Demonstrating the power of extended Masing models for hysteresis through model equivalencies and numerical investigation

Abstract

The extended Masing model (EMM) is a powerful model for hysteresis that is theoretically sound, physically meaningful and computationally efficient. Any such model is defined by specifying a virgin loading curve and is implemented for arbitrary loadings using three simple hysteresis rules. A brief history of the development of these three switching rules is given. They can be accurately and efficiently implemented using a hybrid dynamical system approach where a state event algorithm is seamlessly combined with a time-stepping algorithm for numerical solution of the equations of motion when an EMM is used for the combined restoring and damping force. It is shown why each EMM is equivalent to an Iwan distributed-element model (DEM), which generalizes a multi-linear hysteresis system (a.k.a. Maxwell slip model) that consists of a finite number of elasto-plastic elements in parallel to an infinite number of such elements (countably many or a continuum of them). This model equivalency provides a physical basis for the choice of the three EMM rules. It is also noted that each EMM is also a classical Preisach model, a class of models that is well known in the mathematical literature on hysteresis. The extended Masing model is inherently for softening hysteresis but we show that a simple modification can be used to extend it to hardening hysteresis. It is noted that the EMM can also be extended to model deteriorating hysteresis. The hysteresis behavior of the EMM is further illustrated with examples of single-degree-of-freedom and two-degrees-of-freedom systems under dynamic excitation that use for the restoring force a specific EMM model whose defining virgin loading curve has a quite general parameterized form. It is shown that if the EMM model for the restoring force in a SDOF system that is subjected to earthquake excitation is replaced by a Bouc–Wen model with the same virgin loading curve, the hysteretic response changes dramatically and exhibits substantial drifting of the hysteresis loops. This behavior of the Bouc–Wen model is the result of a physical deficiency that was first noted four decades ago.

Computational details

1.1 SDOF Masing oscillator

In the extended Masing model, there are two modes associated with loading and unloading branches, respectively, while virgin loading is taken as a special case of either the loading or unloading mode. There is a state variable vector $\mathbf {y}= \left[ x, \dot{x}, r \right] ^T$ and an algebraic variable $r^*$ involved in a flow map for each mode as in Eq. (18). In addition, memory parameters $H_r^u$ and $H_r^l$ are also algebraic variables that take care of nonlocal memory. Algebraic variables also include $tag_1$ and $tag_2$ as indicators, which are discussed below. The domain containing the trajectories of the time evolution of the state and algebraic variables within Mode I has $\dot{x} > 0$ while in Mode II it has $\dot{x} < 0$.

A transition diagram illustrating the relationships among the modes is shown in Fig. 25, where transitions between Modes I and II correspond to load reversal, while transitions from Mode I (or II) to itself are caused by closures of minor loops, a manifestation of nonlocal memory.

Fig. 25

Transition diagram for the extended Masing model

Event Type #3 happens when the velocity becomes zero while changing the sign. An Event Type #1 marks a time instant when a minor loop closes smoothly on a more major loading branch, while an Event Type #2 marks a time instant when a minor loop closes smoothly on a more major unloading branch according to EE Rule $2'$. The event functions are given in Eqs. (19) to (21) and Eqs. (30), (31) and (21) for the $H_r$ and $H_r - H_x$ options, respectively, discussed in Sect. 3.3.

The computations were carried out using MATLAB. See Fig. 26 for the organization of all MATLAB m-files. First, main.m runs the ode45 solver, where the “options” via odeset is activated so that Events.m and OutputFcn.m are called upon. The three event functions are defined inside Events.m, while all outputs are recorded by using OutputFcn.m whenever there is a successful time step under ode45. The flow map specified in Eq. (18) is computed in myfun.m and passed back to main.m, as usual. There is strong coupling among all four MATLAB m-files so care is needed in the programming. Fig. 26 uses arrows to show the data flow.

