Building an Analytical Method to Study Cantilever Beam Dynamic Response

Authors

Samaher M.Sarhan Department of Mechatronics Engineering, Al-Khwarizmi College of Engineering, University of Baghdad, Baghdad, 10071, Iraq.";} https://orcid.org/0000-0002-1604-779X

DOI:

https://doi.org/10.30526/36.4.3140

Keywords:

Keywords: analytical solution, cantilever beam, free vibration, mathematical modeling.

Abstract

Cantilever beams are used in many crucial applications in machinery and construction. For example, the airplane wing, the microscopic probe for atomic force measurement, the tower crane overhang and twin overhang folding bridge are typical examples of cantilever beams. The current research aims to develop an analytical solution for the free vibration problem of cantilever beams. The dynamic response of AISI 304 beam represented by the natural frequencies was determined under different working surrounding temperatures ((-100 ℃ to 400 ℃)). A Matlab code was developed to achieve the analytical solution results, considering the effect of some beam geometrical dimensions. The developed analytical solution has been verified successfully with real experimental data and the error was not exceeded 1%.

Keywords: analytical solution, cantilever beam, free vibration, mathematical modeling.

References

Akgöz B, Civalek Ö. Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Composite Structures. 2013 2013/04/01/;98:314-22.

Caliò I, Elishakoff I. Closed-form solutions for axially graded beam-columns. Journal of Sound and Vibration. 2005 2005/02/23/;280(3):1083-94.

Li XF. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. Journal of Sound and Vibration. 2008 2008/12/23/;318(4):1210-29.

Singh KV, Li G. Buckling of functionally graded and elastically restrained non-uniform columns. Composites Part B: Engineering. 2009 2009/07/01/;40(5):393-403.

Huang Y, Li X-F. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. Journal of Sound and Vibration. 2010 2010/05/24/;329(11):2291-303.

Shahba A, Attarnejad R, Marvi MT, Hajilar S. Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Composites Part B: Engineering. 2011 2011/06/01/;42(4):801-8.

Shahba A, Rajasekaran S. Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling. 2012 2012/07/01/;36(7):3094-111.

Stanisław Kukla, Rychlewska J. FREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED BEAMS Journal of Applied Mathematics and Computational Mechanics. 2013;12(2):39-44.

Yilmaz Y, Girgin Z, Evran S. Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method. Mathematical Problems in Engineering. 2013 2013/07/28;2013:793062.

GEETHU CHANDRAN, M.G.RAJENDRAN. STUDY ON BUCKLING OF COLUMN MADE OF FUNCTIONALLY GRADED MATERIAL. International Journal of Mechanical And Production Engineering. 2014;2(2):52-4.

Shafiei N, Kazemi M, Ghadiri M. Nonlinear vibration of axially functionally graded tapered microbeams. International Journal of Engineering Science. 2016 2016/05/01/;102:12-26.

Ranganathan SI, Abed FH, Aldadah MG. Buckling of slender columns with functionally graded microstructures. Mechanics of Advanced Materials and Structures. 2016 2016/11/01;23(11):1360-7.

Elishakoff I, Eisenberger M, Delmas A. Buckling and Vibration of Functionally Graded Material Columns Sharing Duncan's Mode Shape, and New Cases. Structures. 2016 2016/02/01/;5:170-4.

Rezaiee-Pajand M, Masoodi AR. Exact natural frequencies and buckling load of functionally graded material tapered beam-columns considering semi-rigid connections. Journal of Vibration and Control. 2016 2018/05/01;24(9):1787-808.

Lee JK, Lee BK. Free vibration and buckling of tapered columns made of axially functionally graded materials. Applied Mathematical Modelling. 2019 2019/11/01/;75:73-87.

Ghani, Suadad Noori, Neamah, Raghad Azeez, Abdalzahra, Ali Talib, Al-Ansari, Luay S. and Abdulsamad, Husam Jawad. "Analytical and numerical investigation of free vibration for stepped beam with different materials" Open Engineering, vol. 12, no. 1, 2022, pp. 184-196. https://doi.org/10.1515/eng-2022-0031.

Du, Xiaokang, Jing Zhang, Xian Guo, Liang Li, and Dingguo Zhang. 2022. "Dynamics Analysis of Rotating Cantilever Beams with Free End Mass" Applied Sciences 12, no. 15: 7553. https://doi.org/10.3390/app12157553

Carnegie W. Vibrations of Pre-Twisted Cantilever Blading: An Additional Effect Due to Torsion. Proceedings of the Institution of Mechanical Engineers. 1962 1962/06/01;176(1):315-22.

Rao JS, Carnegie W. Solution of the equations of motion of coupled-bending bending torsion vibrations of turbine blades by the method of ritz-galerkin. International Journal of Mechanical Sciences. 1970 1970/10/01/;12(10):875-82.

Abbas BAH, Irretier H. Experimental and theoretical investigations of the effect of root flexibility on the vibration characteristics of cantilever beams. Journal of Sound Vibration. 1989 May 01, 1989;130:353-62.