Pressure-Based and Potential-Based Differential Quadrature Procedures for Free Vibration of Circular Plates in Contact with Fluid

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T10221 Composite Structure Analysis and Optimization

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The differential quadrature method (DQM) has been so far applied to a wide variety of fluid and/or structural problems. The results of many researchers reveal that the DQM is computationally efficient and is applicable to a large class of boundary value problems. However, there is little information about its applications to fluid-structure interaction problems. Therefore, the purpose of this paper is to provide some information in this area and to develop procedures based on the DQM for the numerical solution of fluid-structure interaction problems. First, the governing partial differential equations of motion of the structure and fluid are discretized separately using the DQM. Then, by applying the boundary condition at fluid-structure interface, the governing eigenvalue equations of the coupled system are obtained which can then be solved for the eigenvalues of the system. The applicability of the proposed procedures is shown herein through the free vibration analysis of thin circular plates in contact with a cylindrical fluid-filled cavity. Issues related to the implementation of the regularity conditions at the center of the circular plate and the central line of the cylindrical cavity are addressed. Two new regularity conditions are proposed for the circular cylindrical fluid domain. The accuracy and efficiency of the proposed procedures are demonstrated by comparing the obtained results with those available in the literature. It is shown that highly accurate converged results can be obtained by the proposed procedures using a small number of grid points. Three new dimensionless parameters and variables are also introduced for the free vibration of the coupled system. The influences of these parameters on dynamic behavior of the system are studied.

본문

# Pressure-Based and Potential-Based Differential Quadrature Procedures for Free Vibration of Circular Plates in Contact with Fluid

Abstract
The differential quadrature method (DQM) has been so far applied
to a wide variety of fluid and/or structural problems. The results of
many researchers reveal that the DQM is computationally efficient
and is applicable to a large class of boundary value problems. However, there is little information about its applications to fluidstructure interaction problems. Therefore, the purpose of this paper
is to provide some information in this area and to develop procedures based on the DQM for the numerical solution of fluidstructure interaction problems. First, the governing partial differential equations of motion of the structure and fluid are discretized
separately using the DQM. Then, by applying the boundary condition at fluid-structure interface, the governing eigenvalue equations
of the coupled system are obtained which can then be solved for
the eigenvalues of the system. The applicability of the proposed
procedures is shown herein through the free vibration analysis of
thin circular plates in contact with a cylindrical fluid-filled cavity.
Issues related to the implementation of the regularity conditions at
the center of the circular plate and the central line of the cylindrical cavity are addressed. Two new regularity conditions are proposed for the circular cylindrical fluid domain. The accuracy and
efficiency of the proposed procedures are demonstrated by comparing the obtained results with those available in the literature. It is
shown that highly accurate converged results can be obtained by
the proposed procedures using a small number of grid points. Three
new dimensionless parameters and variables are also introduced for
the free vibration of the coupled system. The influences of these
parameters on dynamic behavior of the system are studied.

Keywords
DQM, fluid-structure interaction, pressure-based DQM, potential-based
DQM, circular plates, cylindrical cavity.

S. A. Eftekhari *

Young Researchers and Elite Club, Karaj
Branch, Islamic Azad University, Karaj,
P.O. Box 31485-313, Iran

*Author email:
aboozar.eftekhari@gmail.com

http://dx.doi.org/10.1590/1679-78252321

Received 30.07.2015
In revised form 07.11.2015
Accepted 04.01.2016
Available online 05.01.2016

1 INTRODUCTION

The problem of plate-like structures interacting with a fluid has been an interesting research subject
for a long time because this problem can be found in many applications such as offshore structures,
nuclear reactors, micro pumps, storage tanks in airplanes and space vehicles and the like. Therefore,
it is of great importance to accurately predict the dynamic behavior and vibration characteristics of
such structures.

Various analytical and numerical methods have been developed in the past to study the behavior of plate-like structures interacting with a fluid. But most of the research efforts mainly focus on
the free vibration analysis of plates in contact with an _incompressible fluid_ . Kwak and Kim (1991)
proposed an analytical procedure using the integral transformation technique in conjunction with
the dual integration equation method to study the effect of fluid on the natural frequencies of circular plates vibrating axisymmetrically in contact with an incompressible fluid. They also introduced
the concept of non-dimensionalized added virtual mass incremental (NAVMI) factor for fluid-loaded
circular plates that reflects the increase of inertia due to the presence of the fluid. Bauer (1995)
developed an analytical procedure to determine the coupled hydroelastic natural frequencies of circular membranes and plates backed by a cylindrical fluid-filled cavity. The fluid was assumed to be
incompressible and frictionless. Amabili and Kwak (1999) proposed an analytical formulation using
the perturbation technique in conjunction with the Hankel transformation method to investigate
effects of free-surface waves on free vibrations of circular plates resting on a free fluid surface. In
their study, the fluid was considered inviscid and incompressible. They showed that the effect of
free-surface waves on the plate’s natural frequencies is significant only when the bulging and sloshing modes of the system have close natural frequencies. Jeong and Kim (2005) developed an analytical algorithm by the Rayleigh–Ritz approach based on the Fourier–Bessel series expansion for the
linear free vibration of a circular plate submerged in a bounded compressible fluid. They also solved
the problem using the finite element method and studied the effects of fluid compressibility on natral frequencies of the coupled system. Chiba (1994), Amabili et al. (1995), Kwak (1997), Kwak and
Han (2000), Jeong (2003), Ergin and Ugurlu (2004), Kim and Lee (2005), Gorman and Horáček
(2007), Gorman et al. (2008), Jeong et al. (2009), Askari and Daneshmand (2010), Askari et al.
(2013), Tariverdilo et al. (2013), and Shafiee et al. (2014) also solved various circular plate–fluid
interaction problems using various analytical or approximate methods.

The differential quadrature method (DQM), which was firstly introduced by Bellman and his
associates (1971, 1972) in the early 1970s, is a powerful numerical method for the direct solution of
partial differential equations that arise in various fields of engineering, mathematics, and physics
(Bert and Malik, 1996; Tornabene et al., 2015). It is simple to use and also straightforward to implement. Since its introduction, the DQM has been successfully applied to a wide variety of fluid
and/or structural problems (Shu and Richards, 1992; Shu et al., 1995; Bert and Malik, 1996; Shu
and Chew, 1999; Shu, 2000; Liu and Wu, 2001; Eftekhari and Jafari 2013; Fantuzzi et al., 2014).
However, there are few studies on its applications to fluid-structure interaction problems.

In the literature, one may refer to the work done by Koohkan et al. (2010) or the recent studies
by Eftekhari and Jafari (2014) and Eftekhari (2016). Koohkan et al. (2010) proposed a combined
application of the DQM and the finite difference method to study the transient responses of damreservoir systems under ground motion. In their study, the dam structure has been modeled as an

Euler-Bernoulli beam while the reservoir modeled as a semi-infinite fluid. Eftekhari and Jafari
(2014) proposed a mixed modal-DQM formulation to study the free and forced vibrations of beams
in contact with fluid. It has been claimed that the mixed method can produce much better accuracy
than the finite element method (FEM) using less computational time. Most recently, Eftekhari
(2015) proposed a coupled Ritz-DQM formulation for the dynamic analysis of Timoshenko beams in
contact with fluid. It has been concluded that the conditioning of the resulting eigenvalue (or ordinary differential) equations of the potential-based formulation is considerably better than that of
the pressure-based formulation. As it can be seen from above literature survey, all of abovementioned works are limited to beam-fluid interaction problems. Besides, the formulations
in Eftekhari and Jafari (2014) and Eftekhari (2015) require the help of another method (modal
technique or Ritz method) to handle the fluid-structure interaction problem. Therefore, the purpose
of this paper is to provide some more information in this area and to develop general procedures
based on the DQM for the numerical solution of fluid-structure interaction problems.

In this study, two general procedures based on the DQM, i.e., the pressure-based and potentialbased DQMs, are proposed to study the free vibration of thin circular plates in contact with a
bounded compressible fluid. First, the governing partial differential equation of motion of the circular plate and that of the fluid are discretized separately using the DQM. Then, by applying the
boundary condition at fluid-structure interface, the governing eigenvalue equations of the coupled
system are obtained which can then be solved for the eigenvalues of the system. The proposed procedures, in general, enjoy from simplicity and high accuracy of the DQM. Their reliability and efficiency are shown through numerical simulations.

