Inertial Forces and Torques Acting on a Spinning Annulus

The many publications related to the gyroscope theory consider the action of the inertial torques on the spinning disc. All of them have simpli ﬁ ed expressions for the inertial torques and mathematical models of the gyroscope motions do not validate by practice. Recent research in the theory of the gyroscopic e ﬀ ects for rotating objects solved problems with mathematical models for interrelated inertial torques generated by the spinning disc, bar, and ring and their motions. Practitioners of engineering designed gyroscopic devices with spinning rotors whose geometry can be an annulus or similar designs. Such spinning annulus generates inertial torques whose expressions di ﬀ er from the disc bar and ring and hence another mathematical model describes the motions of the gyroscopic devices. The value of gyroscopic e ﬀ ects of the devices with spinning annulus is bigger than for the disc-type rotor. This manuscript presents mathematical models for the inertial torques generated by the spinning annulus and the interrelated angular velocities of the gyroscopic devices about axes of rotation.


Introduction
The gyroscopic effects in engineering mechanics are the most complex problems that are being unsolved for a long time [1][2][3][4][5]. Beginning from the Industrial Revolution, scientists yield only partial analytical solutions but did not solve the entire gyroscopic effects. For practical applications were worked out numerical models with the expensive software for gyroscopic effects. The physics of the gyroscopic effects of the simple spinning disc remained unexplained and mysterious for a couple of centuries [6][7][8][9][10]. Scientists and researchers of our time continue to describe gyroscopic effects without success which can be seen in many publications each year [11][12][13][14][15][16][17][18][19]. Recent investigations of the gyroscopic effects showed their nature is more complex and based on several physical principles that were discovered at different times. Scientists of 18-19 centuries could not solve gyroscopic effects in principle because the concept of mechanical energy conservation was formulated at the beginning of twenty century. From this time, researchers have all the analytical tools for formulation of the gyroscope theory but did not do it.
The latest studies of the physics of the gyroscopic effects yield mathematical models for the inertial torques and the interrelated dependency of the angular velocities of the spinning disc motions about axes of rotation. The solution to gyroscopic effects is based on the principle of mechanical energy conservation [20][21][22]. The method of deriving mathematical models for inertial torques shows the action of the system of the centrifugal, Coriolis forces generated by the rotating mass, and the change in the angular momentum of the spinning disc. The dependency of the angular velocities of the spinning disc around axes of motions interrelates the inertial torques. Table 1 presents the expressions of inertial torques acting on the spinning disc and the dependency of the angular velocities of it's motions.
The practice tests validate the action of the system of the interrelated inertial torques on the spinning disc. Practitioners should use the derived method for the modeling of the gyroscopic effects for any designs of the spinning objects [22]. The mathematical models for the inertial torques generated by the centrifugal and Coriolis forces and the change in the angular momentum acting on the spinning flat annulus and its interrelated motions around two axes are present as the contribution of the manuscript.

