5. Conclusions

undamped. The governing equation describing the dynamic behaviour of the system is given

f g Fð Þt is the moving load vector. The deflection of the beam v xð Þ ; t , as the moving load travels

tion at the centre of the simply supported functionally graded beam when subjected to a static load of magnitude P at the centre. The maximum normalised deflection mid-span of the beam

identify the critical velocity for the different variations of the constituent materials as illustrated in Figure 7. The results present are obtained from the BSWI (2 element; m ¼ 4; j ¼ 3; 37 DOFs) WFEM solution. The h-FEM (12 elements; 39 DOFs) and Daubechies (2 elements; L ¼ 12; j ¼ 0; 37 DOFs) WFEM solution gives similar results. The values of the critical moving load velocity and corresponding maximum non-dimensional displacement are presented in Table 2 for the different values of n for all approaches. The results are compared with those presented in [46]. Both the Daubechies and BSWI WFE M solutions very accurately yield the

Figure 7. Variation of the maximum non-dimensional vertical displacement at the centre of a simply supported steel-

alumina beam with respect to moving load velocities, for different n.

and f g Uð Þt represent the system acceleration and displacement vectors at time t.

þ ½ � K f g Uð Þt ¼ f g Fð Þt (86)

<sup>48</sup>ElI is the deflec-

�<sup>1</sup> to

�<sup>1</sup> at increments of 1 m�<sup>s</sup>

½ � <sup>M</sup> <sup>U</sup>€ð Þ<sup>t</sup> n o

72 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

is analysed over a moving load velocity range 0 < c ≤ 300 m�s

across, is normalised as a non-dimensional parameter v xð Þ ; <sup>t</sup> <sup>=</sup>v<sup>0</sup> where <sup>v</sup><sup>0</sup> <sup>¼</sup> Pl<sup>3</sup>

by [45]:

where <sup>U</sup>€ð Þ<sup>t</sup> n o

correct values.

A generalised formulation framework for the construction of an axial rod, Euler Bernoulli beam and functionally graded two-dimensional wavelet based finite elements is presented. The Daubechies and BSWI families are selected due to their desirable properties, particularly compact support, 'two-scale' relation and multiresolution. It is illustrated via a set of numerical examples that the WFEMs perform exceptionally well when compared to conventional h-FEM and p-FEM where high levels of accuracy are achieved with fewer elements required and the approaches converge more rapidly to the exact solution. Furthermore, the methods are able to accurately describe the behaviour of static and dynamic systems with singularities, variation in material properties and loading conditions present. This exhibits the vast potential of the method in the analysis of more complicated systems and the ability to alter the multiresolution scales without affecting the original mesh allows effective and efficient avenues solution accuracy improvement.
