Dynamic Analysis of Maneuvering Flexible Spacecraft Appendage Using Higher Order Sandwich Panel Theory

W2108863371 · 수집 2026-10-04 18:12:21

T10221 Composite Structure Analysis and Optimization

CoT 라벨링

라벨링 전

PDF 파일

자동 다운로드한 PDF PDF 보기

기본 정보

DOI 제목 발행년도 저널 URL 페이지 열기 PDF URL PDF 열기 발행기관 피인용수 라이선스

추가 정보 초록 본문

## Dynamic Analysis of Maneuvering Flexible Spacecraft Appendage Using Higher Order Sandwich Panel Theory

Abstract
This paper is devoted to the vibration of rotating flexible spacecraft.
The maneuver of the spacecraft is modeled by a constant torque
input acting on the hub. For the first time in this paper the
equations of motion of flexible spacecraft are derived based on
higher order sandwich panel theory (HSAPT). Hamilton’s principle
is used for driving the governing partial differential equations of
motion. The generalized differential quadrature method (GDQ) is
utilized to solve the partial differential equations of motion. The
effect of different parameters on vibration of rotating flexible
spacecraft appendage is investigated. It is also investigated the
effect of these parameters on natural frequencies. The result of
HSAPT and Euler Bernoulli theory is compared with each other.
To show the accuracy the natural frequencies of recent paper are
compared with the literature.

Keywords
Flexible Spacecraft, Constant torque, Hamilton’s principle, GDQ,
HSAPT

1 INTRODUCTION

Milad Azimi a

Morteza Shahravi b

Eshagh Farzaneh Joubaneh c

a Malek Ashtar University of
Technology, Tehran, Iran.
azimi.mld@gmail.com
b Malek Ashtar University of
Technology, Tehran,
Iran.Shahravi@aut.ac.ir

c Islamic Azad University, South Tehran
Branch, Tehran, Iran,
Isac.farzaneh@yahoo.com

http://dx.doi.org/10.1590/1679-78252326

Received 01.08.2015
Accepted 01.11.2015
In revised form 03.11.2015
Available online 09.11.2015

There are many studies on flexible spacecraft which is consisted of a rigid hub in the center and
attached flexible appendages. The analysis of these systems is complicated because of coupling
between rigid hub and flexible appendage. In these structures vibration of the flexible part causes the
rotation of the hub and vice versa (Hu, 2009). There have been a lot of intensive researches on
rotating of the flexible spacecraft and finding an appropriate method to suppress the vibrations. They
usually consider each appendages as Euler-Bernoulli beam to simplify the equation of motions. Azadi
et. al (Azadi, et al., 2011)studied on three axes slewing maneuver and the vibration of a flexible
spacecraft. They used an adaptive-robust control scheme to achieve the satellite’s large angel
trajectory tracking and suppress the vibration of the appendages. They also applied an Euler-Bernoulli

beam theory for flexible appendage. Maganti et. al (Maganti & Singh, 2007) present the design of a
new simple adaptive system for the rotational maneuver and vibration suppression of an orbiting
spacecraft with flexible appendages. They choose a control output variable as combination of pitch
angle and its derivative without considering elastic mode. Tugrul Oktay (Oktay, 2015) studied on
bending control of rotating Euler Bernoulli beam. He assumed that the free fixed elastic beam is
attached to a servomotor using variance constrained controller, specifically output variance
constrained controller for vibration suppression. Equations of motion of the system obtained via
Hamilton’s principle and Galerkin method.

Shahravi and azimi studied on vibration control of smart flexible sub-structure of satellite during
attitude maneuver. They compared collocated and non-collocated piezoceramic patches as sensor and
actuator to suppress vibration of flexible substructure. They used finite element method associated
with lagrangian formulation to derive mathematical model of the slewing flexible spacecraft (Shahravi
& Azimi, 2014). Shahravi and Azimi (Shahravi & Azimi, 2014) presented a new control approach for
large angle attitude maneuver of flexible spacecraft. They used singular perturbation theory as a
useful tool for two time rate scale separation of rigid and flexible body dynamics. The resulting slow
and fast subsystems, enabling the use of two control approach for attitude (Modified Sliding Mode)
and vibration Strain Rate Feedback (SRF) control of flexible spacecraft, respectively. Shahravi and
Azimi (Shahravi & Azimi, 2015) studied on multi-axis attitude control of flexible spacecraft using
smart structures and hybrid control scheme . They considered the spacecraft as a rigid hub with two
elastic appendages embedded with piezoelectric sensor/actuator patches. Using a modified sliding
mode control by introducing a synthesized sliding manifold ensures that the spacecraft follows the
shortest possible path to the desired orientation and highly reduce the switching action and excitation
of flexible modes. Zhang e. al (Zhang, et al., 2014), used PZT transducers on flexible links as vibration
sensors and actuators to suppress the vibration. They derived dynamic model of the flexible links
with dynamic of PZT actuators incorporated and discretized the elastic motion of the flexible links
by assumed mode method. An Efficient modal control, in which the feedback forces in different modes
are determined according to the vibration amplitude or energy of their own, is employed to control
the PZT actuators to realize active vibration suppression.

As we know multi body systems with large deflection are being used in many practical engineering
such as aeronautics, aerospace’s and robotics where a flexible appendage is made of sandwich panel.
A sandwich structures with soft cores made of foam or low strength honeycomb, like aramid or nomex,
are being used in various industrial applications such as aerospace and civil engineering structures.
Sandwich panels consist of two thin composite or thick metallic layers which is separated by a thick
lightweight core. The core has different types as Honeycomb, Web core, Balsa Wood, Foamed
polymer. This configuration leads to a sandwich panel with high strength and stiffness, light weight
and high energy absorption capability. Because of these properties, sandwich structures are being
used in aerospace equipment devices widely. It should be added that the core should be strong in
shear and tension in thickness direction and must have a low density to reduce weight. There are
different theories to describe the behavior of sandwich panels due to type of the core. It is common
in the analysis of sandwich panels to neglect the transverse deformation of the core (Vinson, 1999).
This theory is being used as the core is stiffen or rigid. Several people have investigated on these
of sandwich panels. An early theory of sandwich structures, is known as the First-Order Shear

Latin American Journal of Solids and Structures 13 (2016) 296-313

Deformation Theory (FOSDT), which is the beam/plate theory by taking into account the shear
rigidity of the core, but this theory is still assumes that longitudinal deformation is linear in the
thickness coordinate and the core is infinitely rigid in the transverse direction. FOSDT assumes a
uniform shear stain through the height of the panel. Although this model is simple, but it is acceptable
when the sandwich core is too stiff through the thickness and statically loaded. In general, and
especially in modern sandwich panels, the core is flexible in all directions. There are many modified
theories which considered different assumption due to better model the stress, strain, displacement
distribution along the thickness. Hoh et al (Hohe, et al., 2006) investigated on the effect of the
transverse compressibility of the core on the transient dynamic response of structural sandwich panels
under rapid loading conditions. Frostig (Frostig, et al., 1992) is one who has done numerous studies
on the behavior of sandwich panels. He offered a higher order sandwich panel theory (HSAPT) for
compressible cores which in-plane stresses is neglected in it. Regardless of axial stress in the core and
according to the static equilibrium equation, there will be a constant shear distribution within the
core thickness and this type of approximation for the sandwich construction with a soft core would
be a good approximation of the static problems. Static formulation of HSAPT is being used for many
issues. Comparison between HSAPT with elasticity and experiment has shown that HSAPT predicts
displacements and axial strain at the surface and near to the supports and concentrated load regions
accurately. Regarding the core, though HSAPT is a good approximate theory away from supports,
and concentrated load regions, it show inaccurate shear stress and axial strain through-thickness
distributions adjacent to regions of concentrated loads and supports. This theory can be used for the
study of composite beams and composite plates with soft cores. Frostig et. al (Frostig, et al., 2013)
investigated on the free vibration response of a unidirectional sandwich panel with a compressible
and incompressible core using the various computational models and it is presented and compared
with the numerical investigation. Also it should be added that HSAPT theory divides into two
displacement and mixed formulation theories which is different in the formulation of the problem.
Frostig and baruch (Frostig & Baruch, 1994) investigated on free vibrations of sandwich beams with
a transversely flexible core. The analysis presented embodies a general rigorous approach including
higher order effects owing to the non-linearity of the displacement fields of the core caused by its
flexibility in the vertical direction. Swanson and kim (Swanson & Kim, 2000) studied on comparison
of a higher order theory for sandwich beams with finite element and elasticity analysis.

There are different methods to solve these problems. Damanpack and khalili (Damanpack &
Khalili, 2012) is investigated on the higher-order free vibration of sandwich beams with a flexible
core using dynamic stiffness method. One of the methods to solve the vibration analysis is generalized
differential quadrature (GDQ) which is used in different papers. There are a lot of people which used
this method to analysis the problems. Tornabene and Viola (Tornabene & Viola, 2009) studied the
dynamic behavior of functionally graded parabolic and circular panels and shells of revolution. They
used First-order Shear Deformation Theory (FSDT) to study the moderately thick structural
elements. They also investigated on the local GDQ method applied to general higher-order theories
of doubly-curved laminated composite shells and panels (Tornabene & Viola, 2014). Hong et al.
investigated on thermal induced vibration of a thermal sleeve with GDQ method. They used this
method to obtain the numerical results of two-layer cross-ply laminated tubes under thermal vibration
(Hong, et al., 2005).

Latin American Journal of Solids and Structures 13 (2016) 296-313

The present paper provides an analysis to obtain the response of a flexible spacecraft with respect
to a constant torque on the hub for a real sample flexible spacecraft. Unlike the previous papers the
theory which is applied for flexible spacecraft is HSAPT and the result for vibration analysis is
compared with other conventional beams theory.

2 DERIVATION OF THE GOVERNING EQUATIONS

Figure 1 shows the model of a flexible spacecraft, which consists of a rigid hub with radius a, and a
cantilever sandwich beam with length L. The sandwich beam with width b is made of three layers.
Two thin stiff face sheets with thickness �� and a thick soft core with thickness ��.

Figure 1 : Geometry of rigid hub with flexible appendage.

The general assumption for derivation of general equation of motion for a sandwich beam is as
follow

1. All the deformations and strains are very small and the theory of linear elasticity is applied.
2. The face sheets and the core of the beam are made of isotropic and homogeneous materials.
The sandwich beam is assumed to be symmetric.
3. Transverse normal strains are negligible in the face sheets.
4. There is no slippage of delamination between the layers.