Fig. 26

Illustrations of the interactions among all MATLAB m-files for the extended Masing model

Global variables are utilized for passing the values of the tags $tag_1$ and $tag_2$ among different MATLAB mfiles, where $tag_1$ is used to mark each state event for correcting discontinuity sticking, while $tag_2$ is a mode indicator with $tag_2 = +1, -1$ for Mode I and II, respectively. MATLAB’s ode45 uses a pair of Runge-Kutta formulas for time stepping. Numerical integration is done by computing a mesh of time points that are generated adaptively, normally not at a fixed time step. Following Wright and Pei [66], significant care and effort were taken to correct discontinuity sticking, which is a numerical problem that can arise when an algebraic variable changes continuously with time but is not updated automatically at a state event location. In the extended Masing model, there is no discontinuity sticking to correct due to the nonexistence of continuously changing algebraic variables. Nonetheless, $tag_1$ is still used in coding to (i) mark the time instants of any and all events for implementing the reset map, (ii) for better readability of the code, and (iii) as a necessary means to implement the $H_r - H_x$ option regarding nonlocal memory.

The $H_r$ set of event functions given in Eqs. (19) to (21) is defined using state variables $\dot{x}$ and r. The values of $H_r^u$ and $H_r^l$ (and $r^*$ as well) are updated using global variables under MATLAB. The $H_r - H_x$ set of event functions given in Eqs. (30), (31) and (21) were implemented similarly.

For the extended Masing model with only Rules 1 and 2 under the $H_r$ option, the reset map for the state variables is straightforward for the extended Masing model implementation due to no involvement of continuously changing algebraic variables. The reset map for the algebraic variables $tag_1$ and $tag_2$ is straightforward, which is to switch $tag_1$ from 10 to 1 at an event and from 1 to 10 right after a reset, and to extend $tag_2$ by 1 and -1 for loading and unloading branch, respectively. The reset map is more challenging for $H_r^u$ and $H_r^l$, thus affecting $r^*$: (a) At an Event Type #3 at the end of a loading branch, in general, $H_r^u$ is extended with the latest r and $r^*$ is immediately updated with the new end value in $H_r^u$. At an Event Type #3 at the end of an unloading branch, in general, $H_r^l$ is extended with the latest r and $r^*$ is immediately updated with the new end value in $H_r^l$; (b) At an Event Type #1, in general, both $H_r^u$ and $H_r^l$ are updated by removing their end element and $r^*$ is immediately updated with the new end value in $H_r^l$; and (c) At an Event Type #2, in general, both $H_r^u$ and $H_r^l$ are updated by removing their end element and $r^*$ is immediately updated with the new end value in $H_r^u$.

For the extended Masing model with Rules 1 to 3 under the $H_r$ option, the following are the extra steps for adding Rule 3 to the extended Masing model with Rules 1 and 2 under the $H_r$ option: The reset map is challenging for $H_r^u$ and $H_r^l$ thus affecting $r^*$: (a) At an Event Type #3 at the end of a loading branch, in addition, $H_r^l$ is extended with the latest $-r$. At an Event Type #3 at the end of an unloading branch, in addition, $H_r^u$ is extended with the latest $-r$; (b) At an Event Type #1, in addition, $H_r^l$ is removed of that $-r$ added under Event Type #3 if the current r is not equal to that $-r$, and (c) At an Event Type #2, in addition, $H_r^u$ is removed of that $-r$ added under Event Type #3 if the current r is not equal to that $-r$.

The following changes are made going from the $H_r$ to $H_r - H_x$ option: The reset map for the continuous variables x, r and $\dot{x}$ is: (a) For Event Type #3, x and r are updated with their respective current values, while $\dot{x}$ is reset to zero; (b) For Event Type #1, x and r are updated using the end values in $H_x^u$ and $H_r^u$, respectively, while $\dot{x}$ takes the current value; and (c) For Event Type #2, x and r are updated using the end values in $H_x^l$ and $H_r^l$, respectively, while $\dot{x}$ takes the current value. The reset map for the variables in the H vector is: (a) At an Event Type #3, no change is required for the H vectors; (b) At an Event Type #1, in general, all $H_r^u$, $H_r^l$, $H_x^u$ and $H_x^l$ are updated instead by removing their end elements and $r^*$ is immediately updated with the new end value in $H_r^l$; (c) At an Event Type #2, in general, all $H_r^u$, $H_r^l$, $H_x^u$ and $H_x^l$ are updated by removing their end elements and $r^*$ is immediately updated with the new end value in $H_r^u$.

