Skip to main content
SpringerLink
Log in

Vibration control of a transversely excited cantilever beam with tip mass

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler–Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency–response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Thomsen J.J.: Vibration suppression by using self-arranging mass: effects of adding restoring force. J. Sound Vib. 197, 403–425 (1996)

    Article  Google Scholar 

  2. Queini S.S., Nayfeh A.H.: Single-mode control of a cantilever beam under principal parametric excitation. J. Sound Vib. 224, 33–47 (1999)

    Article  Google Scholar 

  3. Eissa M., Amer Y.A.: Vibration control of a cantilever beam subject to both external and parametric excitation. Appl. Math. Comput. 152, 611–619 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Eissa, M., El-Ganaini, W.: Part II, Multi-absorbers for vibration control of non-linear structures to harmonic excitations. In: ISMV Conference. Islamabad, Pakistan (2000)

  5. El-Bassiouny A.F.: Single-mode control and chaos of cantilever beam under primary and principal parametric excitations. Chaos Solitons Fractals 30, 1098–1121 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. El-Bassiouny A.F.: Parametrically excited non-linear systems: a comparison of two methods. Appl. Math. Comput. 132, 385–410 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hegazy U.H.: Single-mode response and control of a hinged–hinged flexible beam. J. Springer Arch. Appl. Mech. 79, 335–345 (2009)

    Article  MATH  Google Scholar 

  8. Alhazza K.A., Daqaq M.F., Nayfeh A.H., Inman D.J.: Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control. Int. J. Non-Linear Mech. 43, 801–812 (2008)

    Article  MATH  Google Scholar 

  9. Yang X.D., Chen Li Q.: Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos Solitons Fractals 23, 249–258 (2005)

    Article  MATH  Google Scholar 

  10. Hyun H., Yoo H.H.: Dynamic modeling and stability analysis of axially oscillating cantilever beams. J. Sound Vib. 228, 543–558 (1999)

    Article  Google Scholar 

  11. Ji J.C., Leung A.Y.T.: Bifurcation control of a parametrically excited duffing system. Nonlinear Dyn. 27, 411–417 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lee H.P.: Stability of a cantilever beam with tip mass subjected to axial sinusoidal excitations. J. Sound Vib. 183, 91–98 (1995)

    Article  MATH  Google Scholar 

  13. Ozkaya E., Pakdemirli M.: Vibrations of an axially accelerating beam with small flexural stiffness. J. Sound Vib. 234, 521–535 (2000)

    Article  MathSciNet  Google Scholar 

  14. Marynowski K.: Non-linear dynamic analysis of an axially moving viscoelastic beam. J. Theor. Appl. Mech. 40, 465–482 (2002)

    MATH  Google Scholar 

  15. Sinha S.C., Redkar S., Deshmukh V., Butcher E.A.: Order reduction of parametrically excited nonlinear systems: techniques and applications. Nonlinear Dyn. 41, 237–273 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Theodore R.J., Arakeri J.H., Ghosal A.: Modelling of axially translating flexible beam. J. Sound Vib. 191, 363–376 (1996)

    Article  Google Scholar 

  17. To C.W.S.: Vibration of a cantilever beam with a base excitation and tip mass. J. Sound Vib. 83, 445–460 (1982)

    Article  Google Scholar 

  18. Yabuno H., Nayfeh A.H.: Nonlinear normal modes of a parametrically excited cantilever beam. Nonlinear Dyn. 25, 65–77 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang W., Wang F., Yao M.: Global bifurcations and chaotic dynamics in nonlinear nonplanar oscillations of a parametrically excited cantilever beam. Nonlinear Dyn. 40, 251–279 (2004)

    Article  MathSciNet  Google Scholar 

  20. Zhou D.: The vibrations of a cantilever beam carrying a heavy tip mass with elastic supports. J. Sound Vib. 206, 275–279 (1997)

    Article  Google Scholar 

  21. Nayfeh A.H., Mook D.T.: Nonlinear Oscillations. Wiley, Canada (1995)

    Book  Google Scholar 

  22. Nayfeh A.H., Balachandran B.: Applied Nonlinear Dynamics-Analytical, Computational and Experimental Methods. Wiley, Canada (1995)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering and Mining Machinery Engineering, Indian School of Mines, Dhanbad, 826004, India

    Barun Pratiher

Authors
  1. Barun Pratiher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barun Pratiher.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pratiher, B. Vibration control of a transversely excited cantilever beam with tip mass. Arch Appl Mech 82, 31–42 (2012). https://doi.org/10.1007/s00419-011-0537-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-011-0537-9

Keywords

Search

Navigation