Passive Control of Differential Algebraic Inclusions - General Method and a Simple Example

Abstract

In this chapter, we consider a master system consisting of a nonlinear differential inclusion and an algebraic equation of constraint (resulting in a Differential Algebraic Inclusion (DAI) system). This system is coupled to a nonlinear energy sink (NES) corresponding to a one degree-of-freedom essentially nonlinear differential equation. We examine how a resonance capture can lead to a reduced order dynamical system. To obtain this reduced order model, we describe a multiple time scale analysis governed by the introduction of multi-timescales via a small parameter $\varepsilon$ that is finite and strictly positive. The mass of the NES is small versus the mass of the master system, and it governs a mass ratio defining the small parameter $\varepsilon$. The first timescale is the fast scale. Introducing the Manevitch complexification leads to the definition of slow time envelope coordinates. These envelope coordinates either do not directly depend on the fast time scale or do not depend on this fast time scale via introduction of the so-called Slow Invariant Manifold (SIM). The slow time dynamics of the master system components is analyzed through introduction of equilibrium points, corresponding to periodic solutions, or singular points (governing bifurcations around the SIM), corresponding to quasi-periodic behaviors. We present a simple example of semi-implicit Differential Algebraic Equation (DAE), including a friction term coupled to a cubic NES. Analytical developments of a 1:1:1 resonance case permit us to predict passive control of a DAI by a NES.

Appendices

APPENDIX 1 - Expressions of F and $H_0$

The function F is defined as

$$ \int _0^{\frac{2 \pi }{\omega }} \rho (\dot{v} + \frac{ \varepsilon \dot{w}}{1 + \varepsilon }) \exp (-i \omega \tau _0) d\tau _0.$$

We have

$$\dot{v} + \frac{ \varepsilon \dot{w}}{1 + \varepsilon } = \frac{1}{2} (\phi _1 + \frac{\varepsilon }{1 + \varepsilon } \phi _2)exp(i \omega \tau _0) + c.c.,$$

and we assume that $\phi _1, \phi _2$ do not depend on $\tau _0$. Using the polar form $\phi _j = N_j \exp (i \delta _j), j = 1, 2$, we also have

$$\dot{v} + \frac{ \varepsilon \dot{w}}{1 + \varepsilon } = N_1 \cos (\omega \tau _0 + \delta _1) + \frac{\varepsilon }{1 + \varepsilon } N_2 \cos (\omega \tau _0 + \delta _2).$$

So, we obtain

$$F \frac{2 i \eta }{\pi } \exp (-i \omega t_1^{\star }),$$

where

$$N_1 \cos (\omega t_1^{\star } + \delta _1) + \frac{\varepsilon }{1 + \varepsilon } N_2 \cos (\omega t_1^{\star } + \delta _2) = 0$$

and

$$ t_1^{\star } \in [0, \frac{\pi }{\omega }],$$

$$\begin{aligned} \eta = \hbox {sign}(N_1 \cos (\delta _1) + \frac{\varepsilon }{1 + \varepsilon } N_2 \cos (\delta _2)). \end{aligned}$$

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Finally, we derive

$$\tan (\omega t_1^{\star }) = \frac{N_1 \cos (\delta _1) + \frac{\varepsilon }{1+ \varepsilon } N_2 \cos (\delta _2) }{N_1 \sin (\delta _1) + \frac{\varepsilon }{1+ \varepsilon } N_2 \sin (\delta _2)},$$

and keeping the main orders of $\varepsilon$, we have

$$\begin{aligned} F (N_1, N_2, \delta _1, \delta _2)=&\frac{2 i}{\pi } \hbox {sign}(\cos (\delta _1)) \exp (-i \delta _1) [1 - \varepsilon i \frac{N_2}{N_1} \sin (\delta _1 - \delta _2) + o(\varepsilon ^2)], \nonumber \\&H_0 (\phi _3, \phi _3^{\star }) = \frac{\beta \phi _3}{2 i \omega _1} = \frac{\beta }{8 i \omega _1^3} \mid \phi _1 \mid ^2 \phi _1. \end{aligned}$$

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APPENDIX 2 - Euler Implicit Numerical Scheme

Let us note that $X_n = \left( \begin{array}{c}x_{1 n} \\ x_{2 n} \\ x_{3 n} \\ x_{4 n} \\ x_{5 n} \end{array}\right)$, $X_{n+1} = \left( \begin{array}{c}x_{1 n+1} \\ x_{2 n+1} \\ x_{3 n+1} \\ x_{4 n+1} \\ x_{5n} \end{array}\right)$. Let $X_0$ be given. For $n \ge 0$, we have

$$\begin{aligned} \begin{array}{c} aux_n = x_{1 n} - \varDelta t L_1 X_n - \varDelta t \varepsilon (h_0(x_{5 n}) + \gamma (x_{2 n} - x_{4 n})^3 -f_0(t_n)),\\ L_1 X_n = \varepsilon (a_0 + \lambda ) x_{1 n} + \omega _1^2 x_{2 n} -\varepsilon \lambda x_{3 n},\\ x_{1 n+1} = \left\{ \begin{array}{cc} &{} 0 \hbox { if } \mid aux_n \mid \le \varepsilon \alpha _0 \varDelta t \\ &{} aux_n - \varepsilon \alpha _0 \varDelta t \hbox { if } aux_n \ge \varepsilon \alpha _0 \varDelta t \\ &{} aux_n + \varepsilon \alpha _0 \varDelta t \hbox { if } aux_n \le - \varepsilon \alpha _0 \varDelta t \end{array},\right. \\ x_{2 n+1} = x_{2 n} + \varDelta t x_{1 n}, \\ x_{3 n+1} = x_{3 n} - \varDelta t (L_2 X_n + \gamma (x_{4 n} - x_{2 n})^3) = x_{3 n} - \varDelta t ( x_{1 n} + \gamma (x_{4 n} - x_{2 n})^3), \\ x_{4 n+1} = x_{4 n} + \varDelta t x_{3 n}, \\ x_{5 n+1} = x_{5 n} + \varDelta t x_{2 n}^2 x_{1 n}. \end{array} \end{aligned}$$