This paper analyzes the dynamic behavior of a fluid-conveying pipe with different pipe end boundary conditions. The pipe is considered to be an Euler-Bernoulli beam, and a motion equation for the pipe is derived using Hamilton's principle. A semi-analytical method, which includes the differential quadrature method (DQM) and the Laplace transform and its inverse, is used to obtain a model for the dynamic behavior of the pipe. The use of DQM provides a solution in terms of pipe length whereas use of the Laplace transform and its inverse produce a solution in terms of time. An examination of the results of sampling pipe displacement at different numbers of sample points along the pipe length shows that the method we developed has a fast convergence rate. The frequency and critical velocity of the fluid-conveying pipe derived by DQM are exactly the same as the exact solution. The numerical results given by the model match well with the result obtained using the Galerkin method. The effect on pipe displacement of the pipe end boundary conditions is investigated, and it increases with an increase in the edge degrees of freedom. The results obtained in this paper can serve as benchmark data in further research.

Approaches to controlling NO x emission have been investigated worldwide. Numerical simulation models describing gas−particle flow, heat transfer, and combustion processes are presented in this paper to provide reasonably accurate predictions of the flow field, temperature field, and the distribution of chemical species in a W-shaped furnace. The predictions indicate that the temperature field of model II is more uniform and the contour is lower due to the complement of nozzles. It results in the decrease of NO x formation. Comparison about NO x emission of two furnace models at different temperatures has been made, it shows that the furnace of model II is better in reducing the emission of NO x .

An analytical method and a semi-analytical method are proposed to analyze the dynamic thermo-elastic behavior of structures resting on a Pasternak foundation. The analytical method employs a finite Fourier integral transform and its inversion, as well as a Laplace transform and its numerical inversion. The semi-analytical method employs the state space method, the differential quadrature method (DQM), and the numerical inversion of the Laplace transform. To demonstrate the two methods, a simply supported Euler–Bernoulli beam of variable length is considered. The governing equations of the beam are derived using Hamilton's principle. A comparison between the results of analytical method and the results of semi-analytical method is carried out, and it is shown that the results of the two methods generally agree with each other, sometimes almost perfectly. A comparison of natural frequencies between the semi-analytical method and the experimental data from relevant literature shows good agreements between the two kinds of results, and the semi-analytical method is validated. Different numbers of sampling points along the axial direction are used to carry out convergence study. It is found that the semi-analytical method converges rapidly. The effects of different beam lengths and heights, thermal stress, and the spring and shear coefficients of the Pasternak medium are also investigated. The results obtained in this paper can serve as benchmark in further research.

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