2 DQM

Let _w_ ( _ζ_ ) be an arbitrary function and _ζ_ 1, _ζ_ 2, …, _ζn_ be a set of grid points in the _ζ_ -direction. According to the DQM, the _q_ th-order derivative of the function _w_ ( _ζ_ ) at any grid point can be approximated by the following formulation

_n_

##  n


## 

_j_  1

## ( q )( ζi )   Aij ( q ) w ( ζ j ) (1)

## w ( q )( ζi )   Aij ( q ) w ( ζ j

_j_

 1

where w(ζj) represents the function value at a grid point ζj and _Aij_ ( _q_ ) are the weighting coefficients

of the qth-order derivative. The first-order weighting coefficients, i.e.,

Chang, 1989)

_Aij_ (1), are given by (Quan and

_ζ_

_i_

 (

( _ζ_ _i_ )



 






_i_ _i_  _k_, _i_, _k_  2,1,...,

( _ζ_ _i_  _ζ_ _k_ )  ( _ζ_ _k_ )

_i_ _i_  _k_, _i_, _k_  2,1,..., _n_

_ζ_ _i_  _ζ_ _k_ )  ( _ζ_ _k_ )

( _ζ_ _i_ ) _i_  _k_, _i_, _k_ 

_i_  _ζ_ _k_ )  ( _ζ_ _k_ )

##  n

,1 _j_  _i_

_ik_ (1)   _i_ _kn_ _k_ (2)

_Aik_ (1)   _i_ _kn_

_Aij_ (1) _i_  _k_, _i_  2,1,..., _n_

(1)

_ik_



##   Aij (1) i  k, i 

_i_ _k_ _k_

, _i_  2,1,...,

_ij_

,1

_j_ ,1 _j_  _i_

where

_n_

(
## 

,1 _j_  _i_

##  ( ζ i )   ( ζ i 

## ( ζ i )   ( ζ i  ζ j ) (3)

## ζ i )   ( ζ i  ζ j

_j_ ,1 _j_  _i_

The weighting coefficients of the higher-order derivatives can be computed from the following
recurrence relationship (Bert and Malik, 1996)

[ _A_ ]( _q_ )  [ _A_ ]( _q_  1)[ _A_ ](1)  [ _A_ ](1)[ _A_ ]( _q_  1), _q_  1 (4)

In this work, the DQM grid points are taken nonuniformly spaced and are given by the following equations (Bert and Malik, 1996)

  ( _i_  )1 

1  cos

  _n_  1 



  ( _i_  )1 

1  cos

  _n_  1 



1   ( _i_  )1
_ζ i_  1  cos
2   _n_  1

, _i_  3,2,1,..., _n_, 0  _ζ_  1 (5)

3 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

The geometry and coordinate system for a circular thin plate backed by a cylindrical fluid-filled
cavity is shown schematically in Figure 1. The circular cylindrical cavity of diameter 2 _a_ and height
_H_ is filled with a compressible and inviscid fluid of density _ρf_ . The cavity bottom at _z_ = _H_ and the
side wall at _r_ = _a_ are considered as rigid walls, while the fluid surface at _z_ = 0 is covered with a
flexible circular plate. The fluid motion is assumed to be irrotational and of small amplitude. The
governing partial differential equations of the coupled system can be expressed in terms of different
variables of the fluid such as displacement, pressure, and velocity potential. In this study, we are
interested to describe them in terms of pressure and velocity potential variables. The details are
given in the following subsections.

Figure 1 : Geometry and coordinate system for a circular plate in contact with a fluid-filled cylindrical cavity.

3.1 Governing Equations and Boundary Conditions for Pressure-Based Procedure

When pressure variable is used to formulate the present fluid–structure interaction problem, the
governing partial differential equations of the coupled system can be expressed as

_D_  4 _w_ ( _r_, , _t_ )  _ρsh_  2 _w_  ( _rt_ ,2  , _t_ )   _p_ ( _r_, , _z_ ,0 _t_ ) (6)

4 _w_ ( _r_, , _t_ )  _ρsh_  2 _w_  ( _rt_ ,2  , _t_ )   _p_ ( _r_, , _z_ ,0 _t_

 2 _w_ ( _r_ ,  , _t_ )

( _r_, , _t_ )  _ρsh_  _t_ 2   _p_ ( _r_, , _z_ ,0 _t_ )

and

 2 _p_ ( _r_, , _z_, _t_ )   2 _p_ ( _r_ ,  2, _z_ , _t_ )  12  2 _p_ ( _r_ ,  2, _z_ , _t_ ) (7)
 _z_ _c_ _f_  _t_

2 _p_ ( _r_, , _z_, _t_ )   2 _p_ ( _r_  , _z_  2, _z_ , _t_ )  _c_ 1 _f_ 2  2 _p_ ( _r_  , _t_  2, _z_ , _t_ )

 2 _p_ ( _r_ ,  , _z_ , _t_

_p_ ( _r_, , _z_, _t_ )  2

,  , _z_ , _t_ ) 1  2 _p_ ( _r_ ,  , _z_ , _t_

_z_ 2  _c_ _f_ 2  _t_ 2

2 _c_ 2

where _D_ is bending (or flexural) stiffness of the plate, _w_ ( _r_, , _t_ ) is the deflection of the plate, _ρs_ is

the mass density of the structure (plate), _h_ is thickness of the plate, _p_ ( _r_, , _z_, _t_ ) is the hydrodynamic

pressure distributions in excess of the hydrostatic pressure, and _cf_ is the speed of sound in fluid.
Furthermore,

 2   _r_ 22  _r_ 1  _r_  _r_ 12  22 (8)

 2 1 

 _r_ 2  _r_  _r_

 1 

 _r_  _r_ 2 

2   _r_ 22  _r_ 1  _r_  _r_ 12  22

2 _r_  _r_ _r_ 2

 2 1 

 _r_ 2  _r_  _r_

  2 1  1  2 

  2   2 2  
  _r_ _r_  _r_ _r_  



 2 1 

 _r_ 2  _r_  _r_

 1 

 _r_  _r_ 2 

 4   2  2     22  1   12  22      22  1   12  22

 _r_ _r_  _r_ _r_   _r_ _r_  _r_ _r_ 

4   2  2     22  1   12  22      22  1   12  22

  _r_ _r_  _r_ _r_    _r_ _r_  _r_ _r_ 



 1 

 _r_  _r_ 2 

  


2 _r_  _r_ _r_ 2



 

   
  _r_

  2 1  1  2 

  2   2 2  
  _r_ _r_  _r_ _r_  

2 _r_  _r_ _r_ 2



(9)

 2 

 2  



 

 2
 _r_  _r_ 



 2 

 _r_ 2  2  _r_ 2

 4 2 

 _r_ 4  _r_  _r_



 4 2  3 1   2  4  1    3  1   2  4

4  3  2   2  2 2 2    3    2 2    4   4 2  4

 _r_ _r_  _r_ _r_   _r_  _r_   _r_   _r_  _r_   _r_   



 _r_ 2 

2  2 _r_ 3

 1  

   3  
 _r_   _r_

 1  

   4   4
 _r_  

 3 1  

3  2  

 _r_ _r_   _r_

3 _r_ 2

  2  4 

  2  2 2 2  
  _r_  _r_  

   3 

   2 2  
  _r_  _r_  

 _r_ 

2 _r_ 4

  2  4 

  4 2  4  
   



 


In this study, the free vibration of fluid-loaded circular plates with clamped edge is considered.
The boundary conditions of the plate for this case are (Rao, 2007)

 _w_ ( _r_  _a_ ,  , _t_ )

_w_ ( _r_  _a_, , _t_ )   0 (10)

 _r_

The boundary conditions of the fluid cavity are assumed to be in the form of Neumann-type
boundary conditions. In other words,

 _p_ ( _r_  _a_ ,  , _z_ , _t_ )  _p_ ( _r_ ,  , _z_ 

 _r_  _z_

_p_ ( _r_  _a_ ,  , _z_ , _t_ )  _p_ ( _r_ ,  ,



_p_ ( _r_ ,  , _z_  ,0 _t_

( _r_  _a_ ,  , _z_ , _t_ )  _p_ ( _r_ ,  , _z_  _H_ , _t_ )

 ,0
 _r_  _z_

 _p_ ( _r_ ,  , _z_  ,0 _t_ )  2 _w_ ( _r_ ,  ,

, _z_  ,0 _t_ )  2 _w_ (

 _z_   _ρ_ _f_ 

_z_  ,0 _t_ )  2 _w_ ( _r_ ,  , _t_

_z_   _ρ_ _f_  _t_ 2

_t_

(11)

_a_ ,  , _z_ , _t_ )  _p_ ( _r_ ,  , _z_  _H_ , _t_



_r_  _z_

( _r_ ,  , _z_  ,0 _t_ )  2 _w_ ( _r_ ,  , _t_ )

 

_z_

where _ρf_ is the mass density of the fluid. For free vibration of the coupled system, the variables _w_
and _p_ are assumed to be harmonic as

_w_ ( _r_, , _t_ )  _w_ ~( _r_,  ) _ei_  _t_, _p_ ( _r_, , _z_, _t_ )  ~ _p_ ( _r_, , _z_ ) _ei_  _t_ (12)

where _ω_ is the circular frequency. It is advantageous to introduce the following parameters and
variables

_λ_  _ρ_ _fa_

_a_ 4 _r_ _z_
_P_  _p_,~ _W_  _w_ ~, _R_ , _Z_ ,
_D_ _a_ _H_

 _ρ_ _fa_,  _H_, _c_  _c_ _f_,

_ρsh_ _a_ _cs_

1 _D_

_cs_ 
_a_ _ρsh_

1 _D_ _a_

_s_ , _P_  _p_,~

_a_ _ρsh_ _D_

_Z_  _z_, Ω  _ωa_ 2 _ρ_ _sh_,

_H_ _D_

 _ωa_ 2 _ρ_ _sh_, (13)

