Nelinearna dinamika satnih mehanizama

Abstract

Satni mehanizmi pripadaju oblasti precizne mehanike, koji visokim stepenom ravnomernosti svoga hoda mere proticanje vremena. Ključni podsklop satnog mehanizma predstavlja zaprečno-impulsni mehanizam koji svojim funkcijama unosi poremećaje u oscilatorni proces tako da oscilacije više nisu sopstvene, već prinudne sa frekvencom koja je podložna promeni. Zato se korektan matematički i fizikalni opis funkcionisanja satnih mehanizama ne može ispravno ostvariti zanemarivanjem malih poremećaja višeg reda, linearizacijama diferencijalnih jednačina i aproksimacijama uobičajenim za klasični pristup u izučavanju mehanizama. Predmet istraživanja ove disertacije jesu upravo ti mali poremećaji višeg reda koji se javljaju u zaprečno-impulsnom mehanizmu i narušavaju ravnomernost hoda satnog mehanizma. Korišćenjem teorije perturbacija biće izvedeni matematički izrazi za izračunavanje grešaka koje generišu dve vrste zaprečno-impulsnih mehanizama. Korektnost kako ovih matematičkih izraza, tako i onih opštih, integralnih formula, proverena je kompjuterskom simulacijom i analizom kretanja 3D modela odgovarajućih zaprečno-impulsnih mehanizama. Prikazana je kvalitativna i kvantitativna analiza grešaka zaprečno-impulsnih mehanizama odnosno, poremećaja frekvence oscilatora časovnika koga izazivaju njihovi zaprečno-impulsni mehanizmi. Pored istraživanja poremećaja rada zaprečno- impulsnog mehanizma, disertacija daje prikaz svih nelinearnih dinamičkih svojstava oscilatora satnog mehanizma kao i opis delova sata koji su korišćeni za simulaciju sata. Numerički rezultati ovih simulacija pokazali su visok stepen slaganja sa rezultatima teorijskih numeričkih proračuna, čime je potvrđena korektnost matematičkih izraza izvedenih primenom perturbacionog računa.

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.