Centrifugal Forces and Torques Acting on a Spinning Annulus
The method for the analytical solution for all inertial torques acting on the spinning disc is described in several publications [21,22]. The inertial torques are generated by the distributed mass elements of the spanning disc disposed on the circle of 2/3 of its radius. This method can be applied to rotating objects of different forms. For the annulus, the radius of the disposition of the distributed mass elements on the circle should be defined. The rotating mass elements produce the plane of centrifugal forces of the spinning annulus around axis oz with an angular velocity of ω in a counter-clockwise direction considered in Figure 1. The mass element m is disposed on the circle of radius r which is perpendicular to the axis of the spinning annulus. The radius r of disposing of the mass elements of the annulus is defined from the truncated sector with the small angle Δδ and the arcs of the external and internal radii R e and R i , respectively. The rotating mass elements generate centrifugal forces. The action of the external torque to the spinning annulus has manifested the inclination of its plane with the rotating centrifugal forces and the turns of the annulus around axes. These motions are presented at the Cartesian 3D coordinate system Σoxyz in Figure 1. The external torque applied to the spinning annulus generates several inertial torques (  (Figure 2(b)). The integrated product of the components' vector of the centrifugal forces f ct:z and their radii relative to axes ox and oy give the torques T ct acting around two axes. The torque acting around axis ox resists the action of the external torque T. The torque acting around axis oy turns the spinning annulus in the counter-clockwise direction.
The mathematical models for the inertial torques generated by the centrifugal forces of the mass elements are expressed as follows: where ΔT ct is the centrifugal torque generated by the mass element; f ct:z is the component of the centrifugal force; y m and x m are the distance from the mass element to axes oy and ox, respectively; m is the mass element of the annulus disposed on the circle of the radius r; a z = rω 2 is the radial acceleration; ω is the angular velocity of the spinning annulus; the signs (+) and (-) mean the counter-clockwise and clockwise direction, respectively. The following equations express the component of the centrifugal force acting along axis oz: where f ct = mrω 2 = ðMrω 2 /2πÞΔδ is the centrifugal force of the mass element m; m = ðM/2πrÞΔδr = ð M/2πÞΔδ in which M is the mass of the annulus; Δ δ is the sector's angle of the mass element's disposition; α is the angle of the mass element's disposition; Δγ is the angle of turn for the annulus plane, sinΔγ = Δγ is the trigonometric identity for the small values of the angle.
The radius r of the circle of disposing of the mass elements m of the annulus is defined from the truncated sector with the small angle Δδ and the arcs of the external and internal radii R e and R i , respectively ( Figure 1). The radius r is expressed where A e = ðπR 2 e × × ΔδÞ/2π and A i = ðπR 2 i × × ΔδÞ/2π are the area of the sectors of radii R e and R i and ð2/3ÞR e and ð2/3ÞR i are the radii of the mass elements disposition, respectively; A ts = ðπðR 2 e − R 2 i Þ × ΔδÞ/2π is the area of the truncated sector.
Substituting the expressions of f ct:z of Equations (3)-(5) into Equations (1) and (2) and transformation yield the expressions of the inertial centrifugal torques produced by the mass element.
where y m = r sin α and x m = r cos α are the distance from the mass element of the annulus relative to axes ox (Figure 2(a)) and to axis oy (Figure 2(b)), respectively; and the other components are as specified above.
The centrifugal torques are distributed on the circle where the mass elements of the annulus are located (Figures 2(a) and 2(b)). The action of the torques is defined by a concentrated load at the centroid points at the semi-circles along axes oy and ox. The known integrated equation calculates the disposition of the centroid (point A of Figure 2(a) and point B of Figure 2(b)) [6][7][8][9]. where Substituting Equations (4) and (5) and other components into Equation (8) 3 Advances in Mathematical Physics (ii) About axis oy Figure 2: Schematic of acting centrifugal forces, torques, and motions around axis ox (a) and oy (b) of the spinning annulus.
Equations (6) and (7) are presented in differential and integral forms. Substituting expressions of Equations (9) and (10) into Equations (6) and (7), replacing sin α = Ð π 0 cos αdα and cos α = − Ð π 0 sin αdα by the integral expressions with defined limits, respectively, the following equations emerge: (ii) About axis oy Solutions of integral Equations (11) and (12) yield the following: (i) About axis ox T ct (ii) About axis oy T ct that gave rise to the following: (i) About axis ox (ii) About axis oy where all components are as specified above.
Equations (15) and (16) are almost identical except for the signs (±) of counter and clockwise action around axes ox and oy. The inertial torque T ct depends on the variable angle γ that expresses the angular velocity ω x of the annulus rotation about axis ox per time t. The differential equation expresses the change in torque T ct per time where t = α/ω is the time taken relative to the angular velocity of the annulus. The differential of time t is dt = dα/ω; the expression d γ/dt = ω x is the angular velocity of the spinning annulus around axis ox. Substituting the defined components into Equation (17) and transforming yield: Separation of the variables of Equation (18), transformation, and presentation by the integral form with defined limits yield: Solution of Equation (19) yields The centrifugal torques act on the upper and lower sides of the annulus about axis ox, and its left and right sides about axis oy. Then, the total torque T ct acting about axes ox and oy is increased double.

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where J = ð1/2ÞMðR 2 e + R 2 i Þ is the annulus moment of inertia, and other components are as specified above.