_ρsh_

_c_  _c_ _f_, _cs_  1 _D_,

_cs_ _a_ _ρsh_

_f_
_cs_

_s_

where _λ_ is the mass ratio, _θ_ is the slenderness ratio, _c_ is the speed ratio, _cs_ is the speed of sound in
structure (circular plate), and Ω is the dimensionless natural frequency of the coupled system. Now,
substituting Eqs. (12) and (13) into Eqs. (6) and (7) gives

 4 _W_ ( _R_,  )   2 _W_ ( _R_,  )   _P_ ( _R_, , _Z_  0) (14)

and

 2 _P_ ( _R_, , _Z_ )   12  2 _P_ (  _RZ_ ,  2 , _Z_ )   _c_ 12 Ω2 _P_ ( _R_, , _Z_ ) (15)

( _R_, , _Z_ )   12  2 _P_ (  _RZ_ ,  2 , _Z_ )   _c_ 12 Ω2 _P_ ( _R_, , _Z_ )

2 _P_ ( _R_, , _Z_ )   12 _P_ (  _RZ_ ,2 , _Z_ )   _c_ 12 Ω2 _P_ ( _R_, , _Z_

Consequently, the boundary conditions of the fluid cavity becomes

 _P_ ( _R_ 

 ,1  , _Z_ )  _p_ ( _R_ ,  , _Z_ 


 _R_  _Z_

(16)

_P_ ( _R_  ,1  , _Z_ )   _p_ ( _R_ ,  , _Z_  )1 ,0  _P_ ( _R_ ,  , _Z_  0)  _λ_  Ω2 _W_ ( _R_,  )

 _R_  _Z_  _Z_

( _R_  ,1  , _Z_ )  _p_ ( _R_ ,  , _Z_  )1

 ,0
 _R_  _Z_

 _P_ ( _R_ ,  , _Z_ 

_P_ ( _R_ ,  , _Z_  0)

 _Z_

,1  , _Z_ )  _p_ ( _R_ ,  , _Z_



_R_  _Z_

_Z_

It can be clearly seen from Eqs. (14)-(16) that the natural frequencies of the coupled system can
be characterized by means of the following three dimensionless parameters

_λ_  _ρ_ _fa_

_c_  _c_ _f_
(17)

_c_

 _f_

_ρsh_

_H_
,

, _a_

_f_
_cs_

_s_

3.2 Governing Equations and Boundary Conditions for Potential-Based Procedure

When velocity potential variable is used to formulate the present fluid–structure interaction problem, the governing partial differential equations of the coupled system can be expressed as

 4 _w_ ( _r_, , _t_ )  _ρsh_  2 _w_  ( _rt_ ,2  , _t_ )  _ρ_ _f_  ( _r_ ,  , _zt_  ,0 _t_ )

4 _w_ ( _r_, , _t_ )  _ρsh_  2 _w_  ( _r_ ,2  , _t_ )  _ρ_ _f_  ( _r_ ,  , _z_  ,0 _t_ ) (18)

_D_  4 _w_ ( _r_, , _t_ )  _ρsh_  2 _w_  ( _rt_ ,2  , _t_ )  _ρ_ _f_  ( _r_ ,  , _zt_

_r_ ,  , _t_ )  ( _r_ ,  , _z_  ,0 _t_

_t_ 2  _ρ_ _f_  _t_

_t_

and

 2  ( _r_, , _z_, _t_ )   2  ( _r_ ,  2, _z_ , _t_ )  12  2  ( _r_ ,  2, _z_ , _t_ ) (19)
 _z_ _c_ _f_  _t_

2  ( _r_, , _z_, _t_ )   2  ( _r_  , _z_  2, _z_ , _t_ )  _c_ 1 _f_ 2  2  ( _r_  , _t_  2, _z_ , _t_ )

 2  ( _r_ ,  , _z_ , _t_

_r_, , _z_, _t_ )  2

,  , _z_ , _t_ ) 1  2  ( _r_ ,  , _z_ , _t_

_z_ 2  _c_ _f_ 2  _t_ 2

2 _c_ 2

where  ( _r_, , _t_ ) is the velocity potential of the fluid. In this case, the boundary conditions of the

fluid cavity are

 ( _r_  _a_ ,  , _z_ , _t_ )  ( _r_ ,  , _z_ 

 _r_  _z_

 ( _r_  _a_ ,  , _z_ , _t_ )  ( _r_ ,  ,



_r_ ,  , _z_  ,0 _t_ )

(20)

( _r_  _a_ ,  , _z_ , _t_ )  ( _r_ ,  , _z_  _H_ , _t_ )

 ,0
 _r_  _z_

 ( _r_ ,  , _z_  ,0 _t_ )  _w_ ( _r_ ,  , _t_ )



, _z_  ,0 _t_ )  _w_ ( _r_


 _z_ 

_z_  ,0 _t_ )  _w_ ( _r_ ,  , _t_



_z_  _t_

_t_

_a_ ,  , _z_ , _t_ )  ( _r_ ,  , _z_  _H_ , _t_


 _r_  _z_

_z_

For free vibration of the coupled system, the variables _w_ and _φ_ are assumed to be harmonic as

_w_ ( _r_, , _t_ )  _w_ ~( _r_,  ) _e_  _t_,

 ( _r_, , _z_, _t_ )   ~( _r_, , _z_ ) _e_  _t_ (21)

Introducing Eqs. (13), (21) and

to Eqs. (18)-(20) gives

_ρ_ _fa_

 _f_  ~ (22)

Θ  _f_  ~

_ρshD_

_s_

 4 _W_ ( _R_,  )  Ω2 _W_ ( _R_,  )  ΩΘ( _R_, , _Z_  0) (23)

 2Θ( _R_, , _Z_ )   12  2Θ(  _RZ_ ,  2 , _Z_ )  _c_ 12 Ω2Θ( _R_, , _Z_ ) (24)

2Θ( _R_, , _Z_ )   12  2Θ(  _RZ_ ,  2 , _Z_ )  _c_ 12 Ω2Θ( _R_, , _Z_ )

2Θ( _R_, , _Z_ )   12 Θ(  _RZ_ ,2 , _Z_ )  _c_ 12 Ω2Θ( _R_, , _Z_

 Θ( _R_ 

 ,1  , _Z_ )  Θ( _R_ ,  , _Z_ 


 _R_  _Z_

_R_  ,1  , _Z_ )  Θ( _R_ ,  , _Z_  )1  Θ( _R_ ,  , _Z_  0)

 ,0  _λ_  Ω _W_ ( _R_,  )
 _R_  _Z_  _Z_

(25)

Θ( _R_  ,1  , _Z_ )  Θ( _R_ ,  , _Z_  )1

 ,0
 _R_  _Z_

 Θ( _R_ ,  , _Z_ 

_R_ ,  , _Z_  0)

 _Z_

,1  , _Z_ )  Θ( _R_ ,  , _Z_



_R_  _Z_

_Z_

At this stage, the DQM can be employed to discretize the dimensionless differential equations of
the fluid and plate. However, there are some important issues related to regularity conditions at _R_
= 0 (or at _r_ = 0) that need to be addressed first. The details on regularity conditions are given in
the following section.

4 REGULARITY CONDITIONS AT R = 0

It can be seen from Eqs. (14) and (15) (or Eqs. 23 and 24) that the governing partial differential
equation of the circular plate is of order _four_ with respect to _R_ coordinate variable while that of the
fluid is of order _two_ with respect to this coordinate variable. The motion of the plate in _R_ -direction
is also subjected to _two boundary conditions_ (see Eq. 10) while that of the fluid is subjected to _one_
_boundary condition_ (see Eq. 16 or 25). It is well known that an arbitrary _q_ th-order differential equation should involve _q_ number of boundary conditions. In the present problem, however, the number
of boundary conditions is smaller than the order of differential equation for both the circular plate
and circular cylindrical fluid-filled cavity. Therefore, difficulty arises when trying to solve such type
of problem using numerical algorithms like the DQM.

The above-mentioned difficulty has also been reported before when solving similar problems using the DQM. For instance, in solving the free vibration problem of circular plates, Wu and Liu
(2001) also Wu et al. (2002) have also addressed this difficulty. They pointed out that some regularity conditions should be applied at the plate center in order to solve such type of problems. They
assumed the deflection of the plate in the form




## 

##  

_k_ 

## W ( R,  )   Wk ( R )cos( k  ) (26)

## ( R,  )   Wk ( R )cos( k  )

_k_  0

_k_

and proposed the following regularity conditions

 0)  d2 _W_ d _k_ ( _RR_ 2  0) 

( _R_  0)  d2 _W_ d _k_ ( _RR_ 2  0)

_Wk_ ( _R_  0)  d2 _Wk_ ( _R_ 2  0) ,0 _k_ ,7,5,3,1  (27)

d _k_ _R_ 2 ,0

 0)  d3 _Wk_ ( _R_ 3  0) 

d _Wk_ ( _R_  0)

d( _RR_  0)  d3 _W_ d _k_ ( _RR_ 3  0)

d _k_ _R_ 3 ,0

_Wk_ d( _RR_  0)  d3 _W_ d _k_ ( _RR_ 3  0) ,0 _k_ ,6,4,2,0  (28)

_R_

The numerical results presented in Wu and Liu (2001) and Wu et al. (2002) showed that the
above regularity conditions can give accurate solutions for natural frequencies of circular plates. In
this study, the above regularity conditions are applied at the center of the fluid-loaded circular
plate. In addition, some regularity conditions should also be applied at the center line of the circular
cylindrical fluid-filled cavity. Let




