Gyroscopic Torques Acting on Crushing Mill

Gyroscopic effects manifest at numerous rotating objects in engineering. Correct computing of and gyroscopic properties enables for the functioning of the gyroscopic devices in engineering. Since the Industrial Revolution published many gyroscope theories as well as many approaches and mathematical solutions that describe the gyroscope properties [1-4]. Numerous publications described the gyroscope effects and applications in engineering [5,6]. All of them describe gyroscope properties only in terms of the law of conservation of energy and the angular momentum. Nevertheless, the nature of gyroscope effects is more complex and known theories do not match the practice of gyroscopic devices [7-9]. Therefore, researchers continue to find true mathematical models of gyroscopic effects [10-14]. New research in the area of the gyroscope theory gives the new mathematical models for inertial forces acting on a gyroscope [15,16]. These publications demonstrate that on rotating objects act the several inertial forces of their mass-elements that express the resistance and precession torques. The centrifugal and Coriolis forces of the rotating masselements result in the resistance torques. The common inertial forces and the change in the angular momentum result in the precession torques. Table 1 represents the equations of the inertial torques acting on rotating objects. The action of the new inertial and external torques of the rotating objects should be considered for the mathematical modeling of the work of mechanism in engineering (Table 1).


Introduction
Gyroscopic effects manifest at numerous rotating objects in engineering. Correct computing of and gyroscopic properties enables for the functioning of the gyroscopic devices in engineering.
Since the Industrial Revolution published many gyroscope theories as well as many approaches and mathematical solutions that describe the gyroscope properties [1][2][3][4]. Numerous publications described the gyroscope effects and applications in engineering [5,6]. All of them describe gyroscope properties only in terms of the law of conservation of energy and the angular momentum.
Nevertheless, the nature of gyroscope effects is more complex and known theories do not match the practice of gyroscopic devices [7][8][9]. Therefore, researchers continue to find true mathematical models of gyroscopic effects [10][11][12][13][14]. New research in the area of the gyroscope theory gives the new mathematical models for inertial forces acting on a gyroscope [15,16]. These publications demonstrate that on rotating objects act the several inertial forces of their mass-elements that express the resistance and precession torques. The centrifugal and Coriolis forces of the rotating masselements result in the resistance torques. The common inertial forces and the change in the angular momentum result in the precession torques. Table 1 represents the equations of the inertial torques acting on rotating objects. The action of the new inertial and external torques of the rotating objects should be considered for the mathematical modeling of the work of mechanism in engineering ( Table 1). Where i is the index for axis; J = (mR 2 /2) is the rotor's mass moment of inertia around spin axis; m is the rotor's mass; R is the external radius of the rotor; ωi is an angular velocity of precession of a spinning rotor around axis i and ω is an angular velocity of a spinning rotor; other components are as specified in Table 1. Analysis of the internal torques that represented in Table   1 demonstrates that their values of are different but proportionally depend on the mass moment of inertia of a rotor and its angular velocity as well as on the angular velocity of precession. These torques represent fundamental principles of the gyroscope theory.
New mathematical models for the inertial torques can describe all gyroscope effects that manifest any rotating objects. This work represents the computing of the torques and power for the crushing mill based on the mathematical models for the inertial torques generated by the flywheel. The mass moment of the flywheel inertia is

Methodology and Working Example
where r = 0.18 m is the radius of the conical flywheel at the point of the center mas, f = 0.2 is crashing coefficient, other parameters are as specified above. The crushing pendulum mill should be equipped with the electric motor of the power 8.7 kW.

Results and Discussion
New mathematical models for the inertial torques acting

Conclusion
The numerous mechanisms in engineering contain rotating objects which manifest gyroscopic effects which analytical solutions for a long time represented the unsolvable problem. New analytical approach to the inertial torques acting on the rotating objects finely resolved all problems in the area of the rotating objects. Several inertial forces generated by the mass of the spinning objects act simultaneously and interdependently on the rotating objects that demonstrate the gyroscopic effects. The new mathematical models for the inertial forces acting on the rotating objects should be used for engineering computing of the working parameters of different mechanisms and machines. The action of the inertial torques generated by the masses of rotating objects demonstrated on the work of the crushing mill.