FREE VIBRATION ANALYSIS OF TIMOSHENKO MULTI-SPAN BEAM CARRYING MULTIPLE POINT MASSES

Bu çalışmada, çok sayıda topaklanmış kütle taşıyan Timoshenko kirişinin doğal frekansları ve mod şekilleri Nümerik Toplama Tekniği (NTT) ve Diferansiyel Transformasyon Metodu (DTM) kullanılarak hesaplanmıştır. İlk olarak, Timoshenko kirişinin sol uç mesnetinin, ara noktada topaklanmış kütlenin, ara mesnetin ve sağ uç mesnetin katsayılar matrisleri elde edilmiştir. Genel katsayılar matrisinin determinantı sıfıra eşitlenerek titreşen sistemin doğal frekansları hesaplanmış ve integrasyon sabitlerinin ilgili özdeğer fonksiyonlarında yerine yazılmasıyla aranan mod şekilleri elde edilmiştir. Analitik çözümden sonra, DTM kullanılarak diferansiyel hareket denklemleri çözülmüştür. Farklı eksenel kuvvet değerleri için çok sayıda topaklanmış kütle taşıyan Timoshenko kirişinin doğal frekans değerleri tablolar halinde sunulmuştur

Anahtar Kelimeler:

Diferansiyel Transformasyon Metodu, serbest titreşim, ara noktalarda topaklanmış kütle, doğal frekans, Nümerik Toplama Tekniği, çok açıklıklı Timoshenko kirişi

ÇOK SAYIDA TOPAKLANMIŞ KÜTLE TAŞIYAN ÇOK AÇIKLIKLI TIMOSHENKO KİRİŞİNİN SERBEST TİTREŞİM ANALİZİ

In this paper, the natural frequencies and mode shapes of Timoshenko multi-span beam carrying multiple point masses are calculated by using Numerical Assembly Technique (NAT) and Differential Transform Method (DTM). At first, the coefficient matrices for leftend support, an intermediate point mass, an intermediate pinned support and right-end support of Timoshenko beam are derived. Equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multi-span beam carrying multiple point masses for the different values of axial force are given in tables

Keywords:

Differential Transform Method, free vibration, intermediate point mass, natural frequency, Numerical Assembly Technique, Timoshenko multi-span beam,

PDF

___

Liu WH, Wu JR, Huang CC. Free Vibration Of Beams With Elastically Restrained Edges And Intermediate Concentrated Masses, Journal of Sound and Vibration, Cilt.122, 1998, s.193-207.
Wu JS, Chou HM., A New Approach For Determining The Natural Frequencies And Mode Shape Of A Uniform Beam Carrying Any Number Of Spring Masses, Journal of Sound and Vibration, Cilt.220, 1999, s.451-468.
Gürgöze M, Erol, H., Determination Of The Frequency Response Function Of A Cantilever Beam Simply Supported In-Span, Journal of Sound and Vibration, Cilt.247, 2001, s.372-378.
Gürgöze M, Erol H., On The Frequency Response Function Of A Damped Cantilever Beam Simply Supported In-Span And Carrying A Tip Mass, Journal of Sound and Vibration, Cilt.255, 2002, s.489-500.
Naguleswaran S. Transverse Vibrations Of An Euler-Bernoulli Uniform Beam Carrying Several Particles, International Journal of Mechanical Science, Cilt.44, 2002, s.2463- 2478.
Naguleswaran S. Transverse Vibration Of An Euler-Bernoulli Uniform Beam On Up O Five Resilient Supports Including Ends, Journal of Sound and Vibration, Cilt.261, 2003, s.372-384.
Lin HY, Tsai YC. On The Natural Frequencies And Mode Shapes Of A Uniform Multi- Span Beam Carrying Multiple Point Masses, Structural Engineering and Mechanics, Cilt.21, 2005, s.351-367.
Lin HY, Tsai YC. On The Natural Frequencies And Mode Shapes Of A Multiple-Step Beam Carrying A Number Of Intermediate Lumped Masses And Rotary Inertias, Structural Engineering and Mechanics, Cilt.22, 2006, s.701-717.
Lin HY, Tsai YC. Free Vibration Analysis Of A Uniform Multi-Span Beam Carrying Multiple Spring-Mass Systems, Journal of Sound and Vibration, Cilt.302, 2007, s.442- 456.
Wang JR, Liu TL, Chen DW. Free Vibration Analysis Of A Timoshenko Beam Carrying Multiple Spring-Mass Systems With The Effects Of Shear Deformation And Rotatory Inertia, Structural Engineering and Mechanics, Cilt.26, 2007, s.1-14.
Yesilce Y, Demirdag O, Catal S. Free Vibrations Of A Multi-Span Timoshenko Beam Carrying Multiple Spring-Mass Systems, Sadhana, Cilt.33, 2008, s.385-401.
Yesilce Y, Demirdag O. Effect Of Axial Force On Free Vibration Of Timoshenko Multi- Span Beam Carrying Multiple Spring-Mass Systems, International Journal of Mechanical Science, Cilt.50, 2008, s.995-1003.
Lin HY. Dynamic Analysis Of A Multi-Span Uniform Beam Carrying A Number Of Various Concentrated Elements, Journal of Sound and Vibration, Cilt.309, 2008, s.262- 275.
Yesilce Y. Effect Of Axial Force On The Free Vibration Of Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems, Journal of Vibration and Control, Cilt.16, 2010, s.11-32.
Lin HY. An Exact Solution For Free Vibrations Of A Non-Uniform Beam Carrying Multiple Elastic-Supported Rigid Bars, Structural Engineering and Mechanics, Cilt.34, 2010, s.399-416.
Zhou JK. Differential transformation and its applications for electrical circuits, Wuhan China: Huazhong University Press, 1986.
Ozgumus OO, Kaya MO. Flapwise Bending Vibration Analysis Of Double Tapered Rotating Euler-Bernoulli Beam By Using The Differential Transform Method, Meccanica, Cilt.41, 2006, s.661-670.
Çatal S. Analysis Of Free Vibration Of Beam On Elastic Soil Using Differential Transform Method, Structural Engineering and Mechanics, Cilt.24, 2006, s.51-62.
Çatal S. Solution Of Free Vibration Equations Of Beam On Elastic Soil By Using Differential Transform Method, Applied Mathematical Modelling, Cilt.32, 2008, s.1744- 1757.
Çatal S, Çatal, HH., Buckling Analysis Of Partially Embedded Pile İn Elastic Soil Using Differential Transform Method, Structural Engineering and Mechanics, Cilt.24, 2006, s.247-268.
Ozgumus OO, Kaya MO., Energy Expressions And Free Vibration Analysis Of A Rotating Double Tapered Timoshenko Beam Featuring Bending-Torsion Coupling, International Journal of Engineering Science, Cilt.45, 2007, s.562-586.
Kaya MO, Ozgumus OO. Flexural-Torsional-Coupled Vibration Analysis Of Axially Loaded Closed-Section Composite Timoshenko Beam By Using DTM, Journal of Sound and Vibration, Cilt.306, 2007, s.495-506.
Yesilce Y, Catal S., Free Vibration Of Axially Loaded Reddy-Bickford Beam On Elastic Soil Using The Differential Transform Method, Structural Engineering and Mechanics, Cilt.31, 2009, s.453-476