2.1 Displacement Theory

The face sheets deform as Bernoulli-Euler beam theory and the normal stress of the core is negligible
and it is soft through the thickness. Here the axial and transverse displacements of the top face sheet
are ����, ��, ����, �� and the transverse and axial displacements of the bottom face sheets are
����, ��, ����, ��, respectively. The axial and transverse displacements of the core layer are
����, ��, ����, �� respectively.
The displacement filed for a sandwich beam assumes as

Latin American Journal of Solids and Structures 13 (2016) 296-313



_h_ 2 _c_  _y_  _h_ 2 _c_  _h_ _f_

_ut_  _ywt_ '




 









_t_ _t_

_c_  _h_ _f_  _y_ _c_

 _hc_  _y_

_hc_  _y_ _hc_

2 2

_c_  _y_ _c_

 1 _y_  2 _y_ 2 

 1 _y_  2 _y_ 2  3

1 _y_  2 _y_ 2  3 _y_ 3

Tope face

Core

Bottom Face

Tope face





Core  





Bottom Face 



_uc_  1 _y_  2 _y_ 2  3 _y_

_c_




 



_c_



_ub_  _ywb_ '

_b_ _b_

 _hc_  _h_ _f_  _y_

2 _hc_  _h_ _f_  _y_ _h_ 2 _c_

_h_ 2 _c_  _y_  _h_ 2 _c_  _h_ _f_

_wt_

(1)

(2)

(3)

_wc_  1 _y_  2 _y_

 1 _y_ 

 1 _y_  2 _y_ 2

_wb_





By using the compatibility conditions as



 


 




 


 


_X_ _Core_ ( _y_  _hc_ )  _X_ _Topface_ ( _y_  _hc_

_Core_

 _c_ )  _X_ _Topface_ ( _y_ 

_c_ )  _X_ _Topface_ ( _y_  _c_

( _y_  _c_ )  _X_ _Topface_ ( _y_  _c_ )
2 2

_X_ _Core_ ( _y_   _hc_ )  _X_ _Bottomface_ ( _y_   _hc_

_Core_

  _c_ )  _X_ _Bottomface_ ( _y_  

_c_ )  _X_ _Bottomface_ ( _y_   _c_

( _y_   _c_ )  _X_ _Bottomface_ ( _y_   _c_ )
2 2

 _hc_  _y_  _hc_

2 2

 2 _hc_  _h_ _f_  _y_  _h_ 2 _c_

_h_ 2 _c_  _y_  _h_ 2 _c_  _h_ _f_

_Y_ _Core_ ( _y_  _hc_ )  _Y_ _Topface_ ( _y_  _hc_

_Core_

 _c_ )  _Y_ _Topface_ ( _y_ 

_c_ )  _Y_ _Topface_ ( _y_  _c_

( _y_  _c_ )  _Y_ _Topface_ ( _y_  _c_ )
2 2

_Y_ _Core_ ( _y_   _hc_ )  _Y_ _Bottomface_ ( _y_   _hc_

_Core_

  _c_ )  _Y_ _Bottomface_ ( _y_  

_c_ )  _Y_ _Bottomface_ ( _y_   _c_

( _y_   _c_ )  _Y_ _Bottomface_ ( _y_   _c_ )
2 2

The displacement fields (Equation 1) change as following



_ut_  _ywt_ '

_t_ _t_

_uc_  1 _y_  ( _ut_  _ub_  _hf_ _wt_ '  _hf_ _wb_ '  2 _uc_ ) 2 _y_ 22

1 _y_  ( _ut_  _ub_  _hf_ _wt_ '  _hf_ _wb_ '  2 _uc_ ) 2 _y_ 22

_c_  1 _y_  ( _ut_  _ub_  _f_ _wt_ '  _f_ _wb_ '  2 _uc_

_hc_

 1 _y_  ( _ut_  _ub_  _f_ _wt_ '  _f_ _wb_ ' 

 1

( _ut_  _ub_  _h_ 2 _f_ _wt_ '  _h_ 2 _f_ _wb_ '  2 _uc_ ) 2 _h_

 _hc_  _y_  _hc_

2 2

Tope face

Core

Bottom Face



















( _ut_  _ub_  _h_ 2 _f_ _wt_ '  _h_ 2 _f_ _wb_ '  _hc_  1) 4 _h_

_ut_  _ub_  _hf_ _wt_ '  _hf_ _wb_ '  _hc_  1) 4 _y_ 33

_c_

_t_  _ub_  _f_ _wt_ '  _f_ _wb_ '  _hc_

 ( _ut_  _ub_  _f_ _wt_ '  _f_ _wb_ ' 

 1

_hc_

_t_ '  _f_ _wb_ '  _hc_  1

_c_



 2 _hc_  _h_ _f_  _y_  _h_ 2 _c_

_ub_  _ywb_ '

_b_ _b_

Latin American Journal of Solids and Structures 13 (2016) 296-313

_wt_

_w_

_h_ 2 _c_  _y_  _h_ 2 _c_  _h_ _f_

 _hc_  _y_  _hc_

2 2

 2 _hc_  _h_ _f_  _y_  _h_ 2 _c_

Tope face

Core

Bottom Face





 










_y_ 2 _y_
_wc_  ( _wt_  _wb_ ) _hc_  ( _wt_  _wb_  2 _wc_ ) _h_ 2

_c_ _t_ _b_ _hc_ _t_ _b_ _c_

_y_

 ( _wt_  _wb_ ) _h_  ( _wt_  _wb_ 

_y_ 2 _y_ 2

( _wt_  _wb_ ) _h_  ( _wt_  _wb_  2 _wc_ ) 2

_hc_

_c_

_wb_

_w_





Because the hub is rigid therefore the total potential energy is just for sandwich panel

1
U  {  _x_  _xdVt_  (  _y_  _y_  _xy_  _xy_ ) _dVc_   _x_  _xdVb_ }
#   
2 _v_ _v_ _v_

 _x_  _xdVt_  (  _y_  _y_  _xy_  _xy_ ) _dVc_   _x_  _xdV_
#  

_x_ _x_ _t_ _y_ _y_ _xy_ _xy_ _c_
#  

_vt_ _vc_

_y_ _y_ _xy_ _xy_ _c_ _x_ _x_ _b_
#  

_vc_ _vb_

_vt_ _vc_ _vb_

U  1 { _E_ _f_  _x_ 2 _dVt_  ( _Ec_  _y_ 2  _Gc_  _xy_ 2 ) _dVc_  _E_ _f_  _x_ 2 _dVb_ }
#   
2 _v_ _v_ _v_

(4)

 1 { _E_ _f_  _x_ 2 _dVt_  ( _Ec_  _y_ 2  _Gc_  _xy_ 2 ) _dVc_  _E_ _f_  _x_ 2
#   

_E_ _f_  _x_ 2 _dVt_  ( _Ec_  _y_ 2  _Gc_  _xy_ 2 ) _dVc_  _E_ _f_  _x_ 2 _dV_
#  

_f_ _x_ _t_ _c_ _y_ _c_ _xy_ _c_
#  

_vt_ _vc_

_c_ _y_ _c_ _xy_ _c_ _f_ _x_ _b_
#  

_vc_ _vb_

_vt_ _vc_ _vb_

Here ��, �� are modulus of elasticity of face sheets and the core respectively and �� is the shear
modulus of the core. By substituting the strain displacement relations (3) in potential energy function
(4)

_l_

2 2
'  _yw_ ' ) _dA_  _E_ ( _u_ '  _yw_ ' )

_l_

1 { _E_ _f_ ( _ut_ '  _ywt_ ' )2 _dAt_  _E_ _f_ ( _ub_ '  _ywb_ ' )
#   
2 0 _A_ _A_

{ _E_ _f_ ( _ut_ '  _ywt_ ' ) _dAt_ 
#   

 1 { _E_ _f_ ( _ut_ '  _ywt_ ' )2 _dAt_  _E_ _f_ ( _ub_ ' 
#   

_U_  { _E_ _f_ ( _ut_ '  _ywt_ ' ) _dAt_  _E_ _f_ ( _ub_ '  _ywb_ ' ) _dAb_
#   

_f_ _t_ _t_

_f_ _b_ _b_ _b_
# 

_Ab_

_f_ _t_ _t_ _t_ _f_ _b_ _b_
#  

_At_ _Ab_

_t_ _Ab_

# 

[( _wt_  _wb_ ) 1  ( _wt_  _wb_  2 _wc_ ) 4 _y_ 2 ]2 _dAc_ }

1

 _Ec_ [( _wt_  _wb_ )  ( _wt_  _wb_ 
# 

_hc_ 2 ]2 _dAc_

2 ]2

#  c t b t b c 2 c

_Ac_ _hc_ _hc_

_c_ _t_ _b_

_c_

1 4 _y_

_Ec_ [( _wt_  _wb_ ) _hc_  ( _wt_  _wb_  2 _wc_ ) _hc_ 2

#  1  l Gc [(  1  cwc ' )  ( ut  ub  2 h f  hc wt '  2 h f  hc wb '  2 uc ) 4 y 2 ] dAc (5)

2 4 4 _h_

_l_

#  Gc [(  1  wc ' )  ( ut  ub  f c wt '  f c wb '  2 uc ) 4 2

_Ac_ 4 4 _hc_

_Gc_ [(  1  _wc_ ' )  ( _ut_  _ub_  2 _h_ _f_  _hc_ _wt_ '  2 _h_ _f_  _hc_ _wb_ '  2 _uc_ ) 4 _y_ 2

# 1  l Gc [(  1  wc ' )  ( ut  ub  2 h f  hc wt '  2 h f  hc wb '  2 uc ) 4 y 2 ] dAc

2 0 _A_ 4 4 _hc_

#  l Gc [(  1  wc ' )  ( ut  ub  2 h f  hc wt '  2 h f  hc wb '  2 uc ) 4 y 2 ]

0 _Ac_ 4 4 _hc_

_Ac_ _c_ 1 _c_ _t_ _b_ 4 _t_ 4 _b_ _c_ _hc_

 1 ( _ut_  _ub_  3 _h_ _f_  _hc_ _wt_ '  3 _h_ _f_  _hc_ _wb_ '  _hc_  1 
# 

2 6 6

# 1  ( ut  ub  3 h f  hc wt '  3 h f  hc wb '  hc  1  hc wc ' ) 12 y 3 2 ]2

2 _A_ 6 6 3 _hc_

_t_ '  3 _h_ _f_  _hc_ _wb_ '  _hc_  1  _hc_ _wc_ ' ) 12 _y_ 3 2

_Ac_

( _ut_  _ub_  3 _h_ _f_ 6  _hc_ _wt_ '  3 _h_ _f_ 6  _hc_ _wb_ '  _hc_  1  _h_ 3 _c_ _wc_ ' ) 12 _hcy_ 3 2 ]2 _dAc_ }

_t_  _ub_  _f_ 6 _c_ _wt_ '  _f_ 6 _c_ _wb_ '  _hc_  1  _h_ 3 _c_ _wc_ ' ) 12 _hcy_ 3 ]2 _dAc_ } _dx_

The kinetic energy for flexible satellite is as

. 2  _Y_ . 2 ) _dV_   ( _X_ . 2  _Y_ . 2 ) _dV_   ( _X_ . 2  _Y_ . 2 ) _dV_ }  1 _J_  .2

1 {  _f_ ( _X_ . 2  _Y_ . 2 ) _dVt_   _f_ ( _X_ . 2  _Y_ . 2 ) _dVb_   _c_ ( _X_ . 2  _Y_ . 2 ) _dVc_ }  1 _J_ _hub_  .2}
#   
2 _V_ _V_ _V_ 2

. . . .

