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\DOI{10.5802/crmeca.226}
\datereceived{2023-05-17}
\daterevised{2023-08-11}
\dateaccepted{2023-09-07}
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\dateposted{2023-11-21}
\begin{document}

\begin{noXML}

\TopicEF{Research article}{Article de recherche}

\title{Dynamics of inclined CNTRC sandwich beams under a moving mass
with influence of CNT agglomeration{\vspace*{-4pt}}}

\alttitle{Dynamique des poutres sandwich CNTRC inclin\'{e}es sous une
masse en mouvement avec influence de l'agglom\'{e}ration des
CNT{\vspace*{-4pt}}}

\author{\firstname{Thi Thom} \lastname{Tran}\CDRorcid{0009-0003-0033-1475}} 
\address{Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi,
Vietnam}

\author{\firstname{Dinh Kien} \lastname{Nguyen}\CDRorcid{0000-0001-9356-8401}\IsCorresp} 
\addressSameAs{1}{Institute of Mechanics, VAST, 18 Hoang Quoc Viet,
Hanoi, Vietnam}
\address{VNU, University of Engineering and Technology, 144 Xuan Thuy,
Cau Giay, Hanoi, Vietnam}
\email[D. K. Nguyen]{ndkien@imech.vast.vn}

\begin{abstract}
Dynamics of inclined carbon nanotube (CNT) reinforced composite beams
under a moving mass with influence of CNT agglomeration is studied. The
beams compose of a homogeneous core and two composite face layers with
effective properties being estimated by Eshelby--Mori--Tanaka approach.
A novel finite element formulation is formulated and used to establish
the equation of motion. Dynamic response is computed for a simply
supported beam by Newmark method. The result reveals the important role
of the CNT agglomeration on the dynamic response. The effects of the
mass velocity, inclined angle and agglomeration parameters~on the
dynamic behavior are investigated in detail.
\vspace*{-1pt}
\end{abstract}

\begin{altabstract}
La dynamique des poutres composites inclin\'{e}es renforc\'{e}es par
des nanotubes de carbone (NTC) sous une masse en mouvement avec
l'influence de l'agglom\'{e}ration des NTC est \'{e}tudi\'{e}e. Les
poutres sont compos\'{e}es d'un noyau homog\`{e}ne et de deux couches
composites frontales dont les propri\'{e}t\'{e}s effectives sont
estim\'{e}es par l'approche Eshelby--Mori--Tanaka. Une nouvelle
formulation par \'{e}l\'{e}ments finis est formul\'{e}e et utilis\'{e}e
pour \'{e}tablir l'\'{e}quation du mouvement. La r\'{e}ponse dynamique
est calcul\'{e}e pour une poutre simplement support\'{e}e par la
m\'{e}thode de Newmark. Les r\'{e}sultats r\'{e}v\`{e}lent le r\^{o}le
important de l'agglom\'{e}ration des NTC sur la r\'{e}ponse dynamique.
Les effets de la vitesse de la masse, de l'angle d'inclinaison et des
param\`{e}tres d'agglom\'{e}ration sur le comportement dynamique sont
\'{e}tudi\'{e}s en d\'{e}tail.
\vspace*{-2pt}
\end{altabstract}

\keywords{\kwd{CNT}\kwd{Inclined sandwich beam}\kwd{Moving
mass}\kwd{Trigonometric shear deformation theory}\kwd{Dynamic
analysis}}

\altkeywords{\kwd{NTC}\kwd{Poutre sandwich inclin\'{e}e}\kwd{Masse 
mobile}\kwd{Th\'{e}orie trigonom\'{e}trique de la d\'{e}formation par
cisaillement}\kwd{Analyse dynamique}}

\thanks{Institute of Mechanics, VAST (Vietnam) (Grant agreement no.
CSCL03.02/22-23){\vspace*{-1pt}}}

\maketitle

\end{noXML}

\section{Introduction}

Since the landmark work of Iijimas~\cite{1}, carbon nanotubes (CNTs)
have been considered as an ideal reinforcement component for new
generation composites. Thanks to superior mechanical, thermal,
thermo-mechanical and vibration characteristics over conventional
composites, CNT reinforced structures have drawn great attention from
researchers. It has been shown that the mechanical properties of a
polymer composite can be considerably enhanced by adding a small amount
of CNTs to the matrix. Most studies on mechanical behavior of CNT
reinforced composite (CNTRC) structures, however, are considered as
aligned single-walled carbon nanotubes (SWCNTs), and the effective
material properties are estimated using a very simple method, namely
the rule of mixture model (ROM)~\cite{2,3,4,5,6,7,8,9,10,11,12}.

Due to the high aspect ratio and low bending stiffness, CNTs tend to be
agglomerated within the polymer matrix. Rubel \etal~\cite{13} discussed
the effects of CNT agglomerations in composites formation for various
CNTs reinforced composites. The authors pointed out that the
agglomeration/clustering of CNTs has a direct effect on the thermal,
electrical, and mechanical properties of composites and reduces the
physical properties. To predict the effective properties such as
Young's modulus and Poisson's ratio of composites reinforced by
agglomerated CNTs, Shi \mbox{\etal}~\cite{14} proposed a two-parameter
micromechanical model that accounts for the influence of CNT
agglomeration. The model in Ref.~\cite{14} was used in conjunction with
Mori--Tanaka scheme by Heshmati and Yas~\cite{15} to derive elastic
moduli of randomly oriented CNTRC in a free vibration analysis of
functionally graded (FG)-CNTRC beams. Finite element method was adopted
in~\cite{15} to obtained the beam frequencies. The generalized
differential quadrature method (GDQM) was adopted by Nejati and
Eslampanah~\cite{16} to evaluate natural frequency of a thick
cantilever CNTRC beam. The influence of agglomerated CNTs on the
natural frequencies of beam reinforced by randomly oriented
agglomerated CNTs is considered by the authors. Based on Timoshenko
beam theory and the GDQM, Kamarian \etal~\cite{17} determined free
vibration characteristics of FG nanocomposite sandwich beams on
Pasternak foundation, considering the agglomeration effect of SWCNTs.
The GDQM was also employed in~\cite{18} to study vibration of
non-uniform agglomerated CNTRC beams with piezoelectric layers. A
two-parameter model based on a combination of self-consistent and
Mori--Tanaka method was presented by Pan and Bian~\cite{19} for
investigating the effect of aggregation on CNT reinforced composites.
It was concluded by the authors that the CNT agglomeration reduces the
elastic stiffness of composites, while the uniformly dispersed CNTs
enhance the reinforcement. Kiani \etal~\cite{20} presented a
thermo-mechanical buckling analysis of CNTRC beams under non-uniform
thermal loading, also considering the CNT agglomeration. Hamilton's
principle was adopted to construct the governing equations, and the
GDQM was used to obtain the solution. Yue \etal~\cite{21} introduced a
quasi-3D beam model in their investigation of the effect of CNT
agglomeration on the nonlinear dynamic stability of CNTRC beams.

The problem of inclined beams under the moving loads is practically
important, and it draws much attention from researchers. As positions
of the dynamic loads vary with time, special techniques are required to
analyze this problem. In an early work, the finite element method was
used by Wu~\cite{22} to study dynamics of an inclined isotropic
Euler--Bernoulli beam subjected to a moving mass. The influence of the
inertia, Coriolis and centrifugal forces was taken into consideration
in the study. The author concluded that the transverse displacement of
the beam decreases considerably by increasing the inclined angle. Also
using the finite element method and Euler--Bernoulli beam theory,
Bahmyari \etal~\cite{23} investigated dynamic response of laminated
composite beams under moving distributed masses. It has been shown that
the deflection of the inclined beams is remarkably influenced by the
friction force, while it is not considerably affected by the Coriolis
and centrifugal forces. An equivalent horizontal beam model in which
the force acting on the inclined beam is split into the transverse and
axial forces, is proposed by Mamandi \etal~\cite{24} for the
geometrically nonlinear dynamic analysis of an inclined
Euler--Bernoulli beam under a moving force. The work was then extended
by Mamandi and Kargarnovin~\cite{25} to investigate the nonlinear
dynamic response of Timoshenko beams subjected to moving masses. The
numerical results in Refs.~\cite{24,25} show that the geometric
nonlinearity stiffens the dynamic response of the beams. Recently,
Nguyen \etal~\cite{26} presented a third-order shear deformation theory
based finite element procedure for dynamic analysis of an inclined
functionally graded sandwich beam carrying a moving mass. The material
properties of the beam are considered to vary in both the axial and the
transverse directions by the power gradation laws. 

