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Primary and secondary resonance analyses of a cantilever beam carrying an intermediate lumped mass with time-delay feedback

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Abstract

Time-delay displacement and velocity feedback of different types of active control in a cantilever beam carrying an lumped mass is investigated in this paper. Based on Euler–Bernoulli beam theory, the nonlinear governing equation is studied with damping, harmonic distribution, displacement delay, velocity delay and two time delays. The multiple scales perturbation method is applied to obtain the frequency response equations near primary, superharmonic and subharmonic resonances. A thorough study on the stability is proposed, with a particular emphasis on delay feedback. The results show that the hardening and softening behaviors of the system depend on the location of lumped mass. Furthermore, the displacement feedback gain coefficient only makes the peak amplitude move to the low frequency, yet velocity feedback coefficient and their time delays can be used to effectively enhance the stability and quench the nonlinear vibration of the cantilever beam. Thus, reasonable selection of the control system parameters can effectively improve the level of vibration control for the mechanical system.

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Acknowledgements

The work described in this paper is funded by the research Grant from the Natural Science Foundation of China (Grant Nos. 11662006 and 11172115). The authors are grateful for their financial support.

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Authors and Affiliations

  1. Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, 650500, Yunnan, People’s Republic of China

    Chun-Xia Liu, Yan Yan & Wen-Quan Wang

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  1. Chun-Xia Liu
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  2. Yan Yan
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Correspondence to Yan Yan or Wen-Quan Wang.