##  k  0

_k_


##  k 

_P_ ( _R_, , _Z_ )  _Pk_ ( _R,Z_ )cos( _k_ ,)
##  0

( _R_, , _Z_ )  _Pk_ ( _R,Z_ )cos( _k_ ,)

## Θ( R, , Z )   Θ k ( R,Z )cos( k  )

_k_  0

## R, , Z )   Θ k ( R,Z )cos( k  ) (29)

##  

_k_ 


## 



_k_

The following regularity conditions are introduced for the fluid domain

_Pk_ ( _R_ ,0 _Z_ )  Θ _k_ ( _R_ ,0 _Z_ ) ,0 _k_ ,7,5,3,1  (30)

 ,0 _Z_ )   Θ _k_ ( _R_ 

 _R_  _R_

 _Pk_ ( _R_ 

( _R_  ,0 _Z_ )   Θ _k_ ( _R_  ,0 _Z_ ) ,0

 _R_  _R_

_Pk_ ( _R_  ,0 _Z_ )   Θ _k_ ( _R_  ,0 _Z_ ) ,0 _k_ ,6,4,2,0  (31)

 ,0 _Z_ )   Θ _k_ ( _R_  ,0 _Z_

 _R_  _R_

_R_

Our numerical experiments showed that the above regularity conditions can produce accurate
solutions for natural frequencies of fluid-loaded circular plates.

5 PRESSURE-BASED DQM PROCEDURE

The governing partial differential equations for free vibration of the coupled system are given in
Eqs. (14) and (15). The solutions to these equations are first assumed to be in the forms of Eqs.
(26) and (29). Substituting Eqs. (26) and (29) into Eqs. (14) and (15) gives

 dd _R_ 4 4  _R_ 2 dd _R_ 3 3  1  _R_ 22 _k_ 2 dd _R_ 2 2  1  _R_ 23 _k_ 2 dd _R_  _k_ 4 _R_  44 _k_ 2  _Wk_ ( _R_ )  Ω2 _Wk_ ( _R_ )   _Pk_ ( _R_, _Z_

 d4 2 d3 1  2 _k_ 2 d2 1  2 _k_ 2 d _k_ 4  4 _k_ 2 

 d _R_ 4  _R_ d _R_ 3  _R_ 2 d _R_ 2  _R_ 3 d _R_  _R_ 4 

4 4  _R_ 2 dd _R_ 3 3  1  _R_ 22 _k_ 2 dd _R_ 2 2  1  _R_ 23 _k_ 2 dd _R_  _k_ 4 _R_  44 _k_ 2  _Wk_ ( _R_ )  Ω2 _Wk_ ( _R_ )   _Pk_ ( _R_, _Z_ 

d2 1  2 _k_ 2 d

d _R_ 2  _R_ 3 d _R_

dd _R_  _k_ 4 _R_  44 _k_ 2  _Wk_ ( _R_ )  Ω2 _Wk_ ( _R_ )   _Pk_ ( _R_, _Z_  0)

d4 2 d

d _R_ 4  _R_ d _R_

d _R_ 4 4  _R_ 2 dd _R_ 3 3  1  _R_ 22 _k_ 2 dd _R_ 2 2  1  _R_ 23 _k_ 2 dd _R_  _k_ 4 _R_  44 _k_ 2  _Wk_ ( _R_ )  Ω2

4  4 _k_ 2

d4 2 d3 1  2 _k_

_R_ 4  _R_ d _R_ 3  _R_ 2

(32)

d3 1  2 _k_ 2 d

d _R_ 3  _R_ 2 d _R_

 2 _k_ 2 d2 1  2 _k_

_R_ 2 d _R_ 2  _R_ 3

 2 _k_ 2 d _k_ 4  4 _k_

_R_ 3 d _R_  _R_ 4

_k_ _k_ _k_

  2 1  _k_ 2 1  2 

 _R_ 2  _R_  _R_  _R_ 2   2  _Z_ 2 



 2 1 

 _R_ 2  _R_  _R_

 _R_ 2 2  _R_ 1  _R_  _Rk_ 22   12  _Z_ 2 2  _Pk_ ( _R_, _Z_ )  _c_ 12 Ω2 _Pk_ ( _R_, _Z_

 2 1  _k_

_R_ 2  _R_  _R_  _R_

2 2  _R_ 1  _R_  _Rk_ 22   12  _Z_ 2 2  _Pk_ ( _R_, _Z_ )  _c_ 12 Ω2 _Pk_ ( _R_, _Z_ ) 

1  _R_  _Rk_ 22   12  _Z_ 2 2  _Pk_ ( _R_, _Z_ )  _c_ 12 Ω2 _Pk_ ( _R_, _Z_ )  0

 _k_ 2 1 

 _R_  _R_ 2   2  _Z_

2  2

 2  _Z_ 2  _Pk_ ( _R_, _Z_ )  _c_ 2 Ω2 _Pk_ ( _R_, _Z_ )  0 (33)



At this stage, the DQM will be used to discretize Eqs. (32) and (33). The details are given in
the following subsections.

5.1 Discretization of Governing Differential Equation of the Structure

Consider _n_ grid points with coordinates _R_ 1, _R_ 2, …, _Rn_ in the _R_ -direction. Satisfying Eq. (32) at any
grid point _R_ = _Ri_ ( _i_ = 1, 2, …, _n_ ) and substituting the quadrature rule into results gives

[ _K_ _ks_ ]{ _W_ _k_ }  Ω2[ _M_ _s_ ]{ _W_ _k_ }   { _P_ _kI_ } (34)

where [ _M_ _s_ ] and [ _K_ _ks_ ] are structural mass and stiffness matrices, { _W_ _k_ } is the displacement vector,

and { _P_ _kI_ } is the load vector due to fluid-structure interaction. [ _M_ _s_ ], [ _K_ _ks_ ], { _W_ _k_ } { _P_ _kI_ } are
and

given by

[ _M_ _s_ ]  [ _I_ ] (35)

[ _K_ _ks_ ]  [ _A_ ](4)  [2 _R_ ](1)[ _A_ ](3)  1(  2 _k_ 2)[ _R_ ](2)[ _A_ ](2)  (1  2 _k_ 2)[ _R_ ](3)[ _A_ ](1)  ( _k_ 4  4 _k_ 2)[ _R_ ](4) (36)

_T_

[ _R_ ](1)  _diag_ { _R_ }(1),

[ _R_ ](2)  _diag_ { _R_ }(2),

[ _R_ ](3)  _diag_ { _R_ }(3),

[ _R_ ](4)  _diag_ { _R_ }(4),

{ _R_ }(1)   1 1  1
 _R_ 1 _R_ 2 _R_

{ _R_ }(4)   14 14  14  (40)

_R_ }(1)   1 1  1  (37)

 1 1 1
  _R_ 14 _R_ 24  _Rn_ 4

1 1 1 

 

_R_ 1 _R_ 2 _Rn_ 

1 1 1 

_R_ 14 _R_ 24  _Rn_ 4 

1 2

1 1 1 

_R_ 12 _R_ 22  _Rn_ 2 

 1 1 1
  _R_ 12 _R_ 22  _Rn_ 2

_T_

{ _R_ }(2)   12 12  12  (38)

 1 1 1
  _R_ 13 _R_ 23  _Rn_ 3

1 1 1 

_R_ 13 _R_ 23  _Rn_ 3 

_T_

{ _R_ }(3)   13 13  13  (39)

_T_

{ _W_ _k_ }   _Wk_ ( _R_ 1) _Wk_ ( _R_ 2)  _Wk_ ( _Rn_ )  _T_ (41)

{ _P_ _kI_ }   _Pk_ ( _R_ 10,) _Pk_ ( _R_ 20,)  _Pk_ ( _Rn_ 0,)  _T_ (42)

where in [ _I_ ] is an identity matrix of order _n_ × _n_, and [ _A_ ] ( _q_ ) ( _q_ = 1, 2, 3, 4) are the DQM weighting
coefficient matrix of the _q_ th-order _R_ -derivative.

