Free vibrations of composite sandwich plates by Chebyshev collocation technique

Composites Part B 165 (2019) 442–455

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Free vibrations of composite sandwich plates by Chebyshev collocation technique

T

Rashmi Rani∗, Roshan Lal Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

ARTICLE INFO

ABSTRACT

Keywords: Chebyshev collocation technique Circular sandwich plate Radially symmetric vibrations Exponentially varying thickness Mode shapes

In this study, free axisymmetric vibrations of composite circular sandwich plates with isotropic core and orthotropic facings have been studied using first-order shear deformation theory. The thickness of the core is assumed to vary exponentially in the radial direction and the face sheets is treated as membrane of constant thickness. The effect of shear deformation and rotatory inertia in the core has been taken into account. The governing differential equations of the present model have been obtained by applying Hamilton's principle. Chebyshev collocation technique has been used to obtain the frequency equations for the plate when it is clamped or simply supported or free at the edge. The first three roots of these equations have been reported as the natural frequencies for the first three modes of vibration. The effect of taper parameter, facing thickness and core thickness on the frequency parameter has been investigated. Comparison of results for some special cases with published results obtained by other approximate methods has been presented which shows an excellent agreement. Mode shapes for a specified plate for all the three boundary conditions have been plotted.

1. Introduction In the present days, composite sandwich structures are found to have enormous application in the field of modern science and technology. Their use in automobile, shipbuilding and transportation industries, geotextile infrastructures, aeronautics and astronautics, marine and offshore oil structures, wind industry and sport goods is growing rapidly due to their specific stiffness, light weight and maximum fatigue resistance. In construction, sandwich structures have been applied as structural elements in foot and vehicular bridges, in the rehabilitation as replacement of concrete bridge, cladding, roofing and also as partition wall elements, sometimes with translucent properties [1,2]. Employment of sandwich structural elements of varying thickness further helps the designer not only in enhancing the performance but also reduce the weight and size of the structure. To use them efficiently, a good knowledge of their constructional and dynamic behavior is essential with a fair amount of accuracy. Numerous studies have appeared in the literature on the development of various theories to investigate the static/dynamic behavior of laminated/composite sandwich structures. The classical lamination theory (CLT) based on the assumption of Kirchhoff's plate theory [3] which neglect the effect of shear deformation, the first order shear deformation theory (FSDT) [4], assume the effect of transverse shear

∗

deformation through the entire thickness, higher order shear deformation theory (HSDT) [5], assume quadraic, cubic or higher variation of surface parallel displacements through the entire thickness, zigzag theory [6], which include the effect of shear train shape function assuming the non-linear distribution of in plane displacement across the thickness. A detailed discussion upon various theories is available in the excellent review articles by Altenbach [7], Carrera [8], Reddy [9], Kreja [10]. Along with different plate theories, the use of efficient numerical techniques is further essential since the analytical techniques such as Bessel functions [11], Levy solution [12], Navier solution [13] etc. are restricted to simple geometry and boundary conditions. The recent developments in computational technologies facilitate the efficient numerical methodologies to solve the coupled differential equations arising for the composite sandwich plates of varying thickness. Among various numerical investigation, spline finite point method used by Zhou and Li [14] to analyze the free vibration of sandwich plates with laminated composite faces. Transient dynamic response of composite and sandwich plates have been studied by Kant et al. [15] using generalized Jacobi method. Rohani and Marcellier [16] have employed Rayleigh-Ritz method for buckling and vibration analysis of anisotropic rectangular sandwich plates with edges elastically restrained against rotation. Finite element method with various considerations has been used in Refs. [17–23] to predict the static/dynamic

Corresponding author. E-mail addresses: [email protected] (R. Rani), [email protected] (R. Lal).

https://doi.org/10.1016/j.compositesb.2019.01.088 Received 5 July 2018; Received in revised form 9 January 2019; Accepted 19 January 2019 Available online 24 January 2019 1359-8368/ © 2019 Elsevier Ltd. All rights reserved.

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

behavior of composite sandwich plates based on different theories. Xia and Shen [24] used perturbation technique to investigate the vibration of post-buckled sandwich plates with FGM face sheets in a thermal environment. Recently, Morozov and Lopatin [25–27] used Galerkin method to predict the fundamental frequencies of rectangular sandwich plates of composite skin based on first-order shear deformation theory with different combinations of clamped and free edge conditions. Isogeometric finite element formulation for static, free vibration and buckling analysis of laminated composite and sandwich plates using a layerwise deformation theory has been presented by Thai et al. [28]. Topal and Uzman [29] used modified feasible direction (MFD) method for the analysis of composite sandwich plates. A layerwise/solid-element method (LW/SE) is proposed by Li et al. [30] to investigate linear static and free vibration of composite sandwich plates. A Generalized Differential Quadrature technique has been used by Ferreira et al. [31] for predicting the static deformations and vibrational behavior of sandwich plates. Free vibration of thermally buckled sandwich plate with embedded pre-strained shape memory alloy (SMA) fibers in temperature dependent laminated composite face sheets is investigated by Samadpour et al. [32] using Galerkin weighted residual method. Wavelet transform (WT) with B-spline wavelets of fractional order has been used by Katunin [33] to analyze vibration-based spatial damage identification in honeycomb-core sandwich composite structures. Alibeigloo and Alizadeh [34] used State space differential quadrature method for the static and free vibration analyses of functionally graded sandwich plates. The classical methods such as Navier's solution, Levy's solution, and Ritz method were successfully extended by Hwu et al. [35] to study the free vibration of sandwich plates and cylindrical shells composed of two composite laminated faces and an ideally orthotropic elastic core. A closed Wang et al. [36] extend the multi-term Kantorovich-Galerkin method (MTKGM) to obtain semi-analytical solutions for vibration response of symmetric CNTRC sandwich plates resting on elastic foundation. The present work analyze the dynamic behavior of composite circular sandwich plates with isotropic core of exponentially varying thickness and orthotropic facings taking into account the contribution of face sheets membrane forces to the stresses of the core. For the present model, the governing differential equations of motion have been obtained using Hamilton's principle which has been further reduced to a set of coupled ordinary differential equations by variable separable technique. Upon these resulting equations, Chebyshev collocation technique has been employed to obtained frequency equations for three classical boundary conditions, namely, clamped, simply supported and free. The lowest three roots of these frequency equations are obtained using MATLAB and reported as the natural frequencies for the first three modes of vibrations. The effect of various plate parameters such as taper parameters, core thickness at the centre and facing thickness on the natural frequencies are studied for all the three boundary conditions. Three-dimensional mode shapes for specified plates have been plotted. A comparison of results has been made. 1.1. Equation of motion Consider a circular sandwich plate of radius a and thickness 2(hc + hf ) referred to cylindrical polar coordinate (r , , z ) , z = 0 being the middle surface of the plate and also the plane of symmetry. The line r = 0 is the axis of the plate. A cross-sectional view of the plate with exponentially varying core thickness hc (r ) , the facing thickness hf ( < < hc ) and facing slope is shown in Fig. 1. Any location in the lower or upper facing is identified by its r - coordinate or by its - coordinate, where = (r ) . The variables and r are connected by the relations

r = R sin

and dr = R cos

d

Fig. 1. Geometry and cross-section of the tapered circular sandwich plate with core of exponentially varying thickness i.e. h c=ho e x .

and the axis of sandwich plate. The thickness variation of core with radial distance is given by: d dh c d2h c = tan , h c=ho e x , and dr = cos2 .where is dr dr 2 taper parameter and ho is the thickness of the core at the centre of the plate. During axisymmetric vibrations, the movement of the line element

(1)

where R is the radius of curvature of the core-facing interface and R is the length of the normal between any point on the core-facing interface 443

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Table 1 Convergence of frequency parameter

for

= 0.5, Ho = 0.3, Hf = 0.02 .

