Vibration analysis of functionally graded materials for cylinder liner used in agricultural engine part

Abstract The engine part is one of the major sources of vibration of agricultural machinery such as a tractor. Therefore, vibration analysis of agricultural engine part will improve the engine efficiency and agricultural performance. The main objective of present work was to study the dynamic behavior of functionally graded (FG) structural material for the application as cylinder liner as agricultural engine part. The vibration analysis of functionally graded (FG) beam was performed using Finite Element Method (FEM). A typical simply-supported FG beam was modeled in COMSOL Software, where the upper portion of the beam was alumina and the lower portion was steel. The basic properties of material such as Young’s Modulus and mass density were varied along the thickness according to the power law. The boundary conditions were also modeled, and parametric study was carried out with mass density and young’s modulus. Eigen value problem was solved and in turn natural frequency and mode shapes were obtained. The frequency ratio was calculated and compared for various boundary conditions. The finding of the results indicated that when power exponent was increased from 0 to 5, the nonlinear reduction in frequency was occurred but when power exponent was increased from 5 to 10, linear reduction in frequency was occurred. Also, the increase in power exponent caused the increase in frequency for Young’s Modulus ratio of 0.25 and 0.5, decrease in frequency for Young’s Modulus ratio of 2 and 4 and no change occurred for Young’s Modulus ratio of 1. The first non-dimensional frequency for Clamped-Clamped boundary condition was comparatively more than other boundary conditions and lowest frequency is obtained for Clamped –Free boundary conditions.


Introduction
Functionally Graded Materials (FGMs) have been developed as innovative materials in various applications such as aerospace, automotive industries and machine elements. The application of FGM in agricultural equipment is new and is very limited as the cost of production of functionally graded materials is high. However, the major features or functional properties of materials suitable for agricultural applications include higher load bearing capacity due to increased machine performance, lighter in weight to avoid soil compaction, more wear resistant, long life, and better ergonomics. In this regard, the better physical, mechanical, thermal and thermo-mechanical properties of FGMs justify their use in important parts and machinery in agriculture. The vibration analysis of agriculture machine and its various parts have been done by researchers (Gialamas et al. 2016;Cutini et al. 2017). The use of FGM as cylinder liner in engine part is a new approach as the function of cylinder liner is to reduce wear on cylinder, piston and transfer heat from piston to coolant. The properties of functionally graded materials change over a varying dimension. In general, FGMs are the advanced composite materials consisting of two or more materials and varying volume fraction along dimension. It can be mixture of metal and metal or ceramics. FGMs are preferred for the structure having very small thickness and subjected to high temperature gradient (Chakrabortya et al. 2003). Under such extreme working condition, composite material leads to fail due to separation of fibers from the matrix called delaminations. Therefore, in the mid-1980s, Japanese researchers worked on hypersonic space plane project to develop thermal barrier coating of less than 10 mm thickness and outside temperature 2000K and inside temperature 1000K. In this regard, they developed a thermal barrier coating of a novel material called Functionally Graded Material (FGM). FGM occur in nature as bones, teeth etc. (Mehta and Balaji https://doi.org/10.1515/opag-2019-0024 received September 27, 2018accepted February 4, 2019 Abstract: The engine part is one of the major sources of vibration of agricultural machinery such as a tractor. Therefore, vibration analysis of agricultural engine part will improve the engine efficiency and agricultural performance. The main objective of present work was to study the dynamic behavior of functionally graded (FG) structural material for the application as cylinder liner as agricultural engine part. The vibration analysis of functionally graded (FG) beam was performed using Finite Element Method (FEM). A typical simply-supported FG beam was modeled in COMSOL Software, where the upper portion of the beam was alumina and the lower portion was steel. The basic properties of material such as Young's Modulus and mass density were varied along the thickness according to the power law. The boundary conditions were also modeled, and parametric study was carried out with mass density and young's modulus. Eigen value problem was solved and in turn natural frequency and mode shapes were obtained. The frequency ratio was calculated and compared for various boundary conditions. The finding of the results indicated that when power exponent was increased from 0 to 5, the nonlinear reduction in frequency was occurred but when power exponent was increased from 5 to 10, linear reduction in frequency was occurred. Also, the increase in power exponent caused the increase in frequency for Young's Modulus ratio of 0.25 and 0.5, decrease in frequency for Young's Modulus ratio of 2 and 4 and no change occurred for Young's Modulus ratio of 1. The first non-dimensional frequency for Clamped-Clamped boundary condition was comparatively more than other boundary conditions and lowest frequency is obtained for Clamped -Free boundary conditions. condition was applied, and finite element mesh size was taken for obtaining optimized result through analysis.