As seen from the above literature review, the dynamic analysis of
inclined beams under moving loads has not been considered sufficiently.
Motivated by this fact, this paper studies dynamics of inclined CNTRC
sandwich beams under a moving mass, taking into account the influence
of CNTs agglomeration. The sandwich beams are composed of a homogeneous
core and two face sheets made from CNTRC material. The effective
properties of the composite face sheets are estimated by
Eshelby--Mori--Tanaka approach. Based on the trigonometric shear
deformation theory, a novel finite element formulation, taking into
account the influence of the inertial, Coriolis and centrifugal forces,
is derived and used to establish the discretized equation of motion for
the beams. To improve efficiency of the formulation, the transverse
shear rotation instead of the conventional section rotation is employed
herein as an independent variable. The effects of various parameters,
including the CNT volume fraction, the agglomeration parameters, the
layer thickness ratio, the mass velocity as well as the inclined angle
on the dynamic behavior of the sandwich beams are investigated in
detail and highlighted.

\section{Inclined sandwich beam reinforced with agglomerated CNTs}

Figure~\ref{fig1} shows an inclined sandwich beam under a moving mass
$m_{c}$ in two Cartesian coordinate systems, a local system $(x,z)$ and
a global one $(\overline{x},\overline{z})$. The sandwich beam has three
layers, a homogeneous core and two face layers made of CNT reinforced
composite. The $x$-axis of the $(x,z)$ system is chosen on the beam's
mid-plane. The mass $m_{c}$ is assumed to move with a constant velocity
$v$, and it is always in contact with the beam. Denoted in
Figure~\ref{fig1}, $L$, $b$, $h$, $h_{0}$, $h_{1}$, $h_{2}$, $h_{3}$
and $\beta$ are, respectively, the length, width and height of the
beam, the vertical coordinates of bottom surface, the interfaces
between the layers, the top surface and the inclined angle.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}A simply supported inclined CNTRC sandwich beam
under a moving mass.}
\end{figure}

The two-parameter micromechanical model of Shi \etal~\cite{14} is
adopted herein to account for the CNT agglomeration. The model is
developed by considering a representative volume element (RVE) with
Eshelby cluster model of CNT agglomeration, as depicted in
Figure~\ref{fig2}. The total volume $V_{r}$ of CNTs in the RVE is
divided into two parts as:
{\begin{equation}\label{1}
V_{r} ={V}_{r}^{\mathrm{cluster}} + {V}_{r}^{m}
\end{equation}}\unskip\noindent
where ${V}_{r}^{\mathrm{cluster}}$ and ${V}_{r}^{m}$ are the volumes of
CNTs inside and outside of the cluster, respectively. The CNT
agglomeration is described by two following parameters:
{\begin{equation}\label{2}
\xi = \dfrac{V_{\mathrm{cluster}}}{V}, \quad
\zeta = \dfrac{{V}_{r}^{\mathrm{cluster}}}{V_{r}}, \quad
0\leq \xi, \zeta \leq 1
\end{equation}}\unskip\noindent
where $V_{\mathrm{cluster}}$ denotes the volume of clusters in the RVE.
Thus, the parameter $\xi$ in Equation~(\ref{2}) is the volume fraction of
clusters with respect to the RVE volume, the parameter $\zeta$ is the
ratio of CNT volume inside the clusters over the total volume of the
CNTs inside the RVE. The case, $\xi=1$ means that CNTs are uniformity
distributed in the matrix; a lower value of $\xi$ is the more
agglomeration of CNTs. When $\zeta=1$, all CNTs are located inside the
clusters. The case $\xi=\zeta$ means that the volume fraction of CNTs
inside the clusters is the same as that of CNTs outside the clusters.
In the case $\zeta > \xi$, the value of $\zeta$ is bigger, the
distribution of CNTs is more heterogeneous. The effective bulk and
shear moduli of the clusters $K_{\mathrm{in}},G_{\mathrm{in}}$, and those of the region
outside the clusters $K_{\mathrm{out}}, G_{\mathrm{out}}$ may be
calculated by~\cite{14}:
{\begin{equation}\label{3}
\begin{array}{@{}rcl@{}}
K_{\mathrm{in}} &=& K_{m} + \dfrac{V_{\mathrm{CNT}} \zeta (\delta _{r} - 3K_{m}
\alpha _{r})}{3(\xi -V_{\mathrm{CNT}}\zeta + V_{\mathrm{CNT}}\zeta 
\alpha _{r})}, \vspace*{7pt} \\
G_{\mathrm{in}} &=& G_{m} + \dfrac{V_{\mathrm{CNT}} \zeta (\eta _{r} -
2G_{m}\beta _{r})}{2(\xi -V_{\mathrm{CNT}}\zeta + V_{\mathrm{CNT}}\zeta
\beta _{r})}, \vspace*{7pt} \\
K_{\mathrm{out}} &=& K_{m} + \dfrac{V_{\mathrm{CNT}} (1-\zeta )(\delta
_{r}-3K_{m}\alpha _{r})}{3[1-\xi
-V_{\mathrm{CNT}}(1-\zeta)+V_{\mathrm{CNT}}(1-\zeta )\alpha _{r}]}, \vspace*{7pt} \\
G_{\mathrm{out}} &=& G_{m} + \dfrac{V_{\mathrm{CNT}}(1-\zeta )(\eta
_{r}-2G_{m}\beta _{r})}{2[1-\xi -V_{\mathrm{CNT}}(1-\zeta
)+V_{\mathrm{CNT}}(1-\zeta )\zeta \beta _{r}]}
\end{array}
\end{equation}}\unskip\noindent
with $V_{\mathrm{CNT}} = \sfrac{V_{r}}{V}$ is CNT volume fraction in
the composite, and
{\begin{equation}\label{4}
\begin{array}{@{}rcl@{}}
\alpha _{r} &=& \dfrac{3(K_{m}+G_{m})+k_{r}-l_{r}}{3(G_{m}+k_{r})}, \vspace*{7pt} \\
\delta _{r} &=& \dfrac{1}{3} \left[n_{r}+2l_{r} + \dfrac{(2k_{r}+l_{r})
(3K_{m}+2G_{m}-l_{r})}{G_{m}+k_{r}}\right], \vspace*{7pt} \\
\beta _{r} &=& \dfrac{1}{5} \left(\dfrac{4G_{m} + 2k_{r} + l_{r}}{3
(G_{m}+k_{r})} + \dfrac{4G_{m}}{G_{m} + p_{r}} + 
\dfrac{2[G_{m}(3K_{m}+G_{m})+G_{m}(3K_{m}+7G_{m})]}
{G_{m}(3K_{m}+G_{m})+m_{r}(3K_{m}+7G_{m})}\right), \vspace*{7pt} \\
\eta _{r} &=& \dfrac{1}{5}\left[\dfrac{2}{3}(n_{r}-l_{r}) + 
\dfrac{8G_{m}p_{r}}{G_{m}+p_{r}} + \dfrac{8m_{r}G_{m}
(3K_{m}+4G_{m})}{3K_{m}(m_{r}+G_{m})+G_{m} (7m_{r}+G_{m})} +
\dfrac{(2k_{r}-l_{r}) (2G_{m}+l_{r})}{3(G_{m}+k_{r})}\right]
\end{array} 
\end{equation}}\unskip\noindent
with $K_{m} = \sfrac{E_{m}}{3(1-2\upsilon _{m})}$, $G_{m} =
\sfrac{E_{m}}{2(1+\upsilon _{m})}$ are the bulk and shear moduli of the
matrix, respectively. In Equations~(\ref{3}) and (\ref{4}), the subscripts $m$
and $r$ stand for the quantities of the matrix and the reinforcing
phase (CNTs); $k_{r},l_{r},m_{r},n_{r},p_{r}$ are the Hill's elastic
moduli for the reinforcing phase. 

