Abstract
The many publications related to the gyroscope theory consider the action of the inertial torques on the spinning disc. All of them have simplified 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 effects 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 differ from the disc bar and ring and hence another mathematical model describes the motions of the gyroscopic devices. The value of gyroscopic effects 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.
1. Introduction
The gyroscopic effects in engineering mechanics are the most complex problems that are being unsolved for a long time [1–5]. 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 [6–10]. Scientists and researchers of our time continue to describe gyroscopic effects without success which can be seen in many publications each year [11–19]. 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 [20–22]. 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.
2. 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 with an angular velocity of in a counter-clockwise direction considered in Figure 1. The mass element is disposed on the circle of radius which is perpendicular to the axis of the spinning annulus.
The radius of disposing of the mass elements of the annulus is defined from the truncated sector with the small angle and the arcs of the external and internal radii and , 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 in Figure 1. The external torque applied to the spinning annulus generates several inertial torques (Figure 2).
(a)
(b)
The plane of the rotating centrifugal forces turns around axis along the diameter line and changes in the directions of centrifugal force vectors of the mass elements. The inclined plane of rotating centrifugal forces on angle around axis is presented in the plane . The action of the centrifugal force vectors is radial and their components produce the torques around axes and . The values of torques acting around axis are maximal at 90o and 270o and zero at 0o and 180o (Figure 2(a)). The values of torques acting around axis are maximal at 0o and 180o and zero at 90o and 270o (Figure 2(b)). The integrated product of the components’ vector of the centrifugal forces and their radii relative to axes and give the torques acting around two axes. The torque acting around axis resists the action of the external torque . The torque acting around axis 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 is the centrifugal torque generated by the mass element; is the component of the centrifugal force; and are the distance from the mass element to axes and , respectively; is the mass element of the annulus disposed on the circle of the radius ; is the radial acceleration; 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 : (i)Around axis (ii)Around axis where is the centrifugal force of the mass element ; in which is the mass of the annulus; is the sector’s angle of the mass element’s disposition; is the angle of the mass element’s disposition; is the angle of turn for the annulus plane, is the trigonometric identity for the small values of the angle.
The radius of the circle of disposing of the mass elements of the annulus is defined from the truncated sector with the small angle and the arcs of the external and internal radii and , respectively (Figure 1). The radius is expressed by the equation of the centroid of the truncated sector where and are the area of the sectors of radii and and and are the radii of the mass elements disposition, respectively; is the area of the truncated sector.
Substituting the expressions of 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 and are the distance from the mass element of the annulus relative to axes (Figure 2(a)) and to axis (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 and . The known integrated equation calculates the disposition of the centroid (point A of Figure 2(a) and point B of Figure 2(b)) [6–9]. where , is the centroid for the semi-circle , and is or .
Substituting Equations (4) and (5) and other components into Equation (8) and transformation yield the following expression. (i)About axis (ii)About axis where the expression is constant of Equations (9) and (10); the expressions and are a trigonometric identity, and other parameters are as specified above.
Equations (6) and (7) are presented in differential and integral forms. Substituting expressions of Equations (9) and (10) into Equations (6) and (7), replacing and by the integral expressions with defined limits, respectively, the following equations emerge: (i)About axis (ii)About axis
Solutions of integral Equations (11) and (12) yield the following: (i)About axis (ii)About axis
that gave rise to the following: (i)About axis (ii)About axis 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 and . The inertial torque depends on the variable angle that expresses the angular velocity of the annulus rotation about axis per time . The differential equation expresses the change in torque per time where is the time taken relative to the angular velocity of the annulus.
The differential of time is ; the expression is the angular velocity of the spinning annulus around axis . 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 , and its left and right sides about axis . Then, the total torque acting about axes and is increased double. where is the annulus moment of inertia, and other components are as specified above.
