<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\makeatletter
\@ifundefined{HCode}                  
{\documentclass[screen,CRMECA,Unicode,biblatex,published]{cedram}
\addbibresource{crmeca20240923.bib}
\newenvironment{noXML}{}{}
\def\tsup#1{$^{{#1}}$}
\def\tsub#1{$_{{#1}}$}
\def\sfrac#1#2{{#1}/{#2}}
\def\thead{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\xtbody{\noalign{\relax}}
\def\tabnote#1{\vskip4pt\parbox{.51\linewidth}{#1}}
\def\xxtabnote#1{\vskip4pt\parbox{.65\linewidth}{#1}}
\def\mathbi#1{\boldsymbol{#1}}
\def\xmathop#1{\mathop{\text{#1}}}
\def\rmDelta{\Delta}
\csdef{Seqnsplit}{\\}
\def\botline{\\\hline}
\newcounter{runlevel}
\usepackage{multirow} 
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\let\MakeYrStrItalic\relax
\def\jobid{crmeca20240923}
%\graphicspath{{/tmp/\jobid_figs/web/}} 
\graphicspath{{./figures/}} 
\def\rmmu{\upmu}
\def\mn{\phantom{$-$}}
\def\refinput#1{}
\def\back#1{}
\def\xmorerows#1#2{#2}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newenvironment{inftab}{}{}
\def\Custref#1#2{\hyperlink{#2}{#1}}
\def\og{\guillemotleft}
\def\fg{\guillemotright}
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\makeatother
\DOI{10.5802/crmeca.295}
\datereceived{2024-11-07}
\daterevised{2025-03-03}
\dateaccepted{2025-04-02}
\ItHasTeXPublished
}
{
\documentclass[crmeca]{article}
\usepackage[T1]{fontenc} 
\def\CDRdoi{10.5802/crmeca.295}
\def\newline{\unskip\break}
\def\sfrac#1#2{{#1}/{#2}}
\def\xxtabnote#1{\tabnote{#1}}
\def\xmathop#1{\mathop{{#1}}}
\def\xmorerows#1#2{\morerows{#1}{#2}}
\def\xtbody{\tbody}
\def\hypertarget#1#2{\label{#1}}
}

\usepackage{upgreek}

\dateposted{2025-04-30}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Delamination identification in beams using an inverse analytical
approach based on the natural frequencies}

\alttitle{Identification du d\'{e}laminage dans les poutres \`{a} l'aide
d'une approche analytique inverse bas\'{e}e sur les fr\'{e}quences
naturelles}

\author{\firstname{Saad} \lastname{Yazir}\CDRorcid{0009-0007-3723-2488}}
\address{Laboratory of Development in Mechanics and Materials (LDMM)
- University of Djelfa, 17000, Algeria}
\email[S. Yazir]{saad.yazir@univ-djelfa.dz}

\author{\firstname{Sadam Houcine} \lastname{Habib}\CDRorcid{0000-0002-1457-7906}\IsCorresp}
\address{Department of Mechanical Engineering, University of Djelfa,
Algeria}
\email[S. H. Habib]{s.habib@univ-djelfa.dz}

\author{\firstname{Cheikh} \lastname{Kezrane}\CDRorcid{0000-0001-7341-0711}}
\addressSameAs{1}{Laboratory of Development in Mechanics and Materials
(LDMM) - University of Djelfa, 17000, Algeria}
\addressSameAs{2}{Department of Mechanical Engineering, University of
Djelfa, Algeria}
\email[C. Kezrane]{c.kezrane@univ-djelfa.dz}

\begin{abstract}
Substantial efforts have been invested in developing diverse methods
for delamination identification.
Among these, the vibration-based
method has garnered considerable attention from researchers and has
demonstrated commendable efficacy. In the present study, we have
implemented a frequency-based methodology to effectively identify
delamination in beams, employing a brief set of only three natural
frequencies. The constrained mode model, an established model available
in the literature, has been adopted as a forward model. Furthermore,
the paper introduces a formulation of a system of nonlinear equations
designed for inversion. The idea is based on nullifying the analytical
expression of the determinant at the resonance condition. The
analytical relationship between the frequencies and delamination
parameters is written in the form of a nonlinear system of equations. A
graphical technique has been used to solve the resulting nonlinear
system of equations. The proposed method has been applied to beams
subjected to three distinct boundary conditions for the sake of testing
its adaptability across various structural configurations. The outcome
of our study proves the successful identification of delamination
parameters with a remarkable accuracy.
\end{abstract}

\begin{altabstract}
D'importants efforts ont \'{e}t\'{e} consacr\'{e}s au d\'{e}veloppement
de diverses m\'{e}thodes pour l'identification du d\'{e}laminage. Parmi
celles-ci, la m\'{e}thode bas\'{e}e sur les vibrations a suscit\'{e} un
vif int\'{e}r\^{e}t chez les chercheurs et a d\'{e}montr\'{e} une
efficacit\'{e} notable. Dans la pr\'{e}sente \'{e}tude, nous avons mis
en \oe{}uvre une m\'{e}thodologie bas\'{e}e sur les fr\'{e}quences pour
identifier efficacement le d\'{e}laminage dans des poutres, en
utilisant un ensemble restreint de seulement trois fr\'{e}quences
naturelles. Le mod\`{e}le des modes contraints, un mod\`{e}le
\'{e}tabli dans la litt\'{e}rature, a \'{e}t\'{e} adopt\'{e} comme
mod\`{e}le direct. En outre, l'article introduit une formulation d'un
syst\`{e}me d'\'{e}quations non lin\'{e}aires con\c{c}u pour
l'inversion. L'id\'{e}e repose sur l'annulation de l'expression
analytique du d\'{e}terminant \`{a} la condition de r\'{e}sonance. La
relation analytique entre les fr\'{e}quences et les param\`{e}tres de
d\'{e}laminage est exprim\'{e}e sous forme d'un syst\`{e}me
d'\'{e}quations non lin\'{e}aires. Une technique graphique a
\'{e}t\'{e} utilis\'{e}e pour r\'{e}soudre ce syst\`{e}me
d'\'{e}quations. La m\'{e}thode propos\'{e}e a \'{e}t\'{e}
appliqu\'{e}e \`{a} des poutres soumises \`{a} trois conditions aux
limites distinctes, afin de tester son adaptabilit\'{e} \`{a}
diff\'{e}rentes configurations structurelles. Les r\'{e}sultats de
notre \'{e}tude prouvent l'identification r\'{e}ussie des
param\`{e}tres de d\'{e}laminage avec une pr\'{e}cision remarquable.
\end{altabstract}

\keywords{\kwd{Inverse method}\kwd{Delamination
identification}\kwd{Natural frequencies}\kwd{Resonance}\kwd{Beam}}

\altkeywords{\kwd{M\'{e}thode inverse}\kwd{Identification du 
d\'{e}laminage}\kwd{Fr\'{e}quences
naturelles}\kwd{R\'{e}sonance}\kwd{Poutre}}

\shortrunauthors

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

Composite materials are used in several applications due to their high
stiffness-to-weight ratio in comparison with metallic materials.
Despite their advantages, composites are subjected to many types of
defects which can originate during the fabrication or in-service phase.
The most known defects are disbands, foreign inclusions, matrix
cracking, porosity, fibre breakage and delamination. The latter defect
is particularly significant for the material's structural integrity, as
it involves the separation of two laminate layers.

