Free vibration analysis of a thick skew composite plate with a circular cutout.

Srividya, K. ; Murthy, V. Bala Krishna ; Satyanarayana, M.R.S. 等

Introduction

Fiber reinforced composites are finding increasing applications in the aerospace, marine, transportation,electrical,chemical,construction and consumer goods industries. In some of these applications the composites are subjected to dynamic loads. The composite structures may some times be provided with different types of holes for the purpose of assembling the components and units inside the structure, for passing the cables and control mechanisms, for inspection, maintenance and attachment to other units. The stresses and deformations of steep gradient are induced around these cutouts. The influence of the thickness parameter is inherent at higher modes of vibration. In this paper we study the dynamic behavior of anisotropic thick skew laminates with circular cutout.

To study the dynamic behavior, different theories were proposed by various authors [1-4]. The higher order bending theories [5-7] and the hybrid-stress finite element [8] have been developed for modeling the behavior of thick laminated plates accurately. Kant and Mallikharjuna [9, 10] used the finite element formulation for higher order theories to evaluate the free vibration frequencies of asymmetric and symmetric laminated plate. A three-dimensional hybrid-stress element was used by Sun and Liou [8] to study the fundamental frequency of cross-ply laminates. Spilker [11] developed an eight-noded hybrid-stress isoparametric element in which intraelement equilibrating stresses and inter-element displacements are interpolated independently. The in-plane displacements are assumed as varying linearity over each layer and the deflection constant over the thickness of entire laminate. Sixty-seven stress parameter fields are chosen.

A few references dealing with dynamic behavior of plates with holes are available. Rao and Pickett [12], Prabhakaran and Rajamani [13] studied the free vibration of rectangular composite plates with circular and square cutouts. Asku and Ali [14], Ali and Atwal [15] studied the natural frequencies of rectangular plates with rectangular holes. A cylinder with cutouts was analyzed by Brogan et al. [16] using the total energy variational functional and finite difference technique. Bicos and Springer [17, 18] applied the higher order theory to study the vibration characteristics of composite panels with cutouts. Advances in vibration analysis of laminated beams and plates were reviewed by Kapania and Raciti [19]. The practical importance of thick laminates with cutouts necessitates further improvement in the analysis. The present investigation intends to apply three-dimensional finite element techniques, based on theory of elasticity, for the free vibration analysis of clamped thick skew laminates with circular cutout. The lowest five natural frequencies are studied by varying the parameters: (i) d/1 ratio (ii) skew angle [alpha]

Problem Modeling

Geometric Modeling

The geometry of the problem is shown in Fig. 1. The sides of the plate 'l' and 'b' are taken equal to 2m and five layers are considered with total thickness of the odd number layers (h/6+h/6+h/6) equal to the thickness of the even number layers (h/4+h/4), Where 'h' is the total thickness of the laminate, which is taken from the length-to-thickness ratio (s=10). The skew angle ([alpha]) is varied from 0[degrees] to 60[degrees]. The circular hole is placed at the geometric centre of the plate. The diameter of the hole is varied as per the d/1 ratio from 0 to 0.55.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Finite Element Modeling

The element used for the present analysis is SOLID 95 of ANSYS [20], which is developed, based on three-dimensional elasticity theory and is defined by 20 nodes having three degrees of freedom at each node, translation in the node x, y and z directions. The model with finite element mesh is shown in Fig.2.

Material Properties

Each layer is unidirectional carbon fiber reinforced plastic possessing the following engineering constants [21].

* Elastic modulus in the longitudinal direction of the fiber, [E.sub.L] = 175 GPa.

* Elastic modulus in the transverse direction of the fiber, [E.sub.T] = 7 GPa.

* Shear modulus in the longitudinal plane of the fiber, [G.sub.LT] = 3.5 GPa.

* Shear modulus in the transverse plane of the fiber, [G.sub.TT] = 1.4 GPa.

* Poisons ratio, [sup.v]LT = [sup.v]TT = 0.25

* Mass density, [rho]=1.6X[10.sup.3] kg/[m.sup.3]

Boundary Conditions

The edges of the skew laminate considered for the present analysis are clamped i.e. all the three degrees of freedom of the nodes along the four edges of the skew plate are constrained.

Analysis of Results

Validation

Sufficient number of convergence tests are made and the finite element model is validated with the results available in the literature and found good agreement. (Table 1.)

Effect of d/1 Ratio

The variation of first five natural frequencies w.r.t d/1 ratio are presented in Figure 37. From the Figure 3 which is observed that the first natural frequency increases with increase in d/1 for all values of [alpha]. The second natural frequency decreases up to 0.35 of d/1 and later increases for [alpha]=0[degrees] and 30[degrees]. In case of [alpha]=60[degrees] this frequency decreases up to 0.25 of d/1 and later increases (Figure 4). The third natural frequency for [alpha]=0[degrees] decreases up to 0.35 of d/1 and then increases. For [alpha]=30[degrees], this frequency decreases up to 0.25 and later increases. At [alpha]=60[degrees] this frequency increases with increase in d/1 (Figure 5). In fourth mode, the frequency is almost constant up to 0.25 of d/1 for [alpha]=0[degrees] and 30[degrees] and beyond this point there is a slight drop for [alpha]=0[degrees] and a rise for [alpha]=30[degrees] is observed. In case of [alpha]=60[degrees] the frequency decreases up to 0.15 of d/1 and later increases (Figure 6). In fifth mode the frequency decreases up to 0.45 of d/1 and then it increases for [alpha]=0[degrees] and 30[degrees].At [alpha]=60[degrees] this frequency decreases up to 0.15 of d/1 and later increases (Figure 7). From the above cases it is observed that the rate of increase of natural frequency is maximum for [alpha]=60[degrees] and minimum for [alpha]=0[degrees].

