This document describes performing a nonlinear time history analysis of a single degree of freedom hysteretic system. It introduces the Vaiana Rosati model for simulating rate-independent hysteretic behavior. The second-order differential equation of motion is numerically solved using an explicit time integration method. Example analyses are presented for a system subjected to sinusoidal and cosine harmonic external forces. Matlab code implementing the numerical method is also provided.

1. Hysteretic Mechanical Systems
and Materials
with Matlab Codes
Version 27 August 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
SDF Hysteretic System 1
VRM AF - CFEMs
NONLINEAR TIME HYSTORY ANALYSIS

3. P21
Introduction
This short report briefly illustrates the main ingredients required to perform Nonlinear Time History Analyses
(NLTHAs) of a Single Degree of Freedom (SDF) system having rate-independent hysteretic behavior.
The Vaiana Rosati Model – Analytical Formulation (VRM AF) is adopted to simulate the behavior of the rate-
independent hysteretic element.
The second-order Ordinary Differential Equation (ODE) of motion is numerically solved by using the Chang’s
Family of Explicit structure-dependent time integration Methods (CFEMs).
Hysteretic Mechanical Systems and Materials
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NONLINEAR TIME HYSTORY ANALYSIS

4. P31
Nonlinear Equilibrium Equation
The nonlinear equilibrium equation of the SDF rate-independent hysteretic system is:
𝑚 ሷ
𝑢(𝑡) + 𝑓(𝑡) = 𝑝 𝑡 ,
where ሷ
𝑢(𝑡) is the acceleration of the mass 𝑚, 𝑓(𝑡) represents the rate-independent hysteretic generalized
force, and 𝑝 𝑡 is the external generalized force.
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NONLINEAR TIME HYSTORY ANALYSIS

5. P41
Rate-Independent Hysteretic Generalized Force
The expression of 𝑓 𝑡 is provided by the Vaiana Rosati Model - Analytical Formulation (VRM AF):
𝑓 𝑡 = 𝑓
𝑒 𝑡 + 𝑘𝑏𝑢(𝑡) + sgn ሶ
𝑢 𝑡 𝑓0 − 𝑓𝑒 𝑡𝑃 + 𝑘𝑏𝑢 𝑡𝑃 + sgn ሶ
𝑢 𝑡 𝑓0 − 𝑓 𝑡𝑃 𝑒−sgn ሶ
𝑢 𝑡 𝛼 𝑢(𝑡)−𝑢(𝑡𝑃)
,
where:
𝑓
𝑒 𝑡 = 𝛽1𝑒𝛽2𝑢(𝑡)
− 𝛽1 +
4𝛾1
1+𝑒−𝛾2 𝑢(𝑡)−𝛾3
− 2𝛾1.
During the generic loading phase ( ሶ
𝑢(𝑡) > 0), the model parameters are:
𝑘𝑏 = 𝑘𝑏
+
, 𝑓0 = 𝑓0
+
, 𝛼 = 𝛼+
, 𝛽1 = 𝛽1
+
, 𝛽2 = 𝛽2
+
, 𝛾1 = 𝛾1
+
, 𝛾2 = 𝛾2
+
, 𝛾3 = 𝛾3
+
,
whereas, during the generic unloading one ( ሶ
𝑢(𝑡) < 0), they are:
𝑘𝑏 = 𝑘𝑏
−
, 𝑓0 = 𝑓0
−
, 𝛼 = 𝛼−
, 𝛽1 = 𝛽1
−
, 𝛽2 = 𝛽2
−
, 𝛾1 = 𝛾1
−
, 𝛾2 = 𝛾2
−
, 𝛾3 = 𝛾3
−
.
Note that the only conditions to be fulfilled are:
𝛼+
> 0, 𝛼−
> 0, 𝑓0
+
> 𝑓0
−
,
since the other parameters can be arbitrary real numbers.
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NONLINEAR TIME HYSTORY ANALYSIS

6. P51
External Generalized Force
In the case of a sinusoidal harmonic generalized force (left), the expression of 𝑝 𝑡 is:
𝑝 𝑡 = 𝑝0 sin 2𝜋𝑓𝑝𝑡 ,
whereas, in the case of a cosine harmonic generalized force (right), it becomes:
𝑝 𝑡 = 𝑝0 cos 2𝜋𝑓𝑝𝑡 ,
where 𝑝0 and 𝑓𝑝 represent the force amplitude and frequency, respectively.
Hysteretic Mechanical Systems and Materials
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NONLINEAR TIME HYSTORY ANALYSIS