The risks associated with damage in mechanical structures can have
severe consequences. These damages can compromise the integrity of
various structures such as bridges, pipelines, aircraft and machinery,
leading to catastrophic failures, financial losses, and potentially
endangering human lives. Identifying defects in these structures is
imperative to ensure their safe operation and longevity. Developing
effective methods for identifying defects is crucial as it allows for
early intervention, maintenance, and repair, preventing the escalation
of minor issues into major problems. Non-destructive testing
techniques, such as ultrasonic testing, radiography, and thermography,
play a pivotal role in this regard, enabling engineers and inspectors
to assess internal structural conditions without causing further harm.


Delamination characterization in composite laminates is a crucial
aspect of structural analysis, aiming to understand the nature and
implications of this form of damage. Various methods are employed to
quantify and assess delamination, including non-destructive techniques
such as ultrasonic testing~\cite{1,2} thermography~\cite{3}, and
acoustic emission monitoring~\cite{4}. These methods allow for the
identification of delamination size, location, and progression without
causing further harm to the material. Additionally, non-destructive
testing methods like microscopy and shearography provide in-depth
insights into the morphology of delamination, aiding in understanding
initiation mechanisms and failure modes. Characterization also involves
employing fracture mechanics concepts, such as assessing the mode of
delamination, to predict crack growth and devise effective repair
strategies. Material property evaluations such as Interlaminar Shear
Strength (ILSS)~\cite{5} contributes to a comprehensive understanding
of delamination behaviour. Ultimately, delamination characterization
plays a pivotal role in ensuring the structural integrity and
reliability of composite laminates, guiding decisions on maintenance,
repair, and future material development.

The field of vibration-based damage identification has seen significant
contributions from several researchers. Doebling et~al.~\cite{5}
provided a comprehensive review of methods to detect, locate, and
characterize structural damage through changes in modal parameters such
as natural frequencies and mode shapes. Their work categorizes various
techniques and discusses challenges and applications.  Gibson~\cite{6}
extended the use of modal testing to fiber-reinforced composites, his
work demonstrates the utility of modal testing in characterizing
elastic moduli, damping, and defect detection under diverse conditions.
Gadelrab~\cite{7} explored the influence of delamination in composite
laminated beams on natural frequencies, it provides insights into
frequency deviations as a diagnostic tool for delamination detection
through finite element analysis. Diaz Valdes and Soutis~\cite{8}
advanced the field by utilizing resonant ultrasound spectroscopy and
piezoelectric sensors to investigate the effect of delamination on
modal frequencies. Keilers et~al.~\cite{9} introduced a delamination
identification method leveraging piezoelectric sensors embedded in
composite structures, employing a simulated annealing algorithm to
estimate delamination size and location by minimizing discrepancies
between measured and modelled responses. Bois and Hoshard~\cite{10}
laid the groundwork for a multi-damage monitoring system by combination
of electro-mechanical impedance measurements with predictive damage
models to identify delamination under static and fatigue loads. Bois
et~al.~\cite{11} extended this work by developing a method to monitor
single delamination using piezoelectric transducers and impedance
measurements, this led to precise localization and dimension estimation
through coupling with bending modes. Zou et~al.~\cite{12} emphasized the
need for health monitoring and presented model-dependent methods that
integrate piezoelectric sensors and actuators with finite element
analysis to locate and estimate damage.

To solve the identification problem of delamination in structures, many
researchers have done important numerical works in the last decades.
For instance, Venkatesan et~al.~\cite{13} have utilized inverse
calculations based on the surrogate-assisted real-coded genetic
algorithm (SARGA) and an artificial neural network (ANN). These inverse
calculations were validated using numerically simulated and
experimentally measured frequencies. Manisha Maurya et~al.~\cite{14}
have used three first natural frequencies and neural networks to
identify delamination in carbon fibre-reinforced polymer beam; they
concluded that ANN can predict delamination starting from natural
frequencies. Saaed Ullah et~al.~\cite{15} have proposed a deep learning
(DL) approach based on lamb waves. They developed two DL models and
trained them on numerical datasets obtained by animation of lamb waves
calculated numerically. The accuracy of these models was evaluated
using unseen numerical and experimental animations of lamb waves. Zhang
et~al.~\cite{16} have examined three vibration-based inverse algorithms
to predict delamination position and size. The algorithms are graphical
method, ANN and surrogate-based optimization. Their inverse models are
based on a numerical FEM forward model. There are many other numerical
studies such as~\cite{17,18,19,20}. In general, numerical techniques
have their own set of disadvantages such as sensitivity to data,
non-uniqueness (multiple solutions), computational complexity, model
uncertainty, noise sensitivity of some experimental input and
incomplete data.


The identification problem of delamination has also been studied
experimentally by different methods, for example, Nakatani
et~al.~\cite{21} have applied the passive electric potential CT
(Computed Tomography) method as a physical phenomenon. The
piezo-electrical film glued to the external surface generates an
electrical potential under a mechanical load, where the potential is
measured by a non-contact device. The variation of this measured
potential indicates the presence of delamination. Wang et~al.~\cite{22}
have applied the wavelet transform method which is based on the
detection of discontinuity in the gradient at the delamination
position. Rakesh et~al.~\cite{23} have used the scattering
characteristics of lamb waves to identify delamination. Similar
experimental works are available in~\cite{24,25,26}. The experimental
methods present many advantages. They allow direct observation of
structure response providing real-time data on its behaviour. They also
provide different quantitative data which can be used for the
assessment of structure integrity. However, experimental methods know
some drawbacks in delamination identification; they can be expensive
and time-consuming, especially for large and complex structures.

The dynamic behaviour of delaminated beams has been investigated
analytically. For example, Wang and Liut~\cite{27} have studied the
free vibration of split beams, Mujumdar and Suryanarayan~\cite{28} have
presented an analytical forward model for isotropic beams with
delamination. There are several analytical forward models available in
the literature.  For example, there are the free mode model and the
constrained mode model. In the free mode model, it is assumed that the
four sub-beams move freely~\cite{27}. In the constrained mode model,
the sub-beams above and below the delamination are constrained to move
together. This model is more accurate, especially for delamination
located on the midplane of the beam~\cite{28}. For identifying
delamination parameters using modal frequencies, an inverse model is
necessarily based on the forward \mbox{analytical} models (constrained and
free mode models). According to our research in the literature, there
are no analytical inverse models available in the literature.