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Appendix A

Appendix A

$$\begin{aligned}&\varepsilon ^{1}:\,\,D_0^2 u_1 +u_1 =+e^{-5i\Omega T_0 }\left( {-\alpha _2 \Lambda ^{5}+2\Lambda ^{5}\beta _2 \Omega ^{2}} \right) \nonumber \\&\qquad +\,e^{5i\Omega T_0 }\left( {-\alpha _2 \Lambda ^{5}+2\Lambda ^{5}\beta _2 \Omega ^{2}} \right) +e^{-5iT_0 }\left( {-\alpha _2 A^{5}+2A^{5}\beta _2 } \right) \nonumber \\&\qquad +\,e^{5iT_0 }\left( {-\alpha _2 A^{5}+2A^{5}\beta _2 } \right) \nonumber \\&\qquad +\,e^{-3i\Omega T_0 }\left( {\begin{array}{l} -\alpha _1 \Lambda ^{3}-20A\bar{{A}}\alpha _2 \Lambda ^{3}-5\alpha _2 \Lambda ^{5}+2\beta _1 \Omega ^{2}\Lambda ^{3}+12A\bar{{A}}\beta _2 \Lambda ^{3}\Omega ^{2} \\ +\,6\beta _2 \Lambda ^{5}\Omega ^{2}+6A\bar{{A}}\beta _2 \Lambda ^{3}\Omega ^{2}+6A\bar{{A}}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{3i\Omega T_0 }\left( {\begin{array}{l} -\alpha _1 \Lambda ^{3}-20A\bar{{A}}\alpha _2 \Lambda ^{3}-5\alpha _2 \Lambda ^{5}+2\beta _1 \Omega ^{2}\Lambda ^{3}+12A\bar{{A}}\beta _2 \Lambda ^{3}\Omega ^{2} \\ +\,6\beta _2 \Lambda ^{5}\Omega ^{2}+6A\bar{{A}}\beta _2 \Lambda ^{3}\Omega ^{2}+6A\bar{{A}}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-i\Omega T_0 }\left( {\begin{array}{l} -\,6A\bar{{A}}\alpha _1 \Lambda -30A^{2}\bar{{A}}^{2}\alpha _2 \Lambda -3\alpha _1 \Lambda ^{3}-60A\bar{{A}}\alpha _2 \Lambda ^{3}-10\alpha _2 \Lambda ^{5} \\ +\,2A\bar{{A}}\beta _1 \Lambda \Omega ^{2}+2\beta _1 \Lambda ^{3}\Omega ^{2}+6A^{2}\bar{{A}}^{2}\beta _2 \Lambda \Omega ^{2}+30A\bar{{A}}\beta _2 \Lambda ^{3}\Omega ^{2} \\ +\,8\beta _2 \Lambda ^{5}\Omega ^{2}+2A\bar{{A}}\beta _1 \Lambda +18A^{2}\bar{{A}}^{2}\beta _2 \Lambda +18A\bar{{A}}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{i\Omega T_0 }\left( {\begin{array}{l} -\,6A\bar{{A}}\alpha _1 \Lambda -30A^{2}\bar{{A}}^{2}\alpha _2 \Lambda -3\alpha _1 \Lambda ^{3}-60A\bar{{A}}\alpha _2 \Lambda ^{3}-10\alpha _2 \Lambda ^{5} \\ +\,2A\bar{{A}}\beta _1 \Lambda \Omega ^{2}+2\beta _1 \Lambda ^{3}\Omega ^{2}+6A^{2}\bar{{A}}^{2}\beta _2 \Lambda \Omega ^{2}+30A\bar{{A}}\beta _2 \Lambda ^{3}\Omega ^{2} \\ +\,8\beta _2 \Lambda ^{5}\Omega ^{2}+2A\bar{{A}}\beta _1 \Lambda +\,18A^{2}\bar{{A}}^{2}\beta _2 \Lambda +18A\bar{{A}}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{4iT_0 }\left( {\begin{array}{l} e^{i\Omega T_0 }\left( {-5A^{4}\alpha _2 \Lambda +A^{4}\beta _2 \Lambda \Omega ^{2}+2A^{4}\beta _2 \Lambda \Omega +7A^{4}\beta _2 \Lambda } \right) \\ +e^{-i\Omega T_0 }\left( {-5A^{4}\alpha _2 \Lambda +A^{4}\beta _2 \Lambda \Omega ^{2}-2A^{4}\beta _2 \Lambda \Omega +7A^{4}\beta _2 \Lambda } \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-4iT_0 }\left( {\begin{array}{l} e^{i\Omega T_0 }\left( {-5A^{4}\alpha _2 \Lambda +A^{4}\beta _2 \Lambda \Omega ^{2}+2A^{4}\beta _2 \Lambda \Omega +7A^{4}\beta _2 \Lambda } \right) \\ +\,e^{-i\Omega