 _f_ ( _X_  _Y_ ) _dVt_   _f_ ( _X_  _Y_ ) _dVb_ 
#   

1

 {  _f_ ( _X_  _Y_ ) _dVt_   _f_ ( _X_  _Y_ ) _dVb_   _c_ ( _X_  _Y_ ) _dVc_ } 
#   

 _f_ ( _X_  _Y_ ) _dVt_   _f_ ( _X_  _Y_ ) _dVb_   _c_ ( _X_  _Y_ ) _dVc_ }  1 _J_ _hub_  2
#  

_T_  {  _f_ ( _X_  _Y_ ) _dVt_   _f_ ( _X_  _Y_ ) _dVb_   _c_ ( _X_  _Y_ ) _dVc_ }  _J_
#   

_f_ _t_ _f_
#  

_Vt_ _Vb_

_f_

_f_ _b_ _c_
#  

_Vb_ _Vc_

_c_ _c_ _hub_
# 

_Vc_ 2

_t_ _Vb_ _Vc_

. 2 . . . 2 . .

 ( _y_  ( _x_  _r_ )  )2 ) _dV_   ( _x_  ( _y_  ( _x_  _r_ )  )2

1 {  _f_ ( _x_ . _t_ 2  ( _y_ . _t_  ( _x_  _r_ )  . )2 ) _dVt_   _f_ ( _x_ . _b_ 2  ( _y_ . _b_  ( _x_  _r_ )  . )2 )
#  
2 _V_ _V_

. . .

 _f_ ( _xt_  ( _y_ _t_  ( _x_  _r_ )  )2 ) _dVt_ 
#  

 1 {  _f_ ( _xt_  ( _y_ _t_  ( _x_  _r_ )  )2 ) _dVt_   _f_ ( _xb_  ( _y_ _b_  ( _x_ 
#  

 _f_ ( _xt_  ( _y_ _t_  ( _x_  _r_ )  )2 ) _dVt_   _f_ ( _xb_  ( _y_ _b_  ( _x_  _r_ )  )
# 

_T_  {  _f_ ( _xt_  ( _y_ _t_  ( _x_  _r_ )  )2 ) _dVt_   _f_ ( _xb_  ( _y_ _b_  ( _x_  _r_ )  )2 ) _dV_
#  

_f_ _t_ _t_ _t_ _f_ _b_ _b_
#  

_Vt_ _Vb_

_f_ _t_ _t_

_f_ _b_ _b_ _b_
# 

_Vb_

(6)

_t_ _Vb_

# 

. 2  ( _y_ .  ( _x_  _r_ )  . )2 ) _dV_ }  1 _J_  .2

  _c_ ( _xc_  ( _y_ _c_  ( _x_  _r_ )  )2 ) _dVc_ } 
# 

 _c_ ( _xc_  ( _y_ _c_  ( _x_  _r_ )  )2 ) _dVc_ }  1 _J_ _hub_  2

_xc_  ( _y_ _c_  ( _x_  _r_ )  )2 ) _dVc_ }  _J_

_c_ _c_ _c_ _c_ _hub_
# 

_Vc_ 2

_c_ _c_ _c_

_c_

( _x_ . _c_ 2  ( _y_ . _c_  ( _x_  _r_ )  . )2 ) _dVc_ }  12

Latin American Journal of Solids and Structures 13 (2016) 296-313

Here �� is the face sheets density, �� is the core density, r is the hub radius, and ���� is moment
inertia of the rigid hub. The energy of external applied torque is as

_F_  _M_ .  (7)

Here M is applied external Torque on the hub. By substituting the relation for displacement into
energy function and using the Hamilton principle the governing partial differential equations of motion
and boundary conditions are being obtained.

 _m_ _f_ ( _h_ _f_  _hc_ ) _w_ ..  _t_  _mc_ ( 1 ( _u_ .. _t_  _u_ .. _b_ )  1 ( _u_ .. _t_  _u_ .. _b_ )  1 _u_ .. _c_  1 _hc_ (  .. 

_m_ _f_ ( _u_ .. _t_ )  _m_ _f_ ( _h_ _f_  _hc_ ) _w_ ..  _t_  _mc_ ( 1 ( _u_ .. _t_  _u_ .. _b_ )  1 ( _u_ .. _t_  _u_ .. _b_ )  1 _u_ .. _c_  1 _hc_

(  )

( _u_ .. _t_ )  _m_ _f_ ( _h_ _f_ 2  _hc_ ) _w_ ..  _t_  _mc_ (201 ( _u_ .. _t_  _u_ .. _b_ )  281 ( _u_ .. _t_  _u_ .. _b_ )  151 _u_ .. _c_  701 _hc_ (  ..  .. )

.. _h_  _h_ .. 1 .. .. 1 .. .. 1 .. 1 .. ..

_f_ ( _u_ _t_ )  _m_ _f_ ( _f_ _c_ ) _w_  _t_  _mc_ ( 1 ( _u_ _t_  _u_ _b_ )  1 ( _u_ _t_  _u_ _b_ )  1 _u_ _c_  1 _hc_

 1 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  1 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  _E_ _f_ _Af_ _ut_ ''  _E_ _f_ _Af_ ( _h_ _f_  _hc_ ) _wt_ '''  _Gc_ 2 _Ac_ (4 ( _ut_ 

_h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  1 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  _E_ _f_ _Af_ _ut_ ''  _E_ _f_ _Af_ ( _h_ _f_  _hc_ ) _wt_ '''  _Gc_ 2 _Ac_

.. ..

'  _w_ '

.. ..

'  _w_ '

.. _b_ ' )  _E_ _f_ _Af_ _ut_ ''  _E_ _f_ _Af_ ( _h_ _f_ 2  _hc_ ) _wt_ '''  _Ghc_ 2 _A_

_hcc_ 2 _c_ (3 ( _ut_  _u_

401 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  561 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  _E_ _f_ _Af_ _ut_ ''  _E_ _f_ _Af_ ( _h_ _f_ 2  _hc_ ) _wt_ '''  _Ghcc_ 2 _Ac_ (43 ( _ut_  _ub_ )

_f_ ( _wt_ '  _wb_ ' )  1 _h_ _f_ ( _wt_ '  _wb_ ' )  _E_ _f_ _Af_ _ut_ ''  _E_ _f_ _Af_ ( _f_ _c_ ) _wt_ '''  _Gc_ 2 _Ac_

..' )  1 _h_ ( _w_ .. '

_c_ 2 3 _t_ _b_

95 ( _ut_  _ub_ )  83 _uc_  54 _hc_  (23 _h_ _f_  13 _hc_ )( _wt_ '  _wb_ ' )  (109 _h_ _f_  103 _hc_ )( _wt_ '  _wb_ ' )  52 _hcwc_ ' )  0

 9 ( _ut_  _ub_ )  8 _uc_  4 _hc_  (2 _h_ _f_  1 _hc_ )( _wt_ '  _wb_ ' )  ( 9 _h_ _f_  3 _hc_ )( _wt_ '  _wb_ ' )  2 _hcwc_ ' ) 



'  _w_ ' )  ( _h_  _h_ )( _w_ '  _w_ ' )  _h_ _w_ '

( _ut_  _ub_ )  _uc_  _hc_  ( _h_ _f_  _hc_ )( _wt_ '  _wb_ ' )  ( _h_ _f_  _hc_ )( _wt_ '  _wb_ ' )  _hcwc_ '

_t_ _b_ _c_ _c_ _f_ _c_ _t_ _b_ _f_ _c_ _t_ _b_ _c_ _c_

_h_ _f_  _hc_ .. 1 .. .. 1 .. .. 1 .. 1 ..

 _m_ _f_ ( ) _w_  _b_  _mc_ ( ( _u_ _t_  _u_ _b_ )  ( _u_ _t_  _u_ _b_ )  _u_ _c_  _hc_ ( 

.. _h_ _f_  _hc_ .. 1 .. .. 1 .. .. 1 .. 1

_m_ _f_ ( _u_ _b_ )  _m_ _f_ ( ) _w_  _b_  _mc_ ( ( _u_ _t_  _u_ _b_ )  ( _u_ _t_  _u_ _b_ )  _u_ _c_  _hc_

 )

.. _h_ _f_  _hc_ .. 1 .. .. 1 .. .. 1 .. 1 .. ..

( _u_ _b_ )  _m_ _f_ ( 2 ) _w_  _b_  _mc_ (20 ( _u_ _t_  _u_ _b_ )  28 ( _u_ _t_  _u_ _b_ )  15 _u_ _c_  70 _hc_ (  )

.. _h_  _h_ .. 1 .. .. 1 .. .. 1 .. 1 .. ..

_f_ _c_ 1 1 1 1
_f_ ( _u_ _b_ )  _m_ _f_ ( ) _w_  _b_  _mc_ ( ( _u_ _t_  _u_ _b_ )  ( _u_ _t_  _u_ _b_ )  _u_ _c_  _hc_

 1 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  1 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' ))  _E_ _f_ _Af_ _ub_ ''  _E_ _f_ _Af_ ( _h_ _f_  _hc_ ) _wb_ '''  _Gc_ 2 _Ac_ (4 ( _ut_ 

_h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  1 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' ))  _E_ _f_ _Af_ _ub_ ''  _E_ _f_ _Af_ ( _h_ _f_  _hc_ ) _wb_ '''  _Gc_ 2 _Ac_

.. ..

'  _w_ '

.. ..

