Nonlinear feedback anti-control of limit cycle and chaos in a mechanical oscillator: theory and experiment

Abstract

Though engineers believe that self-excited oscillation and chaos is detrimental, recent research has revealed that self-excited periodic and chaotic oscillation can be effectively utilized in many engineering processes and devices for significant benefits. Then a simple nonlinear acceleration feedback controller is proposed to generate self-excited periodic and chaotic oscillation in a mechanical oscillator. Analytical expressions relating the amplitude of limit cycles and control parameters are obtained by the method of multiple time scales. Analytical results are verified by numerical simulations performed in MATLAB–SIMULINK. Existence of chaotic oscillations is confirmed by numerical simulations. An extensive parametric study is carried out to reveal the nature of the chaotic oscillations and its dependence on the controller parameters. Theoretical results are finally verified by experiments. It is generally observed that the same controller can be used to induce both periodic and chaotic oscillations just by flipping the phase of the controller. This is possibly the first example where the same controller can be utilized to generate periodic and chaotic oscillation in a mechanical system. The findings of the paper are believed to be used in macro- and micro-mechanical systems.

Appendices

Appendix A

The Jacobian of the system at the static equilibrium is given by

$$ J = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ { - 1} & { - 2\zeta } & {\delta k_{c} } & 0 \\ 0 & 0 & 0 & 1 \\ { - 1} & { - 2\zeta } & {\delta k_{c} - \omega_{f}^{2} } & { - 2\zeta_{f} \omega_{f} } \\ \end{array} } \right]_{(y = 0,u = 0)} $$

(A-1)

The stability at the equilibrium point can be ascertained the eigenvalues of the Jacobian matrix. From this Jacobin matrix, one can write the characteristic equation as.

$$ a_{0} \lambda^{4} + a_{1} \lambda^{3} + a_{2} \lambda^{2} + a_{3} \lambda + a_{4} = 0, $$

(A-2)

where

$$ \begin{aligned} &a_{0} = 1, \hfill \\ &a_{1} = 2\zeta + 2\zeta_{f} \omega_{f} , \hfill \\ &a_{2} = 1 + \omega_{f}^{2} - \delta k_{c} + 4\omega_{f} \zeta_{f} \zeta , \hfill \\ &a_{3} = 2\omega_{f} \zeta_{f} + 2\omega_{f}^{2} \zeta , \hfill \\ &a_{4} =\omega_{f}^{2} \end{aligned} $$

According to the Routh–Hurwitz criterion, the system is called asymptotically stable when all the principal minors are positive and nonzero of the Hurwitz (H) matrix.

$$ H = \left[ {\begin{array}{*{20}c} {a_{1} } & {a_{3} } & 0 & 0 \\ {a_{0} } & {a_{2} } & {a_{4} } & 0 \\ 0 & {a_{1} } & {a_{3} } & 0 \\ 0 & 0 & {a_{2} } & {a_{4} } \\ \end{array} } \right] $$

(A-3)

The principal minors ($\Delta$) of the Hurwitz matrix are

$$ \begin{aligned} &\Delta_{1} = a_{1} , \hfill \\ &\Delta_{2} = a_{1} a_{2} - a_{0} a_{3} , \hfill \\ &\Delta_{3} = a_{1} a_{2} a_{3} - a_{0} a_{3}^{2} - a_{1}^{2} a_{4} , \hfill \\ &\Delta_{4} = a_{1} a_{2} a_{3} a_{4} - a_{1}^{2} a_{4}^{2} - a_{0} a_{3}^{2} a_{4} . \end{aligned} $$

Here, the values of $a_{0} ,a_{1} ,a_{3} ,a_{4}$ are positive for any positive value the $\omega_{f} ,\zeta_{f} ,\zeta$. But the sign of $a_{2}$ depends on the sign of $k_{c}$. If $k_{c} > 0$, $a_{2}$ will be negative (as $\delta \gg 1$) and vice-versa. Thus, all the principal minors are positive for $k_{c} < 0$ which signifies that the static equilibrium of the system is stable. All principal minors are negative except $\Delta_{1}$ for $k_{c} > 0$ signifying the instability of the static equilibrium.

