Abstract
The present paper deals with the experimental and finite element studies carried out on free vibration of isotropic and laminated composite skew plates. The natural frequencies were determined using CQUAD8 finite element of MSC/NASTRAN and comparison made between the experimental values and the finite element solution. The effects of the skew angle and aspect ratio on the natural frequencies of isotropic skew plates were studied. The effects of skew angle, aspect ratio, fiber orientation angle and laminate stacking sequence (keeping total number of layers in the laminate constant) on the natural frequencies of antisymmetric laminated composite skew plates were also studied. The experimental values of the natural frequencies are in good agreement with the finite element solution. The natural frequencies generally increase with an increase in the skew angle for any given value of aspect ratio.
Introduction
The skew plates have wide application in civil, marine, aeronautical and mechanical engineering, some of them being wings of aeroplanes, skew bridges, ship hulls and parallelogram slabs in buildings. The exact solutions to free vibration problems of skew plates are mathematically involved and in many complicated cases not available. Hence, most of the available solutions are based on approximate methods, the most commonly used method being the finite element analysis.
The earlier studies on free vibration characteristics of skew plates are those of Barton (1951), Kaul and Cadambe (1956) and Hasegawa (1957) using Rayleigh–Ritz method. Hamada (1959) applied the Lagrangian multiplier technique to obtain the fundamental frequency of rhombic skew plate. Classen (1963) extended the work of Barton (1951) by adopting a Fourier sine series solution scheme in conjunction with the Rayleigh–Ritz method. Conway and Farnham (1965) employed the point matching method to study the free vibration of triangular, rhombic and parallelogram plates. The frequencies were calculated for different values of skew angle for both simply supported and clamped boundary conditions. Laura and Grosson (1968) obtained fundamental frequencies of vibration for simply supported rhombic plates using conformal mapping and Galerkin’s method and compared their results with those of Conway and Farnham (1965). The discrepancy has been observed to increase with the skew angle.
Monforton (1968) obtained the fundamental frequencies of clamped rhombic plates using finite element method. The frequencies and mode shapes of clamped skew plates were studied by Durvasula (1969) using Galerkin’s method. The deflection function was expressed as a double series of beam characteristic functions in terms of skew coordinates to satisfy the zero deflection and normal slope on all the edges of the plate. The results obtained were compared with the results of Kaul and Cadambe (1956), Hasegawa (1957) and Conway and Farnham (1965). Thangam Babu and Reddy (1971) investigated the free vibration characteristics of orthotropic skew plates with two opposite edges simply supported and the other two edges free. Nair and Durvasula (1973) reported the natural frequencies of isotropic and orthotropic skew plates for simply supported, clamped, free edge boundary conditions and also for combinations of these conditions. Srinivasan and Ramachandran (1975) employed a numerical method to obtain the natural frequencies and mode shapes of orthotropic skew plates. Kuttler and Sigillito (1980) used trial function method to solve the free vibration problem of skew plates.
Mizusuwa et al. (1979, 1980) and Mizusuwa and Kajita (1986, 1987) employed the Rayleigh–Ritz method with Bspline functions and studied the effects of skew angle and location of point supports on natural frequencies of isotropic skew plates. Liew and Lam (1990) used a set of 2D orthogonal plate functions as admissible deflection functions and studied the flexural vibration of skew plates using Rayleigh–Ritz method. They obtained the free vibration characteristics of rhombic plates with different boundary conditions. Bardell (1992) adopted the hierarchical finite element method to determine the natural frequencies and mode shapes of isotropic skew plates. Liew and Wang (1993) employed the Rayleigh–Ritz method and obtained the results for rhombic plates with different boundary conditions by varying the location of internal support, skew angle and aspect ratio. Singh and Chakraverthy (1994) evaluated the first five frequencies of the transverse vibration of skew plates under different boundary conditions using orthogonal polynomials. McGee and Butalia (1994) studied the free vibration of thick and thin cantilever skew plates using C^{0} continuous isoparametric quadrilateral element.
Kamal and Durvasula (1986) studied the free vibration characteristics of laminated composite plates using modified shear deformation layered composite theory and Rayleigh–Ritz energy approach. Malhotra et al. (1988) studied the free vibration characteristics of rhombic orthotropic plates using a parallelogram orthotropic plate finite element for various boundary conditions and skew angles. Krishnan and Deshpande (1992) employed DKT finite element to determine the effects of fiber orientation angle, skew angle, aspect ratio and lengthtothickness ratio on the fundamental frequencies of single layer Graphite/Epoxy and Glass/Epoxy skew plates.
