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Nonlinear dynamics of piezoelectric-based active nonlinear vibration absorber using time delay acceleration feedback

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Abstract

In this paper, dynamic analysis of an active nonlinear vibration absorber using lead zirconate titanate (PZT) stack actuator has been carried out considering time delay in the acceleration feedback. Here the primary system is modelled as a harmonically base excited system with a nonlinear spring, damper and mass which is subjected to an external harmonic force. The smart absorber consists of a nonlinear spring, mass and damper system along with a linear spring connected in series with the PZT stack actuator. In the proposed model, the active control force is produced by the combination of spring and the PZT stack actuator which requires less voltage compared to that of the conventional system where the actuator is directly connected to the primary system and one can tune the frequency ratio of the absorber actively. The nonlinear governing equation of motion of the system is derived and solved by using a modified harmonic balance method. The steady-state response is obtained by using Newton’s method, and the stability of the system is studied using the reduced equations. It has been shown that the proposed novel vibration absorber which has inbuilt fail-safe design can absorb the vibration of the system more effectively with negligible damping than the available passive and active vibration absorbers. Also, the Den Hartog’s equal peaks are achieved when the primary system is subjected to both harmonic force and base excitation.

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  1. Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 781039, India

    S. Mohanty & S. K. Dwivedy

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Appendices

Appendix-I

$$\begin{aligned} a_1= & {} -\,2\Omega \cos \varphi _1 +2\xi _1 \sin \varphi _1 \\&+\, 2F_{c1} \Omega \sin \varphi _1 \sin \Omega \tau \\&-\, 2F_{c1} \Omega \cos \varphi _1 \cos \Omega \tau \\ a_2= & {} 2\xi _1 A\cos \varphi _1 +2\Omega A\sin \varphi _1 \\&+\, 2F_{c1} A\Omega \sin \varphi _1 \cos \Omega _d \\&+\, 2F_{c1} A\Omega \cos \varphi _1 \sin \Omega \tau \\ a_3= & {} -\, 2\xi _2 \sin \varphi _2 \hbox {, }a_4 =-2\xi _2 B\cos \varphi _2 \\ a_5= & {} 2\Omega \sin (\varphi _1 )+2\xi _1 \cos (\varphi _1 ) \\&+\, 2F_{c1} \Omega \sin \varphi _1 \cos \Omega \tau \\&+\, 2F_{c1} \Omega \cos \varphi _1 \sin \Omega \tau \\ a_6= & {} -\, 2\xi _1 A\sin \varphi _1 +2\Omega A\cos \varphi _1 \\&-\, 2F_{c1} A\Omega \sin \varphi _1 \sin \Omega \tau \\&+\, 2F_{c1} A\Omega \cos \varphi _1 \cos \Omega \tau \\ a_7= & {} -\, 2\xi _2 \cos \varphi _2 \hbox {, }a_8 =2\xi _2 B\sin \varphi _2 \\ a_9= & {} -\, 2\mu \Omega \cos \varphi _1 -2F_{c1} \Omega \sin \varphi _1 \sin \Omega \tau \\&+\, 2F_{c1} \Omega \cos \varphi _1 \cos \Omega \tau \\ a_{10}= & {} 2\mu \Omega A\sin \varphi _1 -2F_{c1} A\Omega \sin \varphi _1 \cos \Omega \tau \\&-\, 2F_{c1} A\Omega \cos \varphi _1 \sin \Omega \tau \\ a_{11}= & {} -\,2\mu \Omega \cos \varphi _2 +2\xi _2 \sin \varphi _2 \\ a_{12}= & {} 2\xi _2 B\cos \varphi _2 +2\mu \Omega B\sin \varphi _2 \\ a_{13}= & {} 2\mu \Omega \sin \varphi _1 -2F_{c1} \Omega \sin \varphi _1 \cos \Omega \tau \\&-\, 2F_{c1} \Omega \cos \varphi _1 \sin \Omega \tau \\ a_{14}= & {} 2\Omega \mu A\cos \varphi _1 +2F_{c1} A\Omega \sin \varphi _1 \sin \Omega \tau \\&-\, 2F_{c1} A\Omega \cos \varphi _1 \cos \Omega \tau \\ a_{15}= & {} 2\mu \Omega \sin \varphi _2 +2\xi _2 \cos \varphi _2 \\ a_{16}= & {} -\, 2\xi _2 B\sin \varphi _2 +2\mu \Omega B\cos \varphi _2 \\ b_1= & {} N_1 \cos \varphi _1 +N_2 \sin \varphi _1 \\&+\, N_3 \cos \varphi _2 +N_4 \sin \varphi _2 +N_{b1} \\ b_2= & {} -\, N_1 \sin \varphi _1 +N_2 \cos \varphi _1 \\&-\, N_3 \sin \varphi _2 +N_4 \cos \varphi _2 -F_1 +N_{b2} \\ b_3= & {} N_5 \cos \varphi _2 +N_6 \sin \varphi _2 \\&+\, N_7 \sin \varphi _1 +N_8 \\ b_4= & {} -\, N_5 \sin \varphi _2 +N_6 \cos \varphi _2 \\&+\, N_7 \cos \varphi _1 +N_9 \\ N_1= & {} -\,2\xi _1 A\Omega ,N_2 =\left( {-\Omega ^{2}+1} \right) A, \\ N_3= & {} 2\xi _2 B\Omega ,N_4 =-\left( {\alpha B+0.75\beta B^{3}} \right) , \\ N_{b1}= & {} -\,\alpha _\mathrm{r} B\sin \varphi _2 +0.75\alpha _{13} A^{3}\sin \varphi _1 \\&-\, F_{c1} A\Omega ^{2}\sin \left( {\varphi _1 +\Omega \tau } \right) \\&-\, 0.75A^{2}Y\sin \left( {2\varphi _1 } \right) \cos \left( \gamma \right) \\&+\, 1.5\alpha _{13} AY^{2}\sin \varphi _1 \\&-\, 0.75\alpha _{13} AY^{2}\sin \varphi _1 \cos \left( {2\gamma } \right) \\&-\, Y\sin \left( \gamma \right) -0.75\alpha _{13} Y^{3}\sin \left( \gamma \right) \\&-\, 1.5\alpha _{13} A^{2}Y\sin \left( \gamma \right) \\&+\, 0.75\alpha _{13} A^{2}Y\cos \left( {2\varphi _1 } \right) \sin \left( \gamma \right) \\&-\, 0.75\alpha _{13} AY^{2}\sin \left( {2\gamma } \right) \cos \left( {\varphi _1 } \right) , \\ N_5= & {} -\,2\xi _2 B\Omega ,N_6 =-\mu \Omega ^{2}B+\alpha B+0.75\beta B^{3}, \\ N_7= & {} -\,\mu \Omega ^{2}A, \\ N_{b2}= & {} \alpha _\mathrm{r} B\cos \varphi _2 +0.75\alpha _{13} A^{3}\cos \varphi _1 \\&-\, F_{c1} A\Omega ^{2}\cos \left( {\varphi _1 +\Omega \tau } \right) -Y\cos \left( \gamma \right) \\&-\, 0.75\alpha _{13} Y^{3}\cos \left( \gamma \right) -1.5\alpha _{13} A^{2}Y\cos \left( \gamma \right) \\&-\, 0.75\alpha _{13} A^{2}Y\cos \left( \gamma \right) \cos \left( {2\varphi _1 } \right) \\&+\, 1.5\alpha _{13} AY^{2}\cos \left( {\varphi _1 } \right) \\&+\, 0.75\alpha _{13} AY^{2}\cos \left( {\varphi _1 } \right) \cos \left( {2\gamma } \right) \\&+\, 0.75\alpha _{13} A^{2}Y\sin \left( {2\varphi _1 } \right) \sin \left( \gamma \right) \\&+\, 0.75\alpha _{13} AY^{2}\sin \left( {2\gamma } \right) \sin \left( {\varphi _1 } \right) , \\ N_8= & {} -\,\alpha _\mathrm{r} B\sin \varphi _2 +F_{c1} A\Omega ^{2}\sin \left( {\varphi _1 +\Omega \tau } \right) , \\ N_9= & {} -\,\alpha _\mathrm{r} B\cos \varphi _2 +F_{c1} A\Omega ^{2}\cos \left( {\varphi _1 +\Omega \tau } \right) \end{aligned}$$

