Exact Closed Form Characteristic Equations for Transverse Vibration of Timoshenko Beams

Subject Areas : Structural Mechanics

Shahriar Hosseini-Hashemi ¹ , K Hosseini-Hashemi ² , Reza Nazemnezhad ³

1 - Impact Research Laboratory, Iran University of Science and Technology, Tehran, Iran
2 - School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
3 -

Received: 2023-05-30 Accepted : 2023-07-09 Published : 2023-09-01

Keywords: Bench mark equation, Transverse vibration, characteristic equation, Timoshenko beam,

Abstract :

The dimensionless equations of motion are derived based on the Timoshenko beam theory to study the transverse vibration of beams without further usage of any approximate method. The exact closed form characteristic equations are given within the validity of the Timoshenko beam theory for beams having various boundary conditions. Accurate Eigen frequency parameters are presented for a different length to height ratio for each case. The exact closed form mode shapes related to deflection, slope due to bending and stress resultants are also presented and illustrated for some cases. The modal tests are performed for beams with clamped-Free and Free-Free boundary conditions. Finally, the effect of boundary conditions, length to height ratio on the eigenvalues parameters and vibratory behavior of each distinct case are studied. Validity of the derived closed form characteristic equations are checked through comparison of numerical solutions with the available results. It is believed that in the present work, the exact closed form characteristic equations and their associated Eigen functions, except for the beams with simply supported ends, for the rest of considered cases are obtained for the first time.

References:

1. Rao S.S., 2011, Mechanical Vibrations, Prentice-Hall Inc, New Jersey.


2. Craig R.R., Cordial A.J., 2006, Fundamentals of Structural Dynamics, John Wiley & Sons, New Jersey.


3. Timoshenko S.P., 1921, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine 41(6): 742-746.


4. Timoshenko S. P., 1922, On the transverse vibration of bars of uniform cross section, Philosophical Magazine 43(6): 379-384.


5. Freund J., Karaoke A., 2016, Shear and torsion correction factors of Timoshenko beam model for generic cross sections, Research on Engineering Structures & Materials 2: 19-27.


6. Cowper G.R., 1968, On the accuracy of Timoshenko’s beam theory, ASCE Journal of Engineering Mechanics Division 94(6): 1447-1453.


7. Kocaturk T., Simsek M., 2005, Free vibration analysis of Timoshenko beams under various boundary conditions, Sigma Journal of Engineering and Natural Sciences 2(1): 30-44.


8. Lee J., Schultz W.W., 2004, Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method, Journal of Sound and Vibration 269: 609-621.


9. Civalek O., Kiracioglu O., 2010, Free vibration analysis of Timoshenko beams by DSC method, International Journal for Numerical Methods in Biomedical Engineering 26: 1890-1898.


10. Avcar M., 2014, Free vibration analysis of beams considering different geometric characteristic and boundary conditions, International Journal of Mechanical and Applications 4(3): 94-100.


11. Sayyad A.S., 2011, Comparison of various refined beam theories for the bending and free vibration analysis of thick beams, Applied and Computational Mechanics 5: 217-230.


12. Hosseini Hashemi Sh., 2017, Transverse Vibration Analysis of Euler-Bernoulli and Timoshenko Beams with Classic and Non-Classic Boundary Conditions, BSc. Dissertation, School of Mechanical and Aerospace Engineering, Islamic Azad University, Science and Research Branch.


13. Blevins R.D., 1979, Formula for Natural Frequency and Mode Shape, Van Nostrand Reinhold, Newyork.