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Vibration control of helicopter blade flapping via time-delay absorber

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Abstract

In this study, the controller is used to suppress the vibration due to rotor the helicopter blade flapping motion. The objective of this paper is to investigate the effect of time-delay absorber on the vibrating system when subjected to multi-parametric excitation forces. The equations of motion are described by coupled nonlinear differential equations. The averaging method is applied to obtain the frequency response equations near simultaneous sub-harmonic and internal resonance. The stability of the obtained nonlinear solution is studied and solved numerically. Numerical simulations show the steady state response amplitude versus the detuning parameter and the effects of the parameters system and controller. Effectiveness of the absorber E a is about 2.7×105 of the main system (X).

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Acknowledgements

The authors would like to thank the reviewers for their valuable comments and suggestions for improving the quality of this paper.

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Authors and Affiliations

  1. Department of Basic Sciences, Modern Academy for Engineering and Technology, Maadi, Egypt

    A. T. El-Sayed

  2. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

    H. S. Bauomy

  3. Department of Mathematics, College of Arts and Science in Wadi Addawasir, Salman Bin Abdulaziz University, P.O. Box 54, Wadi Addawasir, 11991, Saudi Arabia

    H. S. Bauomy

Authors
  1. A. T. El-Sayed
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  2. H. S. Bauomy
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Correspondence to A. T. El-Sayed.

Appendix

Appendix

$$\begin{aligned} & L_{1} = \biggl[ - \varepsilon \zeta_{1} \omega_{1} + \varepsilon \frac{F_{1}}{4\omega_{1}}\sin \gamma_{10} \biggr], \\ &L_{2} = \biggl[\varepsilon \frac{F_{1}a_{10}}{4\omega_{1}}\cos \gamma_{10}\biggr], \\ &L_{3} = \biggl[\varepsilon \frac{\beta}{2\omega_{1}}\sin \gamma_{20}\biggr], \\ &L_{4} = \biggl[\varepsilon \frac{\beta a_{20}}{2\omega_{1}}\cos \gamma_{20}\biggr], \\ & L_{5} = \biggl[ - \frac{\varepsilon \sigma_{1}}{a_{10}} - \varepsilon \frac{9\alpha_{1}a_{10}}{4\omega_{1}} + \varepsilon \frac{F_{1}}{2\omega_{1}a_{10}}\cos \gamma_{10} \biggr], \\ &L_{6} = - \biggl[\varepsilon \frac{F_{1}}{2\omega_{1}}\sin \gamma_{10}\biggr], \\ &L_{7} = \biggl[\varepsilon \frac{\beta}{\omega_{1}a_{10}}\cos \gamma_{20}\biggr], \\ &L_{8} = - \biggl[\varepsilon \frac{\beta a_{20}}{\omega_{1}a_{10}}\sin \gamma_{20}\biggr], \\ &L_{9} = - \biggl[\varepsilon \frac{\alpha_{2}}{2\omega_{2}}\{ \sin \gamma_{20}\cos \omega_{1}\tau + \cos \gamma_{20} \sin \omega_{1}\tau \} \biggr], \\ &L_{10} = - \varepsilon \zeta_{2}\omega_{2}, \\ & L_{11} = - \biggl[\varepsilon \frac{\alpha_{2}a_{10}}{2\omega_{2}}\{ \cos \gamma_{20}\cos \omega_{1}\tau - \sin \gamma_{20} \sin \omega_{1}\tau \} \biggr], \\ &L_{12} = - \biggl[\varepsilon \frac{\alpha_{2}}{2\omega_{2}a_{20}}\{ \cos \gamma_{20}\cos \omega_{1}\tau - \sin \gamma_{20} \sin \omega_{1}\tau \} \biggr] \\ & L_{13} = \biggl[\frac{\varepsilon \sigma_{2}}{a_{20}} - \frac{1}{2a_{20}}\varepsilon \sigma_{1}\biggr], \\ &L_{14} = \biggl[\varepsilon \frac{\alpha_{2}a_{10}}{2\omega_{2}a_{20}}\{ \sin \gamma_{20}\cos \omega_{1}\tau + \cos \gamma_{20} \sin \omega_{1}\tau \} \biggr] \end{aligned}$$
$$\begin{aligned} r_{1} &= ( - L_{14} - L_{1} - L_{6} - L_{10}) \\ r_{2} &= (L_{1}L_{6} + L_{1}L_{10} + L_{1}L_{14} + L_{6}L_{10} + L_{6}L_{14} \\ &\quad {} + L_{10}L_{14} - L_{11}L_{13} - L_{5}L_{2} - L_{9}L_{3} - L_{12}L_{4}) \\ r_{3} &= \bigl( - L_{6}L_{10}L_{14} + L_{1}L_{11}L_{13} - L_{1}L_{6}L_{10} \\ &\quad {} - L_{1}L_{6}L_{14} - L_{1}L_{10}L_{14} + L_{5}L_{2}L_{10} \\ &\quad {} - L_{9}L_{13}L_{4} + L_{9}L_{3}L_{14} + L_{6}L_{11}L_{13} - L_{9}L_{2}L_{7} \\ &\quad {}- L_{12}L_{2}L_{8} + L_{5}L_{2}L_{14} + L_{12}L_{6}L_{4} \\ &\quad {} + L_{9}L_{6}L_{3} + L_{12}L_{4}L_{10} - L_{12}L_{3}L_{11}\bigr) \\ r_{4} &= (L_{1}L_{6}L_{10}L_{14} - L_{9}L_{2}L_{13}L_{8} + L_{9}L_{6}L_{13}L_{4} \\ &\quad {} - L_{1}L_{6}L_{11}L_{13} - L_{5}L_{2}L_{10}L_{14} + L_{12}L_{2}L_{8}L_{10}\\ &\quad {}- L_{12}L_{6}L_{4}L_{10} - L_{12}L_{2}L_{7}L_{11} + L_{9}L_{2}L_{7}L_{14}\\ &\quad {} + L_{12}L_{6}L_{3}L_{11} + L_{5}L_{2}L_{11}L_{13} - L_{9}L_{6}L_{3}L_{14}) \end{aligned}$$

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El-Sayed, A.T., Bauomy, H.S. Vibration control of helicopter blade flapping via time-delay absorber. Meccanica 49, 587–600 (2014). https://doi.org/10.1007/s11012-013-9813-9

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