An analytical study of vibrations response and control capabilities of an Electrorheological (ER) material-filled double wall plate to random excitation pressure is presented. The governing equations of motion are developed by incorporating the ER material constitutive relations to thin face plate equations. A three-parameter viscoelastic model is used to characterize ER material behaviour in pre-yield regime. The effect of electric field strength on the response of the double wall sandwich plate with ER material is investigated. Numerical results include response spectral densities and modal frequencies for several ER material core thicknesses and different electric field level

E. Jotautiene

R. Vaicaitis

S. Liu

English

Vaicaitis, R.

Liu, S.

Jotautiene, E.

JVE International

© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716    26 289. Vibrations of a double wall plane adaptive to electrorheological          materials   R. Vaicaitis*, S. Liu*, E. Jotautienė** *Columbia University, Dept. of Civil Engr. & Engr. Mechanics, New York, N.Y. 10027, USA,  E-mail: rimas@civil.columbia.edu   **Lithuanian University of Agriculture, Studentų st. 11, 53361 Kauno dist., Lithuania,  E-mail: egle.jotautiene@lzuu.lt  (Received 24 August 2007; accepted 27 August 2007)          Abstract. An analytical study of vibrations response and control capabilities of an Electrorheological (ER) material-filled double wall plate to random excitation pressure is presented. The governing equations of motion are developed by incorporating the ER material constitutive relations to thin face plate equations. A three-parameter viscoelastic model is used to characterize ER material behaviour in pre-yield regime. The effect of electric field strength on the response of the double wall sandwich plate with ER material is investigated. Numerical results include response spectral densities and modal frequencies for several ER material core thicknesses and different electric field levels.   Keywords: electrorheological materials, sandwich plate, random response.   1. Introduction  Electrorheological effect is considered to be a part of electroviscous effect and is observed in certain colloidal suspensions now generally referred to as ER fluids. Upon the application of an electric field, ER fluids experience reversible changes in rheological properties such as viscosity, elasticity and plasticity. The rheological properties of ER fluids can be increased by several orders of magnitude with electric field strength increments of the order of 1 kV/mm. The ER fluids have capacity to change from a fluid type to that of solid-like gel in milliseconds, which makes them very attractive for precise control of stress transfer in a sandwich configuration [1, 2].  The geometry of the prototype of the adaptive structure considered in this study is shown in Fig. 1. The ER material is sandwiched between two parallel aluminium plates. The face plates are rectangular, simply supported on all four edges, sealed along boundaries to contain the ER material. The top plate is exposed to uniformly distributed random pressure that could arise from various sources such as jet noise, machinery noise or convective flow of a turbulent boundary layer. The electric field is assumed to be uniform and applied across the two face plates. The main objective of this study is to examine how the vibration response and natural frequencies of the sandwich structure change with an increasing electric field. Response of ER material is assumed to be in pre-yield region for all levels of random pressure input considered in this study [3]. Numerical results include displacement response spectral densities and modal frequencies for several ER material core thicknesses and different electric field levels.  2. Equations of Motion and Solution Procedure  The behavior of ER material in pre-yield regime is characterized by linear viscoelastic theory where the shear stress τ is related to shear strain γ by complex shear modulus G* [4]:  γτ *G= .          (1)  The complex shear modulus G* is written in the form:  '''* iGGG += , 1−=i ,        (2)  where G’ is storage modulus, representing the stiffness of ER material, and G” is loss modulus, a measure of dissipative properties of the material. In the present study, we model the ER material behaviour in pre-yield regime by a three-parameter solid model, referred to as the Zener model which is made up of a spring in series with a Kelvin element as shown in Fig. 2 [5]. The stress-strain relation for a three parameter solid in time domain is given by a differential equation [5]:  γγττ 211 qqp +=+ ,         (3)  289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716    27 where:  1211 CKKp+= ,         (4)  1211 CKKq = ,         (5)  22 Kq = ,                       (6)  in which K1, K2, and C1 are functions of electric field strength V in kV/mm. Using  the differential equation for the three-parameter solid given by equation (3) and the complex representation of stress-strain relationship in equation (1), the components of the complex shear modulus can be obtain in frequency domain as:  ( )( ) 22122122122121ΩΩCKKCKKKKKG'++++= ,      (7)  ( ) 221221122ΩΩCKKCKG ''++= ,        (8)  in which Ω is the excitation frequency. Currently experimental data for the values of parameters K1, K2 and C1 as functions of electric field strength V in the three-parameter solid model for ER fluid in pre-yield regime are not available. In the present study, these parameters are calculated using Eqs. 7, 8 and approximate experimental-empirical relations of storage and loss modulus of ER material [3, 5, 6, 7]:  21VcG' ≈ ,                                                                         (9)  32 cVcG'' +≈ ,                                                                 (10)  in which c1, c2, c3 are material constants.  