Free vibration analysis of a rotationally restrained carbon nanotube via nonlocal Timoshenko beam theory

Abstract

In this study an investigation has been performed for a carbon nanotube with rotationally restrained boundary conditions. In order to examine the small size effect, nonlocal elasticity with Timoshenko beam assumptions have been used to solve the problem. Lateral deflection function has been chosen as a Fourier sine series. Stoke transformation has been applied to the higher order boundary conditions. A coefficient matrix including nonlocal and spring parameters is derived after some mathematical calculations. The eigen values of this matrix gives the free vibration frequencies. The capabilities of present coefficient matrix has been tested through several numerical examples and compared the results avaliable in the literature. From the solved samples, the vibration frequencies have been reached as a result of the change depending on the rotational spring constants and the non-local parameter.

Keywords

• Carbon nanotube,rotational restraints,free vibration,Stoke transformation,Fourier sine series

Dönel yaylar ile mesnetlenmiş bir karbon nanotüpün yerel olmayan Timoshenko kiriş teorisine göre serbest titreşim analizi

Abstract

Bu çalışmada dönel yaylar ile mesnetlenmiş bir karbon nanotüpün değişik sınır şartları için çözümü incelenmiştir. Küçük boyut etkisini problemin çözümüne dahil etmek için yerel olmayan elastisite teorisi kullanılmış ve Timoshenko kiriş teorisinde uygulanan kabuller göz önünde tutulmuştur. Yatay yer değiştirme fonksiyonu olarak Fourier sinüs serisi seçilmiştir. Sınır koşullarında esneklik sağlanması bakımından bir matematiksel dönüşüm olarak adlandırılan Stoke dönüşümü yüksek mertebeden sınır koşullarına uygulanmıştır. Matematiksel hesaplardan sonra içinde yay sabitleri ve boyut parametresi de bulunan bir katsayılar matrisi elde edilmiştir. Bu katsayılar matrisinin öz değerleri serbest tireşim frekanslarını vermektedir. Söz konusu katsayılar matrisinin doğal titreşim frekanslarını bulabilme yeteneği bir çok matematiksel örnekte test edilmiş ve literatürde bulunan sonuçlar ile karşılaştırılmıştır.  Çözülen örneklerden titreşim frekanslarının dönel yay sabitlerine ve yerel olmayan parametreye bağlı olarak değiştiği sonucuna ulaşılmıştır. 

Keywords

• Karbon nanotüp; dönel yay; serbest titreşim; Stoke dönüşümü,Fourier sinüs serisi

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