Materials and Method
For vibration analysis, the functionally graded material for hollow cylinder liner has been taken as a functionally graded simply-supported beam of length L, width b (i.e. thickness of liner, different of outer and inner diameter of cylinder), and thickness h (i.e. circumference of outer circle) (Figure 1).

Material properties
The material properties of steel and alumina in presented in Table 2. The material properties such as Young's modulus and mass density varied through the beam thickness in the power-law form as presented in Eq. (1). (1) where P U is material property of the upper surface and P L that of lower surface of the beam and k is the non-negative power-law exponent. k denotes the material variation of properties along the thickness of the beam.

Methodology
The following steps were followed in Eigen frequency analysis through Comsol Multiphysics 4.4 -The parameters (L, b, h and k) were set in global definitions. -The analytic functions of young's modulus and mass density were defined and expression according to power law under component was written. -Geometry was defined, the width, depth, height of the block was set up, also set the position and finally build all objects. -The boundary conditions and the mesh size were set to build the final mesh. 2013). Functionally Graded Beams have many applications in automotive industries, manufacturing, aerospace, and mechanical components. However, in comparison with FG plates and shells, there are few literatures working on FG beams (Aydogdu and Taskin 2007). Rao et al. (Rao et al. 2012) analyzed free vibration of functionally graded (FG) shell structures under thermal and mechanical loading using finite element analysis. They compared the thermal and mechanical properties of FG shells, the homogeneous shells of pure ceramic (Al 2 O 3 ) and pure metal (steel) shells. It was revealed that the thermal and mechanical properties of the FGM shells lie between those of the homogeneous shells.
Ramu and Mohanty (Ramu and Mohanty 2014) evaluated natural frequencies and mode shapes of a functionally graded material (FGM) using Finite Element Method (FEM). It was observed that the mechanical properties of an FGM plate vary continuously from one surface to another surface according to power law. For modal analysis of FGM plate, program has been coded in MATLAB software. The modal shape and natural frequencies of rectangular FGM plate are found at different boundary conditions. In addition, the effects of power law index on the FGM plate natural frequency and mode shapes with different boundary conditions are studied.
Enab (Enab 2014) predicted stress concentration factors for functionally graded material under uniaxial and biaxial loads. Finite element analysis for the plates was done using ANSYS Parametric Design Language (APDL). Various geometric and material parameters were taken as the elliptic hole major axis to plate width ratio, shape factor, the gradation direction of FGM. It was concluded that, by selecting the proper distribution of functionally graded materials, stress concentration factors can be controlled. Some of the important findings of reviews are presented in Table 1.
In comparison with functionally graded plate and shells, studies for functionally graded beams are relatively less and until now most of the works have been done through MATLAB or FEM method. However, in present work, COMSOL 4.4 version software has been used, which gives very effective result in less time as comparable to others. This would help itself to design the dynamically prone structures. The objective of this paper is to study the Structural vibration analysis of functionally graded Beams using Finite Element Method (FEM). The simplysupported FGM beam model was developed where upper surface of the beam was alumina and lower surface was steel, and material properties i.e. (young's modulus and mass density) varied along the thickness of the beam according to the power law. The different boundary They studied free vibration analysis of rectangular plate structures and determined the natural frequencies of an isotropic thin plate using FEM.
By varying the thickness of the plate, it has been concluded that the frequency parameter is constant. Increase in thickness of the plate does not affect the frequency parameter.
2 Xiao and Yue, 2012They studied stress and displacement fields in a functionally graded material of semi-infinite extent induced by rectangular loading. Variation of elastic modulus was varied along the depth but kept constant in lateral directions.
It was found that the heterogeneity of FGM influences elastic fields of the semi-infinite elastic solids 3 Vimal et al. 2014 The functionally graded materials (FGMs) are assumed to be graded across their thicknesses according to a power law distribution of the volume fractions and the Poisson's ratio is taken as constant. Convergence study with respect to the number of nodes has been carried out and the results are compared with those from past investigations available in the literature. The effects of parameters such as cutout, cutout size, volume fraction index, boundary conditions and thickness ratio on the natural frequencies are studied. Free vibration analysis of functionally graded skew plates with circular cut outs based on the finite element approach using ANSYS.
The natural frequency of moderately thick functionally graded skew plates with circular cut outs decreases as the volume fraction index increases and increases as the thickness ratio increases. It is observed that by increasing the distance between the centers of two circular holes in a FG skew plate, the frequency of the first mode, decreases. The variation in the non-dimensional frequencies is less when the skew angle varies from 0˚ to 30˚, but the variation in the non-dimensional frequencies is more when the skew angle rises from 30˚ to 45˚.
4 Anandrao et al. 2012 They studied free vibration analysis of Functionally Graded Beams.
Considering transverse shear in the design and formulation increased the flexibility and reduced the frequencies of short beam.

