This paper firstly introduces a new general solution constructed by double trigonometric cosine series with supplementary terms for the bending and vibration analysis of orthotropic rectangular plates with four free edges on the Winkler foundation subjected to arbitrary vertical force. The general solution, which is fourth-order continuously differentiable with less undetermined coefficients, can be used to solve the bending and vibration problems of orthotropic rectangular plates on the Winkler foundation with various physical parameters requiring no classification and superposition. This makes the bending and vibration analysis of orthotropic rectangular plates with four free edges on the Winkler foundation more unified, simplified and regulated. This paper also gives a Series of analytical example to prove that the method is feasible.

Background: Eucommia ulmoides ‘Oliver’ is an economically important tree species with highly medicinal and ecological values that is naturally distributed in China. E. ulmoides ‘Huazhong No. 12’ (H12) is the only red-leaf genotype of this species. In this study, the pigment contents of H12 and E. ulmoides ‘Huazhong No. 11’ (H11, green leaves) were determined. The differential metabolites in H12 and H11 were detected by UPLC-MS/MS, and the differentially expressed genes were screened by transcriptome. Then the key metabolites and corresponding gene regulation in anthocyanin related metabolic pathway were analyzed.Results: The chlorophyll a, chlorophyll b and carotenoid contents in H12 leaves were lower than H11, while the total anthocyanin content in H12 was 4.06 times higher than in H11. There were 96 up-regulated metabolites in H12, including anthocyanin, proanthocyanidins, flavonoid and flavonol. Among them, four differentially expressed anthocyanins were identified. A total of 8,368 differentially expressed genes were selected from transcriptome between H12 and H11. The flavone and flavonol biosynthesis pathway, anthocyanin pathway and photosynthetic pathway were analyzed. Finally, EuCHI, EuF3'H, EuF3'5'H, EuDFR and Eu3MaT1 were recommended as the key genes. The cyanidin, cyanidin 3-malonyl-glucoside and cyanidin 3, 5-glucoside were responsible for the H12 red leaves.Conclusion: This study revealed the metabolites and gene regulation of anthocyanin synthesis, and the potential function between the anthocyanin and photosynthetic gene expression in E. ulmoides red leaves. Notably, the results also provided a reference for the study of other plant leaf-coloring mechanisms.

In this paper, the analytic solutions of steady vibration of free orthotropic rectangular plate loaded with vertical steady loading on the elastic half-space was given by combining the general solution of double trigonometrically cosine series with supplementary terms with dynamic integral representations for displacements of the elastic half-space loaded with arbitrary vertical steady loading. This solution not only is four-order derivative, but also has less undetermined coefficients. It can be used to solve the problems of bending and steady vibration of orthotropic rectangular plates on the elastic half-space without be classified and be superimposed. This causes this kind of things, bending and steady vibration of orthotropic rectangular plates with four free edges on the elastic half-space, unionization, simplification and systematization. When the material is isotropic, the solutions are turned into analytic solutions of steady vibration of free rectangular plate on the elastic half-space. At last some computational examples are presented and the results are coincided with those in literatures. Then the method in this paper will be of important practical applications.

The modal analysis of pipes conveying fluid considering the influence of Fluid-Structure Interaction (FSI) is always a topic in flow and solid mechanics fields worth study. Three methods of coupling, the direct FSI method, the one-way FSI method and the bidirectional FSI method, with the velocity’s influence on pipe system’s natural frequency were considered in this study. The results of pipe’s dynamic behavior indicate that the two-way FSI method can get a more reasonable result. Furthermore, when velocity changes in a small range far from critical velocity, velocity makes very little influence on system’s nature frequency. The result was compared with conclusion in the literature and very good agreement was found.

In this paper, basing on Hu haichang’s theory and using Fourier transformation, the problem is changed into solving fourth-order and second-order linear ordinary differential equations. The eigenvalue method is used for solving these equations, and then both displacement and stress with undetermined coefficients can be received. Considering the boundary conditions, the exact displacement and stress of the transversely isotropic foundation can also be gotten. Especially on the transversely isotropic elastic half space, with the aid of computer software, a more simple vertical displacement is presented. When the foundation is degenerated into isotropic elastic half space, the displacement is also degenerated into elastic half space displacement in a rectangular coordinate system which is given in the reference. And this exactly proves the result which is received from this paper is right. On this basis, we can analyze a settlement of transversely isotropic foundation and the static characteristics of rectangular plate on the transversely isotropic foundation.