5.2 Discretization of Governing Differential Equation of the Fluid

Consider _n_ grid points with coordinates _R_ 1, _R_ 2, …, _Rn_ in the _R_ -direction and _m_ grid points with
coordinates _Z_ 1, _Z_ 2, …, _Zm_ in the _Z_ -direction. Satisfying Eq. (33) at any grid point _R_ = _Ri_ ( _i_ = 1, 2,
…, _n_ ) and substituting the quadrature rule into results gives

 [ _A_ ](2)  [ _R_ ](1)[ _A_ ](1)  _k_ 2[ _R_ ](2)   12 [ _I_ ]  _Z_ 2 2 



(2)  [ _R_ ](1)[ _A_ ](1)  _k_ 2[ _R_ ](2)   12 [ _I_ ]  _Z_ 2  { _P_ _k_ ( _Z_ )}  _c_ 12 Ω2[ _I_ ]{ _P_ _k_ ( _Z_ )}  { 0

[ _A_ ](2)  [ _R_ ](1)[ _A_ ](1)  _k_ 2[ _R_ ](2)   12 [ _I_ ]  _Z_ 2 2  { _P_ _k_ ( _Z_ )}  _c_ 12 Ω2[ _I_ ]{ _P_ _k_ ( _Z_

[ _A_ ](2)  [ _R_ ](1)[ _A_ ](1)  _k_ 2[ _R_ ](2)   12 [ _I_ ]  _Z_ 2 2  { _P_ _k_ ( _Z_ )}  _c_ 12 Ω2[ _I_ ]{ _P_ _k_ ( _Z_ )}  { 0 }



(43)

where in {0} is a zero vector of order n × 1, and

{ _P_ _k_ ( _Z_ )}   _Pk_ ( _R_ 1, _Z_ ) _Pk_ ( _R_ 2, _Z_ )  _Pk_ ( _Rn_, _Z_ )  _T_ (44)

Now, satisfying Eq. (43) at any grid point _Z_ = _Zi_ ( _i_ = 1, 2, …, _m_ ) and using the quadrature
rule yields

[ _K_ _kf_ ]{ _P_ _k_ }  Ω2[ _M_ _f_ ]{ _P_ _k_ }  {~ 0 } (45)

where [ _M_ _f_ ] and [ _K_ _kf_ ] are the mass and stiffness matrices of the fluid, { _P_ _k_ } is the vector of pressure

values, and {~ 0 } is a zero vector of order nm × 1. The n × n sub-matrices [ _M_ _ijf_ ] and [ _K_ _k,f_ _ij_ ] are

given by

[ _M_ _ijf_ ]  _c_ 12 _I_ _ij_ [ _I_ ], _i_, _j_  2,1,..., _m_ (46)
#### [ K k,f ij ]  I ij  [ A ](2)  [ R ](1)[ A ](1)  k 2[ R ](2)    12 Aij (2)[ I ] (47)

where _I_ _ij_ are the elements of _m_ × _m_ identity matrix, and

of the second-order _Z_ -derivative. Furthermore,

_Aij_ (2) are the DQM weighting coefficients

### { P k }   P k ( Z 1)  T  P k ( Z 2)  T   P k ( Z m )  T  T (48)

where in the vector { _P_ _k_ ( _Z_ )} is defined in Eq. (44).

5.3 Implementation of Boundary and Regularity Conditions

There are different approaches to incorporate the boundary conditions in the DQM resultant matrix
equations. The technique recently proposed by the present author (Eftekhari, 2015) is used here to
implement the boundary conditions of the problem. According to this new approach, the quadrature
analog equations of the boundary conditions are first directly substituted in the resultant matrix
equations of the DQM. The degrees of freedom correspond to the _Dirichlet-type boundary equations_
are then removed from the final matrix equations.

5.3.1 Quadrature Analogs of the Boundary and Regularity Conditions of the Circular Plate

The boundary and regularity conditions of the circular plate are given in Eqs. (10), (27) and (28).
The quadrature analogs of the plate boundary conditions are simply

_Wk_ ( _R_ _n_ ) ,0 _k_ ,4,3,2,1  (49)

_n_

( _Rj_ ) ,0



_n_
##  Anj (1)



_Anj_ (1) _Wk_ ( _Rj_ ) ,0 _k_ ,4,3,2,1  (50)

_j_

 1

Similarly, the quadrature analogs of the regularity conditions are obtained as

_n_

_n_
## 1)   A 1(2 j )

##   A 1(2 j ) Wk ( Rj ) 

## Wk ( R 1)   A 1(2 j ) Wk ( Rj ) ,0 k  7,5,3,1,... (51)

## ( R 1)   A 1(2 j ) Wk ( Rj ) ,0

_j_



_n_

## A 1(1 j ) Wk ( Rj )   A 1(3 j )

1 _j_  1

## A 1(1 j ) Wk ( Rj )   A 1(3 j ) Wk ( Rj ) ,0 k  6,4,2,0,... (52)

## ( Rj )   A 1(3 j ) Wk ( Rj ) ,0

##   A 1(3 j ) Wk ( Rj ) 

_n_

_n_ _n_
##  A 1(1 j ) Wk ( Rj )  

_j_ _k_ _j_

_j_

 1 _j_ 

_j_

5.3.2 Quadrature Analogs of the Boundary and Regularity Conditions of the Fluid

The boundary and regularity conditions of the fluid are given in Eqs. (16), (30) and (31). The quadrature analogs of the fluid boundary conditions are simply

_n_

( _Rj_, _Z_ ) ,0



_n_
##  Anj (1)

_Anj_ (1) _Pk_ ( _Rj_, _Z_ ) ,0 _Z_  _Z_ 1, _Z_ 2,..., _Z_ _m_, _k_  4,3,2,1,... (53)

_j_

 1

_n_

_Amj_ (1){ _P_ _k_ ( _Z_ _j_ )}  { 0 } _n_  1, _k_ ,4,3,2,1  (54)

(1)

_n_

_Amj_ (1){ _P_ _k_ ( _Z_ _j_ )}  { 0 } _n_  1
##  j  1

_j_

_Amj_ (1){ _P_ _k_ ( _Z_ _j_ )}  { 0 } _n_  1,

_n_

{ _P_ _k_ ( _Z_ _j_ )}  _λ_  Ω2{ _W_ _k_ },

_n_
##  A 1( j 1){ P k ( Z j )}  λ  Ω2{ W k

1( _j_ 1){ _P_ _k_ ( _Z_ _j_ )}  _λ_  Ω2{ _W_ _k_

 _λ_  Ω2{ _W_ _k_ }, _k_ ,4,3,2,1  (55)

_j_

 1

Similarly, the quadrature analogs of the regularity conditions are obtained as

_Pk_ ( _R_ 1, _Z_ ) ,0 _Z_  _Z_ 1, _Z_ 2,..., _Z_ _m_, _k_  7,5,3,1,... (56)

_n_

( _Rj_, _Z_ ) ,0



_n_
##  A 1(1 j )

_A_ 1(1 _j_ ) _Pk_ ( _Rj_, _Z_ ) ,0 _Z_  _Z_ 1, _Z_ 2,..., _Z_ _m_, _k_  6,4,2,0,... (57)

_j_



5.4 Solution of Resulting System of Coupled Eigenvalue Equations

Substituting Eq. (55) into Eq. (45) gives

[ _K_ ~ _kf_ ]{ _P_ _k_ }  Ω2[ _M_ ~ _f_ ]{ _P_ _k_ }  Ω2[ _M_ *]{ _W_ _k_ } (58)

where in



 [ _I_

[ 0 ] _n_

_n_  _n_  _nm_  _n_

 [ _I_ ]



_λ_  [ _I_ ] _n_  _n_












_I_

(59)














0 ]

_n_  _n_

[ _M_ *]



0 ]

[ 0 ] _n_

 _n_  _nm_ 

Eq. (34) can also be rewritten as

[ _K_ _ks_ ]{ _W_ _k_ }  Ω2[ _M_ _s_ ]{ _W_ _k_ }   [ _K_ *]{ _P_ _k_ } (60)

where

#### [ K *]  λ 1  [ M *] T   [ I ] n  n [ 0 ] n  n [ 0 ] n  n  [ 0 ] n  n  n  nm

(61)

As it can be seen, the added mass and stiffness matrices due to fluid-structure interaction (i.e.,
the matrices [ _M_ *] and [ _K_ *] ) have very simple forms. Now, combining Eq. (58) with Eq. (60) gives

[ _K_ ~ _kf_ ] _nm_  _nm_ [ 0 ]

 [ _K_ [ _K_ ~ _kf_ *]] _nmn_  _nmnm_ [[ _K_ 0 ] _knms_ ] _n_  _nn_ 



_kf_ ] _nm_  _nm_ [ 0 ] _nm_  _n_

 _nm_ 0 _nm_  _n_

 _nm_ [ _K_ _ks_ ] _n_ 

_K_ 0 ] _knms_ ] _n_  _nn_  {{ _WP_ _k_ _k_ }} _nmn_  1 _1_   Ω2  [ _M_ [~ 0

{{ _WP_ _k_ _k_ }} _nmn_  1 _1_   Ω2  [ _M_ [~ 0 ] _nf_  ] _nmnm_  _nm_ [[ _MM_ - _s_ ]]

 1

(62)

{ _P_ _k_ }

_P_






{ _P_ _k_ }

_P_

_f_ ] _nm_  _nm_ [ _M_ *] _nm_  _n_

_n_  _nm_ [ _M_ _s_ ] _n_  _n_

 _nm_ _M_ _n_ 

[ _K_ ~ _kf_ ] _nm_  _nm_ [ 0 ]

[ _K_ *] _n_  _nm_ [ _K_

  Ω2 
 

2  [ _M_ ~ _f_ ] _nm_  _nm_ [ _M_ *] _nm_  _n_  { _P_ _k_ } _nm_  1

 [ _M_ [~ 0 ] _nf_  ] _nmnm_  _nm_ [[ _MM_ - _s_ ] _nm_ ] _n_  _nn_ 

 _nm_ _M_ _nm_ 

_k_ _nm_  _1_

_k_ _nm_ 

*] _n_  _nm_ [ _K_ _ks_ ] _n_  _n_  { _W_ _k_ } _n_  1   [ 0 ] _n_  _nm_ [ _M_ _s_ ] _n_  _n_  { _W_ _k_ }

_n_  _nm_ [ _K_ _ks_ ] _n_  _n_






_P_ _k_ } _nm_  _1_   Ω2  [ _M_ ~ _f_ ] _nm_  _nm_ [ _M_

_W_ _k_ } _n_  1   [ 0 ] _n_  _nm_ [ _M_

[ _K_ *] _n_  _nm_ [ _K_ _ks_ ]

_k_ _n_







[ 0 ] _n_  _nm_ [ _M_ _s_ ]



_W_

_k_ _n_

After substituting the remaining quadrature analog equations (given in Eqs. 49-52, 53, 54, 56
and 57) into Eq. (62), one can solve the resulting eigenvalue problem for the eigenvalues Ω.