C plate 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S plate

2.2369 2.2267 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278 2.2278

4.5014 4.4674 4.4702 4.4726 4.4725 4.4723 4.4723 4.4723 4.4723 4.4723 4.4723 4.4723 4.4723 4.4723 4.4723

7.4312 6.8617 6.8266 6.8340 6.8386 6.8390 6.8381 6.8381 6.8381 6.8381 6.8381 6.8381 6.8381 6.8381 6.8381

F plate

1.8713 1.8752 1.8738 1.8745 1.8741 1.8743 1.8741 1.8742 1.8742 1.8742 1.8742 1.8742 1.8742 1.8742 1.8742

4.4208 4.4371 4.4328 4.4333 4.4341 4.4336 4.4339 4.4337 4.4338 4.4338 4.4338 4.4338 4.4338 4.4338 4.4338

7.9681 6.7788 6.8762 6.8281 6.8380 6.8382 6.8378 6.8379 6.8380 6.8380 6.8380 6.8380 6.8380 6.8380 6.8380

2.9398 2.9538 2.9525 2.9535 2.9529 2.9533 2.9530 2.9532 2.9531 2.9532 2.9531 2.9531 2.9531 2.9531 2.9531

6.0036 5.3577 5.4413 5.4321 5.4352 5.4337 5.4346 5.4340 5.4344 5.4341 5.4343 5.4341 5.4342 5.4342 5.4342

8.2144 8.0784 7.5934 7.7612 7.7263 7.7366 7.7322 7.7345 7.7330 7.7341 7.7333 7.7339 7.7337 7.7337 7.7337

Bold values shows the convergence of frequency parameter.

Fig. 2. Convergence of frequency parameter Table 2 Comparison of frequency parameter (Hf = 0) circular plate for

c

(

12 Ho

)

for the first three modes of vibration with grid refinement for

uz (r , z , t ) = w (r , t ) to any point in the core and on the line element A B which gives rise the following strains

for uniform ( = 0) isotropic

= 0.3, Ho = 0.001, ks = 1. cr

Boundary condition

C plate

Method/mode

I

II

III

I

II

III

Exact [40] Finite element [41] Rayleigh- Ritz [42] receptance [43] Ritz method [44] symplectic method [45] DQM [46] Present

10.2158 10.2159 10.2160 10.2160 10.2158 10.2160

39.7711 39.7766 39.7710 39.7710 39.7711 39.7710

89.1041 89.1708 89.1030 89.1041 – –

4.9352 4.9352 4.9352 4.9350 4.9351 4.9352

29.7200 29.7222 29.7190 29.7200 29.7200 29.7200

74.1561 74.1938 74.1560 74.1560 – –

10.216 10.2158

39.771 39.7711

89.102 89.1041

4.935 4.9351

29.7200 29.7200

74.155 74.1561

= 0.5, Ho = 0.3, Hf = 0.02 .

S plate

=z

r

,

c

=z , r

crz

=

w + , r

cr

=

c z

= 0,

(2)

in the core. The displacement at the interface of the core and lower facing i.e. at the point B of line element A B in the radial and vertical direction are given by

{ur (r , z , t )} z = hc = hc

,

{uz (r , z, t )} z = hc = w (r , t ) .

(3)

wise ) and normal (n wise ) Using Eq. (3), the tangential ( components of displacement of point B at the interface of the core and the lower facing are given by

u = hc

A B will consist of a rotation (r , t ) about the midpoint in the vertical plane parallel to the r -axis and a vertical displacement w (r , t ) , both assumed to be small. These movements will impart a radial displaceu r (r , z , t ) = z (r , t ) ment and a vertical displacement

cos

w sin

,

un = w cos

+ hc

sin

.

(4)

Following, (Kraus, [37]) and using relations (1) and (4), the meridional and circumferential strains in the lower facing are given by

444

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R. Rani, R. Lal

Table 3 Comparison of frequency parameter plates for ks =

2

Ho Mode

/12,

c

C plate I II III S plate I II III F plate I II III

a b

= 0.3 .

) with uniform (

12 Ho

where, c and f are the densities of the core and the facing, respectively. The length is measured along the facing r = a . Similarly, the strain energy for core and facing of the plate are given by

= 0) circular

for various modes 0.0050

c

(

0.0250

0.0500

0.0750

0.1000

0.1250

10.213 10.2158b 10.2160c 39.7331 39.7706b 39.7710c 88.9264 89.1040c

10.1447 – 10.145c 38.8554 – 38.855c 84.995 84.995c

9.94079 – 9.941c 36.4787 – 36.479c 75.6643 75.664c

9.62856 – 9.629c 33.3934 – 33.393c 65.5507 65.551c

9.24004 – 9.2400c 30.2107 – 30.211c 56.6823 56.682c

8.80683 – 8.8068c 27.2529 – 27.253c 49.4204 49.420c

4.93473 4.9345b 4.9350c 29.7038 29.7198b 29.7200c 74.0543 74.1560c

4.92469 – 4.925c 29.3233 – 29.323c 71.7563 71.756c

4.89382 – 4.894c 28.2400 – 28.240c 65.9424 65.942c

4.84396 – 4.844c 26.7148 – 26.715c 59.0621 59.062c

4.77725 – 4.777c 24.9945 – 24.994c 52.5139 52.514c

4.69633 – 4.696c 23.2541 – 23.254c 46.7745 46.775c

9.00174 9.00279a 9.0030c 38.4162 38.4365a 38.4430c 87.6085 87.7151a 87.7500c

8.96857 – 8.969c 37.7874 – 37.787c 84.4429 – 84.443c

8.86791 – 8.868c 36.0407 – 36.041c 76.6756 – 76.676c

8.70953 – 8.710c 33.6744 – 33.674c 67.8274 – 67.827c

8.5051 – 8.505c 31.1106 – 31.111c 59.645 – 59.645c

8.26736 – 8.267c 28.6055 – 28.605c 52.5842 – 52.584c

f f

= =

1 R

(

u

+ un

(u cot

)

Mr r Qr r

c ( cr

f

=

fr ( f

c

= Ec /(1

+

c c

+

f 2 c)

= Ec /2(1 +

f

),

,

(

dh c dr

+ hc hc r

=

=

r

)

w r

cos2

Tf =

c

1 2 f

a 2

hc

0

f

(

+ f

= E fr /(1

c cr )

+ fr

fr f f

),

,

crz

sin

cos

hf 0

2 0

0

f

= µc

= Ef /(1

( ) ( )

r d dr dz ,

hc2

( ) +( )

rR d d

2

t

2

t

f 0

0

c

+

f

f

+

crz crz )

r d dr dz ,

f

rR d d

dn.

