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 - Differential Formulation (VRM DF) is adopted to simulate the behavior of the rate-independent hysteretic element. The second-order Ordinary Differential Equation (ODE) of motion is replaced by an equivalent system of three coupled first-order ODEs and numerically solved by using the MATLAB® ode45 solver that is based on an explicit fourth-fifth-order Runge Kutta Method (RKM).

1. Hysteretic Mechanical Systems
and Materials
with Matlab Codes
Version 14 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 DF - RKM
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 - Differential Formulation (VRM DF) is adopted to simulate the behavior of the rate-
independent hysteretic element.
The second-order Ordinary Differential Equation (ODE) of motion is replaced by an equivalent system of three
coupled first-order ODEs and numerically solved by using the MATLAB® ode45 solver that is based on an
explicit fourth-fifth-order Runge Kutta Method (RKM).
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.
Such a second-order ODE can be replaced by an equivalent system of coupled first-order ODEs. To this end,
the following state variables are first introduced:
𝑥1 𝑡 = 𝑢 𝑡 ,
𝑥2 𝑡 = ሶ
𝑢 𝑡 ,
𝑥3 𝑡 = 𝑓 𝑡 .
Subsequently, they are differentiated with respect to time 𝑡 thus obtaining:
ሶ
𝑥1 𝑡 = ሶ
𝑢 𝑡 = 𝑥2 𝑡 ,
ሶ
𝑥2 𝑡 = ሷ
𝑢 𝑡 = 𝑚−1
𝑝 𝑡 − 𝑥3(𝑡) ,
ሶ
𝑥3 𝑡 = ሶ
𝑓 𝑡 .
Hysteretic Mechanical Systems and Materials
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NONLINEAR TIME HYSTORY ANALYSIS

5. P41
Rate-Independent Hysteretic Generalized Force
The expression of ሶ
𝑓 𝑡 is provided by the Vaiana Rosati Model - Differential Formulation (VRM DF):
ሶ
𝑓 𝑡 = 𝑘𝑒 𝑡 + 𝑘𝑏 + sgn ሶ
𝑢 𝑡 𝛼 𝑓
𝑒 𝑡 + 𝑘𝑏𝑢 𝑡 + sgn ሶ
𝑢 𝑡 𝑓0 − 𝑓 𝑡 ሶ
𝑢 𝑡 ,
where:
𝑘𝑒 𝑡 = 𝛽1𝛽2𝑒𝛽2𝑢(𝑡)
+
4𝛾1𝛾2 𝑒−𝛾2 𝑢(𝑡)−𝛾3
1+𝑒−𝛾2 𝑢(𝑡)−𝛾3
2 ,
𝑓
𝑒 𝑡 = 𝛽1𝑒𝛽2𝑢(𝑡)
− 𝛽1 +
4𝛾1
1+𝑒−𝛾2 𝑢(𝑡)−𝛾3
− 2𝛾1.
The solution of the differential equation must satisfy the following initial condition:
𝑓 𝑢(𝑡𝑃) = 𝑓 𝑡𝑃 .
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.
Hysteretic Mechanical Systems and Materials
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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. The system of coupled first-order ODEs to be numerically solved is:
ሶ
𝑥1 𝑡 = 𝑥2 𝑡 ,
ሶ
𝑥2 𝑡 = 𝑚−1
𝑝 𝑡 − 𝑥3(𝑡) ,
ሶ
𝑥3 𝑡 = 𝑘𝑒 𝑡 + 𝑘𝑏 + sgn 𝑥2 𝑡 𝛼 𝑓
𝑒 𝑡 + 𝑘𝑏𝑥1 𝑡 + sgn 𝑥2 𝑡 𝑓0 − 𝑥3 𝑡 𝑥2 𝑡 ,
where:
𝑘𝑒 𝑡 = 𝛽1𝛽2𝑒𝛽2𝑥1 𝑡
+
4𝛾1𝛾2 𝑒−𝛾2 𝑥1 𝑡 −𝛾3
1+𝑒−𝛾2 𝑥1 𝑡 −𝛾3
2 ,
𝑓
𝑒 𝑡 = 𝛽1𝑒𝛽2𝑥1 𝑡
− 𝛽1 +
4𝛾1
1+𝑒−𝛾2 𝑥1 𝑡 −𝛾3
− 2𝛾1,
and:
𝑘𝑏 = 𝑘𝑏
+
, 𝑓0 = 𝑓0
+
, 𝛼 = 𝛼+
, 𝛽1 = 𝛽1
+
, 𝛽2 = 𝛽2
+
, 𝛾1 = 𝛾1
+
, 𝛾2 = 𝛾2
+
, 𝛾3 = 𝛾3
+
, if 𝑥2 𝑡 > 0,
𝑘𝑏 = 𝑘𝑏
−
, 𝑓0 = 𝑓0
−
, 𝛼 = 𝛼−
, 𝛽1 = 𝛽1
−
, 𝛽2 = 𝛽2
−
, 𝛾1 = 𝛾1
−
, 𝛾2 = 𝛾2
−
, 𝛾3 = 𝛾3
−
, if 𝑥2 𝑡 < 0.
To this end, it is adopted the MATLAB® ode45 solver that, being based on an explicit fourth-fifth-order Runge
Kutta formula, allows for the evaluation of the solution at time 𝑡 by adopting the solution at the preceding
time 𝑡𝑃 = 𝑡 − ∆𝑡.
P61
Numerical Method
Hysteretic Mechanical Systems and Materials
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NONLINEAR TIME HYSTORY ANALYSIS

