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

Yıl 2018, Cilt: 23 Sayı: 2, 255 - 274, 31.08.2018

Murat Sarıgül

https://doi.org/10.17482/uumfd.315108

Cited By: 1

Öz

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.

Anahtar Kelimeler

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

ELASTİK MESNETLERİN HAFİFÇE EĞRİ BİR KİRİŞİN NONLİNEER TİTREŞİMLERİNE ETKİLERİ

Öz

 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. 

Anahtar Kelimeler

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

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