Free Vibration Analysis of a Functionally Graded Micro-Beam with Tapered Cross Section

Abstract

In this study, the free vibration analysis of a functionally graded (FG) micro-beam with tapered cross-section is carried out theoretically. The beam has material distribution according to the power law throughout the thickness. The governing equation is reduced to an ordinary differential equation for the tapered beam with a cross-sectional geometry whose width varies exponentially. Vibrations of functionally graded tapered microbeams were analyzed analytically in the elastic variation range based on the modified stress couple theory. Motion equations and boundary conditions are derived from the Hamilton principle. Analytical results of natural frequencies are calculated for cantilever exponential FG beams. Solutions for natural frequencies were obtained as the ratio of the beam's characteristic size to the material internal length parameter and according to the FGM distribution function characteristics.

Keywords

• Functionally graded materials
• micro-beam,
• free vibration analysis
• tapered cross-section

Fonksiyonel derecelendirilmiş konik kesitli bir mikro kirişin serbest titreşim analizi

Abstract

Bu çalışmada, konik kesitli fonksiyonel derecelendirilmiş (FD) bir mikro kirişin serbest titreşim analizi teorik olarak gerçekleştirilmiştir. Kiriş, kalınlık boyunca güç yasasına göre malzeme dağılımına sahiptir. Yönetim denklemi, genişliği üstel olarak değişen bir enine kesit geometrisine sahip konik kiriş için sıradan bir diferansiyel denkleme indirgenmiştir. Fonksiyonel derecelendirilmiş konik mikro kirişlerin titreşimleri, modifiye edilmiş gerilim çifti teorisine dayalı olarak elastik varyasyon aralığında analitik olarak analiz edilmiştir. Hareket denklemleri ve sınır koşulları Hamilton ilkesinden türetilmiştir. Doğal frekansların analitik sonuçları, konsol üstel FD kiriş için hesaplanmıştır. Doğal frekanslar için çözümler, kirişin karakteristik boyutunun malzeme iç uzunluk parametresine oranı olarak ve FDM dağılım fonksiyonu özelliklerine göre elde edilmiştir.

Keywords

• Fonksiyonel derecelendirilmiş malzeme
• serbest titreşim analizi
• konik kesit alanı
• mikro-kiriş

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