Nonlinear integral resonant controller for vibration reduction in nonlinear systems

Abstract

A new nonlinear integral resonant controller (NIRC) is introduced in this paper to suppress vibration in nonlinear oscillatory smart structures. The NIRC consists of a first-order resonant integrator that provides additional damping in a closed-loop system response to reduce high-amplitude nonlinear vibration around the fundamental resonance frequency. The method of multiple scales is used to obtain an approximate solution for the closed-loop system. Then closed-loop system stability is investigated using the resulting modulation equation. Finally, the effects of different control system parameters are illustrated and an approximate solution response is verified via numerical simulation results. The advantages and disadvantages of the proposed controller are presented and extensively discussed in the results. The controlled system via the NIRC shows no high-amplitude peaks in the neighboring frequencies of the resonant mode, unlike conventional second-order compensation methods. This makes the NIRC controlled system robust to excitation frequency variations.

Appendix

The coefficients of the time-domain response of Eq. (17) are as follows

$$\begin{aligned} V_1= & {} \left( {\frac{\hat{{\lambda }}U_1 }{4\omega _\mathrm{m}^2 +\omega _N^2 }-\frac{2\hat{{\delta }}^{2}U_1 U_5 }{4\omega _\mathrm{m}^2 +\omega _N^2 }} \right) \left( {\omega _N -2\mathrm{i}\omega _\mathrm{m} } \right) \\&+\frac{\hat{{\delta }}^{2}U_1^2 }{16\omega _\mathrm{m}^2 +\omega _N^2 } \left( {4\mathrm{i}\omega _\mathrm{m} -\omega _{_{N}} } \right) \\&+\frac{\hat{{\delta }}^{2}D_1 A(T_1 )}{( {4\omega _\mathrm{m}^2 +\omega _N^2 } )^{2}}( {\omega _N^2 -4\omega _\mathrm{m}^2 -4\mathrm{i}\omega _\mathrm{m} \omega _{_{N}} } ) ,\\ V_2= & {} \left( {\frac{\hat{{\lambda }}U_2 }{9\omega _\mathrm{m}^2 +\omega _N^2 }-\frac{2\hat{{\delta }}^{2}U_2 U_5 }{9\omega _\mathrm{m}^2 +\omega _N^2 }} \right) \left( {\omega _{_{N}} -3\mathrm{i}\omega _\mathrm{m} } \right) ,\\ V_3= & {} \left( {\frac{\hat{{\lambda }}U_3 }{\Omega ^{2}+\omega _N^2 }-\frac{2\hat{{\delta }}^{2}U_3 U_5 }{\varOmega ^{2}+\omega _N^2 }} \right) \left( {\omega _{_{N}} -\mathrm{i}\varOmega } \right) ,\\ V_4= & {} \frac{\hat{{\lambda }}D_1 A(T_1 )}{( {\omega _\mathrm{m}^2 +\omega _N^2 } )^{2}}( {\omega _\mathrm{m}^2 -\omega _{_{N}}^2 +2\mathrm{i}\omega _\mathrm{m} \omega _{_{N}} } ), \\ V_5= & {} \frac{\hat{{\delta }}^{2}U_2^2 }{81\omega _\mathrm{m}^2 +\omega _N^2 }\left( {9\mathrm{i}\omega _\mathrm{m} -\omega _{_{N}} } \right) ,\\ V_6= & {} \frac{\hat{{\delta }}^{2}U_3^2 }{4\varOmega ^{2}+\omega _N^2 }\left( {2\mathrm{i}\varOmega -\omega _{_{N}} } \right) ,\\ V_7= & {} \frac{\hat{{\delta }}^{2}U_4^2 }{\omega _2 }, \quad V_8 =\frac{2\hat{{\delta }}^{2}U_1 U_2 }{25\omega _\mathrm{m}^2 +\omega _N^2 }\left( {5\mathrm{i}\omega _\mathrm{m} -\omega _{_{N}} } \right) ,\\ V_9= & {} \frac{2\hat{{\delta }}^{2}U_1 U_3 }{\left( {2\omega _\mathrm{m} +\varOmega } \right) ^{2}+\omega _N^2 }\left[ {(2\omega _\mathrm{m} +\varOmega )\mathrm{i}-\omega _{_{N}} } \right] , \\ V_{10}= & {} \frac{\hat{{\delta }}^{2}U_1 U_4 }{\omega _\mathrm{m}^2 +\omega _N^2 }\left( {\mathrm{i}\omega _\mathrm{m} -\omega _{_{N}} } \right) ,\\ V_{11}= & {} \frac{2\hat{{\delta }}^{2}U_2 U_3 }{\left( {3\omega _\mathrm{m} +\varOmega } \right) ^{2}+\omega _N^2 }\left[ {(3\omega _\mathrm{m} +\varOmega )\mathrm{i}-\omega _{_{N}} } \right] , \\ V_{12}= & {} \frac{2\hat{{\delta }}^{2}U_2 U_4 }{9\omega _\mathrm{m}^2 +4\omega _N^2 }\left( {3\mathrm{i}\omega _\mathrm{m} -2\omega _{_{N}} } \right) ,\\ V_{13}= & {} \frac{2\hat{{\delta }}^{2}U_3 U_4 }{\varOmega ^{2}+4\omega _N^2 }\left( {\varOmega \mathrm{i}-2\omega _{_{N}} } \right) , \\ V_{14}= & {} \left( {\hat{{\lambda }}-\frac{D_1 }{\hat{{\tau }}}-\hat{{\delta }}^{2}U_5 } \right) U_5 . \end{aligned}$$