**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32794

##### Experimental Investigation of Natural Frequency and Forced Vibration of Euler-Bernoulli Beam under Displacement of Concentrated Mass and Load

**Authors:**
Aref Aasi,
Sadegh Mehdi Aghaei,
Balaji Panchapakesan

**Abstract:**

This work aims to evaluate the free and forced vibration of a beam with two end joints subjected to a concentrated moving mass and a load using the Euler-Bernoulli method. The natural frequency is calculated for different locations of the concentrated mass and load on the beam. The analytical results are verified by the experimental data. The variations of natural frequency as a function of the location of the mass, the effect of the forced frequency on the vibrational amplitude, and the displacement amplitude versus time are investigated. It is discovered that as the concentrated mass moves toward the center of the beam, the natural frequency of the beam and the relative error between experimental and analytical data decreases. There is a close resemblance between analytical data and experimental observations.

**Keywords:**
Euler-Bernoulli beam,
natural frequency,
forced vibration,
experimental setup.

**References:**

[1] Abramovich, H. and O. Hamburger, Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass. Journal of Sound and Vibration, 1992. 154(1): p. 67-80.

[2] Chun, K., Free vibration of a beam with one end spring-hinged and the other free. 1972.

[3] Laura, P., J. Pombo, and E. Susemihl, A note on the vibrations of a clamped-free beam with a mass at the free end. Journal of Sound and Vibration, 1974. 37(2): p. 161-168.

[4] Goel, R., Free vibrations of a beam-mass system with elastically restrained ends. Journal of Sound and Vibration, 1976. 47(1): p. 9-14.

[5] Parnell, L. and M. Cobble, Lateral displacements of a vibrating cantilever beam with a concentrated mass. Journal of Sound and Vibration, 1976. 44(4): p. 499-511.

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

[7] To, C., Vibration of a cantilever beam with a base excitation and tip mass. Journal of Sound and Vibration, 1982. 83(4): p. 445-460.

[8] Laura, P. and R. Gutierrez, Vibrations of an elastically restrained cantilever beam of varying cross section with tip mass of finite length. Journal of Sound Vibration, 1986. 108: p. 123-131.

[9] Liu, W. and C.-C. Huang, Free vibration of restrained beam carrying concentrated masses. Journal of Sound and Vibration, 1988. 123(1): p. 31-42.

[10] Wang, J.-S. and C.-C. Lin, Dynamic analysis of generally supported beams using Fourier series. Journal of Sound and Vibration, 1996. 196(3): p. 285-293.

[11] Yeih, W., J. Chen, and C. Chang, Applications of dual MRM for determining the natural frequencies and natural modes of an Euler–Bernoulli beam using the singular value decomposition method. Engineering Analysis with Boundary Elements, 1999. 23(4): p. 339-360.

[12] Kim, H. and M. Kim, Vibration of beams with generally restrained boundary conditions using Fourier series. Journal of Sound and Vibration, 2001. 245(5): p. 771-784.

[13] Low, K., Natural frequencies of a beam–mass system in transverse vibration: Rayleigh estimation versus eigenanalysis solutions. International Journal of Mechanical Sciences, 2003. 45(6-7): p. 981-993.

[14] Naguleswaran, S., Transverse vibration of an uniform Euler–Bernoulli beam under linearly varying axial force. Journal of Sound and vibration, 2004. 275(1-2): p. 47-57.

[15] Yaman, M., Finite element vibration analysis of a partially covered cantilever beam with concentrated tip mass. Materials & design, 2006. 27(3): p. 243-250.

[16] Maiz, S., et al., Transverse vibration of Bernoulli–Euler beams carrying point masses and taking into account their rotatory inertia: Exact solution. Journal of Sound and Vibration, 2007. 303(3-5): p. 895-908.

[17] Lai, H.-Y. and J.-C. Hsu, An innovative eigenvalue problem solver for free vibration of Euler–Bernoulli beam by using the Adomian decomposition method. Computers & Mathematics with Applications, 2008. 56(12): p. 3204-3220.

[18] Liu, Y. and C.S. Gurram, The use of He’s variational iteration method for obtaining the free vibration of an Euler–Bernoulli beam. Mathematical and Computer Modelling, 2009. 50(11-12): p. 1545-1552.

[19] Hozhabrossadati, S.M., A. Aftabi Sani, and M. Mofid, Free vibration analysis of a beam with an intermediate sliding connection joined by a mass-spring system. Journal of Vibration and Control, 2016. 22(4): p. 955-964.

[20] Chen, J., et al., An Analytical Study on Forced Vibration of Beams Carrying a Number of Two Degrees-of-Freedom Spring–Damper–Mass Subsystems. Journal of Vibration and Acoustics, 2016. 138(6).

[21] Ganguli, R. and S. Gouravaraju, Damage detection in cantilever beams using spatial Fourier coefficients of augmented modes. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 2016. 230(20): p. 3677-3690.

[22] De Rosa, M., et al., Free vibration of elastically restrained cantilever tapered beams with concentrated viscous damping and mass. Mechanics Research Communications, 2010. 37(2): p. 261-264.

[23] Liu, W. and M.E. Barkey, Nonlinear vibrational response of a single edge cracked beam. Journal of Mechanical Science and Technology, 2017. 31(11): p. 5231-5243.

[24] Rezaiee-Pajand, M., A.A. Sani, and S.M. Hozhabrossadati, Vibration suppression of a double-beam system by a two-degree-of-freedom mass-spring system. Smart Structures and Systems, 2018. 21(3): p. 349-358.

[25] Korayem, M., A. Alipour, and D. Younesian, Vibration suppression of atomic-force microscopy cantilevers covered by a piezoelectric layer with tensile force. Journal of Mechanical Science and Technology, 2018. 32(9): p. 4135-4144.

[26] Ahmadi, M., R. Ansari, and M. Darvizeh, Free and forced vibrations of atomic force microscope piezoelectric cantilevers considering tip-sample nonlinear interactions. Thin-Walled Structures, 2019. 145: p. 106382.

[27] Jazi, A.J., B. Shahriari, and K. Torabi, Exact closed form solution for the analysis of the transverse vibration mode of a nano-Timoshenko beam with multiple concentrated masses. International Journal of Mechanical Sciences, 2017. 131: p. 728-743.

[28] Pouretemad, A., K. Torabi, and H. Afshari, Free Vibration Analysis of a Rotating Non-uniform Nanocantilever Carrying Arbitrary Concentrated Masses Based on the Nonlocal Timoshenko Beam Using DQEM. INAE Letters, 2019. 4(1): p. 45-58.

[29] Nikhil, T., et al., Design and development of a test-rig for determining vibration characteristics of a beam. Procedia Engineering, 2016. 144: p. 312-320.

[30] El Baroudi, A. and F. Razafimahery, Transverse vibration analysis of Euler-Bernoulli beam carrying point masse submerged in fluid media. 2015.

[31] Mahmoud, M., Natural frequency of axially functionally graded, tapered cantilever beams with tip masses. Engineering Structures, 2019. 187: p. 34-42.

[32] Mangussi, F. and D.H. Zanette, Internal resonance in a vibrating beam: a zoo of nonlinear resonance peaks. PloS one, 2016. 11(9): p. e0162365.

[33] Ghannadiasl, A. and S.K. Ajirlou, Forced vibration of multi-span cracked Euler–Bernoulli beams using dynamic Green function formulation. Applied Acoustics, 2019. 148: p. 484-494.