Hysteretic model

Hysteretic models may have a generalized displacement $u$ as input variable and a generalized force $f$ as output variable, or vice versa. In particular, in rate-independent hysteretic models, the output variable does not depend on the rate of variation of the input one.^[1]

Rate-independent hysteretic models can be classified into four different categories depending on the type of equation that needs to be solved to compute the output variable:

Algebraic models
Transcendental models
Differential models
Integral models

Algebraic models

In algebraic models, the output variable is computed by solving algebraic equations.

Bilinear model

Algebraic model by Vaiana et al. (2019)

Model formulation

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

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},

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

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

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

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

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 )}.

Hysteresis loop shapes

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.

Table 1.1 - AM parameters
	$k_{a}$	$k_{b}$	$\alpha$	$\beta _{1}$	$\beta _{2}$
(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

Matlab code

%  =========================================================================================
%  September 2019
%  Algebraic Model Algorithm
%  Nicolò Vaiana, Post-Doctoral Researcher, PhD 
%  Department of Structures for Engineering and Architecture 
%  University of Naples Federico II
%  via Claudio, 21 - 80125, Napoli
%  =========================================================================================

clc; clear all; close all;

%% APPLIED DISPLACEMENT TIME HISTORY

dt = 0.001;                                                                %time step
t  = 0:dt:1.50;                                                            %time interval
a0 = 1;                                                                    %applied displacement amplitude
fr = 1;                                                                    %applied displacement frequency
u  = a0*sin((2*pi*fr)*t(1:length(t)));                                     %applied displacement vector
v  = 2*pi*fr*a0*cos((2*pi*fr)*t(1:length(t)));                             %applied velocity vector
n  = length(u);                                                            %applied displacement vector length

%% 1. INITIAL SETTINGS
%1.1 Set the five model parameters
ka    = 10.0;                                                              %model parameter
kb    = 1.0;                                                               %model parameter
alfa  = 10.0;                                                              %model parameter
beta1 = 0.0;                                                               %model parameter
beta2 = 0.0;                                                               %model parameter
%1.2 Compute the internal model parameters 
u0    = (1/2)*((((ka-kb)/10^-20)^(1/alfa))-1);                             %internal model parameter
f0    = ((ka-kb)/2)*((((1+2*u0)^(1-alfa))-1)/(1-alfa));                    %internal model parameter
%1.3 Initialize the generalized force vector
f     = zeros(1, n);

%% 2. CALCULATIONS AT EACH TIME STEP

for i = 2:n
%2.1 Update the history variable
uj = u(i-1)+sign(v(i))*(1+2*u0)-sign(v(i))*((((sign(v(i))*(1-alfa))/(ka-kb))*(f(i-1)-beta1*u(i-1)^3-beta2*u(i-1)^5-kb*u(i-1)-sign(v(i))*f0+(ka-kb)*(((1+2*u0)^(1-alfa))/(sign(v(i))*(1-alfa)))))^(1/(1-alfa)));
%2.2 Evaluate the generalized force at time t
if (sign(v(i))*uj)-2*u0 < sign(v(i))*u(i) || sign(v(i))*u(i) < sign(v(i))*uj
    f(i) = beta1*u(i)^3+beta2*u(i)^5+kb*u(i)+(ka-kb)*((((1+2*u0+sign(v(i))*(u(i)-uj))^(1-alfa))/(sign(v(i))*(1-alfa)))-(((1+2*u0)^(1-alfa))/(sign(v(i))*(1-alfa))))+sign(v(i))*f0;
else
    f(i) = beta1*u(i)^3+beta2*u(i)^5+kb*u(i)+sign(v(i))*f0;
end
end

%% PLOT
figure
plot(u, f, 'k', 'linewidth', 4)
set(gca, 'FontSize', 28)
set(gca, 'FontName', 'Times New Roman')
xlabel('generalized displacement'), ylabel('generalized force')
grid
zoom off

Transcendental models

In transcendental models, the output variable is computed by solving transcendental equations, namely equations involving trigonometric, inverse trigonometric, exponential, logarithmic, and/or hyperbolic functions.

Exponential models

Exponential model by Vaiana et al. (2018)

Model formulation

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

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},

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

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

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

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

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].

Hysteresis loop shapes

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.