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig02}
\vspace*{-2pt}
\caption{\label{fig2}RVE with Eshelby cluster model of agglomeration of
CNTs.}
\vspace*{-2pt}
\end{figure}

The effective bulk modulus $K$ and shear modulus $G$ of the composite
evaluated by the Mori--Tanaka scheme, scheme that satisfies
Hanshin--Shtrikman bounds~\cite{27}, are of the forms~\cite{14}:
{\begin{equation}\label{5}
K = K_{\mathrm{out}} \left(1 + \dfrac{\xi 
\left(\dfrac{K_{\mathrm{in}}}{K_{\mathrm{out}}}-1\right)}{1 + \alpha (1-\xi)
\left(\dfrac{K_{\mathrm{in}}}{K_{\mathrm{out}}}-1\right)}\right), \quad
G = G_{\mathrm{out}} \left(1+\dfrac{\xi 
\left(\dfrac{G_{\mathrm{in}}}{G_{\mathrm{out}}}-1\right)}{1+\beta (1-\xi)
\left(\dfrac{G_{\mathrm{in}}}{G_{\mathrm{out}}}-1\right)}\right)
\end{equation}}\unskip\noindent
where 
$\alpha = {(1+\nu_{\mathrm{out}})}/ {3 (1-\nu _{\mathrm{out}})}$,
$\beta= {(8-10\nu _{\mathrm{out}})}/{15(1-\nu _{\mathrm{out}})}$,
$\nu _{\mathrm{out}} = {(3K_{\mathrm{out}}-2G_{\mathrm{out}})}/
2(3K_{\mathrm{out}}+G_{\mathrm{out}})$. Noting that the composite is
considered to be isotropic for the case CNTs are oriented completely at
random throughout the matrix. In this case, the bulk and shear moduli
have the following forms~\cite{14}:
{\begin{equation}\label{6}
K = K_{m} + \dfrac{V_{\mathrm{CNT}}(\delta _{r}-3K_{m}\alpha _{r})}
{3(c_{m}+V_{\mathrm{CNT}}\alpha _{r})}, \quad
G = G_{m} + \dfrac{V_{\mathrm{CNT}}(\eta _{r}-2G_{m}\beta _{r})}
{2(c_{m}+V_{\mathrm{CNT}}\beta _{r})}
\end{equation}}\unskip\noindent
where $c_{m} = 1 - V_{\mathrm{CNT}}$, and $\alpha _{r}$, $\delta _{r}$,
$\beta _{r}$, $\eta _{r}$ are given in Equation~(\ref{4}). The effective Young's
modulus $E$ and the Poisson's ratio $\nu$  of the CNT reinforced layer
can be derived by:
{\begin{equation}\label{7}
E = \dfrac{9KG}{3K+G}; \quad
\nu = \dfrac{3K-2G}{6K+2G}.
\end{equation}}\unskip\noindent
Meanwhile, the effective mass density of the CNT reinforced layer can
be estimated by the rule of mixture as~\cite{28}:
{\begin{equation}\label{8}
\rho = (\rho ^{\mathrm{CNT}} - \rho ^{m})V_{\mathrm{CNT}} + \rho ^{m}
\end{equation}}\unskip\noindent
with $\rho ^{\mathrm{CNT}}$, $\rho ^{m}$ are the mass density of CNT
and matrix, respectively.

\vspace*{-2pt}

\section{Mathematical formulation}\label{sec3}

\vspace*{-2pt}

Based on the trigonometric shear deformation theory~\cite{29}, the
displacements in $x$ and $z$ directions of a point in the sandwich are
given as:
{\begin{equation}\label{9}
u(x,z,t) = u_{0} (x,t) - zw_{0,x} + \sin \dfrac{\rmpi z}{h}\theta;
\quad w(x,z,t)=w_{0}(x,t)
\end{equation}}\unskip\noindent
where $u_{0}(x,t)$, $w_{0}(x,t)$ are the displacements of a point on
the $x$ axis; $\theta$ is the cross-sectional rotation, and $t$ is the
time variable. In Equation~(\ref{9}) and hereafter, the subscript comma
indicates the derivative with respect to the variable that followed.

In order to improve the finite element formulation derived in the next
section, the transverse shear rotation $\gamma _{0}=\theta +w_{0,x}$,
not the conventional cross-sectional rotation $\theta$, is employed
herein as the independent variable instead. In this regard, the
displacements in Equation~(\ref{9}) can be rewritten in the form:
{\begin{equation}\label{10}
u(x,z,t) = u_{0} (x,t) - zw_{0,x} + \sin \dfrac{\rmpi z}{h}
(\gamma _{0}-w_{0,x}); \quad w(x,z,t)=w_{0}(x,t).
\end{equation}}\unskip\noindent

The strain field associated with the displacement field in 
Equation~(\ref{10}) is of the form:
{\begin{equation}\label{11}
\varepsilon _{xx} = u_{0,x} - zw_{0,xx} + \sin \dfrac{\rmpi z}{h}
(\gamma _{0,x}-w_{0,xx}); \quad \gamma _{xz} = \dfrac{\rmpi }{h}\cos
\dfrac{\rmpi z}{h}(\gamma _{0}-w_{0,x})
\end{equation}}\unskip\noindent
where $\varepsilon_{xx}$ and $\gamma_{xz}$ are axial and shear strains,
respectively.\pagebreak

