0
Research Papers

Effect of Potential Energy Variation on the Natural Frequency of an Euler–Bernoulli Cantilever Beam Under Lateral Force and Compression

[+] Author and Article Information
B. Béri

Department of Applied Mechanics,
Budapest University of Technology
and Economics,
Budapest 1521, Hungary
e-mail: beri.bence@gmail.com

G. Stépán

Department of Applied Mechanics,
Budapest University of Technology
and Economics,
Budapest 1521, Hungary
e-mail: stepan@mm.bme.hu

S. J. Hogan

Department of Engineering Mathematics,
University of Bristol,
Bristol BS8 1UB, UK
e-mail: s.j.hogan@bristol.ac.uk

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 8, 2017; final manuscript received February 24, 2017; published online March 20, 2017. Assoc. Editor: Nick Aravas.

J. Appl. Mech 84(5), 051002 (Mar 20, 2017) (8 pages) Paper No: JAM-17-1016; doi: 10.1115/1.4036094 History: Received January 08, 2017; Revised February 24, 2017

A cantilever beam is subjected to both lateral force and compression under gravity. By taking into account the potential energy variation of the system, we develop a theoretical result that greatly simplifies the bending vibration frequency calculation in agreement with the experiments.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Cooke, J. , and Rand, R. , 1969, “ Vibratory Fruit Harvesting: A Linear Theory of Fruit-Stem Dynamics,” J. Agric. Eng. Res., 14(3), pp. 195–200. [CrossRef]
Borboni, A. , and De Santis, D. , 2014, “ Large Deflection of a Non-Linear, Elastic, Asymmetric Ludwick Cantilever Beam Subjected to Horizontal Force, Vertical Force and Bending Torque at the Free End,” Meccanica, 49(6), pp. 1327–1336.
Lee, K. , 2002, “ Large Deflections of Cantilever Beams of Non-Linear Elastic Material Under a Combined Loading,” Int. J. Non-Linear Mech., 37(3), pp. 439–443. [CrossRef]
González, C. , and LLorca, J. , 2005, “ Stiffness of a Curved Beam Subjected to Axial Load and Large Displacements,” Int. J. Solids Struct., 42(5–6), pp. 1537–1545. [CrossRef]
Solano-Carrillo, E. , 2009, “ Semi-Exact Solutions for Large Deflections of Cantilever Beams of Non-Linear Elastic Behaviour,” Int. J. Non-Linear Mech., 44(2), pp. 253–256. [CrossRef]
Beléndez, T. , Neipp, C. , and Beléndez, C. , 2002, “ Large and Small Deflections of a Cantilever Beam,” Eur. J. Phys., 23(3), pp. 371–379. [CrossRef]
Bazant, Z. , and Cedolin, L. , 2010, Stability of Structures, World Scientific Publishing Co. Pte. Ltd., Singapore.
Adamowitz, M. , and Stegun, I. , 1972, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, Dover Publications, New York.
Graff, K. , 1975, Wave Motion in Elastic Solids, Dover Publications, New York.
Jureczko, M. , Pawlak, M. , and Mezyk, A. , 2005, “ Optimisation of Wind Turbine Blades,” J. Mater. Process. Technol., 167(2–3), pp. 463–471. [CrossRef]
Kong, C. , Bang, J. , and Sugiyama, Y. , 2005, “ Structural Investigation of Composite Wind Turbine Blade Considering Various Load Cases and Fatigue Life,” Energy, 30(11–12), pp. 2101–2114. [CrossRef]
Lee, D. , Hwang, H. , and Kim, J. , 2003, “ Design and Manufacture of a Carbon Fiber Epoxy Rotating Boring Bar,” Compos. Struct., 60(1), pp. 115–124. [CrossRef]
Ema, S. , and Marui, E. , 2000, “ Suppression of Chatter Vibration of Boring Tools Using Impact Dampers,” Int. J. Mach. Tools Manuf., 40(8), pp. 1141–1156. [CrossRef]
Ema, S. , and Marui, E. , 2003, “ Theoretical Analysis on Chatter Vibration in Drilling and Its Suppression,” J. Mater. Process. Technol., 138(1–3), pp. 572–578. [CrossRef]
Bayly, P. , Lamar, M. , and Calvert, S. , 2002, “ Low-Frequency Regenerative Vibration and the Formation of Lobed Holes in Drilling,” ASME J. Manuf. Sci. Eng., 124(2), pp. 275–285. [CrossRef]
Roukema, J. , and Altintas, Y. , 2007, “ Generalized Modeling of Drilling Vibrations—Part II: Chatter Stability in Frequency Domain,” Int. J. Mach. Tools Manuf., 47(9), pp. 1474–1485. [CrossRef]
Heisig, G. , and Neubert, M. , 2000, “ Lateral Drillstring Vibrations in Extended-Reach Wells,” IADC/SPE Drilling Conference, New Orleans, LA, Feb. 23–25, SPE Paper No. SPE-59235-MS.
Park, J. , Park, H. , Jeong, S. , Lee, S. , Shin, Y. , and Park, J. , 2010, “ Linear Vibration Analysis of Rotating Wind-Turbine Blade,” Curr. Appl. Phys., 10(9), pp. S332–S334. [CrossRef]

Figures

Grahic Jump Location
Fig. 3

Mechanical model of a cantilever beam

Grahic Jump Location
Fig. 2

Approximation of the displacement of the end of the cantilever beam

Grahic Jump Location
Fig. 1

The experimental equipment that consists of a rectangular cross rod with length L and a heavy block with mass m = 0.978 kg

Grahic Jump Location
Fig. 4

Fixed-rolling beam under compression

Grahic Jump Location
Fig. 5

(a) Dimensionless relationship between κL and the relative importance of compression/tension given by Eq. (12). (b)Dimensionless relationship between relative importance of spring stiffness and the relative importance of compression/tension given by Eq. (15).

Grahic Jump Location
Fig. 6

Displacement analysis. The displacements of d11 and d21 are affected by the lateral force F. The d12 and d22 are due to compression P.

Grahic Jump Location
Fig. 8

Dimensionless connection between the normalized works and the relative importance of compression: (a) case 1 and (b) case 2

Grahic Jump Location
Fig. 9

Dynamical model of the two degrees-of-freedom compressed cantilever beam

Grahic Jump Location
Fig. 7

Works of external forces

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In