3. 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 of the annulus that turns on the angle around axis . The turn of the annulus around axis 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 , , and for the small angle (Figure 3). The maximal changes of the velocity vectors are on the line of axis . The tangential velocity whose vector is parallel to axis , i.e., on 90o and 270o does not change. The changes in tangential velocity vectors are presented by the components that are parallel to the annulus axle .
The torque generated by the Coriolis force of the rotating mass element is expressed by where is the torque generated by Coriolis force of the annulus mass element ; is the Coriolis acceleration along with axis ; is the distance from the mass element to axis ; the sigh (-) means the action in the clockwise direction around axis , and the other components are represented in Equation (2).
The differential form of changes in tangential velocity vectors is and change in the angle of rotation around axis is .
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 is point C of Figure 3, which is defined by Equation (8).
Substituting Equation (26) into Equation (25), replacing 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 acts on the upper and lower sides of the annulus. Multiplying Equation (29) by two yields the full expression of : where is the annulus moment of inertia; the sign (-) expresses the clockwise direction.
4. 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 [1–5]. The motion of the spinning annulus around axis (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 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 generates all inertial torques that depend on the mass moment of the annulus’s inertia , its angular velocity , and the angular velocity of the spinning annulus about axi.
The action of all torques around axes and 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 and 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 .
Substituting expressions of the defined torques into Equation (31) yields the following: where the signs (-) and (+) mean the clockwise and counter-clockwise 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 =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.
5. 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 3000 rpm around the axle and precesses with an angular velocity of 0.06 rpm. Determine the values of the torques generated by the centrifugal and Coriolis force, the change in the angular momentum, and the angular velocity of precession. Substituting the data into equations of Table 2 and transformation yields the following results. (i)The torque generated by the centrifugal forces (ii)The torque generated by Coriolis forces (iii)The torque of the change in the angular momentum (iv)The angular velocity of precession
For the disc-type rotor (Table 1), the dependency of the angular velocities is obtained if the =0 in Table 2. Substituting data of the working example and computing yield:
The comparative results of the dependency of the angular velocities rotation around axes of motions for the annulus and disc-type rotor show their big differences. The radius of disposition of the distributed mass of the annulus is bigger than for the disc-type rotor. Analysis of them yields the spinning annulus generates bigger values of the inertial torques and the dependency of the angular velocities than the spinning disc-type rotor.
6. Results and Discussion
Gyroscopic effects are remaining as sophisticated mathematical problems because the inertial torques generated by the rotating mass depend on the geometry of the spinning objects. The practice of engineering designs of the gyroscopic devices shows different geometry of their main components which are the rotors. The value of the inertial torques generated by the spinning rotors depends on their technical data. The working example shows the spinning annulus generates bigger values of the inertial torques and the dependency of the angular velocities than the disc-type rotor. This result gives the tip for the practitioners to design gyroscopic devices with a high value of the inertial torques and the angular velocity of precession. Practitioners can design gyroscopic devices with a necessary output of technical data that depends on the geometry of the spinning rotor. This result opens a new analytical approach to finding optimal designs of the spinning rotor that respond to the quality of gyroscope work.
7. Conclusion
Aerospace and engineering industries try to solve gyroscopic problems and find optimal designs for gyroscopic devices that give the necessary technical data. The new theory of gyroscopic effects for rotating objects opens new methods for computing the technical data for gyroscopic devices. The method for deriving mathematical models for the gyroscopic effects enables finding the output data of gyroscopic devices that depend on the geometry of the main unit which is the spinning rotor. Optimization of the rotor’s design by criteria of the aerospace industry is a new direction for investigations in engineering. The fundamental principles of the theory of gyroscopic effects for rotating objects and methods enable solving them in engineering. The known numerical modeling for gyroscopic effects with expensive software is going past and engineering science receives new analytical methods.
Data Availability
The authors declare that the data supporting the findings of this study are available within the article. R. Usubamatov wrote the mathematical models of inertial torques. S. Kapayea corrected the text and references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.