Nonlinear dynamic of clock mechanisms

Miša D. Stojićević
Doctoral Dissertation
UNIVERSITY OF BELGRADE
FACULTY OF MECHANICAL ENGINEERING
Belgrade, 2018.

NOTE: Thesis was translated from Serbian to English using Google Translate and has not yet been corrected for any translation errors.

ABSTRACT: Clock mechanisms belongs in the field of precision mechanics, and they measures the flow of time with a precision uniformity of its work. A key subassembly of the clock mechanism is a escapement mechanism that, by its functions, creates disruptions in the oscillatory process so that the oscillations are no longer their own, but are forced with a frequency that is subjective to change. Therefore, the correct mathematical and physical description of the functioning of clock mechanisms cannot be correctly achieved by neglecting small disorders of higher order, linearization of differential equations and approximations common to the classical approach in the study of mechanisms.

The subject of the study of this dissertation are precisely these small disorders of higher order that occur in the escapement mechanism and undermine the uniformity of the work of the clock mechanism. Using the theory of perturbation, mathematical expressions for the calculation of errors are derived for two types of impulse mechanisms. The correctness of both these mathematical expressions and those general integral formulas was verified by computer simulation and the analysis of the motion of the 3D model of the corresponding escapement mechanisms. A qualitative and quantitative analysis of errors of the escapement mechanisms, i.e. frequency disturbances of the clock oscillator caused by their escapement mechanisms are presented. In addition to investigating the disturbance of the work of the escapement mechanism, the dissertation provides an overview of all the non-linear dynamic properties of the clock mechanism oscillator as well as a description of the parts of the clock that were used to simulate the clock. The numerical results of these simulations showed a high degree of agreement with the results of the theoretical numerical calculations, which confirmed the correctness of mathematical expressions performed using the perturbation calculus.