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Finite element dynamic analysis of rotating orthotropic sandwich annular plates
Composite Structures 62 (2003) 205–212 www.elsevier.com/locate/compstruct
Finite element dynamic analysis of rotating orthotropic sandwich annular plates Horng-Jou Wang, Yu-Ren Chen, Lien-Wen Chen
*
Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
Abstract The finite element dynamic analysis of rotating orthotropic sandwich annular plates with a viscoelastic central layer and two polar orthotropic laminated face layers are analysed in the present paper. The axisymmetric discrete layer annular element and HamiltonÕs principle are employed to derive the finite element equations of motion for a rotating sandwich plate including transverse shear effects. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved, and the modal frequencies and loss factors of the sandwich plate are extracted. The results of the symmetric and non-symmetric sandwich annular plates are both presented. The effects of outer radius to thickness ratio, orthotropic properties of face layers, the thickness of the viscoelastic central layer, and thickness of face layers on the natural frequencies and modal loss factors of sandwich annular plates are discussed. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Rotating; Orthotropic; Sandwich; Annular plate; Viscoelastic; Discrete layer annular finite element
1. Introduction It is well known that a sandwich structure with a high damping viscoelastic central layer has high damping capacity and high resistance to excessive vibration and noise. The major damping mechanisms in vibration are due to the extensional and shear deformations of viscoelastic materials. The damping property of a sandwich structure with a viscoelastic core is usually expressed as the modal loss factor for the corresponding vibration mode. The early investigations on the vibration and damping properties of basic structures, such as beams and rectangular plates, with a viscoelastic middle layer can be found in Refs. [1–4]. The annular plates have received a great deal of attention because of their widely used in many mechanical and electronic devices, such as computer disk memory units, circular saws, and turbine rotors. The vibration behavior of single-layer annular plates has been studied by many investigators. Irie et al. [5] discussed the natural frequencies of thick annular plates, Wang et al. [6] investigated free vibration analysis of annular plates by the differential quadrature method, and Lin and Tseng *
Corresponding author. Tel.: +886-6-2757575x62143; fax: +886-62352973. E-mail address: [email protected] (L.-W. Chen). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00115-6
[7] studied the free vibration of polar orthotropic laminated circular and annular plates. The free vibration problems of polar orthotropic annular plates were solved by Ramaiah and Vijayakumar [8], Narita [9] presented the natural frequencies of completely free annular and circular plates having polar orthotropy, and Liu et al. [10] investigated the axisymmetric vibrations of rotating annular plates by the finite element method. The vibration and stress analysis of thin rotating discs using annular finite elements were studied by Kirkhope and Wilson [11]. Liang et al. [12] worked on the vibration and stability problems of rotating polar orthotropic annular disks subjected to a stationary concentrated transverse load. As for sandwich annular plates, Roy and Ganesan [13] developed a finite element method for vibration and damping analyses of circular plates with the constrained damping layer treatment. Yu and Huang [14] derived the equations of motion of a three-layered circular plate based on thin shell theory to handle the very thin viscoelastic layer problem. They both worked on a stationary sandwich plate with isotropic face layers. The studies of vibration and damping properties of a stationary three-layered annular plate with a viscoelastic middle layer and two polar orthotropic laminated face layers were presented by Wang and Chen [15]. Seubert et al. [16] investigated the rotating disks with two
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isotropic face layers and a viscoelastic core by using a commercial finite element analysis program ANSYSâ and experimental measurements. The damping properties of laminated composites are also of importance. Saravanos and Pereira [17] studied the dynamic characteristics of composite plates with interply damping layers. The dynamic properties of E-glass fiber reinforced composites were discussed by Gibson and Plunkett [18]. It has been found that the loss factor for epoxy matrix is very low compared with that of high damping viscoelastic materials. It is thus reasonable to assume that the two laminated surface layers are undamped in the present analysis. The dynamic behavior and damping properties of rotating orthotropic sandwich annular plates with a viscoelastic central layer and two polar orthotropic laminated face layers are studied here. The discrete layer annular elements and the Ritz finite element method are adopted to obtain the finite element equations of motion for the sandwich plate. The extensional and shear moduli of the linear isotropic viscoelastic material layer are described by complex quantities. The natural frequencies and modal loss factors of the sandwich plate are obtained by solving the complex eigenvalue problem. The effects of outer radius to thickness ratio, orthotropic properties of face layers, the thickness of the viscoelastic core layer, and thickness of face layers on the modal natural frequencies and loss factors of sandwich annular plates are investigated.
2. Formulation As shown in Fig. 1, the rotating sandwich annular plate of inner radius a and outer radius b is considered. The sandwich plate is composed of two polar orthotropic face layers and a viscoelastic core layer, and rotates about the z-axis at a constant angular speed X. The polar orthotropic face layers, designated as the Layer 1 and the Layer 3 respectively, are pure elastic and homogeneous (e.g. high-modulus graphite, high-strength graphite epoxy, and ultrahigh modulus graphite epoxy [19]). Also, the face layers are assumed to be undamped, because the loss factors for epoxy matrix are very small compared with those of the high damping viscoelastic materials in the middle layer [18]. The middle layer, designated as the Layer 2, is a linear viscoelastic material layer (e.g. rubber) and is capable of adhesively dissipating vibratory motions. The interfaces between layers are assumed to be perfectly bonding. The thicknesses of three layers are h1 , h2 , and h3 , respectively. Because the thickness changes of three layers have insignificant influences on natural frequencies and modal loss factors of the system [20], it is assumed that the thicknesses of three layers are all constant during deformation.
Fig. 1. The rotating three-layered composite annular plate with two polar orthotropic face layers and a viscoelastic core layer.