Jacobian matrix obtained as

$$\begin{aligned} J=\left[ {{\begin{array}{llll} {\frac{\partial f_1 }{\partial A}}&{} {\frac{\partial f_1 }{\partial \varphi _1 }}&{} {\frac{\partial f_1 }{\partial B}}&{} {\frac{\partial f_1 }{\partial \varphi _2 }} \\ {\frac{\partial f_2 }{\partial A}}&{} {\frac{\partial f_2 }{\partial \varphi _1 }}&{} {\frac{\partial f_2 }{\partial B}}&{} {\frac{\partial f_2 }{\partial \varphi _2 }} \\ {\frac{\partial f_3 }{\partial A}}&{} {\frac{\partial f_3 }{\partial \varphi _1 }}&{} {\frac{\partial f_3 }{\partial B}}&{} {\frac{\partial f_3 }{\partial \varphi _2 }} \\ {\frac{\partial f_4 }{\partial A}}&{} {\frac{\partial f_4 }{\partial \varphi _1 }}&{} {\frac{\partial f_4 }{\partial B}}&{} {\frac{\partial f_4 }{\partial \varphi _2 }} \\ \end{array} }} \right] , \end{aligned}$$

Appendix-II

Fig. 21
figure 21

a MATLAB SIMULINK model of the system b subsystem model for passive nonlinear vibration absorber and linear primary system with external harmonic excitation. c Subsystem model for active nonlinear vibration absorber and nonlinear primary system under harmonic external force and base excitation by time delay acceleration feedback

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The SIMULINK model for the passive and active system are shown in Fig. 21

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Mohanty, S., Dwivedy, S.K. Nonlinear dynamics of piezoelectric-based active nonlinear vibration absorber using time delay acceleration feedback. Nonlinear Dyn 98, 1465–1490 (2019). https://doi.org/10.1007/s11071-019-05271-4

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