Acta Mechanica

, Volume 216, Issue 1–4, pp 207–223 | Cite as

Dynamic stability of linear parametrically excited twisted Timoshenko beams under periodic axial loads

Article
First Online:
Received:
Revised:

Abstract

In this paper, the parametric instability of twisted Timoshenko beams with various end conditions and under an axial pulsating force is studied. The equations of motion in the twisted frame are derived using a finite element method. Based on Bolotin’s method, a set of second-order ordinary differential equations with periodic coefficients of Mathieu–Hill type is formed to determine the instability regions for twisted Timoshenko beams. A dynamic instability index is defined and used as an instability measure to study the influence of various parameters. The effects of beam length, inertia ratio, pre-twist angle, dynamic component of axial force and restraint condition on the instability regions and dynamic instability index of the twisted beam are investigated and discussed.

Keywords

Dynamic Stability Timoshenko Beam Instability Region Timoshenko Beam Theory Bernoulli Beam Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

A

Cross-sectional area

b

Width of the beam

DII

Dynamic instability index

d

Displacement matrix in frame ξηZ

d(e)

Displacement function of the beam

E

Young’s modulus

Fz(t)

Time-dependent axial load

F

Global stiffness matrix due to unit axial force

F(e)

Element stiffness matrix due to unit axial force

G

Shear modulus

IXX, IYY, IXY

Area moments and product of inertia in frame XYZ

Iξ, Iη

Principal area moments of inertia in frame ξηZ

JXX, JYY, JXY

Mass moments and product of inertia per unit length in frame XYZ

Jξ, Jη

Principal mass moments of inertia per unit length in frame ξηZ

K

Global stiffness matrix

\({\overline{\bf{K}}_1, \overline{\bf{K}}_2, \overline{\bf{K}}_3, \overline{\bf{K}}_4}\)

Stiffness coefficient matrices due to the bending and shear effects

KB

Global stiffness matrix due to the bending and shear effects

\({{\boldsymbol K}_B^{(e)}}\)

Element stiffness matrix due to the bending and shear effects

\({{\boldsymbol K}_F^{(e)}}\)

Element stiffness matrix due to axial force

L

Beam length

Le

Beam element length

Lref

Reference beam length

m

beam mass per unit length

M

Global inertia matrix

M(e)

Element inertia matrix

\({\overline{\bf{M}}}\)

Inertia coefficient matrix

N

Transformation matrix between displacement function and nodal displacements

N1, N2

Shape functions of linear beam element

Pcr

Critical static buckling load

\({\overline{{P}}_{\rm cr}}\)

Dimensionless critical buckling load

p

Global displacement matrix

p(e)

Element displacement matrix

q

Constant vector

R

Inertia ratio of beam cross-section

T

Kinetic energy

t

Thickness of the beam

uX, uY

Total transverse displacements in frame XYZ

uξ, uη

Transverse displacements in frame ξηZ

uξ1, uη1, uξ 2, uη2

Nodal transverse displacements in frame ξηZ

V

Potential energy

W

Work produced by the axial load

Z

Axial coordinate

κ

Shear correction factor

ρ

Density of the beam

ω

Natural frequency

Ω

Disturbing or boundary frequency

\({\overline{{\Omega}}}\)

Boundary frequency ratio

Δ Ω

Opening of instability region

α

Static load factor

β

Dynamic load factor

βo

Twist angle per unit length

\({\phi}\)

Total twist angle

\({\varphi_{x}, \varphi_{y}}\)

Angles of rotation in frame XYZ

\({\varphi_{\xi}, \varphi_{\eta}}\)

Angles of rotation in frame ξηZ

\({\varphi_{\xi 1}, \varphi_{\eta 1}, \varphi_{\xi 2}, \varphi_{\eta 2}}\)