The discrete layer annular finite element [21], as shown in Fig. 2(a), is employed to model the sandwich annular plate system. The axisymmetric discrete layer annular finite element of inner radius ri and outer radius ro for the Layer i has eight degrees of freedom: the A B displacements in the r-direction, UiA , Uiþ1 , UiB and Uiþ1 , A B the transverse displacements, W and W , and the rotation angles, HA and HB . When the thicknesses of three layers are all constant during deformation, the nodal degrees of freedom for the three-layered discrete layer annular finite element are shown in Fig. 2(b). The displacement field in the basic element for the Layer i can be expressed in terms of the nodal degrees of freedom as ui ðr; z; tÞ ¼ L1;i ðzÞL2 ðrÞUei ðtÞ;
ð1Þ
where ui is the displacement field vector in the basic element, L1;i is the transverse thickness interpolation matrix (the second subscript i means layer i hereafter), L2 is the interpolation matrix in the r-direction, and Uei is the vector of nodal displacements of the element. These vectors and matrices are given by ui ðr; z; tÞ ui ðr; z; tÞ ¼ ; ð2Þ wi ðr; tÞ " # 1 1 hzi þ hzi 0 2 2 L1;i ðzÞ ¼ ; ð3Þ 0 0 1 2 A 3 0 0 0 nBu 0 0 0 nu ð4Þ L2 ðrÞ ¼ 4 0 nA 0 0 0 nBu 0 0 5; u A A 0 0 nw nH 0 0 nBw nBH T A B Uei ðtÞ ¼ UiA Uiþ1 ð5Þ W A HA UiB Uiþ1 W B HB
H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
3
2
o 6 or 6 61 D¼6 6 r 6 4o oz
207
07 7 7 07 7: 7 o5
ð8Þ
or
The stress–strain relations for the Layer i can be expressed as 8 9 2 9 38 C11;i C12;i 0 < er;i = < rr;i = rh;i ¼ 4 C21;i C22;i 0 5 eh;i ð9aÞ : ; : ; crz;i 0 0 C44;i srz;i or ri ¼ Ci i ;
ð9bÞ
where Ci is the elasticity matrix. For the polar orthotropic material in the face layers ði ¼ 1; 3Þ, the components of the elasticity matrix are Er;i ; 1 mrh;i mhr;i Eh;i ¼ ; 1 mrh;i mhr;i
C11;i ¼ C22;i
C44;i ¼ j2 Grz;i ;
C12;i ¼ C21;i ¼
mhr;i Er;i ; 1 mrh;i mhr;i ð10a–cÞ
j2 ¼
p2 ; 12
ð10d;eÞ
where E is the YoungÕs modulus, G is the shear modulus, m is the PoissonÕs ratio, and j2 is the shear correction factor. For the isotropic linear viscoelastic material in the central layer, assuming that viscoelastic material is almost incompressible [22], the components of the elasticity matrix are given by C11;2 ¼ C22;2 ¼ Fig. 2. The discrete layer annular finite element: (a) basic element; (b) three-layered element.
and nA u
¼ ð1 nÞ;
nA w
2
nBu
¼ n;
¼ ð1 3n þ 2n3 Þ;
2 3 nA H ¼ ðn 2n þ n Þ; r ri : n¼ ro ri
nBw ¼ ð3n2 2n3 Þ;
ð6a–dÞ
nBH ¼ ðn2 þ n3 Þ; ð6e–gÞ
The linear strains in the Layer i of the annular plate can be written in terms of the displacement i ¼ Dui ;
ð7Þ T
where the strain vector i ¼ fer;i eh;i crz;i g and D is the differential operator matrix
E2 ; 1 m22
E2 ; 2ð1 þ m2 Þ m2 ¼ 0:5 d;
C44;2 ¼
C12;2 ¼ C21;2 ¼
m 2 E2 ; 1 m22
E2 ¼ Ev ð1 þ jgv Þ; ð11a–eÞ
where gv is the loss factor of the viscoelastic material, d is assumed to be a small constant real value (e.g. d ¼ 0:01) which is introduced to avoid material stiffness pffiffiffiffiffiffiffi singularities, and j ¼ 1. The kinetic energy of the element e of the Layer i can be written as [10] I n o 1 2 Tie ¼ qi u_ 2i þ ½ðr þ ui ÞX þ w_ 2i dV ð12aÞ 2 Ve or in the matrix form I h T 1 e Ti ¼ qi u_ Ti u_ i þ X2 L3 rð0Þ L3 rð0Þ þ X2 ðL3 ui ÞT ðL3 ui Þ 2 Ve i þ 2X2 ðL3 ui ÞT L3 rð0Þ dV ; ð12bÞ where the matrix L3 and the position vector rð0Þ are
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1 L3 ¼ 0
0 ; 0
r
r ¼ ; z
ð0Þ
ð13a;bÞ
Gei
I ¼2 ðD1 L4 L1;i L2 ÞT b r ei ðD2 L4 L1;i L2 Þ Ve
þ ðD1 L5 L1;i L2 ÞT b r ei ðD2 L5 L1;i L2 Þ
H
q is the mass density and Ve represents a volume integral. The first term of kinetic energy in Eq. (12b) is due to vibratory motions of the plate. The second term is due to the rigid body motion and actually no contribution to the equations of motion. The third term is the supplementary strain energy due to displacement dependent centrifugal force. The fourth term is resulting from the initial in-plane stresses that are developed by the centrifugal force. The strain energy of the annular element e of the Layer i is I I 1 T Uie ¼ rTi i dV þ rei N ð14Þ i dV : 2 Ve Ve The second term in Eq. (14) is the additional strain energy induced by rotation [10]. The rotation-induced stress vector rei and the non-linear strain vector N i are T e rei ¼ rer;i reh;i serz;i ¼ Ci DðL1;i L2 ÞUi ð15Þ and
N i ¼
8 > 1 > > > > > 2 > > <
oui or
!2
1 owi þ 2 or 1 u 2 i
> > > > > > > > :
2 r oui oui or oz
!2 9 > > > > > > > > = > > > > > > > > ;
1 T e r i ðD3 L5 L1;i L2 Þ dV ; þ ðD3 L5 L1;i L2 Þ b 2
Fei
¼
I h
i qi X2 ðL3 L1;i L2 ÞT L3 rð0Þ dV
ð16Þ
e
in which Ui is the equilibrium nodal displacement vector of the rotating annular element for the Layer i and can be evaluated from the solutions of static problems. The details of this section are listed in Appendix A. Substituting Eqs. (1), (7) and (9) into Eqs. (12) and (14), the HamiltonÕs principle [23], Z e e d ðTi Ui Þdt ¼ 0; ð17Þ t
is used to derive element dynamic equilibrium equations. The element differential equations are expressed in matrix form as € e þ ðKe þ Ge ÞUe ¼ Fe ; Mei U i i i i i
ð18Þ
ð22Þ
Ve
and D1 ¼
81 > <2
9
o or > =
0 ; > ; : o > 2 eor rr;i 0 6 0 re e bi ¼ 4 r h;i 0 0
L4 ¼ ½ 1
0 ;
8o9 > = < or > D2 ¼ 0 ; > ; :o> 3 oz 0 0 7 5; serz;i L5 ¼ ½ 0
D3 ¼
1 :
8 9 > = <0> 1
r > ; : > 0
;
ð23a–dÞ ð23e;fÞ
Gei
is usually called the geometric The stiffness matrix stiffness matrix that increases the plate stiffness. Assembling the contribution of all elements, the global finite element equation can be obtained € þ ðK þ GÞU ¼ F; MU
;
ð21Þ
ð24Þ