In our work, we seek to identify delamination in beams using an
analytical inverse model. In this model, the first three natural
frequencies, which correspond to a real delaminated beam, are given as
input to find the delamination position and size. Based on the forward
constrained mode model, we compose a system of nonlinear equations from
three characteristic equations. Each characteristic equation is the
determinant of the continuity and boundary conditions matrix. These
characteristic equations are in terms of four independent parameters:
horizontal position \textit{a}, vertical position \textit{d},
delamination size \textit{c} and natural frequency $\lambda $, where
the fourth parameter $\lambda $ is a given parameter. The solution of
the system of nonlinear equations represents the desired delamination
parameters (\textit{a}, \textit{d} and \textit{c}). We use a graphical
technique in order to find the solution as in~\cite{16}. The main
advantage of this inverse model is to find an accurate solution rapidly
without using frequencies of pristine beams or comparing with modal
models.


\section{Methodology}\label{sec2}

The natural frequency-based method is a well-known technique for
identifying delamination in materials. It operates on the principle
that the presence of delamination alters the stiffness and mass
distribution of the structure, leading to changes in its natural
frequencies. Our new inverse approach uses this same principle by means
of a set of natural frequencies to identify delamination analytically.
This approach consists of two stages: forward modelling and inverse
modelling.

The forward modelling stage begins by defining the geometry, material
properties, and environmental conditions surrounding the structure. The
presence of delamination is accounted for in the forward modelling. The
calculation of natural frequencies involves solving an equation where
the determinant of a coefficient matrix equals zero. This determinant
is a function of the delamination parameters and the frequency. The
forward modelling concludes when the natural frequencies are
determined.

The inverse modelling begins by considering a set of natural
frequencies obtained from experimental measurements. Using the same
determinant function built in the forward modelling stage, an equation
is formed for each frequency. Solving the system of equations in terms
of the delamination parameters allows for the identification of these
parameters. More details are provided in the synoptic scheme in
Figure~\ref{fig1}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Synoptic scheme of the proposed approach.}
\end{figure}


Note that when additional delaminations are introduced, identifying
their locations and sizes requires additional equations. Specifically,
for each new delamination, three additional equations are needed to
ensure a well-posed system. Higher-order modes can provide more
information, but they also tend to be more sensitive to modeling
inaccuracies and experimental uncertainties, which could affect the
reliability of the identification process. Therefore, the
generalization must be approached carefully, ensuring that sufficient
independent equations are available to accurately determine the
additional unknowns introduced by multiple delaminations.



\section{Application example} \label{sec3}

To validate the proposed inverse approach, an isotropic beam with a
single delamination is selected as the test structure. The choice of a
homogeneous isotropic beam model was made to simplify the problem and
focus on validating the proposed analytical approach for inverse
identification of delamination. This model provides a foundational
framework that ensures analytical representability and eliminates
additional complexities introduced by anisotropy or heterogeneity.
While we acknowledge this as a limitation, the methodology is designed
to be extendable to composite materials in future work, where
anisotropic and heterogeneous properties will be incorporated.

In the forward modelling phase, the constrained mode model is employed,
and three different boundary conditions are examined: clamped--clamped,
cantilever, and clamped-simply supported beams. The displacement
solution is expressed as in Section~\ref{sec3.1.2}. For this model, eight
continuity conditions are considered: continuity of displacement,
slope, bending moment, and internal shear force. Each side of the
delamination introduces four continuity conditions, resulting in a
total of eight conditions for both sides. Additionally, boundary
conditions are applied, leading to a system of twelve equations in
total. The details of direct and inverse modelling are explained in the
following sections.

\subsection{Forward modelling}\label{sec3.1}

The investigation of the free vibration behaviour of delaminated  beams
has garnered scholarly attention spanning over five decades, 
commencing with important works documented in references~\cite{29,30},
and up to contemporary research  efforts~\cite{31,32}. Consequently,
various analytical models have been proposed and discussed in the
literature~\cite{27,28}. Among these models, the constrained mode model
presented in reference~\cite{24} appears to be a particularly simple
but accurate model. This section presents key aspects of the
constrained mode model's development, starting from foundational
assumptions and ending with the delaminated beam natural frequencies.

\subsubsection{Assumptions}\label{sec3.1.1}

To avoid complexities in modelling and analysis, the following
assumptions are made: 
\begin{itemize}
\item The beam is homogeneous and isotropic.
\item Only a single delamination is considered.
\item There is no gap between layers at the delamination region.
\item The beam is subdivided into four segments as shown in
Figure~\ref{fig2}, where each segment is considered as an Euler beam.
\item The two segments of the delaminated region have the same
transverse displacement. 
\item The plane cross sections of the beam remain plane and
perpendicular to the deformed beam.
\end{itemize}


\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Geometry and coordinate systems of the delaminated
beam~\cite{28}.}
\end{figure}

\subsubsection{Modelling}\label{sec3.1.2}

The beam has three regions, the delamination region and two integral
regions, as shown in Figure~\ref{fig2}. The left integral region is the
segment 1, and the right integral region is the segment 4. The
delamination region contains two segments: the segment 2 above the
delamination plane, and the segment 3 below the delamination plane. The
calculation of the natural frequency of the delaminated beam involves
satisfying the boundary conditions at the ends of the beam and the
continuity conditions (displacement, slope, bending moment, and shear
force) at the junctions between the integral and delaminated regions as
shown in Figure~\ref{fig3}.


\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Compatibility force system.}
\end{figure}


In the constrained mode model, a contact pressure distributed between
adjacent layers is introduced to prevent overlapping of one layer over
the other. This pressure distribution $P$ ensures compatibility by
constraining the two adjacent layers to deform simultaneously. This
consideration aligns with the penultimate assumption of the model,
which states that the two segments of the delaminated region have the
same transverse displacement, thereby maintaining consistent
deformation across the layers.


