscope motions under the action of the external and internal torques is validated by practical tests.

Gyroscope, Theory, Prorpety, Test, Torque.

More than two centuries ago famous mathematician L. Euler first lay out the mathematical foundations for the gyroscope theory in his work on the dynamics of rigid bodies. At those time other brilliant scientists such as I. Newton, J-L. Lagrange, L. Poinsot, J.L.R. D’Alembert, P-S. Laplace, L. Foucault, and others analyzed and interpreted the gyroscope property of maintaining the axis of a spinning rotor in a space. The industrial revolution and following engineering activity in the twentieth century developed the applied theory of gyroscopic devices and systems [1-5]. Since those times, probably tons of manuscripts and several dozens of gyroscope theories were published. These publications explain the gyroscope effects in terms of the conservation of kinetic energy and by the action of the change in the angular momentum of the spinning rotor.

Today, gyroscopic devices are primary units for navigation and control systems that are widely used in engineering industries [6,7]. The textbooks of classical mechanics contain the chapters dedicated to the gyroscope theory [8-10]. Nevertheless, the known textbooks and manuscripts do not explain fully and adequately the physics of acting forces on the gyroscope and its motions in a space [11,12].

For practical application, the forces and motions of gyroscopes formulated by severe numerical mathematics based on Lagrangian dynamics that is solved with computer’s software [13]. These theories and other publications explain gyroscope effects based on deductions, assumptions and simplifications [14-16]. Unexplained gyroscopic effects, researchers intuitively pointed on the action of the inertial forces of the rotating disc. Nevertheless, this true intuition did not formulate the mathematical models of the action of inertial forces on a gyroscope, except only the principle of the change in the angular momentum [17,18]. From this, researchers created several artificial terms like gyroscope resistance, gyroscope effects, gyroscope couple, and so on that contradict known terms of classical mechanics. These unsolved gyroscope problems have represented the challenge for researchers to formulate the mathematical models for acting forces and motions for gyroscopic devices [19,20].

New investigations of the physical principles of gyroscopic devices demonstrate that the origin of gyroscope properties and effects is more complex than seemed. The inertial forces generated by the spinning mass elements and center mass of the rotor, produce the several torques based on the action of centrifugal, common inertial and Coriolis forces as well as changes in the angular momentum. The mathematical models and physical principles of these inertial torques are well described and present the basis for all gyroscope effects and properties [21].

New analytical investigations of the gyroscope’s internal torques demonstrate that the actions of them are interrelated and occur simultaneously around two gyroscope axes. The action of the change in the angular momentum represents the small component in gyroscope properties. The results of new studies make clear why the known gyroscope theories contain too many simplifications and assumptions.

New mathematical models for gyroscope motions accurately describe gyroscope effects. Moreover, these models discover new gyroscope properties and interpret known one. The basic mathematical models for motions of the gyroscope validated by practical tests that conducted on Super Precision Gyroscope (Brightfusion Ltd, Abbeymead, UK). These new fundamental principles for gyroscope theory can solve all gyroscope problems and represent new challenges for future studies of gyroscopic devices [22-25].

In engineering, most of these gyroscopic devices run with the frictional forces that are results of the action of external and internal torques on the supports and pivots. The frictional torques are external ones, which values can be commensurable with the values of the resulting torques acting on the gyroscope. Practice demonstrates the frictional forces have an effect upon on gyroscope motions, which velocities of rotation around axes in increased. At first sight, this phenomenon of the gyroscope motions contradicts to the rules of classical mechanics that state the velocity of the object decreases with the action of the frictional force. This work represents a mathematical model for motions of a gyroscope with one side support under the action of the external and internal torques, and explain the physics of the unusual properties.

Recent analytic investigations of the gyroscope motions have presented new mathematical models of inertial forces acting in a gyroscope. The action of the external load on a gyroscope generates several inertial resistance and precession torques, whose physics well described and represented in (Table 1), [23]. Research demonstrated that the centrifugal and Coriolis forces generated by the mass elements of the spinning rotor produce the resistance torque. The common inertial forces and the change in the angular momentum generate the precession torque. This resistance and precession torques act perpendicular to each other around their axes and simultaneously interrelated [13]. The action of the load and inertial torques on the supports and pivots of the gyroscopes generate the frictional forces. The frictional torques are considered the external load that has an appreciable influence on the motions of gyroscopic devices.