(8)

0

1

+

1 Q r r

M)

(

Qr = 2

= 2( c hc +

h 3c c 3

+

hf sec

f

2w

)

)

hf h 2c sec

f t2

2

t2

,

,

(9)

2N f sin

)=

Mcr , Mc

,

,

hc

M = Mc + 2hc N f sec

Qr = Qcr +

(ks

crz ,

z

dh 2 drc N f

cr ,

z

c

cos

,

,

) dz ,

(10)

(N f ,

)=

hf

(

f

,

f

) dn

(11)

= c / µc , R fr = fr /µc , R f = f / µc , gether with relations (10) and (11), the equations of motion (9) can be written as follows: (Rc Hc2 + 3Rfr Hf Hc cos3

+

w 2 t

w 2 t

c

crz ,

9Rfr Hf Hc cos2 fr

f

)

µc

(

sin

)x2 + 2x

d x 2} x dx

d2H

3 Hc Hf R sec 9Rfr Hf

dHc dx

3Rfr Hf x 2 sin

3k s

cos2 cos2

3 Rfr Hf x 2 cos d

) dx }

W x

2

+ R c Hc2 x + 3 Rc Hc

x2 dHc dx

+ 3 Rc c Hc

3Rfr Hf Hc

dHc dx

R c Hc2

x

)

+ x d xc cos3 x2

dHc 2 x dx

)

dH

+ 3Rfr Hf x 2 d x 2c cos3

2 sin2

dn ,

(H

+ 3Rfr Hf x cos3

),

z2

0

2

c

Qcr , Mcr , Mc are the stress resultants and ks is the shear constant. By using non-dimensional variables x = r / a W = w /a , Hf = hf /a , Hc = hc / a , Ho = ho / a , Rc to-

c)

hc

+

cr cr

hc

where Ec , Ef , µc and c , fr , f are Young's moduli, shear modulus and Poisson's ratio of the core and the facings, respectively. For isotropic core and orthotropic facing the kinetic energy of the plate are given by 1 2

(

0

+ r (Mr

(Qcr ,

(6)

Tc =

hf

Qr = Qcr

.

c( c

=

f fr

0

0

+ un )

c

hc

Mr = Mcr + 2hc N f cos

Nf

=

),

a 2

where

The corresponding strains in the upper facing are f and f . The suffixes c and f are used for core and face sheet. For homogeneous isotropic core and orthotropic face sheets, the stress-strain relations are given by (Sodel, [38]):

=

hc

Using the expressions T = Tc + 2 Tf and U = Uc + 2 Uf for total kinetic and potential energy of the plate together with Eqs. (2), (5)–(8) in Hamilton's energy principle (Soedel, 38]), one obtains the governing differential equations of motion for the plate as follows:

(5)

cr

1 2

Uf =

Irie et al. [47]. Wang et al. [48]. Mohammadi et al. [49]. 1 R

1 2

Uc =

f

sin

d x dx

d x 2} dx

sin 2W

x2

+ { 3 Hf x sin

( Rf

Rfr cos2

fr

)

3ks x 2

(cos2

(x 2a2Hc / µc ) Hc

c

+ 3Hf

f

sec

2

t2

=0 ,

(12)

(7)

445

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Table 4 Values of frequency parameter

with varying values of taper parameter

for Hf = 0.005.

FOR VARIOUS MODES

Ho = 0.1

C plate −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 S plate −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 F plate −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5