where M is the global mass matrix, K is the global stiffness matrices, G is the global geometry stiffness matrix and F is the global centrifugal force vector due to rotation. To study the free vibration of rotating annular plates, the forcing term in Eq. (24) is neglected [10]. Substituting UðtÞ ¼ U0 ejkt into Eq. (24), the following eigenvalue equation can be obtained: ðK þ GÞU0 ¼ k2 MU0 ;
ð25Þ
where k is a complex number because of complex-valued terms of the stiffness matrices. The circular natural frequency and modal loss factor of the sandwich annular plate system can be calculated from the formulae [15] qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Imðk2 Þ 2 x ¼ ReðkÞ ; g¼ : ð26a;bÞ Reðk2 Þ
3. Numerical results and discussion
Mei ,
stiffness matrices where the elemental mass matrix Kei and Gei , and force vector Fei of the Layer i are I h i T Mei ¼ qi ðL1;i L2 Þ ðL1;i L2 Þ dV ; ð19Þ Ve
I h Kei ¼ ðDL1;i L2 ÞT CTi ðDL1;i L2 Þ Ve
i T qi X2 ðL3 L1;i L2 Þ ðL3 L1;i L2 Þ dV ;
ð20Þ
The finite element method described in above section is used to calculate the dynamic behavior of rotating orthotropic sandwich annular plates with a viscoelastic core layer. To validate the proposed algorithm and calculation, a special FE code was developed in the software Mathematicaâ . In order to make clear presentations, the following non-dimensional quantities are introduced:
H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
a b h2 h3 e n ¼ ; b~ ¼ ; h~2 ¼ ; h~3 ¼ ; b h1 h1 h1 q2 q3 e q3 ¼ ; q2 ¼ ; e q1 q1 e rz;1 ¼ Grz;1 ; e h;1 ¼ Eh;1 ; G E Er;1 Er;1 ReðE Þ 2 e2 ¼ ; E Er;1 e r;3 ¼ Er;3 ; E e h;3 ¼ Eh;3 ; E Er;1 Er;1 G rz;3 e rz;3 ¼ G ; Er;1 rffiffiffiffiffiffiffiffiffi q 1 h1 Er;1 h31 e ¼ xb2 : x ; D11 ¼ D11 12ð1 mrh;1 mhr;1 Þ
209
Table 2 Non-dimensional natural frequencies of the axisymmetric transverse vibration mode of the rotating annular plate made of single material (e n ¼ 0:25, b=h ¼ 10, mhr;1 ¼ 0:3, Nr ¼ 16, Nz ¼ 3) e x e X
ð27Þ
The meanings of symbols can be found in Appendix A. To validate the proposed algorithm and calculations, comparisons between the present results and results of an existing simplified model [7] are made first. The nondimensional natural frequencies of polar orthotropic laminated annular plates are shown in Table 1. The boundary conditions employed are free at inner radius r ¼ a and clamped at outer radius r ¼ b. The threelayered elements are used in the present computations. The grids of the finite element meshes for the threelayered element in the thickness direction are equally spaced. Good convergence is observed. It is also found that the present results agree well with those obtained by Lin and Tseng [7]. The non-dimensional natural frequencies of the first and second axisymmetric transverse vibration modes of rotating single material thick annular plates are presented in Table 2. The material properties of plates are assumed to be linear elastic, homogenous and isotropic. The boundary conditions adopted are free at outer radius r ¼ b and clamped at inner radius r ¼ a. The number of elements in the r-direction is taken to be 32. It is also shown that the results agree well with those obtained by Liu et al. [10] for the three-layered element model at various rotational speeds. e on the nonThe effects of b~ and rotational speed X dimensional frequencies and the modal loss factors
0 2 4 8 12 16
First mode
Second mode
Present
Liu et al. [10]
Present
Liu et al. [10]
5.724 8.103 12.71 23.54 36.29 52.52
5.747 8.140 12.78 23.65 36.41 52.59
33.27 36.65 45.40 71.40 104.8 147.9
33.49 36.93 45.80 72.06 105.7 149.1
Nr : number of elements in the r-direction, Nz : number of elements in the z-direction.
of symmetric sandwich annular plates are shown in Fig. 3. The boundary conditions of the annular plates are taken to be clamped at the inner radius and free at the outer radius. As expected, the results show that the non-dimensional frequencies increase with increasing
Table 1 Natural frequencies of polar orthotropic laminated annular plates (e n ¼ 0:1, Eh;1 =Er;1 ¼ 5, Ghz;1 =Er;1 ¼ 0:292, mhr;1 ¼ 0:3, q ¼ 1) e x Nr
b=h ¼ 20
b=h ¼ 100
8 16 32 64
14.132 14.064 14.047 14.042
14.356 14.289 14.271 14.267
Lin et al. [7]
13.936
14.178
Fig. 3. The effects of b~ on the non-dimensional frequencies and the modal loss factors of symmetric composite annular plates versus roe : (a) first mode; (b) second mode. Key: (––) non-ditational speed X mensional frequency: (- - -) modal loss factor (e n ¼ 0:25, h~2 ¼ 0:25, e rz;1 ¼ G e rz;3 ¼ 0:292, e r;3 ¼ 1, E e h;1 ¼ E e h;3 ¼ 5, G h~3 ¼ 1, e q2 ¼ e q 3 ¼ 1, E mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5).
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H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
rotational speed and increasing b~. The present results coincide well with those of Wang and Chen [15]. As for the modal loss factors, it can be seen that the modal loss factors decrease with increasing rotational speeds. This can also be seen that the thin plate has higher loss factors. It implies that a larger value of stiffness of the vie 2 has to be chosen to achieve sufficient scoelastic layer E damping for a thicker host plate. The results for the first and second modes are similar. e h;1 are The effects of the modulus ratio of face layers E plotted in Fig. 4. As the modulus ratio of face layers becomes larger, the non-dimensional frequencies increase at the lower rotational speeds. But above a certain rotational speed, the opposite effect occurs. Lower modulus ratios result in greater sensitivity of the natural frequencies to increasing the rotational speed. The results are similar to those of rotating single-layered polar orthotropic circular plates obtained by Liang et al. [12]. The modal loss factors decrease with increasing rotational speed. It can also be seen that large modulus ratios result in high modal loss factors.
