This document describes hysteretic mechanical systems and materials using a generalized force-displacement model. It presents the analytical formulation of the model, including expressions for the generalized force and tangent stiffness during loading and unloading. It defines the model parameters and describes four possible hysteresis loop shapes (S1-S4) depending on the parameter values. Examples of each shape type are given through simulated hysteresis loops. The implementation algorithm is also outlined.

1. Hysteretic Mechanical Systems
and Materials
with Matlab Codes
Version 09 July 2023 Nicolò Vaiana, Ph.D.
University of Naples Federico II
Polytechnic and Basic Sciences School
Department of Structures for Engineering and Architecture

2. 1
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Hysteretic Mechanical Systems and Materials
Vaiana-Rosati Model
Analytical Formulation
SIMULATION OF COMPLEX HYSTERESIS LOOPS

3. P1
P21
Model formulation - Generalized force
The generalized force 𝑓, during the generic loading phase ( ሶ
𝑢 > 0), is evaluated as:
𝑓+ 𝑢, 𝑢𝑃, 𝑓𝑃 = 𝑓𝑒
+
𝑢 + 𝑘𝑏
+
𝑢 + 𝑓0
+
− 𝑓𝑒
+
𝑢𝑃 + 𝑘𝑏
+
𝑢𝑃 + 𝑓0
+
− 𝑓𝑃 𝑒−𝛼+ 𝑢−𝑢𝑃 ,
with:
𝑓𝑒
+
𝑢 = 𝛽1
+
𝑒𝛽2
+
𝑢 − 𝛽1
+
+
4 𝛾1
+
1+𝑒−𝛾2
+ 𝑢−𝛾3
+ − 2𝛾1
+ .
Similarly, during the generic unloading one ( ሶ
𝑢 < 0), it is computed as:
𝑓− 𝑢, 𝑢𝑃, 𝑓𝑃 = 𝑓𝑒
−
𝑢 + 𝑘𝑏
−
𝑢 − 𝑓0
−
− 𝑓𝑒
−
𝑢𝑃 + 𝑘𝑏
−
𝑢𝑃 − 𝑓0
−
− 𝑓𝑃 𝑒+𝛼− 𝑢−𝑢𝑃 ,
with:
𝑓𝑒
−
𝑢 = 𝛽1
−
𝑒𝛽2
−
𝑢 − 𝛽1
−
+
4 𝛾1
−
1+𝑒−𝛾2
− 𝑢−𝛾3
− − 2𝛾1
−.
In the previous expressions, 𝑢 is the generalized displacement, 𝑢𝑃 and 𝑓𝑃 are the coordinates of the current
point 𝑃, whereas the other terms represent the model parameters.
Note that the graph of 𝑓+
(𝑓−
) asymptotically approaches the upper (lower) limiting curve 𝑐𝑢 (𝑐𝑙).
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Hysteretic Mechanical Systems and Materials

4. P1
P31
The generalized tangent stiffness 𝑘𝑡, during the generic loading phase ( ሶ
𝑢 > 0), is evaluated as:
𝑘𝑡
+
𝑢, 𝑢𝑃, 𝑓𝑃 = 𝑘𝑒
+
𝑢 + 𝑘𝑏
+
+ 𝑓𝑒
+
𝑢𝑃 + 𝑘𝑏
+
𝑢𝑃 + 𝑓0
+
− 𝑓𝑃 𝛼+𝑒−𝛼+ 𝑢−𝑢𝑃 ,
with:
𝑘𝑒
+
𝑢 = 𝛽1
+
𝛽2
+
𝑒𝛽2
+
𝑢 +
4 𝛾1
+𝛾2
+𝑒−𝛾2
+ 𝑢−𝛾3
+
1+𝑒−𝛾2
+ 𝑢−𝛾3
+ 2 .
Similarly, during the generic unloading one ( ሶ
𝑢 < 0), it is computed as:
𝑘𝑡
−
𝑢, 𝑢𝑃, 𝑓𝑃 = 𝑘𝑒
−
𝑢 + 𝑘𝑏
−
− 𝑓𝑒
−
𝑢𝑃 + 𝑘𝑏
−
𝑢𝑃 − 𝑓0
−
− 𝑓𝑃 𝛼−𝑒+𝛼− 𝑢−𝑢𝑃 ,
with:
𝑘𝑒
−
𝑢 = 𝛽1
−
𝛽2
−
𝑒𝛽2
−
𝑢 +
4 𝛾1
−𝛾2
−𝑒−𝛾2
− 𝑢−𝛾3
−
1+𝑒−𝛾2
− 𝑢−𝛾3
− 2 .
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Model formulation - Generalized tangent stiffness
Hysteretic Mechanical Systems and Materials