Coriolis Forces and Torques Acting on a Spinning Annulus
The Coriolis acceleration and force generated by mass elements are revealed when the spinning annulus turns around axis perpendicular to its axle. The integrated Coriolis force produced by the rotating mass elements of the annulus generates the torque counteracting the external torque [10]. Figure 3 shows the rotation of the mass element m of the annulus that turns on the angle Δγ around axis ox. The turn of the annulus around axis ox changes the direction of the tangential velocity vectors of mass elements. The change in the tangential velocity of mass elements produces the acceleration in which a product with a mass yields the Coriolis forces of the mass elements, where ΔV = V sin Δγ, V = r cos α × ω, and sin Δγ = Δγ for the small angle ( Figure 3). The maximal changes of the velocity vectors V * are on the line of axis ox. The tangential velocity V whose vector is parallel to axis ox, i.e., on 90 o and 270 o does not change.
The changes in tangential velocity vectors are presented by the components V z that are parallel to the annulus axle oz. The torque generated by the Coriolis force of the rotating mass element is expressed by where ΔT cr is the torque generated by Coriolis force f cr of the annulus mass element m; a z is the Coriolis acceleration along with axis oz; y m = r sin α is the distance from the mass element to axis ox; the sigh (-) means the action in the clockwise direction around axis ox, and the other components are represented in Equation (2). The differential form of changes in tangential velocity vectors is dV/dt = a z and change in the angle of rotation around axis ox is dγ/dt = ω x .
Then, the Coriolis acceleration is: The Coriolis force of the mass element is presented: Then, the Coriolis torque is: where all parameters are as specified above. The centroid for the torque ΔT cr is point C of Figure 3, which is defined by Equation (8). Figure 3: Schematic of the acting forces, torques, and motions of the spinning annulus. 6 Advances in Mathematical Physics Substituting Equation (26) into Equation (25), replacing cos α = Ð π 0 − sin αdα by the integral expression with defined limits, and presenting other components by the integral forms, the following equation emerges: Solution of integral Equation (27) yields: that gave rise to the following: where all components are as specified above. The inertial torque T cr acts on the upper and lower sides of the annulus. Multiplying Equation (29) by two yields the full expression of T cr : where J = ð1/2ÞMðR 2 e + R 2 i Þ is the annulus moment of inertia; the sign (-) expresses the clockwise direction.

Attributes of Inertial Torques Acting on a Spinning Annulus
The load torque applied to the spinning annulus produces the system of the inertial torques generated by the rotating mass [22]. Among them is the change in the angular momentum of a spinning disc, which is Euler's fundamental principle of gyroscope theory [1][2][3][4][5]. The motion of the spinning annulus around axis oy (Figure 3) manifests the change in the annulus angular momentum in the counter-clockwise direction which is called precession. The expression of the change in the angular momentum is T am = Jωω x where all components are presented above. The system of the inertial torques produces the resistance and precession torques and motions of the spinning annulus around axes of the rotation. Table 2 represents expressions of the inertial torques generated by pseudo forces of the spinning annulus. The external torque T generates all inertial torques that depend on the mass moment of the annulus's inertia J, its angular velocity ω, and the angular velocity ω x of the spinning annulus about axis ox.
The action of all torques around axes ox and oy on the spinning object is shown in Figure 4 for the given symmetrical disposing of its supports [22]. The interrelated action of the inertial torques is considered for the horizontal disposition of the spinning object.
The action of the inertial torques of the spinning annulus around axes ox and oy expresses analytically the equality of their mechanical energies that enables for deriving of the dependency of its interrelated angular velocities around axes of motions (Figure 4) [22].
where expressions of the inertial torques are presented by Equations (21) and (30), and T am = Jωω x .

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Substituting expressions of the defined torques into Equation (31) yields the following: where the signs (-) and (+) mean the clockwise and counterclockwise directions of the action of the inertial torques around two axes, respectively.
Simplification and transformation of Equation (32) yield the following: The expressions of the inertial torques and their action, and the dependency of interrelated angular velocities of the spinning annulus around axes are presented in Table 2.
The expressions of torques and the dependency angular velocities (Table 2) formulate the mathematical models for the motions of the gyroscopic devices with the annulus rotor. Comparative analysis of formulas for the spinning annulus shows when its internal radius R i =0, all of them are converted to the formulas for the disc-type rotor presented in Table 1. This fact validates the mathematical correctness of the formulas. The values of formulas in Table 2 give results bigger than the formulas in Table 1.

Working Example
The annulus has a mass of 1.0 kg and the external and internal radii of 0.1 m and 0.06 m, respectively. The annulus rotates at    Dependency of angular velocities of spinning disc rotations about axes