The effects of elastic supports on nonlinear vibrations of a slightly curved beam

Abstract

In this study, nonlinear vibrations of a slightly curved beam having arbitrary rising function are handled. The beam is restricted in longitudinal direction using elastic supports on both ends. Sag-to-span ratio of the beam, which is assumed to have sinusoidal curvature function at the beginning, is taken as 1/10. Beam being of Euler-Bernoulli type rests on Winkler elastic foundation and carries an arbitrarily placed concentrated mass. Equations of motion are obtained by using Hamilton Principle. Cubic and quadratic nonlinear terms have been aroused at the mathematical model because of the foundation and the beam's elongation. The Method of Multiple Scales (MMS), a perturbation technique, is used to solve the equations of motion analytically. The primary resonance case is taken into account during steady-state vibrations. The natural frequencies are obtained exactly for different control parameters such as supports' types, locations of the masses and linear coefficient of foundation. Frequency-amplitude and frequency-response graphs are drawn by using amplitude-phase modulation equations.

Keywords

• Nonlinear vibrations,Slightly curved beam,Elastic supports,Elastic foundation.

ELASTİK MESNETLERİN HAFİFÇE EĞRİ BİR KİRİŞİN NONLİNEER TİTREŞİMLERİNE ETKİLERİ

Abstract

 Bu çalışmada, keyfi başlangıç fonksiyonuna sahip hafifçe eğri bir kirişin lineer olmayan titreşimleri ele alınmaktadır. Her iki ucundan elastik mesnetler kullanılarak kiriş, boyuna yönünde kısıtlanmıştır. Başlangıçta sinüsoidal eğrilik fonksiyonuna sahip olduğu varsayılan kiriş için, ulaşılan eğrilik yüksekliğinin izdüşüme oranı 1/10 alınmaktadır. Euler-Bernoulli tipinde olan kiriş Winkler elastik zemini üzerine oturmakta ve üzerinde keyfi olarak yerleştirilmiş kütleler taşımaktadır. Hamilton prensibi kullanılarak hareket denklemleri elde edilmiştir. Zeminden ve kiriş uzamasından dolayı matematiksel modelde kübik ve quadratik lineer olmayan terimler ortaya çıkmaktadır. Hareket denklemlerini analitik olarak çözümlemek için bir Pertürbasyon tekniği olan Çok Ölçekli Metod(MMS) kullanılmaktadır. Geçici-durum titreşimleri süresince baskın rezonans durumu dikkate alınmaktadır. Mesnetlerin tipleri, kütlelerin konumları ve zeminin lineer bileşeni gibi farklı mukayese parametreleri için doğal frekanslar elde edilmektedir. Genlik-faz modülasyon denklemleri kullanılarak frekans-genlik ve frekans-cevap grafikleri çizilmiştir. 

Keywords

• Lineer olmayan titreşimler,Hafifçe eğri kiriş,Elastik mesnetler,Elastik zemin.

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