II

III

I

II

III

I

II

III

1.2680 1.3189 1.3715 1.4261 1.4829 1.5422 1.6043 1.6694 1.7380 1.8103 1.8868

3.5224 3.5936 3.6647 3.7361 3.8079 3.8803 3.9535 4.0278 4.1035 4.1808 4.2600

5.8196 5.9159 6.0111 6.1054 6.1989 6.2916 6.3837 6.4755 6.5670 6.6585 6.7502

1.6172 1.6667 1.7180 1.7715 1.8276 1.8869 1.9501 2.0177 2.0904 2.1688 2.2530

4.2756 4.3267 4.3782 4.4306 4.4844 4.5400 4.5979 4.6587 4.7229 4.7912 4.8642

6.8987 6.9677 7.0362 7.1044 7.1729 7.2419 7.3121 7.3839 7.4578 7.5346 7.6148

1.7866 1.8319 1.8790 1.9286 1.9814 2.0385 2.1006 2.1688 2.2438 2.3259 2.4150

4.6253 4.6609 4.6976 4.7362 4.7775 4.8222 4.8711 4.9251 4.9851 5.0518 5.1261

7.3897 7.4386 7.4879 7.5382 7.5904 7.6452 7.7035 7.7661 7.8341 7.9084 7.9900

0.9753 1.0129 1.0512 1.0900 1.1296 1.1699 1.2111 1.2533 1.2966 1.3414 1.3878

3.3483 3.4184 3.4883 3.5580 3.6279 3.6980 3.7685 3.8397 3.9117 3.9849 4.0594

5.7563 5.8539 5.9505 6.0461 6.1409 6.2349 6.3285 6.4217 6.5148 6.6081 6.7018

1.3862 1.4241 1.4618 1.4994 1.5372 1.5755 1.6146 1.6550 1.6971 1.7417 1.7894

4.1546 4.2061 4.2580 4.3106 4.3645 4.4202 4.4781 4.5388 4.6029 4.6709 4.7436

6.8691 6.9393 7.0089 7.0785 7.1484 7.2192 7.2915 7.3658 7.4428 7.5234 7.6084

1.6168 1.6506 1.6838 1.7168 1.7500 1.7839 1.8192 1.8567 1.8973 1.9423 1.9926

4.5280 4.5648 4.6029 4.6430 4.6860 4.7327 4.7839 4.8406 4.9036 4.9738 5.0519

7.3734 7.4232 7.4734 7.5249 7.5782 7.6341 7.6882 7.5205 7.2669 7.0207 6.7834

1.8210 1.8669 1.9125 1.9578 2.0030 2.0484 2.0940 2.1403 2.1874 2.2358 2.2857

4.1719 4.2492 4.3253 4.4003 4.4743 4.5474 4.6195 4.6909 4.7615 4.8315 4.9008

6.6330 6.7364 6.8374 6.9361 7.0323 7.1260 7.2171 7.3054 7.3908 7.4727 7.5507

2.5107 2.5413 2.5710 2.6001 2.6292 2.6586 2.6890 2.7210 2.7551 2.7920 2.8323

5.1439 5.1962 5.2474 5.2976 5.3470 5.3958 5.4442 5.4919 5.5390 5.5849 5.6290

7.8596 7.9161 7.9665 8.0094 8.0427 8.0630 8.0657 8.0453 7.9971 7.9202 7.8200

2.8763 2.8915 2.9059 2.9201 2.9349 2.9511 2.9695 2.9909 3.0163 3.0463 3.0815

5.5791 5.6101 5.6399 5.6690 5.6974 5.7250 5.7513 5.7754 5.7956 5.8092 5.8124

8.2648 8.2372 8.1844 8.1033 7.9957 7.8672 7.7263 7.5817 7.4416 7.3140 7.2066

+

(

dHc dx

(x

)+R

(

dHc dx

d2Hc d x2

+ ks Hc x

(2x

dHc dx

U5

cos2

d

) dx }

fr Hf

cos2

(xa2 / µc ) ( c Hc +

+

dHc dx

cos2

(cos2

+

2 sin2

sin (2 cos2 sin2

f Hf sec )

dHc dx

)+ d

)

2W

t2

sin2

d

) dx

W x

x2

3Rfr Hf Hc

=0

W (x , t ) = W ( x )

and (x , t ) =

(x )

ei

P2 =

d2 d + U1 + (U2 d x2 dx

2P ) 2

+ U3

(16)

Hc + 2x

dHc dx

t

d 2W dW + U4 =0, dx 2 dx

f

d x dx

sin

( Rf

fr

9Rfr Hf

R fr

cos2

3 R fr Hf x 2 cos (cos2 U5 = R fr Hf Hc x sin cos2

(14)

U6 = ks Hc x

R fr Hf sin

+ R fr Hf Hc x cos

(15) 446

d 2Hc cos3 d x2

+x

dHc cos3 dx

3k s x 2 dHc cos2 dx

x 2Hc (Hc + 3Hf R sec ) ,

3 Hf x sin

Substituting equation (14) in equations (12) and (13), we get,

U0

=0,

) x 2,

R c Hc2 + 3R fr Hf x 2

3 Hc Hf R sec

For harmonic vibrations with circular frequency , we assume that t

dHc dx

2W

)}

2P W 10

Hc cos ) dx }

(13)

ei

d 2W dW + U9 dx dx

dHc 2 x + 3R fr Hf x cos3 dx d 2 9R fr Hf Hc cos2 sin x dx

U2 = 3 Rc c Hc

+ { x (ks Hc + R fr Hf cos

+ U8

U1 = R c Hc2 x + 3 Rc Hc

x

sin2

)+R

d2 d + U6 + U7 d x2 dx

where, U0 = (Rc Hc2 + 3R fr Hf Hc cos3

Hf (x sin

fr

R fr Hf sin

fr Hf

2 sin2

+ { R fr Hf cos + ks Hc + x

x2

)+R

(cos2

R fr Hf sin

ks Hc + x

2

cos2

Hc + Hc cos2

Hc x cos

Ho = 0.3

I

R fr Hf Hc x sin +

Ho = 0.2

)

3ks

2x (cos2

,

x2

d x, dx

3R fr Hf x 2 sin

U3 =

2 sin2 ,

sin

d

) dx

dHc cos2 dx 2 sin2

+ )

d dx

Hc + Hc cos2

cos2

,U4 =

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Table 5 Values of frequency parameter

with varying values of core thickness at centre Ho for

FOR VARIOUS MODES

Ho

Hf = 0.005

C plate 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 S plate 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 F plate 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300