The effects of the thickness of the viscoelastic central layer h~2 are sketched in Fig. 5. The sandwich plates with thinner viscoelastic central layer have slightly higher natural frequencies, but the reverse effects can be seen for the modal loss factors. It can also be seen that the non-dimensional frequencies increase with increasing rotational speeds. Finally, non-symmetric sandwich annular plates are considered. Unequal thickness of face layers is discussed first. The effects of the thickness of Layer 3 on the nondimensional frequencies and modal loss factors are shown in Fig. 6. Note that the non-dimensional frequencies increase with increasing thickness of Layer 3. This is because the stiffness of the plates increases with increasing thickness of Layer 3. It can also be seen that the increasing thickness of h~3 results in higher modal loss factors when h~3 < 1. For the cases that h~3 > 1, the trends are similar at higher rotational speeds, however the phenomena are reverse at lower rotational speeds. Although the face layers are assumed to be undamped, the variation of face layer thickness will induce the rel-
e h;1 ¼ E e h;3 on the non-dimensional frequencies Fig. 4. The effects of E and the modal loss factors of composite annular plates versus rotae : (a) first mode; (b) second mode. Key: (––) nontional speed X dimensional frequency: (- - -) modal loss factor ( e n ¼ 0:25, b~ ¼ 200, e rh;1 ¼ G e rh;3 ¼ 0:35, G e rz;1 ¼ e r;3 ¼ 1, G h~2 ¼ 0:25, h~3 ¼ 1, e q2 ¼ e q 3 ¼ 1, E e rz;3 ¼ 0:292, G e hz;1 ¼ G e hz;3 ¼ 0:292, mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5). G
Fig. 5. The effects of h~2 on the non-dimensional frequencies and the modal loss factors of composite annular plates versus rotational speed e : (a) first mode; (b) second mode. Key: (––) non-dimensional freX quency; (- - -) modal loss factor (e n ¼ 0:25, b~ ¼ 200, h~3 ¼ 1, e rh;1 ¼ G e rh;3 ¼ 0:35, G e rz;1 ¼ e r;3 ¼ 1, E e h;1 ¼ E e h;3 ¼ 5, G e q2 ¼ e q 3 ¼ 1, E e hz;1 ¼ G e hz;3 ¼ 0:292, mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5). e rz;3 ¼ 0:292, G G
H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
211
e : (a) first Fig. 6. The effects of h~3 on the non-dimensional frequencies and the modal loss factors of composite annular plates versus rotational speed X mode; (b) second mode; (c) first mode; (d) second mode. Key: (––) non-dimensional frequency; (- - -) modal loss factor (e n ¼ 0:25, b~ ¼ 200, h~2 ¼ 0:25, e rh;1 ¼ G e rh;3 ¼ 0:35, G e rz;1 ¼ G e rz;3 ¼ 0:292, G e hz;1 ¼ G e hz;3 ¼ 0:292, mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5). e r;3 ¼ 1, E e h;1 ¼ E e h;3 ¼ 5, G e q2 ¼ e q 3 ¼ 1, E
ative stiffness between three layers and the modal loss factors of the system will vary. The details of this section can be found in Ref. [15].
Appendix A. Evaluation of elemental equilibrium nodal displacement vector The elemental equilibrium nodal displacement vector e Ui is obtained from
4. Conclusion
Ui ¼ Tei U;
Finite element dynamic analyses of rotating orthotropic sandwich annular plates with a viscoelastic central layer are presented. The Rayleigh–Ritz finite element model with a discrete layer element is used to calculate the natural frequencies and modal loss factors. A complex description of the viscoelastic material is adopted. From the previous results, the following conclusions can be drawn:
where Tei is the transformation matrix and U is the global equilibrium nodal coordinate evaluated form the static problem of the whole system when the centrifugal force is applied,
1. The frequencies increase with increasing rotational speed, but the modal loss factors decrease. 2. Lower modulus ratios result in greater sensitivity of the natural frequencies to rotational speed. 3. The larger thickness of the viscoelastic layer will cause larger modal loss factors. When the thickness of core layer stays constant, the variation of face layer thicknesses will cause the relative stiffness of three layers various and the modal loss factors of the system will also change.
e
ReðKÞU ¼ F;
ðA:1Þ
ðA:2Þ
K is the stiffness matrix and F is the centrifugal force vector.