Fig. 1. Configuration of ER material sandwich plate  The equations of motion of sandwich plate vibrations shown in Fig. 1 are developed using a thin plate theory [8]. The two plates are made from isotropic elastic material with thickness of h1 each. The core layer is the ER material of thickness h2. An electric field used to tune the rheological properties of ER material is applied across the two face plates. It is assumed that the elastic and shear moduli of the face plates are much larger than the storage modulus of the ER core material. In addition, it is assumed that there is no slipping between face plates and the core, and that continuity of transverse displacement across the thickness of the sandwich plate is guaranteed. The effects of ER material action are incorporated into equations of motion through equilibrium equations and stress-displacement relations [8]. The vibrations of two face plates are coupled through the interaction with the constitutive relations of the ER core material. The four coupled partial differential equations are developed in terms of four geometric variables of the sandwich plate:  2bt www+= ,                                                                  (11)  as effective transverse displacement of the middle plane,  2bt wwe−= ,                                                                    (12)  as effective transverse deformation of the ER material core layer,  huu bt −=α , hvv bt −=β ,                                              (13) as effective rotations of the normal to the undeformed middle plane. In these equations, the superscripts t and b indicate top and bottom plates. The variables w, u, and v denote transverse, longitudinal and lateral deformations, respectively. The final form of the equations of motion in terms of the four described variables are [8]:  ( ) ,22222214twhhPyVxVwCwDcfryxff∂∂+==+∂∂+∂∂+−∆−ρρ            (14)  ( )22124 2422tehPeCeEheCeD frccff ∂∂=++−−∆− ρ ,  (15)  ,12 232211⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂∂∂+∂∂+××⎟⎠⎞⎜⎝⎛∂∂+=+yVxVxEhhxwtqqhVpVyxcxxα   (16)  ,12 232211⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂∂∂+∂∂+××⎟⎠⎞⎜⎝⎛∂∂+=+yVxVyEhhywtqqhVpVyxcyyβ    (17)  289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716    28 where:  ⎥⎥⎦⎤⎢⎢⎣⎡∂∂∂++⎟⎟⎠⎞⎜⎜⎝⎛∂∂−+∂∂=yxvyvxDVfcfxβα222222121,    (18)  ⎥⎥⎦⎤⎢⎢⎣⎡∂∂∂++⎟⎟⎠⎞⎜⎜⎝⎛∂∂−+∂∂=yxvxvyDVfcfyαβ222222121,    (19)  the biharmonic operator ∆4:  44224444 2yyxx ∂∂+∂∂∂+∂∂=∆ .     (20)  The other parameters appearing in these equations are defined as follows:  ( )221121ffvEhhD−= ,       (21)  ( )231112 fffvhED−= ,       (22)  where D is effective stiffness of sandwich plate, Df is stiffness of face plates. Cf, Cc, Ef, Ec, vf, ρf, ρc are damping coefficient of the face plate, damping coefficient of the core layer, modulus of elasticity of face plates, modulus of elasticity of core layer, Poisson’s ratio of face plate material, density of the face plates, density of the core material, respectively, and h=h1+h2. The details of the derivation of the equations of motion for linear and nonlinear cases can be found in [8]. When subjected to an electric field, the rheological properties of the ER material in the core layer of the sandwich structure will change so that stiffness, damping and natural frequencies will also change accordingly. To examine these changes, the linear response of the sandwich plate can be analyzed in frequency domain. First, the system of governing partial differential equations (14-17) is solved by Galerkin method, i. e. for the simply supported sandwich plate, solutions of effective transverse displacement and rotations are expressed as:  ( ) ( )bynaxmtAtyxwm nmnππsinsin,, ∑∑= ,                       (23)  ( ) ( )bynaxmtZtyxem nmnππsinsin,, ∑∑= ,                        (24)  ( ) ( )bynaxmtBtyxm nmnππα sincos,, ∑∑= ,                      (25)  ( ) ( )bynaxmtCtyxm nmnππβ cossin,, ∑∑= ,                      (26) where Amn, Zmn, Bmn, Cmn are modal amplitudes. Substituting the assumed modal solutions into equations (14-17) and making use of orthogonality of modal functions, a system of coupled differential equations for Amn, Zmn, Bmn, Cmn is obtained. Taking a Fourier transformation of these equations results in a system of algebraic equations that can be solved for the transformed modal amplitude. Then, using these results and Fourier transforms of equations (23-26), solutions for vibration response in frequency domain are determined. Since the input pressure acting on the top plate is random, the response spectral density of the effective transverse displacement w of the middle plate is determined [8, 9]:  ( ) ( )ωωω mnrsrsmnm n r s*rsmn SXXHH,y,xS ∑∑∑∑= ,   (27)  where Hmn are the frequency response functions, Smnrs are the cross-spectral densities of the generalized random pressure and Xmn are the vibration modes of simply supported plate as specified in equation (23). Similar solutions can be developed for effective displacement e and rotations α, β. The expression for the frequency response function Hmn is rather lengthy and it is not included in this paper. However, the details of frequency response function and of natural frequencies of the sandwich construction can be found in [8].  For random pressure uniformly distributed over the top plate surface, the cross-spectral densities of the generalized random forces are:  ( )( )⎪⎩⎪⎨⎧=,0,,,,2564otherwiseoddallsrnmmnrsSSpmnrs πωω      (28)  where Sp(ω) is the power spectral density of random sound pressure acting on the top plate. In the present study, the random pressure is assumed to be band limited Gaussian white noise with spectral amplitude computed from:  ( ) 10/200 10SPLppSSωω∆== ,                                     (29)  where p0=2x10-5 N/m2 is the reference pressure, SPL is the sound pressure level expressed in decibels and ω∆  is the selected frequency bandwidth. The frequency range of the selected band limited Gaussian white noise pressure is 0-1000 Hz.   