6 POTENTIAL-BASED DQM PROCEDURE

Substituting Eqs. (26) and (29) into Eqs. (23) and (24) gives

 d4 2 d3 1  2 _k_ 2 d2 1  2 _k_ 2 d _k_ 4  4 _k_ 2 

 d _R_ 4  _R_ d _R_ 3  _R_ 2 d _R_ 2  _R_ 3 d _R_  _R_ 4 

_R_ 4 4  _R_ 2 dd _R_ 3 3  1  _R_ 22 _k_ 2 dd _R_ 2 2  1  _R_ 23 _k_ 2 dd _R_  _k_ 4 _R_  44 _k_ 2  _Wk_ ( _R_ )  Ω2 _Wk_ ( _R_ )  ΩΘ _k_ ( _R_, _Z_ 

d2 1  2 _k_ 2 d

d _R_ 2  _R_ 3 d _R_

dd _R_  _k_ 4 _R_  44 _k_ 2  _Wk_ ( _R_ )  Ω2 _Wk_ ( _R_ )  ΩΘ _k_ ( _R_, _Z_  0)

d4 2 d

d _R_ 4  _R_ d _R_

4  4 _k_ 2

d4 2 d3 1  2 _k_

_R_ 4  _R_ d _R_ 3  _R_ 2

(63)

d3 1  2 _k_ 2 d

d _R_ 3  _R_ 2 d _R_

 2 _k_ 2 d2 1  2 _k_

_R_ 2 d _R_ 2  _R_ 3

 2 _k_ 2 d _k_ 4  4 _k_

_R_ 3 d _R_  _R_ 4

_k_ _k_ _k_

  2 1  _k_ 2 1  2 

 _R_ 2  _R_  _R_  _R_ 2   2  _Z_ 2 



 2 1 

 _R_ 2  _R_  _R_

 _R_ 2 2  _R_ 1  _R_  _Rk_ 22   12  _Z_ 2 2  Θ _k_ ( _R_, _Z_ )  _c_ 12 Ω2Θ _k_ ( _R_, _Z_

 2 1  _k_

_R_ 2  _R_  _R_  _R_

2 2  _R_ 1  _R_  _Rk_ 22   12  _Z_ 2 2  Θ _k_ ( _R_, _Z_ )  _c_ 12 Ω2Θ _k_ ( _R_, _Z_ ) 

 2  _Z_ 2  Θ _k_ ( _R_, _Z_ )  _c_ 2 Ω2Θ _k_ ( _R_, _Z_ )  0 (64)

1  _R_  _Rk_ 22   12  _Z_ 2 2  Θ _k_ ( _R_, _Z_ )  _c_ 12 Ω2Θ _k_ ( _R_, _Z_ )  0

 _k_ 2 1 

 _R_  _R_ 2   2  _Z_

2  2



The procedure for discretizing Eqs. (63) and (64) and the corresponding regularity and boundary conditions is similar to that described in Section 5. To save the space, only the resultant system
of eigenvalue equations is given in this section. That is,



2  [ _M_ ~ _f_ ] _nm_  _nm_ [ 0 ] _nm_  _n_   Ω  [ 0 ] _nm_  _nm_ [ _C_

 [ _M_ [~ 0 ] _nf_  ] _nmnm_  _nm_ [ _M_ [ 0 ] _nms_ ] _n_  _nn_ 



[ _M_ ~

   Ω2  [ _M_ [~ 0 ] _nf_  ] _nmnm_  _nm_ [ _M_ [ 0 ] _nms_ ] _n_  _nn_   Ω  [ _C_ [ 0 ]* _nm_ *] _n_  _nmnm_ [ _C_ [ 0 *]] _nnm_  _n_  _n_    [ _K_ ~[ 0 _kf_ ]] _nnm_  _nm_  _nm_ [[ _K_ 0 ] _knms_ ] _n_  _nn_    

Ω2  [ _M_ ~ _f_ ] _nm_  _nm_ [ 0 ]

[ _M_ [~ 0 ] _nf_  ] _nmnm_  _nm_ [ _M_ [ 0 ] _nms_ ] _n_  _nn_   Ω  [ _C_ [ 0 ]* _nm_ *] _n_  _nmnm_ [ _C_ [ 0 *]]

[ _C_ [ 0 ]* _nm_ *] _n_  _nmnm_ [ _C_ [ 0 *]] _nnm_  _n_  _n_    [ _K_ ~[ 0 _kf_ ]] _nnm_  _nm_  _nm_ [[ _K_ 0 ]

 [ _K_ ~[ 0 _kf_ ]] _nnm_  _nm_  _nm_ [[ _K_ 0 ] _knms_ ] _n_  _nn_ 

(65)



 [ _C_ [ 0 ]* _nm_ *] _n_  _nmnm_ [ _C_ [ 0 *]] _nnm_  _n_  _n_ 

_nm_  _nm_ _nm_  _n_

 _nm_ _nm_  _n_


 


 1

_C_

 

  
 

_kf_ ] _nm_  _nm_ [ 0 ] _nm_  _n_

 _nm_ _nm_ 




 


{ Θ _k_ }

{ _W_ _k_ }

{ Θ _k_

_k_ _nm_

_k_ } _n_ 

 _nm_ _C_ _nm_ 

_n_  _nm_ [ 0 ] _n_  _n_

 _nm_ 0 _n_ 

*] _nm_  _n_    [ _K_ ~ _kf_ ] _nm_  _nm_ [ 0 ]

0 ] _n_  _n_   [ 0 ] _n_  _nm_ [ _K_




_f_ ] _nm_  _nm_ [ 0 ] _nm_  _n_

_n_  _nm_ [ _M_ _s_ ] _n_  _n_

_M_ [ 0 ] _nms_ ] _n_  _nn_   Ω  [ _C_ [ 0 ]* _nm_ *] _n_  _nmnm_ [ _C_ [ 0 *]] _nnm_  _n_  _n_    [ _K_ ~[ 0 _kf_ ]] _nnm_  _nm_  _nm_ [[ _K_ 0 ] _knms_ ] _n_  _nn_     {{ Θ _W_ _k_

_n_  _nm_ [ _K_ _ks_ ] _n_  _n_

 _nm_ _K_ _k_ _n_ 

_M_

[ 0 ]

[ 0 ] _n_  _nm_ [ _K_ _ks_ ]

 _nm_ _M_ _n_ 

 

  Ω 
  [

[ _C_ [ 0 ]* _nm_ *] _n_  _nmnm_ [ _C_ [ 0 *]] _nnm_  _n_  _n_    [ _K_ ~[





{ 0 }



{ 0 }







 1

{ 0

_nm_

} _n_ 

where the matrices [ _M_ ~ _f_ ],[ _M_ _s_ ],[ _K_ ~ _kf_ ] and [ _K_ _ks_ ] are exactly the same as those defined in Section 5.

Furthermore,

[ _C_ *]  [ _M_ *], [ _C_ ]  [ _K_** *] (66)
### { Θ k }   Θ k ( Z 1)  T  Θ k ( Z 2)  T   Θ k ( Z m )  T  T (67)

{ Θ _k_ ( _Z_ )}   _k_ ( _R_ 1, _Z_ )  _k_ ( _R_ 2, _Z_ )   _k_ ( _Rn_, _Z_ )  _T_ (68)

where in the matrices [ _M_ *] and [ _K_ *] are defined in Section 5.

7 COMPARISON OF THE TWO PROCEDURES

It can be seen from Eq. (62) that the effect of fluid-structure interaction in pressure-based formulation is appeared in the form of added mass and stiffness matrices [ _M_ *] and [ _K_ *] . In general, the

elements of matrix [ _K_ *] are not in the same order of magnitude as compared with those of matrix

[ _K_ _ks_ ] . For instance, they may be much smaller than those of matrix [ _K_ _ks_ ] . Similarly, the order of

elements of matrix [ _M_ *] may be very different in magnitude as compared with that of matrix

[ _M_ ~ _f_ ] . As a result, the eigenvalue problem (62) may become ill-conditioned particularly when the

_dimensional_ differential equations of the fluid and structure are handled using the DQM.

The above-mentioned difficulty can be easily overcome by the help of potential-based formulation. This can be clearly seen from Eq. (65) wherein the effect of fluid-structure interaction is appeared in the form of added damping matrices [ _C_ *] and [ _C_ **] . As a result, the problem of ill

conditioning is fully eliminated in the potential-based formulation even if the _dimensional_ differential equations of the problem are discretized using the DQM.