T_0 }\left( {-5A^{4}\alpha _2 \Lambda +A^{4}\beta _2 \Lambda \Omega ^{2}-2A^{4}\beta _2 \Lambda \Omega +7A^{4}\beta _2 \Lambda } \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{3iT_0 }\left( {\begin{array}{l} -A^{3}\alpha _1 -5A^{4}\bar{{A}}\alpha _2 -20A^{3}\alpha _2 \Lambda ^{2}+6A^{3}\beta _2 \Lambda ^{2}\Omega ^{2}+2A^{3}\beta _1 +6A^{4}\bar{{A}}\beta _2 \\ +\,18A^{3}\beta _2 \Lambda ^{2} \\ +e^{2i\Omega T_0 }\left( {-10A^{3}\alpha _2 \Lambda ^{2}+5A^{3}\Lambda ^{2}\beta _2 \Omega ^{2}+6A^{3}\beta _2 \Lambda ^{2}\Omega +9A^{3}\beta _2 \Lambda ^{2}} \right) \\ +e^{-2i\Omega T_0 }\left( {-10A^{3}\alpha _2 \Lambda ^{2}+5A^{3}\Lambda ^{2}\beta _2 \Omega ^{2}-6A^{3}\beta _2 \Lambda ^{2}\Omega +9A^{3}\beta _2 \Lambda ^{2}} \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-3iT_0 }\left( {\begin{array}{l} -A^{3}\alpha _1 -5A^{4}\bar{{A}}\alpha _2 -20A^{3}\alpha _2 \Lambda ^{2}+6A^{3}\beta _2 \Lambda ^{2}\Omega ^{2}+2A^{3}\beta _1 +6A^{4}\bar{{A}}\beta _2 \\ +\,18A^{3}\beta _2 \Lambda ^{2} \\ +e^{2i\Omega T_0 }\left( {-10A^{3}\alpha _2 \Lambda ^{2}+5A^{3}\Lambda ^{2}\beta _2 \Omega ^{2}+6A^{3}\beta _2 \Lambda ^{2}\Omega +9A^{3}\beta _2 \Lambda ^{2}} \right) \\ +e^{-2i\Omega T_0 }\left( {-10A^{3}\alpha _2 \Lambda ^{2}+5A^{3}\Lambda ^{2}\beta _2 \Omega ^{2}-6A^{3}\beta _2 \Lambda ^{2}\Omega +9A^{3}\beta _2 \Lambda ^{2}} \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{iT_0 }\left( {\begin{array}{l} -\,3A^{2}\alpha _1 \bar{{A}}-10A^{3}\bar{{A}}^{2}\alpha _2 -6A\alpha _1 \Lambda ^{2}-60A^{2}\bar{{A}}\alpha _2 \Lambda ^{2}-30A\alpha _2 \Lambda ^{4}+2A\Lambda ^{2}\beta _1 \Omega ^{2} \\ +\,18A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega ^{2}+18A\Lambda ^{4}\beta _2 \Omega ^{2}+2A^{2}\bar{{A}}\beta _1 +2A\Lambda ^{2}\beta _1 +8A^{3}\bar{{A}}^{2}\beta _2 \\ +\,30A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \omega _0^2 +6A\Lambda ^{4}\beta _2 +g_p Ae^{-i\tau _1 }+ig_d Ae^{-i\tau _2 } \\ +e^{4i\Omega T_0 }\left( {-5A\alpha _2 \Lambda ^{4}+7A\Lambda ^{4}\beta _2 \Omega ^{2}+2A\beta _2 \Lambda ^{4}\Omega +A\beta _2 \Lambda ^{4}} \right) \\ +e^{-4i\Omega T_0 }\left( {-5A\alpha _2 \Lambda ^{4}+7A\Lambda ^{4}\beta _2 \Omega ^{2}-2A\beta _2 \Lambda ^{4}\Omega +A\beta _2 \Lambda ^{4}} \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{iT_0 }e^{2i\Omega T_0 }\left( {\begin{array}{l} -\,3A\alpha _1 \Lambda ^{2}-30A^{2}\bar{{A}}\alpha _2 \Lambda ^{2}-20A\alpha _2 \Lambda ^{4}+3A\Lambda ^{2}\beta _1 \Omega ^{2}+15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega ^{2} \\ 16A\Lambda ^{4}\beta _2 \Omega ^{2}+2A\Lambda ^{2}\beta _1 \Omega +6A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega +4A\Lambda ^{4}\beta _2 \Omega +A\Lambda ^{2}\beta _1 \\ +\,15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 +4A\Lambda ^{4}\beta _2 \\ \end{array}} \right) \end{aligned}$$