'  _w_ '

.. _b_ ' ))  _E_ _f_ _Af_ _ub_ ''  _E_ _f_ _Af_ ( _h_ _f_ 2  _hc_ ) _wb_ '''  _Ghc_ 2 _A_

_hc_ _c_ 2 _c_ (3 ( _ut_  _u_

401 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' )  561 _h_ _f_ ( _w_ .. _t_ '  _w_ .. _b_ ' ))  _E_ _f_ _Af_ _ub_ ''  _E_ _f_ _Af_ ( _h_ _f_ 2  _hc_ ) _wb_ '''  _Ghc_ _c_ 2 _Ac_ (43 ( _ut_  _ub_ )

_f_ ( _wt_ '  _wb_ ' )  1 _h_ _f_ ( _wt_ '  _wb_ ' ))  _E_ _f_ _Af_ _ub_ ''  _E_ _f_ _Af_ ( _f_ _c_ ) _wb_ '''  _Gc_ 2 _Ac_

..' )  1 _h_ ( _w_ .. '

_c_ 2 3 _t_ _b_

95 ( _ut_  _ub_ )  83 _uc_  54 _hc_  (23 _h_ _f_  13 _hc_ )( _wt_ '  _wb_ ' )  (109 _h_ _f_  103 _hc_ )( _wt_ '  _wb_ ' )  52 _hc_ _wc_ ' )  0

(8)

(9)

(10)

 9 ( _ut_  _ub_ )  8 _uc_  4 _hc_  (2 _h_ _f_  1 _hc_ )( _wt_ '  _wb_ ' )  ( 9 _h_ _f_  3 _hc_ )( _wt_ '  _wb_ ' )  2 _hc_ _wc_ ' ) 



'  _w_ ' )  ( _h_  _h_ )( _w_ '  _w_ ' )  _h_ _w_ '

( _ut_  _ub_ )  _uc_  _hc_  ( _h_ _f_  _hc_ )( _wt_ '  _wb_ ' )  ( _h_ _f_  _hc_ )( _wt_ '  _wb_ ' )  _hc_ _wc_ '

_t_ _b_ _c_ _c_ _f_ _c_ _t_ _b_ _f_ _c_ _t_ _b_ _c_ _c_

 ( _hc_  _h_ _f_ ) _u_ .. _t_  ( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _w_ .. _t_  ( _r_  _l_ )  .. )  _mc_ (  1 _h_ _f_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  1 _h_ _f_ ( _u_ .. _t_ ' 

..  ( _hc_  _h_ _f_ ) _u_ ..  ( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _w_ ..  ( _r_  _l_ )  .. )  _m_ (  1 _h_ ( _u_ ..'  _u_ ..' )  1 _h_ ( _u_ ..'  _u_ ..'

_m_ _f_ ( _w_ .. _t_  ( _hc_  _h_ _f_ ) _u_ .. _t_  ( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _w_ .. _t_  ( _r_  _l_ )  .. )  _mc_ (  1 _h_ _f_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  1 _h_ _f_ ( _u_ .. _t_ '  _u_ ..

_f_ ( _w_ .. _t_  ( _hc_  2 _h_ _f_ ) _u_ .. _t_  ( _h_ 4 _c_ 2  _h hc_ 2 _f_  _h_ 32 _f_ ) _w_ .. _t_  ( _r_  2 _l_ )  .. )  _mc_ (  401 _h_ _f_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  561 _h_ _f_ ( _u_ .. _t_ '  _u_ .. _b_ ' )

301 _h_ _f_ _u_ .. _c_ '  1401 _h_ _f_ _hc_  .. '  801 _h_ _f_ 2 ( _w_ .. _t_ ''  _w_ .. _b_ '' )  121 ( _w_ .. _t_  _w_ .. _b_ )  1121 _h_ _f_ 2 ( _w_ .. _t_ ''  _w_ .. _b_ '' )  201 ( _w_ .. _t_  _w_ .. _b_ )

_f_ ( _wt_  ( _c_ _f_ ) _ut_  ( _hc_  _c_ _f_  _f_ ) _wt_  ( _r_  _l_ )  )  _mc_ (  1 _h_ _f_ ( _ut_ '  _ub_ ' )  1 _h_ _f_ ( _ut_ '  _ub_ '

 )

..'  1 _h_ _h_  .. '  1 _h_ 2 ( _w_ .. ''  _w_ ..'' )  1 ( _w_ ..  _w_ .. )  1 _h_ 2 ( _w_ .. ''  _w_ ..'' )  1 ( _w_ ..  _w_ ..

 1 _h_ _f_ _uc_ '  1 _h_ _f_ _hc_  '  1 _h_ _f_ 2 ( _wt_ ''  _wb_ '' )  1 ( _wt_  _wb_ )  1 _h_ _f_ 2 ( _wt_ ''  _wb_ '' )  1 ( _wt_ 

 '

_h_ _f_ _uc_ '  _h_ _f_ _hc_  '  _h_ _f_ 2 ( _wt_ ''  _wb_ '' )  ( _wt_  _wb_ )  _h_ _f_ 2 ( _wt_ ''  _wb_ '' )  ( _wt_  _wb_

_f_ _c_ _f_ _c_ _f_ _t_ _b_ _t_ _b_ _f_ _t_ _b_ _t_ _b_

_w_ .. _c_  ( 1  1 )( _r_  _l_ )  .. )  _E_ _f_ _Af_ [( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _wt_ ''''  ( _hc_  _h_ _f_ ) _ut_

..  ( 1  1 )( _r_  _l_ )  .. )  _E_ _A_ [( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _w_ ''''  ( _hc_  _h_ _f_ ) _u_ '''

 ( 1  1 )( _r_  _l_ )  .. )  _E_ _f_ _Af_ [( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _wt_ ''''  ( _hc_ 



1
15

15 _w_ .. _c_  (101  151 )( _r_  2 _l_ )  .. )  _E_ _f_ _Af_ [( _h_ 4 _c_ 2  _h hc_ 2 _f_  _h_ 32 _f_ ) _wt_ ''''  ( _hc_  2 _h_ _f_ ) _ut_ ''']

_GAhc_ 2 _c_ [  ( 23 _h_ _f_  13 _hc_ )( _ut_ '  _ub_ ' )  (109 _h_ _f_  103 _hc_ )( _ut_ '  _ub_ ' )  ( 43 _h_ _f_  23 _hc_ ) _uc_ '

_c_  ( 1  1 )( _r_  _l_ )  )  _E_ _f_ _Af_ [( _hc_  _c_ _f_  _f_ ) _wt_ ''''  ( _c_ _f_ ) _ut_

 )

_GAhc_ 2 _c_ [  ( 23 _h_ _f_  13 _hc_ )( _ut_ '  _ub_ ' )  (109 _h_ _f_  103 _hc_ )( _ut_ '  _ub_ ' )  ( 43 _h_ _f_  23 _hc_ ) _u_

 _GA_ 2 _c_ [  ( 2 _h_ _f_  1 _hc_ )( _ut_ '  _ub_ ' )  ( 9 _h_ _f_  3 _hc_ )( _ut_ '  _ub_ ' )  ( 4 _h_ _f_ 

2 _c_ [  ( 3 _h_ _f_  3 _hc_ )( _ut_ '  _ub_ ' )  (10 _h_ _f_  10 _hc_ )( _ut_ '  _ub_ ' )  ( 3 _h_ _f_  3 _hc_ ) _uc_ '

_c_

_hc_ 2 3 _f_ 3 _c_ _t_ _b_ 10 _f_ 10 _c_ _t_ _b_ 3 _f_ 3 _c_ _c_

 ( 52 _h_ _f_ _hc_  152 _hc_ 2 )(  ')  (13 _h_ _f_ _hc_  13 _h_ _f_ 2  121 _hc_ 2 )( _wt_ ''  _wb_ '' )  ( 209 _h_ _f_ 2  130 _h_ _f_ _hc_  201 _hc_ 2 )

( _wt_ ''  _wb_ '' )  (15 _h_ _f_ _hc_  151 _hc_ 2 ) _wc_ '' ]  _EhccA_ 2 _c_ [ 34 ( _wt_  _wb_ )  ( _wt_  _wb_ )  83 _wc_ ]  0

( 52 _h_ _f_ _hc_  152 _hc_ 2 )(  ')  (13 _h_ _f_ _hc_  13 _h_ _f_ 2  121 _hc_ 2 )( _wt_ ''  _wb_ '' )  ( 209 _h_ _f_ 2  130

 '

 ( 2 _h_ _f_ _hc_  2 _hc_ 2 )(  ')  (1 _h_ _f_ _hc_  1 _h_ _f_ 2  1 _hc_ 2 )( _wt_ ''  _wb_ '' )  ( 9 _h_ _f_ 2  3 _h_ _f_ _hc_  1 _hc_ 2

2 )(  ')  ( _h_ _h_  _h_ 2  _h_ 2 )( _w_ ''  _w_ '' )  ( _h_ 2

_h_ _f_ _hc_  _hc_



_h_ _f_ _hc_  _hc_ 2 )(  ')  ( _h_ _f_ _hc_  _h_ _f_ 2  _hc_ 2 )( _wt_ ''  _wb_ '' )  ( _h_ _f_

_f_ _c_ _c_ _f_ _c_ _f_ _c_ _t_ _b_ _f_

_f_ _c_ _c_

_wt_ ''  _wb_ '' )  (15 _h_ _f_ _hc_  151 _hc_ 2 ) _wc_ '' ]  _EhccA_ 2 _c_ [ 34 ( _wt_  _wb_ )  ( _wt_  _wb_ )  83 _wc_

 _wb_ '' )  (1 _h_ _f_ _hc_  1 _hc_ 2 ) _wc_ '' ]  _Ec_ _A_ 2 _c_ [ 4 ( _wt_  _wb_ )  ( _wt_  _wb_ )  8 _wc_ ] 

''  _w_ '' )  ( _h_ _h_  _h_ 2 ) _w_ ''

_t_ ''  _wb_ '' )  (5 _h_ _f_ _hc_  15 _hc_ 2 ) _wc_ '' ]  _hcc_ 2 _c_ [ 3 ( _wt_  _wb_ )  ( _wt_  _wb_ )  3 _wc_

Latin American Journal of Solids and Structures 13 (2016) 296-313

 ( _hc_  _hf_ ) _u_ .. _b_   ( _hc_ 2  _h hc_ _f_ _h_ 2 _f_ ) _w_ .. _b_  ( _r_  _l_ )  .. )  _mc_ ( 1 _hf_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  1 _hf_ ( _u_ .. _t_ '  _u_ .. _b_ ' ) 

_w_ ..  ( _hc_  _hf_ ) _u_ ..   ( _hc_ 2  _h hc_ _f_ _h_ 2 _f_ ) _w_ ..  ( _r_  _l_ )  .. )  _m_ ( 1 _h_ ( _u_ ..'  _u_ ..' )  1 _h_ ( _u_ ..'  _u_ ..' )  1 _h_ _u_ ..'

