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Journals / Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi / 2026 / Cilt: 18

Nonlinear free vibration analysis of a functionally graded multi-layered beam

Fonksiyonel derecelendirilmiş tabakalı kirişin lineer olmayan serbest titreşim analizi

Kübra Erdem(KONYA TEKNİK ÜNİVERSİTESİ)0000-0003-3780-8209
Mahmud Sami Döndüren(KONYA TEKNİK ÜNİVERSİTESİ)0000-0001-9686-9322
Abdullah Avey(İSTANBUL TİCARET ÜNİVERSİTESİ)0000-0000-0000-0000
Pages
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Publication Date
2026-07-20
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The effects of various factors, such as the dimensionless amplitude parameter and the volume fraction index, on the nonlinear frequency ratio were investigated through parametric studies.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, within the framework of Timoshenko beam theory, the nonlinear natural frequencies of three-layer beams with simply supported ends and functionally graded surfaces, whose core consists of homogeneous ceramic, are analyzed.
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Key Findings & Data:It is assumed that the effective properties of the functionally graded material (FDM) vary along the thickness according to a simple mixing rule known as the Voigt model.
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Conclusion & Impact:The effects of various factors, such as the dimensionless amplitude parameter and the volume fraction index, on the nonlinear frequency ratio were investigated through parametric studies.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Üç tabakalı kiriş · Lineer olmayan serbest titreşim · Timoshenko kiriş teorisi

Abstract

In this paper, within the framework of Timoshenko beam theory, the nonlinear natural frequencies of three-layer beams with simply supported ends and functionally graded surfaces, whose core consists of homogeneous ceramic, are analyzed. It is assumed that the effective properties of the functionally graded material (FDM) vary along the thickness according to a simple mixing rule known as the Voigt model. The theoretical formulation of the problem was developed using von Karman-type geometric nonlinearity and stress-strain relations, while the equations of motion were derived via Hamilton’s principle. Under the assumption that axial and torsional inertia terms are neglected, based on Timoshenko beam theory, the partial differential equations describing the nonlinear motion of the laminated beam were reduced to ordinary differential equations using the Galerkin method. Nonlinear frequencies were calculated using the Poincaré–Lindstedt method. The effects of various factors, such as the dimensionless amplitude parameter and the volume fraction index, on the nonlinear frequency ratio were investigated through parametric studies.

Özet

Bu makalede, Timoshenko kiriş teorisi kapsamında, basit mesnetli ve yüzeyleri fonksiyonel derecelendirilmiş, çekirdeği homojen seramikten oluşan üç tabakalı kirişlerin lineer olmayan serbest titreşim frekansları analiz edilmiştir. Fonksiyonel derecelendirilmiş malzemenin (FDM) etkin özelliklerinin, Voigt modeli adı verilen basit bir karışım kuralı ile kalınlık boyunca değiştiği kabul edilmiştir. Problemin teorik formülasyonu, von-Karman tipi geometrik lineer olmama ve gerilme-şekil değiştirme bağıntıları kullanılarak yapılmış, hareket denklemleri ise Hamilton ilkesi aracılığıyla türetilmiştir. Timoshenko kiriş teorisine dayanan eksenel ve dönme eylemsizliği terimlerinin ihmal edildiği varsayımı altında, tabakalı kirişin lineer olmayan hareketini tanımlayan kısmi diferansiyel denklemler Galerkin yöntemi ile adi diferansiyel denklemlere indirgenmiştir. Lineer olmayan frekanslar, Poincaré–Lindstedt yöntemi ile hesaplanmıştır. Boyutsuz genlik parametresi ve hacim kesir indeksi gibi çeşitli faktörlerin lineer olmayan frekans oranı üzerindeki etkileri parametrik çalışmalarla incelenmiştir.

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Cite This Article

Erdem, K., Döndüren, M. S., Avey, A. (2026). Nonlinear free vibration analysis of a functionally graded multi-layered beam. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, Cilt: 18. https://doi.org/10.28948/ngumuh.1935506

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