7. P61
Numerical Method
Hysteretic Mechanical Systems and Materials
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NONLINEAR TIME HYSTORY ANALYSIS
1 Initial settings
1.1 Evaluate the following scalars:
𝑚 and 𝑘0.
1.2 Initialize 𝑢𝑡=0 and ሶ
𝑢𝑡=0; then evaluate the initial generalized acceleration:
ሷ
𝑢𝑡=0 = 𝑚−1
𝑝𝑡=0 − 𝑓𝑡=0 .
2 Calculations at each time step
2.1 Compute the generalized displacement:
2.2 Evaluate the generalized velocity:
𝑢𝑡 = 𝑢𝑡−∆𝑡 + ∆𝑡 ሶ
𝑢𝑡−∆𝑡 + 𝑠0
−1
𝑚 ∆𝑡 2
ሷ
𝑢𝑡−∆𝑡 + α ∆𝑡 2
𝑝𝑡 − 𝑝𝑡−∆𝑡 .
ሶ
𝑢𝑡 = ሶ
𝑢𝑡−∆𝑡 + 𝑠0
−1
𝑚 ∆𝑡 ሷ
𝑢𝑡−∆𝑡.
2.4 Evaluate the generalized acceleration:
ሷ
𝑢𝑡 = 𝑚−1
𝑝𝑡 − 𝑓𝑡 .
1.3 Select time step ∆𝑡, set 𝛼 = 1/4, and compute 𝑠0:
𝑠0 = 𝑚 + α ∆𝑡 2
𝑘0.
2.3 Compute the rate-independent hysteretic generalized force:
𝑓𝑡 = 𝑓𝑒 𝑡 + 𝑘𝑏 𝑢𝑡 + 𝑠𝑡 𝑓0 − 𝑓𝑒 𝑡−∆𝑡 + 𝑘𝑏 𝑢𝑡−∆𝑡 + 𝑠𝑡 𝑓0 − 𝑓𝑡−∆𝑡 𝑒−𝑠𝑡𝛼 𝑢𝑡−𝑢𝑡−∆𝑡 ,
with:
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,
and:
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).
The adopted numerical method, whose implementation algorithm is illustrated above, belongs to the Chang’s
Family of Explicit structure-dependent time integration Methods (CFEMs). Such a method, obtained by setting
𝛼 = 1/4, exhibits excellent accuracy and stability properties. More details are available in [3].

8. P71
Results – Sinusoidal Generalized Force
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NONLINEAR TIME HYSTORY ANALYSIS
mass applied force VRM parameters
𝑚 𝑝0 𝑓𝑝 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
Ns2m−1
N Hz Nm−1
N m−1
N m−1
N m−1
m
10 14 1 + 0 1.2 80 0.01 35 2 80 0.006
− 0 1.2 80 - 0.01 - 35 2 80 - 0.006

9. P81
Results – Cosine Generalized Force
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NONLINEAR TIME HYSTORY ANALYSIS
mass applied force VRM parameters
𝑚 𝑝0 𝑓𝑝 𝑘𝑏 𝑓0 𝛼 𝛽1 𝛽2 𝛾1 𝛾2 𝛾3
Ns2m−1
N Hz Nm−1
N m−1
N m−1
N m−1
m
10 14 1 + 0 1.2 80 0.01 35 2 80 0.006
− 0 1.2 80 - 0.01 - 35 2 80 - 0.006

