From Geometry, Kinematics and Dynamics of Billiards to the Extended Theory of Skew Collision between Two Rolling Bodies and Methodology of Vibro-Impact Dynamics

Starting from explanation of geometry, kinematics and dynamics of game billiards, and phenomena of impact a rolling ball into different types of curved surfaces and direct and skew central collision of two rolling, same dimension, balls we open question of collision of two rolling axial symmetrically bodies with different dimensions and different forms. Use elementary approach and Petrovic's theory presented in two books “Elements of mathematical phenomenology” and “Phenomenological mappings”, extended theory of direct and skew central collision of two rolling, axially symmetric, but different dimensions and forms, bodies is formulated with all additional and new analytical expressions, theorems , to define all pre- and post- collision kinetic states. Use these new results complete methodology of vibro-impact system dynamics is formulated and applied for investigation kinetic parameters and phenomena in vibro-impact systems with successive collisions between two or a finite number of rolling bodies. Energy jumps in collisions between rolling bodies in vibroimpact system dynamics are indicated and analytically described in a number of these systems

A New Collision Model for Analysing the Vibro-Impact of Spur Gears

A model of a central centric collision of two fictive rolling disks, with radii equal to the radii of pitch diameters of coupled gears, is used to describe the vibro-impact dynamics of spur gears. Total deformations vary during the operation of spur gears with involute profile, particularly as a result of variable number of pairs of teeth in contact. These variations, as well as tooth profile misalignments or damages, lead to the backlash, the appearance of internal dynamic forces, and impact. Particular values of tooth profile parameters and transmission ratio create the conditions for a vibro-impact vibration regime. The vibro-impact gear dynamics is characterized by vibro-impact vibrations in the contact between the pinion tooth and the wheel tooth during a short period of time after the collision. New formulas for the calculation of the time periods between two tooth collisions, as well as between two successive tooth collision impacts, are devised. Also, a set of equations for the calculation of the disturbance angular velocity of the pinion called the transmission error is developed; the transmission error has a role of an excitation for vibro-impact vibrations in the gear tooth contact

Discrete fractional order system vibrations

A theory of free vibrations of discrete fractional order (FO) systems with a finite number of degrees of freedom (dof) is developed. A FO system with a finite number of dof is defined by means of three matrices: mass inertia, system rigidity and FO elements. By adopting a matrix formulation, a mathematical description of FO discrete system free vibrations is determined in the form of coupled fractional order differential equations (FODE). The corresponding solutions in analytical form, for the special case of the matrix of FO properties elements, are determined and expressed as a polynomial series along time. For the eigen characteristic numbers, the system eigen main coordinates and the independent eigen FO modes are determined. A generalized function of visoelastic creep FO dissipation of energy and generalized forces of system with no ideal visoelastic creep FO dissipation of energy for generalized coordinates are formulated. Extended Lagrange FODE of second kind, for FO system dynamics, are also introduced. Two examples of FO chain systems are analyzed and the corresponding eigen characteristic numbers determined. It is shown that the oscillatory phenomena of a FO mechanical chain have analogies to electrical FO circuits. A FO electrical resistor is introduced and its constitutive voltage–current is formulated. Also a function of thermal energy FO dissipation of a FO electrical relation is discussed

Multi-frequency analysis of the double circular plate system non-linear dynamics

This paper presents a multi-frequency analysis of non-linear dynamics in a double circular plate system. The original series of the amplitude–frequency and phase–frequency graphs as well as eigen forced time functions–frequency graphs are obtained and analyzed for stationary resonant states. The series of the frequency characteristic of the forced time non-linear harmonics are presented first. The analyses identify the presence of singularities and triggers of coupled singularities, as well as resonant jumps. Furthermore, the analogies between non-linear phenomena of dynamics in particular multi-frequency stationary resonant regimes of multi-circular plate system, multi-beam system and according regimes in the chain system are commented upon

Structural analogies on systems of deformable bodies coupled with non-linear layers

The paper is addressed at phenomenological mapping and mathematical analogies of oscillatory regimes in systems of coupled deformable bodies. Systems consist of coupled deformable bodies like plates, beams, belts or membranes that are connected through visco-elastic non-linear layer, modeled by continuously distributed elements of Kelvin–Voigt type with nonlinearity of third order. Using the mathematical analogies the similarities of structural models in systems of plates, beams, belts or membranes are obvious. The structural models consist by a set of two coupled non-homogenous partial non-linear differential equations. The problems to solve are divided into space and time domains by the classical Bernoulli–Fourier method. In the time domains the systems of coupled ordinary non-linear differential equations are completely analog for different systems of deformable bodies and are solved by using the Krilov–Bogolyubov–Mitropolskiy asymptotic method. This paper presents the beauty of mathematical analytical calculus which could be the same even for physically different systems. The mathematical numerical calculus is a powerful and useful tool for making the final conclusions between many input and output values. The conclusions about nonlinear phenomena in multi-body systems dynamics have been revealed from the particular example of double plate׳s system stationary and non-stationary oscillatory regimes