Mem-models and state event location algorithm for a prototypical aerospace system

Abstract

This paper presents a suite of dynamic models enhanced by mem-models. A prototypical pogo stick model might mimic a space hopper or a segmented telescope mirror actuator, while an extension of that prototype could simulate the terramechanics problem of a deformable soil model interacting with a penetrator. The inclusion of mem-models, including memristor and memcapacitor models, introduces an increased level of fidelity and a new and effective means of modeling memory effects and path dependence of nonlinear interacting dynamical systems. This challenging class of problems, which involves non-smooth dynamical systems posed as differential-algebraic equations, is formulated here as a hybrid dynamical system with both continuous and discrete states, and a state event location algorithm is successfully applied to track discontinuous changes in the system states.

Appendix A: Programming details

This Appendix summarizes programming details related to Sects. 3, 4 and 5. The code was adapted from the mixed formulation (M) for the bilinear hysteresis model discussed in [39], given that there are two modes to switch between. Nonetheless, the number of details that must be implemented is significant. As an adaptive time stepping scheme, ode45 in MATLAB does not have fixed time steps. Control of time stepping is done through specifying ${{\,\mathrm{RelTol}\,}}$ plus ${{\,\mathrm{MaxStep}\,}}$ and ${{\,\mathrm{InitialStep}\,}}$ via odeset.

1.1 A.1 1-D pogo stick basic model

The flow map in Eqs. (41) and (42) is obtained from Eqs. (18) and (19) into the following states and state equations:

$$\begin{aligned} \mathbf {y}_F&= \left\{ \begin{array}{c} y_F(1) \\ y_F(2) \\ y_F(3) \\ y_F(4) \\ y_F(5) \end{array} \right\} = \left\{ \begin{array}{l} {\bar{x}} \\ {\dot{x}} \\ y \\ {\dot{y}} \\ z \end{array} \right\} , \end{aligned}$$

(41)

$$\begin{aligned} {\dot{\mathbf {y}}}_F&= \left\{ \begin{array}{c} {\dot{y}}_F(1) \\ {\dot{y}}_F(2) \\ {\dot{y}}_F(3) \\ {\dot{y}}_F(4) \\ {\dot{y}}_F(5) \end{array} \right\} = \left\{ \begin{array}{l} {\dot{x}} \\ \ddot{x} \\ {\dot{y}} \\ \ddot{y} \\ {\dot{z}} \end{array} \right\} \end{aligned}$$

(42)

$$\begin{aligned}&= \left\{ \begin{array}{l} y_F(2) \\ s \left( - \frac{c}{m} y_F(2) - \frac{k }{m} ( y_F(1) - x_i) - g \right) \\ y_F(4) \\ (1 - s) (- g) + s \left( - \frac{c}{m} y_F(4) - \frac{k }{m} ( y_F(3) - x_i) - g \right) \\ (1 - s) (y_F(4) - y_F(2)) \end{array} \right\} \end{aligned}$$

(43)

where s is the mode indicator: $s = 0$ for non-contact mode, $s = 1$ for contact mode.

The following detail concerns the domain of this hybrid system: Since a rigid surface is impenetrable, $y(t) > 0$ must hold for all t for physically meaningful results; similarly $y > z$ and ${\bar{x}} > 0$. Also, the spring never extends, so $x(t) \le 0$. Such physically implied conditions were established initially and maintained throughout all simulations.

See Figs. 21 and 22 for the assessment of computational efficiency and stability.

Fig. 21

Tolerance proportionality (upper left) and work-accuracy diagrams (the rest) for the linear dashpot combined with linear spring in parallel simulated using the “event option” under MATLAB ode45 as given in Fig. 3, where ${{\,\mathrm{AbsTol}\,}}= 10^{-12}$ and ${{\,\mathrm{MaxStep}\,}}= 0.01$, $m = 1$, $k = 100$, $c = 0.5$, $x_i= 1$, ${\bar{x}}(0) = 1$, ${\dot{x}}(0) = 0$, $y(0) = 1.5$, ${\dot{y}}(0) = 0$, $z(0) = 0.5$. Refer to Table 2 for the definitions of the two types of events. GE and FE stand for global error and function evaluation, respectively

Fig. 22

Number of state events and their event time values plotted against MATLAB ode45 ${{\,\mathrm{RelTol}\,}}$ for the linear dashpot combined with linear spring in parallel simulated using the “event option” under MATLAB ode45 as given in Fig. 3, where ${{\,\mathrm{AbsTol}\,}}= 10^{-12}$ and ${{\,\mathrm{MaxStep}\,}}= 0.01$, $m = 1$, $k = 100$, $c = 0.5$, $x_i= 1$, ${\bar{x}}(0) = 1$, ${\dot{x}}(0) = 0$, $y(0) = 1.5$, ${\dot{y}}(0) = 0$, $z(0) = 0.5$. Refer to Table 2 for the definitions of the two types of events

1.2 A.2 1-D pogo stick with mem-models

For the pogo stick with mem-models in Sect. 4, we generalized the basic model by implementing a module that includes all combinations of linear elements and mem-elements where the restoring force r is defined as described in Sect. 4 (the basic model being a special case here).

