Vibration Analysis of a Cantilevered Beam with Spring Loading at the Tip as a Generic Elastic Structure


Article Preview

Although fundamental, vibration of a cantilevered Euler-Bernoulli beam with spring attached at the tip is not found in literatures and is here solved analytically and numerically using finite element approach. The equation of motion of the beam is obtained by using Hamilton’s principle. Finite element method is utilized to write in-house program for the free vibration of the beam. Results show plausible agreements.



Edited by:

R. Varatharajoo, F.I. Romli, K.A. Ahmad, D.L. Majid and F. Mustapha




M. Jafari et al., "Vibration Analysis of a Cantilevered Beam with Spring Loading at the Tip as a Generic Elastic Structure", Applied Mechanics and Materials, Vol. 629, pp. 407-413, 2014

Online since:

October 2014




* - Corresponding Author

[1] H. Djojodihardjo and P. M. Ng, Numerical Simulation Of Impact Loading On Elastic Structure And Case Studies, Paper IAC-08-C2. 6. 1, Proceedings, 590th International Astronautical Congress, (29 Sep – 3 Oct 2008), Glasgow, United Kingdom.

[2] H. Djojodihardjo and A. Shokrani, Generic Study And Finite Element Analysis Of Impact Loading On Elastic Panel Structure, Paper IAC-10. C2. 6. 2, Proceedings, 61th International Astronautical Congress, (Sep – Oct 2010) Prague, The Czech Republic.

[3] H. Djojodihardjo, 4733-Computational simulation for analysis and synthesis of impact resilient structure, Acta Astronautica, 91 (Oct-Nov 2013) 283-301.

[4] H. Djojodihardjo, Vibro-acoustic analysis of random vibration response of a flexible structure due to acoustic forcing, Beijing Conference, Space Technology & Systems Development , (2013), Beijing, China.

[5] C.A. Rossit, P.A.A. Laura, Free vibrations of a cantilever beam with a spring–mass system attached to the free end, Ocean Engineering, 28 (2001) 933–939.

[6] W. L. Li , Free vibrations of beams with general boundary conditions. Journal of Sound and Vibration, (2000).

[7] S.S. Rao, Vibration of Continuous System, John Wiley & Sons, (2007).

[8] S.K. Tso, T.W. Yang, W.L. Xu, Z.Q. Sun, Vibration control for a flexible-link robot arm with deflection feedback, International Journal of Non-Linear Mechanics, (2003).


[9] P. Gasbarri,R. Monti,C. De Angelisand M. Sabatini, Second Order Effects Of The Flexibility on the Control of a Spacecraft Full-Coupled Model, IAA-AAS-DyCoSS1-10-08, in Advances in Astronautical Sciences, Ed.P. Bainum, J-M. Contant and A. Guerman, published by Univelt (2012).

[10] H. Baruh , Analytical dynamics, McGraw-Hill, Singapore, (1999).

[11] L. Meirovitch, Elements of vibration analysis, McGraw-Hill, New York, (1986).

[12] L. Meirovitch Fundamentals of vibration, McGraw-Hill Higher Education, (2001).

[13] H. Djojodihardjo and M. Jafari, Vibration Analysis of a Cantilevered Beam with Piezo-Electric Actuator at the Tip as a Controllable Elastic Structure, to be presented at the International Astronautical Congress, Toronto, (2014).

[14] E. Kreyszig, Advanced engineering mathematics, 10th ed., John Wiley & Sons, New York (2010).

[15] M. J. Maurizi, R. E. Rossi, J. A. Reyes, Vibration frequencies for a uniform beam with one end spring- hinged and subjected to a translational restraintat the other end, Journal of Sound and Vibration, 48(4) (1976) 565-568.