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Abstract(s)
A equação de Du¢ ng-Holmes é uma equação diferencial não linear de segunda ordem que apresenta um termo de rigidez não linear polinomial de ordem três e um termo dissipativo de natureza viscosa. Esta equação, introduzida por Georg Du¢ ng, em 1918, no contexto do estudo de sistemas com amortecimento não linear, apresenta propriedades físico-matemáticas extremamente interessantes. A equação de Du¢ ng-Holmes quando forçada harmonicamente revela a existência de órbitas sub-harmónicas, bifurcações associadas à duplicação do período, ciclos limite, estados periódicos críticos e comportamento caótico. A equação de Du¢ ng-Holmes é atualmente utilizada para desenvolver algoritmos de identificação de sinais periódicos ténues
na presença de ruído. Os algoritmos desenvolvidos para esse efeito baseiam-se na sensibilidade da resposta desta equação. As aplicações dos algoritmos referidos têm tido lugar em áreas relacionadas com a identi cação e análise de assinaturas acústicas de navios, identi cação e análise de sinais de GPS,
deteção precoce de dano, entre muitas outras. Neste trabalho efetuar-se-á o estudo exploratório desta equação e das aplicações práticas que têm sido desenvolvidas até ao presente. O estudo analítico e as simulações preliminares, servirão para escolher de entre as diversas configurações da equação de
Du¢ ng-Holmes, aquela que é capaz de realizar a deteção de sinais periódicos acústicos mesmo na presença de ruído. Os resultados das simulações foram satisfatórios, sendo possível concluir que as ferramentas criadas poderão ser usadas futuramente em sinais hidrofónicos acústicos reais.
The Du¢ ng equation is a second order non-linear di¤erential equation with a cubic polynomial sti¤ness term as well as a linear viscous type damping term. This nonlinear equation was introduced by Georg Du¢ ng in 1918 as a result of his work in systems with nonlinear damping, and it has extremely interesting physical and mathematical properties. The Du¢ ng-Holmes equation, when forced harmonically reveals the existence of subharmonic orbits, bifurcations associated with the doubling of the period, limit cycles, critical periodic states and chaotic behavior. The Du¢ ng-Holmes equation is currently used to develop algorithms for identifying weak periodic signals in the presence of noise. The algorithms developed for this purpose are based on the sensitivity of the response of this equation. The applications of the aforementioned algorithms have taken place in areas related to the identi - cation and analysis of acoustic signatures of ships, identi cation and analysis of GPS signals, early damage control, among others. In this work we will carry out an exploratory study of this equation and the practical applications that have been developed to date. The analytical study and preliminary simulations, will serve to choose, from among the di¤erent con gurations of the Du¢ ng-Holmes equation, the one that is capable of detecting periodic acoustic signals even in the presence of noise. The results of the simulations were satisfactory, and it is possible to conclude that the tools created may be used in the future in real acoustic signals.
The Du¢ ng equation is a second order non-linear di¤erential equation with a cubic polynomial sti¤ness term as well as a linear viscous type damping term. This nonlinear equation was introduced by Georg Du¢ ng in 1918 as a result of his work in systems with nonlinear damping, and it has extremely interesting physical and mathematical properties. The Du¢ ng-Holmes equation, when forced harmonically reveals the existence of subharmonic orbits, bifurcations associated with the doubling of the period, limit cycles, critical periodic states and chaotic behavior. The Du¢ ng-Holmes equation is currently used to develop algorithms for identifying weak periodic signals in the presence of noise. The algorithms developed for this purpose are based on the sensitivity of the response of this equation. The applications of the aforementioned algorithms have taken place in areas related to the identi - cation and analysis of acoustic signatures of ships, identi cation and analysis of GPS signals, early damage control, among others. In this work we will carry out an exploratory study of this equation and the practical applications that have been developed to date. The analytical study and preliminary simulations, will serve to choose, from among the di¤erent con gurations of the Du¢ ng-Holmes equation, the one that is capable of detecting periodic acoustic signals even in the presence of noise. The results of the simulations were satisfactory, and it is possible to conclude that the tools created may be used in the future in real acoustic signals.
Description
Keywords
Equação de Du¢ ng-Holmes Deteção sinais periódicos Ruído Acústica submarina