Fig. 2. Three parameter Solid Model   289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716    29 3. Numerical Results  Numerical results are presented for a sandwich plate construction shown in Fig. 1 with the following dimensions: a=500 mm, b=250mm, h1=0,4 mm, h2=2,0 mm. The density of ER material is 1060 kg/m3. The face plates are made of aluminium with the following material properties: material density=2768 kg/m3, elastic modulus=6,898x1010 N/m2, Poisson’s ratio=0,3. The values of the ER core material parameters K1, K2, C1, Ec, Cc are given in Table 1 for different electric field strengths V.  Table 1. Values of K1, K2, C1, Ec, Cc  Electric Field Strength (kV/mm) Parameter V=0 V=1 V=2 V=3 K1 (N/m2) 282,0 68814,0 486090,0 1445240,0 K2 (N/m2) 587,0 181001,0 338828,0 652529,0 C1 (Ns/m2) 10,0 50,0 250,0 600,0 Ec (N/m2) 500,0 44265,9 106965,0 169788,0 Cc (Ns/m2) 8,0 6,0 4,0 2,0  Modal damping coefficients of the face plates are determined from:  ⎟⎟⎠⎞⎜⎜⎝⎛=klkl ωωξξ 1111 ,                                                          (30)  where ωkl are the modal frequencies and ξ11=0,01. Displacement response spectral densities corresponding to SPL=110 dB are presented in Fig. 3 for different electric field strengths. These results indicate that the dominant resonant frequencies increase with increasing electric field and consequently the displacement spectral density levels decrease. The sandwich plate becomes more stiff and the dissipation energy of the ER core increases with increasing electric strength. Fig. 4 shows more clearly the trend of change of natural modal frequencies with variation of an applied electric field for the three selected dominant modes of the sandwich structure. The results given in Fig. 5 illustrate the change of first modal frequency at electric field levels of 0.0, 1.0, 2.0, 3.0 kV/mm as a function of ratio of ER material core thickness to the thickness of face plates. As can be observed from these results, the first modal frequency decreased with increasing ratio of h2/h1 for cases of no voltage and for low voltage levels. This decrease is mostly due to the added mass effect of the core material. For voltage levels of larger than 1,35 kV/mm, the first modal frequency shows tendency to increase with increasing ratio h2/h1. This phenomenon can be explained as follows: the ER material core contributes both mass inertia and stiffness to the sandwich structure upon application of the electric field. The contribution of mass inertia is fixed while the contribution of stiffness increases with increasing electric field. At a certain electric field level, the stiffening effects of the core become more pronounced than the added mass and natural frequency increases with increasing thickness ratio. The results presented in Fig. 5 could be useful in choosing the optimal core thickness for active vibration control with ER materials. The results presented in Fig. 6 illustrate the change in the first modal frequency with increasing electric field strength for several values of core to face plate thickness ratio h2/h1. These results also illustrate that for the chosen geometry and material properties of the sandwich construction, the first modal frequency increases more rapidly with increasing ratio h2/h1 after electric field strength applied to ER material exceeds about 1,35 kV/mm.    Fig. 3. Displacement spectral density for SPL=110 dB    Fig. 4. Modal natural frequencies vs. electric field levels   Fig. 5. First modal frequency vs. ER core thickness ratio for different electric field levels   289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716    30  Fig. 6. First modal frequency vs. electric field for different core thickness ratios  4. Conclusions   Displacement vibration amplitudes were significantly reduced as applied electric field to the ER core increased. Modal frequencies shifted to higher values with an increased electric field, which indicated that the sandwich structure becomes more stiff upon application of the electric field. The effect of different ER core layer thickness on structural response could be useful in choosing optimal core thickness for vibration control. Depending on geometric and material properties of the sandwich plate construction, a thicker ER core might not be beneficial for vibration response control at low levels of electric field strength.                             References   [1] Preumont A. Vibration Control of Active Structures: An Introduction, 2nd Edition. The Netherlands. - Kluwer Academic Publishers, 2002. [2] Weiss K. D., Coulter J. P., Carlson J. D. Material Aspects of Electrorheological Systems. Journal of Intelligent Material Systems and Structures, 1993, Vol. 4, p. 13-33. [3] Yalcintas M., Coulter J. L., Don D. L. Structural Modeling and Optimal Control of Electrorheological Material Based Adaptive Beams. Smart Material Structures, 1995, Vol. 4, p. 207-214. [4] Flugge W. Viscoelasticity, 2nd edition. - Springer-Verlag, 1975. [5] Gamota D. R., Filisko F. E. Dynamic Mechanical Studies of Electrorheological Materials: Moderate Frequencies. J. Rheology, 1991, Vol. 35(3), p. 399-425. [6] Yalcintas M., Coulter J. L. Magnetorheological and Electrorheological Materials in Adaptive Structures and Their Perfomance Comparison. Smart Mater. Struct., 1999, No 8, p. 560-573. [7] Liu S. An Analytical Study of Electrorheological Material Based Adaptive Structures. Doctoral Dissertation. - Columbia University, New York, N. Y., 2005. [8] Lin Y. K. Probabilistic Theory of Structural Dynamics. McGraw-Hill, N. Y., 1967. 

