Modal Characterization of Sandwich Skew Plates


 The current work focuses on the experimental and finite element free vibration studies of laminated composite sandwich skew plates. The comparison was made between the experimental values obtained by the Fast Fourier transform (FFT) analyzer and a finite element solution obtained from CQUAD8 finite element of The MacNeal-Schwendler Corporation (MSC) / NASA STRucture Analysis (NASTRAN) software. The influence of parameters such as aspect ratio (AR) (a/b), skew angle (α), edge condition, laminate stacking sequence, and fiber orientation angle (θ°) on the natural frequencies of sandwich skew plates was studied. The values obtained by both the finite element and experiment approaches are in good agreement. The natural frequencies increase with an increase in the skew angle for all given ARs.


INTRODUCTION
Due to the reduced weight and high stiffness, sandwich structures have a wide area of applications in engineering science. With anisotropy and considering all parameters involved in the sandwich structure, it is difficult to evaluate the dynamic response analytically. In the literature review, various theories, methods, and techniques were cited (Pavan et al., 2021). However, with the evolution of technology in real engineering applications, it is possible to predict the dynamic response of any structure accurately.
First,an experimental approach to the determination of natural frequencies of sandwich plates was made (Raville et al., 1967). Verification was made (Barkanov et al., 2005) of the numerical results of laminated composite and different sandwich panels with pulse and noncontact laser techniques. Static deformation and free vibration study was conducted on sandwich plates with variable thickness using both numerical and experimental holographic interferometry techniques (Chang et al., 2006). A theoretical and experimental study was conducted on the vibration and acoustical properties of sandwich composite materials (Zhuang, 2006). The vibration testing approach was used to identify the material constants (Lee et al., 2007)  ). An extensive survey on the analysis of sandwich FGM structures under different loading conditions, effects of porosities, hygrothermal loadings, and structures resting on elastic foundations was made .
A vast amount of literature was reported in detail on free vibration studies for laminated composite skew sandwich plates using analytical and numerical approaches. Limited literature was found related to detailed experimental work on the dynamic response of laminated composite sandwich skew plates. Literature on finite element solutions validated by the experimental method is very scarce. The present paper is an attempt to address this issue in some detail. Fast Fourier transform (FFT) analyzer is an instrument best suited for dynamic applications. In the current work, an FFT analyzer is used for the prediction of natural frequencies of sandwich skew plates.

TEST SPECIMEN PREPARATION
The sandwich skew panels were prepared using glass/epoxy laminated composites as face sheets and aluminum honeycomb as core materials. Aluminum honeycomb (Al-3003) panels are used as the core with a cell size of 6.35 mm, foil thickness of 50 microns, and height of 6 mm. Laminated glass/epoxy reinforced polymer composites are used as face sheets. Unidirectional glass fibers for [±0°/Core/±0°], [±90°/Core/±90°] and bidirectional glass fibers for [±45°/Core/±45°], [(0°/90°)5/Core/(0°/90°)5] of 220 gsm and Lapox L-12 (Epoxy) along with Lapox K-6 hardener were used in fabricating the laminates. Continuous hand lay-up technique was employed for fabricating the sandwich plates, during which the excess resin was removed from the laminate by steel roller. The sandwich laminates are cured at room temperature for a period of 48 h placing weights over them. The test specimens were prepared with fiber weight percentage 50:50 according to relevant American Society for Testing and Materials ASTM standards. The

EXPERIMENTAL SETUP
The arrangement for conducting the experimentation is shown in Fig. 1. The test specimen was held in the fixtures, imposing Clamped-Free-Clamped-Free and Clamped-Free-Free-Free edge conditions. A piezoelectric accelerometer sensor was placed at the center of the test specimen using glue and was connected to the FFT analyzer (signal conditioning and amplifying unit). An impact hammer was also connected to the FFT analyzer, dedicated to exciting the test specimen on selected points five times. For each test specimen, five trials were made and an average value was adopted.
Soon, the impact hammer excites the test specimen with a strike (impact); the accelerometer sensor captures the vibration signals and exports them to the FFT analyzer for further pro-cessing. The FFT analyzer gives the output in terms of Frequency Response Function (FRF) using the pulse lab software. It was ensured that the strike of the impact hammer was normal to the test specimen's surface and no other sources of excitations (surrounding and floor vibrations nearby) were present other than the impact hammer.