Nodal angles of rotation in frame ξηZ

This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Leissa A.: Vibrational aspects of rotating turbomachinery blades. Appl. Mech. Rev. 34, 629–635 (1981)Google Scholar
  2. 2.
    Rosen A.: Structural and dynamic behavior of pretwisted rods and beams. Appl. Mech. Rev. 44, 483–515 (1991)CrossRefGoogle Scholar
  3. 3.
    Carnegie W.: Vibrations of pre-twisted cantilever blading. Proc. Inst. Mech. Eng. 173, 343–374 (1959)CrossRefGoogle Scholar
  4. 4.
    Dawson B., Carnegie W.: Modal curves of pre-twisted beams of rectangular cross-section. J. Mech. Eng. Sci. 11, 1–13 (1969)CrossRefGoogle Scholar
  5. 5.
    Sabuncu M.: Coupled vibration analysis of blades with angular pretwist of cubic distribution. AIAA J. 23, 1424–1430 (1985)CrossRefGoogle Scholar
  6. 6.
    Carnegie W.: Vibrations of pre-twisted cantilever blading allowing for rotary inertia and shear deflection. J. Mech. Eng. Sci. 6, 105–109 (1964)CrossRefGoogle Scholar
  7. 7.
    Dawson B., Ghosh N.G., Carnegie W.: Effect of slenderness ratio on the natural frequencies of pre-twisted cantilever beam of uniform rectangular cross-section. J. Mech. Eng. Sci. 13, 51–59 (1971)CrossRefGoogle Scholar
  8. 8.
    Gupta R.S., Rao S.S.: Finite element eigenvalue analysis of tapered and twisted Timoshenko beams. J. Sound Vib. 56, 187–200 (1978)MATHCrossRefGoogle Scholar
  9. 9.
    Abbas B.A.H.: Simple finite elements for dynamic analysis of thick pre-twisted blades. Aeronaut. J. 83, 450–453 (1979)Google Scholar
  10. 10.
    Subrahamanyam K.B., Kulkami S.V., Rao J.S.: Coupled bending-bending vibrations of pre-twisted cantilever blading allowing for shear deflection and rotary by the reissner method. Int. J. Mech. Sci. 23, 517–530 (1981)CrossRefGoogle Scholar
  11. 11.
    Lin S.M., Wang W.R., Lee S.Y.: The dynamic analysis of nonuniformly pre-twisted Timoshenko beams with elastic boundary conditions. Int. J. Mech. Sci. 43, 2385–2405 (2001)MATHCrossRefGoogle Scholar
  12. 12.
    Banerjee J.R.: Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam. J. Sound Vib. 270, 379–401 (2004)CrossRefGoogle Scholar
  13. 13.
    Yardimoglu B., Yildirim T.: Finite element model for vibration analysis of pre-twisted Timoshenko beam. J. Sound Vib. 273, 741–754 (2004)CrossRefGoogle Scholar
  14. 14.
    Fu C.C.: Computer analysis of a rotating axial turbomachine blade in coupled bending-bending-torsion vibrations. Int. J. Num. Methods Eng. 8, 569–588 (1974)MATHCrossRefGoogle Scholar
  15. 15.
    Rao J.S., Carnegie W.: A numerical procedure for the determination of the frequencies and mode shapes of lateral vibration of blades allowing for the effect of pre-twisting and rotation. Int. J. Mech. Eng. Edu. 1, 37–47 (1973)Google Scholar
  16. 16.
    Subrahamanyam K.B., Kaza K.R.V.: Vibration and buckling of rotating, pretwisted, preconed beams including Coriolis effects. ASME J. Vib. Acoust. Str. Rel. Des. 108, 140–149 (1981)Google Scholar
  17. 17.
    Rao S.S., Gupta R.S.: Finite element vibration analysis of rotating Timoshenko beams. J. Sound Vib. 242, 103–124 (2001)CrossRefGoogle Scholar
  18. 18.
    Subuncu M., Evran K.: Dynamic stability of a rotating pre-twisted asymmetric cross-section Timoshenko beam subjected to an axial periodic force. Int. J. Mech. Sci. 48, 579–590 (2006)CrossRefGoogle Scholar
  19. 19.
    Tekinalp O., Ulsoy A.G.: Modeling and finite element analysis of drill bit vibrations. ASME J. Vib. Acoust. Str. Rel. Des. 111, 148–155 (1989)Google Scholar
  20. 20.
    Tekinalp O., Ulsoy A.G.: Effects of geometric and process parameters on drill transverse vibrations. ASME J. Eng. Ind. 112, 189–194 (1990)CrossRefGoogle Scholar
  21. 21.
    Lee H.P.: Dynamic stability of spinning pre-twisted beams. Int. J. Sol. Struct. 31, 2509–2517 (1994)MATHCrossRefGoogle Scholar