Vibrations of a double wall plane adaptive to electrorheological materials

Journal of Mechatronics and Artificial Intelligence in Engineering

Journal of Vibroengineering

© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716 
 
 
 
26 
289. Vibrations of a double wall plane adaptive to electrorheological  
        materials  
 
R. Vaicaitis*, S. Liu*, E. Jotautienė** 
*Columbia University, Dept. of Civil Engr. & Engr. Mechanics, New York, N.Y. 10027, USA,  
E-mail: rimas@civil.columbia.edu   
**Lithuanian University of Agriculture, Studentų st. 11, 53361 Kauno dist., Lithuania,  
E-mail: egle.jotautiene@lzuu.lt 
 
(Received 24 August 2007; accepted 27 August 2007) 
 
 
 
 
 
 
 
 
 
Abstract. An analytical study of vibrations response and control capabilities of an Electrorheological (ER) material-filled 
double wall plate to random excitation pressure is presented. The governing equations of motion are developed by 
incorporating the ER material constitutive relations to thin face plate equations. A three-parameter viscoelastic model is 
used to characterize ER material behaviour in pre-yield regime. The effect of electric field strength on the response of the 
double wall sandwich plate with ER material is investigated. Numerical results include response spectral densities and 
modal frequencies for several ER material core thicknesses and different electric field levels.  
 