Appendix B. List of symbols a b b~ D11 Er;i , Eh;i
inner radius of the plate outer radius of the plate b=h1 , outer radius to thickness ratio of the Layer 1 Er;1 h31 =½12ð1 mrh;1 mhr;1 Þ , flexural rigidity of the host plate YoungÕs moduli of face layers, i ¼ 1; 3
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modulus of the viscoelastic material ReðE2 Þ=Er;1 , non-dimensional modulus of the viscoelastic core e r;3 , E e h;1 , E e h;3 Er;3 =Er;1 , Eh;1 =Er;1 , Eh;3 =Er;1 , non-dimenE sional moduli of face layers Grz;i , Ghz;i shear moduli of face layers, i ¼ 1; 3 e rz;i G Grz;i =Er;1 , non-dimensional shear moduli of face layers, i ¼ 1; 3 hi thickness of the Layer i, i ¼ 1; 2; 3 h~i hi =h1 , non-dimensional thickness of the Layer i, i ¼ 2; 3 i index for layer, i ¼ 1; 2; 3 Nr number of elements in the r-direction Nz number of elements in the z-direction g modal loss factor of the composite plate gv loss factor of the viscoelastic material mi , mrh;i , mhr;i PoissonÕs ratio of the Layer i e n a=b, non-dimensional inner radius of the plate qi mass density of the Layer i, i ¼ 1; 2; 3 e qi qi =q1 , non-dimensional mass density of the Layer i, i ¼ 2; 3 x natural frequency of the composite plate X rotational speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e x xb2 q1 h1 =D11 , non-dimensional natural frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e X Xb2 q1 h1 =8D11 , non-dimensional rotational speed
Ev e2 E
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Finite element dynamic analysis of rotating orthotropic sandwich annular plates Horng-Jou Wang, Yu-Ren Chen, Lien-Wen Chen
*
Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC
Abstract The finite element dynamic analysis of rotating orthotropic sandwich annular plates with a viscoelastic central layer and two polar orthotropic laminated face layers are analysed in the present paper. The axisymmetric discrete layer annular element and HamiltonÕs principle are employed to derive the finite element equations of motion for a rotating sandwich plate including transverse shear effects. The viscoelastic material in the central layer is assumed to be incompressible, and the extensional and shear moduli are described by complex quantities. Complex-eigenvalued problems are then solved, and the modal frequencies and loss factors of the sandwich plate are extracted. The results of the symmetric and non-symmetric sandwich annular plates are both presented. The effects of outer radius to thickness ratio, orthotropic properties of face layers, the thickness of the viscoelastic central layer, and thickness of face layers on the natural frequencies and modal loss factors of sandwich annular plates are discussed. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Rotating; Orthotropic; Sandwich; Annular plate; Viscoelastic; Discrete layer annular finite element
1. Introduction It is well known that a sandwich structure with a high damping viscoelastic central layer has high damping capacity and high resistance to excessive vibration and noise. The major damping mechanisms in vibration are due to the extensional and shear deformations of viscoelastic materials. The damping property of a sandwich structure with a viscoelastic core is usually expressed as the modal loss factor for the corresponding vibration mode. The early investigations on the vibration and damping properties of basic structures, such as beams and rectangular plates, with a viscoelastic middle layer can be found in Refs. [1–4]. The annular plates have received a great deal of attention because of their widely used in many mechanical and electronic devices, such as computer disk memory units, circular saws, and turbine rotors. The vibration behavior of single-layer annular plates has been studied by many investigators. Irie et al. [5] discussed the natural frequencies of thick annular plates, Wang et al. [6] investigated free vibration analysis of annular plates by the differential quadrature method, and Lin and Tseng *
Corresponding author. Tel.: +886-6-2757575x62143; fax: +886-62352973. E-mail address: [email protected] (L.-W. Chen). 0263-8223/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0263-8223(03)00115-6
[7] studied the free vibration of polar orthotropic laminated circular and annular plates. The free vibration problems of polar orthotropic annular plates were solved by Ramaiah and Vijayakumar [8], Narita [9] presented the natural frequencies of completely free annular and circular plates having polar orthotropy, and Liu et al. [10] investigated the axisymmetric vibrations of rotating annular plates by the finite element method. The vibration and stress analysis of thin rotating discs using annular finite elements were studied by Kirkhope and Wilson [11]. Liang et al. [12] worked on the vibration and stability problems of rotating polar orthotropic annular disks subjected to a stationary concentrated transverse load. As for sandwich annular plates, Roy and Ganesan [13] developed a finite element method for vibration and damping analyses of circular plates with the constrained damping layer treatment. Yu and Huang [14] derived the equations of motion of a three-layered circular plate based on thin shell theory to handle the very thin viscoelastic layer problem. They both worked on a stationary sandwich plate with isotropic face layers. The studies of vibration and damping properties of a stationary three-layered annular plate with a viscoelastic middle layer and two polar orthotropic laminated face layers were presented by Wang and Chen [15]. Seubert et al. [16] investigated the rotating disks with two
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isotropic face layers and a viscoelastic core by using a commercial finite element analysis program ANSYSâ and experimental measurements. The damping properties of laminated composites are also of importance. Saravanos and Pereira [17] studied the dynamic characteristics of composite plates with interply damping layers. The dynamic properties of E-glass fiber reinforced composites were discussed by Gibson and Plunkett [18]. It has been found that the loss factor for epoxy matrix is very low compared with that of high damping viscoelastic materials. It is thus reasonable to assume that the two laminated surface layers are undamped in the present analysis. The dynamic behavior and damping properties of rotating orthotropic sandwich annular plates with a viscoelastic central layer and two polar orthotropic laminated face layers are studied here. The discrete layer annular elements and the Ritz finite element method are adopted to obtain the finite element equations of motion for the sandwich plate. The extensional and shear moduli of the linear isotropic viscoelastic material layer are described by complex quantities. The natural frequencies and modal loss factors of the sandwich plate are obtained by solving the complex eigenvalue problem. The effects of outer radius to thickness ratio, orthotropic properties of face layers, the thickness of the viscoelastic core layer, and thickness of face layers on the modal natural frequencies and loss factors of sandwich annular plates are investigated.
2. Formulation As shown in Fig. 1, the rotating sandwich annular plate of inner radius a and outer radius b is considered. The sandwich plate is composed of two polar orthotropic face layers and a viscoelastic core layer, and rotates about the z-axis at a constant angular speed X. The polar orthotropic face layers, designated as the Layer 1 and the Layer 3 respectively, are pure elastic and homogeneous (e.g. high-modulus graphite, high-strength graphite epoxy, and ultrahigh modulus graphite epoxy [19]). Also, the face layers are assumed to be undamped, because the loss factors for epoxy matrix are very small compared with those of the high damping viscoelastic materials in the middle layer [18]. The middle layer, designated as the Layer 2, is a linear viscoelastic material layer (e.g. rubber) and is capable of adhesively dissipating vibratory motions. The interfaces between layers are assumed to be perfectly bonding. The thicknesses of three layers are h1 , h2 , and h3 , respectively. Because the thickness changes of three layers have insignificant influences on natural frequencies and modal loss factors of the system [20], it is assumed that the thicknesses of three layers are all constant during deformation.
Fig. 1. The rotating three-layered composite annular plate with two polar orthotropic face layers and a viscoelastic core layer.