For each segment, the general solution is given as
{\begin{equation} \label{eq1}
%\begin{array}{l}
\overline{u}_{i}=\overline{u}_{i}(\overline{z}_{i})\sin (wt),
\text{and }\overline{u}_{i}(\overline{z}_{i})=A_{i}\sinh 
(\lambda ^{i}\overline{z}_{i})+B_{i}\cosh 
(\lambda ^{i}\overline{z}_{i})+C_{i}\sin 
(\lambda ^{i}\overline{z}_{i})+D_{i}\cos 
(\lambda ^{i}\overline{z}_{i}),
%\end{array}
\end{equation}}\unskip
where, ${A}_{i}, {B}_{{i}}, {C}_{{i}}$ and ${D}_{{i}}$ are coefficients
associated to each segment (${i} = 1$ to 4). The frequency of the beam
segments are given as
{\begin{eqnarray*}
(\lambda ^{1})^{4}=(\lambda ^{4})^{4}=\rho A_{c}\omega ^{2}L^{4}/EI,
(\lambda ^{2})^{4}=(\lambda ^{3})^{4}=\rho (A_{c2}+A_{c3})\omega ^{2}
L^{4}/E(I_{2}+I_{3})=\lambda ^{4}/(\overline{d}_{2}^{3}+
\overline{d}_{3}^{3}),
\end{eqnarray*}}\unskip
with ${\rho}$ is the density, ${\omega}$ is the frequency, \textit{L}
is the beam length, \textit{E} is the Young's modulus, ${A}_{c}$
is the cross sectional area and \textit{I} is its moment of inertia.


\subsubsection{Boundary conditions}\label{sec3.1.3}

The type of boundary condition applied depends on the end support.
Table~\ref{tab1} presents the appropriate boundary conditions.


\begin{table}
\caption{\label{tab1} Support types and their corresponding boundary conditions}
\begin{tabular}{cc}
\thead
Support type &amp; Boundary condition \\
\endthead
Free {\raisebox{1.89em}{}} &amp; $\displaystyle\frac{\partial ^{2}u_{i}}{\partial z^{2}}=0,\frac{\partial ^{3}u_{i}}{\partial z^{3}}=0${\vspace*{4pt}} \\ 
Simple &amp; $u_{i}=0,~ ~ ~ \displaystyle\frac{\partial ^{2}u_{i}}{\partial z^{2}}=0${\vspace*{4pt}} \\ 
Sliding &amp; $\displaystyle\frac{\partial u_{i}}{\partial z}=0,~ \frac{\partial ^{3}u_{i}}{\partial z^{3}}=0${\vspace*{4pt}} \\ 
Clamped &amp; $u_{i}=0,~ ~ \displaystyle\frac{\partial u_{i}}{\partial z}=0${\vspace*{2pt}}
\botline
\end{tabular}
\end{table}



For example, let's consider a clamped--clamped beam. The boundary
conditions of this beam are clamped at both ends. Using Table~\ref{tab1} above,
we can write:

\begin{itemize}
\item At the left side ${z}_{1}=0$ we have
{\[
u_{1}(0)=0,\frac{\partial u_{1}}{\partial z}(0)=0.
\]}\unskip

\item At the right side ${z}_{4}=0$, (see Figure~\ref{fig1}) we have
{\begin{equation*}
u_{4}(0)=0,\frac{\partial u_{4}}{\partial z}(0)=0.
\end{equation*}}\unskip
\end{itemize}

By substituting in (\ref{eq1}), we obtain:
{\begin{eqnarray}
 B_{1}+D_{1}=0 ,  \label{eq2}\Seqnsplit
A_{1}+C_{1}=0,\label{eq3}\Seqnsplit
B_{4}+D_{4}=0,\label{eq4}\Seqnsplit
A_{4}+C_{4}=0.\label{eq5}
\end{eqnarray}}\unskip

\subsubsection{Continuity conditions}\label{sec3.1.4}

The continuity conditions are applied between the delaminated region
and integral regions. These conditions are displacement continuity,
slope continuity, shear force and bending moment continuity. The
continuity conditions are expressed for the continuity between the
segments 1, 2 and 3. By similarity, the continuity conditions between
segment 4 segments 2 and 3 are found.


For the same example of a clamped--clamped beam, the displacement
continuity
{\begin{equation}
\label{eq6}
\overline{u}_{1}\left(\overline{a}\right)=\overline{u}_{2}\left(-\frac{\overline{b}}{2}\right),
\end{equation}}\unskip
at the point
$(z_{1}=\overline{a},z_{2}={-\overline{b}}/{2})$ can be
written as
{\begin{eqnarray*}
\overline{u}_{1}(\overline{a})-\overline{u}_{2}
\left(-\frac{\overline{b}}{2}\right)&amp;=&amp;A_{1}\sinh 
(\lambda _{1}\overline{a})+B_{1}\cosh (\lambda _{1}\overline{a})+
C_{1}\sin (\lambda _{1}\overline{a})+D_{1}\cos (\lambda _{1}\overline{a})
-A_{2}\sinh \left(-\lambda _{2}\frac{\overline{b}}{2}\right)\\
&amp;&amp;-\,B_{2}\cosh \left(-\lambda _{2}\frac{\overline{b}}{2}\right)-
C_{2}\sin \left(-\lambda _{2}\frac{\overline{b}}{2}\right)-
D_{2}\cos \left(-\lambda _{2}\frac{\overline{b}}{2}\right)=0.
\end{eqnarray*}}\unskip
Using the same method, the displacement continuity for a
clamped--clamped at the point ($z_{2}={\overline{b}}/{2}$,
$-z_{4}=\overline{c}$) can be expressed for slope continuity
{\begin{equation}\label{eq7}
\frac{\partial \overline{u}_{1}}{\partial \overline{z}}(\overline{a})
=\frac{\partial \overline{u}_{2}}{\partial \overline{z}}\left(-\frac{\overline{b}}{2}\right).
\end{equation}}\unskip
Shear forces continuity
{\begin{equation}
\label{eq8}
-EI \frac{\partial ^{3}u_{1}}{\partial z^{3}}(\overline{a})=-E(I_{2}+I_{3})
\frac{\partial ^{3}u_{2}}{\partial z^{3}}\left(-\frac{\overline{b}}{2}\right),
\end{equation}}\unskip
and bending moment continuity
{\begin{equation}
\label{eq9}
\frac{\partial ^{2}\overline{u}_{1}}{\partial \overline{z}^{2}}
(\overline{a})=\left({\overline{d}}_{2}^{3}+{\overline{d}}_{3}^{3}\right)
\frac{\partial ^{2}\overline{u}_{2}}{\partial \overline{z}^{2}}
\left(-\frac{\overline{b}}{2}\right)-\frac{P_{d}L^{2}}{2EI},
\end{equation}}\unskip
with
{\begin{equation}\label{eq10}
\frac{P_{d}L^{2}}{2EI}
=3(\overline{d}_{2}\overline{d}_{3}/
\overline{b})
\left(\frac{\partial \overline{u}_{1}}{\partial \overline{z}}
(\overline{a})
- \frac{\partial \overline{u}_{4}}{\partial \overline{z}}
(-\overline{c})\right).
\end{equation}}\unskip



In conclusion, our delaminated beam model involves four boundary
equations at the two ends. Alongside these, we have eight continuity
equations, making a total of twelve equations with twelve unknowns.

