The convergence test results for the first three natural frequencies of the quadratic tapered wing are presented in Figures 6-4 to 6-6. The comparison is made between the numerical results obtained from the ‘DFE with no deviators’, ‘RDFE incorporating the deviator terms’ and the reference natural frequencies were obtained from 120 conventional beam Finite Elements. The FEM model is based on cubic Hermite and linear approximation for bending and torsion displacements, respectively, and a constant mass matrix.

 

Figure  6‑5: Convergence of dually quadratic tapered wing-box for the first natural frequency.

 

Figure  6‑6: Convergence of dually quadratic tapered wing-box for the second natural frequency

 

Figure  6‑7: Convergence of dually quadratic tapered wing-box for the third natural frequency.

 

     As it can be seen, in this case, the FEM converges faster than the DFE when deviators are not used. By including the deviator terms the convergence rates for all three frequencies increases significantly. This consistent convergence using the deviators shows that there are no apparent limitations on these terms. Referring to chapter 5, the deviators became more effective for higher taper angles. The quadratic tapered wing is now more complex such that the degrading effects resulting from numerical error is so small that they do not affect the positive refining results of the deviators.

     A comparison is made between the fundamental natural frequencies of the graphite/epoxy composite wing obtained from FEM and DFE methods using different meshes. It is observed that the FEM errors for the first, second, and third natural frequencies, respectively, are approximately 20, 50 and 50 times higher than the corresponding DFE errors (see Table 6-2).

 

Table  6‑2: Fundamental frequencies in Hz for a graphite/epoxy quadratic tapered composite wing

 

6.4.2        Cubic tapered wing.

Let us consider a dually cubic tapered wing-box with the same mechanical properties as in the previous example. The FEM and DFE convergence results for the wing’s first 5 natural frequencies are presented in Figures 6-7 through 6-11. By implementing a cubic variation the deviators associated with the DFE method amplify the convergence in contrast to a linearly varying cross-section of low taper ratio seen previously in Chapter 5.    

 

Figure  6‑8: Convergence of dually cubic tapered wing-box for the first natural frequency.

Figure  6‑9: Convergence of dually cubic tapered wing-box for the second natural frequency.

 

Figure  6‑10: Convergence of dually cubic tapered wing-box for the third natural frequency.

 

Figure 6‑11: Convergence of dually cubic tapered wing-box for the fourth natural frequency.

 

Only for the fourth natural frequency (Figure 6‑11) greater convergence rates are obtained from the FEM, which is irregular since all other convergence tests favoured the DFE. In order to further investigate these results, the numerical values for frequencies are presented in Table 6-3.

 

Figure  6‑12: Convergence of dually cubic tapered wing-box for the fifth natural frequency.

 

Table  6‑3: Natural frequencies for a dually cubic tapered graphite/epoxy composite wing.

 

It is observed from the tabulated results that the DFE is significantly more accurate than the FEM by a factor of greater than 10. These results are expected as the DFE formulation is designed to be more accurate for complex elements such as the present dual cubic tapered model. The natural modes for the cubic tapered graphite/epoxy wing are displayed in Figure 6‑13 to Figure 6‑17. The modes of deformation have been plotted in both 2-D and 3-D spaces and have been normalized to better distinguish the modes as bending, torsion or coupled bending-torsion.

 

Figure  6‑13: First predominantly bending mode of vibration for a composite graphite/epoxy cubic tapered wing in both 2-D and 3-D plots. For the 2-D plot the bending displacenment is represented by a solid line (-) and torsion is represented by a dashed line (--).

 

Figure 6‑14: Second predominantly bending mode of vibration for a composite graphite/epoxy cubic tapered wing in both 2D and 3-D plots. For the 2-D plot the bending displacenment is represented by a solid line (-) and torsion is represented by a dashed line (--).

 

From the first two plots in Figure 6‑13 and Figure 6‑14  it can be seen that the modes are predominantly bending with slight influence of twist. For the higher modes a stronger influence of torsion is observed particularly for the third mode in Figure 6‑15 where the mode is primarily torsion.

  

Figure  6‑15: Third predominantly torsion mode of vibration for a composite graphite/epoxy cubic tapered wing in both 2D and 3-D plots. For the 2-D plot the bending displacenment is represented by a solid line (-) and torsion is represented by a dashed line (--).

Figure  6‑16: Fourth bending-torsion mode of vibration for a composite graphite/epoxy cubic tapered wing in both 2D and 3-D plots. For the 2-D plot the bending displacenment is represented by a solid line (-) and torsion is represented by a dashed line (--).

 

The bending-torsion coupling is apparent in the last two modes extracted, in Figure 6‑16 and Figure 6‑17 for the fourth and fifth free vibration modes. Although the interpolated surface plot used in MATLAB^®  is exceptionally useful in visualizing these modes the cubic taper has been stretched into a rectangular surface such that the 3-D surface plots are not necessarily to scale, but can still be useful differentiating the modes as bending or torsion.

 

Figure  6‑17:  Fifth bending-torsion mode of vibration for a composite graphite/epoxy cubic tapered wing in both 2D and 3-D plots. For the 2-D plot the bending displacenment is represented by a solid line (-) and torsion is represented by a dashed line (--).

 

6.6    Conclusion

    The free vibration analysis of thin-walled composite wing-boxes with quadratic and cubic tapers is presented. By implementing the CAS configuration and non-coincident mass and shear axes, the wing exhibits dually coupled vibration. The natural frequencies and modes of deformation have been extracted using the three methods, conventional FEM, DFE, and the refined DFE (DFE with deviators). These deviators take into account the variable geometric and/or material parameters of the wing model over each DFE. The convergence of the refined DFE (RDFE) is validated in comparison with the FEM method for multiple tapered geometries and ply orientations. The RDFE method provides a much higher convergence rate than classical finite elements. The corresponding natural modes of vibration were also evaluated and plotted using the advanced plotting features in MATLAB^®. The programs coded in MATLAB^® are discussed in the Appendix.