10. 9
Matlab Code - NLTHA_SYSTEM_1_VRM_AF_CFEMs.m
% =========================================================================================
% August 2023
% Nonlinear Time History Analysis of SDF Rate-Independent Hysteretic Systems
% 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;
%% SDF RATE-INDEPEDENT HYSTERETIC SYSTEM MASS
m = 10; % Ns^2/m
%% VAIANA ROSATI MODEL PARAMETERS
kbp = 0; kbm = 0; % N/m
f0p = 1.2; f0m = 1.2; % N
alfap = 80; alfam = 80; % 1/m
beta1p = 0.01; beta1m = -0.01; % N
beta2p = 35; beta2m = -35; % 1/m
gamma1p = 2; gamma1m = 2; % N
gamma2p = 80; gamma2m = 80; % 1/m
gamma3p = 0.006; gamma3m = -0.006; % m
parp = [kbp f0p alfap beta1p beta2p gamma1p gamma2p gamma3p]; % -
parm = [kbm f0m alfam beta1m beta2m gamma1m gamma2m gamma3m]; % -
%% INITIAL CONDITIONS
u0 = 0; % m
ud0 = 0; % m/s
%% EXTERNAL GENERALIZED FORCE
dt = 0.001; % s
tv = 0:dt:10; % s
fp = 1; % Hz
p0 = 14; % N
p = p0*sin(2*pi*fp*tv); % N
Ns = length(tv); % -
%% CHANG'S FAMILY OF EXPLICIT METHODS
%% 1 INITIAL SETTINGS
% 1.1 Evaluate the following scalars:
[f(1),kt(1)] = VRM_AF(u0,ud0,0,0,parp,parm); % -
k0 = kt(1); % N/m
% 1.2 Initialize u0 and ud0; then evaluate the initial generalized acceleration:
u(1) = u0; % m
ud(1) = ud0; % m/s
udd(1) = m(p(1)-f(1)); % m/s^2
% 1.3 Set alfa = 1/4 and compute s0:
alfa = 1/4; % -
s0 = m+alfa*dt^2*k0; % Ns^2/m
%% 2 CALCULATIONS AT EACH TIME STEP
for i = 2:Ns
% 2.1 Compute the generalized displacement:
u(i) = u(i-1)+dt*ud(i-1)+s0(m*dt^2*udd(i-1)+alfa*dt^2*(p(i)-p(i-1))); % m
% 2.2 Evaluate the generalized velocity:
ud(i) = ud(i-1)+s0(m*dt*udd(i-1)); % m/s
% 2.3 Compute the rate-independent hysteretic generalized force:
f(i) = VRM_AF(u(i),sign(ud(i)),u(i-1),f(i-1),parp,parm); % N
% 2.4 Evaluate the generalized acceleration:
udd(i) = m(p(i)-f(i)); % m/s^2
end
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11. 10
%% PLOTS
figure('Color',[0.949019610881805 0.949019610881805 0.949019610881805]);
subplot('Position',[0.05 0.58 0.2 0.4]);
grid on; box on;
xlabel('time [s]');
ylabel('applied force [N]');
axis([0 10 -20 20]);
set(gca,'XTick',[0 2 4 6 8 10]);
set(gca,'YTick',[-20 -10 0 10 20]);
set(gca,'GridLineStyle','--');
set(gca,'FontName','Times New Roman');
set(gca,'FontSize',16);
plot1 = line(tv,p,'Color','[0.584313750267029 0.168627455830574 0.294117659330368]','LineWidth',3);
subplot('Position',[0.30 0.58 0.2 0.4]);
grid on; box on;
xlabel('time [s]');
ylabel('displacement [m]');
axis([0 10 -0.2 0.2]);
set(gca,'XTick',[0 2 4 6 8 10]);
set(gca,'YTick',[-0.2 -0.1 0 0.1 0.2]);
set(gca,'GridLineStyle','--');
set(gca,'FontName','Times New Roman');
set(gca,'FontSize',16);
plot2 = line(tv,u,'Color','[0.204, 0.302, 0.494]','LineWidth',3);
subplot('Position',[0.05 0.08 0.2 0.4]);
grid on; box on;
xlabel('time [s]');
ylabel('velocity [m/s]');
axis([0 10 -0.8 0.8]);
set(gca,'XTick',[0 2 4 6 8 10]);
set(gca,'YTick',[-0.8 -0.4 0 0.4 0.8]);
set(gca,'GridLineStyle','--');
set(gca,'FontName','Times New Roman');
set(gca,'FontSize',16);
plot3 = line(tv,ud,'Color','[0.204, 0.302, 0.494]','LineWidth',3);
subplot('Position',[0.30 0.08 0.2 0.4]);
grid on; box on;
xlabel('displacement [m]');
ylabel('force [N]');
axis([-0.2 0.2 -8 8]);
set(gca,'XTick',[-0.2 -0.1 0 0.1 0.2]);
set(gca,'YTick',[-8.0 -4.0 0 4.0 8.0]);
set(gca,'GridLineStyle','--');
set(gca,'FontName','Times New Roman');
set(gca,'FontSize',16);
plot4 = line(u,f,'Color','[0.204, 0.302, 0.494]','LineWidth',3);
subplot('Position',[0.56 0.12 0.4 0.8]);
grid on; box on;
xlabel('d [m]');
ylabel('v [m/s]');
zlabel('f [N]');
axis([-0.2 0.2 -0.8 0.8 -8 8]);
set(gca,'XTick',[-0.2 -0.1 0 0.1 0.2]);
set(gca,'YTick',[-0.8 -0.4 0 0.4 0.8]);
set(gca,'ZTick',[-8.0 -4.0 0 4.0 8.0]);
set(gca,'GridLineStyle','--');
set(gca,'FontName','Times New Roman');
set(gca,'FontSize',16);
set(gca,'BoxStyle','full');
view([229.572533907569 40.0908387200157]);
plot5 = line(u,ud,f,'Color','[0.204, 0.302, 0.494]','Linewidth',3);
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Matlab Code - NLTHA_SYSTEM_1_VRM_AF_CFEMs.m

