Hysteretic model

In the bilinear model formulated by Vaiana et al. (2018),[4] the generalized force at time t, representing the output variable, is evaluated as a function of the generalized displacement as follows:

${\displaystyle f_{t}=k_{a}\left(u_{t}-u_{j}\right)+k_{b}u_{j}+s_{t}f_{0}\quad {\text{when}}\quad u_{t}s_{t}<u_{j}s_{t},}$

${\displaystyle f_{t}=k_{b}u_{t}+s_{t}f_{0}\quad {\text{when}}\quad u_{t}s_{t}>u_{j}s_{t},}$

where ${\displaystyle k_{a},k_{b},}$ and ${\displaystyle f_{0}}$ are the three 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, ${\displaystyle u_{0}}$is an internal model parameter evaluated as:

${\displaystyle u_{0}={\frac {f_{0}}{k_{a}-k_{b}}},}$

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

${\displaystyle u_{j}={\frac {k_{a}u_{t-\Delta t}+s_{t}f_{0}-f_{t-\Delta t}}{k_{a}-k_{b}}}}$.

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

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

Table 1.1 - BM parameters

	${\displaystyle k_{a}}$	${\displaystyle k_{b}}$	${\displaystyle f_{0}}$
(a)	10.0	1.0	0.5
(b)	10.0	-1.0	0.5

In the asymmetric bilinear model formulated by Vaiana et al. (2020),[3] the generalized force at time t, representing the output variable, is evaluated as a function of the generalized displacement as follows:

${\displaystyle f_{t}=k_{a}\left(u_{t}-u_{j}\right)+k_{b}u_{j}+s_{t}f_{0}\quad {\text{when}}\quad u_{t}s_{t}<u_{j}s_{t},}$

${\displaystyle f_{t}=k_{b}u_{t}+s_{t}f_{0}\quad {\text{when}}\quad u_{t}s_{t}>u_{j}s_{t},}$

where ${\displaystyle k_{a}=k_{a}^{+}(k_{a}^{-}),k_{b}=k_{b}^{+}(k_{b}^{-}),}$ and ${\displaystyle f_{0}=f_{0}^{+}(f_{0}^{-})}$ are the three model parameters of the generic loading (unloading) case 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, ${\displaystyle u_{0}}$ is an internal model parameter evaluated as:

${\displaystyle u_{0}={\frac {(k_{b}^{+}-k_{b}^{-})u_{j}+f_{0}^{+}+f_{0}^{-}}{2(k_{a}^{+}-k_{b}^{-})}}\quad {\Biggl (}{\displaystyle {\frac {(k_{b}^{+}-k_{b}^{-})u_{j}+f_{0}^{+}+f_{0}^{-}}{2(k_{a}^{-}-k_{b}^{+})}}}{\Biggr )}\quad {\text{when}}\quad s_{t}>0\quad (s_{t}<0),}$

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

${\displaystyle u_{j}={\frac {k_{a}^{+}u_{t-\Delta t}+s_{t}f_{0}^{+}-f_{t-\Delta t}}{k_{a}^{+}-k_{b}^{+}}}\quad {\Biggl (}{\displaystyle {\frac {k_{a}^{-}u_{t-\Delta t}+s_{t}f_{0}^{-}-f_{t-\Delta t}}{k_{a}^{-}-k_{b}^{-}}}}{\Biggr )}\quad {\text{when}}\quad s_{t}>0\quad (s_{t}<0).}$

The following gif shows the nonlinear response of a single-degree-of-freedom (SDOF) mechanical system, with unit mass and asymmetric rate-independent hysteretic behavior, subjected to an external random force. To simulate its response, the following ABM parameters have been used: ${\displaystyle k_{a}^{+}=50\,\,(k_{a}^{-}=50),k_{b}^{+}=10\,\,(k_{b}^{-}=1),f_{0}^{+}=1\,\,(f_{0}^{-}=2)}$.

Nonlinear dynamic response of a SDOF hysteretic system modelled by means of the ABM

In the algebraic model developed by Vaiana et al. (2019),[5] the generalized force at time ${\displaystyle t}$, representing the output variable, is evaluated as a function of the generalized displacement as follows:

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

${\displaystyle f_{t}=\beta _{1}u_{t}^{3}+\beta _{2}u_{t}^{5}+k_{b}u_{t}+s_{t}f_{0}\quad {\text{when}}\quad u_{t}s_{t}>u_{j}s_{t},}$

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 ${\displaystyle f_{0}}$ are two internal model parameters evaluated as:

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

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

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

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

Figure 1.2 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 1.2.

Figure 1.2. Hysteresis Loops reproduced by using the AM model parameters in Table 1.2

Table 1.2 - 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),[4] the generalized force at time ${\displaystyle t}$, representing the output variable, is evaluated as a function of the generalized displacement as follows:

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

${\displaystyle f_{t}=-2\beta u_{t}+e^{\beta u_{t}}-e^{-\beta u_{t}}+k_{b}u_{t}+f_{0}s_{t}\quad {\text{when}}\quad u_{t}s_{t}>u_{j}s_{t},}$

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 ${\displaystyle f_{0}}$ are two internal model parameters evaluated as:

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

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

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

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

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