Acta Mechanica Sinica

, Volume 34, Issue 4, pp 676–688

# Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

• Hai-Sheng Zhao
• Yao Zhang
• Seng-Tjhen Lie
Research Paper

## Abstract

Considerations of nonlocal elasticity and surface effects in micro- and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short, explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.

## Keywords

Fredholm integral equation Natural frequency Nonlocal elasticity Surface effects Timoshenko beam

## Notes

### Acknowledgements

The authors would like to thank the School of Civil and Environmental Engineering at Nanyang Technological University, Singapore for kindly supporting this research topic.

## References

1. 1.
Wang, G.F., Feng, X.Q.: Timoshenko beam model for buckling and vibration of nanowires with surface effects. J. Phys. D Appl. Phys. 42, 155411 (2009)
2. 2.
Li, Y., Song, J., Fang, B., et al.: Surface effects on the postbuckling of nanowires. J. Phys. D Appl. Phys. 44, 425304 (2011)
3. 3.
Wang, G.F., Feng, X.Q.: Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett. 90, 231904 (2007)
4. 4.
He, J., Lilley, C.M.: Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl. Phys. Lett. 93, 263108 (2008)
5. 5.
Farshi, B., Assadi, A., Alinia-ziazi, A.: Frequency analysis of nanotubes with consideration of surface effects. Appl. Phys. Lett. 96, 093105 (2010)
6. 6.
Gurtin, M.E., Weissmüller, J., Larché, F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)
7. 7.
Timoshenko, S.P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921)
8. 8.
Timoshenko, S.P.: On the transverse vibrations of bars of uniform cross-section. Philos. Mag. 43, 125–131 (1922)
9. 9.
Ansari, R., Gholami, R., Norouzzadeh, A., et al.: Surface stress effect on the vibration and instability of nanoscale pipes conveying fluid based on a size-dependent Timoshenko beam model. Acta Mech. Sin. 31, 708–719 (2015)
10. 10.
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)
11. 11.
Wang, C.M., Zhang, Y.Y., He, X.Q.: Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007)
12. 12.
Li, X.-F., Wang, B.-L.: Vibrational modes of Timoshenko beams at small scales. Appl. Phys. Lett. 94, 101903 (2009)
13. 13.
Ansari, R., Torabi, J.: Nonlocal vibration analysis of circular double-layered graphene sheets resting on an elastic foundation subjected to thermal loading. Acta Mech. Sin. 32, 841–853 (2016)
14. 14.
Fernández-Sáez, J., Zaera, R., Loya, J.A., et al.: Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int. J. Eng. Sci. 99, 107–116 (2016)
15. 15.
Karličić, D., Kozić, P., Pavlović, R.: Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium. Appl. Math. Model. 40, 1599–1614 (2016)
16. 16.
Nejad, M.Z., Hadi, A.: Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams. Int. J. Eng. Sci. 105, 1–11 (2016)
17. 17.
Wang, L.F., Hu, H.Y.: Flexural wave propagation in single-walled carbon nanotubes. Phys. Rev. B 71, 195412 (2005)
18. 18.
Reddy, J.N., El-Borgi, S.: Eringen’s nonlocal theories of beams accounting for moderate rotations. Int. J. Eng. Sci. 82, 159–177 (2014)
19. 19.
Zhang, Y.: Frequency spectra of nonlocal Timoshenko beams and an effective method of determining nonlocal effect. Int. J. Mech. Sci. 128–129, 572–582 (2017)
20. 20.
Maranganti, R., Sharma, P.: Length scales at which classical elasticity breaks down for various materials. Phys. Rev. Lett. 98, 195504 (2007)
21. 21.
Zhang, Y., Zhao, Y.P.: Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies. Int. J. Solids Struct. 102–103, 259–266 (2016)
22. 22.
Huang, Y., Li, X.F.: A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J. Sound Vib. 329, 2291–2303 (2010)
23. 23.
Shi, W.C., Li, X.F., Lee, K.Y.: Transverse vibration of free–free beams carrying two unequal end masses. Int. J. Mech. Sci. 90, 251–257 (2015)
24. 24.
Shi, W.C., Shen, Z.B., Peng, X.L., et al.: Frequency equation and resonant frequencies of free–free Timoshenko beams with unequal end masses. Int. J. Mech. Sci. 115–116, 406–415 (2016)
25. 25.
Zhang, Z., Wang, C.M., Challamel, N., et al.: Obtaining Eringenxs length scale coefficient for vibrating nonlocal beams via continualization method. J. Sound Vib. 333, 4977–4990 (2014)
26. 26.
Lei, Y., Adhikari, S., Friswell, M.I.: Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams. Int. J. Eng. Sci. 66–67, 1–13 (2013)
27. 27.
Downs, B.: Transverse vibration of a uniform simply supported timoshenko beam without transverse deflection. J. Appl. Mech. 43, 671–674 (1976)
28. 28.
Abbas, B.A.H., Thomas, J.: The second frequency spectrum of timoshenko beams. J. Sound Vib. 51, 123–137 (1977)
29. 29.
Bhashyam, G.R., Prathap, G.: The second frequency spectrum of timoshenko beams. J. Sound Vib. 76, 407–420 (1981)
30. 30.
Levinson, M., Cooke, D.W.: On the two frequency spectra of timoshenko beams. J. Sound Vib. 84, 319–326 (1982)
31. 31.
Stephen, N.G.: The second spectrum of Timoshenko beam theory—further assessment. J. Sound Vib. 292, 372–389 (2006)
32. 32.
Lu, P., Lee, H.P., Lu, C., et al.: Dynamic properties of flexural beams using a nonlocal elasticity model. J. Appl. Phys. 99, 073510 (2006)
33. 33.
Chang, T.C., Craig, R.R.: Normal modes of uniform beams. J. Eng. Mech. 195, 1027–1031 (1969)Google Scholar