Krishna Reedy and Palaninathan (1999) used a high precision triangular plate bending element to study the free vibration characteristics of laminated composite skew plates. A consistent mass matrix has been used in the study. Singha and Ganapathi (2004) studied the free flexural vibration characteristics of laminated composite skew plates using finite element approach. Garg et al. (2006) have carried out free vibration studies on isotropic, orthotropic, and layered anisotropic composite and sandwich skew plates using isoparametric finite element model.
Clary (1975) investigated theoretically and experimentally the effect of fiber orientation on the first five flexural modes of free vibration of rectangular unidirectional Boron/Epoxy panels. The theoretical and experimental values of frequencies were in good agreement except in the case of thinner panels where there was appreciable error. Cawley and Adams (1978) used finite element method which included the effect of transverse shear deformation to predict the natural modes of freefree CFRP plates. Maruyama et al. (1983) used the realtime technique of holographic interferometry to determine the natural frequencies and the corresponding mode shapes of transverse vibration of clamped trapezoidal plates. Chakraborty et al. (2000) made studies on free vibration response of FRP composite plates using experimental and numerical techniques. Dutt and Shivanand (2011) studied the free vibration response of woven carbon composite laminated plates with C–F–F–F (one edge clamped and the other three free) and C–F–C–F (two opposite edges clamped and the other two edges free) boundary conditions using a FFT analyzer and compared the results obtained with finite element solution. Today the skew plate problem is widely recognized as a benchmark in the testing of a newly developed finite element. Very few experimental studies have been made on laminated composite skew plates and the finite element solution experimentally validated. The present paper is an attempt to address this aspect at least in a partial manner.
Test specimens and experimental setup
Test specimens
In this study, isotropic plates made of Aluminum 7075T6 were used. The material was supplied by the RioTinto Alcon, Canada (Pechiney Aluminum, France). The laminated composite plate specimens were fabricated by hand layup technique using unidirectional Glass fibers, Epoxy556 Resin and the Hardener (HY951) supplied by CibaGeigy India Ltd. The fiber weight percentage is 50:50. The test specimens were prepared in accordance with the relevant ASTM provisions. The material properties of the isotropic plates made of Aluminum 7075T6 are: E = 71.7 GPa, μ = 0.33 and ρ = 2,800 kg/m^{3} and these were supplied by the manufacturer RioTinto Alcon, Canada. For laminated Glass/Epoxy composite plate, the material constants E_{1} and E_{2} were evaluated experimentally using Instron Universal Testing Machine as per ASTM Standard D 3039/D 3039 M (2006). The average of three experimental determinations was adopted. For the determination of Poisson’s ratio υ_{12}, two strain gages were bonded to the specimen, one in the direction of the loading and the other at right angles to it. The strains were measured in longitudinal and transverse directions using strain indicator. The ratio of transverse to longitudinal strain gives the Poisson’s ratio within the elastic range. The average of three experimental determinations was adopted. The shear modulus G_{12} was computed using standard expression available in Jones (1975). The adopted material properties are: E_{1} = 38.07 GPa, E_{2} = 8.1 GPa, G_{12} = 3.05 GPa, υ_{12} = 0.22, ρ = 2,200 kg/m^{3}. The aspect ratio of the test specimens was varied from 1.0 to 2.5.
Experimental setup
The experimental setup is shown in Fig. 1. First, the test specimen was placed in the fixture shown in Fig. 1 with two opposite edges fully clamped and the remaining two edges completely free. The piezoelectric accelerometer was directly mounted on the test specimen at the geometric center using an adhesive. The accelerometer was then connected to signalconditioning unit (Fast Fourier Transform Analyzer), where the signal goes through the charge amplifier and an AnalogtoDigital Converter. The plate was excited in a selected point by means of impact hammer. The impact hammer was struck at the selected point five times and the average value of the frequency response function (FRF) was input to the computer through an USB port. Sufficient precautions were taken for ensuring that the strike of the impact hammer was normal to the surface of the plate. The pulse lab software accompanying the equipment was used for recording the signals directly in the memory of the computer. The signal was then read and processed to extract different features including frequencies. The frequencies were measured by moving the cursor to the peaks of the FRF. Five separate experimental determinations were done for the natural frequency of each specimen and then the average value was adopted.