Keywords: electrorheological materials, sandwich plate, random response. 
 
 
1. Introduction 
 
Electrorheological effect is considered to be a part of 
electroviscous effect and is observed in certain colloidal 
suspensions now generally referred to as ER fluids. Upon 
the application of an electric field, ER fluids experience 
reversible changes in rheological properties such as 
viscosity, elasticity and plasticity. The rheological 
properties of ER fluids can be increased by several orders 
of magnitude with electric field strength increments of the 
order of 1 kV/mm. The ER fluids have capacity to change 
from a fluid type to that of solid-like gel in milliseconds, 
which makes them very attractive for precise control of 
stress transfer in a sandwich configuration [1, 2].  
The geometry of the prototype of the adaptive structure 
considered in this study is shown in Fig. 1. The ER 
material is sandwiched between two parallel aluminium 
plates. The face plates are rectangular, simply supported on 
all four edges, sealed along boundaries to contain the ER 
material. The top plate is exposed to uniformly distributed 
random pressure that could arise from various sources such 
as jet noise, machinery noise or convective flow of a 
turbulent boundary layer. The electric field is assumed to 
be uniform and applied across the two face plates. The 
main objective of this study is to examine how the 
vibration response and natural frequencies of the sandwich 
structure change with an increasing electric field. Response 
of ER material is assumed to be in pre-yield region for all 
levels of random pressure input considered in this study 
[3]. Numerical results include displacement response 
spectral densities and modal frequencies for several ER 
material core thicknesses and different electric field levels. 
 
2. Equations of Motion and Solution Procedure 
 
The behavior of ER material in pre-yield regime is 
characterized by linear viscoelastic theory where the shear 
stress τ is related to shear strain γ by complex shear 
modulus G* [4]: 
 
γτ *G= .          (1) 
 
The complex shear modulus G* is written in the form: 
 
'''* iGGG += , 1−=i ,        (2) 
 
where G’ is storage modulus, representing the stiffness of 
ER material, and G” is loss modulus, a measure of 
dissipative properties of the material. In the present study, 
we model the ER material behaviour in pre-yield regime by 
a three-parameter solid model, referred to as the Zener 
model which is made up of a spring in series with a Kelvin 
element as shown in Fig. 2 [5]. The stress-strain relation for 
a three parameter solid in time domain is given by a 
differential equation [5]: 
 
γγττ ?? 211 qqp +=+ ,         (3) 
 289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716 
 
 
 
27 
where: 
 
1
21
1 C
KK
p
+= ,         (4) 
 
1
21
1 C
KKq = ,         (5) 
 
22 Kq = ,                       (6) 
 
in which K1, K2, and C1 are functions of electric field 
strength V in kV/mm. 
Using  the differential equation for the three-parameter 
solid given by equation (3) and the complex representation 
of stress-strain relationship in equation (1), the components 
of the complex shear modulus can be obtain in frequency 
domain as: 
 
( )
( ) 221221
22
122121
Ω
Ω
CKK
CKKKKKG' ++
++= ,      (7) 
 
( ) 221221
1
2
2
Ω
Ω
CKK
CKG '' ++= ,        (8) 
 
in which Ω is the excitation frequency. Currently 
experimental data for the values of parameters K1, K2 and 
C1 as functions of electric field strength V in the three-
parameter solid model for ER fluid in pre-yield regime are 
not available. In the present study, these parameters are 
calculated using Eqs. 7, 8 and approximate experimental-
empirical relations of storage and loss modulus of ER 
material [3, 5, 6, 7]: 
 
2
1VcG
' ≈ ,                                                                         (9) 
 
32 cVcG
'' +≈ ,                                                                 (10) 
 
in which c1, c2, c3 are material constants. 
 
