Hysteretic model

In the algebraic model developed by Vaiana et al. (2019),[2] the generalized force at time ${\displaystyle t}$, representing the output variable, is evaluated as a function of the generalized displacement as follows:

${\displaystyle f_{t}=\beta _{1}u_{t}^{3}+\beta _{2}u_{t}^{5}+k_{b}u_{t}+\left(k_{a}-k_{b}\right)\left[{\frac {(1+s_{t}u_{t}-s_{t}u_{j}+2u_{0})^{1-\alpha }}{s_{t}(1-\alpha )}}-{\frac {(1+2u_{0})^{1-\alpha }}{s_{t}(1-\alpha )}}\right]+s_{t}{\bar {f}}{\text{ when }}u_{t}s_{t}<u_{j}s_{t},}$

${\displaystyle f_{t}=\beta _{1}u_{t}^{3}+\beta _{2}u_{t}^{5}+k_{b}u_{t}+s_{t}{\bar {f}}{\text{ when }}u_{t}s_{t}>u_{j}s_{t},}$

where ${\displaystyle k_{a},k_{b},\alpha }$, ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$ are the five model parameters to be calibrated from experimental or numerical tests, whereas ${\displaystyle s_{t}}$ is the sign of the generalized velocity at time ${\displaystyle t}$, that is, ${\displaystyle s_{t}=\operatorname {sign} ({\dot {u}}_{t})}$. Furthermore, ${\displaystyle u_{0}}$and ${\displaystyle {\bar {f}}}$ are two internal model parameters evaluated as:

${\displaystyle u_{0}={\frac {1}{2}}\left[\left({\frac {k_{a}-k_{b}}{10^{-20}}}\right)^{1/\alpha }-1\right],}$

${\displaystyle {\bar {f}}={\frac {k_{a}-k_{b}}{2}}\left[{\frac {\left(1+2u_{0}\right)^{1-\alpha }-1}{1-\alpha }}\right],}$

whereas ${\displaystyle u_{j}}$ is the history variable:

${\displaystyle u_{j}=u_{t-\Delta t}+s_{t}(1+2u_{0})-s_{t}\left\lbrace {\frac {s_{t}(1-\alpha )}{k_{a}-k_{b}}}\left[f_{t-\Delta t}-\beta _{1}u_{t-\Delta t}^{3}-\beta _{2}u_{t-\Delta t}^{5}-k_{b}u_{t-\Delta t}-s_{t}{\bar {f}}+(k_{a}-k_{b}){\frac {(1+2u_{0})^{1-\alpha }}{s_{t}(1-\alpha )}}\right]\right\rbrace ^{1/(1-\alpha )}.}$

Figure 1.1 shows four different hysteresis loop shapes obtained by applying a sinusoidal generalized displacement having unit amplitude and frequency and simulated by adopting the Algebraic Model (AM) parameters listed in Table 1.1.

Figure 1.1. Hysteresis Loops reproduced by using the AM model parameters in Table 1.1

Table 1.1 - AM parameters

	${\displaystyle k_{a}}$	${\displaystyle k_{b}}$	${\displaystyle \alpha }$	${\displaystyle \beta _{1}}$	${\displaystyle \beta _{2}}$
(a)	10.0	1.0	10.0	0.0	0.0
(b)	10.0	1.0	10.0	0.2	0.2
(c)	10.0	1.0	10.0	−0.2	−0.2
(d)	10.0	1.0	10.0	−1.2	1.2

In the exponential model developed by Vaiana et al. (2018),[3] the generalized force at time ${\displaystyle t}$, representing the output variable, is evaluated as a function of the generalized displacement as follows:

${\displaystyle f_{t}=-2\beta u_{t}+e^{\beta u_{t}}-e^{-\beta u_{t}}+k_{b}u_{t}-s_{t}{\frac {k_{a}-k_{b}}{\alpha }}\left[e^{-\alpha (u_{t}s_{t}-u_{j}s_{t}+2u_{0})}-e^{-2\alpha u_{0}}\right]+{\bar {f}}s_{t}{\text{ when }}u_{t}s_{t}<u_{j}s_{t},}$

${\displaystyle f_{t}=-2\beta u_{t}+e^{\beta u_{t}}-e^{-\beta u_{t}}+k_{b}u_{t}+{\bar {f}}s_{t}{\text{ when }}u_{t}s_{t}>u_{j}s_{t},}$

where ${\displaystyle k_{a},k_{b},\alpha }$ and ${\displaystyle \beta }$ are the four model parameters to be calibrated from experimental or numerical tests, whereas ${\displaystyle s_{t}}$ is the sign of the generalized velocity at time ${\displaystyle t}$, that is, ${\displaystyle s_{t}=\operatorname {sign} ({\dot {u}}_{t})}$. Furthermore, ${\displaystyle u_{0}}$and ${\displaystyle {\bar {f}}}$ are two internal model parameters evaluated as:

${\displaystyle u_{0}=-{\frac {1}{2\alpha }}\ln \left({\frac {10^{-20}}{k_{a}-k_{b}}}\right),}$

${\displaystyle {\bar {f}}={\frac {k_{a}-k_{b}}{2\alpha }}\left(1-e^{-2\alpha u_{0}}\right),}$

whereas ${\displaystyle u_{j}}$ is the history variable:

${\displaystyle u_{j}=u_{t-\Delta t}+2u_{0}s_{t}+{\frac {s_{t}}{\alpha }}\ln \left[{\frac {\alpha s_{t}}{k_{a}-k_{b}}}\left(-2\beta u_{t-\Delta t}+e^{\beta u_{t-\Delta t}}-e^{-\beta u_{t-\Delta t}}+k_{b}u_{t-\Delta t}+{\frac {k_{a}-k_{b}}{\alpha }}s_{t}e^{-2\alpha u_{0}}+{\bar {f}}s_{t}-f_{t-\Delta t}\right)\right].}$

Figure 2.1 shows four different hysteresis loop shapes obtained by applying a sinusoidal generalized displacement having unit amplitude and frequency and simulated by adopting the Exponential Model (EM) parameters listed in Table 2.1.

Figure 2.1. Hysteresis Loops reproduced by using the EM model parameters in Table 2.1.

Table 2.1 - EM parameters

	${\displaystyle k_{a}}$	${\displaystyle k_{b}}$	${\displaystyle \alpha }$	${\displaystyle \beta }$
(a)	5.0	0.5	5.0	0.0
(b)	5.0	−0.5	5.0	0.0
(c)	5.0	0.5	5.0	1.0
(d)	5.0	0.5	5.0	−1.0