Finite element solution
A finite element analysis was made for obtaining the first three natural frequencies using MSC/NASTRAN software. CQUAD8 (eightnoded isoparametric curved shell element) was employed as it yields better results compared to CQUAD4 element of the said software as revealed by the investigation reported by Srinivasa et al. (2012). Several options exist in the software in respect of real eigenvalue extraction, and the Lanczos method was used in the present study as it combines the best features of the other methods and computes accurate eigenvalues and eigenvectors.
Results and discussion
The results of the present work are presented in tables and figures in terms of nondimensional frequency coefficient (K_{f}) defined by ${K}_{\text{f}}=\frac{\mathit{\omega}{a}^{2}}{{\mathit{\pi}}^{2}}\sqrt{\frac{\mathit{\rho}\phantom{\rule{0.166667em}{0ex}}t}{D}}$ for isotropic plates and by ${K}_{\text{f}}=\mathit{\omega}{a}^{2}\sqrt{\frac{{\mathit{\rho}}_{1}}{{E}_{2}\phantom{\rule{0.166667em}{0ex}}{t}^{3}}}$ for laminated composite plates (Srinivasa et al. 2013).
Isotropic skew plates
The thickness of the isotropic skew plates was taken as 2.0 mm. The aspect ratio was varied from 1.0 to 2.5 and the skew angle from 0^{o} to 45^{o}. Table 1 and Fig. 2 show the variation of nondimensional frequency coefficient K_{f} with aspect ratio and skew angle for isotropic skew plates. The variation of first natural frequency with an increase in the aspect ratio is not appreciable up to a skew angle of about 30°. The variation of second and third natural frequencies with an increase in the aspect ratio for a given skew angle is considerable. The first, second and third natural frequencies increase monotonically with an increase in the skew angle for any given value of aspect ratio. The experimental values of the nondimensional frequency coefficient K_{f} are in very good agreement with those of the finite element solution. The first three mode shapes obtained by finite element analysis are shown in Fig. 3 for aspect ratio = 1.0.
Laminated composite skew plates
The total thickness of the laminate was maintained constant at 2 mm, the number of layers being 20. The aspect ratio was varied from 1.0 to 2.5 and the skew angle from 0^{o} to 45^{o}. Tables 2, 3, 4, 5 and Figs. 4, 5, 6, 7 show the variation of nondimensional frequency coefficient K_{f} with aspect ratio and laminate stacking sequence for various values of skew angle. Four laminate stacking sequences viz., antisymmetric angleply [+0^{o}/−0^{o}/…/−0^{o}], antisymmetric angleply [+45^{o}/−45^{o}/…/−45^{o}], antisymmetric angleply [+90^{o}/−90^{o}/…/−90^{o}] and antisymmetric crossply [0^{o}/90^{o}/…/90^{o}] were considered. It is seen that the first natural frequency decreases with an increase in the aspect ratio for all laminate stacking sequences. The second natural frequency increases with an increase in the aspect ratio for all laminate stacking sequences. For laminate stacking sequence [+0^{o}/−0^{o}/…/−0^{o}], the third natural frequency increases with an increase in the aspect ratio except in the case of skew angle = 45° where it increases up to aspect ratio = 2.0 and later decreases. For laminate stacking sequence [+45^{o}/−45^{o}/…/−45^{o}], the third natural frequency first increases and then decreases with an increase in the aspect ratio for all skew angles. For laminate stacking sequence [+90^{o}/−90^{o}/…/−90^{o}], the third natural frequency decreases monotonically with an increase in the aspect ratio for skew angle = 0 whereas for other skew angles, it first increases and then decreases. For laminate stacking sequence [0^{o}/90^{o}/…/90^{o}], the third natural frequency first increases and later decreases with an increase in the aspect ratio for all the skew angles considered. Other parameters being the same, the antisymmetric angleply [+0^{o}/−0^{o}/…/−0^{o}] yields the highest value for the first natural frequency and the antisymmetric angleply [+90^{o}/−90^{o}/…/−90^{o}] yields the lowest value. The first natural frequency for the other laminate stacking sequences lies in between the above two extreme values. The first, second and third natural frequencies generally increase with an increase in the skew angle for any given value of aspect ratio for all the laminate stacking sequences. The first three mode shapes obtained by finite element analysis are shown in Fig. 8 for antisymmetric crossply [0^{o}/90^{o}/…/90^{o}] for aspect ratio = 1.0. The natural frequency depends on the stiffness of the crosssection of the laminate among other factors such as boundary conditions etc. The stiffness of the crosssection depends upon the contribution made by extensional stiffness, coupling stiffness and bending stiffness terms (Jones 1975).
Conclusions
The following conclusions are made based on the above study.

