Vibration Analysis of Fibrous Composite Plates with Central Circular Hole
Dr. N. K. Jain
Department of Mechanical Engineering, National Institute of Technology, Raipur (CG), India-492010
*Corresponding Author E-mail: nkjmanit@rediffmail.com
ABSTRACT:
A finite element study is made for the analysis of effect of fibre orientation on natural frequency (ω) for different mode shape in the simply supported and fixed rectangular fibrous composite plates with central circular hole. Study is carried out for three different D/A ratios (where D is hole diameter and A is plate width) to analysis the variation of natural frequency with respect to D/A ratio. The analytical treatment of such type of problem is very difficult; hence, finite element method is adopted for whole analysis. The finite element formulation is carried out in ANSYS package. The work is most appreciated in aerospace and marine industry where the study will provide the guidelines technique to designer for laminate configuration as per their requirements.
KEYWORDS: Vibration, Natural Frequency, Mode Shape, Finite element method, Plate, Composite, Elastic constants, Fibre orientation.
1 INTRODUCTION:
The fibrous and laminated composite plates with central circular hole have found widespread applications in various fields of engineering such as aerospace, marine, automobile and mechanical. For the design of such composite plates with central circular hole, accurate knowledge of stress and vibration is required. In order to cater the wide applications of such plates, dynamic analysis especially, knowledge of vibration characteristics become important to improve upon the reliability of their design. An exhaustive literature review on vibration of plates has been given by Leissa [1] in his monograph. The conventional Navier and Levy type solution methods [2, 3] can be extended to orthotropic plates having either all edges simply supported or two opposite edges simply supported. Rayleigh-Ritz method, one of the well-known methods is widely used to provide approximate solutions to natural frequencies of orthotropic plates [4-7]. Shastry and Raj [8] have analyzed the effect of fiber orientation for a unidirectional composite laminate with finite element method by assuming a plane stress problem under in plane static loading. Jain and Mittal [9] have studied a rectangular laminated composite plate with central hole upon the effect of fiber orientation on stress concentration factor under transverse static loading, made of different composite materials. Literature on vibration analysis shows that the natural frequencies are affected by the presence of defects in the form of cracks or holes.
In this article a study of rectangular fibrous composite plate with central hole upon the effect of fiber orientation on natural frequency is made. The analytical treatment for such type of problem is very difficult and hence the finite element method adopt for whole analysis.
2 Description of Problem
To study the influence of fibre orientation upon natural frequency a fibrous composite plate of dimension 200 mm X 100 mm X 1 mm with a circular hole of diameter D at center for all boundary conditions is analyzed by finite element method. The analysis is carried out for three different D/A ratios. Figure 1 shows the basic model of the problem.
Fig. 1 Details of model analyzed in study (A fibrous plate with central hole)
3 Finite Element Analysis
An eight nodded Linear Layered Structural 3-D Shell Element with six degrees of freedom at each node (specified as Shell99 in ANSYS package) was selected based on convergence test and used throughout the study. Each node has six degrees of freedom, making a total 48 degrees of freedom per element. In order to construct the graphical image of the geometries of the three different models for different D/A ratios, a laminated plate examined using the ANSYS. It was necessary to input the basic geometric elements such as points, lines and arcs. Mapped meshing are used for all models so that more elements are employed near the hole boundary. Due to the un-symmetric nature of different models investigated, it was necessary to discretize the full laminated plate for finite element analysis. Main task in finite element analysis is selection of suitable element type. Numbers of checks and convergence test are made for selection of suitable element type from different available elements and to decide the element length. Figure 2 provides the example of the discretized models for D/A =0.2, used in study.
Fig. 2 Finite element mesh for D/A=0.2
4 RESULTS AND CONCLUSIONS:
The numerical results are presented solid and hollow plates with three different D/A ratio as 0.1, 0.2 and 0.5. The material for the composite plate is taken as Boron Epoxy with properties, Ex = 208 x 109 Pa, Ey = 18.9 x 109 Pa, νx = 0.23, Gxy = 5.7 x 109 Pa, ρ = 2000 kg/m3. Where; E, G, ν and ρ represent modulus of elasticity, modulus of rigidity, poisson’s ratio and density respectively. Results are calculated for two different boundary conditions. In simply supported plate, all edges are simply supported and in fixed plate, all edges are fixed. The effect of fibre orientation on natural frequencies for simply supported and fixed fibrous composite plates with different D/A ratios for 1st, 2nd and 3rd mode are shown figures 3 to 8 respectively. Following important observations can be made from figures 3-8. For 1st mode natural frequency increase with fibre orientation and attaining a maximum value at θ=900 for solid and hollow plates both and for all boundary conditions. For 1st mode, it is also clear that at zero degree fibre orientation maximum natural frequency is obtaining for D/A=0.5, but at θ=900 natural frequency in solid plate is much higher in compare to hollow plate for all D/A ratios. In case of mode 2 and 3, there is also considerable variation in natural variation with fibre orientation. However, the natural frequency for 1st mode is most important in compare to higher modes.
In general, the maximum natural frequency is always obtained maximum in case of fix plates for all modes. It is also observed that natural frequency follow the symmetric trend with respect to 900 fibre orientation. It has been also seen that the natural frequency is most sensitive to material properties and directly depend on the ratio of Ex/Ey and Ex/Gxy. It is also clear from results that at θ=900, production of hole or any other types of discontinuity decreases the natural frequency of solid plate very much, and hence strength of plate is also decrease.
5. REFERENCES
1. Leissa AW., 1969, Vibration of Plates (NASA SP-160), Washington DC, Government Printing Office.
2. Szilard R (2004) Theories and applications of plate analysis. New jersey: John Wileyand Sons.
3. Xing YF and Liu B (2009) New exact solutions for free vibrations of thin orthotropic rectangular plates. Composite structures 89: 567-574.
4. Bazely NW, Fox DW and Stadter JT (1965) Upper and lower bounds for frequencies of rectangular clamped plates. Applied Physics Laboratory, Technical Memo, TG-626. The John Hopkins University, Baltimore.
5. Bhat RB (1985) Natural frequencies of rectangular plates using characteristics orthogonal polynomials in the Rayleigh -Ritz method. Journal of Sound and Vibration 102(4):493-499.
6. Dickinson SM and Di Blasio A (1986) On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates. Journal of Sound and Vibration 108(1): 51-62.
7. Marangoni RD, Cook LM and Basavanhally N (1978) Upper and lower bounds to the natural frequencies of vibration of clamped rectangular orthotropic plates. International Journal of Solids and Structures 14: 611-623.
8. Shastry BP and Raj GV (1977) Effect of fibre orientation on stress concentration in a unidirectional tensile laminate of finite width with a central circular hole. Fibre Science and Technology, 10: 151-154.
9. Jain NK and Mittal ND (2008) Effect of fibre orientation on stress concentration factor in a laminate with central hole under transverse static loading. Indian Journal of Engineering and Material Science, 15: 452-458.
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Received on 12.01.2014 Accepted on 29.01.2014 © EnggResearch.net All Right Reserved Int. J. Tech. 4(1): Jan.-June. 2014; Page 145-147 |