Table 1 contains the several symbols that marked by subscript signs indicating the axis of action, where for instance T^rx is the resistance torque acting around axis ox, ω_y is the angular velocity of precession around axis oy, etc., J is the mass moment of inertia of the gyroscope’s rotor around the axle. ω_i is the angular velocity of the gyroscope around axis i and ω is the angular velocity of a spinning rotor.

Mathematical models for motions of the gyroscope with one side support consider the interrelated action of external and inertial torques on the supports and pivot around two axes. The values of any external torques generate proportionally the values of inertial one acting around two axes. The action of the internal torques around axes expresses the internal kinetic energies of the spinning rotor that are equal along axes [23]. Hence, any change in the value of the internal torques around one axis leads to the reciprocal change in the value of the internal torques around another axis. The analytic models for the motions of the gyroscope are considered on a stand represented in (Figure 1). A detailed picture of the gyroscope’s stand and its geometrical parameters is shown in (Figure 2). Technical data of the test stand with Super Precision Gyroscope, “Bright fusion LTD” is represented in (Table 2).

Table 2 contains the following symbols: Jx = Jy = (MR²/4) + Ml² is the mass moment of the gyroscope’s inertia around axes ox and oy respectively; R is the external radius of the gyroscope; l is the overhang of the centre of gravity of the gyroscope from the centre beam; M is the mass of the gyroscope; Jrx = Jry = (mrR² _c/4) is rotor’s mass moment of inertia around axis ox and oy; Rc is the conditional radius of the rotor; J = (mrR² _c/2) is rotor’s mass moment of inertia around axis oz. mf is the mass of the gyroscope’s frame; rf is the radius of the frame; The computed mass moments of the gyroscope’s components inertia around axes ox and oy is presented in [23].

The gyroscope was assembled on the centre beam b with the ability freely rotate around axis ox on the spherical journals of sliding bearings B and D. The centre beam b is located on the two vertical arms of the frame, which is assembled on horizontal bar bs (Figures 1 and 2). The bar has the ability to rotate around the pivot C (axis oy) on the platform.

The mathematical model of motions for the gyroscope with one side support is formulated for the common case when its axle is inclined on the angle γ. The basis of this model is described in [23] and used with the modification that includes the action of the frictional torques on the supports and pivot. The change in the values of the external and inertial torques is expressed by the correction coefficient η.

The action of the frictional torques around axes ox and oy decreases the values of the torque generated by the gyroscope weight and the precession torques respectively. The resistance torques acting around axis oy are depended on the precession torques. Hence, the value of the inertial torques acting around axis ox also is decreased proportionally. The changes in values of inertial torques acting around two axes are equal because the inertial torques are expressed the inertial kinetic energies of the spinning rotor that are equal along axes. Then, the mathematical models for motions of the gyroscope are represented by the following Euler’s differential equations:

$$J_{Ex}\frac{d{\omega }_x}{dt}=T\cos \gamma +T_{x.ct.y}-T_{f.x}^{}-(T_{ct.x}+T_{cr.x}+T_{in.y}\cos \gamma +T_{am.y}\cos \gamma )\eta \left(1\right)$$

$$J_{Ey}\frac{d{\omega }_y}{dt}=(T_{in.x}\cos \gamma +T_{am.x}\cos \gamma -T_{ct.y}\cos \gamma -T_{cr.y}\cos \gamma )\eta -T_{y.cr.y}-T_{f.y}\left(2\right)$$

$${\omega }_y=-\left(\frac{2{\pi }^2+8}{\cos \gamma }+2{\pi }^2+9\right){\omega }_x\left(3\right)$$

where ω_x and ω_y is the angular velocity of the gyroscope around axes ox and oy, respectively; T_ct.x, T_ct.y, T_cr.x, T_cr.y, T_in.x, T_in.y, T_am.x and T_am.y are inertial torques generated by the centrifugal, Coriolis, inertial forces and the change in the angular momentum acting around axes ox and oy, respectively (Table 1); η is the coefficient of the decreasing of the inertial torques due to the action of the frictional torques on gyroscope’s pivot; other components are as specified above.

