On choosing state variables for piecewise-smooth dynamical system simulations

Abstract

Choosing state variables in a state-space representation of a nonlinear dynamical system is a nonunique procedure for a given input–output relationship and therefore a potentially challenging task. It can be even more challenging when there are piecewise-defined restoring forces, as in bilinear hysteresis or Bouc–Wen models, which are just two of many such engineering mechanics models. Using various piecewise-smooth models, we make use of flow- and effort-controlled system concepts, common to bond graph theory, to initiate our state variable selection task, and we view numerical simulation as being within the framework of hybrid dynamical systems. In order to develop accurate and efficient time integration, we incorporate MATLAB’s state event location algorithm, which is a mathematically sound numerical solver that deserves to be better known in the engineering mechanics community. We show that different choices of state variables can affect state event implementation, which in turn can significantly affect accuracy and efficiency, as judged by tolerance proportionality and work–accuracy diagrams. Programming details of state event location are included to facilitate application to other models involving piecewise-defined restoring forces. In particular, one version of the Bouc–Wen–Baber–Noori (BWBN) class of models is implemented as a demonstration.

Appendices

A Appendix for Sect. 2.2

Tables 2 and 3 list nonunique state-space representations of three- and four-parameter linear models, respectively.

Table 2 State variables for the three-element models each connected with a mass

Table 3 State variables for the four-element models each connected with a mass

Table 4 Events, event functions and reset maps for F and M formulations

Table 5 Algebraic and state equations in flow maps obtained from piecewise restoring force expressions for F and M formulations, respectively

B Appendix for Sect. 3

For the F formulation (flow-controlled) and according to the programming framework in [34], our previous work, global variables are utilized for passing values of restoring force r, memory parameters $H_l$, $H_u$ and O, and tags $\hbox {tag}_1$ and $\hbox {tag}_2$ among different MATLAB m-files. In particular, the memory parameters are obtained and updated by tagging the entire solution history H in order to switch according to the correct logic and to correct discontinuity sticking. $H_l$, $H_u$ and O are for the lower (Mode II), upper (Mode IV) bound values in the event functions as in Table 4, and a constant in the algebraic equations as in Table 5, respectively. $\hbox {tag}_1$ is used to mark each state event for correcting discontinuity sticking, while $\hbox {tag}_2$ is an indicator for the mode. In terms of variables, x(t), $\dot{x}(t)$ and r(t) are continuous variables, while $H_l$, $H_u$, O, $\hbox {tag}_1$ and $\hbox {tag}_2$ are discrete variables.

For the M formulation (mixed, partially effort-controlled) and as in our previous work, a global variable is utilized for passing values of $\hbox {tag}_2$ among different MATLAB m-files. $\hbox {tag}_2$ is an indicator for the mode and is the only discrete variable to be taken care of.