Fig. 1. Configuration of ER material sandwich plate 
 
The equations of motion of sandwich plate vibrations 
shown in Fig. 1 are developed using a thin plate theory [8]. 
The two plates are made from isotropic elastic material 
with thickness of h1 each. The core layer is the ER material 
of thickness h2. An electric field used to tune the 
rheological properties of ER material is applied across the 
two face plates. It is assumed that the elastic and shear 
moduli of the face plates are much larger than the storage 
modulus of the ER core material. In addition, it is assumed 
that there is no slipping between face plates and the core, 
and that continuity of transverse displacement across the 
thickness of the sandwich plate is guaranteed. The effects 
of ER material action are incorporated into equations of 
motion through equilibrium equations and stress-
displacement relations [8]. The vibrations of two face 
plates are coupled through the interaction with the 
constitutive relations of the ER core material. The four 
coupled partial differential equations are developed in 
terms of four geometric variables of the sandwich plate: 
 
2
bt ww
w
+= ,                                                                  (11) 
 
as effective transverse displacement of the middle plane, 
 
2
bt ww
e
−= ,                                                                    (12) 
 
as effective transverse deformation of the ER material core 
layer, 
 
h
uu
bt −=α , 
h
vv bt −=β ,                                              (13) 
as effective rotations of the normal to the undeformed 
middle plane. In these equations, the superscripts t and b 
indicate top and bottom plates. The variables w, u, and v 
denote transverse, longitudinal and lateral deformations, 
respectively. The final form of the equations of motion in 
terms of the four described variables are [8]: 
 
( ) ,2
22
2
2
21
4
t
whh
P
y
V
x
V
wCwD
cf
ryx
ff
∂
∂+=
=+∂
∂+∂
∂+−∆−
ρρ
?
            (14) 
 
( ) 2
2
1
2
4 2422
t
ehPeCeE
h
eCeD f
r
ccff ∂
∂=++−−∆− ρ?? ,  (15) 
 
,
12 2
3
2
211
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂
∂
∂+∂
∂+×
×⎟⎠
⎞⎜⎝
⎛
∂
∂+=+
y
V
x
V
xEh
h
x
w
t
qqhVpV
yx
c
xx
α
?
   (16) 
 
,
12 2
3
2
211
⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂+∂
∂
∂
∂+∂
∂+×
×⎟⎠
⎞⎜⎝
⎛
∂
∂+=+
y
V
x
V
yEh
h
y
w
t
qqhVpV
yx
c
yy
β
?
    (17) 
 289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716 
 
 
 
28 
where: 
 
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
∂++⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂−+∂
∂=
yx
v
y
v
x
DV f
c
f
x
βα
2
2
2
2
2
2
1
2
1
,    (18) 
 
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
∂++⎟⎟⎠
⎞
⎜⎜⎝
⎛
∂
∂−+∂
∂=
yx
v
x
v
y
DV f
c
f
y
αβ
2
2
2
2
2
2
1
2
1
,    (19) 
 
the biharmonic operator ∆4: 
 
4
4
22
4
4
4
4 2
yyxx ∂
∂+∂∂
∂+∂
∂=∆ .     (20) 
 
The other parameters appearing in these equations are 
defined as follows: 
 
( )2
2
1
1
2
1
f
f
v
Ehh
D −= ,       (21) 
 
( )2
3
1
112 f
f
f
v
hE
D −= ,       (22) 
 
where D is effective stiffness of sandwich plate, Df is 
stiffness of face plates. Cf, Cc, Ef, Ec, vf, ρf, ρc are damping 
coefficient of the face plate, damping coefficient of the 
core layer, modulus of elasticity of face plates, modulus of 
elasticity of core layer, Poisson’s ratio of face plate 
material, density of the face plates, density of the core 
material, respectively, and h=h1+h2. 
The details of the derivation of the equations of motion 
for linear and nonlinear cases can be found in [8]. 
When subjected to an electric field, the rheological 
properties of the ER material in the core layer of the 
sandwich structure will change so that stiffness, damping 
and natural frequencies will also change accordingly. To 
examine these changes, the linear response of the sandwich 
plate can be analyzed in frequency domain. First, the 
system of governing partial differential equations (14-17) is 
solved by Galerkin method, i. e. for the simply supported 
sandwich plate, solutions of effective transverse 
displacement and rotations are expressed as: 
 
( ) ( )
b
yn
a
xm
tAtyxw
m n
mn
ππ
sinsin,, ∑∑= ,                       (23) 
 
( ) ( )
b
yn
a
xm
tZtyxe
m n
mn
ππ
sinsin,, ∑∑= ,                        (24) 
 
( ) ( )
b
yn
a
xm
tBtyx
m n
mn
ππα sincos,, ∑∑= ,                      (25) 
 