While other hybrid dynamical system components follow those in the basic model, the states and state equations in Eqs. (44) and (45) were implemented as the flow map:

$$\begin{aligned} \mathbf {y}_F&= \left\{ \begin{array}{c} y_F(1) \\ y_F(2) \\ y_F(3) \\ y_F(4) \\ y_F(5) \\ y_F(6) \end{array} \right\} = \left\{ \begin{array}{l} a \\ {\bar{x}} \\ {\dot{x}} \\ y \\ {\dot{y}} \\ z \end{array} \right\} , \end{aligned}$$

(44)

$$\begin{aligned} {\dot{\mathbf {y}}}_F&= \left\{ \begin{array}{c} {\dot{y}}_F(1) \\ {\dot{y}}_F(2) \\ {\dot{y}}_F(3) \\ {\dot{y}}_F(4) \\ {\dot{y}}_F(5) \\ {\dot{y}}_F(6) \end{array} \right\} = \left\{ \begin{array}{l} x \\ {\dot{x}} \\ \ddot{x} \\ {\dot{y}} \\ \ddot{y} \\ {\dot{z}} \end{array} \right\} = \left\{ \begin{array}{l} y_F(2) - x_i \\ y_F(3) \\ s \left( \frac{-r }{m} - g \right) \\ y_F(5) \\ (1 - s) (- g) + s \left( \frac{-r }{m} - g \right) \\ (1 - s) (y_F(5) - y_F(3)) \end{array} \right\} \end{aligned}$$

(45)

where s is the mode indicator, defined previously for the basic model.

Fig. 23

A linear dashpot combined with a softening mem-spring defined in Eq. (25) in parallel simulated using the “event option” under MATLAB ode45, where ${{\,\mathrm{RelTol}\,}}= 10^{-3}$, ${{\,\mathrm{AbsTol}\,}}= 10^{-12}$ and ${{\,\mathrm{MaxStep}\,}}= 0.01$: time histories of solved continuous and discrete variable indicating an equivalent coefficient of restitution factor smaller than the linear counterpart with a drift in y, where $m = 1$, $k = 100$, $c = 0.5$, $x_i= 1$, ${\bar{x}}(0) = 1$, ${\dot{x}}(0) = 0$, $y(0) = 1.5$, ${\dot{y}}(0) = 0$, $z(0) = 0.5$. Refer to Table 2 for the definitions of the two types of events

The SMA model was a good test of allowable forms of state event functions. We developed two different techniques. For Event Type #3, we have:

$$\begin{aligned}&y_F(3) + 2*\mathrm{abs}(y_F(2) - x_i -x_1)/(y_F(2)\nonumber \\&\qquad - x_i -x_1) + 2 = 0 \end{aligned}$$

(46)

so that we could satisfy both terms equal to zero simultaneously. The factor of 2 was decided in a trial-and-error fashion. The other technique used the built-in, i.e., the type number of the state events as follows. For Event Type #5, we have:

$$\begin{aligned} y_F(2)-x_i-H_l-(x_3-x_1)+{ie}(\hbox {end})-3 = 0 \end{aligned}$$

(47)

$H_l$ stores all displacement values when Event Type #3 occurs and was made a global variable. For Event Type #6, we have:

$$\begin{aligned} y_F(2)-x_i-x_3+{ie}(\hbox {end})-5 = 0 \end{aligned}$$

(48)

1.3 A.3 1-D pogo stick with HC soil model

For the pogo stick with soil model in Sect. 5, the states and state equations in Eqs. (49) and (50) were implemented:

$$\begin{aligned}&\mathbf {y}_F = \left\{ \begin{array}{c} y_F(1) \\ y_F(2) \\ y_F(3) \end{array} \right\} = \left\{ \begin{array}{l} y \\ {\dot{y}} \\ \delta \end{array} \right\} ,~~~~~~~~~~~~~~~~~~\end{aligned}$$

(49)

$$\begin{aligned}&{\dot{\mathbf {y}}}_F = \left\{ \begin{array}{c} {\dot{y}}_F(1) \\ {\dot{y}}_F(2) \\ {\dot{y}}_F(3) \end{array} \right\} = \left\{ \begin{array}{l} {\dot{y}} \\ \ddot{y} \\ {\dot{\delta }} \end{array} \right\} = \left\{ \begin{array}{l} y_F(2) \\ (1 - s) (- g) + s \left( -g - \frac{k}{m} (y_F(1) + y_F(3)) + \frac{k}{m} x_i - \frac{c}{m} \left( y_F(2) + \right. \right. \\ \left. \left. \left( \frac{-k y_F(1) - c y_F(2) - K \max (0,y_F(3))^n - k y_F(3) + k x_i}{K \frac{3}{2} \frac{1-c_r}{{\dot{\delta }}_0} \max (0,y_F(3))^n + c} \right) \right) \right) \\ s \left( \frac{-k y_F(1) - c y_F(2) - K \max (0,y_F(3))^n - k y_F(3) + k x_i}{K \frac{3}{2} \frac{1-c_r}{{\dot{\delta }}_0} \max (0,y_F(3))^n + c} \right) \end{array} \right\} ~~~~~~~~~~~~~~~~~ \end{aligned}$$

(50)

It was a non-trivial challenge to calculate $\delta ^n$ with $n = \frac{3}{2}$ (or other non-integer exponent) as $\delta$ approaches zero during a state event with continuous extension, which is the situation with the HC soil model during a transition from contact to non-contact. There were a couple of options tested by us, all aimed at avoiding $\delta$ becoming negative during extension. Replacing $\delta ^n$ by $\max (0, \delta )^n$ (coded as $\max (0, y_F(3))^n$) worked well and was adopted in this study. Continuous approximations of sign and abs functions lead to the loss of the switching behavior, which is what the event function tries to capture. For this reason, we do not recommend such a course of action.