The discrete layer annular finite element [21], as shown in Fig. 2(a), is employed to model the sandwich annular plate system. The axisymmetric discrete layer annular finite element of inner radius ri and outer radius ro for the Layer i has eight degrees of freedom: the A B displacements in the r-direction, UiA , Uiþ1 , UiB and Uiþ1 , A B the transverse displacements, W and W , and the rotation angles, HA and HB . When the thicknesses of three layers are all constant during deformation, the nodal degrees of freedom for the three-layered discrete layer annular finite element are shown in Fig. 2(b). The displacement field in the basic element for the Layer i can be expressed in terms of the nodal degrees of freedom as ui ðr; z; tÞ ¼ L1;i ðzÞL2 ðrÞUei ðtÞ;
ð1Þ
where ui is the displacement field vector in the basic element, L1;i is the transverse thickness interpolation matrix (the second subscript i means layer i hereafter), L2 is the interpolation matrix in the r-direction, and Uei is the vector of nodal displacements of the element. These vectors and matrices are given by ui ðr; z; tÞ ui ðr; z; tÞ ¼ ; ð2Þ wi ðr; tÞ " # 1 1 hzi þ hzi 0 2 2 L1;i ðzÞ ¼ ; ð3Þ 0 0 1 2 A 3 0 0 0 nBu 0 0 0 nu ð4Þ L2 ðrÞ ¼ 4 0 nA 0 0 0 nBu 0 0 5; u A A 0 0 nw nH 0 0 nBw nBH T A B Uei ðtÞ ¼ UiA Uiþ1 ð5Þ W A HA UiB Uiþ1 W B HB
H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
3
2
o 6 or 6 61 D¼6 6 r 6 4o oz
207
07 7 7 07 7: 7 o5
ð8Þ
or
The stress–strain relations for the Layer i can be expressed as 8 9 2 9 38 C11;i C12;i 0 < er;i = < rr;i = rh;i ¼ 4 C21;i C22;i 0 5 eh;i ð9aÞ : ; : ; crz;i 0 0 C44;i srz;i or ri ¼ Ci i ;
ð9bÞ
where Ci is the elasticity matrix. For the polar orthotropic material in the face layers ði ¼ 1; 3Þ, the components of the elasticity matrix are Er;i ; 1 mrh;i mhr;i Eh;i ¼ ; 1 mrh;i mhr;i
C11;i ¼ C22;i
C44;i ¼ j2 Grz;i ;
C12;i ¼ C21;i ¼
mhr;i Er;i ; 1 mrh;i mhr;i ð10a–cÞ
j2 ¼
p2 ; 12
ð10d;eÞ
where E is the YoungÕs modulus, G is the shear modulus, m is the PoissonÕs ratio, and j2 is the shear correction factor. For the isotropic linear viscoelastic material in the central layer, assuming that viscoelastic material is almost incompressible [22], the components of the elasticity matrix are given by C11;2 ¼ C22;2 ¼ Fig. 2. The discrete layer annular finite element: (a) basic element; (b) three-layered element.
and nA u
¼ ð1 nÞ;
nA w
2
nBu
¼ n;
¼ ð1 3n þ 2n3 Þ;
2 3 nA H ¼ ðn 2n þ n Þ; r ri : n¼ ro ri
nBw ¼ ð3n2 2n3 Þ;
ð6a–dÞ
nBH ¼ ðn2 þ n3 Þ; ð6e–gÞ
The linear strains in the Layer i of the annular plate can be written in terms of the displacement i ¼ Dui ;
ð7Þ T
where the strain vector i ¼ fer;i eh;i crz;i g and D is the differential operator matrix
E2 ; 1 m22
E2 ; 2ð1 þ m2 Þ m2 ¼ 0:5 d;
C44;2 ¼
C12;2 ¼ C21;2 ¼
m 2 E2 ; 1 m22
E2 ¼ Ev ð1 þ jgv Þ; ð11a–eÞ
where gv is the loss factor of the viscoelastic material, d is assumed to be a small constant real value (e.g. d ¼ 0:01) which is introduced to avoid material stiffness pffiffiffiffiffiffiffi singularities, and j ¼ 1. The kinetic energy of the element e of the Layer i can be written as [10] I n o 1 2 Tie ¼ qi u_ 2i þ ½ðr þ ui ÞX þ w_ 2i dV ð12aÞ 2 Ve or in the matrix form I h T 1 e Ti ¼ qi u_ Ti u_ i þ X2 L3 rð0Þ L3 rð0Þ þ X2 ðL3 ui ÞT ðL3 ui Þ 2 Ve i þ 2X2 ðL3 ui ÞT L3 rð0Þ dV ; ð12bÞ where the matrix L3 and the position vector rð0Þ are
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1 L3 ¼ 0
0 ; 0
r
r ¼ ; z
ð0Þ
ð13a;bÞ
Gei
I ¼2 ðD1 L4 L1;i L2 ÞT b r ei ðD2 L4 L1;i L2 Þ Ve
þ ðD1 L5 L1;i L2 ÞT b r ei ðD2 L5 L1;i L2 Þ
H
q is the mass density and Ve represents a volume integral. The first term of kinetic energy in Eq. (12b) is due to vibratory motions of the plate. The second term is due to the rigid body motion and actually no contribution to the equations of motion. The third term is the supplementary strain energy due to displacement dependent centrifugal force. The fourth term is resulting from the initial in-plane stresses that are developed by the centrifugal force. The strain energy of the annular element e of the Layer i is I I 1 T Uie ¼ rTi i dV þ rei N ð14Þ i dV : 2 Ve Ve The second term in Eq. (14) is the additional strain energy induced by rotation [10]. The rotation-induced stress vector rei and the non-linear strain vector N i are T e rei ¼ rer;i reh;i serz;i ¼ Ci DðL1;i L2 ÞUi ð15Þ and
N i ¼
8 > 1 > > > > > 2 > > <
oui or
!2
1 owi þ 2 or 1 u 2 i
> > > > > > > > :
2 r oui oui or oz
!2 9 > > > > > > > > = > > > > > > > > ;
1 T e r i ðD3 L5 L1;i L2 Þ dV ; þ ðD3 L5 L1;i L2 Þ b 2
Fei
¼
I h
i qi X2 ðL3 L1;i L2 ÞT L3 rð0Þ dV
ð16Þ
e
in which Ui is the equilibrium nodal displacement vector of the rotating annular element for the Layer i and can be evaluated from the solutions of static problems. The details of this section are listed in Appendix A. Substituting Eqs. (1), (7) and (9) into Eqs. (12) and (14), the HamiltonÕs principle [23], Z e e d ðTi Ui Þdt ¼ 0; ð17Þ t
is used to derive element dynamic equilibrium equations. The element differential equations are expressed in matrix form as € e þ ðKe þ Ge ÞUe ¼ Fe ; Mei U i i i i i
ð18Þ
ð22Þ
Ve
and D1 ¼
81 > <2
9
o or > =
0 ; > ; : o > 2 eor rr;i 0 6 0 re e bi ¼ 4 r h;i 0 0
L4 ¼ ½ 1
0 ;
8o9 > = < or > D2 ¼ 0 ; > ; :o> 3 oz 0 0 7 5; serz;i L5 ¼ ½ 0
D3 ¼
1 :
8 9 > = <0> 1
r > ; : > 0
;
ð23a–dÞ ð23e;fÞ
Gei
is usually called the geometric The stiffness matrix stiffness matrix that increases the plate stiffness. Assembling the contribution of all elements, the global finite element equation can be obtained € þ ðK þ GÞU ¼ F; MU
;
ð21Þ
ð24Þ