5
Alshorbagy, 2011 They studied the effect of material-temperature dependent on the vibration characteristics of a functionally graded thick beam using FEM. The beam was modeled by higher order shear deformation theory (HOBT), which was accommodated for a thick beam.
The natural frequencies increase with an increase in power exponent, and decrease with an increase in power exponent.
6 Wei et al. 2012 Obtained an analytical method for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation The presence of cracks reduces the frequencies and changes the vibration mode shapes of FGM beams. Increase in crack diminishes the natural frequency ratios. The slenderness ratio and Young's modulus ratio are more sensitive to the free vibration of cracked FGM beams than the presence of cracks.
7 Khan and Parhi, 2013 Estimated the effects of crack depth on natural frequency and mode shape of beam Natural frequency increases and Mode shape decreases as the crack depth increases.
8 Rao et al. 2018 Proposed a new extended isogeometric hybrid collocation-Galerkin method to obtain natural frequencies of a homogeneous and functionally graded material plate with internal defects of varying gradient index, sizes and shapes and compared with FEM result using COMSOL.
The obtained results from the proposed method are found to be in good agreement with FEM results (COMSOL).
Where δW s and δW i virtual work done by the stress field and inertia forces respectively. The virtual work done by a stress field δW s on a virtual stain field is given by (8) By substituting Eqs. (5) and (6) into Eq. (8), yields (9) The virtual work done by the inertia forces through the virtual displacement field i.e. δW i can be represented as Eq. (10).

The finite element analysis (Alshorbagy 2011)
The displacement function of functionally graded beam at the mid-plane can be represented in plane component ( Figure 2) (12) Where, N denotes the shape functions or interpolating function.
-Set the parameter sweep and desired Eigen frequency under study tab. -Final step was to compute. -Different Eigen frequencies and mode shapes for different parameter were obtained.

The condition of geometric fit (Alshorbagy 2011)
If u o be the axial and w o the transverse displacement of any point on the mid-plane at time t, then, the axial and transverse displacement of any point of the beam are given by Euler-Bernoulli beam theory (Eq. (2) and Eq. (3) respectively). (2) Eqs. (2) and (3) can be rewritten as Eq. (4) to define displacement vector d s.

The law of material (Alshorbagy 2011)
According to Hooke's law, the relationship between stress σ xx and strain (∈ xx ) is given by Eq. (6).

(6)
Where young modulus varies through the thickness direction.

Equilibrium condition (Alshorbagy 2011)
According to the principle of virtual work, the equilibrium condition of functionally graded beam for free vibration can be stated by Eq. (7).