U7 =

U8 = x

= 0.010

Hf = 0.020

II

III

I

II

III

I

II

III

1.4298 1.7046 1.8868 2.0169 2.1147 2.1913 2.2530 2.3039 2.3467 2.3833 2.4150

3.4539 3.9470 4.2600 4.4780 4.6393 4.7642 4.8642 4.9464 5.0155 5.0747 5.1261

5.5900 6.3009 6.7502 7.0624 7.2932 7.4718 7.6148 7.7324 7.8314 7.9162 7.9900

1.2992 1.5596 1.7425 1.8800 1.9884 2.0767 2.1505 2.2134 2.2679 2.3158 2.3582

2.9690 3.4586 3.7958 4.0458 4.2403 4.3970 4.5266 4.6363 4.7307 4.8132 4.8861

4.7087 5.4388 5.9410 6.3126 6.6010 6.8328 7.0242 7.1858 7.3245 7.4455 7.5524

1.0813 1.3146 1.4924 1.6362 1.7566 1.8601 1.9506 2.0309 2.1030 2.1683 2.2278

2.3733 2.8234 3.1606 3.4282 3.6485 3.8346 3.9950 4.1354 4.2598 4.3714 4.4723

3.7153 4.3945 4.9014 5.3018 5.6298 5.9055 6.1420 6.3480 6.5298 6.6920 6.8382

0.9653 1.2104 1.3878 1.5230 1.6298 1.7169 1.7894 1.8511 1.9044 1.9511 1.9926

3.1850 3.7152 4.0594 4.3029 4.4854 4.6282 4.7436 4.8394 4.9206 4.9906 5.0519

5.4880 6.2315 6.7018 7.0288 7.2708 7.4582 7.6084 7.6091 7.2923 7.0201 6.7834

0.9570 1.1996 1.3761 1.5113 1.6189 1.7072 1.7813 1.8447 1.9000 1.9488 1.9925

2.8161 3.3247 3.6774 3.9403 4.1457 4.3120 4.4503 4.5677 4.6693 4.7584 4.8375

4.6615 5.4048 5.9158 6.2939 6.5875 6.8237 7.0188 7.1834 7.3243 7.4441 7.4738

0.8552 1.0695 1.2317 1.3611 1.4680 1.5588 1.6373 1.7064 1.7680 1.8236 1.8742

2.2927 2.7496 3.0922 3.3644 3.5889 3.7789 3.9429 4.0869 4.2148 4.3296 4.4338

3.6935 4.3771 4.8870 5.2900 5.6203 5.8981 6.1364 6.3441 6.5275 6.6910 6.8380

1.6521 2.0263 2.2873 2.4762 2.6224 2.7382 2.8323 2.9104 2.9814 3.0382 3.0874

3.8807 4.5033 4.9020 5.1750 5.3732 5.5198 5.6290 5.7088 5.7669 5.8008 5.8152

6.2444 7.0623 7.5512 7.8411 7.9715 7.9503 7.8200 7.6489 7.4768 7.3280 7.2044

1.6095 1.9777 2.2388 2.4353 2.5896 2.7146 2.8259 2.9064 2.9822 3.0583 3.1069

3.4498 4.0688 4.4966 4.8133 5.0586 5.2547 5.4201 5.5488 5.6615 5.7638 5.8386

5.3376 6.1848 6.7590 7.1715 7.4726 7.6823 7.7998 7.8206 7.7614 7.6564 7.5481

1.4102 1.7382 1.9901 2.1808 2.3427 2.4900 2.6099 2.7153 2.8091 2.8779 2.9531

2.8283 3.3940 3.8200 4.1504 4.4243 4.6615 4.8602 5.0335 5.1866 5.3123 5.4342

4.2558 5.0469 5.6371 6.0968 6.4726 6.7896 7.0543 7.2792 7.4689 7.6187 7.7336

R fr Hf sin

+ R fr Hf (x sin

f

I

x

+ ks Hc + x

d 2Hc cos2 d x2

(ks Hc + R fr Hf

+ R fr Hf sin

+

dHc cos2 dx

+

the range 0 x 1 is transformed to 1 y 1, the applicability range of the Chebyshev collocation technique. Eqs. (15) and (16) now reduce to

dHc dx

dHc dx

(cos2

U9 = R fr Hf cos

2 sin2

cos

sin2

)

Hc cos )

d , dx

),

A5

d2 d + A6 + A7 d y2 dy

2q ) 2

+ A8

+ A3

d 2W dW + A4 =0, dy 2 dy

d 2W dW + A9 dy dy

2q W 10

(18)

=0,

A 1 = 2 U1 ,

A 2 = U2 ,

q2 = P2 ,

A 3 = 4 U3 ,

A4

= 2 U4 , A5 = 4 U5 , A 6 = 2 U6 , A 7 = U7 A 8 = 4 U8 A 9 = 2 U9 and q 10 = P10 . According to Chebyshev collocation technique, we assume d2 = d y2

m 3

ak + 3 Tk

(20)

k=0

and

d 2W = d y2

m 3

bk + 3 Tk

(21)

k=0

where aj and bj ( j = 3, 4...m ) are the unknown constants and Tj (j = 0, 1 ... m 3 ) are the Chebyshev polynomials,

By taking a new independent variable

1

d2 d +A 1 + (A 2 d y2 dy

where A 0 = 4 U0 ,

1.2. Method of solution

2x

A0

(19)

dH + ks Hc + x c dx d 2 2 (2 cos sin ) dx

sin2

P10 = x (Hc + Hf R sec ) , 2 = c a2 2 / µc and R = f / c . The solution of equations (15) and (16) together with regularity Qr = 0 at the centre x = 0 and boundary condicondition = 0 ; tions at the edge x = 1 gives rise to a two point boundary value problem with variable coefficients whose closed form solution is not possible. Keeping this in view, an approximate solution is obtained by employing Chebyshev collocation technique.

y

= 0.5 .

T0 = 1 ,

(17) 447

T1 = y,

Tj = 2 y Tj

1

Tj

2

,

j

2.

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Table 6 Values of frequency parameter

with varying values of core thickness at centre Ho for

Hf = 0.005

C plate 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 S plate 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300 F plate 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 0.300

f

= 0.010

Hf = 0.020

I

II

III

I

II

III

I

II

III

0.8895 1.1101 1.2680 1.3870 1.4802 1.5552 1.6172 1.6694 1.7140 1.7527 1.7866

2.6446 3.1655 3.5224 3.7842 3.9853 4.1451 4.2756 4.3844 4.4767 4.5561 4.6253

4.5236 5.2981 5.8196 6.1978 6.4861 6.7138 6.8987 7.0522 7.1819 7.2931 7.3897

0.8167 1.0167 1.1643 1.2791 1.3717 1.4485 1.5136 1.5697 1.6187 1.6621 1.7009

2.2615 2.7308 3.0727 3.3369 3.5489 3.7239 3.8713 3.9977 4.1075 4.2041 4.2899

3.7435 4.4537 4.9687 5.3654 5.6831 5.9447 6.1648 6.3530 6.5163 6.6596 6.7867

0.6831 0.8489 0.9777 1.0834 1.1728 1.2504 1.3187 1.3797 1.4348 1.4850 1.5311

1.7973 2.1884 2.4914 2.7385 2.9467 3.1259 3.2829 3.4220 3.5467 3.6595 3.7621

2.9084 3.5071 3.9702 4.3474 4.6646 4.9372 5.1752 5.3859 5.5741 5.7439 5.8981

0.6154 0.8157 0.9753 1.1058 1.2146 1.3069 1.3862 1.4552 1.5156 1.5691 1.6168

2.4260 2.9698 3.3483 3.6278 3.8433 4.0147 4.1546 4.2711 4.3698 4.4545 4.5280

4.4158 5.2170 5.7563 6.1469 6.4442 6.6788 6.8691 7.0269 7.1601 7.2743 7.3734

0.6253 0.8306 0.9938 1.1268 1.2374 1.3308 1.4109 1.4803 1.5411 1.5948 1.6427

2.1514 2.6360 2.9892 3.2617 3.4801 3.6600 3.8114 3.9409 4.0532 4.1519 4.2393

3.7035 4.4248 4.9464 5.3474 5.6681 5.9319 6.1537 6.3432 6.5075 6.6517 6.7794

0.5823 0.7650 0.9110 1.0314 1.1331 1.2206 1.2970 1.3643 1.4244 1.4784 1.5272

1.7503 2.1489 2.4567 2.7073 2.9179 3.0991 3.2574 3.3977 3.5233 3.6368 3.7399

2.8958 3.4983 3.9636 4.3422 4.6603 4.9335 5.1720 5.3830 5.5716 5.7415 5.8959

1.1753 1.5393 1.8210 2.0456 2.2290 2.3816 2.5107 2.6214 2.7175 2.8018 2.8763

3.0484 3.7122 4.1719 4.5104 4.7705 4.9766 5.1439 5.2820 5.3977 5.4956 5.5791

5.1031 6.0189 6.6330 7.0744 7.4055 7.6607 7.8596 8.0143 8.1316 8.2147 8.2648

1.1642 1.5224 1.8003 2.0230 2.2060 2.3594 2.4901 2.6031 2.7018 2.7890 2.8667

2.6955 3.2985 3.7402 4.0824 4.3573 4.5842 4.7752 4.9385 5.0800 5.2039 5.3133

4.2915 5.1305 5.7379 6.2043 6.5761 6.8800 7.1330 7.3464 7.5275 7.6816 7.8120

1.0522 1.3599 1.6024 1.8016 1.9697 2.1147 2.2417 2.3542 2.4551 2.5462 2.6292

2.1983 2.7042 3.0967 3.4169 3.6864 3.9180 4.1205 4.2996 4.4597 4.6041 4.7352

3.3717 4.0812 4.6299 5.0765 5.4514 5.7731 6.0535 6.3009 6.5215 6.7197 6.8990

Successive integration of Eqs. (20) and (21) leads to

The satisfaction of Eqs. (24) and (25) at (m 2) collocation points yi , i = 1 , ..., (m 2) together with the regularity condition: = 0 ; Qr = 0 (Wu et al., [39]), at the centre of the plate provides a set of (2m 2) equations in terms of unknowns aj and bj . The resulting system of equations can be written in matrix form as