8. P71
Results – Sinusoidal Generalized Force
Hysteretic Mechanical Systems and Materials
P2
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
Hysteretic Mechanical Systems and Materials
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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_DF_RKM.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]; % -
%% EXTERNAL GENERALIZED FORCE
tv = 0:0.001:10; % s
fp = 1; % Hz
p0 = 14; % N
p = p0*sin(2*pi*fp*tv(1:length(tv))); % N
%% RUNGE-KUTTA METHOD
%% INITIAL SETTING
neq = 3; % - number of equations
IC = [0 0 0]; % - initial conditions [x1 x2 x3]
%% CALCULATIONS AT EACH TIME STEP
options = odeset('RelTol',1e-10,'AbsTol',1e-10);
[t,x] = ode45(@(t,x) ODEs(t, x, neq, m, parp, parm, p, tv), tv, IC, options);
Hysteretic Mechanical Systems and Materials
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NONLINEAR TIME HYSTORY ANALYSIS

11. 10
Matlab Code - NLTHA_SYSTEM_1_VRM_DF_RKM.m
%% 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(t,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(t,x(:,1),'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(t,x(:,2),'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(x(:,1),x(:,3),'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(x(:,1),x(:,2),x(:,3),'Color','[0.204, 0.302, 0.494]','Linewidth',3);
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NONLINEAR TIME HYSTORY ANALYSIS

12. 11
Matlab Code - ODEs.m
function xd = ODEs(t, x, neq, m, parp, parm, pv, tv)
%% EXTERNAL GENERALIZED FORCE
p = interp1(tv,pv,t); % N
%% STATE VARIABLES
u = x(1); % m displacement
ud = x(2); % m/s velocity
f = x(3); % N hysteretic force
%% VAIANA ROSATI MODEL PARAMETERS
if ud > 0
kb = parp(1); f0 = parp(2); alfa = parp(3); beta1 = parp(4);
beta2 = parp(5); gamma1 = parp(6); gamma2 = parp(7); gamma3 = parp(8);
else
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
%% SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS
xd = zeros(neq,1);
xd(1) = ud;
xd(2) = (p-f)/m;
fe = beta1*exp(beta2*u)-beta1+(4*gamma1/(1+exp(-gamma2*(u-gamma3))))-2*gamma1;
ke = beta1*beta2*exp(beta2*u)+(4*gamma1*gamma2*exp(-gamma2*(u-gamma3)))/(1+exp(-gamma2*(u-gamma3)))^2;
xd(3) = (ke+kb+sign(ud)*alfa*(fe+kb*u+sign(ud)*f0-f))*ud;
end
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NONLINEAR TIME HYSTORY ANALYSIS

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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