In-plane Free Vibration Analysis Of FGM Arches With Variable Curvature

Based on the Euler-Bernoulli curved beam theory in this paper, there are two parts that the in-plane free vibration of plane arches with variable curvature and Functionally Graded Materials(Functionally Graded Materials, abbreviated as FGM) arches with variable curvature are studied, the various factors that effected the in-plane free vibration frequencies of the variable curvature plane arches that the curvatures are linear variable with the opening angle, the material is uniform and the variable curvature FGM arches that the curvatures are linear variable with the opening angle, the materials are distributed in a power law along the thickness of the arches are analysed. The conclusions of the relationship between the in-plane free vibration frequencies and the curvature variable coefficients in plane arches with variable curvature and the relationship between the in-plane free vibration frequencies and the curvature variable coefficients, the in-plane free vibration frequencies and the material volume fraction variable coefficients in FGM arches with variable curvature are obtained.The first chapter mainly introduces the background of the topic, the meaning of this study for the topic, the research content, the basic assumption, the research status of arch vibration as well as the theoretical basis and research methods of this paper, for these are aims to illustrate the necessity and feasibility for the study.The second chapter is based on the theory of the Euler-Bernoulli curved beam and Transfer Matrix Method(Transfer Matrix Method, abbreviated as TMM), the free vibration problems of the plane arches with variable curvatures were investigated. Considering the curvature of the arch axis is linearly varies according to the opening angle, the plane arch with variable curvature was divided into a number of arch elements in the arc direction, and using Hamilton variational principle and TMM, the unit transfer matrix of the arch elements and the free vibration characteristic equation of the plane arch with variable curvature was derived. The free vibration natural frequencies of the plane arch with variable curvature under the different boundary conditions were calculated, and considering the effect of curvature variation coefficient on the free vibration frequencies of the arch, comparing the results of the two ends clamped circular arch with the previously reported, and proved that TMM is effected for solving these problems. The influence for the free vibration frequencies of the plane arches with variable curvature owing to the variable coefficient of curvature was analyzed.The third chapter is based on the theory of the Euler-Bernoulli curved beam, the shifting of neutral surface is considered when the materials are gradually distributed along the arch thickness, and the FGM arches with variable curvature are discretized into a number of curved arch elements along the arc direction. Every curved arch elements are considered as the circular arch elements with constant radiuses, and according to the Hamilton variational principle, the in-plane free vibration equation of the FGM circular arch element is derived, and the unite transfer matrix is obtained. Then using TMM, the in-plane free vibration characteristic equation of the FGM arches with variable curvature is derived, the in-plane free vibration natural frequencies of the FGM arches with variable curvature under different boundary conditions are solved, compared the results of the two ends clamped circular arch with the previously reported and it is proved that TMM is effected for solving these problems. The influence for in-plane free vibration frequencies of the FGM arches with variable curvature owing to the variable coefficient of curvature and the variable coefficient of materials volume fraction are analyzed.The fourth chapter summarizes the conclusions of this study and looks in to the future of the study for this topic, the conclusions shows that: When the curvatures of the arch axis are linear variable with the opening angle, each orders free vibration frequencies of the plane arches with variable curvatures are increased with the increasing of the curvature variable coefficients in different boundary conditions, and the amplitudes of free vibration frequencies are increased with the increasing of free vibration frequencies orders in the same amplification of the curvature variable coefficients; when the curvatures of the arch axis are linear variable with the opening angle, the FGM that composite with ceramics and metal are distributed along the arches thickness in power law, each orders in-plane free vibration frequencies of the FGM arches with variable curvature not only conform to the relationship about the curvature variable coefficients of the plane arches with variable curvature in the above, but also satisfied the relationship that each orders free vibration frequencies are increased with the decrease of the metal phase volume fraction variation coefficients, and the relationships are severer when the metal phase volume fraction variation coefficients are smaller. The road is twists and turns and the prospect is broad for this topic.Research the in-plane free vibration characteristics of the plane arches with variable curvature and the FGM arches with variable curvature, given the influences conclusion of the curvature variation coefficients and the variable coefficient of materials volume fraction to the plane arches with variable curvatures and the FGM arches with variable curvature, it is very important to design, optimize and control the dynamic characteristics of the structure.