m 3

ak + 3 Tk2 ,

= a1 + a2 T1 +

(22)

k=0 m 3

bk + 3 Tk2 ,

W = b1 + b2 T1 +

where A and C are the matrices of orders (2m 2) × 2m and 2m × 1, respectively. The above (m 2) internal grid points chosen for collocation are the zeros of Chebyshev polynomial of order (m 2) with orthogonality range ( 1, 1) , given by

where a1 a2 , b1 and b2 are the constants of integration and Tkj represents the jth integral of Tk defined as:

T01 = T1 ,

T11 =

T2 + T0 4

,

T1j =

and T ij = T ij 1 d y . Substituting the values of and (19), we get 2

(A 2 + +

Tj d y

=

1 2

Tj + 1 j+1

Tj 1 j 1

,

j>1

yk = cos

, W and their derivatives in Eqs. (18)

a1 A7 + a2 ( A6 + A7 T1) +

m 3 k=0

b1 + b2 (A9

2

(i) = 0 ; (ii) W = 0 ; (iii) Mr = 0;

q10 T1)+

2

q10 Tk2 bk + 3 = 0

1 2 2

),

k = 1, 2, ..., (m

2) .

W = 0 : for clamped edge (C-plate), Mr = 0 : for simply supported edge (S-plate), Qr = 0 : for free edge (F-plate),

a set of two homogeneous equations is obtained. These equations together with the field Eq. (26) give a complete set of 2m equations in terms of 2m unknowns. For a clamped plate, the set of these 2m homogeneous equations can be written as

(24)

A8 Tk + A9 Tk1

2k m

By satisfying the relations:

q2 ) Tk2 ) ak + 3 + 0. b1 + b2 A 4

(A5 Tk + A6 Tk1 + A7 Tk2) ak + 3

2q 10

(

1.3. Boundary conditions and frequency equations

2

q2) a1 + (A1 + (A2 q2 ) T1) m 3 1 2 (A0 Tk + A1 Tk + (A2 k=0 m 3 1 ( A3 Tk + A 4 Tk ) bk + 3 = 0 k=0

(26)

[A] [C] = [0] ,

(23)

k=0

k=0

0.5.

FOR VARIOUS MODES

Ho

m 3

=

(25) 448

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Table 7 Values of frequency parameter

o

= 0.10

0.5 .

o

= 0.20

o

= 0.30

I

II

III

I

II

III

I

II

III

1.2468 1.2680 1.2210 1.1643 1.1100 1.0608 1.0169 0.9777

3.7047 3.5224 3.2845 3.0727 2.8922 2.7384 2.6062 2.4914

6.3097 5.8196 5.3535 4.9687 4.6519 4.3875 4.1632 3.9702

1.6045 1.6172 1.5703 1.5136 1.4584 1.4072 1.3608 1.3187

4.4272 4.2756 4.0676 3.8713 3.6962 3.5416 3.4047 3.2829

7.2751 6.8987 6.5094 6.1648 5.8656 5.6053 5.3770 5.1752

1.7800 1.7866 1.7480 1.7009 1.6539 1.6097 1.5687 1.5311

4.7456 4.6253 4.4556 4.2899 4.1377 3.9997 3.8751 3.7621

7.6765 7.3897 7.0761 6.7867 6.5269 6.2945 6.0860 5.8981

0.8697 0.9753 0.9988 0.9938 0.9773 0.9563 0.9337 0.9110

3.4048 3.3483 3.1689 2.9892 2.8284 2.6877 2.5648 2.4567

6.1626 5.7563 5.3183 4.9464 4.6365 4.3763 4.1547 3.9636

1.2486 1.3862 1.4177 1.4109 1.3884 1.3594 1.3283 1.2970

4.2181 4.1546 3.9858 3.8114 3.6500 3.5046 3.3743 3.2574

7.2076 6.8691 6.4923 6.1537 5.8579 5.5997 5.3728 5.1720

1.4798 1.6168 1.6489 1.6427 1.6204 1.5913 1.5595 1.5272

4.5857 4.5280 4.3878 4.2393 4.0981 3.9678 3.8487 3.7399

7.6461 7.3734 7.0656 6.7794 6.5216 6.2906 6.0831 5.8959

1.6650 1.8210 1.8329 1.8003 1.7530 1.7018 1.6510 1.6024

4.2503 4.1719 3.9555 3.7402 3.5471 3.3774 3.2283 3.0967

7.0745 6.6330 6.1510 5.7379 5.3912 5.0981 4.8473 4.6299

2.3131 2.5107 2.5290 2.4901 2.4316 2.3674 2.3033 2.2417

5.1835 5.1439 4.9686 4.7752 4.5896 4.4183 4.2622 4.1205

8.1661 7.8596 7.4852 7.1330 6.8157 6.5325 6.2798 6.0535

2.6758 2.8763 2.9002 2.8667 2.8125 2.7515 2.6896 2.6292

5.5624 5.5791 5.4633 5.3133 5.1582 5.0082 4.8669 4.7352

8.3886 8.2648 8.0539 7.8120 7.5655 7.3281 7.1055 6.8990

Table 8 Values of frequency parameter

with varying values of face thickness Hf for

= 0.5 .

FOR VARIOUS MODES

Hf o

C plate 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 S plate 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 F plate 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200

