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Free vibration of a Timoshenko beam carrying three dimensional tip mass: Analytical solution and experimental modal testing

Freie Vibration eines Timoshenko-Balkens mit dreidimensionaler Endmasse: Analytische Lösung und experimentelle modale Prüfung
Hilal Doğanay Katı and Hakan Gökdağ
From the journal Materials Testing

Abstract

In this work, free vibration of a Timoshenko beam carrying three dimensional tip mass whose center of gravity is not coincident with beam end is dealt with. The beam performs bending in two orthogonal planes plus torsional deformation about the beam axis. Since the tip mass center of gravity is not coincident with beam end point, these deformations are coupled through the boundary conditions. First, an analytical model is developed for the considered system and solved. Later, the analytical results are compared with the experimental and finite element results. Experimental values are obtained by the impact testing, and peak picking approach is employed to extract modal data. Furthermore, finite element results are obtained by ANSYS, a famous finite element software. According to the relevant literature, this is the first study including experimental data for the considered system. It is observed that analytical results, in general, agree with the numerical and experimental results. Thus, the analytical model developed in this work may safely be used for later studies where lower modal data of the structure is required.

Kurzfassung

In der diesem Beitrag zugrunde liegenden Forschungsarbeit wurde die freie Vibration eines Timoshenko-Balkens mit einer dreidimensionalen Endlast untersucht, dessen Schwerpunkt nicht mit dem behandelten Balkenende koinzident ist. Der Balken führt eine Biegung in zwei orthogonalen Ebenen mit einer zusätzlichen Torsionsdeformation um die Balkenachse aus. Da der Schwerpunkt am Balkenende nicht mit dem Balkenendpunkt übereinstimmt, werden diese Deformationen mittels Randbedingungen gekoppelt. Hierzu wurde zuerst ein analytisches Modell für das entsprechende System entwickelt und gelöst. Danach wurden die analytischen Ergebnisse mit experimentellen Resultaten sowie denen aus Finite Elemente Analysen verglichen. Die experimentellen Werte wurden mittels Impaktversuchen ermittelt und es wurde der Spitzenwertansatz angewandt, um die modalen Daten herauszufiltern. Darüber hinaus wurden Finite Elemente Analysen mittels der bekannten Software ANSYS durchgeführt. Entsprechend der relevanten Literatur stellt dies die erste Studie dar, die auch experimentelle Ergebnisse für das betrachtete System enthält. Es wurde beobachtet, dass die analytischen Ergebnisse allgemein mit den numerischen und experimentellen Resultaten übereinstimmen. Daher kann das in dieser Studie entwickelte analytische Modell sicher für weitere Studien angewandt werden, in denen niedrigere modale Daten der Struktur erforderlich sind.


*Correspondence Address, Associate Prof. Dr. Hakan Gökdağ, Mechanical Engineering Department, Bursa Technical University, 152 Evler Mh. Eğitim Cd. 1. Damla Sk. No: 2/10, 16330 Yıldırım, Bursa, Turkey, E-mail:

Hilal Doğanay Katı received her Bachelor's degree from the Department of Mathematics and Mechanical Engineering, Ataturk University in Erzurum, Turkey in 2008 and 2010, respectively. She received her Master's degree from the Department of Mechanical Engineering, University College London (UCL) in UK. She is currently a PhD student and research assistant in the Mechanical Engineering Department,t Bursa Technical University, Turkey.

Hakan Gökdağ has been Associated Professor of Mechanical Engineering at Bursa Technical University, in Bursa, Turkey since 2014. Before that, he was research assistant at Uludağ University, Turkey where he received his MSc on applications of linear theory of vibrations in 2005, and his PhD on wavelet transform based structural damage detection in 2010. From 2009 to 2010, he was a visiting researcher in Mechanical Engineering, Imperial College London, UK. His research interests include applications of mechanical vibrations, experimental modal analysis, structural damage detection, wavelet transform and its applications, applied mathematics, and signal processing for sound and vibration applications.