Table 2.1 - EM parameters
	$k_{a}$	$k_{b}$	$\alpha$	$\beta$
(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

Matlab code

%  =========================================================================================
%  September 2019
%  Exponential Model Algorithm
%  Nicolò Vaiana, Post-Doctoral Researcher, PhD 
%  Department of Structures for Engineering and Architecture 
%  University of Naples Federico II
%  via Claudio, 21 - 80125, Napoli
%  =========================================================================================

clc; clear all; close all;

%% APPLIED DISPLACEMENT TIME HISTORY

dt = 0.001;                                                                %time step
t  = 0:dt:1.50;                                                            %time interval
a0 = 1;                                                                    %applied displacement amplitude
fr = 1;                                                                    %applied displacement frequency
u  = a0*sin((2*pi*fr)*t(1:length(t)));                                     %applied displacement vector
v  = 2*pi*fr*a0*cos((2*pi*fr)*t(1:length(t)));                             %applied velocity vector
n  = length(u);                                                            %applied displacement vector length

%% 1. INITIAL SETTINGS
%1.1 Set the four model parameters
ka    = 5.0;                                                               %model parameter
kb    = 0.5;                                                               %model parameter
alfa  = 5.0;                                                               %model parameter
beta  = 1.0;                                                               %model parameter
%1.2 Compute the internal model parameters 
u0    = -(1/(2*alfa))*log(10^-20/(ka-kb));                                 %internal model parameter
f0    = ((ka-kb)/(2*alfa))*(1-exp(-2*alfa*u0));                            %internal model parameter
%1.3 Initialize the generalized force vector
f     = zeros(1, n);

%% 2. CALCULATIONS AT EACH TIME STEP

for i = 2:n
%2.1 Update the history variable
uj = u(i-1)+2*u0*sign(v(i))+sign(v(i))*(1/alfa)*log(sign(v(i))*(alfa/(ka-kb))*(-2*beta*u(i-1)+exp(beta*u(i-1))-exp(-beta*u(i-1))+kb*u(i-1)+sign(v(i))*((ka-kb)/alfa)*exp(-2*alfa*u0)+sign(v(i))*f0-f(i-1)));
%2.2 Evaluate the generalized force at time t
if (sign(v(i))*uj)-2*u0 < sign(v(i))*u(i) || sign(v(i))*u(i) < sign(v(i))*uj
    f(i) = -2*beta*u(i)+exp(beta*u(i))-exp(-beta*u(i))+kb*u(i)-sign(v(i))*((ka-kb)/alfa)*(exp(-alfa*(sign(v(i))*(u(i)-uj)+2*u0))-exp(-2*alfa*u0))+sign(v(i))*f0;
else
    f(i) = -2*beta*u(i)+exp(beta*u(i))-exp(-beta*u(i))+kb*u(i)+sign(v(i))*f0;
end
end

%% PLOT
figure
plot(u ,f, 'k', 'linewidth', 4)
set(gca, 'FontSize', 28)
set(gca, 'FontName', 'Times New Roman')
xlabel('generalized displacement'), ylabel('generalized force')
grid
zoom off

Differential models

Integral models

References

^ Dimian, Mihai; Andrei, Petru (4 November 2013). Noise-driven phenomena in hysteretic systems. ISBN 9781461413745.
^ Vaiana, Nicolò; Sessa, Salvatore; Marmo, Francesco; Rosati, Luciano (March 2019). "An accurate and computationally efficient uniaxial phenomenological model for steel and fiber reinforced elastomeric bearings". Composite Structures. 211: 196–212. doi:10.1016/j.compstruct.2018.12.017.
^ Vaiana, Nicolò; Sessa, Salvatore; Marmo, Francesco; Rosati, Luciano (26 April 2018). "A class of uniaxial phenomenological models for simulating hysteretic phenomena in rate-independent mechanical systems and materials". Nonlinear Dynamics. 93 (3): 1647–1669. doi:10.1007/s11071-018-4282-2.

[1] Dimian, Mihai; Andrei, Petru (4 November 2013). Noise-driven phenomena in hysteretic systems. ISBN 9781461413745.

[2] Vaiana, Nicolò; Sessa, Salvatore; Marmo, Francesco; Rosati, Luciano (March 2019). "An accurate and computationally efficient uniaxial phenomenological model for steel and fiber reinforced elastomeric bearings". Composite Structures. 211: 196–212. doi:10.1016/j.compstruct.2018.12.017.

[3] Vaiana, Nicolò; Sessa, Salvatore; Marmo, Francesco; Rosati, Luciano (26 April 2018). "A class of uniaxial phenomenological models for simulating hysteretic phenomena in rate-independent mechanical systems and materials". Nonlinear Dynamics. 93 (3): 1647–1669. doi:10.1007/s11071-018-4282-2.

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