=

FOR VARIOUS MODES

Hf

C plate 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 S plate 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 F plate 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200

with varying values of face thickness Hf for

= 0.10

o

= 0.20

o

= 0.20

I

II

III

I

II

III

I

II

III

1.8823 1.8868 1.8191 1.7425 1.6699 1.6041 1.5451 1.4924

4.4541 4.2600 4.0170 3.7958 3.6030 3.4355 3.2893 3.1606

7.1835 6.7502 6.3161 5.9410 5.6214 5.3472 5.1095 4.9014

2.2495 2.2530 2.2064 2.1505 2.0951 2.0429 1.9948 1.9506

4.9907 4.8642 4.6930 4.5266 4.3736 4.2348 4.1090 3.9950

7.8920 7.6148 7.3086 7.0242 6.7677 6.5372 6.3297 6.1420

2.4028 2.4150 2.3911 2.3582 2.3237 2.2899 2.2579 2.2278

5.2060 5.1261 5.0072 4.8861 4.7710 4.6637 4.5643 4.4723

8.1719 7.9900 7.7679 7.5524 7.3517 7.1664 6.9957 6.8382

1.2700 1.3878 1.3988 1.3761 1.3420 1.3046 1.2674 1.2317

4.1419 4.0594 3.8687 3.6774 3.5039 3.3499 3.2134 3.0922

7.0898 6.7018 6.2832 5.9158 5.6006 5.3293 5.0937 4.8870

1.6723 1.7894 1.8021 1.7813 1.7484 1.7114 1.6739 1.6373

4.8062 4.7436 4.6010 4.4503 4.3070 4.1747 4.0536 3.9429

7.5840 7.6084 7.3030 7.0188 6.7623 6.5318 6.3242 6.1364

1.8880 1.9926 2.0071 1.9925 1.9667 1.9366 1.9053 1.8742

5.0839 5.0519 4.9503 4.8375 4.7270 4.6224 4.5247 4.4338

6.2033 6.7834 7.1925 7.4738 7.3498 7.1663 6.9957 6.8380

2.1120 2.2857 2.2873 2.2388 2.1753 2.1092 2.0451 1.9847

4.9450 4.9008 4.7062 4.4966 4.2997 4.1211 3.9606 3.8164

7.8829 7.5507 7.1397 6.7590 6.4220 6.1260 5.8655 5.6348

2.6408 2.8323 2.8517 2.8185 2.7672 2.7105 2.6535 2.5983

5.5191 5.6290 5.5502 5.4152 5.2670 5.1209 4.9821 4.8524

7.6855 7.8200 7.8656 7.8004 7.6441 7.4480 7.2453 7.0496

2.8816 3.0815 3.1196 3.1069 3.0750 3.0359 2.9945 2.9531

5.4845 5.8124 5.8848 5.8386 5.7484 5.6443 5.5380 5.4342

7.0887 7.2066 7.3750 7.5481 7.6885 7.7760 7.7921 7.7336

449

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

been reported as the first three natural frequencies of vibration to investigate the influence of the taper parameters, core thickness at the centre and face thickness for all the three boundary conditions. In the work reported here, the values of various plate parameters are taken as = 0.7 (0.1) 0.7, = 0.7 (0.1) 0.7, Ho = 0.05 (0.05 ) 0.30, follows: Hf = 0.0025 (0.0025) 0.0200 . The material for the core and the facings are taken to be PVC (Poly vinyl chloride) and glass polyester resin, respectively. The values of various constants are taking from Ref. [21] as follows: R c = 2.3090 , R fr = 155.7000, R f = 491.1462, R = 13.8462, f = 0.078 together with ks = 1. c = 0.32 and fr = 0.0247, To choose the appropriate number of collocation pointsm , a computer program used to evaluate the frequencies, was run for m = 5 (1) 20 for different sets of plate parameters for all the three boundary conditions. The numerical values showed a consistent improvement with the increase in the number of collocation pointsm . In all the computations, the number of grid points has been taken as 12 for clamped plate, 14 for simply supported plate and 18 for free plate, since further increase in m does not improve the result even at the fourth place of decimal for all the three modes. The number of collocation points required for the same accuracy increases with the increase in the number of modes as given in Table 1. In this regard, normalized frequency parameter / with the number of grid points m for first three modes of vibration for specified plate i.e. = 0.5, Ho = 0.3, Hf = 0.02 have been presented in Fig. 2, as maximum deviations were observed for this data. The present analysis can be extended to higher modes by increasing the number of collocation of points for same accuracy up to an extent otherwise the concept of ill-conditioning may occur or the effect of round-off errors will increase. A comparison of results for specified uniform isotropic circular plate ( = 0, Hf = 0, c = 0.3, Ho = 0.001, ks = 1) with exact solution [40] and obtained by Finite element method [41], Rayleigh-Ritz method [42], receptance method [43], Ritz method [44], symplectic method [45], DQM [46] for clamped and simply supported boundary conditions 2 has been presented in Table 2. For varying values of Ho and ks = 12 frequencies have been compared in Table 3 for all the three boundary conditions with Irie et al. [47], Wang et al. [48] and Mohammadi et al. [49]. A close agreement of results shows the versatility of present technique. The numerical results are given in Tables (4–9) and Figure (3-8). It is found that the values of frequency parameter for a clamped plate are greater than those for a simply supported plate but less than that for

Table 9 Percentage variation in the values of frequency parameter : p→represents the variation of Ho from 0.0 to −0.5; q→represents the variation of Ho from 0.0 to 0.5. Hf

Ho

0.005

0.1 I C plate p −17.7798 q 22.3447 S plate p −16.6339 q 18.6255 F plate p −11.1013 q 11.5847

II

III

0.2 I

II

III

−9.2235 9.7853

−7.5021 7.2891

−14.2933 19.4022

−5.8238 7.1410

−4.7391 5.1492

−9.4565 9.7729

−7.6761 7.4885

−12.0152 13.5766

−6.0088 7.3164

−4.8496 5.3912

−8.2575 7.7715

−6.9183 5.9599

−5.5631 6.5335

−4.6684 4.3219

−2.5226 −3.0138

A [C] = [0] , Ac

(27)

where Ac is a matrix of order 2 × 2m . For a non-trivial solution of Eq. (30), the frequency determinant must vanish and hence

A Ac

= 0.

(28)

Similarly, for simply supported and free plates, the frequency determinants can be written as

A As

A AF

= 0

(29)

=0 ,

(30)

respectively. 1.4. Numerical results and discussion The frequency Eqs. (28)–(30) provide the values of the frequency parameter and have been solved using MATLAB for various values of plate parameters. The numerical values of the lowest three roots have

Fig. 3. Taper parameter

vs. Frequency parameter 450

for Hf = 0.005 : Ho = 0.1, Ho = 0.2 .

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Fig. 4. Core thickness at the centre Ho vs. Frequency parameter

Fig. 5. Core thickness at the centre Ho vs. Frequency parameter

a free plate for the same set of plate parameters. Fig. 3 depict the effect of taper parameter on the frequency parameter for Hf = 0.005 and two values of Ho = 0.1, 0.2 for all the three modes of vibration for all the three plates. It is observed that frequency parameter increases with the increasing values of taper parameter for all the three modes of vibration and for all the three boundary conditions except in the third mode for simply supported and free plates. For these cases, there exist a local maxima in the vicinity of = 0.6 for simply supported plate which shifts towards = 0.1 in case of free plate. The frequency parameter also increases with the increasing values of Ho for all the three plates. The rate of increase of frequency parameter with the increasing values of taper parameter is in the order of boundary conditions C > F > S. This rate increases with the increase in the number of modes and for decreasing values of core thickness at the centre Ho for all the three boundary conditions.

for

for

= 0.5, Hf = 0.005; Hf = 0.020 .

=

0.5, Hf = 0.005 ; Hf = 0.020 .