References

1 D. C. D.Oguamanam: Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling, International Journal of Mechanical Sciences45 (2003), pp. 96397910.1016/j.ijmecsci.2003.09.014Search in Google Scholar

2 M. S.Allen, H.Sumali, P. C.Penegor: DMCMN: Experimental/analytical evaluation of the effect of tip mass on atomic force microscope cantilever calibration, ASME Journal of Dynamic Systems, Measurement and Control131 (2009), pp. 064501064501-10 10.1115/1.4000160Search in Google Scholar

3 M.Ansari, E.Esmailzadeh, N.Jalili: Coupled vibration and parameter sensitivity analysis of rocking-mass vibrating gyroscopes, Journal of Sound and Vibration327 (2009), pp. 56458310.1016/j.jsv.2009.06.021Search in Google Scholar

4 M.Grobbelaar-van Dalsen: Uniform stability for the Timoshenko beam with tip load, Journal of Mathematical Analysis and Applications361 (2010), pp. 39240010.1016/j.jmaa.2009.06.059Search in Google Scholar

5 M.Eftekhari, M.Mahzoon, S.Ziaei-Rad: Effect of added tip mass on the nonlinear flapwise and chordwise vibration of cantilever composite beam under base excitation, International Journal of Structural Stability and Dynamics12 (2012), No. 2, pp. 28531010.1142/S0219455412500046Search in Google Scholar

6 A. B. AlaminDow, M.Schneider, D.Koo, H. A.Al-Rubaye, A.Bittner, U.Schmid, N.Kherani: Modeling the performance of a micromachined piezoelectric energy harvester, Microsystem Technologies18 (2012), pp. 1035104310.1007/s00542-012-1436-xSearch in Google Scholar

7 D.Upadrashta, Y.Yang, L.Tang: Material strength consideration in the design optimization of nonlinear energy harvester, Journal of Intelligent Material Systems and Structures26 (2014), pp. 1980199410.1177/1045389X14546651Search in Google Scholar

8 I.Mehdipour, A.Barari: Why the center-point of bridged carbon nanotube length is the most mass sensitive location for mass attachment?, Computational Material Science55 (2012), pp. 13614110.1016/j.commatsci.2011.11.036Search in Google Scholar

9 K. T.Lee: An analytical solution for dynamic behavior of a beam-column frame with a tip body, Applied Mathematical Modeling37 (2013), pp. 9086910010.1016/j.apm.2013.04.017Search in Google Scholar

10 K.Torabi, A.Jafarzadeh, E.Zafari: Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses, Applied Mathematics and Computation238 (2014), pp. 34235710.1016/j.amc.2014.04.019Search in Google Scholar

11 H.Gökdağ, O.Kopmaz: Coupled bending and torsional vibration of a beam with in-span and tip attachments, Journal of Sound and Vibration287 (2005), pp. 59161010.1016/j.jsv.2004.11.019Search in Google Scholar

12 D. C. D.Oguamanam, M.Arshad: On the natural frequencies of a flexible manipulator with a tip load, Proceedings of the Institution of Mechanical Engineers, 219 (2005), pp. 1199120510.1243/095440605X32039Search in Google Scholar

13 H.Salarieh, M.Ghorashi: Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling, International Journal of Mechanics48 (2006), pp. 76377910.1016/j.ijmecsci.2006.01.008Search in Google Scholar

14 O.Demirdag, Y.Yesilce: Solution of free vibration of elastically supported Timoshenko columns with a tip mass by differential transform method, Advances in Engineering Software42 (2011), pp. 86086710.1016/j.advengsoft.2011.06.002Search in Google Scholar

15 M.Ansari, E.Esmailzadeh: Exact frequency analysis of a rotating cantilever beam with tip mass subjected to torsional-bending vibrations, ASME Journal of Vibration and Acoustics133 (2011), pp. 0410031-9 10.1115/1.4003398Search in Google Scholar

16 M.Gürgöze, S.Zeren: The influence of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam, Meccanica46 (2011), pp. 1401141210.1007/s11012-010-9396-7Search in Google Scholar

17 C. F. T.Matt: Simulation of transverse vibrations of a cantilever beam with an eccentric tip mass in the axial direction using integral transforms, Applied Mathematical Modeling37 (2013), pp. 9338935410.1016/j.apm.2013.04.038Search in Google Scholar

18 H.Malaeke, H.Moeenfard: Analytical modeling of large amplitude free vibration of non-uniform beams carrying a both transversely and axially eccentric tip mass, Journal of Sound and Vibration366 (2016), pp. 21122910.1016/j.jsv.2015.12.003Search in Google Scholar

19 J.He, Z.-F.Fu: Modal Analysis, Butterworth-Heinemann, Oxford, UK (2001)Search in Google Scholar

Published Online: 2017-06-19
Published in Print: 2017-06-01

© 2017, Carl Hanser Verlag, München