8 NUMERICAL RESULTS AND DISCUSSION

8.1 Verification and Validation of Numerical Results

To validate the proposed procedures for free vibration analysis of fluid-loaded circular plates, application is made to a numerical example given by Tariverdilo et al. (2013). Tariverdilo et al. (2013)
proposed two analytical approaches, i.e., the Fourier–Bessel series method and the variational
method, for asymmetric free vibration of circular plates in contact with a bounded _incompressible_
fluid. The parameters used in their study are as follows:

_ρs_ = 7800 Kg/m 3, _a_ = 100 mm, _h_ = 2 mm, _ρf_ = 1000 Kg/m 3, _H_ = 100 mm, _E_ = 210 GPa, μ = 0.3

where _E_ is modulus of elasticity while μ is the Poisson’s ratio. Although the proposed techniques are
developed for free vibration analysis of circular plates in contact with a _compressible_ fluid, they are
also available for that of circular plates in contact with an _incompressible_ fluid if the sound speed _cf_
is assumed to be infinity or the mass matrix of the fluid is assumed to be zero matrix.

Figure 2 presents the convergence of solutions for axisymmetric and asymmetric natural frequencies of the fluid-loaded circular clamped plate. The results are obtained using both the pressure-based
and potential-based DQM procedures. It can be seen that the solutions of the pressure-based procedure are exactly identical to those of the potential-based procedure. Besides, both procedures predict
zero value for the fundamental frequency ( _ω_ 0,1) of the coupled system. The existence of zero-frequency
mode for Neumann problem has been discussed at length in Eftekhari and Jafari (2014) and is not
repeated here again. The present results show that such phenomenon may also be observed in free
vibration problem of fluid-loaded circular plates with Neumann boundary conditions.

3500

3000

2500

2000

1500

1000

500

5000

4500

4000

3500

3000

2500

2000

1500

1000

Results for axisymmetric vibration
of the system

0,1 (Pressure-based DQM)

0,1 (Potential-based DQM)

0,2 (Pressure-based DQM)

0,2 (Potential-based DQM)

0,3 (Pressure-based DQM)

0,3 (Potential-based DQM)

Results for asymmetric vibration
of the system

1,1 (Pressure-based DQM)

1,1 (Potential-based DQM)

1,2 (Pressure-based DQM)

1,2 (Potential-based DQM)

1,3 (Pressure-based DQM)

1,3 (Potential-based DQM)

~~0~~ 5 10 15 20

Number of grid points, n = m

50 ~~0~~ 5 10 15 20

Number of grid points, n = m

Figure 2 : Convergence of axisymmetric and asymmetric natural frequencies of a fluid-loaded circular clamped plate.

On the other hand, from Figure 2 one sees that the obtained solutions show an excellent convergence behavior with increasing number grid points. Besides, a reasonably converged solution can
be achieved by the proposed procedures using a considerably small number of grid points. It is clear
that the accurate simulation of the higher modes of the coupled system requires a larger number of
grid points.

In Table 1, the results of present simulations (pressure-based and potential-based DQMs) are
compared with the approximate analytical solutions of Tariverdilo et al. (2013). It can be seen that
the results generated by the present methodologies converge very quickly and agree well with the
analytical solutions of Tariverdilo et al. (2013). Besides, the present results have closer agreement
with the results of the variational formulation than those of the Fourier–Bessel series. It is interesting to note that, almost in most cases, the present procedures can produce a reasonably accurate
solution for natural frequencies of the system using only 11 grid points ( _n_ = _m_ = 11).

On the other hand, from Table 1 one sees that the solutions of the pressure-based procedure are
exactly identical to those of the potential-based procedure. As we discussed before in Eftekhari
(2015), this is due to physical equivalence of the pressure-based formulation and the potential-based
formulation (i.e., physical equivalence of the governing differential equations and boundary conditions for both formulations).

Fourier–Bessel
_ω_ k,l Method _n_ = 11 a _n_ = 13 _n_ = 15 _n_ = 17 _n_ = 19 _n_ = 21

series b

_ω_ 1,1 Present c 578.086 578.086 578.086 578.086 578.086 578.086

Variational
formulation b

Present

578.086 578.086 578.086 578.086 578.086 578.086 578.11 578.07

_ω_ 1,2 Present c 2051.67 2052.04 2052.03 2052.03 2052.03 2052.03

Present

2051.67 2052.04 2052.03 2052.03 2052.03 2052.03 2052.61 2052.03

_ω_ 2,1 Present c 1097.45 1108.05 1107.68 1107.70 1107.70 1107.70

Present

1097.45 1108.05 1107.68 1107.70 1107.70 1107.70 1107.84 1107.83

_ω_ 2,2 Present c 3281.02 3031.31 3066.74 3063.96 3064.08 3064.07

Present

3281.02 3031.31 3066.74 3063.96 3064.08 3064.07 3077.72 3086.97

_ω_ 3,1 Present c 1752.58 1752.68 1752.68 1752.68 1752.68 1752.68

Present

1752.58 1752.68 1752.68 1752.68 1752.68 1752.68 1753.16 1753.27

_ω_ 3,2 Present c 4196.92 4203.71 4203.36 4203.38 4203.38 4203.38

Present

4196.92 4203.71 4203.36 4203.38 4203.38 4203.38 4204.67 4203.44

_ω_ 4,1 Present c 2518.10 2518.66 2518.65 2518.65 2518.65 2518.65

Present

2518.10 2518.66 2518.65 2518.65 2518.65 2518.65 2518.79 2518.40

_ω_ 4,2 Present c 5470.73 5474.10 5474.04 5474.04 5474.04 5474.04

Present

~~d~~

5470.73 5474.10 5474.04 5474.04 5474.04 5474.04 5476.53 5475.36

Table 1 : Convergence and accuracy of asymmetric natural frequencies ( _ωk_, _l_ (Hz)) of a fluid-loaded circular clamped

plate. a _m_ = _n_, b Tariverdilo et al. (2013), c Solutions of pressure-based DQM, d Solutions of potential-based DQM.

Finally, it should be noted that the resulting eigenvalue problems of the proposed procedures
are found to be well-conditioned and thus they can be easily and directly solved using various
standard eigenvalue solvers. But the resultant matrices of analytical solutions of Tariverdilo et al.
(2013) were reported to be highly ill-conditioned. This can lead to large errors in the calculation of
the eigenvalues particularly in the calculation of higher modes of the coupled system. Tariverdilo et
al. (2013) mentioned that such an eigenvalue problem can only be solved by the help of precondi

tioner algorithms such as the diagonal scaling method. Therefore, it can be concluded that the proposed procedures are superior over the analytical formulations of Tariverdilo et al. (2013) in terms
of accuracy and reliability. Most importantly, the higher modes of the system can be easily and
accurately computed by the proposed procedures without adopting any preconditioner algorithm.

8.2 Free Vibration of Circular Plates in Contact with a Bounded Incompressible Fluid

It can be seen from Section 3 that the natural frequencies of the coupled system in the case of incompressible fluid can be described by means of the following dimensionless parameters

 _ρ_ _fa_,  _H_ (69)

_ρ_ _h_ _a_

_λ_  _ρ_ _fa_

_f_,

_ρsh_

_s_

where _λ_ is the mass ratio and _θ_ is the slenderness ratio. Table 2 gives the dimensionless asymmetric
natural frequencies ( Ω _k_, _l_  _ωk,la_ 2 _ρsh_ / _D_ ) of the coupled system for different values of _λ_ and _θ_ . It

can be seen from Table 2 that the natural frequencies of the system decrease as the value of the
parameter _λ_ increases. Besides, they increase as the value of _θ_ increases. Note that the solutions of
the pressure-based DQM are found to be exactly identical to those of the potential-based DQM.

The variations of dimensionless axisymmetric natural frequencies ( Ω _k_, _l_  _ωk,la_ 2 _ρsh_ / _D_ ) of the

fluid-loaded circular clamped plate with _θ_ are shown in Figure 3 for different values of _λ_ . Since the value
of the fundamental frequency of the coupled system (i.e., Ω0 _,_ 1) is zero, the results are shown for Ω _k,l_ Ω0 _,_ 1. It can be seen from Figure 3 that the natural frequencies of the system are initially increasing with
increasing the value of _θ_ and after a certain value of _θ_ (say, _θ_ * ) they are remaining constant. The value
of _θ_ * is almost independent of values of _λ_, but it decreases as the mode number increases.

_θ_ Ω _k,l_ _λ_ = 1 _λ_ = 5 _λ_ = 10 _λ_ = 15 _λ_ = 20 _λ_ = 25

0.25 Ω1,1 16.2327 9.9032 7.4092 6.1735 5.4025 4.8630

Ω1,2 53.7736 41.1425 33.6662 29.2149 26.1681 23.9127

Ω1,3 111.2275 92.3947 79.0128 70.2162 63.8418 58.9437

Ω1,4 188.4292 163.2182 143.3200 129.3945 118.9007 110.6164

0.65 Ω1,1 17.9172 12.1703 9.4040 7.9389 6.9970 6.3264

Ω1,2 55.3268 43.3404 35.7823 31.1903 28.0137 25.6465

Ω1,3 112.6166 94.1575 80.6896 71.7889 65.3207 60.3410

Ω1,4 189.7289 164.7183 144.6984 130.6702 120.0930 111.7395

1 Ω1,1 18.1699 12.5842 9.7915 8.2907 7.3192 6.6248

Ω1,2 55.5570 43.6753 36.0879 31.4660 28.2657 25.8797

Ω1,3 112.8366 94.4495 80.9484 72.0198 65.5306 60.5347

Ω1,4 189.9426 164.9851 144.9296 130.8748 120.2781 111.9098

Table 2 : Dimensionless asymmetric natural frequencies of a fluid-loaded circular clamped plate. Note that the

solutions are obtained using both the pressure-based and potential-based DQMs with _n_ = 21 and _m_ = 17.