Appendix B

Here the stability of the limit cycle oscillation is established. The steady-state solution of the modulation equations Eqs. (24)–(26) is

$$ A_{s} = \frac{{( - 2\overline{{k_{c} }} \sin \Psi_{s} )}}{{\pi \overline{\zeta } }} $$

(B-1)

$$ B_{s} = \frac{{( - A\overline{{k_{s} }} \sin \Psi_{s} )}}{{2\overline{{\zeta_{f} }} \omega_{f} }} $$

(B-2)

$$ \sin \Psi_{s} = \frac{ - 1}{{\sqrt {1 + \left( {\frac{{2\zeta + 2\zeta_{f} \omega_{f} }}{{(\omega_{f}^{2} - 1)}}} \right)^{2} } }} $$

(B-3)

The Jacobian of the flow at the steady-state solution is obtained from Eqs. (27)–(29) as

$$ J = \left[ {\begin{array}{*{20}c} { - \zeta } & 0 & {\frac{{ - 2k_{c} \cos \Psi_{s} }}{\pi }} \\ {\frac{{ - \sin \Psi_{s} }}{2}} & { - \zeta_{f} \omega_{f} } & {\frac{{ - A_{s} \cos \Psi_{s} }}{2}} \\ {\frac{{2k_{c} \cos \Psi_{s} }}{{A_{s}^{2} \pi }} - \frac{{\cos \Psi_{s} }}{{2B_{s} }}} & {\frac{{A\cos \Psi_{s} }}{{2B_{s}^{2} }}} & {\frac{{2k_{c} \sin \Psi_{s} }}{{A_{s} \pi }} + \frac{{A_{s} \sin \Psi_{s} }}{{2B_{s} }}} \\ \end{array} } \right]_{{(A = A_{s} ,B = B_{s} ,\Psi = \Psi_{s} )}} $$

(B-4)

The characteristic equation of the Jacobian matrix can be written as

$$ b_{0} \lambda^{3} + b_{1} \lambda^{2} + b_{2} \lambda + b_{3} = 0 $$

(B-5)

where

$$ \begin{aligned} &b_{0} = 1 \hfill \\ &b_{1} = 2(\zeta + \omega_{f} \zeta_{f} ) \hfill \\ &b_{2} = (\zeta + \zeta_{f} \omega_{f} )^{2} + \zeta \zeta_{f} \omega_{f} + \{ (\zeta + \zeta_{f} \omega_{f} )^{2} + \zeta \zeta_{f} \} \{ \omega_{f}^{2} - 1\}^{2} /(4(\zeta + \zeta_{f} \omega_{f} )^{2} ) \hfill \\ &b_{3} = \omega_{f} \zeta \zeta_{f} \{ (\omega_{f}^{2} - 1)^{2} + (2\omega_{f} \zeta_{f} + 2\zeta )^{2} \} /(4(\zeta + \omega_{f} \zeta_{f} )) \end{aligned} $$

According to the Routh–Hurwitz criterion, the system response is asymptotically stable when all the principal minors of the Hurwitz (H) matrix are positive and nonzero. The Hurwitz matrix is written as

$$ H = \left[ {\begin{array}{*{20}c} {b_{1} } & {b_{3} } & 0 \\ {b_{0} } & {b_{2} } & 0 \\ 0 & {b_{1} } & {b_{3} } \\ \end{array} } \right] $$

(B-6)

and the principal minors are

$$ \begin{aligned} &\Delta_{1} = b_{1} , \hfill \\ &\Delta_{2} = b_{1} b_{2} - b_{0} b_{3} , \hfill \\ &\Delta_{3} = b_{1} b_{2} b_{3} - b_{0} b_{3}^{2} . \end{aligned} $$

All the principal minors are positive for $\omega_{f} ,\zeta ,\zeta_{f} > 0$. Thus, it can be concluded that for $k_{c} > 0$, the system response is stable.