  22. 22.
    Lee H.P.: Buckling and dynamic stability of spinning pre-twisted beams under compressive axial loads. Int. J. Mech. Sci. 36, 1011–1026 (1994)MATHCrossRefGoogle Scholar
  23. 23.
    Liao C.L., Dang Y.H.: Structural characteristics of spinning pretwisted orthotropic beams. Comput. Struct. 45, 715–731 (1992)CrossRefGoogle Scholar
  24. 24.
    Chen W.R., Keer L.M.: Transverse vibrations of a rotating twisted Timoshenko beam under axial loading. ASME J. Vib. Acoust. 115, 285–294 (1992)CrossRefGoogle Scholar
  25. 25.
    Chen W.R.: Parametric studies on buckling loads and critical speeds of microdrill bits. Int. J. Mech. Sci. 49, 935–949 (2007)CrossRefGoogle Scholar
  26. 26.
    Chen C.K., Ho S.H.: Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform. Int. J. Mech. Sci. 41, 1339–1356 (1999)MATHCrossRefGoogle Scholar
  27. 27.
    Ho S.H., Chen C.K.: Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform. Int. J. Mech. Sci. 48, 1323–1331 (2006)MATHCrossRefGoogle Scholar
  28. 28.
    Bolotin V.V.: The Dynamic Stability of Elastic Systems. Holden-Day, San Francisco (1964)MATHGoogle Scholar
  29. 29.
    Evan-Ivanowski R.M.: Resonance Oscillations in Mechanical Systems. Elsevier, New York (1976)Google Scholar
  30. 30.
    Celep Z.: Dynamic stability of pretwisted columns under periodic axial loads. J. Sound Vib. 103, 35–48 (1985)CrossRefGoogle Scholar
  31. 31.
    Gürgöze M.: On the dynamic stability of a pre-twisted beam subject to a pulsating axial load. J. Sound Vib. 102, 415–422 (1985)CrossRefGoogle Scholar
  32. 32.
    Lee H.P.: Dynamic stability of spinning pre-twisted beams subject to axial pulsating loads. Comput. Methods Appl. Mech. Eng. 127, 115–126 (1995)MATHCrossRefGoogle Scholar
  33. 33.
    Liao C.L., Huang B.W.: Parametric instability of a spinning pretwisted beam under periodic axial force. Int. J. Mech. Sci. 37, 423–439 (1995)MATHCrossRefGoogle Scholar
  34. 34.
    Tan T.H., Lee H.P., Leng G.S.B.: Parametric instability of spinning pretwisted beams subjected to sinusoidal compressive axial loads. Comput. Struct. 6, 745–764 (1998)CrossRefGoogle Scholar
  35. 35.
    Young T.H., Gau C.Y.: Dynamic stability of pre-twisted beams with non-constant spin rates under axial random forces. Int. J. Solids Struct. 18, 4675–4698 (2003)CrossRefGoogle Scholar
  36. 36.
    Meirovitch L.: Analytical Method in Vibrations. The Macmillan Co., London (1967)Google Scholar
  37. 37.
    Bathe K.J.: Finite Element Procedures. Prentice-Hall, New Jersey (1996)Google Scholar
  38. 38.
    Hughes T.J.R., Taylor R.L., Kanoknukulchai W.: A simple and efficient finite element for plate bending. Int. J. Num. Methods Eng. 11, 1529–1543 (1977)MATHCrossRefGoogle Scholar
  39. 39.
    Anderson E., Bai Z., Bischof C., Blackford S., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., McKenney A., Sorensen D.: LAPACK Users’ Guide. 3rd edn. SIAM, Philadelphia (1999)Google Scholar
  40. 40.
    Mohanty, S.C.: Dynamic Stability of Beams Under Parametric Excitation. PhD thesis, National Institute of Technology (2005)Google Scholar
  41. 41.
    Briseghella G., Majorana C.E., Pellegrino C.: Dynamic stability of elastic structures: a finite element approach. Comput. Struct. 69, 11–25 (1998)MATHCrossRefGoogle Scholar
  42. 42.
    Shastry B.P., Rao G.V.: Stability boundaries of a cantilever column subjected to an intermediate periodic concentrated axial load. J. Sound Vib. 116, 195–198 (1987)CrossRefGoogle Scholar
  43. 43.
    Wang S., Dawe D.J.: Dynamic instability of composite laminated rectangular plates and prismatic plate structures. Comput. Methods Appl. Mech. Eng. 191, 1791–1826 (2002)MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringChinese Culture UniversityTaipeiTaiwan, ROC

Personalised recommendations