_m_ _f_ ( _w_ .. _b_  ( _hc_  _hf_ ) _u_ .. _b_   ( _hc_ 2  _h hc_ _f_ _h_ 2 _f_ ) _w_ .. _b_  ( _r_  _l_ )  .. )  _mc_ ( 1 _hf_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  1 _hf_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  1 _hf_ _u_ .. _c_ '

_f_ ( _w_ .. _b_  ( _hc_  2 _hf_ ) _u_ .. _b_   ( _h_ 4 _c_ 2  _h hc_ 2 _f_ _h_ 32 _f_ ) _w_ .. _b_  ( _r_  2 _l_ )  .. )  _mc_ (401 _hf_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  561 _hf_ ( _u_ .. _t_ '  _u_ .. _b_ ' )  301 _hf_ _u_ .. _c_ ' )

1401 _hf_ _hc_  .. '  801 _hf_ 2 ( _w_ .. _t_ ''  _w_ .. _b_ '' )  121 ( _w_ .. _t_  _w_ .. _b_ )  1121 _hf_ 2 ( _w_ .. _t_ ''  _w_ .. _b_ '' )  201 ( _w_ .. _t_  _w_ .. _b_ ) 

_f_ ( _wb_  ( _c_ _f_ ) _ub_   ( _hc_  _c_ _f_ _f_ ) _wb_  ( _r_  _l_ )  )  _mc_ ( 1 _hf_ ( _ut_ '  _ub_ ' )  1 _hf_ ( _ut_ '  _ub_ ' )  1 _hf_ _uc_ '

 )

.. '  1 _h_ 2 ( _w_ .. ''  _w_ ..'' )  1 ( _w_ ..  _w_ .. )  1 _h_ 2 ( _w_ .. ''  _w_ ..'' )  1 ( _w_ ..  _w_ ..

 '

 1 _hf_ _hc_  '  1 _hf_ 2 ( _wt_ ''  _wb_ '' )  1 ( _wt_  _wb_ )  1 _hf_ 2 ( _wt_ ''  _wb_ '' )  1 ( _wt_  _wb_ ) 

_hf_ _hc_  '  _hf_ 2 ( _wt_ ''  _wb_ '' )  ( _wt_  _wb_ )  _hf_ 2 ( _wt_ ''  _wb_ '' )  ( _wt_  _wb_

_f_ _c_ _f_ _t_ _b_ _t_ _b_ _f_ _t_ _b_ _t_ _b_

_w_ .. _c_  ( 1  1 )( _r_  _l_ )  .. )  _E_ _f_ _Af_ [( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _wb_ ''''  ( _hc_  _hf_ ) _ub_

1
15

..  ( 1  1 )( _r_  _l_ )  .. )  _E_ _A_ [( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _w_ ''''  ( _hc_  _hf_ ) _u_ '''

 ( 1  1 )( _r_  _l_ )  .. )  _E_ _f_ _Af_ [( _hc_ 2  _h hc_ _f_  _h_ 2 _f_ ) _wb_ ''''  ( _hc_ 

 (101  151 )( _r_  2 _l_ )  .. )  _E_ _f_ _Af_ [( _h_ 4 _c_ 2  _h hc_ 2 _f_  _h_ 32 _f_ ) _wb_ ''''  ( _hc_  2 _hf_ ) _ub_ ''']

[(23 _hf_  13 _hc_ )( _ut_ '  _ub_ ' )  (109 _hf_  103 _hc_ )( _ut_ '  _ub_ ' )  ( 43 _hf_  23 _hc_ ) _uc_ '

_wc_  ( 1  1 )( _r_  _l_ )  )  _E_ _f_ _Af_ [( _hc_  _c_ _f_  _f_ ) _wb_ ''''  ( _c_ _f_ ) _ub_

)  )

_GAhc_ 2 _c_ [(23 _hf_  13 _hc_ )( _ut_ '  _ub_ ' )  (109 _hf_  103 _hc_ )( _ut_ '  _ub_ ' )  ( 43 _hf_  23 _hc_ ) _uc_

 _GA_ 2 _c_ [(2 _hf_  1 _hc_ )( _ut_ '  _ub_ ' )  ( 9 _hf_  3 _hc_ )( _ut_ '  _ub_ ' )  ( 4 _hf_ 

2 _c_ [(3 _hf_  3 _hc_ )( _ut_ '  _ub_ ' )  (10 _hf_  10 _hc_ )( _ut_ '  _ub_ ' )  ( 3 _hf_  3 _hc_ ) _uc_ '

_c_

_hc_ 2 3 _f_ 3 _c_ _t_ _b_ 10 _f_ 10 _c_ _t_ _b_ 3 _f_ 3 _c_ _c_

(52 _hf_ _hc_  152 _hc_ 2 )(  '')  (13 _hf_ _hc_  13 _hf_ 2  121 _hc_ 2 )( _wt_ ''  _wb_ '' )  ( 209 _hf_ 2  130

(52 _hf_ _hc_  152 _hc_ 2 )(  '')  (13 _hf_ _hc_  13 _hf_ 2  121 _hc_ 2 )( _wt_ ''  _wb_ '' )  ( 209 _hf_ 2  130 _hf_ _hc_  201 _hc_ 2 )( _wt_ ''  _wb_ '' )

(15 _hf_ _hc_  151 _hc_ 2 ) _wc_ '' ]  _EhccA_ 2 _c_ [34 ( _wt_  _wb_ )  ( _wt_  _wb_ )  83 _wc_ ]  0

)(  ''

 ( 2 _hf_ _hc_  2 _hc_ 2 )(  '')  (1 _hf_ _hc_  1 _hf_ 2  1 _hc_ 2 )( _wt_ ''  _wb_ '' )  ( 9 _hf_ 2  3 _hf_ _hc_  1 _hc_ 2 )( _wt_ ''  _wb_ ''

2 )(  '')  ( _h_ _h_  _h_ 2  _h_ 2 )( _w_ ''  _w_ '' )  ( _h_ 2

_hf_ _hc_  _hc_ 2 )( _wt_ ''  _wb_ ''

 1 _hc_ 2 )( _wt_ '' 

_hf_ _hc_  _hc_ 2 )(  '')  ( _hf_ _hc_  _hf_ 2  _hc_ 2 )( _wt_ ''  _wb_ '' )  ( _hf_

_f_ _c_ _c_ _f_ _c_ _f_ _c_ _t_ _b_ _f_

_f_ _c_ _c_ _t_ _b_

_hf_ _hc_  151 _hc_ 2 ) _wc_ '' ]  _EhccA_ 2 _c_ [34 ( _wt_  _wb_ )  ( _wt_  _wb_ )  83 _wc_

 (1 _hf_ _hc_  1 _hc_ 2 ) _wc_ '' ]  _Ec_ _A_ 2 _c_ [4 ( _wt_  _wb_ )  ( _wt_  _wb_ )  8 _wc_ ] 

2 ) _w_ ''

_f_ _hc_  15 _hc_ 2 ) _wc_ '' ]  _hcc_ 2 _c_ [3 ( _wt_  _wb_ )  ( _wt_  _wb_ )  3 _wc_

8 .. 1 .. .. 1 ' '
(15 _u_ _c_  15 ( _ub_  _ut_ )  30 _h_ _f_ ( _wt_  _wb_ )

..  1 ( _u_ ..  _u_ .. )  1 _h_ ( _w_ '  _w_ '

 1 ( _ub_  _ut_ )  1 _h_ _f_ ( _wt_ ' 

_mc_ ( _u_ _c_  ( _ub_  _ut_ )  _h_ _f_ ( _wt_  _w_

_c_ ( _u_ _c_  ( _ub_  _ut_ )  _h_ _f_ ( _wt_  _wb_

_Gc_ _Ac_

 _Gc_ _A_ 2 _c_ [  8 ( _ut_  _ub_ )  16 _uc_  ( 4 _h_ _f_  2 _hc_ )( _wt_ '  _wb_ ' )] 

2 _c_ [  3 ( _ut_  _ub_ )  3 _uc_  ( 3 _h_ _f_  3 _hc_ )( _wt_ '  _wb_ '

_c_ _c_

[  83 ( _ut_  _ub_ )  163 _uc_  ( 34 _h_ _f_  23 _hc_ )( _wt_ '  _wb_ ' )]  0

_hc_ _c_ 2 _c_ [  3 ( _ut_  _ub_ )  3 _uc_  ( 3 _h_ _f_  3 _hc_ )( _wt_ '  _wb_ '

_hc_ 2 3 _t_ _b_ 3 _c_ 3 _f_ 3 _c_ _t_ _b_

(1052 _hc_ 2 (  ..  .. )  701 _hc_ ( _u_ .. _t_  _u_ .. _b_ )  1401 _h_ _f_ _hc_ ( _w_ .. _t_ '  _w_ .. _b_ ' )

2 (  ..  .. )  1 _h_ ( _u_ ..  _u_ .. )  1 _h_ _h_ ( _w_ .. '

 )

 )  1 _hc_ ( _ut_  _ub_ )  1 _h_ _f_ _hc_ ( _wt_ ' 

.. ..

'  _w_ '

_mc_ ( _hc_ 2 (  )  _hc_ ( _ut_  _ub_ )  _h_ _f_ _hc_ ( _wt_ '  _wb_ '

_c_ _c_ _c_ _t_ _b_ _f_ _c_ _t_ _b_

_Gc_ _Ac_

 _Gc_ 2 _Ac_ [  4 _hc_ ( _ut_  _ub_ )  4 _hc_ 2  ( 2 _hc_ 2  2 _hch_ _f_ )( _wt_ '  _wb_ ' )  4 _hc_ 2 _wc_ ' ] 

2 _c_ [  5 _hc_ ( _ut_  _ub_ )  5 _hc_ 2  (15 _hc_ 2  5 _hch_ _f_ )( _wt_ '  _wb_ ' )  15 _hc_ 2 _wc_ '

_c_ _c_

[  54 _hc_ ( _ut_  _ub_ )  54 _hc_ 2  (152 _hc_ 2  52 _hch_ _f_ )( _wt_ '  _wb_ ' )  154 _hc_ 2 _wc_ ' ]  0



_hcc_ 2 _c_ [  5 _hc_ ( _ut_  _ub_ )  5 _hc_ 2  (15 _hc_ 2  5 _hch_ _f_ )( _wt_ '  _wb_ ' )  15 _hc_ 2 _wc_ '

_c_ 2 5 _c_ _t_ _b_ 5 _c_ 15 _c_ 5 _c_ _f_ _t_ _b_ 15 _c_ _c_

_mc_ ( 8 _w_ .. _c_  1 _hc_ ( _w_ .. _b_  _w_ .. _t_ )  ( 8  2 )( _r_  _l_ )  .. )  _Gc_ 2 _Ac_ [  2 _hc_ ( _ut_ '  _ub_ ' )  4 _hc_ 2  '

(158 _w_ .. _c_  151 _hc_ ( _w_ .. _b_  _w_ .. _t_ )  (158  152 )( _r_  2 _l_ )  .. )  _Ghcc_ 2 _Ac_ [  52 _hc_ ( _ut_ '  _ub_ ' )  154

.. 1 .. .. 8 2 _l_ ..

