Generating self-excited oscillation in a class of mechanical systems by relay-feedback

Abstract

Recently, a few research efforts are made to utilize artificially generated self-excited vibration in several mechanical and micromechanical applications. The present paper considers some important theoretical aspects in connection with the efficacy of the relay-feedback in generating and controlling self-excited oscillation in a class of mechanical systems. The force applied by the relay-feedback is essentially constant and acts in the direction of the measured quantity. Mathematically, an ideal relay-feedback is represented by the signum function of the measured variable. Detailed theoretical analyses, both analytic and numerical, are presented for single, two, and three degrees-of-freedom spring–mass–damper systems under relay-feedback with underactuated, collocated, and noncollocated control configurations. It is shown that relay-feedback, if used in a suitable way, can be effective in selectively generating a particular mode of oscillation in a multi degrees-of-freedom mechanical system. It is also possible to change the mode of oscillation and its amplitude by suitably selecting the control gains.

Appendix 1

The coefficients $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, and $a_{6}$ used in Eq. (67) are given as follows:

$$\begin{aligned} \begin{array}{ll} a_6 &{}=\pi \mu _1 \mu _2 , \\ a_5 &{}=\pi ( {\mu _2 h_2 +\mu _2 \mu _1 h_1 +\mu _1 h_3 }), \\ a_4 &{}=\pi ( \mu _2 \alpha _1 +\mu _2 \alpha _2 +\mu _2 h_2 h_1 +\mu _1 \mu _2\\ &{}\quad +\mu _1 \mu _2 \alpha _1 +h_2 h_3 +h_3 \mu _1 h_1 +\mu _1 \alpha _2 ), \\ \end{array} \end{aligned}$$

$$\begin{aligned} a_3 =a_{30} +\left( {\frac{1}{A_3 \omega }}\right) a_{31} , \end{aligned}$$

$$\begin{aligned} a_2 =a_{20} +\left( {\frac{1}{A_3 \omega }}\right) a_{21} , \end{aligned}$$

$$\begin{aligned} a_1 =a_{10} +\left( {\frac{1}{A_3 \omega }}\right) a_{11} , \end{aligned}$$

$a_{0}$ = $\pi \alpha _{1}\alpha _{2}$,

where

$$\begin{aligned} a_{30} =\pi \!\left( {\begin{array}{l} \mu _2 h_1 \alpha _1 +\mu _2 h_1 \alpha _2 +\mu _2 h_2 +\mu _2 h_2 \alpha _1\\ \quad +h_3 \alpha _1 +h_3 \alpha _2 + \\ h_1 h_2 h_3 \!+\!\mu _1 h_3 \!+\!\mu _1 h_3 \alpha _1 \!+\!h_2 \alpha _2 \!+\!\mu _1 h_1 \alpha _2 \\ \end{array}}\!\right) \!, \end{aligned}$$

$$\begin{aligned} a_{31} =-4\alpha _2 k_c2 , \end{aligned}$$

$$\begin{aligned} a_{20} =\pi \!\left( {\begin{array}{l} \mu _2 \alpha _1 +\mu _2 \alpha _2 +\mu _2 \alpha _2 \alpha _1 +h_3 h_1 \alpha _1\\ \quad +h_3 h_1 \alpha _2 +h_3 h_2 + \\ h_3 h_2 \alpha _1 \!+\!\alpha _2 \alpha _1 \!+\!\alpha _2 h_2 h_1 \!+\!\mu _1 \alpha _2 \!+\!\mu _1 \alpha _2 \alpha _1 \\ \end{array}}\!\right) \!, \end{aligned}$$

$$\begin{aligned} \begin{array}{ll} a_{21} &{}=-\,4\alpha _2 k_c2 h_1 , \\ a_{10} &{}=\pi ( h_3 \alpha _1 +h_3 \alpha _2 +h_3 \alpha _2 \alpha _1 +\alpha _2 h_1 \alpha _1MYAMP]\quad +\,h_2 \alpha _2 +\alpha _2 h_2 \alpha _1 ), \\ a_{11} &{}=-\,4( {\alpha _2 k_c1 \alpha _1 +\alpha _2 k_c2 +\alpha _2 k_c2 \alpha _1 }). \\ \end{array} \end{aligned}$$