The stress--strain relations in the $k${th} layer with the Hook's law
assumption are as follows:
{\begin{equation}\label{12}
{\sigma}_{xx}^{(k)} = E^{(k)}\varepsilon _{xx}; \quad 
{\tau}_{xz}^{(k)} = G^{(k)}\gamma _{xz} \quad (k=1,2,3).
\end{equation}}\unskip\noindent
One can verify that the adopted theory satisfies the free transverse
shear stress conditions on the top and bottom surfaces of the beam. The
elastic strain energy of the sandwich beam $U$ is given by:
{\begin{eqnarray}
U &=& \dfrac{1}{2}{\displaystyle \int }_{0}^{L} \displaystyle \int _{A}
({\sigma }_{xx}^{(k)}\varepsilon _{xx}+{\tau }_{xz}^{(k)}\gamma _{xz})
\,\mathrm{d}A\,\mathrm{d} x  \nonumber \\
&=& \dfrac{1}{2}{\displaystyle \int }_{0}^{L}
\left[\begin{array}{@{}l@{}}
A_{1}{u}_{0,x}^{2}-2A_{2}u_{0,x}w_{0,xx}+2A_{3}u_{0,x}
(\gamma _{0,x}-w_{0,xx})-2A_{4}w_{0,xx}(\gamma _{0,x}-w_{0,xx})\\
+ A_{5}(\gamma _{0,x}-w_{0,xx})^{2}+A_{6}{w}_{0,xx}^{2}+A_{7}
(\gamma _{0}-w_{0,x})^{2}\end{array}\right]\,\mathrm{d}x \label{13}
\end{eqnarray}}\unskip\noindent
where $A=bh$ is the area of cross section of the beam;
$A_{1},A_{2},\ldots,A_{7}$ are the beam rigidities, defined as follows:
{\begin{eqnarray}
(A_{1},A_{2},A_{3},A_{4},A_{5},A_{6}) &=& b{\displaystyle \int
}_{h_{0}}^{h_{3}}E \left(1,z,\sin \dfrac{\rmpi z}{h}, z\sin
\dfrac{\rmpi z}{h}, \sin ^{2}\dfrac{\rmpi z}{h},z^{2}\right)\mathrm{d}z
\nonumber \\
&=& b\displaystyle \sum _{k=1}^{3}{\displaystyle \int }_{h_{k-1}}^{h_{k}}
E^{(k)}\left(1,z,\sin \dfrac{\rmpi z}{h},z\sin \dfrac{\rmpi z}{h},
\sin ^{2}\dfrac{\rmpi z}{h},z^{2}\right)\mathrm{d}z; \label{14}
\end{eqnarray}}\unskip\noindent
%{\vspace*{-1pc}}
{\begin{equation*}
A_{7} = b\dfrac{\rmpi ^{2}}{h^{2}}{\displaystyle \int }_{h_{0}}^{h_{3}}
G\cos ^{2}\dfrac{\rmpi z}{h}\,\mathrm{d}z = b\dfrac{\rmpi ^{2}}{h^{2}}
\sum _{k=1}^{3}{\displaystyle \int }_{h_{k-1}}^{h_{k}}G^{(k)}\cos ^{2}
\dfrac{\rmpi z}{h}\,\mathrm{d}z.
\end{equation*}}\unskip\noindent
The kinetic energy $T$ of the sandwich beam can be expressed as:
{\begin{eqnarray}\label{15}
T &=& \dfrac{1}{2}{\displaystyle \int }_{0}^{L}\displaystyle \int
_{A}\rho (\dot{u}^{2}+\dot{w}^{2})\,\mathrm{d}A\,\mathrm{d}x \nonumber \\
&=& \dfrac{1}{2}{\displaystyle \int }_{0}^{L}
\left[\begin{array}{@{}l@{}}
I_{1}({\dot{u}}_{0}^{2}+{\dot{w}}_{0}^{2}) - 2I_{2}\dot{u}_{0}
\dot{w}_{0,x}+2I_{3}\dot{u}_{0}(\dot{\gamma }_{0}-\dot{w}_{0,x})\\
-2I_{4}\dot{w}_{0,x}(\dot{\gamma }_{0}-\dot{w}_{0,x})+I_{5}
(\dot{\gamma }_{0}-\dot{w}_{0,x})^{2}+I_{6}{\dot{w}}_{0,x}^{2}
\end{array}\right]\mathrm{d}x.
\end{eqnarray}}\unskip\noindent
The over dot in Equation~(\ref{15}), and hereafter, denotes the derivative with
respect to time variable $t$;  $I_{1},I_{2},\ldots,I_{6}$ are mass
moments, defined as:
{\begin{eqnarray}
(I_{1},I_{2},I_{3},I_{4},I_{5},I_{6}) 
&=& b{\displaystyle \int }_{h_{0}}^{h_{3}}\rho 
\left(1,z,\sin \dfrac{\rmpi z}{h},z\sin \dfrac{\rmpi z}{h},\sin ^{2}
\dfrac{\rmpi z}{h},z^{2}\right)\mathrm{d}z \nonumber \\
&=& b\displaystyle \sum _{k=1}^{3}{\displaystyle \int }_{h_{k-1}}^{h_{k}}
\rho ^{(k)}\left(1,z,\sin \dfrac{\rmpi z}{h},z\sin 
\dfrac{\rmpi z}{h},\sin ^{2}\dfrac{\rmpi z}{h},z^{2}\right)\mathrm{d}z.
\label{16}
\end{eqnarray}}\unskip\noindent
When the beam is inclined an angle $\beta$ and subjected to the moving
mass $m_{c}$, the potential energy is calculated as follows~\cite{25}:
{\begin{equation}\label{17}
V = {\displaystyle \int }_{0}^{L}[(-m_{c}g\cos \beta +m_{c}\ddot{w}_{0}
+ 2m_{c}v\dot{w}_{0,x}+m_{c}v^{2}w_{0,xx})w_{0} + (-m_{c}g\sin \beta
+m_{c} \ddot{u}_{0})u_{0}]\delta (x-vt)\,\mathrm{d}x
\end{equation}}\unskip\noindent
where $P_{x}=-m_{c}g\sin \beta$, $P_{z}=-m_{c}g\cos \beta$ are force
components induced by the mass $m_{c}$ with $g=9.81$ m/s$^{2}$ is the
acceleration of gravity; $m_{c}\ddot{u}_{0}$ and $m_{c}\ddot{w}_{0}$
are components of the inertial force; $2m_{c}v\dot{w}_{0,x}$ is the
Coriolis force and $m_{c}v^{2}w_{0,xx}$ is centrifugal force; $\delta
(.)$ is the Dirac delta function; $x$ is the abscissa of the moving
mass, respect to the left end of the beam. One can obtain the
differential equations of motion for the beam by applying Hamilton's
principle to Equations~(\ref{13}), (\ref{15}) and (\ref{17}). However, due to the
material heterogeneity as well as the non-traditional inclined beam
configuration, a closed-form solution for such equations is hardly
obtained. Finite element formulation derived in the next section is a
reasonable choice to compute the dynamic response of the beam. 

\section{Finite element formulation}\label{sec4}

A two-node beam element, (1, 2), with length $l$ is firstly derived in
the local system $(x,z)$ and then transferred to the global one in this
work. Each node of the element has four degrees of freedom, namely the
axial displacement, transverse displacement, derivative of the
transverse displacement and the shear rotation. Thus, the element
vector of nodal displacements $(\mathbf{d})$ has the form:
{\begin{equation}\label{18}
\mathbf{d} = \{
\begin{array}{@{}ccc@{}}
{\mathbf{d}_{u}}&
{\mathbf{d}_{w}}&
{\mathbf{d}_{\gamma}}
\end{array}
\}^{T}
\end{equation}}\unskip\noindent
where
{\begin{equation}\label{19}
{\mathbf{d}_{u}} = {\{u_{01}u_{02}\}}^{T};\quad 
{\mathbf{d}_{w}} = {\{w_{01}w_{0x1}w_{02}w_{0x2}\}}^{T};\quad
{\mathbf{d}_{\gamma}} = {\{\gamma _{01}\gamma _{02}\}}^{T}
\end{equation}}\unskip\noindent
are, respectively, the vectors of nodal displacement for $u_{0}$,
$w_{0}$ and $\gamma _{0}$ at the nodes 1 and 2. In the above equations
and hereafter, a superscript ``${T}$'' denotes the transpose of a
vector or a matrix. 