_hcc_ 2 _c_ [  5 _hc_ ( _ut_ '  _ub_ ' )  15 _hc_

2 _Ac_ [  52 _hc_ ( _ut_ '  _ub_ ' )  154 _hc_ 2  '

 1 _hc_ ( _wb_  _wt_ )  ( 8  2 )( _r_  _l_ )  )  _Gc_ 2 _Ac_ [  2 _hc_ ( _ut_ '  _ub_ ' ) 

_c_ ( _wc_  _hc_ ( _wb_  _wt_ )  (  )( _r_  )  )  _c_ 2 _c_

_c_ 2 5 _c_ _t_ _b_ 15 _c_

_h_ _f_ _hc_  1 _hc_ 2 )( _wt_ ''  _wb_ '' )  8 _hc_ 2 _wc_ '' ]  _Ec_ _A_ 2 _c_

(15 _h_ _f_ _hc_  151 _hc_ 2 )( _wt_ ''  _wb_ '' )  158 _hc_ 2 _wc_ '' ]  _Ehc_ _cA_ 2 _c_ [  83 ( _wt_  _wb_ )  163 _wc_ ]  0

 (1 _h_ _f_ _hc_  1 _hc_ 2 )( _wt_ ''  _wb_ '' )  8 _hc_ 2 _wc_ '' ]  _Ec_ _A_ 2 _c_ [  8 ( _wt_  _wb_ )  16 _wc_ ] 

_c_ 2 )( _wt_ ''  _wb_ '' )  15 _hc_ 2 _wc_ '' ]  _hc_ 2

_f_ _hc_  _hc_ 2 )( _wt_ ''  _wb_ '' )  _hc_ 2 _wc_ '' ]  _c_ 2 _c_

_hc_ _c_ 2 _c_ [  3 ( _wt_  _wb_ )  3 _wc_

_c_ 2 3 _t_ _b_ 3 _c_

.. 1 .. _m_ _l_ .. 1 .. ' 1 .. '

.. 1 .. _mc_ _l_ .. 1 .. ' 1 .. '

_mchc_ _u_ _t_  _mchc_ _u_ _b_  ( _m_ _f_  )( _r_  ) _wt_  _mchch_ _f_ _wt_  _mchch_ _f_ _wb_

1 .. 1 .. _mc_ _l_ .. 1 .. ' 1
70 _mchc_ _u_ _t_  70 _mchc_ _u_ _b_  ( _m_ _f_  6 )( _r_  2) _wt_  140 _mchch_ _f_ _wt_  140 _m_
2 _l_ .. 2 2 .. 2 2 .. 2 _l_ 2

 3 _mc_ ( _r_  2) _wc_  105 _mchc_  105 _mchc_  (2 _m_ _f_  _mc_ )( _r_  3  _rl_ )

1 _mc_ _l_ 1

 _mchc_ _u_ _b_  ( _m_ _f_  )( _r_  ) _wt_  _mchch_ _f_ _wt_ 

_c_
_chc_ _u_ _t_  _mchc_ _u_ _b_  ( _m_ _f_  )( _r_  ) _wt_  _mchch_ _f_ _wt_  _mchch_ _f_ _wb_

..  2 _m_ _h_ 2  ..  2 _m_ _h_ 2  ..  (2 _m_  _m_ )( _r_ 2  _l_ 2  _rl_ )  ..  _J_ ..

_l_ .. 2 2 .. 2 2 .. 2 _l_ 2 .. ..
_mc_ ( _r_  ) _wc_  _mchc_  _mchc_  (2 _m_ _f_  _mc_ )( _r_   _rl_ )  _J_ _h_  _M_

(11)

(12)

(13)

(14)

(15)

2 _l_ 2 2 2 2 2 _l_

 _mc_ ( _r_  ) _wc_  _mchc_  _mchc_  (2 _m_ _f_  _mc_ )( _r_   _rl_ )  _J_ _h_ 

 2 _mchc_ 2  (2 _m_ _f_  _mc_ )( _r_ 2  _l_  _rl_ )  _J_ _h_ 

_c_ ( _r_  ) _wc_  _mchc_  _mchc_  (2 _m_ _f_  _mc_ )( _r_   _rl_ )  _J_ _h_

The axial forces ��,���, ��, the shear forces ��,�,���, �� and the bending moments��,���, ��:

_Pt_   _E_ _f_ _A_ _f_ (  _uxt_  ( _hc_  2 _h_ _f_ )  2 _xw_ 2 _t_

_Pb_   _E_ _f_ _A_ _f_ (  _uxb_  ( _hc_  2 _h_ _f_ )  2 _xw_ 2 _b_

_hc_  _h_ _f_  2 _w_

( _t_  ( _c_ _f_ ) _t_ )

 _x_ 2  _x_ 2

_x_

(  _ub_  ( _hc_  _h_ _f_ )  2 _wb_ ) (16)

 _x_ 2  _x_ 2

_hc_  _h_ _f_  2 _w_

_x_

Latin American Journal of Solids and Structures 13 (2016) 296-313

 _m_ _f_ (( _h_ 2 _f_  _hc_ 2  _hf_ _hc_ ) _w_ .. _t_ '  ( _h_ _f_ 

2 _f_  _hc_ 2  _hf_ _hc_ ) _w_ .. '  ( _h_ _f_  _hc_ ) _u_ ..

_h_ 2 _f_ _hc_ 2 _hf_ _hc_ .. ' _h_ _f_  _hc_ ..
_Vt_  _m_ _f_ ((   ) _wt_  ( ) _ut_

_f_ _hc_ _f_ _c_ ' _f_ _c_
_t_  _m_ _f_ ((   ) _wt_  ( ) _ut_

_f_ _c_ _f_ _c_ ' _f_ _c_

(( 3  4  2 ) _wt_  ( 2 ) _ut_ )

_mc_ (  36 _h_ _f_ _u_ .. _t_  6 _h_ _f_ _u_ .. _b_  28 _h_ _f_ _u_ .. _c_  6 _hf_ _hc_  ..  3 _h_ 2 _f_ _w_ .. _b_ '  18 _h_ 2 _f_ _w_ .. _t_ '

 _mc_ (  36 _h_ _f_ _ut_  6 _h_ _f_ _ub_  28 _h_ _f_ _uc_  6 _hf_ _hc_  3 _h_ 2 _f_ _wb_ ' 

_c_

_f_ _t_ _f_ _b_ _f_ _c_ _f_ _c_ _f_ _b_ _f_ _t_

.. .. .. .. .. ..

 6 _h_ _u_  28 _h_ _u_  6 _h_ _h_  3 _h_ 2 _w_ '  18 _h_ 2 _w_ '



840 _c_ (  36 _h_ _f_ _ut_  6 _h_ _f_ _ub_  28 _h_ _f_ _uc_  6 _hf_ _hc_  3 _h_ 2 _f_ _wb_ '  18 _h_ 2 _f_ _wt_ ' )

 _E_ _f_ _Af_ (( _hc_ 2  _hf_ _hc_  _h_ 2 _f_ ) _wt_ '''  ( _hf_ 

_c_ 2  _hf_ _hc_  _h_ 2 _f_ ) _w_ '''  ( _hf_  _hc_ ) _u_ ''

_E_ _f_ _Af_ (( _hc_ 2  _hf_ _hc_  _h_ 2 _f_ ) _wt_ '''  ( _hf_  _hc_ ) _ut_

_f_ _Af_ (( _hc_  _f_ _c_  _f_ ) _wt_ '''  ( _f_ _c_ ) _ut_

(( 4 _c_  _f_ 2 _c_  3 _f_ ) _wt_ '''  ( _f_ 2 _c_ ) _ut_ '' )

840 _Gc_ _Ahcc_ 2 [(196 _hf_  28 _hc_ ) _ub_  (1316 _h_ _f_  532 _hc_ ) _ut_  (560 _hc_

 _Gc_ _Ac_ 2 [(196 _hf_  28 _hc_ ) _ub_  (1316 _h_ _f_  532 _hc_ ) _ut_  (560 _hc_ 

_f_ _c_

_c_ _c_

840 _c_ _hcc_ 2 [(196 _hf_  28 _hc_ ) _ub_  (1316 _h_ _f_  532 _hc_ ) _ut_  (560

_hc_ 2 _f_ _c_ _b_ _f_ _c_ _t_ _c_

1120 _h_ _f_ )

_h_ _f_ ) _u_

 (28 _hc_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wb_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2 

2  98 _h_ 2  28 _h_ _h_ ) _w_ ''  (112 _h_ 2  658 _h_ 2  532 _h_ 2  532 _h_ _h_ ) _w_ '

(28 _hc_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wb_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2  532 _h_ _f_ _hc_ )

_hc_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wb_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2  532 _h_ _f_ _hc_ ) _wt_ '

_c_ _f_ _f_ _c_ _b_ _c_ _f_ _f_ _f_ _c_ _t_

 (336 _hf_ _hc_  112 _hc_ 2 )  (56 _hc_ 2 

_hf_ _hc_  112 _hc_ 2 )  (56 _hc_ 2  168 _hf_ _hc_ ) _wc_ '

2 )  (56 _h_ 2  168 _h_ _h_ ) _w_ '

(336 _hf_ _hc_  112 _hc_ 2 )  (56 _hc_ 2  168 _hf_ _hc_ ) _wc_ ' ]

_f_ _c_ _c_ _c_ _f_ _c_ _c_

 _m_ _f_ (( _h_ 2 _f_  _hc_ 2  _hf_ _hc_ ) _w_ .. _b_ '  ( _h_ _f_ 

2 _f_  _hc_ 2  _hf_ _hc_ ) _w_ .. '  ( _h_ _f_  _hc_ ) _u_ ..

_Vb_  _m_ _f_ (( _h_ 2 _f_  _hc_ 2  _hf_ _hc_ ) _w_ .. _b_ '  ( _h_ _f_  _hc_ ) _u_ ..

_f_ _hc_ _f_ _c_ ' _f_ _c_
_b_  _m_ _f_ ((   ) _wb_  ( ) _ub_

_f_ _c_ _f_ _c_ ' _f_ _c_

(( 3  4  2 ) _wb_  ( 2 ) _ub_ )

_mc_ (6 _h_ _f_ _u_ .. _t_  36 _hf_  28 _hf_ _u_ .. _c_  6 _h_ _f_ _hc_  ..  3 _h_ 2 _f_ _w_ .. _t_ '  18 _h_ 2 _f_ _w_ .. _b_ '

 _mc_ (6 _h_ _f_ _ut_  36 _hf_  28 _hf_ _uc_  6 _h_ _f_ _hc_  3 _h_ 2 _f_ _wt_ ' 

_c_

_f_ _t_ _f_ _f_ _c_ _f_ _c_ _f_ _t_ _f_ _b_

.. .. .. .. ..