12. 11
Matlab Code - VRM_AF.m
function [f,kt] = VRM_AF(u,s,up,fp,parp,parm)
% Update the model parameters
kb = parp(1); f0 = parp(2); alfa = parp(3); beta1 = parp(4);
beta2 = parp(5); gamma1 = parp(6); gamma2 = parp(7); gamma3 = parp(8);
if s < 0
kb = parm(1); f0 = parm(2); alfa = parm(3); beta1 = parm(4);
beta2 = parm(5); gamma1 = parm(6); gamma2 = parm(7); gamma3 = parm(8);
end
% Evaluate the generalized force
fep = beta1*exp(beta2*up)-beta1+(4*gamma1/(1+exp(-gamma2*(up-gamma3))))-2*gamma1;
fe = beta1*exp(beta2*u) -beta1+(4*gamma1/(1+exp(-gamma2*(u -gamma3))))-2*gamma1;
f = fe+kb*u+s*f0-(fep+kb*up+s*f0-fp)*exp(-s*alfa*(u-up));
% Compute the generalized tangent stiffness
ke = beta1*beta2*exp(beta2*u)+(4*gamma1*gamma2*exp(-gamma2*(u-gamma3)))/(1+exp(-gamma2*(u-gamma3)))^2;
kt = ke+kb+s*alfa*(fep+kb*up+s*f0-fp)*exp(-s*alfa*(u-up));
end
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13. 12
References
[1] Vaiana N, Sessa S, Marmo F, Rosati L (2018) A class of uniaxial phenomenological models for simulating hysteretic
phenomena in rate-independent mechanical systems and materials. Nonlinear Dynamics 93(3): 1647-1669.
[2] Vaiana N, Sessa S, Marmo F, Rosati L (2019) An accurate and computationally efficient uniaxial phenomenological model for
steel and fiber reinforced elastomeric bearings. Composite Structures 211: 196-212.
[3] Vaiana N, Sessa S, Marmo F, Rosati L (2019) Nonlinear dynamic analysis of hysteretic mechanical systems by combining a
novel rate-independent model and an explicit time integration method. Nonlinear Dynamics 98(4): 2879-2901.
[4] Vaiana N, Sessa S, Rosati L (2021) A generalized class of uniaxial rate-independent models for simulating asymmetric
mechanical hysteresis phenomena. Mechanical Systems and Signal Processing 146: 106984.
[5] Vaiana N, Rosati L (2023) Classification and unified phenomenological modeling of complex uniaxial rate-independent
hysteretic responses. Mechanical Systems and Signal Processing 182: 109539.
[6] Vaiana N, Capuano R, Rosati L (2023) Evaluation of path-dependent work and internal energy change for hysteretic
mechanical systems. Mechanical Systems and Signal Processing 186: 109862.
[7] Vaiana N, Rosati L (2023) Analytical and differential reformulations of the Vaiana–Rosati model for complex rate-independent
mechanical hysteresis phenomena. Mechanical Systems and Signal Processing 199: 110448.
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