The displacements and shear rotation inside the element are
interpolated through their nodal values according to: 
{\begin{equation}\label{20}
u_{0} = \mathbf{N}{\mathbf{d}_{u}};\quad
w_{0} = \mathbf{H}{\mathbf{d}_{w}};\quad
\gamma _{0} = \mathbf{N}{\mathbf{d}_{\gamma}}
\end{equation}}\unskip\noindent
where $\mathbf{N} = \{N_{1}N_{2}\}$ is the matrix of the following
linear interpolation functions
{\begin{equation}\label{21}
N_{1} = 1 - \dfrac{x}{l}, \quad
~ ~ N_{2}=\dfrac{x}{l}
\end{equation}}\unskip\noindent
and $\mathbf{H} = \{\begin{array}{@{}cccc@{}}H_{1}&H_{2}&H_{3}&H_{4}
\end{array}\}$ is the matrix of Hermite
shape functions with 
{\begin{equation}\label{22}
H_{1} = 1 - 3\left(\dfrac{x}{l}\right)^{2} + 2\left(\dfrac{x}{l}\right)^{3}, \quad
H_{2} = x - 2\dfrac{x^{2}}{l} + \dfrac{x^{3}}{l^{2}},\quad
H_{3} = 3 \left(\dfrac{x}{l}\right)^{2}-2\left(\dfrac{x}{l}\right)^{3}, \quad
H_{4} = -\dfrac{x^{2}}{l} +\dfrac{x^{3}}{l^{2}}.
\end{equation}}\unskip\noindent
Using the interpolations, one can write the strain energy in
Equation~(\ref{13}) in the form:
{\begin{equation}\label{23}
U = \dfrac{1}{2}\displaystyle \sum _{i=1}^{ne}
{\mathbf{d}}_{i}^{T}\mathbf{k}_{i}\,\mathbf{d}_{i}
\end{equation}}\unskip\noindent
where $ne$ is the total number of elements used to discretize the
sandwich beam, and $\mathbf{k}$ is the element stiffness matrix with
the following form:
{\begin{equation}\label{24}
\mathbf{k} = \left[\begin{array}{@{}lll@{}}
\mathbf{k}_{{u_{0}}{u_{0}}} & \mathbf{k}_{{u_{0}}{w_{0}}} & \mathbf{k}_{{u_{0}}{\gamma _{0}}}\vspace*{5pt} \\
\mathbf{k}_{u_{0}w_{0}}^{T} & \mathbf{k}_{{w_{0}}{w_{0}}} & \mathbf{k}_{{w_{0}}{\gamma _{0}}}\vspace*{5pt} \\
\mathbf{k}_{u_{0}\gamma _{0}}^{T} & {\mathbf{k}}_{w_{0}\gamma _{0}}^{T} & \mathbf{k}_{{\gamma _{0}}{\gamma _{0}}}
\end{array}\right].
\end{equation}}\unskip\noindent
In Equation~(\ref{24}),
$\mathbf{k}_{{u_{0}}{u_{0}}},\mathbf{k}_{{u_{0}}{w_{0}}},\ldots,
\mathbf{k}_{{\gamma _{0}}{\gamma _{0}}}$ are, respectively, the element
stiffness matrices stemming from the axial, bending, shear deformations
and their couplings with the following expressions:
{\begin{equation}\label{25}
\begin{array}{@{}rcl@{}}
\mathbf{k}_{{u_{0}}{u_{0}}} &=&  {\displaystyle \int
}_{0}^{l}{\mathbf{N}}_{,x}^{T}A_{1}
\mathbf{N}_{,x}\,\mathrm{d}x, \quad 
\mathbf{k}_{{w_{0}}{w_{0}}} = {\displaystyle \int
}_{0}^{l} [{\mathbf{H}}_{,xx}^{T}
(2A_{4}+A_{5}+A_{6})\mathbf{H}_{,xx}+{\mathbf{H}}_{,x}^{T}
A_{7}\mathbf{H}_{,x}]\,\mathrm{d}x, \vspace*{7pt} \\
\mathbf{k}_{{\gamma _{0}}{\gamma _{0}}} &=& {\displaystyle \int
}_{0}^{l} ({\mathbf{N}}_{,x}^{T}A_{5}\mathbf{N}_{,x}+
\mathbf{N}^{T}A_{7}\mathbf{N})\,\mathrm{d}x, \vspace*{7pt} \\
\mathbf{k}_{{u_{0}}{w_{0}}} &=& -{\displaystyle \int
}_{0}^{l}{\mathbf{N}}_{,x}^{T} (A_{2}+A_{3})
\mathbf{H}_{,xx}\,\mathrm{d}x, \quad \mathbf{k}_{{u_{0}}{\gamma _{0}}}
= {\displaystyle \int
}_{0}^{l}{\mathbf{N}}_{,x}^{T}A_{3}\mathbf{N}_{,x}\,\mathrm{d}x,\vspace*{7pt} \\
\mathbf{k}_{{w_{0}}{\gamma _{0}}} &=& -{\displaystyle \int
}_{0}^{l} [{\mathbf{H}}_{,xx}^{T}(A_{4}+A_{5})\mathbf{N}_{,x}+
{\mathbf{H}}_{,x}^{T}A_{7}\mathbf{N}]\,\mathrm{d}x.
\end{array}
\end{equation}}\unskip\noindent
Similarly, the kinetic energy of the beam in Equation~(\ref{15}) can be
rewritten as
{\begin{equation}\label{26}
T = \dfrac{1}{2}\displaystyle \sum _{i=1}^{ne}
{\dot{\mathbf{d}}}_{i}^{T}\mathbf{m}_{i}\,\dot{\mathbf{d}}_{i}
\end{equation}}\unskip\noindent
where the element mass matrix of the beam $\mathbf{m}$ can be written in
sub-matrices as
{\begin{equation}\label{27}
\mathbf{m} = \left[\begin{array}{@{}lll@{}}
\mathbf{m}_{{u_{0}}{u_{0}}} & \mathbf{m}_{{u_{0}}{w_{0}}} & \mathbf{m}_{{u_{0}}{\gamma _{0}}}\vspace*{5pt} \\
\mathbf{m}_{u_{0}w_{0}}^{T} & \mathbf{m}_{{w_{0}}{w_{0}}} & \mathbf{m}_{{w_{0}}{\gamma _{0}}}\vspace*{5pt} \\
\mathbf{m}_{u_{0}\gamma _{0}}^{T} & {\mathbf{m}}_{w_{0}\gamma _{0}}^{T} & \mathbf{m}_{{\gamma _{0}}{\gamma _{0}}}
\end{array}\right]
\end{equation}}\unskip\noindent
in which
{\begin{equation}\label{28}
\begin{array}{@{}rcl@{}}
\mathbf{m}_{{u_{0}}{u_{0}}} &=& 
{\displaystyle \int }_{0}^{l}\mathbf{N}^{T}A_{1}\mathbf{N}\,\mathrm{d}x,
\quad
\mathbf{m}_{{w_{0}}{w_{0}}} = 
{\displaystyle \int }_{0}^{l} [{\mathbf{H}}_{,x}^{T}
(2I_{4}+I_{5}+I_{6})\mathbf{H}_{,x}+\mathbf{H}^{T}I_{1}\mathbf{H}]
\,\mathrm{d}x,\vspace*{7pt} \\
\mathbf{m}_{{\gamma _{0}}{\gamma _{0}}} &=& 
{\displaystyle \int }_{0}^{l}(\mathbf{N}^{T}I_{5}\mathbf{N})\,\mathrm{d}x,
\quad
\mathbf{m}_{{u_{0}}{w_{0}}} = -{\displaystyle \int }_{0}^{l}
\mathbf{N}^{T}(I_{2}+I_{3})\mathbf{H}_{,x}\,\mathrm{d}x,\vspace*{7pt} \\
\mathbf{m}_{{u_{0}}{\gamma _{0}}} &=& 
{\displaystyle \int }_{0}^{l}\mathbf{N}^{T}I_{3}\mathbf{N}\,\mathrm{d}x,
\quad
\mathbf{m}_{{w_{0}}{\gamma _{0}}} = -{\displaystyle \int }_{0}^{l}
{\mathbf{H}}_{,xx}^{T}(I_{4}+I_{5})\mathbf{N}_{,x}\,\mathrm{d}x.
\end{array}
\end{equation}}\unskip\noindent
The potential energy in Equation~(\ref{17}) is now of the form:
{\begin{equation}\label{29}
V = \displaystyle \sum ^{ne} (\ddot{\mathbf{d}}^{T}\mathbf{m}_{m}\,
\ddot{\mathbf{d}} + \dot{\mathbf{d}}^{T}\mathbf{c}_{m}\,
\dot{\mathbf{d}}+\mathbf{d}^{T}\mathbf{k}_{m}\,\mathbf{d}-
\mathbf{d}^{T}\mathbf{f}_{m}),
\end{equation}}\unskip\noindent
where $\mathbf{m}_{m}$, $\mathbf{c}_{m}$ and $\mathbf{k}_{m}$ are,
respectively, the element mass, damping and stiffness matrices due to
the effects of the inertia, Coriolis and centrifugal forces of the
moving mass; $\mathbf{f}_{m}$ is the time-dependent element nodal load
vector generated by the moving mass. The expressions for these matrices
and vector are as follows:
{\begin{equation}\label{30}
\underset{8\times 8}{\mathbf{m}_{m}} = 
m_{c}\left[\begin{array}{@{}ccc@{}}
\mathbf{N}^{T}\mathbf{N} & \mathbf{0} & \mathbf{0}\\
\mathbf{0} & \mathbf{H}^{T}\mathbf{H} & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{array}\right]_{| {x_{c}}},
\quad
\underset{8\times 8}{\mathbf{c}_{m}} = 
2m_{c}v\left[\begin{array}{@{}ccc@{}}
\mathbf{0} & \mathbf{0} & \mathbf{0}\\
\mathbf{0} & \mathbf{H}^{T}\mathbf{H}_{,x} & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{array}\right]_{| {x_{c}}}, 
\quad
\underset{8\times 8}{\mathbf{k}_{m}} = 
m_{c}v^{2}\left[\begin{array}{@{}ccc@{}}
\mathbf{0} & \mathbf{0} & \mathbf{0}\\
\mathbf{0} & \mathbf{H}^{T}\mathbf{H}_{,xx} & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & \mathbf{0}
\end{array}\right]_{| {x_{c}}},
\end{equation}
\begin{equation}\label{31}
\underset{8\times 1}{\mathbf{f}_{m}} = 
{\{P_{x}\mathbf{N}^{T}\,\,P_{z}\mathbf{H}^{T}\,\,\mathbf{0}\}}_{| x_{c}}^{T}.
\end{equation}}\unskip\noindent
In Equations~(\ref{30}) and (\ref{31}), the notation $(.)_{| {x_{c}}}$ means that
the expression $(.)$ is evaluated at $x_{c}$---the current abscissa of
the moving mass with respect to the element left node.