 36 _h_  28 _h_ _u_  6 _h_ _h_  3 _h_ 2 _w_ '  18 _h_ 2 _w_ '



840 _c_ (6 _h_ _f_ _ut_  36 _hf_  28 _hf_ _uc_  6 _h_ _f_ _hc_  3 _h_ 2 _f_ _wt_ '  18 _h_ 2 _f_ _wb_ ' )

 _E_ _f_ _Af_ (( _hc_ 2  _hf_ _hc_  _h_ 2 _f_ ) _wb_ '''  ( _hf_ 

_c_ 2  _hf_ _hc_  _h_ 2 _f_ ) _w_ '''  ( _hf_  _hc_ ) _u_ ''

_E_ _f_ _Af_ (( _hc_ 2  _hf_ _hc_  _h_ 2 _f_ ) _wb_ '''  ( _hf_  _hc_ ) _u_

_f_ _Af_ (( _hc_  _f_ _c_  _f_ ) _wb_ '''  ( _f_ _c_ ) _ub_

(( 4 _c_  _f_ 2 _c_  3 _f_ ) _wb_ '''  ( _f_ 2 _c_ ) _ub_ '')

840 _Gc_ _Ahcc_ 2 [(  196 _hf_  28 _hc_ ) _ut_  (1316 _hf_  532 _hc_ ) _ub_  (560 _hc_

 _Gc_ _Ac_ 2 [(  196 _hf_  28 _hc_ ) _ut_  (1316 _hf_  532 _hc_ ) _ub_  (560 _hc_ 

_f_ _c_

_c_ _c_

840 _c_ _hcc_ 2 [(  196 _hf_  28 _hc_ ) _ut_  (1316 _hf_  532 _hc_ ) _ub_  (560 _hc_  1120

)

_hc_ 2 _f_ _c_ _t_ _f_ _c_ _b_ _c_

_h_ _f_ ) _u_

 (28 _hc_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wt_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2 

_c_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wt_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2  532 _hf_ _hc_ ) _wb_ '

(28 _hc_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wt_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2  532 _hf_ _hc_ )

_hc_ 2  98 _h_ 2 _f_  28 _h_ _f_ _hc_ ) _wt_ ''  (112 _hc_ 2  658 _h_ 2 _f_  532 _h_ _f_ 2  532 _hf_ _hc_ ) _wb_ '

 (336 _h_ _f_ _hc_  112 _hc_ 2 )  (56 _hc_ 2 

_h_ _f_ _hc_  112 _hc_ 2 )  (56 _hc_ 2  168 _hf_ _hc_ ) _wc_ '

2 )  (56 _h_ 2  168 _h_ _h_ ) _w_ '

(336 _h_ _f_ _hc_  112 _hc_ 2 )  (56 _hc_ 2  168 _hf_ _hc_ ) _wc_ ' ]

_f_ _c_ _c_ _c_ _f_ _c_ _c_

_Vc_  _Gc_ _Ac_ 2 [  336 _hf_ ( _ut_  _ub_ ) _c_  (224 _hc_ 2 )  (56 _hc_ 2  168 _h_ _f_ _hc_ )( _wt_ '  _wb_ ' )  448 _hc_ 2 _wc_ '

840 _hc_

_c_ _c_ 2 [  336 _hf_ ( _ut_  _ub_ ) _c_  (224 _hc_ 2 )  (56 _hc_ 2  168 _h_ _f_ _hc_ )( _wt_ '  _wb_ ' )  448 _hc_ 2 _wc_ ' ]

840 _hc_

_c_  _c_ _c_ 2 [  336 _hf_ ( _ut_  _ub_ ) _c_  (224 _hc_ 2 )  (56 _hc_ 2  168 _h_ _f_ _hc_ )( _wt_ '  _wb_ ' )  448 _hc_ 2 _wc_ '

840 _hc_

_M_ _t_  _E_ _f_ _Af_ (( _h_ 4 _c_ 2  _hf_ 2 _hc_  _h_ 32 _f_ ) _wt_ ''  ( _hf_ 2  _hc_ ) _ut_ ')

_M_ _b_   _E_ _f_ _A_ _f_ (( _hc_ 2  _h_ _f_ _hc_  _h_ 2 _f_ ) _wb_ ''  ( _h_ _f_  _hc_ ) _ub_ '

_c_ 2 _h_ _f_ _hc_ _h_ 2 _f_ '' _h_ _f_  _hc_ '

(( 4 _c_  _f_ 2 _c_  3 _f_ ) _wb_ ''  ( _f_ 2 _c_ ) _ub_ ' )

The boundary conditions for a sandwich beam:

Latin American Journal of Solids and Structures 13 (2016) 296-313

 _u_

 _w_

 _u_

 _w_

_b_ _b_

,0 _Pt_ ( _l_, _t_ ) 

,0 _Vt_ ( _l_, _t_ ) 

_ut_ (,0 _t_ ) ,0 _Pt_ ( _l_, _t_ )

_wt_ (,0 _t_ ) ,0 _Vt_ ( _l_, _t_

(,0 _t_ ) ,0 _Pt_ ( _l_, _t_ )  0

_t_ (,0 _t_ ) ,0 _Vt_ ( _l_, _t_ )  0

_t_ _t_

,0 _M_ _t_ ( _l_, _t_ ) 

_wt_ ' (,0 _t_ ) ,0 _M_ _t_ ( _l_, _t_

' (,0 _t_ ) ,0 _M_ _t_ ( _l_, _t_ )  0

_t_ _t_

,0 _Vc_ ( _l_, _t_ ) 

,0 _Pb_ ( _l_, _t_ ) 

,0 _Vb_ ( _l_, _t_ ) 

_wc_ (,0 _t_ ) ,0 _Vc_ ( _l_, _t_

_ub_ (,0 _t_ ) ,0 _Pb_ ( _l_, _t_

_wb_ (,0 _t_ ) ,0 _Vb_ ( _l_, _t_

(,0 _t_ ) ,0 _Vc_ ( _l_, _t_ )  0

(17)

_c_ _c_

(,0 _t_ ) ,0 _Pb_ ( _l_, _t_ )  0

_b_ _b_

(,0 _t_ ) ,0 _Vb_ ( _l_, _t_ )  0

_b_ _b_

(,0 _t_ ) ,0 _M_ _b_ ( _l_, _t_ )  0

,0 _M_ _b_ ( _l_, _t_ ) 

_wb_ ' (,0 _t_ ) ,0 _M_ _b_ ( _l_, _t_

3 NUMERICAL RESULTS AND DISCUSSION

There are different arrangements of grid points which are being used in GDQ method. Equal spacing
sample of grid points used in earlier papers give some inaccurate results. Here the Chebyshev-GaussLobatto distribution is utilized to discretize the spatial domain as follows (Zong & Zhang, 2009)

 (

and             i=1,…,N (18)

_l_  ( _i_ 

 1[  cos ~~(~~

2 _N_ 

_l_  ( _i_
_xi_  1[  cos ~~(~~

_l_  ( _i_  )1

1[  cos ~~(~~ )]
2 _N_  1

Natural frequencies (rad/s) Present paper Ref. (Damanpack & Khalili, 2012) Ref. (Sokolinsky & Nutt, 2004)

1 1031.02 1030.962 1036.725

2 3195.114 3194.948 3210.708

3 5671.588 5617.640 5717.698

4 8517.974 8517.640 8658.229

5 11910.109 11909.750 12189.379

10 15945.013 15944.648 16411.680

Table 1 : The lowest natural frequencies for a cantilever sandwich beam (Antisymmetric Modes).

Natural frequencies (rad/s) Present paper Ref. (Damanpack & Khalili, 2012) Ref. (Sokolinsky & Nutt, 2004)

6 14473.978 14473.978 15029.379

7 14490.414 14490.419 15048.228

8 14500.200 14664.758 15236.724

9 15248.520 14664.739 15858.759

Table 2 : The lowest natural frequencies for a cantilever sandwich beam (Symmetric Modes).

Because in vibration of a flexible structure natural frequencies are so important in this analysis,
first it should be found accurate amounts for natural frequencies. If there is no awareness of exact
amount of natural frequencies and it is applied a harmonic load with frequency near to the structure’s
natural frequency it might lead to the colapse of structure. Therefore finding exact natural frequencies
is vital in vibration analysis of structures. To examine the accuracy of the present paper it is compared

Latin American Journal of Solids and Structures 13 (2016) 296-313

the natural frequencies of recent paper with the literatures in Tables 1 and 2 and it can be seen a
good accuracy of recent analysis with respect to literatures. In figures 2 three diffrenet shape modes
of sandwich beam with GDQ method are also compared with Damanpak and Khalili (Damanpack &
khalili, 2012) for assuraness of the results. As it can be seen in seventh shape mode which is a
symmetric one, the core deformes more than the asymmetric ones.

Figure 2 : Shape modes of flexible sandwich appendage.

Latin American Journal of Solids and Structures 13 (2016) 296-313

3.1 Applied Torque on the Hub

In this section it is investigated on vibration behaviors of the flexible spacecraft with physical
parameters presented in table 4 for a constant applied torque on the hub����� �10 �. ��.

Parameters
Face sheets Honeycomb Core Hub

Young’s Modulus/Shear Modulus (GPa) � - �70 �� �0.26

Density �

�� �� �2700 �� �83.3

- � ~~�~~

��

Thickness (m) � - �0.0005 � - �0.02

Width (m) �=0.5 �=0.5

Length (m) ��,� �2 �� �2

Hub dimension (m) a=0.3

Hub Inertia �kgm � - �� �27.2

Table 3 : Parameters of flexible spacecraft.

Figure 3 shows that by applying a constant torque on the hub the rotation of the hub changes as
parabolic with respect to time which is in comply with the mathematic of the problem. For more
clarification it shoud be added that, it is clear that the torque is a kind of acceleration �Torque ≅
�acceleration�� � �Constant �C� and by one time integrating of a constant acceleration, linear

velocity �� � ������ C�� and two times integrating of that a parabolic rotation will be concluded

�θ �������� C� � ���. This rotation in figure 4 give us information about the behavior of hub

and satellite and how it’s direction with respect to earth changes. By having this information about
the rotation of hub (satellite) with respect to time, and by using an appropriate control algurithm it
is possible to fix the sattelite in a special direction for sending singal to the earth.

Figure 3 : Rotation of the hub and tip deflection of flexible appendage.

Latin American Journal of Solids and Structures 13 (2016) 296-313

As it can be seen in figure 3 that the deflection of the beam is up to 0.003 (m) and this deflection
is logical with respect to the 0.021 (m) thickness of the beam.

