Abstract: Free vibration analysis of homogenous and axially functionally graded simply supported beams within the context of Euler-Bernoulli beam theory is presented in this paper. The material properties of the beams are assumed to obey the linear law distribution. The effective elastic modulus of the composite was predicted by using the rule of mixture. Here, the complexities which appear in solving differential equation of transverse vibration of composite beams which limit the analytical solution to some special cases are overcome using a relatively new approach called the Differential Transformation Method. This technique is applied for solving differential equation of transverse vibration of axially functionally graded beams. Natural frequencies and corresponding normalized mode shapes are calculated for different Young’s modulus ratios. MATLAB code is designed to solve the transformed differential equation of the beam. Comparison of the present results with the exact solutions proves the effectiveness, the accuracy, the simplicity, and computational stability of the differential transformation method. The effect of the Young’s modulus ratio on the normalized natural frequencies and mode shapes is found to be very important.
Abstract: Radiative heat transfer in participating medium was
carried out using the finite volume method. The radiative transfer
equations are formulated for absorbing and anisotropically scattering
and emitting medium. The solution strategy is discussed and the
conditions for computational stability are conferred. The equations
have been solved for transient radiative medium and transient
radiation incorporated with transient conduction. Results have been
obtained for irradiation and corresponding heat fluxes for both the
cases. The solutions can be used to conclude incident energy and
surface heat flux. Transient solutions were obtained for a slab of heat
conducting in slab and by thermal radiation. The effect of heat
conduction during the transient phase is to partially equalize the
internal temperature distribution. The solution procedure provides
accurate temperature distributions in these regions. A finite volume
procedure with variable space and time increments is used to solve
the transient radiation equation. The medium in the enclosure
absorbs, emits, and anisotropically scatters radiative energy. The
incident radiations and the radiative heat fluxes are presented in
graphical forms. The phase function anisotropy plays a significant
role in the radiation heat transfer when the boundary condition is
non-symmetric.
Abstract: In this paper, we research the standard 13-point difference schemes for solving the biharmonic equation. Heuristic method is applied to judging the stability of multi-level difference schemes of the biharmonic equation. It is showed that the standard 13-point difference schemes are stable.