The above element matrices and vector are derived in the local
coordinate system and they are necessarily transferred to the global
system. To this end, we considered the displacements of a point in the
beam with respect to the global $\overline{x}$ and $\overline{z}$
directions, $\overline{u}$ and $\overline{w}$. These displacements are
related to the ones in the local $x$ and $z$ directions, $u$ and $w$
by:
{\begin{equation}\label{32}
\overline{u} = u\cos \beta -w\sin \beta;
\quad
\overline{w} = u\sin \beta +w\cos \beta. 
\end{equation}}\unskip\noindent
Because the local rotation and the global one are identical, the vector
of local degrees of freedom ($\mathbf{d}$) is related to the global one
$\overline{\mathbf{d}} = {\{\begin{array}{@{}ccc@{}}
\overline{\mathbf{d}}_{u} &
\overline{\mathbf{d}}_{w}& \overline{\mathbf{d}}_{\gamma }
\end{array}\}}^{T}$ by
$\mathbf{d} = \mathbf{T}\overline{\mathbf{d}}$ where the transformation
matrix $\mathbf{T}$ having the form:
{\begin{equation}\label{33}
\mathbf{T} = \left[\begin{array}{@{}cccccccc@{}}
\cos \,\beta & 0 & \sin \,\beta & 0 & 0 & 0 & 0 & 0\\
0 & \cos \,\beta & 0 & 0 & \sin \,\beta & 0 & 0 & 0\\
-\sin \,\beta & 0 & \cos \,\beta & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & -\sin \,\beta & 0 & 0 & \cos \,\beta & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right].
\end{equation}}\unskip\noindent
The global element stiffness and mass matrices are finally computed as:
{\begin{equation}\label{34}
\overline{\mathbf{k}} = \mathbf{T}^{T}\mathbf{k}\mathbf{T},
\quad\mbox{and}\quad 
\overline{\mathbf{m}} = \mathbf{T}^{T}\mathbf{m}\mathbf{T}.
\end{equation}}\unskip\noindent

Similarly, the element mass, damping, stiffness matrices and nodal load
vector in Equations~(\ref{30}) and (\ref{31}) written in global coordinate system
are as follows:
{\begin{equation}\label{35}
\overline{\mathbf{m}}_{m} = \mathbf{T}^{T}\mathbf{m}_{m}\mathbf{T};
\quad
\overline{\mathbf{c}}_{m} = \mathbf{T}^{T}\mathbf{c}_{m}\mathbf{T};
\quad
\overline{\mathbf{k}}_{m} = \mathbf{T}^{T}\mathbf{k}_{m}\mathbf{T};
\quad
\overline{\mathbf{f}}_{m} = \mathbf{T}^{T}\mathbf{f}_{m}.
\end{equation}}\unskip\noindent

The equation for the dynamic analysis of the inclined sandwich beam can
now be written as:
{\begin{equation}\label{36}
\overline{\mathbf{M}}\; \ddot{\overline{\mathbf{D}}} +
\overline{\mathbf{C}}\; \dot{\overline{\mathbf{D}}} + 
\overline{\mathbf{K}}\; \overline{\mathbf{D}} = 
\overline{\mathbf{F}}^{\mathrm{ex}}
\end{equation}}\unskip\noindent
where $\overline{\mathbf{M}}$, $\overline{\mathbf{K}}$ are the
instantaneous overall mass and stiffness matrices, respectively. These
matrices composed of the constant overall mass and stiffness matrices
of the inclined beam itself and the time-dependent element property
matrices due to the moving mass. The overall damping matrix
$\overline{\mathbf{C}}$ is obtained by adding the element damping
matrix $\overline{\mathbf{c}}_{m}$ to the damping matrix of the
inclined beam itself. The overall damping matrix of the inclined beam
is proportional to the overall mass and stiffness matrices by using the
theory of Rayleigh damping with a damping ratio of 5\%.

\section{Numerical results and discussion}\label{sec5}

This section presents the numerical results on dynamic response of the
inclined sandwich beam subjected to the moving mass. Otherwise stated,
a sandwich beam with $L/h=20$, $b=0.4~\mathrm{m}$, $h=1~\mathrm{m}$ is
considered. The beam is simply supported at both ends, and the
geometric boundary conditions are as follows:
{\begin{equation*}
\begin{array}{@{}l@{}}
\mbox{At }x = 0\colon \overline{u}_{0}(0,t) = \overline{w}_{0}(0,t)=0,\\
\mbox{At }x = L\colon \overline{u}_{0}(L,t) = \overline{w}_{0}(L,t)=0
\end{array}
\end{equation*}}\unskip\noindent
with $\overline{u}_{0}$ and $\overline{w}_{0}$ are the displacements in
$\overline{x}$ and $\overline{z}$ directions of a point on the
$x$-axis, respectively. The beam is asumed initially at rest, and all
the displacements and velocities at time $t=0$ equal to zero. 

The material properties of the matrix are $E_{m} = 10$~GPa, $\rho_{m} =
1150$~kg/m$^{3}$, $\nu _{m} = 0.3$. The armchair (10, 10) SWCNTs with
$\rho^{\mathrm{CNT}} = 1400$ kg/m$^{3}$ and elastic constants tabulated
in Table~\ref{tab1} are used as the reinforcements. The core of the
sandwich beam is assumed pure matrix material.

%tab1
\begin{table}
\caption{\label{tab1}Hill's elastic modulus for the CNTs~\cite{28}}
\begin{tabular}{cccccc}
\thead
CNT radius (\AA) & $k_{r}$ (GPa) & $l_{r}$ (GPa) & $m_{r}$
(GPa) & $n_{r}$ (GPa) & $p_{r}$ (GPa) \\
\endthead
10 & 30 & 10 & 1 & 450 & 1
\botline
\end{tabular}
\end{table}

Three numbers in parentheses, e.g. (2-1-1), are used below to denote
the layer thickness ratio of the beam layers, from the bottom layer to
the top layer. A moving mass $m_{c}=0.5\rho _{m}AL$ is employed in all
computations reported below. To facilitate the discussion, the
dimensionless parameter is introduced for dynamic magnification factor
(DMF) $D_{d}$ as follows:
{\begin{equation}\label{37}
D_{d} = \max \left(\dfrac{w_{0}(L/2,t)}{w_{st}}\right),
\end{equation}}\unskip\noindent
where $w_{st}=L^{3}m_{c}g/48E_{m}I$ is the static deflection of a
horizontal beam made of fully core material under mid-span concentrated
load $P=m_{c}g$, and $I$ is the inertia moment of area of the
cross-section. A uniform increment time step $\Updelta t = \Updelta
T/200$ with $\Updelta T$ is the total time needed for the mass crossing
the beam, is used for the Newmark procedure.

\subsection{Formulation verification}\label{ssec51}

In order to examine the accuracy and reliability of the present
formulation, the effective Young's modulus and Poisson's ratio of an
agglomerated randomly oriented CNTRC beam obtained in the present work
are firstly compared with that of Daghigh \etal~\cite{28} in
Figures~\ref{fig3} and~\ref{fig4}, respectively. The result in the
figures is obtained for a matrix phase with $E_{m} = 2.5$~{GPa}, the
Hill's elastic moduli for the CNTs listed in Table~\ref{tab1} and CNT
volume fraction $V_{\mathrm{CNT}}=0.1$. A good agreement between the
results of the present work with that of Ref.~\cite{28} is seen from
the figures. The figures show the significant influence of the
agglomeration parameters on the effective elastic coefficients of the
CNTRC material. Specifically, the effective Young's modulus increases
with the increase of $\xi$, and it approaches the largest value at
$\xi=\zeta$, corresponding to the full dispersion of CNTs. On the
contrary, the effective Poisson's ratio decreases by increasing the
parameter $\xi$, and then reaches the smallest when $\xi=\zeta$.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Comparison of effective Young's modulus of CNTRC
for $V_{\mathrm{CNT}}=0.1$ and different agglomeration parameters.}
\end{figure}

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Comparison of Poisson's ratio of CNTRC for
$V_{\mathrm{CNT}}=0.1$ and different agglomeration parameters.}
\end{figure}

In Figure~\ref{fig5}, the time histories for mid-span deflection of an
inclined homogeneous beam under a moving mass obtained by present
formulation are compared with that of Mamandi and Kargarnovin~\cite{25}
for $\beta = \sfrac{\rmpi}{5}$ and various values of the mass velocity
parameter, namely $\alpha=0.1$, 0.25 and 0.5. The velocity parameter 
$\alpha$ is defined as $\alpha =v/v_{cr}$, with $v_{cr} =
({\rmpi}/{L}) \sqrt{EI/\rho A}$ is the critical velocity of a
moving force on a simply  supported beam. Good agreement between the
time histories of the present work with that of Ref.~\cite{25} is noted
from Figure~\ref{fig5}. 

