Vibration Analysis of Timoshenko Beams Under General Elastic Boundary Conditions by Using Fourier Series
Abstract
In this study, the free vibration analysis of
beams with general elastic boundary conditions is investigated based on
Timoshenko beam theory. This theory considers effect on elastic curve. So, this theory has more accurate result than
Euler-Bernoulli beam theory. In this study, lateral displacement function is
chosen as a Fourier cosine series.
Similarly, slope function is chosen as a Fourier sine series. These functions are used in the equation of
motion to calculate the Fourier coefficients.
Then Stoke transformation
is applicated to boundary conditions to obtain to the linear equations. A coefficients matrix is created by using the
linear systems of equations. The eigenvalues
of this coefficient matrix gives the angular frequencies. Results of this study are compared with other
studies in the literature. Moreover, calculated results are
presented in a series of figures and tables.
References
- Banerjee, J. R. (1998). Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method, Computres & Structures, 69, 197-208. doi:10.1016/S0045-7949(98)00114-X
- Bozyiğit, B., Çatal, S. and Çatal H. H. (2015) Timoshenko kirişlerinin serbest titreşim analizinin diferansiyel transformasyon methodu ile incelenmesi, 3. Türkiye Deprem Mühendisliği ve Sismoloji Konferansı, D.E.Ü., İzmir.
- Demirdag, O. and Yesilce, Y. (2011). Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method, Advances in Engineering Software, 42(10), 860-867. doi: 10.1016/j.advengsoft.2011.06.002
- De Rosa, M. A. (1995) Free vibrations of Timoshenko beams on two-parameter elastic foundation, Computers & Structures, 57(1), 151-156. doi: 10.1016/0045-7949(94)00594-S
- Develi, A. G. (2007). Elastik zemin üzerine oturan Timoshenko kirişlerinde titreşim problemi, Yüksek Lisans Tezi, İstanbul Teknik Üniversitesi Fen Bilimleri Enstitüsü, İstanbul.
- Farghaly, S. H. (1994). Vibration and stability analysis Timoshenko beams with discontinuities in cross-section, Journal of Sound and Vibration, 174, 591-605. doi: 10.1006/jsvi.1994.1296
- Gül, U. and Aydoğdu, M. (2015). Elastik zemin üzerinde oturan Timoshenko kirişlerinde dalga yayınımı, Uluslararası Katılımlı 17. Makina Teorisi Sempozyumu, İzmir.
- Kim, H. K. and Kim, M. S. (2001). Vibration of beams with generally restrained boundary conditions using Fourier series, Journal of Sound and Vibration, 245(5), 771-784. doi: 10.1006/jsvi.2001.3615
- Kocatürk, T. and Şimşek, M. (2005). Free vibration analysis of elastically supported Timoshenko beams, Sigma, (3) 79-93. Doi: 53A40, 74H45.
- Yesilce, Y. and Catal, H. H. (2011). Solution of free vibration equations of semi-rigid connected Reddy-Bickford beams resting on elastic soil using the differential transform method, Archive of Applied Mechanics, 81(2), 199-213. doi: 10.1007/s00419-010-0405-z
- Zhou, D. (2001). Free vibration of multi-span Timoshenko beams using static Timoshenko beam functions, Journal of Sound and Vibration, 241, 725-734.
- Wolfram Mathematica
TIMOSHENKO KİRİŞLERİNİN GENEL ELASTİK SINIR KOŞULLARINDA FOURİER SERİLERİ KULLANILARAK TİTREŞİM ANALİZİ
Abstract
bu
çalışmada kirişlerin genel elastik sınır koşullarında titreşim analizi,
Timoshenko kiriş teorisiyle incelenmiştir. Timoshenko kiriş teorisi kayma
etkilerinin de elastik eğriye katkısının dikkate alındığı bir teori olup,
Euler-Bernoulli kiriş teorisine göre daha doğru sonuçlar vermektedir. Problemin
çözümünde yer değiştirme fonksiyonu olarak Fourier sinüs serisi seçilmiştir.
Dönme fonksiyonu olarak ise Fourier kosinüs serileri seçilmiştir. Seçilen bu
fonksiyonlar problemi yöneten denklemlerde yerine yazılarak; Fourier
katsayıları hesaplanmıştır. Modeli kurulan kiriş için Timoshenko kiriş teorisine
göre sınır koşulları tespit edilmiştir. Bulunmuş olan Fourier katsayıları ve
Stoke dönüşümüyle elde edilmiş yüksek mertebeden türevler yardımıyla; lineer
diferansiyel denklem takımları elde edilmiştir. Bu denklemlerin katsayılar
matrisi oluşturulmuştur ve katsayılar matrisinin determinantından açısal
frekanslar elde edilmiştir. Bulunan sonuçlar litaretürde bulunan diğer sonuçlar
ile karşılaştırılmış, değerlendirilmiş, tablolar ve grafikler halinde
sunulmuştur.
References
- Banerjee, J. R. (1998). Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method, Computres & Structures, 69, 197-208. doi:10.1016/S0045-7949(98)00114-X
- Bozyiğit, B., Çatal, S. and Çatal H. H. (2015) Timoshenko kirişlerinin serbest titreşim analizinin diferansiyel transformasyon methodu ile incelenmesi, 3. Türkiye Deprem Mühendisliği ve Sismoloji Konferansı, D.E.Ü., İzmir.
- Demirdag, O. and Yesilce, Y. (2011). Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method, Advances in Engineering Software, 42(10), 860-867. doi: 10.1016/j.advengsoft.2011.06.002
- De Rosa, M. A. (1995) Free vibrations of Timoshenko beams on two-parameter elastic foundation, Computers & Structures, 57(1), 151-156. doi: 10.1016/0045-7949(94)00594-S
- Develi, A. G. (2007). Elastik zemin üzerine oturan Timoshenko kirişlerinde titreşim problemi, Yüksek Lisans Tezi, İstanbul Teknik Üniversitesi Fen Bilimleri Enstitüsü, İstanbul.
- Farghaly, S. H. (1994). Vibration and stability analysis Timoshenko beams with discontinuities in cross-section, Journal of Sound and Vibration, 174, 591-605. doi: 10.1006/jsvi.1994.1296
- Gül, U. and Aydoğdu, M. (2015). Elastik zemin üzerinde oturan Timoshenko kirişlerinde dalga yayınımı, Uluslararası Katılımlı 17. Makina Teorisi Sempozyumu, İzmir.
- Kim, H. K. and Kim, M. S. (2001). Vibration of beams with generally restrained boundary conditions using Fourier series, Journal of Sound and Vibration, 245(5), 771-784. doi: 10.1006/jsvi.2001.3615
- Kocatürk, T. and Şimşek, M. (2005). Free vibration analysis of elastically supported Timoshenko beams, Sigma, (3) 79-93. Doi: 53A40, 74H45.
- Yesilce, Y. and Catal, H. H. (2011). Solution of free vibration equations of semi-rigid connected Reddy-Bickford beams resting on elastic soil using the differential transform method, Archive of Applied Mechanics, 81(2), 199-213. doi: 10.1007/s00419-010-0405-z
- Zhou, D. (2001). Free vibration of multi-span Timoshenko beams using static Timoshenko beam functions, Journal of Sound and Vibration, 241, 725-734.
- Wolfram Mathematica
Details
| Subjects | Engineering |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | December 1, 2017 |
| Submission Date | May 4, 2017 |
| Acceptance Date | December 31, 2017 |
| Published in Issue | Year 2017 Volume: 22 Issue: 3 |
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