INAE Letters

pp 1–14 | Cite as

Free Vibration Analysis of a Rotating Non-uniform Nanocantilever Carrying Arbitrary Concentrated Masses Based on the Nonlocal Timoshenko Beam Using DQEM

  • Alireza PouretemadEmail author
  • Keivan Torabi
  • Hassan Afshari
Original Article


In this paper, a differential quadrature element method (DQEM) is proposed for free vibration analysis of rotating non-uniform nanocantilevers carrying multiple concentrated masses. Employing Hamilton’s principle, the governing equations of rotating nanoblades, modeled by the nonlocal Timoshenko beam theory, are derived. The differential quadrature (DQ) analogs of the governing equations of motion, compatibility conditions at the positions of masses, and related boundary conditions are established, and then, an eigen-value problem is obtained. The vibration characteristics of the problem are investigated with various conditions of number, positions, and magnitudes of the masses, while the cross section, rotational velocity, hub radius, and nonlocal parameters are arbitrary. The accuracy of the results is confirmed by the exact data available in the literature. In comparison with the other applicable methods, DQEM is less time-consuming and also enables the researcher to analyze the problem under arbitrary conditions, which are complex or even sometimes impossible to solve. The studies show that rotation rates, geometric properties, and masses conditions can contribute significantly in the dynamic characteristics of rotating N/MEMS devices such as nanoturbines, nanoscale molecular bearings, shaft and gear, and nanosensors.


