Dynamic Behavior Analysis Of Functionally Graded Material Beams Considering Longitudinal Vibration

Functionally Graded Materials (FGM) is a new type of heterogeneous composite materials. It is a combination of two or several different materials in accordance with certain laws of the composition of the material. It would be eliminate the abrupt change of physical properties and stress concentration between the two materials and can set the advantages of two materials in one. So it is widely used in aerospace,medical, mechanical, electronic engineering and other fields. Beam is one of the most common structural members in practical engineering, study the static and dynamic behavior of the system in temperature load, mechanical load and other environments is one of the most important research contents in the field of mechanics. At present,most of the researches on the dynamic vibration of functionally graded materials are carried out neglecting the longitudinal vibration. In this paper, the nonlinear dynamic mechanical behavior of the functionally graded material beam under mechanical load is studied and analyzed. The main contents include the following two aspects are as follows:1.Dynamic behavior analysis of functionally graded materials Euler beam considering the longitudinal vibration.Based on the classical beam theory, the nonlinear vibration problem of functionally graded material beam under transverse distributed load is established,than use the vibrational principle establish the geometric nonlinear dynamic differential equation under the action of mechanical load when consider the longitudinal vibration. Using the shooting method solved the boundary value problem consist of the equation and the corresponding boundary conditions. Firstly, the transverse vibration of the Euler beam is studied, and obtained the natural frequency of the transverse vibration of the functionally graded material beam. On the basis of this, the vibration problem of functionally graded material Euler beam is studied when the longitudinal vibration is taken into account. The results show that the frequency of functionally graded beam decreases with the consideration of the longitudinal vibration, but the degree of reduction is within the acceptable range. After that, the influence of gradient parameters, height ratio and boundary conditions on the dynamic behavior of FGM beam is discussed. The results show that when the longitudinal vibration is taken into account, the frequency of functionally graded Euler beam will be reduced, but the degree of reduction can be within the acceptable range.2. Analysis of dynamic behavior of functionally graded Sine shear beam.Based on the Sine shear beam theory, a mathematical model of the Sine shear beam vibration of functionally graded materials is established, than use the vibrational principle establish the dynamic differential equation. Using the shooting method solved the boundary value problem consist of the equation and the corresponding boundary conditions. Obtained the natural frequency of the Sine shear beam when considering the longitudinal vibration, the results are coincide with the existing literature. Than compared the results of the Sine shear with the first two sections of the same condition, the results show that the Sine shear theory frequency of FGM beam is lower than that of the classical beam model. Finally, discussed the vibration characteristics of functionally graded Sine shear beams with different slenderness ratios and boundary conditions.