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Comparison of time histories for mid-span
deflection of inclined homogeneous beam with $\beta =
\sfrac{\rmpi}{5}$.}
\end{figure}

Convergence of the derived formulation in evaluating the fundamental
frequency parameter $\lambda ^{2} = \omega L^{2}\sqrt{\sfrac{\rho
_{m}A}{E_{m}I}}$ ($I=bh^{3}/12$) of a CNTRC beam is shown in
Table~\ref{tab2} for $V_{\mathrm{CNT}}=0.075$ and two types of boundary
conditions, namely fixed at one end and free at the other (CF), and
simply supported at both ends (SS). For the sake of comparison, the
result obtained by 100 Timoshenko beam elements of Yas and 
Heshmati~\cite{30} is also given in the table. The table shows the fast
convergence of the present formulation, and it needs only 14 and 18
elements for the CF and SS beams to converge, respectively. The
difference between the frequency parameters of the present work with
that of Ref.~\cite{30} may be resulted from the different beam theories
employed in the two works, and more important is that the effect of CNT
agglomeration is not considered in~\cite{30}.

%tab2
\begin{table}
\caption{\label{tab2}Convergence of the formulation in evaluating
fundamental frequency parameters of a composite beam reinforced with
randomly oriented CNTs for different boundary conditions}
\tabcolsep=4pt
\begin{tabular}{ccccccccc}
\thead
\morerows{1}{\parbox[t]{4.5pc}{\centering Boundary conditions}}\vspace*{2pt} & 
\multicolumn{6}{c}{Present} & \xmorerows{1}{Ref.~\cite{30}} & 
\xmorerows{1}{Error (\%)} \\\cline{2-7}
& $ne = 10$ & $ne = 12$ & $ne = 14$ & $ne = 16$ & $ne = 18$ & $ne = 20$
& & \\
\endthead
CF & 2.0573 & 2.0571 & 2.0570 & 2.0570 & 2.0570 & 2.0570 & 2.151246 & 4.58 \\
SS & 3.4432 & 3.4428 & 3.4426 & 3.4425 & 3.4424 & 3.4424 & 3.574603 & 3.84
\botline
\end{tabular}
\end{table}

Table~\ref{tab3} shows the convergence of the present formulation in
computing the dynamic magnification factor of the (2-1-2) and (2-2-1)
sandwich beams for $\beta = \rmpi/5$, $V_{\mathrm{CNT}}=0.1$, $\zeta=1$
and $v=20$~m/s. As can be seen from the table, for all considered cases
the convergence is achieved by using 28 elements. In this regards, a
mesh of 28 elements is employed in all computations reported below.  

%tab3
\begin{table}
\caption{\label{tab3}Convergence of the formulation in computing DMF of
the sandwich beam reinforced agglomerated CNTs for $\beta=\rmpi/5$,
$V_{\mathrm{CNT}}=0.1$, $\zeta=1$ and $v=20$~m/s}
\begin{tabular}{ccccccc}
\thead
& & $ne = 16$ & $ne = 20$ & $ne = 24$ & $ne = 28$ & $ne = 32$ \\
\endthead
\morerows{2}{(2-1-2)} & $\xi=0.2$ & 0.7292 & 0.7292 & 0.7292 & 0.7292 & 0.7292 \\
        & $\xi=0.5$ & 0.6339 & 0.6341 & 0.6342 & 0.6342 & 0.6342 \\
        & $\xi=1$   & 0.5685 & 0.5687 & 0.5687 & 0.5687\vspace*{5pt} & 0.5687 \\

\morerows{2}{(2-2-1)} & $\xi=0.2$ & 0.7448 & 0.7448 & 0.7448 & 0.7448 & 0.7448 \\
        & $\xi=0.5$ & 0.6581 & 0.6583 & 0.6583 & 0.6584 & 0.6584 \\
        & $\xi=1$   & 0.5975 & 0.5977 & 0.5977 & 0.5977 & 0.5977 
\botline
\end{tabular}
\end{table}

\subsection{Dynamic response}\label{ssec52}

The dynamic response of the inclined CNTRC sandwich beam under the
moving mass is investigated in this sub-section. In Table~\ref{tab4},
the DMFs of the sandwich beam with an inclined angle $\beta
=\sfrac{\rmpi}{5}$ are tabulated for $v=50$~m/s, $\zeta=1$ and various
values of the agglomeration parameter $\xi$ and the CNT volume fraction
$V_{\mathrm{CNT}}$ as well as the layer thickness ratio. One can
observe from the table the important role of the agglomeration
parameter $\xi$ on the DMF. At the given value of the parameter
$\zeta$,  the DMF decreases when increasing the parameter $\xi$, and
the decrease is more significant when the beam associated with a large
CNT volume fraction value of $V_{\mathrm{CNT}}$. As expected, the
increase of the CNT volume fraction $V_{\mathrm{CNT}}$ and the face
thickness leads to a significant decrease in the DMF. The influence of
the CNT volume fraction and the face thickness on the DMF, however is
governed by the agglomeration parameters. Looking at (1-1-1) sandwich
beam with $\xi=\zeta=1$ (uniformly distributed or no agglomeration),
one can see from Table~\ref{tab4} that the factor $D_{d}$ decreases
71.45\% when increasing the $V_{\mathrm{CNT}}$ from 0 to 0.3, while the
corresponding value is just 17.09\% for the sandwich beam with $\xi
=0.1$, $\zeta=1$ (severe CNT agglomeration). Examining the table in
more detail one can see that the dependence of the DMF $D_{d}$ upon the
layer thickness ratio is significantly influenced by the agglomeration
as well. Thus, one can conclude from the table that the increase of the
CNT volume fraction and the reinforced face thickness is the most
effective only when the CNTs agglomeration does not occur.

%tab4
\begin{table}
\caption{\label{tab4}DMFs of sandwich beam with $\beta=\rmpi/5$ for
$v=50$ m/s, $\zeta=1$ and different values of agglomeration parameter
$\xi$ and CNT volume fraction}
\begin{tabular}{cccccccc}
\thead
& $\zeta = 1$ & $V_{\mathrm{CNT}} = 0$ & $V_{\mathrm{CNT}} = 0.02$ & 
$V_{\mathrm{CNT}} = 0.05$ & $V_{\mathrm{CNT}} = 0.1$ & 
$V_{\mathrm{CNT}} = 0.2$ & $V_{\mathrm{CNT}} = 0.3$ \\
\endthead
\morerows{2}{(1-0-1)} & $\xi =0.1$ & 1.4320 & 1.3066 & 1.2453 & 1.2096 & 1.1880 & 1.1815 \\
 & $\xi =0.5$ & 1.4320 & 1.2460 & 1.0677 & 0.8970 & 0.7337 & 0.6560 \\
 & $\xi =1$ & 1.4320 & 1.2341 & 1.0189 & 0.7839 & 0.5289{\vspace*{5pt}} & 0.3955 \\ 

\morerows{2}{(2-1-2)} & $\xi =0.1$ & 1.4320 & 1.3077 & 1.2466 & 1.2110 & 1.1889 & 1.1819 \\
 & $\xi =0.5$ & 1.4320 & 1.2476 & 1.0703 & 0.9001 & 0.7369 & 0.6589 \\
 & $\xi =1$ & 1.4320 & 1.2358 & 1.0218 & 0.7875 & 0.5323{\vspace*{5pt}} & 0.3982 \\ 

\morerows{2}{(1-1-1)} & $\xi =0.1$ & 1.4320 & 1.3111 & 1.2515 & 1.2165 & 1.1945 & 1.1873 \\
 & $\xi =0.5$ & 1.4320 & 1.2525 & 1.0787 & 0.9109 & 0.7486 & 0.6705 \\
 & $\xi =1$ & 1.4320 & 1.2409 & 1.0310 & 0.7992 & 0.5442{\vspace*{5pt}} & 0.4088 \\ 

\morerows{2}{(2-2-1)} & $\xi =0.1$ & 1.4320 & 1.3205 & 1.2652 & 1.2326 & 1.2120 & 1.2052 \\
 & $\xi =0.5$ & 1.4320 & 1.2662 & 1.1033 & 0.9436 & 0.7864 & 0.7096 \\
 & $\xi =1$ & 1.4320 & 1.2554 & 1.0582 & 0.8360 & 0.5842{\vspace*{5pt}} & 0.4465 \\ 

\morerows{2}{(1-2-1)} & $\xi =0.1$ & 1.4320 & 1.3214 & 1.2662 & 1.2335 & 1.2127 & 1.2056 \\
 & $\xi =0.5$ & 1.4320 & 1.2673 & 1.1044 & 0.9440 & 0.7856 & 0.7080 \\
 & $\xi =1$ & 1.4320 & 1.2565 & 1.0592 & 0.8357 & 0.5820{\vspace*{5pt}} & 0.4433 \\ 