where M is the global mass matrix, K is the global stiffness matrices, G is the global geometry stiffness matrix and F is the global centrifugal force vector due to rotation. To study the free vibration of rotating annular plates, the forcing term in Eq. (24) is neglected [10]. Substituting UðtÞ ¼ U0 ejkt into Eq. (24), the following eigenvalue equation can be obtained: ðK þ GÞU0 ¼ k2 MU0 ;
ð25Þ
where k is a complex number because of complex-valued terms of the stiffness matrices. The circular natural frequency and modal loss factor of the sandwich annular plate system can be calculated from the formulae [15] qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Imðk2 Þ 2 x ¼ ReðkÞ ; g¼ : ð26a;bÞ Reðk2 Þ
3. Numerical results and discussion
Mei ,
stiffness matrices where the elemental mass matrix Kei and Gei , and force vector Fei of the Layer i are I h i T Mei ¼ qi ðL1;i L2 Þ ðL1;i L2 Þ dV ; ð19Þ Ve
I h Kei ¼ ðDL1;i L2 ÞT CTi ðDL1;i L2 Þ Ve
i T qi X2 ðL3 L1;i L2 Þ ðL3 L1;i L2 Þ dV ;
ð20Þ
The finite element method described in above section is used to calculate the dynamic behavior of rotating orthotropic sandwich annular plates with a viscoelastic core layer. To validate the proposed algorithm and calculation, a special FE code was developed in the software Mathematicaâ . In order to make clear presentations, the following non-dimensional quantities are introduced:
H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
a b h2 h3 e n ¼ ; b~ ¼ ; h~2 ¼ ; h~3 ¼ ; b h1 h1 h1 q2 q3 e q3 ¼ ; q2 ¼ ; e q1 q1 e rz;1 ¼ Grz;1 ; e h;1 ¼ Eh;1 ; G E Er;1 Er;1 ReðE Þ 2 e2 ¼ ; E Er;1 e r;3 ¼ Er;3 ; E e h;3 ¼ Eh;3 ; E Er;1 Er;1 G rz;3 e rz;3 ¼ G ; Er;1 rffiffiffiffiffiffiffiffiffi q 1 h1 Er;1 h31 e ¼ xb2 : x ; D11 ¼ D11 12ð1 mrh;1 mhr;1 Þ
209
Table 2 Non-dimensional natural frequencies of the axisymmetric transverse vibration mode of the rotating annular plate made of single material (e n ¼ 0:25, b=h ¼ 10, mhr;1 ¼ 0:3, Nr ¼ 16, Nz ¼ 3) e x e X
ð27Þ
The meanings of symbols can be found in Appendix A. To validate the proposed algorithm and calculations, comparisons between the present results and results of an existing simplified model [7] are made first. The nondimensional natural frequencies of polar orthotropic laminated annular plates are shown in Table 1. The boundary conditions employed are free at inner radius r ¼ a and clamped at outer radius r ¼ b. The threelayered elements are used in the present computations. The grids of the finite element meshes for the threelayered element in the thickness direction are equally spaced. Good convergence is observed. It is also found that the present results agree well with those obtained by Lin and Tseng [7]. The non-dimensional natural frequencies of the first and second axisymmetric transverse vibration modes of rotating single material thick annular plates are presented in Table 2. The material properties of plates are assumed to be linear elastic, homogenous and isotropic. The boundary conditions adopted are free at outer radius r ¼ b and clamped at inner radius r ¼ a. The number of elements in the r-direction is taken to be 32. It is also shown that the results agree well with those obtained by Liu et al. [10] for the three-layered element model at various rotational speeds. e on the nonThe effects of b~ and rotational speed X dimensional frequencies and the modal loss factors
0 2 4 8 12 16
First mode
Second mode
Present
Liu et al. [10]
Present
Liu et al. [10]
5.724 8.103 12.71 23.54 36.29 52.52
5.747 8.140 12.78 23.65 36.41 52.59
33.27 36.65 45.40 71.40 104.8 147.9
33.49 36.93 45.80 72.06 105.7 149.1
Nr : number of elements in the r-direction, Nz : number of elements in the z-direction.
of symmetric sandwich annular plates are shown in Fig. 3. The boundary conditions of the annular plates are taken to be clamped at the inner radius and free at the outer radius. As expected, the results show that the non-dimensional frequencies increase with increasing
Table 1 Natural frequencies of polar orthotropic laminated annular plates (e n ¼ 0:1, Eh;1 =Er;1 ¼ 5, Ghz;1 =Er;1 ¼ 0:292, mhr;1 ¼ 0:3, q ¼ 1) e x Nr
b=h ¼ 20
b=h ¼ 100
8 16 32 64
14.132 14.064 14.047 14.042
14.356 14.289 14.271 14.267
Lin et al. [7]
13.936
14.178
Fig. 3. The effects of b~ on the non-dimensional frequencies and the modal loss factors of symmetric composite annular plates versus roe : (a) first mode; (b) second mode. Key: (––) non-ditational speed X mensional frequency: (- - -) modal loss factor (e n ¼ 0:25, h~2 ¼ 0:25, e rz;1 ¼ G e rz;3 ¼ 0:292, e r;3 ¼ 1, E e h;1 ¼ E e h;3 ¼ 5, G h~3 ¼ 1, e q2 ¼ e q 3 ¼ 1, E mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5).
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rotational speed and increasing b~. The present results coincide well with those of Wang and Chen [15]. As for the modal loss factors, it can be seen that the modal loss factors decrease with increasing rotational speeds. This can also be seen that the thin plate has higher loss factors. It implies that a larger value of stiffness of the vie 2 has to be chosen to achieve sufficient scoelastic layer E damping for a thicker host plate. The results for the first and second modes are similar. e h;1 are The effects of the modulus ratio of face layers E plotted in Fig. 4. As the modulus ratio of face layers becomes larger, the non-dimensional frequencies increase at the lower rotational speeds. But above a certain rotational speed, the opposite effect occurs. Lower modulus ratios result in greater sensitivity of the natural frequencies to increasing the rotational speed. The results are similar to those of rotating single-layered polar orthotropic circular plates obtained by Liang et al. [12]. The modal loss factors decrease with increasing rotational speed. It can also be seen that large modulus ratios result in high modal loss factors.