5. shape type limiting curves subtype obtained for
S1 straight lines −
𝛽1
+
= 𝛽2
+
= 0
𝛽1
−
= 𝛽2
−
= 0
𝛾1
+
= 𝛾2
+
= 0
𝛾1
−
= 𝛾2
−
= 0
S2
curves with no
inflection point
S2.1
𝛽1
+
> 0, 𝛽2
+
> 0
𝛽1
−
> 0, 𝛽2
−
> 0
𝛾1
+
= 𝛾2
+
= 0
𝛾1
−
= 𝛾2
−
= 0
S2.2
𝛽1
+
> 0, 𝛽2
+
> 0
𝛽1
−
< 0, 𝛽2
−
< 0
𝛾1
+
= 𝛾2
+
= 0
𝛾1
−
= 𝛾2
−
= 0
S2.3
𝛽1
+
> 0, 𝛽2
+
> 0
𝛽1
−
< 0, 𝛽2
−
< 0
𝛾1
+
> 0, 𝛾2
+
< 0
𝛾1
−
> 0, 𝛾2
−
< 0
S3
curves with one
inflection point
S3.1
𝛽1
+
= 𝛽2
+
= 0
𝛽1
−
= 𝛽2
−
= 0
𝛾1
+
> 0, 𝛾2
+
> 0
𝛾1
−
> 0, 𝛾2
−
> 0
S3.2
𝛽1
+
= 𝛽2
+
= 0
𝛽1
−
= 𝛽2
−
= 0
𝛾1
+
> 0, 𝛾2
+
> 0
𝛾1
−
> 0, 𝛾2
−
< 0
S3.3
𝛽1
+
> 0, 𝛽2
+
> 0
𝛽1
−
< 0, 𝛽2
−
< 0
𝛾1
+
> 0, 𝛾2
+
< 0
𝛾1
−
> 0, 𝛾2
−
< 0
S4
curves with two
inflection points
−
𝛽1
+
> 0, 𝛽2
+
> 0
𝛽1
−
< 0, 𝛽2
−
< 0
𝛾1
+
> 0, 𝛾2
+
> 0
𝛾1
−
> 0, 𝛾2
−
> 0
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P41
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Model parameters
The model parameters governing the generic loading phase ( ሶ
𝑢 > 0) are:
𝑘𝑏
+
, 𝑓0
+
, 𝛼+, 𝛽1
+
, 𝛽2
+
, 𝛾1
+, 𝛾2
+, 𝛾3
+,
whereas those governing the generic unloading one ( ሶ
𝑢 < 0) are:
𝑘𝑏
−
, 𝑓0
−
, 𝛼−, 𝛽1
−
, 𝛽2
−
, 𝛾1
−, 𝛾2
−, 𝛾3
−.
The only conditions to be satisfied are:
𝛼+ > 0, 𝛼− > 0, 𝑓0
+
> 𝑓0
−
,
since the other parameters can be arbitrary real numbers.
The model is capable of reproducing four types of hysteresis loop shapes depending on the values assumed
by the parameters 𝛽1
+
, 𝛽2
+
, 𝛾1
+, 𝛾2
+ and 𝛽1
−
, 𝛽2
−
, 𝛾1
−, 𝛾2
−, as shown in the above table.
Hysteretic Mechanical Systems and Materials

6. P1
P51
Simulated hysteresis loops – Shape type S1
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Examples of hysteresis loops limited by two straight lines
shape type superscript 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
S1a + 0.5 2 10 0 0 0 0 0
− 0 2 10 0 0 0 0 0
S1b + 0.5 4 10 0 0 0 0 0
− 0.5 2 10 0 0 0 0 0
S1c + 0.5 2 2 0 0 0 0 0
− 0.5 2 10 0 0 0 0 0
Hysteretic Mechanical Systems and Materials

7. P1
P61
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Simulated hysteresis loops – Shape type S2
Examples of hysteresis loops limited by two curves with no inflection point
shape type superscript 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
S2.1 + 0.5 1 10 0.5 1.2 0 0 0
− 0 1 10 0.5 0.8 0 0 0
S2.2 + 0.5 1 10 0.5 1.2 0 0 0
− 0 1 10 -0.5 -0.8 0 0 0
S2.3 + 0.5 4 10 0.5 1.2 1.5 -2 -2
− 0 4 10 -0.5 -0.8 1.5 -2 2
Hysteretic Mechanical Systems and Materials

