Dynamic Analysis of Shaft with Variable Mechanical Property under Dynamic Load by FEM Method

: The paper introduces the dynamic responses of a circular shaft with variable mechanical properties under dynamic loads. The direct construction of finite element formulation for dynamic problems, which include stiffness matrix, and mass matrix for circular axis. The Functionally Graded Material (FGM) shaft with material properties is assumed to vary continuously in the longitudinal direction investigated. Building a numerical program for the problem based on Bernoulli beam theory, and Newmark numerical algorithm. The obtained numerical results have found the kinematic response of the shaft with the longitudinal variation of mechanical properties under the action of dynamic loads. The results were also compared with the analytical results in some special cases to clarify the reliability of the calculation method.


Introduction
The shaft is a common type of construction in most machine equipment. The oscillation of the shaft and beam with constant mechanical properties under the effect of dynamic loads has been proposed in many previous studies [1][2][3][4][5][6][7]. Functionally graded material (FGM) is created by continuously varying the volume fractions of the component materials, often in spatial coordinates. Thus, FGM does not have the disadvantages commonly encountered in layered and fiber-reinforced composites such as delamination or stress concentration. Combining the advantages of component materials, FGM materials are increasingly studied and widely applied in practice. Many researchers have investigated different aspects of the dynamic behavior of a rotating shaft by assuming it as a rotating Euler Bernoulli's theory. The traditional analytical methods, especially the Galerkin method are employed by researchers in studying the mechanical behavior of FGM shafts, Finite element method (FEM) is also widely used to study the behavior of FGM shafts and beams. Several finite-element beam models for the analysis of FGM beams have been proposed [8,9,10]. However, in addition to torsional strain, the shaft is also subjected to flexural deformation at the same time, which is rarely mentioned in previous studies. This study focuses on the kinematic analysis of shafts whose longitudinal mechanical properties vary by finite element method. Which includes building a finite element formulation based on a 2-node axis element with mechanical properties varying along the axis.

Elastic energy function and kinetic energy of a shaft with variable mechanical properties
Consider the circular shaft with the coordinate system being the x-axis in the direction of the circular axis and the y-and z-axes on the cross-section. The effective properties such as Young's modulus (E), shear modulus (G), and mass density (ρ) of the ceramic (c) and metal (m), respectively. The effective properties of the axially FGM beams evaluated by Voigt mode are as follows where , cm EE are the volume elastic modulus of ceramic and metal, respectively, , cm GG the shear modulus of elasticity of ceramic and metal, , cm  are the mass density of ceramic and metal , cm vv are the volume fraction of ceramic and metal (1 ) n m where n is the nonnegative grading index, and L is the length of the shaft. Based on the Euler-Bernoulli beam theory, the displacements u and ω of an arbitrary point in the x and z directions, respectively are given by , , where u(x, t) and ω(x,t) are respectively the axial and transverse displacements of a point on the x-axis at the midplane, φ(x,t) is the cross-sectional rotation, and t is the time variable, z is the distance of the midplane, θ is the angle of twist of the cross-sectional, the strain ,  and the stress can be obtained as follows [15]: Hooke's law: r is the distance from any point of stress to the x-axis of the shaft The strain energy of torsion and bending shafts for an element with an initial length of with variable mechanical properties having a form:

Materials Plus
In which where V e is the volume of an element of the shaft, and A is the area of the cross-section of the shaft.
The kinetic energy of an element with an initial length of having a form where ρ(z) is mass density, θ is twist angle.

Element stiffness matrix
Considering the shaft element with two typical nodes (node i, node j) with length L, due to the deformation characteristics of the shaft, there will be four node displacements at each node. The global element vector of nodal displacements for a two-node genetic FGM element (i, j) has eight components as The displacements u(x, t), ω(x, t), rotation φ(x, t), and twist θ(x, t) are interpolated from the nodal displacements as (10) where Nu, Nω, Nφ and Nθ are the matrices of shape functions.
The element stiffness matrix is derived from the strain energy of an element U, and it has the form

Element mass matrix
The element mass matrix is derived from the shape functions for the displacement field. we write the kinetic energy (8) in the following form  The equations of motion for 2D-FGM shaft can be written in the context of the finite element analysis as [11][12][13] MD + KD = F (16) where , D D are the total vectors of nodal displacements and accelerations. M, K and F are the mass matrix, the stiffness matrices and the nodal load vector of the shaft, respectively.