140

120

100

 = 5
 = 10
 = 15
 = 20

220

200

180

160

140

120

100

 = 5
 = 10
 = 15
 = 20

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slenderness ratio, 

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slenderness ratio, 

 = 5
 = 10
 = 15
 = 20

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slenderness ratio, 

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slenderness ratio, 

Figure 3 : Variations of axisymmetric natural frequencies of a fluid-loaded circular

clamped plate with _θ_ for different values of _λ_ .

Figure 4 presents the variations of dimensionless axisymmetric natural frequencies of the fluidloaded circular clamped plate with _λ_ for different values of _θ_ . Note that the results are shown for
Ω _k,l_ - Ω0 _,_ 1. As it can be seen, the natural frequencies of the system are initially decreasing with
increasing value of _λ_ and after a certain value of _λ_ (say, _λ_ * ) they are remaining constant. The value
of _λ_ * is almost independent of values of _θ_, but it increases as the mode number increases. On the
other hand, from Figure 4 one sees the curve for _θ_ = 0.1 approaches to the one for _θ_ = 0.5 as the
mode number increases. This implies that the higher modes of the coupled system are less influenced by the values of the parameter _θ_ (this can also be clearly observed from the numerical results
presented in Figure 3).

The numerical results for asymmetric natural frequencies of the coupled system are shown in
Figures 5 and 6. Note that the numerical results are shown for Ω1 _,_ 1 and Ω1 _,_ 4. As it can be seen, the
behavior of results in this case (asymmetric vibration) is very similar to that of the axisymmetric
vibration.

 = 0.1
 = 0.2
 = 0.35
 = 0.5

00 100 200 300 400 500 600 700 800 900 1000

Mass ratio, 

00 100 200 300 400 500 600 700 800 900 1000

Mass ratio, 

160

140

120

100

 = 0.1
 = 0.2
 = 0.35
 = 0.5

250

200

150

100

 = 0.1
 = 0.2
 = 0.35
 = 0.5

00 100 200 300 400 500 600 700 800 900 1000

Mass ratio, 

00 100 200 300 400 500 600 700 800 900 1000

Mass ratio, 

Figure 4 : Variations of axisymmetric natural frequencies of a fluid-loaded circular

clamped plate with λ for different values of θ.

180

 = 5
 = 10
 = 15
 = 20

160

140

120

100

 = 5
 = 10
 = 15
 = 20

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slenderness ratio, 

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Slenderness ratio, 

Figure 5 : Variations of asymmetric natural frequencies of a fluid-loaded circular

clamped plate with _θ_ for different values of _λ_ .

 = 0.1
 = 0.2
 = 0.35
 = 0.5

200

180

160

140

120

100

 = 0.1
 = 0.2
 = 0.35
 = 0.5

00 100 200 300 400 500 600 700 800 900 1000

Mass ratio, 

00 100 200 300 400 500 600 700 800 900 1000

Mass ratio, 

Figure 6 : Variations of asymmetric natural frequencies of a fluid-loaded circular

clamped plate with _λ_ for different values of _θ_ .

8.3 Free Vibration of Circular Plates in Contact with a Bounded Compressible Fluid

In Section 3 we showed that the natural frequencies of the coupled system in the case of compressible fluid can be expressed in terms of the following dimensionless parameters

_λ_  _ρ_ _fa_

_c_  _c_ _f_

 _f_

_ρsh_

_H_
,

, _a_

_f_
_cs_

_s_

(70)

where _cf_ is the sound speed in fluid, _cs_ is the sound speed in structure (circular plate) and _c_ is the
speed ratio. Table 3 gives the dimensionless asymmetric natural frequencies ( Ω _k_, _l_  _ωk,la_ 2 _ρsh_ / _D_ )

of the coupled system for different values of _λ_, _θ_ and _c_ . Note that the numerical results are obtained
using both the pressure-based and potential-based DQMs and are given for Ω1 _,_ 1 and Ω1 _,_ 3.

It can be seen from Table 3 that the natural frequencies of the system are increasing as the value of the parameter c increases. Besides, they approach to a constant value as the magnitude of this
parameter increases. Again, the solutions of the pressure-based DQM are found to be exactly identical to those of the potential-based DQM. On the other hand, by comparing these results with those
of Table 2, one sees the results of compressible fluid model approach to those of the incompressible
fluid model as the value of c increases. Furthermore, the lower modes of the system are less influenced by compressibility of the fluid as compared with the higher modes of the system.

The effects of parameters _λ_ and _θ_ on free vibration of the coupled system have been studied in
Section 8.2. Therefore, in this section we only investigate the effects of compressibility of the fluid
on free vibration of the system. Theoretically, when speed ratio ( _c_ ) approaches to infinity while the
other parameters of the problem are fixed, the solutions of the compressible fluid model should approach to those of the incompressible fluid model. To investigate this, the axisymmetric and asymmetric natural frequencies of the coupled system are plotted against _c_ in Figures 7 and 8. These
results are obtained using _λ_ = _θ_ = 1.

It can be seen from Figures 7 and 8 that the natural frequencies of the coupled system are initially increasing with increasing the magnitude of _c_ and after a certain value of _c_ (say, _c_ * ) they are

remaining the same. Note that the higher modes of the coupled system may have several _c_ * (in general, the _k_ th mode of the system has _k_ number of _c_ * ). But when the speed ratio is large enough (i.e.,
when _c_ = _c_ * _f_ ), the natural frequencies of the system do not change by increasing the magnitude of
_c_ . The magnitude of _c_ * _f_ is found to be dependent on the mode number and increases as the mode
number increases. This implies that the higher modes of the coupled system are more affected by
compressibility of the fluid.

_c_ _θ_ Ω _k,l_ _λ_ = 1 _λ_ = 5 _λ_ = 10 _λ_ = 15 _λ_ = 20

50 0.25 Ω1,1 16.1463 9.8667 7.3921 6.1632 5.3954

Ω1,3 103.5712 90.0745 77.6572 69.2687 63.1257

1.0 Ω1,1 18.1256 12.5491 9.7714 8.2777 7.3099

Ω1,3 97.1864 90.9275 79.1996 70.8643 64.6812

100 0.25 Ω1,1 16.2113 9.8941 7.4049 6.1709 5.4008

Ω1,3 110.3537 91.8885 78.6926 69.9872 63.6669

1.0 Ω1,1 18.1591 12.5755 9.7865 8.2875 7.3169

Ω1,3 112.3034 93.9600 80.6144 71.7757 65.3424

1000 0.25 Ω1,1 16.2325 9.9031 7.4092 6.1735 5.4025

Ω1,3 111.2203 92.3899 79.0096 70.2140 63.8401

1.0 Ω1,1 18.1698 12.5841 9.7914 8.2907 7.3192

Ω1,3 112.8326 94.4451 80.9453 72.0175 65.5287

Table 3 : Dimensionless asymmetric natural frequencies of a fluid-loaded circular clamped plate. Note that the

solutions are obtained using both the pressure-based and potential-based DQMs with _n_ = 21 and _m_ = 17.

150

100

0,1
0,2
0,3
0,4

200

180

160

140

120

100

1,1
1,2
1,3
1,4

0100 101 102 103 104

Speed ratio, c

0100 101 102 103 104

Speed ratio, c

Figure 7 : Variations of axisymmetric and asymmetric natural frequencies of a fluid-loaded

circular clamped plate with _c_ ( _λ_ = _θ_ = 1).

2500

2000

1500

1000

500

Results for axisymmetric vibration of the system

0,15

2500

2000

1500

1000

500

Results for asymmetric vibration of the system

1,15

0100 101 102 103 104

Speed ratio, c

0100 101 102 103 104

Speed ratio, c

Figure 8 : Variations of higher-order natural frequencies of a fluid-loaded circular clamped plate with _c_ ( _λ_ = _θ_ = 1).

Note that Ω0 _,_ 1 ≤ Ω _k,l_ ≤ Ω0 _,_ 15 for the case of axisymmetric vibration of the system and Ω1 _,_ 1 ≤ Ω _k,l_ ≤ Ω1 _,_ 15

for the case of asymmetric vibration of the system.

9 CONCLUSIONS

Tow general procedures based on the DQM are developed to study the free vibration of circular
plates in contact with fluid. The accuracy and reliability of the proposed procedures are examined
through numerical experiments. Three new dimensionless parameters and variables are also introduced for the free vibration of the coupled system. The influences of these parameters on dynamic
behavior of the system are studied.

The main advantages of the proposed procedures are their simplicity and high accuracy. The
proposed procedures can also produce highly accurate converged results for natural frequencies of
the coupled system by using a considerably small number of grid points. Besides, the resulting matrices of the proposed procedures are found to be well-conditioned and thus there is no need to use
any preconditioner algorithm to solve the resultant eigenvalue problems.

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