NUMERICAL INVESTIGATION OF NATURAL FREQUENCIES FOR CLAMPED LONGITUDINAL COMPOSITE PLATES

ABSTRACT In this paper, two finite element models were performed. The fiber and matrix represented as two different materials in the first plate, while the second showed as a composite plate. The boundary conditions included clamped the plates on four ends and the dimensions of the composite plates were changed in this study. The finite element was performed and ANSYS 16.1 was employed in modeling. By comparing the results between the frequency ratio and mode numbers for different plate thickness, the results showed that natural frequencies calculated by these two model thickness of the (length/width) ratio will be more uniform and the error will be small. The numerical equations that performed from this study used to investigate the natural frequencies for longitudinal clamped composite plate.


INTRODUCTION
Laminated composite plates are used as a main component in primary aerospace and aircraft structures and marine structures because of their properties. Properties such as high specific strength and stiffness, high fatigue and corrosion resistance have attracted researchers and companies to use them instead of conventional materials. In the last few decades, the use of composite materials has been increased significantly (Sharma and Mittal, 2010). Generally, composite materials consist of two (or more) phases. The first one has is the strongest phase which is call fiber. Fiber is used to reinforce the other phase which it is usually polymer.
Depending on the application and manufacturing process, fiber can be continuous or discontinuous and longitudinal or randomly oriented in the matrix. The distributed fibers in the matrix are used to transfer the loads to fibers (Gay et al., 2003;William and David, 2010).
Because of the requirement of high performance, the resonance behavior of the laminated structures materials in aerospace structures have been studied by many researchers. For example, Leissa (Leissa, 1973) studied the vibration of plate with various geometries. Crawly, 1979 showed experimentally the natural frequency and mode shapes of aluminum plates and compared the results with finite element method.
Recently, finite element method have been employed in analyzing the engineering structure.
For instance, Reddy, 1979 used the finite element method to study the free vibration of simplysupported plates. Han and Petyt, 1996 estimated the natural frequency of laminated rectangular plates by extending the p-version finite element method. Also, different boundary conditions are using to study the free vibration for rectangular plates by Hsu, 2003. Pandit et al., 2007 analyzed the free vibration of laminated composite rectangular plate using finite element method. Sang and Sang-Hyun, 2008 proposed new analysis of vibration for simply supported composite plates. Latheswary et al., 2004 investigated the free vibration analysis of laminated composite plates in linear and non-linear. Akavci, 2007 examined the analysis of free vibration for simply supported composite plates. Andrzej and Gawryluk, 2016 studied the modal analysis for three composite blades and the results of natural frequencies and mode shapes were compared with the modal analysis for the cantilever composite beam and the fixed rotor with one composite blade. The free vibration analysis of composite plates with various boundary conditions were discussed in several research using several mathematical techniques (Nayak, 2008;Alexander et al., 2012;Lopatin and Morozov, 2011).
In this project, two models are performed to calculate the natural frequencies of clamped longitudinal composite plates with different dimensions. ANSYS 16.1 was used in analyzing 94 Firas T. Al-Maliky these two models. In the first model, perfect bonding between fibers and matrix was assumed.
While, the equivalent mechanical properties of composite plate were taken into account. The effects of dimensions of composite plate, volume fractions and mechanical properties of the composite plates were studied in order to find the correction factor between the two models.

MATERIAL PROPERTIES AND VOLUME FRACTIONS
The mechanical properties of fiber and matrix that are used in this work are presented in Table  1.

Table 1. Fiber and Matrix Properties
The effective mechanical properties of the composite plates depend on mechanical properties of matrix and fiber. In order to calculate the equivalent mechanical properties, the following procedure can be used (Gay et al., 2003): 1. The total volume fraction and the matrix volume fraction: in this work , the fiber volume fraction changes from (10%) to (40%) by increasing the volume fraction by (5%) and according to the following equation, the matrix volume fraction can be calculated. According to these steps, the equivalent mechanical and physical properties of composite plates for different fiber volume fractions can be listed in Table 2.

FIRST FINITE ELEMENT MODEL
In this model, the fibers and matrices were represented as two different materials which are In this model, the size of element is one of the most important factor in order to get an accurate result. Therefore, the suitable size of element is found for each volume fraction.

SECOND FINITE ELEMENT MODEL
In this model, the composite plate is represented as one material with equivalent mechanical and physical properties calculated previously (see Table 2). The element Layer 99 is also used in this model and the thickness of each layer in this model does not exceed (1 mm) as shown in Fig. 1. = ).

DIMENSIONS AND BOUNDARY CONDITIONS OF COMPOSITE PLATE
The length and width of composite plates considered in this work are changed in order to study the effect of dimensions of plate on the natural frequency. On the other hand , the thickness of plate is changed and its values are (2 ,3 , 4 and 5) mm. These dimensions of plates can be summarized in Table 3. The clamped of the four ends of plates is the boundary condition used in this work. 0.3 0.3 2, 3, 4 and 5

RESULTS
In this work, several cases were studied including two finite element models of composite plates for different dimensions and thickness. Natural frequency ratio of the two models of composite plates can be predicted from Table 4

CONCLUSION
From the conducted results, it has been observed that the natural frequency increases when the value of fiber volume fraction increases. The increasing in the fiber volume fraction leads to increase in the modules of elasticity of composite plate.
It has been found that when the plate is fixed from all the sides, the highest natural frequency is achieved compared to other considered boundary conditions. Furthermore, the numerical equations that derived from these relations as a function of the (length/width) ratio was used to investigate the natural frequencies for longitudinal composite plates.