Parçalı Sabit Argümana Fonksiyonel Bagımlı Kuvvetli Bir Yay-Kütle Sisteminin Green ˘ Fonksiyonu ve Periyodik Çözümleri

Bu çalı¸smada, genelle¸stirilmi¸s parçalı sabit argümanlı ve genelle¸stirilmi¸s parçalı sabit argümana fonksiyonel bagımlı sönümlü yay-kütle sistemleri dü¸sünülmü¸stür. Bu ˘ yay-kütle sistemleri sırasıyla Ax(γ(t)) ve Ax(γ(t))+h(t, xt , xγ(t) ) formlarında parçalı sabit kuvvetlere sahiptirler. Bu yay-kütle sistemleri ayrık denklemlere dönü¸stürülmeden incelenmi¸stir. Bu inceleme yapılırken, genelle¸stirilmi¸s parçalı sabit argümana fonksiyonel bagımlı ˘ diferansiyel denklemler için elde edilen sonuçlardan [1] faydalanılmı¸stır. Genelle¸stirilmi¸s parçalı sabit argümana fonksiyonel bagımlı yay-kütle sisteminin çözümlerinin varlık ve ˘ tekligi için yeterli ko¸sullar verilmi¸stir. Green fonksiyonu yardımıyla fonksiyonel kuvvetli ˘ yay-kütle sisteminin periyodik çözümü olu¸sturulmu¸stur ve tekligi ispatlanmı¸stır. Elde ˘ edilen teorik sonuçlar örneklendirilmi¸stir. Bu örnek, belli parametreler için genelle¸stirilmi¸s parçalı sabit argümana fonksiyonel bagımlı sönümlü yay-kütle sisteminin Green ˘ fonksiyonu ile ifade edilebilen tek bir periyodik çözüme sahip oldugunu göstermi¸stir.

Green’s Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument

In this paper, damped spring-mass systems with generalized piecewise constant argument and with functional dependence on generalized piecewise constant argument are considered. These spring-mass systems have piecewise constant forces of the forms Ax(γ(t)) and Ax(γ(t)) +h(t, xt , xγ(t) ), respectively. These spring-mass systems are examined without reducing them into discrete equations. While doing this examination, we make use of the results which have been obtained for differential equations with functional dependence on generalized piecewise constant argument in [1]. Sufficient conditions for the existence and uniqueness of solutions of the spring-mass system with functional dependence on generalized piecewise constant argument are given. The periodic solution of the spring-mass system which has functional force is created with the help of the Green’s function, and its uniqueness is proved. The obtained theoretical results are illustrated by an example. This illustration shows that the damped spring-mass systems with functional dependence on generalized piecewise constant argument with proper parameters has a unique periodic solution which can be expressed by Green’s function.

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