Effect of mass density and young modulus variation
The variation in young modulus and mass density of functionally graded material with respect to thickness directions as well as in power law exponent were shown in Figure 3 and Figure 4 respectively. From Figure 3, it can be revealed that at the value of z/h as -0.5, young modulus is 210 GPa and at z/h as 0.5, young modulus is 390Gpa. Similarly, from Figure 4, it can be revealed that at z/h as -0.5, the mass density is 4000 kg/m 3 and at z/h as 0.5, the young modulus is 7800 kg/m 3 . Different value of power law exponent (k) showed different physical significance given below: (1) If the value of power law exponent (k) is zero, then the beam is full alumina i.e. (E(z) = E alumina , ρ(z) = ρ alumina ) for (2) If the value of power law exponent (k) is 1000, then the beam is full steel i.e. (E(z) = E steel , ρ(z) = ρ steel ). (3) If the value of power exponent (k) is between 0 and 1000, the beam is functionally graded beam and their composition changes from alumina to steel.
Thus, the properties of material vary in proportion in the direction of thickness. In the present work, functionally graded material (FGM) consisted of steel (lower part) and alumina (upper part). Therefore, the lowest part is pure The non-dimensional quantities such as young modulus ratio, mass density ratio and frequency can be obtained from Eq. (17) (Alshorbagy 2011): Where I = bh 3 /12 is the moment of inertia of the crosssection of the beam.
Therefore, the first three non-dimensional frequencies (λ) for varying power exponent (k) are presented in Table 3. Figure 5 shows the graph between the first Eigen frequency and power law exponent, in which the X-axis is the power law exponent and Y-axis is the first Eigen frequency. It is observed that, as the value of k was 0, the beam was pure alumina. At k = 1000, it was pure steel. This deviation from alumina to steel resulted in decrease in the Eigen frequency. The non-dimensional frequency was deceased nonlinearly for value of k from 0 to 5 and decreased linearly for further value of k from 5 to 10.
The effects of varying power law exponent (k) and young modulus (E ratio ) ratio on non-dimensional frequency (λ) is presented on Table 4. It can be observed that for young's modulus less than one, Eigen frequencies   was increased with the increase in power exponent. Further, for young's modulus greater than one, the Eigen frequencies was decreased with an increase in power exponent. However, no change was obtained for young's modulus ratio equals to 1. Above conclusions are also valid for second and third non dimensional frequencies with different young modulus ratio variation and nonnegative power law index. Figure 6 showed the graph between the first nondimensional frequency and power law exponent for different young's modulus ratio. X-axis represents power law exponent and Y-axis represents first non-dimensional frequency. The increase in power exponent causes the increase in frequency for E ratio = 0.25 and 0.5, decrease in frequency for E ratio = 2 and 4 and no change occurs for E ratio = 1.For both situations when E ratio is less than one or greater than one it is clear that the first Eigen frequency of the FG beam approaches the first Eigen frequency of the homogeneous beam as the power-law exponent k increases. Figure 7 showed the graph between the first nondimensional frequency and young's modulus ratio for different power law exponent. X-axis represents young's modulus ratio and Y-axis represents first non-dimensional frequency. The frequency was increased significantly with increase in young's modulus ratio for lower value of power exponent. Conversely, there were no significantly changes obtained in frequency for different value of young's modulus ratios for higher value of power exponent.

The effect of boundary conditions
The boundary conditions (BCs) affects the vibration behavior of functionally graded beam materials. Table 5 indicates the effects of four different boundary conditions on the fundamental frequencies of the functionally graded beam. Four different boundary conditions are simplyclamped (SC), clamped-clamped (CC), simply-supported (SS) and clamped-free (CF). is characterized by the power law exponent. The natural frequency and stiffness elevated when power law exponent approached zero shows the interdependency of natural frequency and stiffness. Hence, performance of FGM beam was similar to the alumina beam.
Conversely, the behavior of FGM beam was similar to the steel beam, when power law exponent was equal to or greater than 1000. The natural frequency increases with the increase of power exponent only when the Young's modulus ratio is less than one.
In contrast to this, the natural frequency decreased as the power exponent was increased, when the Young's modulus ratio was greater than one. The natural frequency was invariant, when Young's modulus ratio was unity. The initial non-dimensional frequency for CC boundary condition was relatively more compared to other BCs and lowest frequency was obtained for CF BCs. As the fixed-fixed beam has lesser moments and deflection at mid-span in comparison to other boundary conditions. The frequency is inversely related to the deflection. The frequency of CC boundary condition is moderately more than other BCs.

Conflict of interest:
Authors declare no conflict of interest. Figure 8 showed the graph between the first nondimensional frequency and power law exponent for different boundary conditions. X-axis represents first nondimensional frequency and Y-axis represents first power law exponent. It can be observed that for all boundary conditions, increase in power law exponent (k) led to decrease in frequencies. The reduction in frequencies was attributed to increase in steel part as compared to alumina part in the beam composition. The first non-dimensional frequency for Clamped-Clamped boundary condition is comparatively more than other boundary conditions and lowest frequency is obtained for Clamped-Free boundary conditions.

Conclusions
Using Finite Element Analysis, the vibration analysis of functionally graded beams for cylinder liner in agricultural application has been studied. The natural frequency of the FGM beam is affected by two parameters such as Young's modulus ratio and power law exponent. It can be concluded from the above analysis that the effect of Young's modulus ratio is more profound than the effect of power law exponent. The material behavior