Nonlocal Rotating Nonocantilever DQEM Vibration Masses 



  1. Abramovich H, Hamburger O (1991) Vibration of a cantilever Timoshenko beam with a tip mass. J Sound Vib 148:162–170. CrossRefGoogle Scholar
  2. Abramovich H, Hamburger O (1992) Vibration of a uniform cantilever Timoshenko beam with translational and rotational springs and with a tip mass. J Sound Vib 154:67–80. CrossRefzbMATHGoogle Scholar
  3. Afshari H, Irani Rahaghi M (2018) Whirling analysis of multi-span multi-stepped rotating shafts. J Braz Soc Mech Sci Eng 40:424. CrossRefGoogle Scholar
  4. Aranda-Ruiz J, Loya J, Fernández-Sáez J (2011) Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory. Comput Struct 94:2990–3000. CrossRefGoogle Scholar
  5. Bakhshi Kaniki H (2018) Vibration analysis of rotating nanobeam systems using Eringen’s two-phase local/nonlocal model. Phys E 92:310–319. CrossRefGoogle Scholar
  6. Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49(1):1–28. CrossRefGoogle Scholar
  7. Brunch JC Jr, Mitchell TP (1987) Vibrations of a mass-loaded clamped-free Timoshenko beam. J Sound Vib 114:341–345. CrossRefzbMATHGoogle Scholar
  8. De Rosa MA, Lippiello M, Maurizi MJ et al (2010) Free vibration of elastically restrained cantilever tapered beams with concentrated viscous damping and mass. Mech Res Commun 37:261–264. CrossRefzbMATHGoogle Scholar
  9. Elishakoff I, Challamel N, Soret C et al (2013) Virus sensor based on single-walled carbon nanotube: improved theory incorporating surface effects. Philos Trans R Soc A Math Phys Eng Sci 371:20120424. MathSciNetCrossRefzbMATHGoogle Scholar
  10. Eringen AC (1972a) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16. MathSciNetCrossRefzbMATHGoogle Scholar
  11. Eringen AC (1972b) Linear theory of nonlocal elasticity and dispersion of plane-waves. Int J Eng Sci 10(5):233–248. MathSciNetCrossRefzbMATHGoogle Scholar
  12. Eringen AC (1983) On differential-equations of nonlocal elasticity and solutions of screw dislocation and surface-waves. J Appl Phys 54(9):4703–4710. CrossRefGoogle Scholar
  13. Goel RP (1976) Free vibrations of a beam-mass system with elastically restrained ends. J Sound Vib 47:9–14. CrossRefzbMATHGoogle Scholar
  14. Grant DA (1978) The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass. J Sound Vib 57:357–365. CrossRefzbMATHGoogle Scholar
  15. Jafarzadeh Jazi A, Shahriari B, Torabi K (2017) Exact closed form solution for the analysis of the transverse vibration mode of a nano-Timoshenko beam with multiple concentrated masses. Int J Mech Sci 131–132:728–743. CrossRefGoogle Scholar
  16. Karami G, Malekzadeh P (2002) A new differential quadrature methodology for beam analysis and the associated differential quadrature element method. Comput Methods Appl Mech Eng 191(32):3509–3526. CrossRefzbMATHGoogle Scholar
  17. Karami G, Malekzadeh P, Shahpari SA (2003) A DQEM for vibration of shear deformable nonuniform beams with general boundary conditions. Eng Struct 25:1169–1178. CrossRefGoogle Scholar
  18. Kaya MO (2006) Free vibration analysis of a rotating Timoshenko beam by differential transform method. Aircr Eng Aerosp Technol 78:194–203. CrossRefGoogle Scholar
  19. Kral P, Sadeghpour HR (2002) Laser spinning of nanotubes: a path to fast-rotating microdevices. Phys Rev B 1:65. Google Scholar
  20. Krim J, Solina DH, Chiarello R (1991) Nanotribology of a Kr monolayer: a quartz-crystal microbalance study of atomic-scale friction. Phys Rev Lett 66(2):181–184. CrossRefGoogle Scholar
  21. Laura PAA, Pombo JA, Susemihl EA (1974) A note on the vibration of a clamped-free beam with a mass at the free end. J Sound Vib 37:161–168. CrossRefGoogle Scholar
  22. Lee SY, Lin SM (1995) Vibration of elastically restrained non-uniform Timoshenko beams. J Sound Vib 183:403–415. CrossRefzbMATHGoogle Scholar
  23. Li XF, Tang GJ, Shen ZB et al (2015) Resonance frequency and mass identification of zeptogram-scale nanosensor based on the nonlocal beam theory. Ultrasonics 55:75–84. CrossRefGoogle Scholar
  24. Lin HY, Tsai YC (2007) Free vibration analysis of a uniform multi-span carrying multiple spring-mass systems. J Sound Vib 302:442–456. CrossRefGoogle Scholar
  25. Matsuda H, Morita C, Sakiyama T (1992) A method for vibration analysis of a tapered Timoshenko beam with constraint at any points and carrying a heavy tip body. J Sound Vib 158:331–339. CrossRefzbMATHGoogle Scholar
  26. Murmu T, Adhikari S (2010) Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation. J Appl Phys. Google Scholar
  27. Murmu T, Adhikari S (2011) Nonlocal vibration of carbon nanotubes with attached buckyballs at tip. Mech Res Commun 38:62–67. CrossRefzbMATHGoogle Scholar
  28. Murmu T, Adhikari S (2012) Nonlocal frequency analysis of nanoscale biosensors. Sens Actuators A Phys 173:41–48. CrossRefGoogle Scholar
  29. Narendar S, Gopalakrishnan S (2011) Nonlocal wave propagation in rotating nanotube. Physics 1(1):17–25. Google Scholar
  30. Natsuki T, Matsuyama N, Shi J-X et al (2014) Vibration analysis of nanomechanical mass sensor using carbon nanotubes under axial tensile loads. Appl Phys A. Google Scholar
  31. Parnell LA, Cobble MH (1976) Lateral displacements of a vibrating cantilever beam with a concentrated mass. J Sound Vib 44:499–511. CrossRefzbMATHGoogle Scholar
  32. Pradhan SC, Murmu T (2010) Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Phys E 42(7):1944–1949. CrossRefGoogle Scholar
  33. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307. CrossRefzbMATHGoogle Scholar
  34. Rossi RE, Laura PAA, Avalos DR et al (1993) Free vibrations of Timoshenko beams carrying elastically mounted concentrated masses. J Sound Vib 165:209–223. CrossRefzbMATHGoogle Scholar
  35. Salarieh H, Ghorashi M (2006) Free vibration of Timoshenko beam with finite mass rigid tip load and flexural-torsional coupling. Int J Mech Sci 48:763–779. CrossRefzbMATHGoogle Scholar
  36. Shen ZB, Li XF, Sheng LP et al (2012a) Transverse vibration of nanotube-based micro-mass sensor via nonlocal Timoshenko beam theory. Comput Mater Sci 53:340–346. CrossRefGoogle Scholar
  37. Shen ZB, Sheng LP, Li XF et al (2012b) Nonlocal Timoshenko beam theory for vibration of carbon nanotube-based biosensor. Phys E 44:1169–1175. CrossRefGoogle Scholar
  38. Shen ZB, Tang GJ, Zhang L, Li XF (2012c) Vibration of double-walled carbon nanotube based nanomechanical sensor with initial axial stress. Comput Mater Sci 58:51–58. CrossRefGoogle Scholar
  39. Souayeh S, Kacem N (2014) Computational models for large amplitude nonlinear vibrations of electrostatically actuated carbon nanotube-based mass sensors. Sens Actuators A Phys 208:10–20. CrossRefGoogle Scholar
  40. To CWS (1982) Vibration of a cantilever beam with a base excitation and tip mass. J Sound Vib 83:445–460. CrossRefGoogle Scholar
  41. Torabi K, Jafarzadeh Jazi A, Zafari E (2014a) Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses. Appl Math Comput 238:342–357. MathSciNetzbMATHGoogle Scholar
  42. Torabi K, Afshari H, Heidari-Rarani M (2014b) Free vibration analysis of a rotating non-uniform blade with multiple open cracks using DQEM. Univers J Mech Eng 2(3):101–111. CrossRefGoogle Scholar
  43. Wu JS, Chen CT (2007) A lumped-mass TMM for free vibration analysis of a multi-step Timoshenko beam carrying eccentric lumped masses with rotary inertias. J Sound Vib 301:878–897. CrossRefGoogle Scholar
  44. Wu JS, Hsu SH (2007) The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia. Ocean Eng 34:54–68. CrossRefGoogle Scholar

Copyright information

© Indian National Academy of Engineering 2019

Authors and Affiliations

  • Alireza Pouretemad
    • 1
    Email author
  • Keivan Torabi
    • 2
  • Hassan Afshari
    • 3
  1. 1.Department of Mechanical EngineeringUniversity of KashanKashanIran
  2. 2.Department of Mechanical Engineering, Faculty of EngineeringUniversity of IsfahanIsfahanIran
  3. 3.Department of Mechanical Engineering, Khomeinishahr BranchIslamic Azad UniversityKhomeinishahrIran

Personalised recommendations