The unknowns are:  ${Y} =({A}_{1}{,B}_{{1}},C_{{1}},D_{{1}},A_{{2}},
B_{{2}},C_{{2}},D_{{2}},A_{{4}},B_{{4}},C_{{4}},D_{{4}})^{{T}}$.

The system of equations is 
{\begin{equation}
\label{eq11}
AY=0.
\end{equation}}\unskip
Such that matrix \textit{A} is a $12\times12$ which depends on
\textit{a}, \textit{b}, \textit{d}, and ${\lambda}_{{i}}$, for the same example
(clamped--clamped beam), the non-zero values of the first line of
\textit{A} are: ${A}(1,2)=1;
A(1,4)=1$; the non-zero values of lines 2, 3, 4, and
5 are
{\begin{eqnarray*}
{A}(2,1) &amp;=&amp; 1; {A}(2,3) = 1; {A}(3,10) = 1; {A}(3,12)=1; 
{A}(4,9)=1; {A}(4,11)=1;\\ 
{A}(5,1) &amp;=&amp; \sinh(\lambda_{{1}}{a}); {A}(5,2) = 
\cosh(\lambda_{{1}}{a}); {A}(5,3) = \sin(\lambda_{{1}}{a}); 
{A}(5,4) = \cos(\lambda_{{1}}{a});\\ 
{A}(5,5) &amp;=&amp; -{\sinh}({\lambda}_{{2}}(-{b/2})); 
{A}(5,6) = -\cosh({\lambda}_{{2}}(-{b}/2)); 
{A}(5,7) = -\sin({\lambda}_{{2}}(-{b/2})); {A}(5,8)\\
&amp;=&amp; - \cos({\lambda}_{{2}}(-{b/2}));
\end{eqnarray*}}\unskip

The other lines of \textit{A} are found by the same way, while the
frequency ${\uplambda}_{2}$ is calculated by (\ref{eq10}).

\subsection{Resonance}\label{sec3.2}

\vspace*{-2pt}

The mechanical resonance can be defined as the tendency of a mechanical
system to respond with greater amplitude when the frequency of its
oscillations matches the system's natural frequency of vibration (its
resonance frequency).  At the resonance frequency, the determinant of
the matrix \textit{A} equals zero
\vspace*{-2pt}
{\begin{equation}\label{eq12}
\mathrm{Det}(A(a,b,d,\lambda _{i}))=0.
\end{equation}}\unskip

\vspace*{-2pt}

\subsection{Inverse modelling}\label{sec3.3}

\vspace*{-2pt}

The inversion procedure involves determining the delamination
parameters (\textit{a}, \textit{b} and \textit{d}) based on natural
frequencies ($\lambda _{1}$, ${\lambda}_{2}$ and ${\lambda}_{3}$). In
the forward modelling phase, it is stated that at resonance, the
determinant of the matrix \textit{A} equals zero. This determinant
depends on the delamination parameters and the frequency. Since the
three delamination parameters are the unknowns of the inverse model,
only three independent equations are needed. The characteristic
equations corresponding to the first three frequencies can be used such
that
\vspace*{-2pt}
{\begin{equation}\label{eq13}
\begin{cases} \det (A(a,b,d,\lambda _{1}))=0\\
\det (A(a,b,d,\lambda _{2}))=0\\
\det (A(a,b,d,\lambda _{3}))=0.
\end{cases} 
\end{equation}}\unskip

It is important to note that with this approach, we do not need the
natural frequencies of the pristine beam to identify delamination
parameters. Additionally, there is no need for a database representing
frequency shifts as a function of position and size. 

\vspace*{-2pt}
\subsection{Graphical technique}\label{sec3.4}

\vspace*{-2pt}

The graphical technique for solving a system of nonlinear equations
involves visualizing the solutions by plotting the equations and
identifying the points of intersection, which represent the solutions
to the system. In this method, each equation is treated as a curve or
surface in a multidimensional space, depending on the number of
variables involved. By plotting the curves on the same graph, the
graphical technique allows us to observe where the curves intersect,
providing solutions to the system of nonlinear equations. It is
particularly useful when analytical methods become impractical, and it
gives an intuitive understanding of how the solutions behave. However,
it may require iterative refinement to achieve higher accuracy,
especially in systems with complex interactions between variables.
Figure~\ref{fig4} displays an example for two values of \textit{d}
showing two different cases.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Curves of intersection of the determinant with
plane ${z} = 0$. (a) \textit{d} does not correspond to a
delamination, (b) \textit{d} corresponds to a delamination.}
\end{figure}

To solve the system (\ref{eq13}) for given modal parameters
(${\lambda}_{{1}}, {\lambda}_{{2}}, {\lambda}_{{3}}$), the
first step is to choose a value of \textit{d} such that $0 &lt; d &lt; 0.5$.
The system~(\ref{eq13}) then depends only on \textit{a} and
\textit{b}, which are the unknowns of the problem. Since \textit{a} and
\textit{b} vary between 0 and 1, the three surfaces ${S}_{{i }}=
\mathrm{det}({a,b})$ can be plotted. Each surface intersects the plane ${S}_{{i
}}= 0$, forming an intersection curve. If the three resulting curves
intersect at a single point, this point is the solution to the system.
Otherwise, if the curves do not intersect at a single point, the value
of \textit{d} needs to be adjusted. This process is repeated until a
single intersection point is found. Figure~\ref{fig5} presents a plot
of the surface ${S}_{2}= \det({a},{b})$ for a cantilever beam in the
second mode ${\lambda}_{{2  }}= 4.690$ with ${d} = 0.5$. The
intersection curve with the plane ${S}_{{2 }}= 0$ (in yellow) is shown
in red.


\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Curve of intersection of the determinant with
plane ${S}_{{2 }} = 0$.}
\end{figure}

\vspace*{-2pt}
\subsection{Uniqueness of the solution}\label{sec3.5}

\vspace*{-2pt}

The uniqueness of the solution in inverse problems is a critical
consideration, particularly in delamination identification using
natural frequencies. By employing the graphical technique during the
inversion phase, it is possible to visualize whether multiple solutions
exist for a given set of frequencies. For symmetric configurations,
such as clamped--clamped beams, this technique clearly reveals the
presence of two solutions due to the inherent symmetry, as illustrated
in Figure~\ref{fig4}. In such cases, mirrored delamination locations
produce identical sets of natural frequencies, making it impossible to
distinguish between them based solely on frequency data. For
non-symmetric configurations and methods that do not use the graphical
technique, the determinant of the Jacobian matrix of the system
(\ref{eq13}) provides a mathematical framework to assess uniqueness. If
the determinant is non-zero for all values of the delamination
parameters (\textit{a}, \textit{b}, \textit{d}), the solution is likely
unique. However, if the determinant vanishes at any point, multiple
delamination configurations may yield the same set of frequencies,
indicating non-uniqueness. Additionally, challenges such as measurement
noise can further complicate the uniqueness of the solution. Small
changes in frequency measurements can lead to significantly different
delamination patterns. Multiple delamination configurations can also
produce similar frequency shifts, making it difficult to identify a
unique solution.