The experimental values of the first, second and third natural frequencies are in good agreement with those of the finite element solution in the case of both isotropic and laminated composite skew plates.

In the case of isotropic skew plates, the variation of first natural frequency with an increase in the aspect ratio is not appreciable up to a skew angle of about 30°. The second natural frequency increases with an increase in the aspect ratio for all laminate stacking sequences. The first, second and third natural frequencies increase monotonically with an increase in the skew angle for any given value of aspect ratio.

In the case of laminated composite skew plates, the first natural frequency decreases with an increase in the aspect ratio for all laminate stacking sequences. The variation of second and third natural frequencies with aspect ratio is considerable in the range of skew angle considered. Other parameters being the same, the antisymmetric angleply [+0^{o}/−0^{o}/…/−0^{o}] yields the highest value for the first natural frequency and the antisymmetric angleply [+90^{o}/−90^{o}/…/−90^{o}] yields the lowest value. The first natural frequencies for the other laminate stacking sequences lie in between these two extreme values. The first, second and third natural frequencies generally increase with an increase in the skew angle for any given value of aspect ratio.
Abbreviations
 a :

Plate length
 b :

Plate width
 t :

Plate thickness
 NL :

Number of layers in the laminate
 E :

Modulus of elasticity of the material of isotropic plate
 E _{1} :

Young’s modulus of the lamina in the longitudinal direction
 E _{2} :

Young’s modulus of the lamina in the transverse direction
 G _{12} :

Inplane shear modulus of the lamina
 α :

Skew angle of the plate
 θ :

Fiber orientation angle of the lamina
 μ :

Poisson’s ratio of the material of isotropic plate
 ν _{12} :

Major Poisson’s ratio of the lamina
 ρ :

Mass density of the material of the plate
 ρ _{1} :

Mass density per unit area
 D :

Flexural rigidity of isotropic plate, Et^{3}/12(1 − μ^{2})
 ω :

Natural angular frequency of plate
 K _{f} :

Nondimensional frequency coefficient
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Acknowledgments
The first author would like to thank the Management and Principal Dr. S.G. Hiremath of GM Institute of Technology, Davangere, Karnataka, India for the kind encouragement and support provided. The second author would like to thank the Management of Jawaharlal Nehru College of Engineering, Shivamogga, Karnataka, India for the kind encouragement and support provided. The third author would like to thank the Management, Principal Dr. N Ranaprathap Reddy and Head of the Department of Civil Engineering Dr. Y. Ramalinga Reddy, Reva Institute of Technology and Management, Bangalore, Karnataka, India for the kind encouragement and support provided.
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Srinivasa, C.V., Suresh, Y.J. & Prema Kumar, W.P. Experimental and finite element studies on free vibration of skew plates. Int J Adv Struct Eng 6, 48 (2014). https://doi.org/10.1007/s4009101400483
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Keywords
 Skew plate
 Natural frequency
 Aspect ratio
 Fibre orientation angle
 Finite element analysis