\morerows{2}{(1-4-1)} & $\xi =0.1$ & 1.4320 & 1.3417 & 1.2958 & 1.2681 & 1.2502 & 1.2440 \\
 & $\xi =0.5$ & 1.4320 & 1.2967 & 1.1575 & 1.0143 & 0.8670 & 0.7918 \\
 & $\xi =1$ & 1.4320 & 1.2878 & 1.1177 & 0.9147 & 0.6678 & 0.5244
\botline
\end{tabular}
\end{table}

In Figure~\ref{fig6}, the relation between the DMF $D_{d}$ with the
moving mass velocity $v$ of the (1-2-1) and (2-1-2) beams are depicted
for $\beta=\sfrac{\rmpi}{5}$ and different CNT volume fractions and the
layer thickness ratios. Various remarks can be drawn from
Figure~\ref{fig6}. The CNT reinforcement, even with a small amount of
CNT volume fraction of 0.02 can reduce the DMF significantly. However,
the increase in the amount of reinforced CNTs is highly dependent on
the CNTs agglomeration, regardless of the layer thickness ratio. For
the case of severe agglomeration, that is when the two agglomeration
parameters are far from each other such as $\xi=0.1$, $\zeta=1$ in
Figure~\ref{fig6}a,c, the DMF does not decrease considerably when
increasing the CNT volume fraction $V_{\mathrm{CNT}}$, and this
tendency is more clearly for the beam associated with a larger
$V_{\mathrm{CNT}}$. In contrast, in case of no agglomeration, that is
$\xi =\zeta =1$ in Figure~\ref{fig6}b,d, the increase of the CNT
volume fraction leads to a sharp decrease in the DMF. By comparing
Figure~\ref{fig6}b with Figure~\ref{fig6}d, one can see that the
effect of the layer thickness ratio on the DMF is really evident only
for the case of no CNT agglomeration. The dependence of the DMF upon
the moving mass velocity is similar to that of the FGM beams subjected
a moving load~\cite{31}, and the DMF repeatedly increases and decreases
when increasing the moving mass velocity $v$, and it then approaches a
maximum value. The moving mass velocity at which the DMF attains the
maximum value is higher for the sandwich beam reinforced with a higher
CNT volume fraction, and this tendency is more visible for the beam
with a larger face layer thickness and no CNT agglomeration in the beam
(Figure~\ref{fig6}d).

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Relation between DMF with moving mass velocity of
sandwich beam for $\beta=\rmpi/5$ and different CNT volume fractions
and layer thickness ratios.}
\end{figure}

The effect of the inclined angle on the dynamic response of the CNTRC
sandwich beam is shown in Figure~\ref{fig7}, where the relation between
the DMF with the moving mass velocity of the (2-1-2) beam is depicted
for $V_{\mathrm{CNT}}=0.1$ and different agglomeration parameters. A
significant decrease of the DMF by increasing the inclined angle is
seen from the figure, regardless of the agglomeration parameters and
the moving mass velocity. The influence of the inclined angle on the
dynamic response is more significant for the case of CNT agglomeration
(Figure~\ref{fig7}a) than for the case of full dispersion
(Figure~\ref{fig7}b). Looking closely at Figure~\ref{fig7} one can see
that although the inclined angle can significantly change the DMF, the
angle hardly alter the shape of the curves in the figure as well as the
velocity value at which the DMF attains the maximum value. The figure
also confirms again the above remark on the influence of the CNT
agglomeration, namely the DMF obtained for the beam without CNT
agglomeration (Figure~\ref{fig7}b) is considerably lower than that of
the beam with severe CNT agglomeration (Figure~\ref{fig7}a). 

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}The effect of inclined angle on the relation
between the DMF and moving mass velocity of (2-1-2) beam with
$V_{\mathrm{CNT}}=0.1$ and different agglomeration parameters.}
\end{figure}

The influence of the two agglomeration parameters on the dynamic
response of the sandwich beam is shown in Figure~\ref{fig8}, where the
variation of the DMF with the first agglomeration parameter $\xi$ of
the (2-1-2) beam is depicted for $\beta = \rmpi/5$,
$V_{\mathrm{CNT}}=0.1$ and various values of agglomeration parameter
$\zeta$. For $\xi < \zeta$, the DMF decreases with an increase of the
parameter $\xi$, and it attains the lowest value when $\xi =\zeta$,
which corresponds to the case of uniform distribution of CNTs. For
$\xi>\zeta$, the agglomeration degree is more serious when $\xi$ is far
from $\zeta$, and the DMF increases again when increasing the parameter
$\xi$. One can see from Figure~\ref{fig8} that the larger difference
between the two agglomeration parameters is the higher DMF the beam
has. Thus, the DMF is clearly underestimated by ignoring the
agglomeration of CNTs. 

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}Variation of DMF with agglomeration parameter
$\xi$ of (2-1-2) beam with $\beta =\rmpi /5$, $V_{\mathrm{CNT}}=0.1$
and various values of agglomeration parameter $\zeta$.}
\end{figure}

It would be remiss not to consider the influence of the agglomeration
on the time history of the inclined sandwich beam. Figure~\ref{fig9}
displays the time histories for mid-span deflection of (2-1-2) sandwich
beam for $\beta =\rmpi/5$, $V_{\mathrm{CNT}}=0.1$, ${\xi}=1$ and
various values of the agglomeration parameters. It can be seen from the
figure that the mid-span deflection of the beam reaches the smallest
value when $\zeta=\xi=1$, and it attains the maximum value when the two
agglomeration parameters are farthest apart, regardless of the moving
mass velocity. In addition, the beam tends to execute less vibration
cycles when it is subjected to the mass moving with a higher velocity.
The time at which the deflection reaches its maximum value is earlier
as the two parameters closer to each other. In addition, it seems that
the change of the second agglomeration parameter $\zeta$ has a stronger
effect on the mid-span deflection of the beam, and this can be
confirmed by comparing Figure~\ref{fig9}b,d with 
Figure~\ref{fig9}a,c.

\begin{figure}
{\vspace*{-4pt}}
\includegraphics{fig09}
{\vspace*{-4pt}}
\caption{\label{fig9}Time histories for mid-span deflection of (2-1-2)
beam for $\beta=\rmpi/5$ and various values of two agglomeration
parameters.}
{\vspace*{-4pt}}
\end{figure}

\section{Conclusions}\label{sec6}

The dynamics of the inclined sandwich beams with agglomerated carbon
nanotube reinforced face sheets under a moving mass has been studied in
the basis of the trigonometric shear deformation theory. The
Mori--Tanaka approach, taking into consideration of CNT agglomeration is
employed to estimate the material properties of the CNTRC layers. A
novel finite element formulation in which the transverse shear rotation
is employed as an independent variable is derived and used to establish
the discretized equation of motion for the sandwich beams. Dynamic
characteristics, including the dynamic magnification factors and the
time histories for mid-span deflections are obtained for a simply
supported sandwich beam with the aid of Newmark method. The effects of
the inclined angle, the CNT agglomeration, the mass velocity as well as
the layer thickness ratio on the dynamic response have been
investigated in detail. The present study yields the following main
conclusions:
\begin{enumerate}[(1)]
\item The CNT reinforcement significantly improves the dynamic response
of the inclined sandwich beams. An increase in the CNTs volume fraction
leads to a sharp decrease in the dynamic magnification factor,
especially for the beam with a low degree of the CNT agglomeration.

\item CNT agglomeration has an impact role in dynamic response of the
sandwich composite beams. The more severe agglomeration of the CNTs is,
the higher dynamic magnification factor the beam has.

\item The inclined angle plays an important role on the dynamic
response of the sandwich composite beams, and the dynamic magnification
factors of the beams are sharply decreased by increasing the inclined
angle.

\item The layer thickness ratio has a pronounced effect on the dynamic
behavior of the sandwich beam. An increase of the CNT reinforced
surface layer thickness leads to a significant decrease in the dynamic
magnification factor, but this reduction is highly dependent on the
CNTs volume fraction and the CNT agglomeration.
\end{enumerate}

\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.

\section*{Acknowledgement}

This research was made possible by the grant number
CSCL03.02/22-23, Institute of Mechanics, VAST (Vietnam).

\CDRGrant[VAST]{CSCL03.02/22-23}

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