The first-in-last-out (FILO) rule is applied for the stored $r^*$ values upon the closing of minor loops. Treating a virgin loading curve as a special case of a loading or an unloading branch requires care in coding. One way of doing this is to first introduce an arbitrarily large value for $r^*$ and assign this value and its negative to the virgin loading curve in both directions. These special values allow the accumulation in both $H_r^u$ and $H_r^l$ to start with an indicator for the virgin loading curve and further facilitate an automated execution of the reset map involving the closing of minor loops. This is because the closing of minor loops could lead to a return to the virgin loading curve, which may be treated as either Mode I or Mode II.

As an adaptive time stepping scheme, ode45 in MATLAB does not have fixed time steps. The control of time step is done through specifying AbsTol, RelTol, MaxStep, and InitialStep via odeset. Following Wright and Pei [66], the accuracy of our numerical solutions is assessed by studying the global error (${{\,\mathrm{GE}\,}}$) of the displacement at a specific time $t_n$:

$$\begin{aligned} {{\,\mathrm{GE}\,}}(t_n) = |\hat{x} (t_n) - x(t_n)| \end{aligned}$$

(33)

where $x(t_n)$ is the exact displacement at $t_n$, and $\hat{x}(t_n)$ is the approximated displacement at that time. Since the exact displacement is unknown, we obtain a converged solution by using a very small value of ${{\,\mathrm{RelTol}\,}}$ in MATLAB. In each numerical solution in this study, we fix the value of ${{\,\mathrm{AbsTol}\,}}$, thus allowing the value of ${{\,\mathrm{RelTol}\,}}$ to control the approximation accuracy. This is the so-called tolerance proportionality (TP) property (see [7] for a recent review).

The algorithm behind the event option under ode45 has been considered superb according to studies by Park and Barton [41]; Esposito and Kumar [15] - as reviewed in Wright and Pei [66]. Following Pei et al. [45], we also examined the robustness of the identified numbers of events (for each type) and convergence of the identified timing of events (for each type) regardless of the choice of ${{\,\mathrm{RelTol}\,}}$, which is an indicator for the size of time step for this time-varying scheme. The results are satisfactory for all SDOF extended Masing models exercised in this work.

1.2 2DOF Masing oscillator

We select the following state variables:

$$\begin{aligned} \mathbf {y}&= \left\{ \begin{array}{c} y(1) \\ y(2) \\ y(3) \\ y(4) \\ y(5) \\ y(6) \end{array} \right\} = \left\{ \begin{array}{l} x_1 \\ \dot{x}_1 \\ r_1 \\ x_2 - x_1 \\ \dot{x}_2 - \dot{x}_1 \\ r_2 \end{array} \right\} \nonumber \\ \dot{\mathbf {y}}&= \left\{ \begin{array}{l} y(2) \\ \text {see Eq.}~(35) \text { for } \ddot{x}_1 \\ \left\{ \begin{array}{ll} K_1 \left[ 1 - \left( \frac{ y(3) tag_{21} }{r_{u,1}}\right) ^n \right] y(2) &{} \text {initial loading} \\ K_1 \left[ 1 - \left( \frac{ (y(3) - r_1^*) tag_{21} }{2{r_{u,1}}}\right) ^n\right] y(2) &{} \text {other branches} \end{array}\right\} \\ y(5) \\ \text {see Eq.}~(35) \text { for } \ddot{x}_2 \\ \left\{ \begin{array}{ll} K_2 \left[ 1 - \left( \frac{ y(6) tag_{22} }{r_{u,2}}\right) ^n \right] y(5) &{} \text {initial loading} \\ K_2 \left[ 1 - \left( \frac{ (y(6) - r_2^*) tag_{22} }{2{r_{u,2}}}\right) ^n \right] y(5) &{} \text {other branches} \end{array}\right\} \end{array} \right\} \end{aligned}$$