The effects of the thickness of the viscoelastic central layer h~2 are sketched in Fig. 5. The sandwich plates with thinner viscoelastic central layer have slightly higher natural frequencies, but the reverse effects can be seen for the modal loss factors. It can also be seen that the non-dimensional frequencies increase with increasing rotational speeds. Finally, non-symmetric sandwich annular plates are considered. Unequal thickness of face layers is discussed first. The effects of the thickness of Layer 3 on the nondimensional frequencies and modal loss factors are shown in Fig. 6. Note that the non-dimensional frequencies increase with increasing thickness of Layer 3. This is because the stiffness of the plates increases with increasing thickness of Layer 3. It can also be seen that the increasing thickness of h~3 results in higher modal loss factors when h~3 < 1. For the cases that h~3 > 1, the trends are similar at higher rotational speeds, however the phenomena are reverse at lower rotational speeds. Although the face layers are assumed to be undamped, the variation of face layer thickness will induce the rel-
e h;1 ¼ E e h;3 on the non-dimensional frequencies Fig. 4. The effects of E and the modal loss factors of composite annular plates versus rotae : (a) first mode; (b) second mode. Key: (––) nontional speed X dimensional frequency: (- - -) modal loss factor ( e n ¼ 0:25, b~ ¼ 200, e rh;1 ¼ G e rh;3 ¼ 0:35, G e rz;1 ¼ e r;3 ¼ 1, G h~2 ¼ 0:25, h~3 ¼ 1, e q2 ¼ e q 3 ¼ 1, E e rz;3 ¼ 0:292, G e hz;1 ¼ G e hz;3 ¼ 0:292, mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5). G
Fig. 5. The effects of h~2 on the non-dimensional frequencies and the modal loss factors of composite annular plates versus rotational speed e : (a) first mode; (b) second mode. Key: (––) non-dimensional freX quency; (- - -) modal loss factor (e n ¼ 0:25, b~ ¼ 200, h~3 ¼ 1, e rh;1 ¼ G e rh;3 ¼ 0:35, G e rz;1 ¼ e r;3 ¼ 1, E e h;1 ¼ E e h;3 ¼ 5, G e q2 ¼ e q 3 ¼ 1, E e hz;1 ¼ G e hz;3 ¼ 0:292, mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5). e rz;3 ¼ 0:292, G G
H.-J. Wang et al. / Composite Structures 62 (2003) 205–212
211
e : (a) first Fig. 6. The effects of h~3 on the non-dimensional frequencies and the modal loss factors of composite annular plates versus rotational speed X mode; (b) second mode; (c) first mode; (d) second mode. Key: (––) non-dimensional frequency; (- - -) modal loss factor (e n ¼ 0:25, b~ ¼ 200, h~2 ¼ 0:25, e rh;1 ¼ G e rh;3 ¼ 0:35, G e rz;1 ¼ G e rz;3 ¼ 0:292, G e hz;1 ¼ G e hz;3 ¼ 0:292, mhr;1 ¼ mhr;3 ¼ 0:3, gv ¼ 0:5). e r;3 ¼ 1, E e h;1 ¼ E e h;3 ¼ 5, G e q2 ¼ e q 3 ¼ 1, E
ative stiffness between three layers and the modal loss factors of the system will vary. The details of this section can be found in Ref. [15].
Appendix A. Evaluation of elemental equilibrium nodal displacement vector The elemental equilibrium nodal displacement vector e Ui is obtained from
4. Conclusion
Ui ¼ Tei U;
Finite element dynamic analyses of rotating orthotropic sandwich annular plates with a viscoelastic central layer are presented. The Rayleigh–Ritz finite element model with a discrete layer element is used to calculate the natural frequencies and modal loss factors. A complex description of the viscoelastic material is adopted. From the previous results, the following conclusions can be drawn:
where Tei is the transformation matrix and U is the global equilibrium nodal coordinate evaluated form the static problem of the whole system when the centrifugal force is applied,
1. The frequencies increase with increasing rotational speed, but the modal loss factors decrease. 2. Lower modulus ratios result in greater sensitivity of the natural frequencies to rotational speed. 3. The larger thickness of the viscoelastic layer will cause larger modal loss factors. When the thickness of core layer stays constant, the variation of face layer thicknesses will cause the relative stiffness of three layers various and the modal loss factors of the system will also change.
e
ReðKÞU ¼ F;
ðA:1Þ
ðA:2Þ
K is the stiffness matrix and F is the centrifugal force vector.
Appendix B. List of symbols a b b~ D11 Er;i , Eh;i
inner radius of the plate outer radius of the plate b=h1 , outer radius to thickness ratio of the Layer 1 Er;1 h31 =½12ð1 mrh;1 mhr;1 Þ , flexural rigidity of the host plate YoungÕs moduli of face layers, i ¼ 1; 3
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modulus of the viscoelastic material ReðE2 Þ=Er;1 , non-dimensional modulus of the viscoelastic core e r;3 , E e h;1 , E e h;3 Er;3 =Er;1 , Eh;1 =Er;1 , Eh;3 =Er;1 , non-dimenE sional moduli of face layers Grz;i , Ghz;i shear moduli of face layers, i ¼ 1; 3 e rz;i G Grz;i =Er;1 , non-dimensional shear moduli of face layers, i ¼ 1; 3 hi thickness of the Layer i, i ¼ 1; 2; 3 h~i hi =h1 , non-dimensional thickness of the Layer i, i ¼ 2; 3 i index for layer, i ¼ 1; 2; 3 Nr number of elements in the r-direction Nz number of elements in the z-direction g modal loss factor of the composite plate gv loss factor of the viscoelastic material mi , mrh;i , mhr;i PoissonÕs ratio of the Layer i e n a=b, non-dimensional inner radius of the plate qi mass density of the Layer i, i ¼ 1; 2; 3 e qi qi =q1 , non-dimensional mass density of the Layer i, i ¼ 2; 3 x natural frequency of the composite plate X rotational speed pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e x xb2 q1 h1 =D11 , non-dimensional natural frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e X Xb2 q1 h1 =8D11 , non-dimensional rotational speed
Ev e2 E
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