( ) ( )
b
yn
a
xm
tCtyx
m n
mn
ππβ cossin,, ∑∑= ,                      (26) 
where Amn, Zmn, Bmn, Cmn are modal amplitudes. 
Substituting the assumed modal solutions into equations 
(14-17) and making use of orthogonality of modal 
functions, a system of coupled differential equations for 
Amn, Zmn, Bmn, Cmn is obtained. Taking a Fourier 
transformation of these equations results in a system of 
algebraic equations that can be solved for the transformed 
modal amplitude. Then, using these results and Fourier 
transforms of equations (23-26), solutions for vibration 
response in frequency domain are determined. Since the 
input pressure acting on the top plate is random, the 
response spectral density of the effective transverse 
displacement w of the middle plate is determined [8, 9]: 
 
( ) ( )ωωω mnrsrsmn
m n r s
*
rsmn SXXHH,y,xS ∑∑∑∑= ,   (27) 
 
where Hmn are the frequency response functions, Smnrs are 
the cross-spectral densities of the generalized random 
pressure and Xmn are the vibration modes of simply 
supported plate as specified in equation (23). Similar 
solutions can be developed for effective displacement e and 
rotations α, β. The expression for the frequency response 
function Hmn is rather lengthy and it is not included in this 
paper. However, the details of frequency response function 
and of natural frequencies of the sandwich construction can 
be found in [8].  
For random pressure uniformly distributed over the top 
plate surface, the cross-spectral densities of the generalized 
random forces are: 
 
( ) ( )
⎪⎩
⎪⎨
⎧
=
,0
,,,,
256
4
otherwise
oddallsrnm
mnrs
S
S
p
mnrs π
ω
ω      (28) 
 
where Sp(ω) is the power spectral density of random sound 
pressure acting on the top plate. In the present study, the 
random pressure is assumed to be band limited Gaussian 
white noise with spectral amplitude computed from: 
 
( ) 10/200 10 SPLp pSS ωω ∆== ,                                     (29) 
 
where p0=2x10-5 N/m2 is the reference pressure, SPL is the 
sound pressure level expressed in decibels and ω∆  is the 
selected frequency bandwidth. The frequency range of the 
selected band limited Gaussian white noise pressure is 0-
1000 Hz. 
 
 
Fig. 2. Three parameter Solid Model 
 
 289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716 
 
 
 
29 
3. Numerical Results 
 
Numerical results are presented for a sandwich plate 
construction shown in Fig. 1 with the following 
dimensions: a=500 mm, b=250mm, h1=0,4 mm, h2=2,0 
mm. The density of ER material is 1060 kg/m3. 
The face plates are made of aluminium with the 
following material properties: material density=2768 
kg/m3, elastic modulus=6,898x1010 N/m2, Poisson’s 
ratio=0,3. The values of the ER core material parameters 
K1, K2, C1, Ec, Cc are given in Table 1 for different electric 
field strengths V. 
 
Table 1. Values of K1, K2, C1, Ec, Cc 
 
Electric Field Strength (kV/mm) Parameter 
V=0 V=1 V=2 V=3 
K1 (N/m2) 282,0 68814,0 486090,0 1445240,0 
K2 (N/m2) 587,0 181001,0 338828,0 652529,0 
C1 (Ns/m2) 10,0 50,0 250,0 600,0 
Ec (N/m2) 500,0 44265,9 106965,0 169788,0 
Cc (Ns/m2) 8,0 6,0 4,0 2,0 
 
Modal damping coefficients of the face plates are 
determined from: 
 
⎟⎟⎠
⎞
⎜⎜⎝
⎛=
kl
kl ω
ωξξ 1111 ,                                                          (30) 
 
