Vibration analysis of beams with intermediate supports under moving loads

Abstract

In this study, transverse vibration of a beam under a moving singular load and a moving moment is investigated. A simply supported beam with intermediate vertical supports modeled according to Euler-Bernoulli beam theory. The intermediate supports are modeled as consisting of a linear spring and a linear damper. The moving force and the moment, the spring force and the damper force are expressed using Dirac delta functions in the equations of motion. Obtaining the exact solution for this problem with classical methods are quite lengthy and complicated. Beam must be divided into spans between each support. Each span must be solved separately with different set of coordinates having same boundary conditions on support points. As the number of support increases, solution becomes more complicated. However, the present method can be used to solve the problem for the whole beam length without having to separate into various spans regardless of number of supports. Dirac delta functions are converted to series expansions which allows us to get exact solution in form of series expansion. This solution than can be easily calculated by a computer. Dynamic responses of several cases such as various number of supports; different support points; various moving load, moving moment and axial load combinations are examined.

Keywords

• Moving load
• Moving moment
• Simply supported beam
• Intermediate supports

Hareketli yüklere maruz ayaklı kirişlerin titreşiminin incelenmesi

Öz

Bu çalışmada, bir kirişin hareketli bir yük ve moment altındaki titreşimi incelenmiştir. Destek ayakları içeren basit mesnetli bir kiriş Euler-Bernoulli kiriş teorisine göre modellenmiştir. Destek ayakları doğrusal bir yay ve doğrusal bir damperden oluştuğu varsayılarak modellenmiştir. Hareketli yük, hareketli moment, yay kuvveti ve damper kuvveti Dirac delta fonksiyonu kullanılarak hareket denklemine dâhil edilmiştir. Klasik yöntemler ile bu problemin kesin çözümünü elde etmek oldukça uzun ve karmaşıktır. Her bir ayak için kiriş, ayaklar arası parçalara bölünerek ele alınmalıdır. Bu parçaların her biri, sınır şartları destek ayak noktalarında olacak şekilde ayrı koordinat eksenlerinde çözümlenmelidir. Ayak sayısı arttıkça çözüm daha da zorlaşacaktır. Mevcut çalışmaya konu yöntemle ayak sayısından bağımsız olarak, kirişi parçalara ayırmadan tüm kiriş boyunca tek bir koordinat ekseninde çözüm elde etmek mümkündür. Dirac delta fonksiyonu seri açılımlarına dönüştürülerek seri açılımı halinde kesin çözüm elde edilmiştir. Çözüm, basit bir bilgisayar programı yazarak hesaplanabilir. Bu çözümle kirişin farklı ayak sayıları; farklı konumlarındaki ayaklar; farklı hareketli yük, hareketli moment ve eksenel yük gibi çeşitli durumlardaki dinamik davranışı incelenmiştir.

Anahtar Kelimeler

• Hareketli yük
• Hareketli moment
• Basit mesnetli kiriş
• Destek ayakları

References

1. [1] Frýba L. Vibration of Solids and Structures Under Moving Loads. 1st ed. Groningen, Netherlands, Noordhoff International Publishing, 1972.
2. [2] Kameswara R.C. “Frequency analysis of clamped-clamped uniform beams with intermediate elastic support”. Journal of Sound and Vibration, 133(3), 502-509, 1989.
3. [3] Lee HP. “Dynamic response of a beam with intermediate point constraints subject to a moving load”. Journal of Sound and Vibration, 171(3), 361-368, 1994.
4. [4] Esmailzadeh E, Ghorashi M. “Vibration analysis of a Timoshenko beam subjected to a moving mass”. Journal of Sound and Vibration, 199(4), 615-628, 1997.
5. [5] Reis M, Pala Y, Karadere G. “Dynamic analysis of a bridge supported with many vertical supports under moving load”. The Baltic Journal of Road and Bridge Engineering, 3(1), 14-20, 2008.
6. [6] Uzzal RUA, Bhat RB, Ahmed W. “Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation”. Shock and Vibration, 19, 205-220, 2012.
7. [7] Senalp AD, Arikoglu A, Ozkol I, Dogan VZ. “Dynamic response of a finite length Euler-Bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force”. Journal of Mechanical Science and Technology, 24 (10), 1957-1961, 2010.
8. [8] Zhang B, Shepard S. “Dynamic responses of supported beams with intermediate supports under moving loads”. Shock and Vibration, 19, 1403-413, 2012.

9. [9] Chawda D, Murugan S. “Dynamic response of a cantilevered beam under combined moving moment, torque and force”. International Journal of Structural Stability and Dynamics, 20(5), 2050065, 2020.
10. [10] Luo J, Zhu S, Zhai W. “Exact closed-form solution for free vibration of Euler-Bernoulli and Timoshenko beams with intermediate elastic supports”. International Journal of Mechanical Sciences 213, 106842, 2022.
11. [11] Rao SS. Mechanical Vibrations. 6th ed. Malaysia, Pearson Prentice Hall, 2018.
12. [12] Kreyszig E, Advanced Engineering Mathematics. 10th ed. Jefferson City, USA, Wiley, 2011.
13. [13] Case J, Chilver AH, Ross CTF. Strength of Materials and Structures. 4th ed. London, England, John Wiley & Sons Inc., 1999.