FINITE ELEMENT ANALYSIS
MSC/NASTRAN software package was employed for finite element analysis in obtaining the first three fundamental frequencies of skew sandwich plates. Eight-noded isoparametric curved shell elements, i.e., CQUAD8 and CQUAD4, are used in the analysis. A study  disclosed that the CQUAD8 element produces more converging and accurate results than the CQUAD4 element. Accordingly, the CQUAD8 element was employed in the present study. To evaluate the real eigenvalues and eigenvectors, the Lanczos method of extraction was imposed for accurate results.

RESULTS AND DISCUSSION
The current work focuses on the influence of AR, skew angle, laminate stacking sequence, and edge conditions on the sandwich skew plates. To obtain higher natural frequencies of sandwich skew plates, considering the influence of tc/tf and a/h as discussed (Pavan et al., 2021) and minimizing the production cost, twenty-one layer, antisymmetric, angle-ply, and cross-ply laminated sandwich plates are designed and prepared. The thickness ratio of core to face sheet is kept constant in the whole study as tc/tf=6 and the ratio of length overall thickness a/h is varied to 12. The results are tabulated for the C-F-F-F edge condition in Tab. 4 and also graphically presented in Figs. 6-9. The mode shapes are presented in Tab. 5.

Tab. 2. Natural frequencies of Clamped-Free-Clamped-Free laminated composite sandwich skew plates
Tab. 3. Natural frequencies of Clamped-Free-Free-Free laminated composite sandwich skew plates        the first natural frequency decrease with an increase in the AR. AR is the ratio of length to width, where width is kept constant and only length is varied. As the AR increases, the length of the plate increases, which makes the plate less stiff, leading to a decrease in the natural frequency. For the second natural frequency, it decreases with an increase in AR except for skew = 45°. For skew 0° to 45°, as the skew angle increases, the width of the plate decreases due to the skew, and also the overall area of the plate goes on reducing. This makes the plate lighter and at the same time stiffer. For the third natural frequency, it increases from AR = 1 to AR = 1.5, then it decreases, it is due to the combined effect of fiber orientation, mode shape, mass density, and stiffness of the plate.  For angle ply [(±45°)5/Core/(±45°)5] the first and third natural frequencies decrease with an increase in the AR for a given skew angle. However, in the second natural frequency, variation is negligible for skew = 0°; it is considerable for other skew angles. Here the plate produces more stiffness due to the fiber orientation and it overshadows the effect of plate stiffness due to the change in the AR compared to other skew angles.  In angle ply [(±90°)5/Core/(±90°)5] stacking sequence, for all three frequencies, the Kf value decreases as the increase in the AR except skew = 45°. For skew = 45° the Kf value decreases from AR = 1 to AR = 1.5 and then it increases. It is exactly opposite as in the case of angle ply [(±0°)5/Core/(±0°)5].  The Kf value decreases with an increase in the AR in all the three modes of cross-ply [(0°/90°)5/Core/(0°/90°)5] stacking sequence from skew = 0° to 30°. The only exception is for skew = 45° in which the second natural frequency goes on increasing as the AR increased. The third natural frequency first increases from AR = 1 to AR = 1.5 and then decreases.

CONCLUSION
Free vibration investigation was made on laminated sandwich skew plates adopting both experimental and finite element methods. Glass epoxy laminated composites are used as face sheets and aluminum honeycomb (Al3003) is used as the core in the current study. Two types of edge conditions were used. i.e., C-F-C-F and C-F-F-F. The experimentally obtained results were then validated by finite element values, and the experimental values are promising and close to the finite element values. Influences of various parameters, for instance, skew angle, AR, laminate stacking sequence, and edge conditions are studied. The thickness ratio of core to face sheet tc/tf = 6 was kept constant all through the study. As the AR increases for any skew angle, the Kf value decreases irrespective of the edge condition. When the AR increases, the length of the sandwich plate also increases. This reduces the stiffness of the sandwich skew plate. The skew angle plays a major role in the dynamic response of the sandwich skew plate. When the skew angle is increased, the Kf value increases for all ARs. The fiber orientation of the lamina plays a significant role in deciding the vibration response of the sandwich skew plates. When the fiber angle is 0°i.e., the fiber is placed parallel to the length of the plate, the plate produces higher natural frequency due to more stiffness in the longitudinal direction. In addition, the lowest natural frequency is observed when the fiber angle is 90°i.e., the fiber angle is perpendicular to the length of the plate. The natural frequencies for the other laminate stacking sequences lie in between these two extreme values. The sandwich skew plates with C-F-C-F edge conditions produce a higher natural frequency than plates with C-F-F-F edge conditions.