To address these challenges, several strategies can be employed to
prove the uniqueness of the solution. Incorporating additional modal
parameters, such as mode shapes, curvature modes, or damping ratios,
can provide more information and reduce ambiguity. Increasing the
number of measured natural frequencies can also enhance the likelihood
of a unique solution, although it does not always guarantee uniqueness
in complex configurations. Hybrid approaches, such as combining
vibration-based methods with ultrasonic or thermographic techniques,
can improve accuracy by leveraging complementary data. Advanced
optimization algorithms and machine learning techniques can further
reduce ambiguity by identifying patterns and correlations that are not
apparent through traditional methods. Additionally, varying
environmental or loading conditions during testing can provide
additional data points, helping to distinguish between similar
delamination scenarios. While non-uniqueness cannot always be entirely
eliminated, these strategies significantly improve the reliability and
accuracy of the inverse problem solution, paving the way for more
robust delamination identification methods.

\section{Results and discussion}\label{sec4}

In this section, the previously developed mathematical model is used to
determine the size and position of delamination in isotropic beams.
First, the forward model is validated using the finite element method
existing in the literature. Then, the inverse model is used to identify
delamination in three beams, each with different boundary condition.

\subsection{Forward model validation}\label{sec4.1}

The clamped--clamped beam shown in Figure~\ref{fig6} is considered in
this section to validate the forward model. Our results of this model
are compared with those obtained by Wang et~al.~\cite{27} and
Lee~\cite{33} using an analytical model and FEM, respectively. Firstly,
the dimensionless frequency of the first three vibration modes is
calculated for fixed central delamination with different sizes, as
shown in Table~\ref{tab2}. Then, the horizontal delamination position
and size are changed to calculate the dimensionless frequency of the
first mode, as presented in Table~\ref{tab3}. To conclude the forward
model validation, the vertical delamination position is varied along
the symmetric axis for different sizes, as displayed in
Table~\ref{tab4}. Figures~\ref{fig7}, \ref{fig8}, and~\ref{fig9} show
the frequency in terms of size and horizontal position for different
modes and vertical positions.


\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Clamped--clamped beam.}
\end{figure}


\begin{table}
\caption{\label{tab2} The first three frequencies for a clamped--clamped beam}
\fontsize{9.5}{12}\selectfont\tabcolsep 2.7pt
\begin{tabular}{cccccccccc}
\thead
\morerows{2}{\parbox[t]{1.9cm}{\centering Delamination size (\textit{b/L})}}  &amp; 
\multicolumn{9}{c}{${\lambda}^{2}$} \\\cline{2-10}
 &amp; \multicolumn{3}{c}{1st mode} &amp; 
 \multicolumn{3}{c}{2nd mode} &amp; 
 \multicolumn{3}{c}{3rd mode} \\\cline{2-4} \cline{5-7}\cline{8-10}
 &amp; \parbox[t]{1.2cm}{\centering Wang et~al.~\cite{27}} &amp; Lee \cite{33} &amp; 
 \parbox[t]{1.1cm}{\centering Present model} &amp; 
\parbox[t]{1.2cm}{\centering  Wang et~al.~\cite{27}} &amp; 
Lee \cite{33} &amp; \parbox[t]{1.1cm}{\centering Present model} &amp; 
\parbox[t]{1.2cm}{\centering Wang et~al.~\cite{27}} &amp; Lee \cite{33} &amp; 
\parbox[t]{1.1cm}{\centering Present model}\vspace*{2pt} \\
\endthead
0.1 &amp; 22.37 &amp; 22.36 &amp; 22.37 &amp; 60.76 &amp; 60.74 &amp; 60.81 &amp; 120.81 &amp; 120.62 &amp; 120.83 \\
0.2 &amp; 22.36 &amp; 22.35 &amp; 22.36 &amp; 55.97 &amp; 55.95 &amp; 56.00 &amp; 118.76 &amp; 118.69 &amp; 118.87 \\
0.3 &amp; 22.23 &amp; 22.23 &amp; 22.24 &amp; 49.00 &amp; 48.97 &amp; 49.00 &amp; 109.04 &amp; 109.03 &amp; 109.16 \\
0.4 &amp; 21.83 &amp; 21.82 &amp; 21.83 &amp; 43.87 &amp; 43.86 &amp; 43.89 &amp; 93.57 &amp; 93.51 &amp; 93.59 \\
0.5 &amp; 20.88 &amp; 20.88 &amp; 20.89 &amp; 41.45 &amp; 41.50 &amp; 41.52 &amp; 82.29 &amp; 82.23 &amp; 82.29 \\
0.6 &amp; 19.29 &amp; 19.28 &amp; 19.30 &amp; 40.93 &amp; 41.01 &amp; 41.04 &amp; 77.64 &amp; 77.64 &amp; 77.69 \\
0.7 &amp; 17.23 &amp; 17.22 &amp; 17.23 &amp; 40.72 &amp; 40.80 &amp; 40.82 &amp; 77.05 &amp; 77.12 &amp; 77.18 \\
0.8 &amp; 15.05 &amp; 15.05 &amp; 15.05 &amp; 39.01 &amp; 39.04 &amp; 39.07 &amp; 75.33 &amp; 75.39 &amp; 75.43 \\
0.9 &amp; 13.00 &amp; 12.99 &amp; 13.00 &amp; 35.38 &amp; 35.38 &amp; 35.39 &amp; 69.17 &amp; 69.16 &amp; 69.19 
\botline
\end{tabular}
\end{table}