The effect of Ho is more pronounced when the plate is thin at the outer edge i.e. changes from 0.7 to −0.7 for all the three plates. This effect increases with the increase in the number of modes for all the three plates in the order of boundary conditions F > S > C when Ho increases from 0.1 to 0.2. Fig. 4 demonstrate the effect of core thickness at the centre Ho on the frequency parameter for = 0.5 and two different values of face thickness Hf = 0.005, 0.020 for all the three plates and all the three modes of vibration. It is observed that the frequency parameter increases monotonically with the increasing values of Ho for all the three plates except the third mode of simply supported and free plates. For these cases, there exist a local maxima in the vicinity of Ho = 0.225 for simply supported plate which shifts towards Ho = 0.150 in case of free plate. The values of frequency parameter are also found to increase with the decreasing values of Hf keeping other parameters fixed. This 451

Composites Part B 165 (2019) 442–455

R. Rani, R. Lal

Fig. 6. Face thickness Hf vs. Frequency parameter

Fig. 7. Face thickness Hf vs. Frequency parameter

effect is more pronounced for smaller values as compared to the higher values of Ho . The rate of increase in the values of frequency parameter with Ho is noticed in the order of boundary conditions F > S > C which increases with the increasing values of Hf . Fig. 5 shows the effect of core thickness at the centre Ho on the frequency parameter for = 0.5 and two different values of face thickness Hf = 0.005, 0.020 for all the three plates for the first three modes of vibration. It is observed that the frequency parameter increases monotonically with the increasing values of Ho for all the three modes of vibration and for all the three plates. The rate of increase in the values of frequency parameter with the increasing values of Ho is found in the order of boundary conditions F > S > C which increases with the increase in the number of modes and the decreasing values of Hf . The values of frequency parameter are also found to increase with

for

for

= 0.5,: Ho = 0.1, Ho = 0.3.

=

0.5,: Ho = 0.1, Ho = 0.3.

the decreasing values of Hf keeping other parameters fixed. This effect is more pronounced for higher values as compared to smaller values of Ho in the order of boundary conditions C > F > S. Fig. 6 illustrate the behavior of facing thickness on the frequency parameter for two different values of Ho = 0.1, 0.3 and = 0.5. It is found that there is a local maxima in the vicinity of Hf = 0.005 for clamped and Hf = 0.0175 in case of simply supported and free plates when the plate is vibrating in the first mode of vibration. In case of decreases with the second mode the values of frequency parameter increasing values Hf except for free plate when Ho = 0.3. In this case, there is a local maxima in the vicinity of Hf = 0.0075. The value of frequency parameter decreases continuously with the increasing values of Hf for both the values of Ho = 0.1, 0.3 in case of clamped plate vibrating in third the mode of vibration. However, for simply 452

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Fig. 8. First three mode shapes for (a) Clamped (b) Simply supported, and (c) Free plate for Ho = 0.3, Hf = 0.02 and

supported and free plates the values of frequency parameter decreases continuously with increasing values of Hf for Ho = 0.1, but for Ho = 0.3, there exist a local maxima in the vicinity of Hf = 0.01 for simply supported plate and Hf = 0.0175 for free plate. The effect of Ho is more pronounced for higher values of Hf as compared to the smaller values.

= 0.5.

Fig. 7 delineate the effect of facing thickness Hf on the frequency parameter for two different values of Ho = 0.1, 0.3 and = 0.5. It is observed that there is a local maxima in the vicinity of Hf = 0.005 for clamped plate and Hf = 0.0075 in case of simply supported and free plates when the plate vibrating in the first mode of vibration. In case of second mode, there exist a local maxima in the vicinity of Hf = 0.0075 453

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for Ho = 0.3 for free plate. This may be attributed the increased stiffness of the plate due to increasing values of Hf . The value of frequency parameter decreases continuously with the increasing values of Hf for both the values of Ho = 0.1, 0.3 for all the three plates vibrating in third the mode of vibration. . The effect of Ho is more pronounced for higher values of face thickness Hf as compare to smaller values and increases with the increase in the number of modes in the order of boundary conditions F > S > C. Three dimensional mode shapes for a specified composite circular sandwich plate taking Ho = 0.3, Hf = 0.02 and = 0.5 have been plotted and shown in Fig. 8.

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2. Conclusion The radially symmetric vibrations of composite circular sandwich plates with core of exponentially varying thickness have been analyzed employing Chebyshev collocation technique retaining the effect of transverse shear deformation and rotatory inertia in the core and the face sheets are treated as membrane. Following observations have been carried out from the obtained results: 1. It is observed that the frequency parameter for clamped plate is higher than that for simply supported plate but lower than that for free plate for the same set of the values of plate parameters. 2. The frequency parameter increases with the increasing values of taper parameters as well as core thickness at the centre Ho but it decreases with the increasing values of face thickness parameter Hf . 3. The percentage variation in the values of frequency parameter for the varying values of taper parameter with Hf = 0.005 and two values of Ho = 0.1, 0.2 have been given in Table 8 and it can be seen that as the plate becomes thinner and thinner towards the boundary i.e. changes from −0.5 to 0.0 for Hf = 0.005 and Ho = 0.1, 0.2 . 4. The percentage variation in frequency parameter is in the order of boundary conditions F > S > C while it becomes C > S > F when changes from 0.0 to 0.5. This may be attributed to the increased/decreased inertia as well as stiffness of the plate. The present analysis will be of great use to the design engineers in obtaining the desired frequency by varying one or more plate parameters considered here. Acknowledgment The authors wish to express their sincere thanks to the learned reviewers for their constructive suggestions for improving the paper. One of the authors, Rashmi Rani is thankful to Ministry of Human Resources and Development (MHRD), India for the financial support to carry out this research work. References [1] Almeida MIA. Structural behaviour of composite sandwich panels for applications in the construction industry. October 2009. [2] Mantari JL, Oktem AS, Soares CG. A new trigonometric layerwise shear deformation theory for the finite element analysis of laminated composite and sandwich plates. Comput Struct 2012;94–95:45–53. [3] Whitney JM, Leissa AW. Analysis of simply supported laminated anisotropic plates. AIAA J 1970;8:28–33. [4] Reissner E. A consistent treatment of transverse shear deformation in laminated anisotropic plates. AIAA J 1972;10(5):716–8. [5] Ferreira AJM, Roque CMC, Jorge RMN. Analysis of composite plates by trigonometric shear deformation theory and multiquadrics. Compos Struct 2005;83:2225–37. [6] Sahoo R, Singh BN. A new trigonometric zigzag theory for static analysis of laminated composite and sandwich plates. Aerosp Sci Technol 2014;35:15–28. [7] Altenbach H. Theories for laminated and sandwich plates. A Rev Mech Compos Mats 1998;34:243–52. [8] Carrera E. An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates. Compos Struct 2000;50:183–98.

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