Analysis of the torques acting on the gyroscope demonstrates the following peculiarities. The external load torque T that is produced by the gyroscope weight W generates the frictional and inertial torques acting around axes ox and oy (Figure 2). These torques are represented by the following components:

The peculiarity of the action of the gyroscope’s inertial torques is as follows:

The running gyroscope produces several external and inertial torques acting around two axes that are expressed by the following equations:

(a) The weight of the gyroscope inclined on the angle γ produces the torque T acting around axis ox in the counter clockwise direction:

$$\text{T = Mgl}\left(4\right)$$

where M is the gyroscope mass; g is the gravity acceleration, l is the distance between the centre mass of the gyroscope and axis of the centre beam (Figure 2), γ is the angle of the axle inclination.

(b) The torque generated by the centrifugal force of the rotating gyroscope centre mass around axis oy acting in the counter clockwise direction around axis ox:

$$T_{x.ct.y}=Ml(\cos \gamma ){\omega }_y^2(l\sin \gamma )=M{l}^2{\omega }_y^2\cos \gamma \sin \gamma \left(5\right)$$

where ω_y is the angular velocity of the gyroscope around axis oy and other components are as specified above.

(c) The centrifugal force of the rotating gyroscope centre mass around axis ox acting along axis oz (Figure 2)

$$F_{z.ct.x}=Ml{\omega }_x^2\left(6\right)$$

where ω_x is the angular velocity of the gyroscope around axis ox and other components are as specified above.

(d) The frictional torques acting on the supports B and D is represented by the following equation:

$$T_{f.x}=T_{f.x.E.x}+T_{f.x.cr.y}++T_{f.z.ct.x}+T_{f.x.ct.y}\left(7\right)$$

Where:

(e) The frictional torque acting on the supports B and D (Fig. 2) in the clockwise direction around axis ox generated by the centrifugal force of the rotating gyroscope centre mass around axis oy:

$$T_{f.x.ct.y}=Ml{\omega }_y^2\cos \gamma \left(\frac{df}{2\cos \delta }\right)\left(8\right)$$

where d is the diameter of the centre beam, f is the frictional sliding coefficient, δ = 45º is the angle of the cone of sliding bearing of the supports (Figure 2) and other components are as specified above.

(f) The frictional torque acting on the supports B and D in the clockwise direction around axis ox generated by the centrifugal force of the rotating gyroscope centre mass around axis ox:

$$T_{f.z.ct.x}=Ml{\omega }_x^2\frac{df}{2\cos \delta }\left(9\right)$$

where all components are as specified above.

(g) The frictional torque acting on the supports B and D in the clockwise direction around axis ox generated by the Coriolis force of the rotating gyroscope centre mass around axes oy and ox:

$$T_{f.x.cr.y}=Ml{\omega }_y^{}{\omega }_x^{}\sin \gamma \left[\frac{l}{h}\frac{df}{2\cos (\delta -\tau )}\right]\left(10\right)$$

where h = 56.925 mm is the distance between the centre of the gyroscope and the support (Figure 2); τ = arctan (l/ [c/2]) = 38.581º is the angle of the action of Coriolis force on the sliding bearing and other components are as specified above.

(h) The frictional torque acting on the sliding bearing of the supports B and D in the clockwise direction around axis ox generated by the gyroscope weight with the centre beam E:

$$T_{f.x.E.x}=(Eg-Ml{\omega }_x^2\sin \gamma )\frac{df}{2\cos \delta }\left(11\right)$$

where E is the mass of the gyroscope with the centre beam (Table 2), 2 Fy.ct.x = Mlωx sinγ is the centrifugal force of the rotating gyroscope centre mass around axis ox acting along axis oy (Figure 2) and other components are as specified above.

The expression of the total frictional torque (Equations (8)-(11)) acting on the supports B and D is represented by the following equation:

$$\begin{array}{l}{T}_{f.x}=(Eg-Ml{\omega }_x^2\sin \gamma )\frac{df}{2\cos \delta }+Ml{\omega }_y^{}{\omega }_x^{}(\sin \gamma )\frac{l}{h}\frac{df}{2\cos (\delta -\tau )}+\\ Ml{\omega }_x^2\frac{df}{2\cos \delta }+Ml{\omega }_y^2(\cos \gamma \sin \gamma )\frac{df}{2\cos \delta }\left(12\right)\end{array}$$