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Effect of Skew Angle [alpha]

The variation of first five natural frequencies w.r.t the skew angle [alpha] is shown in Figs. 8-12. It is observed that the frequency in all the modes increases with increase in [alpha] and the rate of increase is observed to be more beyond [alpha]=40[degrees]. The first two natural frequencies are minimum for d/1=0 and maximum for d/1=0.55 (Figure 8 & 9). In the third natural frequency is maximum for d/1=0.55 and minimum for d/1=0.35 up to 40[degrees] of [alpha] and for d/1=0 between [alpha]=40[degrees] and 60[degrees] (Figure10). In the fourth mode, also the frequency is maximum for d/1=0.55. The frequency is minimum for d/1=0.05 up to 27[degrees] of [alpha] and later it is minimum for d/1=0 (Figure11). In fifth mode, the frequencies are maximum for d/1=0[degrees] up to 30[degrees] of [alpha]. Beyond [alpha]=30[degrees] the frequency is maximum for d/1=0.55up to [alpha]=50[degrees] the frequencies are minimum for d/1=0.35 and for [alpha]=60[degrees] the frequency is minimum for d/1=0 (Figure 12).

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

The increase in frequency in any case is due to the increase in stiffness of the plate and/or due to the decrease in mass of the plate for any change in the geometry of the plate. The decrease in frequency at any position is due to the decrease in stiffness of the plate. In some of the modes it is observed that there is no significant variation in frequency w.r.t d/1. This is due to the balancing of the factors that influence the

Conclusions

Free vibration analysis of a thick five layered symmetric cross-ply skew laminate with circular cutout has been taken up in the present work. From the present analysis the following conclusions are drawn. The effect of skew angle is more on the frequencies than the effect of d/1 ratio .The frequencies are maximum for all the values of d/1 at [alpha]=0[degrees]. The frequencies are maximum at d/1=0.55 for all the values of [alpha] except at in the fifth mode up to [alpha]=30[degrees], in which case the frequency is maximum at d/1=0. The skew plate with higher values of [alpha] and d/1 is preferred since it has the highest natural frequencies in all the modes. For a particular value of d/1 the plate with highest skew angle is preferred. For a plate with specific skew angle higher d/1 values are preferred.

References

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[10] Mallikarjuna, and Kant, T., 1989, "Free vibration of symmetrically laminated plates using a higher order theory with finite element technique", Int. J.number.Meth.Engg., 28, pp.1975-1989

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[12] Joga Rao, C.V., and Picket, G., 1961, "Vibrations of plates of irregular shapes and plates with holes", J.Aeronaut.Soc. India., 13, pp.83-88

[13] Prabhakaran, H., and Rajamani, A., 1978, "Free vibration characteristics of clamped-clamped composite plates with circular holes and square cutouts", Second Proc.Int. Conf. Composite materials (ICCM-2), pp.367-375

[14] Aksu, G., and Ali, R.,1976, "Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation", J.Sound and Vibration, 44, pp.147-158

[15] Ali, R., and Atwal, S.J., 1980, "Prediction of natural frequencies of vibration of rectangular plates with rectangular cutouts", comput.struct.,12, pp.819-823

[16] Brogan, F., Forseberg, K., and Smith, S., 1969, "Dynamic behavior of a cylinder with a cutout", AIAA Jnl.,7, pp.903-911

[17] Bicos, A.S., and Springer, G.S., 1989, "Analysis of free damped vibration of laminated composite plates and shells", Int. J.Solids Struct., 25, pp. 129-149

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[20] ANSYS reference Manuals, 2007.

[21] Ramakrishna, S., rao, M.K., and Rao, N.S., 1993, "Dynamic analysis of laminates with elliptical cutouts using the hybrid-stress finite element", Computers & Structures, 47, No.2, pp.281-287

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K. Srividya (1), V. Bala Krishna Murthy (1), M.R.S. Satyanarayana (2) G. Sambasiva Rao (1) and K. Mohana Rao (3)

(1) Mech. Engg. Dept., P.V.P.Siddhartha Institute of Technology, Vijayawada

(2) Mech. Engg. Dept., GITAM Institute of Technology, Visakhapatnam

(3) Mech. Engg. Dept., V.R. Siddhartha Engineering College, Vijayawada E-mail: Srividya.kode@yahoo.in, vbkm64@yahoo.co.in

Table 1: Validation of the finite element results.
([bar.[omega]] of three-layered cross-ply square plate)

[E.sub.L]/[E.sub.T]      3         10        20

Present               0.26112   0.3223    0.3591
Ramakrishna [21]      0.26461   0.32451   0.37717
Noor [22]             0.26474   0.32841   0.38241
CLT [23]                   --   0.41264   0.54043

Where the normalized frequency, [bar.[omega]] = [omega]h
[([[rho].sub.3]/E[T.sub.T3]).sup.1/2] [omega] = Natural Frequency