The coefficients used in Eq. (69) are given as follows:

$$\begin{aligned} p_8 =4\pi \mu _1 \mu _2 \alpha _2 k_c2 , \end{aligned}$$

$$\begin{aligned} p_6 =4\pi \alpha _2 \!\left( \!\!{\begin{array}{l} -h_3 h_2 k_{c2} -\mu _2 \alpha _1 k_{c2} -\alpha _2 \mu _1 k_{c2} -\mu _2 \alpha _2 k_{c2}\\ \quad -2\mu _1 \mu _2 k_{c2} -2\mu _1 \mu _2 k_{c2} \alpha _1 -\mu _1 \mu _2 k_{c1} \alpha _1\\ \quad +\mu _1 \mu _2 k_{c2} h_1^2 \\ \end{array}}\right) , \end{aligned}$$

$$\begin{aligned} p_4 =-4\pi \alpha _2 \left( {\begin{array}{l} -2\mu _1 \alpha _2 k_{c2} -\mu _1 \mu _2 k_{c1} \alpha _1 -2\mu _1 \mu _2 k_{c2} \alpha _1 \\ -\mu _1 \mu _2 \alpha _1^2 k_{c1} -\mu _1 \mu _2 \alpha _1^2 k_{c2} -\mu _1 \mu _2 k_{c2}\\ -h_3 h_2 k_{c1} \alpha _1 -2h_3 h_2 k_{c2} -2h_3 h_2 k_{c2} \alpha _1\\ -\mu _1 h_3 h_1 k_{c1} \alpha _1 -\mu _2 \alpha _2 k_{c1} \alpha _1 -\mu _1 \alpha _2 \alpha _1 k_{c1}\\ -\alpha _2 \alpha _1 k_{c2} -2\mu _1 \alpha _2 k_{c2} \alpha _1 +\alpha _2 k_{c2} h_1^2 \mu _1\\ +k_{c2} h_1^2 \mu _2 \alpha _1 +\alpha _2 k_{c2} h_1^2 \mu _2 +k_{c2} h_1^2 h_3 h_2\\ -\mu _2 h_2 h_1 k_{c1} \alpha _1 -k_{c1} \alpha _1^2 \mu _2 -2\mu _2 \alpha _1 k_{c2}\\ -\mu _2 \alpha _1^2 k_{c2} -2\mu _2 \alpha _2 k_{c2} -2\mu _2 \alpha _2 k_{c2} \alpha _1 \\ \end{array}}\right) , \end{aligned}$$

$$\begin{aligned} p_2 =-4\pi \alpha _2 \!\left( {\begin{array}{l} 2\mu _2 \alpha _2 k_{c2} \alpha _1 +\mu _1 \alpha _2 k_{c1} \alpha _1 +\mu _2 \alpha _1^2 k_{c1}\\ +2\alpha _2 \alpha _1 k_{c2} +h_3 h_2 k_{c1} \alpha _1 \!+\!\alpha _2 \alpha _1^2 k_{c2} \!+\!\alpha _2 \alpha _1^2 k_{c1}\\ +\mu _2 \alpha _1^2 k_{c2} +2h_3 h_2 k_{c2} \alpha _1 +\mu _2 \alpha _2 \alpha _1^2 k_{c2}\\ +\mu _2 \alpha _2 \alpha _1^2 k_{c1} +h_3 h_2 \alpha _1^2 k_{c1} +h_3 h_1 \alpha _1^2 k_{c1}\\ +h_3 h_1 \alpha _1^2 k_{c2} +h_3 h_1 \alpha _2 k_{c1} \alpha _1 -\alpha _2 k_{c2} h_1^2 \alpha _1\\ +h_3 h_2 \alpha _1^2 k_{c2} +\alpha _2 h_2 h_1 k_{c1} \alpha _1 +\alpha _2 \mu _1 \alpha _1^2 k_{c1}\\ +\alpha _2 \mu _1 \alpha _1^2 k_{c2} +h_3 h_2 k_{c2} \!+\!\mu _2 \alpha _1 k_{c2}\!+\!\mu _1 \alpha _2 k_{c2}\\ +\mu _2 \alpha _2 k_{c2} +2\mu _1 \alpha _2 k_{c2} \alpha _1 +\mu _2 \alpha _2 k_{c1} \alpha _1 \\ \end{array}}\!\!\!\right) \!, \end{aligned}$$

$$\begin{aligned} p_0 =4\pi \alpha _2 ^2\alpha _1 ( {k_c2 +\alpha _1 k_c2 +\alpha _1 k_c1 }). \end{aligned}$$