8. P1
P71
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Simulated hysteresis loops – Shape type S3
Examples of hysteresis loops limited by two curves with one inflection point
shape type superscript 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
S3.1a + 0.5 1 10 0 0 2 2 0
− 0.5 1 10 0 0 2 2 0
S3.1b + 0.5 2 10 0 0 0.5 4 0.5
− 0.5 2 10 0 0 0.5 4 -0.5
S3.1c + 0.5 2 10 0 0 0.5 4 0.5
− 0.5 2 10 0 0 0.5 8 -1
Hysteretic Mechanical Systems and Materials

9. P1
P81
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Simulated hysteresis loops – Shape type S3
Examples of hysteresis loops limited by two curves with one inflection point
shape type superscript 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
S3.1d + 0.5 1 10 0 0 2 40 0
− 0.5 1 10 0 0 2 40 0
S3.2 + 0.5 3.5 10 0 0 1.5 2 0.5
− 0.5 3.5 10 0 0 2 -1 0.5
S3.3 + 0.5 0.5 100 0.5 0.8 4 -2 0
− 0.5 0.5 100 -0.5 -0.8 4 -2 0
Hysteretic Mechanical Systems and Materials

10. P1
P91
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Simulated hysteresis loops – Shape type S4
Examples of hysteresis loops limited by two curves with two inflection points
shape type superscript 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
S4a + 0 1 10 0.1 2 1 4 0
− 0 1 10 -0.1 -2 1 4 0
S4b + 0 0.5 20 0.001 5 1 8 -0.05
− 0 0.5 20 -0.001 -5 1 8 0.05
S4c + 0.5 0.5 10 0.1 2 1 40 0
− 0.5 0.5 10 -0.1 -2 1 40 0
Hysteretic Mechanical Systems and Materials

11. P101
Implementation algorithm
1 Initial setting
1.1 Set the model parameters
𝑘𝑏
+
, 𝑓0
+
, 𝛼+
, 𝛽1
+
, 𝛽2
+
, 𝛾1
+
, 𝛾2
+
, 𝛾3
+
and 𝑘𝑏
−
, 𝑓0
−
, 𝛼−
, 𝛽1
−
, 𝛽2
−
, 𝛾1
−
, 𝛾2
−
, 𝛾3
−
.
1.2 Define initial values of generalized force and tangent stiffness
𝑓𝑡=0 and 𝑘𝑡 𝑡=0.
2 Calculations at each time step
2.1 Update the model parameters
2.2 Evaluate the generalized force at time 𝑡
𝑘𝑏 = 𝑘𝑏
+
𝑘𝑏
−
, 𝑓0 = 𝑓0
+
𝑓0
−
, 𝛼 = 𝛼+ 𝛼− , 𝛽1 = 𝛽1
+
𝛽1
−
, 𝛽2 = 𝛽2
+
𝛽2
−
,
𝛾1 = 𝛾1
+ 𝛾1
− , 𝛾2 = 𝛾2
+ 𝛾2
− , 𝛾3 = 𝛾3
+ 𝛾3
− , if 𝑠𝑡 > 0 (𝑠𝑡 < 0).
𝑓𝑒 𝑡−∆𝑡 = 𝛽1𝑒𝛽2𝑢𝑡−∆𝑡 − 𝛽1 +
4𝛾1
1+𝑒−𝛾2 𝑢𝑡−∆𝑡−𝛾3
− 2𝛾1,
𝑓𝑒 𝑡 = 𝛽1𝑒𝛽2𝑢𝑡 − 𝛽1 +
4𝛾1
1+𝑒−𝛾2 𝑢𝑡−𝛾3
− 2𝛾1,
𝑓𝑡 = 𝑓𝑒 𝑡 + 𝑘𝑏 𝑢𝑡 + 𝑠𝑡 𝑓0 − 𝑓𝑒 𝑡−∆𝑡 + 𝑘𝑏 𝑢𝑡−∆𝑡 + 𝑠𝑡 𝑓0 − 𝑓𝑡−∆𝑡 𝑒−𝑠𝑡𝛼 𝑢𝑡−𝑢𝑡−∆𝑡 .
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SIMULATION OF COMPLEX HYSTERESIS LOOPS
2.3 Compute the generalized tangent stiffness at time 𝑡
𝑘𝑒 𝑡 = 𝛽1𝛽2𝑒𝛽2𝑢𝑡 +
4𝛾1𝛾2𝑒−𝛾2 𝑢𝑡−𝛾3
1+𝑒−𝛾2 𝑢𝑡−𝛾3
2,
𝑘𝑡 𝑡 = 𝑘𝑒 𝑡 + 𝑘𝑏 + 𝑠𝑡𝛼 𝑓𝑒 𝑡−∆𝑡 + 𝑘𝑏 𝑢𝑡−∆𝑡 + 𝑠𝑡 𝑓0 − 𝑓𝑡−∆𝑡 𝑒−𝑠𝑡𝛼 𝑢𝑡−𝑢𝑡−∆𝑡 .
Hysteretic Mechanical Systems and Materials