(34)

where we have:

$$\begin{aligned} \left[ \begin{array}{c} \ddot{x}_1 \\ \ddot{x}_2 \end{array} \right]&= - \mathbf {M}^{-1}\mathbf {C}\left[ \begin{array}{c} y(2) \\ y(2) + y(5) \end{array} \right] \nonumber \\&\quad - \mathbf {M}^{-1} \left[ \begin{array}{c} y(3) - y(6) \\ y(6) \end{array} \right] -\left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \ddot{u}(t) \end{aligned}$$

(35)

Fig. 27

The extended Masing model given in Eq. (34) with $m_1 = m_2 = 1.25 \times 10^5$ kg, $K_1 = K_2 = 2.5 \times 10^8$ N/m, $r_{u1} = r_{u2} = 1.75 \times 10^6$ N, and $n_1 = n_2 = 4$ simulated using the “event option” under MATLAB ode45, where ${{\,\mathrm{RelTol}\,}}= 10^{-3}$, and ${{\,\mathrm{AbsTol}\,}}= 10^{-12}$: tolerance proportionality (TP) and work-accuracy diagrams. The results from three different value of ${{\,\mathrm{MaxStep}\,}}$ are compared here, where GE and FE stand for global error and function evaluation, respectively

The event functions are as follows:

$$\begin{aligned} \text {Event Type}~\#1: \ \&y(3) - H_{r1}^u (\text {end}) = 0, \nonumber \\&\quad \text {when } y(3) \text { is ascending} \end{aligned}$$

(36)

$$\begin{aligned} \text {Event Type}~ \#2: \ \&y(3) - H_{r1}^l (\text {end}) = 0, \nonumber \\&\quad \text {when } y(3) \text { is descending} \end{aligned}$$

(37)

$$\begin{aligned} \text {Event Type}~ \#3: \ \&y(2) = 0, \nonumber \\&\quad \text {when } y(2)\nonumber \\&\quad \text { is either ascending or descending} \end{aligned}$$

(38)

$$\begin{aligned} \text {Event Type}~ \#4: \ \&y(6) - H_{r2}^u (\text {end}) = 0, \nonumber \\&\quad \text {when } y(6) \text { is ascending} \end{aligned}$$

(39)

$$\begin{aligned} \text {Event Type}~ \#5: \ \&y(6) - H_{r2}^l (\text {end}) = 0, \nonumber \\&\quad \text {when } y(6) \text { is descending} \end{aligned}$$

(40)

$$\begin{aligned} \text {Event Type}~ \#6: \ \&y(5) = 0, \nonumber \\&\quad \text {when } y(5)\nonumber \\&\quad \text { is either ascending or descending} \end{aligned}$$

(41)

where $H_{r1}^u$ and $H_{r1}^l$ are for the reset map involving the closing of minor loops associated with the first extended Masing spring and $H_{r2}^u$ and $H_{r2}^l$ are for the reset map involving the closing of minor loops associated with the second extended Masing spring.

The effect of tuning ${{\,\mathrm{MaxStep}\,}}$ is studied in Fig. 27. The effect of ${{\,\mathrm{MaxStep}\,}}$ (and ${{\,\mathrm{InitialStep}\,}}$, which is not shown here) becomes essential when dealing with MDOF simulation due to the growing closeness of events belonging to different DOF. When the value of ${{\,\mathrm{RelTol}\,}}$ is large, a smaller value of ${{\,\mathrm{MaxStep}\,}}$ tends to saturate the value of ${{\,\mathrm{GE}\,}}$ leading to a longer CPU time and a larger number of successful function calls.