$$\begin{aligned}&\qquad +\,e^{iT_0 }e^{-2i\Omega T_0 }\left( {\begin{array}{l} -\,3A\alpha _1 \Lambda ^{2}-30A^{2}\bar{{A}}\alpha _2 \Lambda ^{2}-20A\alpha _2 \Lambda ^{4}+3A\Lambda ^{2}\beta _1 \Omega ^{2}+15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega ^{2} \\ 16A\Lambda ^{4}\beta _2 \Omega ^{2}-2A\Lambda ^{2}\beta _1 \Omega -6A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega -4A\Lambda ^{4}\beta _2 \Omega +A\Lambda ^{2}\beta _1 \\ +\,15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 +4A\Lambda ^{4}\beta _2 \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-iT_0 }\left( {\begin{array}{l} -3A^{2}\alpha _1 \bar{{A}}-10A^{3}\bar{{A}}^{2}\alpha _2 -6A\alpha _1 \Lambda ^{2}-60A^{2}\bar{{A}}\alpha _2 \Lambda ^{2}-30A\alpha _2 \Lambda ^{4} \\ +\,2A\Lambda ^{2}\beta _1 \Omega ^{2}+18A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega ^{2}+18A\Lambda ^{4}\beta _2 \Omega ^{2}+2A^{2}\bar{{A}}\beta _1 \\ +\,2A\Lambda ^{2}\beta _1 +8A^{3}\bar{{A}}^{2}\beta _2 +30A^{2}\bar{{A}}\Lambda ^{2}\beta _2 +6A\Lambda ^{4}\beta _2 \\ +e^{4i\Omega T_0 }\left( {-5A\alpha _2 \Lambda ^{4}+7A\Lambda ^{4}\beta _2 \Omega ^{2}+2A\beta _2 \Lambda ^{4}\Omega +A\beta _2 \Lambda ^{4}} \right) \\ +e^{-4i\Omega T_0 }\left( {-5A\alpha _2 \Lambda ^{4}+7A\Lambda ^{4}\beta _2 \Omega ^{2}-2A\beta _2 \Lambda ^{4}\Omega +A\beta _2 \Lambda ^{4}} \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-iT_0 }e^{2i\Omega T_0 }\left( {\begin{array}{l} -\,3A\alpha _1 \Lambda ^{2}-30A^{2}\bar{{A}}\alpha _2 \Lambda ^{2}-20A\alpha _2 \Lambda ^{4}+3A\Lambda ^{2}\beta _1 \Omega ^{2} \\ +\,15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega ^{2}+16A\Lambda ^{4}\beta _2 \Omega ^{2}+2A\Lambda ^{2}\beta _1 \Omega \\ +6A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega +4A\Lambda ^{4}\beta _2 \Omega +A\Lambda ^{2}\beta _1 \\ +\,15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 +4A\Lambda ^{4}\beta _2 \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-iT_0 }e^{-2i\Omega T_0 }\left( {\begin{array}{l} -\,3A\alpha _1 \Lambda ^{2}-30A^{2}\bar{{A}}\alpha _2 \Lambda ^{2}-20A\alpha _2 \Lambda ^{4}+3A\Lambda ^{2}\beta _1 \Omega ^{2} \\ +15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega ^{2}+16A\Lambda ^{4}\beta _2 \Omega ^{2}-2A\Lambda ^{2}\beta _1 \Omega \\ -\,6A^{2}\bar{{A}}\Lambda ^{2}\beta _2 \Omega -4A\Lambda ^{4}\beta _2 \Omega +A\Lambda ^{2}\beta _1 \\ +15A^{2}\bar{{A}}\Lambda ^{2}\beta _2 +4A\Lambda ^{4}\beta _2 \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{2iT_0 }\left( {\begin{array}{l} e^{i\Omega T_0 }\left( {\begin{array}{l} -\,3A^{2}\alpha _1 \Lambda -20A^{3}\bar{{A}}\alpha _2 \Lambda -30A^{2}\alpha _2 \Lambda ^{3}+A^{2}\beta _1 \Lambda \Omega ^{2}+4A^{3}\bar{{A}}\beta _2 \Lambda \Omega ^{2} \\ +\,15A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}+2A^{2}\beta _1 \Lambda \Omega +4A^{3}\bar{{A}}\beta _2 \Lambda \Omega +6A^{2}\Lambda ^{3}\beta _2 \Omega \\ +\,3A^{2}\beta _1 \Lambda +16A^{3}\bar{{A}}\beta _2 \Lambda +15A^{2}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \\ +e^{-i\Omega T_0 }\left( {\begin{array}{l} -\,3A^{2}\alpha _1 \Lambda -20A^{3}\bar{{A}}\alpha _2 \Lambda -30A^{2}\alpha _2 \Lambda ^{3}+A^{2}\beta _1 \Lambda \Omega ^{2}+4A^{3}\bar{{A}}\beta _2 \Lambda \Omega ^{2} \\ +\,15A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}-2A^{2}\beta _1 \Lambda \Omega -4A^{3}\bar{{A}}\beta _2 \Lambda \Omega -6A^{2}\Lambda ^{3}\beta _2 \Omega \\ +\,3A^{2}\beta _1 \Lambda +16A^{3}\bar{{A}}\beta _2 \Lambda +15A^{2}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \\ +e^{3i\Omega T_0 }\left( {-10A^{2}\alpha _2 \Lambda ^{3}+9A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}+6A^{2}\Lambda ^{3}\beta _2 \Omega +5A^{2}\Lambda ^{3}\beta _2 } \right) \\ +e^{-3i\Omega T_0 }\left( {-10A^{2}\alpha _2 \Lambda ^{3}+9A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}-6A^{2}\Lambda ^{3}\beta _2 \Omega +5A^{2}\Lambda ^{3}\beta _2 } \right) \\ \end{array}} \right) \nonumber \\&\qquad +\,e^{-2iT_0 }\left( {\begin{array}{l} e^{i\Omega T_0 }\left( {\begin{array}{l} -\,3A^{2}\alpha _1 \Lambda -20A^{3}\bar{{A}}\alpha _2 \Lambda -30A^{2}\alpha _2 \Lambda ^{3}+A^{2}\beta _1 \Lambda \Omega ^{2}+4A^{3}\bar{{A}}\beta _2 \Lambda \Omega ^{2} \\ +\,15A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}+2A^{2}\beta _1 \Lambda \Omega +4A^{3}\bar{{A}}\beta _2 \Lambda \Omega +6A^{2}\Lambda ^{3}\beta _2 \Omega \\ +\,3A^{2}\beta _1 \Lambda +16A^{3}\bar{{A}}\beta _2 \Lambda +15A^{2}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \\ +e^{-i\Omega T_0 }\left( {\begin{array}{l} -3A^{2}\alpha _1 \Lambda -20A^{3}\bar{{A}}\alpha _2 \Lambda -30A^{2}\alpha _2 \Lambda ^{3}+A^{2}\beta _1 \Lambda \Omega ^{2}+4A^{3}\bar{{A}}\beta _2 \Lambda \Omega ^{2} \\ +\,15A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}-2A^{2}\beta _1 \Lambda \Omega -4A^{3}\bar{{A}}\beta _2 \Lambda \Omega -6A^{2}\Lambda ^{3}\beta _2 \Omega \\ +\,3A^{2}\beta _1 \Lambda +16A^{3}\bar{{A}}\beta _2 \Lambda +15A^{2}\beta _2 \Lambda ^{3} \\ \end{array}} \right) \\ +e^{3i\Omega T_0 }\left( {-10A^{2}\alpha _2 \Lambda ^{3}+9A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}+6A^{2}\Lambda ^{3}\beta _2 \Omega +5A^{2}\Lambda ^{3}\beta _2 } \right) \\ +e^{-3i\Omega T_0 }\left( {-10A^{2}\alpha _2 \Lambda ^{3}+9A^{2}\Lambda ^{3}\beta _2 \Omega ^{2}-6A^{2}\Lambda ^{3}\beta _2 \Omega +5A^{2}\Lambda ^{3}\beta _2 } \right) \\ \end{array}} \right) \end{aligned}$$
(32)

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Liu, CX., Yan, Y. & Wang, WQ. Primary and secondary resonance analyses of a cantilever beam carrying an intermediate lumped mass with time-delay feedback. Nonlinear Dyn 97, 1175–1195 (2019). https://doi.org/10.1007/s11071-019-05039-w

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