Figure 4 shows lateral deflection of a rotating appendage for three different types of beam; an
ordinary beam made of just face sheet’s material (stiff beam), an ordinary beam made of just core’s
material (soft material), and a sandwich beam. But the physical dimention for three cases is equal. It
can be seen that deflection of flexible appendage for sandwich beam is less than stiff and soft ones.
This behavior is being related to parameter EI. The comparison of modulus of elasticity for three
different kinds of materials is as ����� ���������� �E����� . Modulus of elasticity for stiff beam is
more than two other cases. Comparison of moment of inertia is as������ ������ ����������. Because
in sandwich beam the natural axis of stiffen faces is far from the natural axis of core therefore moment
of inertia for sandwich beam is more than the two other cases with just on layer (stiff beam or soft
beam) and therefore the moment of inertia for sandwich beam is much more than the amount of this
parameter for stiffen beam with the same geometry. By multiplication of modulus of elasticity (E)
and moment of inertia (I) and by considering the physical parameters from table 3 this comparison
changes as ���������� ������������� ��������� .Whatever this product is smaller the deflection is
more. Therefore although sandwich panels are softer than stiff beams but their moment of inertia is
significantly more than the moment of inertia of stiff beams and as a result this production (EI) for
sandwich beam is more than two other cases, and therefore the deflection of sandwich panel with
lower weight is less than a stiff beam and this is one of the advantages of sandwich structures. Figures
5-7 show the behavior of a hub appendage system for different thicknesses and widths of the sandwich
beam. As it is mentioned above there are two important parameters which are effective on deflection
of sandwich structures. Stiffness ������ and moment inertia of sandwich beam (����). As it is clear
the stiffness of the core is less than the face sheet’s. Therefore by increasing the core thickness we are
increasing the proportion of softer part and therefore it is expected more deflection. But it can be
seen different behavior in figure 5 that by increasing the core thickness the sandwich beam tends to
lower deflection. This behavior depends on other parameter which is called moment of inertia (����).
In this figure it can be seen that by increasing the core thickness we are increasing the distance of
face sheets with respect to core’s natural axis. This increases the moment of inertia (����) of sandwich
panel and therefore the production of these two parameters ���������) increases. In sandwich panels
the face sheets are so thin and as it is mentioned before moment of inertia parameter (I) is depend
on the distance between natural axis of face sheets and the natural axis of the core. Because facesheets
are so thin increasing the thickness of these parts don’t have a significant effect on moment of inertia
(����), but it increases the proportion of stiff part with respect to softer part (core) (���� ���������)
and therefore sandwich beam’s deformation decreases. It can be concluded that there are two ways
for decreasing the deflection of the sandwich beam. First by increasing the thickness of thick soft core
(increasing the moment of inertia I), and second increasing the thin stiff layers (increasing the stiffness
E).

Latin American Journal of Solids and Structures 13 (2016) 296-313

Figure 4 : Tip deflection of flexible appendage for different types of beam.

Figure 5 : Tip deflection of flexible appendage for three different core thiknesses.

Latin American Journal of Solids and Structures 13 (2016) 296-313

Figure 6 : Tip deflection of flexible appendage for three different face sheet thicknesses.

Figure 7 : Tip deflection of flexible appendage for three different widths of the beam.

Figure 6 shows that by increasing the sandwich panel’s width the deflection will decrease. By
comparing figures 5-7 it can be seen the difference of sandwich beam’s behavior with respect to
variation of thickness and width of the beam. In figures 5 and 6 it can be seen that by increasing the
thickness of face sheets or core thickness of the beam the frequency will change too. The natural

Latin American Journal of Solids and Structures 13 (2016) 296-313

���� ������� �����

��

�� ���� � ~~�~~ ���

��� � ������� ���� �

�� ~~�~~ ���

���� � (Rao,, n.d.).

frequency for a lateral vibration of beam is as ω �� � ~~�~~

As it can be seen from this relation the frequency will increase by the factor� ��

- but it remains

constant with respect to width of the beam (b). These behaviors can be seen in figure 5-7. As it is
mentioned in previous sections, in this paper the HSAPT method is used for the first time for flexible
part of the satellite instead of using Euler Bernoulli theory. Therefore here in Figure 8 it is compared
these two deformation theories for flexible appendage. It can be seen that the deflection of sandwich
beam for HSAPT is more than the Euler Bernoulli. Because in HSAPT it is concerened different
parameters such as Shear strain and rotation, therefore the energy function here is more than the
Euler Bernoulli theroy and this results in more deflection for HSAPT as it can be seen in figure 8.

Figure 8 : Tip deflection of flexible appendage for HSAPT and euler bernulli theory.

4 CONCLUSION

In the present article, the energy method is used for the vibration analysis of rotating flexible
spacecraft made of sandwich beam. The governing equations of motion are derived using Hamilton
principle. The GDQ method is developed to transfer the governing partial differential equations to
the algebraic equations. To certify the accuracy of the present formulations, natural frequencies
obtained by the present analysis are compared with the literatures. It is presented three different
shape modes for flexible appendage in which can be seen a good accordance with the literatures.
Finally the effect of different geometrical parameters on vibration of flexible appendage are studied.
Several salient points should be mentioned here:

Latin American Journal of Solids and Structures 13 (2016) 296-313

1. The accuracy of GDQ method with respect to other numerical methods can be discovered
by comparison of the natural frequencies of this paper with the literatures.
2. The effect of compressible core in sandwich panel structure can be seen in higher frequencies
and symmetric shape modes. In these frequencies the core sustains the significant part of
deformation.
3. The deflection of the sandwich beam is affected by the important parameter EI. Whatever
this product is more the deflection is smaller.
4. Although increasing the core thickness of sandwich beam, increases the proportional of
flexible part but it can be seen that the tip deflection decreases. Actually this behavior
returns to the effect of moment of inertia (I) on the tip deflection of the beam. Because we
are increasing the distance between face sheets and natural axis by increasing the core
thickness. As this parameter is more the moment of inertia is more and as a consequent the
deflection of sandwich beam decreases.
5. Since the face sheets of sandwich structures are so thin therefore increasing their thickness
hasn’t significant effect on the moment of inertia of sandwich beam. But we are actually
increasing the proportion of stiff part of the beam by increasing the face sheet’s thickness
and this decreases the tip deflection of the beam.
6. Increasing the width of the beam decreases the deflection of the sandwich beam with this
difference that the natural frequency remains constant.
7. HSAPT shows higher deflection for sandwich beam with respect to Euler Bernoulli theory.

References

Azadi, E., Fazelzadeh, S., Eghtesad, M. & Azadi, M., 2011. Vibration suppression and adaptive-robust control of a
smart flexible satellite with three axes maneuvering. Volume 69, pp. 307-322.

Damanpack, A. & Khalili, S., 2012. Higher-order free vibration analysis of sandwich beams with a flexible core using
dynamic stiffness method. _Composite Structures,_ Volume 94, pp. 1503-1514.

Frostig, Y. & Baruch, M., 1994. Free vibrations of sandwich beams with a transversely flexible core: a high order
approach, Journal of sound and vibration. Volume 176, pp. 195-208.

Frostig, Y., Baruch, M., Vilnay, O. & Sheinman, I., 1992. High order theory for sandwich beam behavior with
transversely flexible core. _Journal of Engineering Mechanics,_ Volume 118, pp. 1026-1043.

Frostig, Y., Phan, C. & Kardomateas, G., 2013. Free vibration of unidirectional sandwich panel, part I: Compressible
core. _Journal of Sandwich Structures and Materials,_ Volume 15, pp. 377-411.

Hohe, J., Librescu, L. & Yong Oh, S., 2006. Dynamic buckling of flat and curved sandwich panels with transversely
compressible core. _Composite structures,_ Volume 74, pp. 10-24.

Hong, C. et al., 2005. Thermally induced vibration of a thermal sleeve with the GDQ method. _Int J Mech Sci,_ Volume
27, pp. 1789-1806.

Hu, Q., 2009. A composite control scheme for attitude maneuvering and elastic mode stabilization of flexible spacecraft
with measurable output feedback. _Aerosp Sci Technol,_ Volume 13, pp. 81-91.

Maganti, G. B. & Singh, S. N., 2007. Simplified adaptive control of an orbiting flexible spacecraft. _Acta Astonautica,_
Volume 61, pp. 575-589.

Oktay, T., 2015. Output variance constrained bending control of rotating Euler-Bernoulli beam. _Journal of Process_
_Mechanical Engineering._

Rao, S. S., n.d. Mechanical Vibrations. _Purdue University._

Latin American Journal of Solids and Structures 13 (2016) 296-313

Shahravi, M. & Azimi, M., 2014. A comparative study for collocated and non-collocated sensor/actuator placement in
vibration control of a maneuvering flexible satellite. _Proceedings of the institution of Mechanical Engineers, Part C:_
_Journal of mechanical engineering science._

Shahravi, M. & Azimi, M., 2014. A Hybrid Scheme of Synthesized Sliding Mode/Strain Rate Feedback Control Design
for Flexible Spacecraft Stabilization Using Time Scale Decomposition. _International Journal of structural Stability and_
_dynamics,_ Volume 16, pp. 1-18.

Shahravi, M. & Azimi, M., 2015. Multi-Axis Attitude Control of Flexible Spacecraft Using Smart Structures and
Hybrid Control Scheme. _J. of control Engineering and applied informatics,_ Volume 17, pp. 12-21.

Sokolinsky, V. S. & Nutt, S. R., 2004. Consistent higher-order dynamic equations for soft-core sandwich beams. _AIAA_
_J.,_ Volume 42, pp. 74-82.

Swanson, S. & Kim, J., 2000. Comparison of a higher order theory for sandwich beams with finite element and elasticity
analyses. _Journal of Sandwich Structures and Materials,_ Volume 2, pp. 33-49.

Tornabene, F. & Viola, E., 2009. Free vibration analysis of functionally graded panels and shells of revolution.
_Meccanica,_ Volume 44, pp. 255-281.

Tornabene, F. & Viola, E., 2014. The Local GDQ method applied to general higher-order theories of doubly-curved
laminated composite shells and panels. _The free vibration analysis, Composite Structures,_ Volume 116, pp. 637-660.

Vinson, J., 1999. The behavior of sandwich structures of isotropic and composite materials. _CRC._

Zhang, Q., Jin, J., Zhang, J. & Zhao, C., 2014. Active vibration suppression of a 3-Dof flexible parallel manipulator
using efficient modal control. _Shock and vibration,_ pp. 1-10.

Zong, Z. & Zhang, Y., 2009. Advanced differential quadrature methods.. _New York, CRC Press._

Latin American Journal of Solids and Structures 13 (2016) 296-313

본문 출처 PDF: PDF 보기 · 다른 PDF는 위 [PDF 업로드]로 올릴 수 있습니다.

활용가능성

이력 0건

아직 편집 이력이 없습니다. 저장할 때마다 이전 내용이 여기에 백업됩니다.