where ωkl are the modal frequencies and ξ11=0,01. 
Displacement response spectral densities corresponding to 
SPL=110 dB are presented in Fig. 3 for different electric 
field strengths. These results indicate that the dominant 
resonant frequencies increase with increasing electric field 
and consequently the displacement spectral density levels 
decrease. The sandwich plate becomes more stiff and the 
dissipation energy of the ER core increases with increasing 
electric strength. Fig. 4 shows more clearly the trend of 
change of natural modal frequencies with variation of an 
applied electric field for the three selected dominant modes 
of the sandwich structure. The results given in Fig. 5 
illustrate the change of first modal frequency at electric 
field levels of 0.0, 1.0, 2.0, 3.0 kV/mm as a function of 
ratio of ER material core thickness to the thickness of face 
plates. As can be observed from these results, the first 
modal frequency decreased with increasing ratio of h2/h1 
for cases of no voltage and for low voltage levels. This 
decrease is mostly due to the added mass effect of the core 
material. For voltage levels of larger than 1,35 kV/mm, the 
first modal frequency shows tendency to increase with 
increasing ratio h2/h1. This phenomenon can be explained 
as follows: the ER material core contributes both mass 
inertia and stiffness to the sandwich structure upon 
application of the electric field. The contribution of mass 
inertia is fixed while the contribution of stiffness increases 
with increasing electric field. At a certain electric field 
level, the stiffening effects of the core become more 
pronounced than the added mass and natural frequency 
increases with increasing thickness ratio. The results 
presented in Fig. 5 could be useful in choosing the optimal 
core thickness for active vibration control with ER 
materials. The results presented in Fig. 6 illustrate the 
change in the first modal frequency with increasing electric 
field strength for several values of core to face plate 
thickness ratio h2/h1. These results also illustrate that for 
the chosen geometry and material properties of the 
sandwich construction, the first modal frequency increases 
more rapidly with increasing ratio h2/h1 after electric field 
strength applied to ER material exceeds about 1,35 kV/mm.
  
 
 
Fig. 3. Displacement spectral density for SPL=110 dB  
 
 
Fig. 4. Modal natural frequencies vs. electric field levels  
 
Fig. 5. First modal frequency vs. ER core thickness ratio for 
different electric field levels  
 289. VIBRATIONS OF A DOUBLE WALL PLANE ADAPTIVE TO ELECTRORHEOLOGICAL MATERIALS. R. VAICAITIS, S. LIU, E. JOTAUTIENĖ 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 JULY/SEPTEMBER,   VOLUME  9,  ISSUE  3,  ISSN 1392-8716 
 
 
 
30 
 
Fig. 6. First modal frequency vs. electric field for different core 
thickness ratios 
 
4. Conclusions  
 
Displacement vibration amplitudes were significantly 
reduced as applied electric field to the ER core increased. 
Modal frequencies shifted to higher values with an 
increased electric field, which indicated that the sandwich 
structure becomes more stiff upon application of the 
electric field. The effect of different ER core layer 
thickness on structural response could be useful in 
choosing optimal core thickness for vibration control. 
Depending on geometric and material properties of the 
sandwich plate construction, a thicker ER core might not be 
beneficial for vibration response control at low levels of 
electric field strength. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
References  
 
[1] Preumont A. Vibration Control of Active Structures: An 
Introduction, 2nd Edition. The Netherlands. - Kluwer 
Academic Publishers, 2002. 
[2] Weiss K. D., Coulter J. P., Carlson J. D. Material Aspects 
of Electrorheological Systems. Journal of Intelligent Material 
Systems and Structures, 1993, Vol. 4, p. 13-33. 
[3] Yalcintas M., Coulter J. L., Don D. L. Structural Modeling 
and Optimal Control of Electrorheological Material Based 
Adaptive Beams. Smart Material Structures, 1995, Vol. 4, p. 
207-214. 
[4] Flugge W. Viscoelasticity, 2nd edition. - Springer-Verlag, 
1975. 
[5] Gamota D. R., Filisko F. E. Dynamic Mechanical Studies of 
Electrorheological Materials: Moderate Frequencies. J. 
Rheology, 1991, Vol. 35(3), p. 399-425. 
[6] Yalcintas M., Coulter J. L. Magnetorheological and 
Electrorheological Materials in Adaptive Structures and Their 
Perfomance Comparison. Smart Mater. Struct., 1999, No 8, p. 
560-573. 
[7] Liu S. An Analytical Study of Electrorheological Material 
Based Adaptive Structures. Doctoral Dissertation. - Columbia 
University, New York, N. Y., 2005. 
[8] Lin Y. K. Probabilistic Theory of Structural Dynamics. 
McGraw-Hill, N. Y., 1967. 


https://www.extrica.com/article/10120/pdf

Vibrations of a double wall plane adaptive to electrorheological materials

Abstract

Similar works

Full text

Available Versions

JVE International

Journal of Mechatronics and Artificial Intelligence in Engineering

Journal of Vibroengineering