Classical Mechanics

Section 1.1 is devoted to the study of dynamical processes in electric circuits. It includes derivations of the constitutive relations of elements of electric circuits (capacitors, inductors) and describes current and voltage sources and Kirchhoff’s law. Section 1.2 deals with dynamical processes in mechatronic systems (transducers) and the electromagnetomechanical circuit. In Sect. 1.3, the dynamics and control of a mass levitating in magnetic and gravitational fields is discussed. Two cases of numerical control are considered and verified experimentally. In Sect. 1.4, combined analytical and numerical analyses of vibrations in string-type generators is carried out. The vibrations of a string are governed by a PDE, whereas the dynamics of an amplifier is governed by an ODE with a time delay. The voltage generated on the string ends depends on both electromagnetic induction and string vibration speed. An averaged set of equations is derived and numerically studied. Finally, in Sect.1.5, a 2-DOF nonlinear dynamics of a rotor supported by a magnetohydrodynamic bearing is investigated using perturbation analysis. Two modes corresponding to the vertical and horizontal vibrations of the rotor are coupled. The non-resonant case and the various resonant cases (with and without an internal resonance) are considered. Frequency-response curves are obtained. When the amplitude of the external harmonic excitation is near one of the natural frequencies of the vibrations and the system experiencing internal resonance, a saturation phenomenon occurs.

In this chapter, we introduce basic equations of dynamics of a rigid body during motion about a fixed pivot point. On the basis of these equations, later in this work, we will describe gyroscopic phenomena.

In this chapter, first a historical outline of the theory of gyroscopes is given. Elements of gyroscope classification are introduced, and then the evolution of the gyroscope concept is presented. In particular, the following gyroscope-type devices are considered: the directional gyroscope, the gyroscopic vertical, the stabilized gyroscopic platform, the laser gyroscope, the fiber-optic gyroscope, the piezoelectric gyroscope, the fork gyroscope, and the microgyroscope with a spinning disk and with a vibrating ring. Examples of devices for gyroscopic navigation and an observation device with a built-in gyroscope are provided. Finally, new challenges for gyroscopes are briefly summarized.

In this chapter theoretical investigations and the results of computer simulations are presented to show that the following factors affect the accuracy of realization of the required motion of a controlled gyroscope axis: 1 Compliance of initial conditions of the gyroscope motion with imposed initial conditions. In order to guide the gyroscope axis to the appropriate initial position one can apply additional time-independent control. 2. Values of the resistant-force coefficients in the bearings of gyroscope frames. Too small values of these coefficients, during external disturbance or kinematic excitation of the base, cause dynamical effects to arise and decrease the accuracy of realization of the preset motion. However, large values make the gyroscope axis drift off the preset position in space. Thus, one needs to minimalize the friction coefficients in the bearings of the gyroscope suspension and, additionally, to apply optimally selected dampers. 3. Influence of non-linearities in the model of gyroscope motion, which manifests especially at large angular deviations of the gyroscope axis. 4. Additional deviations of gyroscope—which, independently of the numerous technological tricks, always emerge during gyroscope operation—need to be reduced by means of the gyroscope’s automatic control system. The proper position of the gyroscope axis is maintained by the automatic control system on the basis of the real position obtained from measurements and the required position of the gyroscope axis worked out by a digital machine.

In this chapter a gyroscopic control in self-guidance systems of flying objects (FOs) is presented, and a gyroscopic control in an unmanned aerial vehicle is studied. First, the navigational kinematics of an unmanned aerial vehicle (UAV) is analyzed, and then the control of a gyroscope fixed on its board as well as its full control are discussed. Furthermore, a gyroscope in a guided aerial bomb is studied. It includes analysis of kinematics of a bomb self-guided motion to a ground target, equations of motion of a guided bomb, and a description of a gyroscopic system designed for bomb control including automatic pilot control.