12. 11
Matlab code
% =========================================================================================
% July 2023
% Vaiana Rosati Model Algorithm
% Nicolo' Vaiana, Assistant Professor in Structural Mechanics and Dynamics
% Department of Structures for Engineering and Architecture
% University of Naples Federico II
% via Claudio 21, 80125, Napoli, Italy
% e-mail: nicolo.vaiana@unina.it, nicolovaiana@outlook.it
% =========================================================================================
clc; clear all; close all;
%% APPLIED GENERALIZED DISPLACEMENT
dt = 0.001; % s time step
t = 0:dt:1.5; % s time interval
u0 = 1.0; % m displacement amplitude
fr = 1; % Hz displacement frequency
u = u0*sin((2*pi*fr)*t(1:length(t))); % m displacement vector
ud = 2*pi*fr*u0*cos((2*pi*fr)*t(1:length(t))); % m/s velocity vector
Ns = length(u); % - number of time steps
%% 1 INITIAL SETTINGS
% 1.1 Set the model parameters
kbp = 2.5; kbm = 0; % N/m
f0p = 4; f0m = 4; % N
alfap = 10; alfam = 10; % 1/m
beta1p = 0; beta1m = -2; % N
beta2p = 0; beta2m = 1; % 1/m
gamma1p = 1; gamma1m = 0; % N
gamma2p = 3.5; gamma2m = 0; % 1/m
gamma3p = 0; gamma3m = 0; % m
% 1.2 Define initial values of generalized force and tangent stiffness
f(1) = 0.0; % N
kt(1) = 0.0; % N/m
%% 2 CALCULATIONS AT EACH TIME STEP
for i = 2:Ns
% 2.1 Update the model parameters
kb = kbp; f0 = f0p; alfa = alfap; beta1 = beta1p; beta2 = beta2p; gamma1 = gamma1p; gamma2 = gamma2p; gamma3 =
gamma3p;
if sign(ud(i)) < 0
kb = kbm; f0 = f0m; alfa = alfam; beta1 = beta1m; beta2 = beta2m; gamma1 = gamma1m; gamma2 = gamma2m; gamma3 =
gamma3m;
end
% 2.2 Evaluate the generalized force
fe(i-1) = beta1*exp(beta2*u(i-1))-beta1+(4*gamma1/(1+exp(-gamma2*(u(i-1)-gamma3))))-2*gamma1;
fe(i) = beta1*exp(beta2*u(i)) -beta1+(4*gamma1/(1+exp(-gamma2*(u(i) -gamma3))))-2*gamma1;
f(i) = fe(i)+kb*u(i)+sign(ud(i))*f0-(fe(i-1)+kb*u(i-1)+sign(ud(i))*f0-f(i-1))*exp(-sign(ud(i))*alfa*(u(i)-u(i-
1)));
% 2.3 Compute the generalized tangent stiffness
ke(i) = beta1*beta2*exp(beta2*u(i))+(4*gamma1*gamma2*exp(-gamma2*(u(i)-gamma3)))/(1+exp(-gamma2*(u(i)-
gamma3)))^2;
kt(i) = ke(i)+kb+sign(ud(i))*alfa*(fe(i-1)+kb*u(i-1)+sign(ud(i))*f0-f(i-1))*exp(-sign(ud(i))*alfa*(u(i)-u(i-
1)));
end
%% PLOT
figure
plot(u,f,'k','linewidth',4);
set(gca,'FontSize',28)
set(gca,'FontName','Times New Roman')
grid('on');
xlabel('displacement');
ylabel('force');
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SIMULATION OF COMPLEX HYSTERESIS LOOPS
Hysteretic Mechanical Systems and Materials

13. 12
References
P1
SIMULATION OF COMPLEX HYSTERESIS LOOPS
Hysteretic Mechanical Systems and Materials