\begin{table}
\caption{\label{tab3} First mode variation of frequencies
${\lambda}^{2}$ in function of delamination size and horizontal
position for a clamped--clamped beam}
\tabcolsep 3.7pt
\begin{tabular}{ccccccccc}
\thead
\morerows{3}{\parbox[t]{2cm}{\centering Delamination size (\textit{b/L})}} 
&amp; \multicolumn{8}{c}{First mode frequency ($\lambda^{2}$)} \\\cline{2-9}
 &amp; \multicolumn{8}{c}{Delamination position (\textit{a/L})} \\\cline{2-9}
 &amp; \multicolumn{2}{c}{0.1} &amp; \multicolumn{2}{c}{0.2} &amp; 
 \multicolumn{2}{c}{0.3} &amp; \multicolumn{2}{c}{0.4} \\\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9}
 &amp; \parbox[t]{1.2cm}{\centering Wang et~al.~\cite{27}} &amp; 
 \parbox[t]{1.2cm}{\centering Present model} &amp; 
 \parbox[t]{1.2cm}{\centering Wang et~al.~\cite{27}} &amp; 
 \parbox[t]{1.2cm}{\centering Present model} &amp; 
 \parbox[t]{1.2cm}{\centering Wang et~al.~\cite{27}} &amp; 
 \parbox[t]{1.2cm}{\centering Present model} &amp; 
 \parbox[t]{1.2cm}{\centering Wang et~al.~\cite{27}} &amp; 
 \parbox[t]{1.2cm}{\centering Forward model}\vspace*{2pt} \\
 \endthead
0.1 &amp; 22.15 &amp; 22.16 &amp; 22.21 &amp; 22.23 &amp; 22.30 &amp; 22.31 &amp; 22.36 &amp; 22.37 \\ 
0.2 &amp; 21.00 &amp; 21.00 &amp; 21.53 &amp; 21.55 &amp; 22.10 &amp; 22.11 &amp; 22.35 &amp; 22.36 \\ 
0.3 &amp; 19.34 &amp; 19.33 &amp; 20.78 &amp; 20.78 &amp; 22.02 &amp; 22.04 &amp; 22.02 &amp; 22.04 \\ 
0.4 &amp; 18.18 &amp; 18.17 &amp; 20.48 &amp; 20.49 &amp; 21.83 &amp; 21.83 &amp; 20.48 &amp; 20.49
\botline
\end{tabular}
\end{table}



\begin{table}
\caption{\label{tab4} First mode variation of frequencies
${\lambda}^{2}$ in function of size and vertical position for a
clamped--clamped beam}
\begin{tabular}{ccccc}
\thead
\morerows{2}{Delamination size (\textit{b/L)}}  &amp; 
\multicolumn{4}{c}{First mode frequency ($\lambda^{2}$)} \\\cline{2-5}
 &amp; \multicolumn{2}{c}{$d = 0.25$} &amp; 
 \multicolumn{2}{c}{$d = 0.5$} \\\cline{2-5}
 &amp; Present model &amp; Wang et~al.~\cite{27} &amp; Present model &amp; 
 Wang et~al.~\cite{27} \\
 \endthead
0.05 &amp; 22.37 &amp; 22.37 &amp; 22.37 &amp; 22.37 \\ 
0.1 &amp; 22.37 &amp; 22.37 &amp; 22.37 &amp; 22.37 \\ 
0.2 &amp; 22.36 &amp; 22.35 &amp; 22.35 &amp; 22.35 \\ 
0.3 &amp; 22.31 &amp; 22.16 &amp; 22.24 &amp; 22.24 \\ 
0.4 &amp; 22.14 &amp; 21.37 &amp; 21.83 &amp; 21.83 
\botline
\end{tabular}
\end{table}


\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}First three dimensionless frequency modes
${\lambda}$~for ${d} = 0.1$. (a) Mode I. (b) Mode II. (c) Mode
III.}
\end{figure}


\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}First three dimensionless frequency modes
${\lambda}$ for ${d} = 0.3$. (a) Mode I. (b) Mode II. (c) Mode
III.}
\end{figure}


\begin{figure}
\includegraphics{fig09}
\caption{\label{fig9}First three dimensionless frequency modes
${\lambda}$ for ${d} = 0.5$. (a) Mode I. (b) Mode II. (c) Mode
III.}
\end{figure}


A first observation is that the forward model results are in good
agreement with the results of Lee~\cite{33} as shown in
Table~\ref{tab2}, and Wang et~al.~\cite{27} as shown in
Tables~\ref{tab2} and~\ref{tab3}. In Table~\ref{tab4}, the difference
is notable with~\cite{27} when the delamination vertical position
\textit{d} is near zero and the delamination size is important. The
complete agreement with~\cite{33} and the observed discrepancy
with~\cite{27} can be attributed to the fact that the FEM in~\cite{33}
is based on the same assumptions, whereas the model in~\cite{27} does
not assume that beam segments 2 and 3 experience the same displacement
(assuming delamination remains always closed). In reality, experimental
tests show that the delamination breathes (opens and closes), with each
segment having its own displacement. On the other hand, the frequency
decreases with the increase of delamination size, as shown in all
tables and  Figures~\ref{fig7}--\ref{fig9}.
Additionally, the horizontal delamination position has a smaller effect
on the frequency for small delamination sizes compared to larger sizes
(see~Table~\ref{tab3}). In Figures~\ref{fig7}--\ref{fig9}, 
we observe that the frequency has semi-sinusoidal
shapes in terms of delamination position when the delamination size is
more significant. Moreover, there is one hump for the first mode, two
humps for the second mode, and three humps for the third mode.


\subsection{Application of the present inverse model}\label{sec4.2}

Three examples are considered to apply the present inverse model. The
first example is the previous clamped--clamped beam. The second and
third examples are cantilever and \mbox{clamped-simply} supported beams as
presented in Figure~\ref{fig10}. The delamination in the
clamped--clamped beam is studied by Wang et~al.~\cite{27} where the
delamination size and position are given to calculate the three first
frequencies. These frequencies are used in Example 1 as inputs in order
to find the delamination parameters which are shown in
Table~\ref{tab5}. For Examples 2 and 3, the three frequencies are
decreased starting from the frequencies of the pristine beam.
Tables~\ref{tab6} and~\ref{tab7} showcase the corresponding results.


\begin{figure}
\includegraphics{fig10}
\caption{\label{fig10}Geometry and boundary conditions. (a) Example 2. (b) Example 3.}
\end{figure}


\begin{table}
\caption{\label{tab5} Results for clamped--clamped beam}
%\tabcolsep9.5pt
\begin{tabular}{cccccccccc}
\thead
\morerows{1}{Case} &amp; \multicolumn{3}{c}{${\lambda}^{2}$} &amp;
\multicolumn{3}{c}{\parbox[t]{3cm}{\centering Delamination parameters \cite{27}}} &amp;
\multicolumn{3}{c}{\parbox[t]{4cm}{\centering Predicted delamination parameters}}\vspace*{2pt} \\\cline{2-4}\cline{5-7}\cline{8-10}
&amp; ${\lambda }_{1}^{2}$\raisebox{1.2em}{} &amp; ${\lambda }_{2}^{2}$ &amp; ${\lambda }_{3}^{2}$ &amp;
\textit{a/L} &amp; \textit{b/L} &amp; \textit{d/h} &amp; \textit{a/L} &amp; \textit{b/L} &amp; \textit{d/h} \\
\endthead
1 &amp; 22.370 &amp; 60.810 &amp; 120.830 &amp; 0.50 &amp; 0.45 &amp; 0.10 &amp; 0.50 &amp; 0.45 &amp; 0.10 \\
2 &amp; 22.350 &amp; 56.000 &amp; 118.870 &amp; 0.50 &amp; 0.40 &amp; 0.20 &amp; 0.50 &amp; 0.40 &amp; 0.20 \\
3 &amp; 22.240 &amp; 49.000 &amp; 109.160 &amp; 0.50 &amp; 0.35 &amp; 0.30 &amp; 0.50 &amp; 0.35 &amp; 0.30 \\
4 &amp; 21.830 &amp; 43.870 &amp; 93.570 &amp; 0.50 &amp; 0.30 &amp; 0.40 &amp; 0.50 &amp; 0.30 &amp; 0.40 \\
5 &amp; 20.880 &amp; 41.450 &amp; 82.080 &amp; 0.50 &amp; 0.25 &amp; 0.50 &amp; 0.50 &amp; 0.25 &amp; 0.50
\botline
\end{tabular}
\end{table}