Numerical results and discussion
A computer code based on the developed formulation (16) and Newmark numerical algorithm [13,14] is developed and employed to study the dynamic behavior of the shaft in this Section Computations are carried out on FGM shaft made of steel SUS304 and aluminium Al 2 O 3 . Assume the mechanical properties of the material change from the left to end of the shaft, the left of the shaft is steel material, and the right end of the shaft is purely ceramic (Fig.1). To investigate the effect of shear strain and twist strain, for illustration, there is considered the shaft with constant cross-section, and shaft containing gear in the middle. The average radius of the gear is R, and the shaft has the following parameters: L = 0,5 (m), d = 0,02 (m), R = 0,06 m, the forces acting on the shaft at the midpoint of the shaft have the form P = P 0 .cos(Ωt); M = R.P 0 .cos(Ωt); P 0 = 2000 (N). The material data for constituents are as follows: E m = 2,1.  The results in Tables (1-2) with different frequencies when comparison of deflection and torsion angle calculated by the FEM method and traditional analytical method shows the reliability and accuracy of the FEM. However, in the case of mechanical properties of variable materials (or FGM materials), analytical results are not available, so using the FEM is the best method to simulate the kinematic behavior of the shaft structure.

Relationship between maximum shaft deflection and time
The relationships between the shaft deflection over time with the different frequencies of the excitation forces are shown in the Figure 2. From these results, it can be seen that the dependence of the maximum deflection on the frequency of the excitation force is quite clear. The number of vibrations which the shaft executes increase with increasing the excitation frequency. When paying attention to the vibration amplitude over time of the largest deflection, we also see the dependence of the vibration amplitude on the material parameter (n), specifically when n is small (n = 1), the deflection has a larger amplitude than the amplitude with high material parameter (n = 5). That means, the vibration amplitude decreases when increasing the material parameter (n). However, the number of cycles performing the oscillation does not depend on the material parameter but on the frequency of the excitation force, the material parameter affects the amplitude of the oscillation.

Relationship between the maximum torsion angle of the shaft and time
The effect of the material parameter (n) and the oscillation frequency (Ω) on the torsional vibration of the FGM shaft can be seen in Fig. 3. Similar to the horizontal oscillation described above, the number of the torsional vibrations which the shaft executes increase with increasing the excitation frequency and the vibration amplitude decreases when increasing the material parameter (n).

Natural frequency of oscillation
We consider the fundamental frequency parameter of the dimensionless horizontal oscillation of the shaft, which defined by [2]: where ψ 1 is the fundamental torsional frequency of the shaft.
The convergence of the fundamental frequency parameters received in Tables (3÷4) can be seen in that the numerical program converges quite quickly with a grid of 30 elements with all variable values of the material parameters. It is also possible to see the effect of the material parameter on the fundamental vibrational frequencies of the shaft. As the material parameter n increases, the fundamental frequencies also increase, which is consistent with the fact that the stiffness of the shaft increases as the frequency n increases.

Conclusions
The article has built a FEM model for the circular shaft element under the effect of harmonic force with mechanical properties changing along the longitudinal shaft. The finite element model is built based on Bernoulli's theory, which is quite commonly used in engineering and gives high reliability.
Building elastic deformation energy functions, and kinetic energy functions for variable mechanical shafts. From there, it is possible to build FEM formulas such as stiffness matrix, and mass matrix. Based on the FEM method combined with the Newmark digital algorithm has built a numerical program by Matlab software, which helps to simulate the kinematic behavior of the shaft.
The results obtained from the FEM equation using numerical algorithms and Matlab software give reliable results when compared with the results of the analytical method and the results of the convergence of the frequency parameter. Through the simulation results, it is possible to select the material parameter along with the excitation force frequency suitable for each problem with the structure or axial detail, thereby increasing or decreasing the vibration according to the wishes of the user.