The results of Example 1 in Table~\ref{tab5} show total agreement
between the inverse model and Ref.~\cite{27},
proving that the delamination position and size can be identified
accurately by the present inverse model. In Table~\ref{tab6}, the
corresponding delamination size shows an increasing trend, whereas the
vertical position is predominantly located around the median plane of
the beam. The vertical position drops from 0.5 to 0.17, indicating that
the delamination is \mbox{approaching} the edge of the beam. The horizontal
position decreases, representing the movement of the delamination
towards the left side of the beam. As the first three frequencies
decrease simultaneously, the delamination size increases while the
horizontal and vertical positions fluctuate in Table~\ref{tab7}.
According to Figures~\ref{fig11}, \ref{fig12} and~\ref{fig13}, the
intersection curves are nonlinear with high orders.

\begin{table}
\caption{\label{tab6}Results for a cantilever beam}
\tabcolsep 11pt
\begin{tabular}{ccccccc}
\thead
\morerows{1}{Case}  &amp; \multicolumn{3}{c}{First three frequencies} &amp; 
\multicolumn{3}{c}{Delamination parameters} \\\cline{2-4}\cline{5-7}
 &amp; ${\lambda}_{1}$ &amp; ${\lambda}_{2}$ &amp; ${\lambda}_{3}$ &amp; 
 \textit{a}/\textit{L} &amp; \textit{b}/\textit{L} &amp; \textit{d}/\textit{h}\\
 \endthead
1 &amp; 1.875 &amp; 4.690 &amp; 7.840 &amp; 0.76 &amp; 0.10 &amp; 0.31 \\ 
2 &amp; 1.855 &amp; 4.600 &amp; 7.120 &amp; 0.45 &amp; 0.36 &amp; 0.50 \\ 
3 &amp; 1.845 &amp; 4.550 &amp; 6.930 &amp; 0.42 &amp; 0.42 &amp; 0.50 \\ 
4 &amp; 1.835 &amp; 4.500 &amp; 6.790 &amp; 0.40 &amp; 0.46 &amp; 0.50 \\ 
5 &amp; 1.825 &amp; 4.450 &amp; 7.100 &amp; 0.09 &amp; 0.69 &amp; 0.17 
\botline
\end{tabular}
\end{table}


\begin{table}
\caption{\label{tab7}Results for clamped-simply supported beam}
\tabcolsep 11pt
\begin{tabular}{ccccccc}
\thead
\morerows{1}{Case}  &amp; \multicolumn{3}{c}{First three frequencies} &amp; 
\multicolumn{3}{c}{Delamination parameters} \\\cline{2-4}\cline{5-7}
 &amp; ${\lambda}_{1}$ &amp; ${\lambda}_{2}$ &amp; ${\lambda}_{3}$ &amp; 
 \textit{a}/\textit{L} &amp; \textit{b}/\textit{L} &amp; \textit{d}/\textit{h} \\ 
\endthead
1 &amp; 3.910 &amp; 7.000 &amp; 10.110 &amp; 0.10 &amp; 0.11 &amp; 0.40 \\ 
2 &amp; 3.900 &amp; 6.950 &amp; 10.000 &amp; 0.08 &amp; 0.12 &amp; 0.50 \\ 
3 &amp; 3.890 &amp; 6.900 &amp; 9.900 &amp; 0.05 &amp; 0.14 &amp; 0.40 \\ 
4 &amp; 3.880 &amp; 6.800 &amp; 9.700 &amp; 0.29 &amp; 0.35 &amp; 0.22 \\ 
5 &amp; 3.870 &amp; 6.970 &amp; 9.600 &amp; 0.23 &amp; 0.33 &amp; 0.20 
\botline
\end{tabular}
\end{table}


\begin{figure}
\includegraphics{fig11}
{\vspace*{-4pt}}
\caption{\label{fig11}Intersection curve for Example 1
(clamped--clamped beam).}
{\vspace*{8pt}}
\end{figure}



\begin{figure}
\includegraphics{fig12}
{\vspace*{-4pt}}
\caption{\label{fig12}Intersection curve for Example 2 (cantilever beam).}
{\vspace*{8pt}}
\end{figure}


\begin{figure}
\includegraphics{fig13}
{\vspace*{-4pt}}
\caption{\label{fig13}Intersection curve for Example 3 (clamped--simply
supported beam).}
\end{figure}


\section{Conclusion}\label{sec5}

By comparing the present approach with the other inverse technics
existing in the literature, we can distinguish the simplicity and
rapidity of this present approach. The novelty in this work exists in
the formulating of the new system of nonlinear equations. This work can
be applied to structures of simple mechanical beams and bridges. Also,
this work can be applied easily to composite beams.

Despite the using of advanced tools like DL and ANN the delamination
identification is still a hard task. This can be explained by the
present work, where the system of equation formulated contains at least
three highly nonlinear equations, each equation is the determinant of
$12 \times12$ matrix which is a sum of hundreds of nonlinear terms.
This fact leads to the high sensitivity of the phenomena.

For more complex structures, the analytical forward model become
complex and may not represent the real dynamic behaviour. For
structures that present symmetry multiple solutions are found. To
resolve this problem, we propose to add equations by changing boundary
conditions or by adding supports along the beam and modify the forward
model, this will shift the natural frequencies and allow delamination
identification. 

Another difficulty may be encountered will be in the resolution of the
system of equations. The numeric solvers used depend on initial
solution, and the highly nonlinear system of equations presents a great
number of local picks that can cause convergence to local minima. This
problem can be resolved by integrating ANN algorithms for selecting
initial solution, by combining ANN with numerical solvers allow to
eliminate this problem and make a fast and accurate solver.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\back{}

\printbibliography
\refinput{crmeca20240923-reference.tex}

\end{document}
</pre></body></html>