Dynamic Analysis of Functionally Graded Shaft

This paper deals with modal analysis of a functionally graded composite rotor shaft. Material properties distribution is assumed to vary along the radial direction of the Functionally Graded shaft according to power law distribution. The rotor shaft has been modelled and modal analysis is carried out. Functionally Graded Rotor shaft is analyzed by applying the loads and boundary conditions to obtain the vibration response and Campbell diagram is plotted. Various results have been obtained such as Campbell diagram, stability speed limit and time responses for functionally graded shaft and also compared with conventional steel shaft. It has been found that the responses of the functionally graded shaft are significantly influenced by material properties, radial thickness and power law index. The results obtained also show the benefits of functionally graded shaft over conventional steel shaft.


INTRODUCTION
Vibrational losses in machines are unavoidable where there are rotating parts involved.It is important to have a clear knowledge about modal frequencies of rotating systems to avoid resonance phenomenon leading to complete breakdown.Special precautions need to be taken even while crossing these modal frequencies.For example, if first modal frequency is 1550 rpm at which resonance occurs and we need to attain a frequency of 2000 rpm we need to add vibration arrestors to go through the first modal frequency.Another important aspect to reduce the vibrational losses in machines is the use of more effective materials.
Composites have a diverse range of applications in mechanical structures owing to lower weight to strength ratio.Weakness at the interface between adjacent layers is one of the major drawbacks in composites; this phenomenon is commonly known as delamination and this is the main reason behind breakdown of machinery.To deal with this shortcoming, a different type of material was brought forth, namely Functionally Graded Materials (FGMs).FGMs are those materials whose characteristics vary with respect to certain directions and parameters and this is how it removes the problem of weakness at interfaces.In FGMs, volume proportions of two or more than two materials are changed gradually with respect to specific directions to obtain desired properties.This gradual change allowed for a more uniform stress distribution in the component, and as a result, they improved in quality and durability.Use of FGMs results in smart resistance and better performance in extreme conditions such as those involving high pressure and high temperature.FGMs exhibit higher strength, lower density and higher stiffness and material characteristics than their metallic counterparts; this fact led to the concept of replacing metallic shafts with FG shafts in many fields like aerospace, automobile, defence, energy, marine, optoelectronics, and thermo-electronics; in machines like turbine shafts, thermo-electric generators and other manufacturing/machining equipment's [1].In FGMs the local physical and chemical properties of materials are different, with the changes in properties taking place along certain predetermined directions.Abrupt changes in material properties, which is the phenomenon taking place in composites, can be eliminated by using power law gradation in FGMs; this gradation would prevent delamination phenomenon, reduce stress between laminates and give better strength within material.
Many basic machine parts are designed using FGMs.Machine parts are more likely to fail by large deflections due to combined effect of thermo mechanical loading.For such boundary conditions FGMs are used in high temperature variations along the length of the material.FGMs are mostly man-made from isotropic materials like metals and ceramics.A comparative study was done by Sino et al. [2] between the proposed Simplified Homogenized Beam Theory (SHBT) and the existing EMBT, and the former is shown to be superior to the latter in that the SHBT considers the distance between composite layers and neutral axis.In addition to this, it also considers the internal damping by using the specific damping capacity of each ply of the composite assembly, and takes into account the transversal flexural shear.Hosseini et al. [3] analyzed the combination resonance in combination mode of a simple rotating shaft by using harmonic balancing method, and they studied the effects of, eccentricity, diametrical mass of inertia and external damping on the steady state response of the rotating shaft.Experimental vibrational analysis was carried out by Jain et al. [4] on multi-cracked rotor shafts.Variation in stiffness of material was observed with variation in crack depth.Stiffness variation was observed in the presence of second crack as well.Nonlinear dynamic behavior was studied while considering effects of crack location and shaft speed too.Khadem et al. [5] studied and analyzed a geometrically non-linear inextensible rotor shaft to obtain critical speeds by applying method of complex differential equation.It was shown that the results obtained, like the frequency curves and amplitude diagrams, were the same from application of the two approaches.Dasgupta and Rajan [6] analyses the dynamic responses of a compound rotor system subjected to the external and internal damping.
Montagnier et al. [7] investigated the composite shaft to obtain instabilities like unstable whirl oscillations triggered by damping in the supercritical regime.As a result, they were able to obtain the whirling speeds of the supercritical shaft mounted on elastic supports in the analytical form.It was found that shafts having ± 30 0 or ± 45 0 orientation of plies cannot be used in supercritical zone because of extremely high losses.Gayen et al. [8], through FE formulations, determined the effects of temperature variation, crack dimensions, power law gradient index, slenderness ratio on the dynamic characteristics of the FG rotating shaft system.They found that the whirl frequencies vary significantly with the gradient index for cracked and uncracked FG rotor shaft systems.Further they calculated the Local Flexibility Coefficients (LFCs) for varying crack depths and varying crack closer lines for a cracked FG shaft and they studied the effect of crack depth and crack closer line position on the LFCs.They concluded that, for a particular crack configuration, LFCs values attain their maximum for metallic shafts and minimum for ceramic shafts.Furthermore, it was found that LFCs values increase with the increasing power law gradient indices [9].Liu et al. [10] studied the un-damped vibration of exponential FG beams with only one delamination, by developing an analytical solution for the same.They concluded that shifting of neutral axis should be taken into account while formulating the continuity conditions for delaminated beam, because of asymmetrical distribution of material property in FGMs.Badour et al. [11] on realizing that the rapid and short period Fourier transforms could not efficiently analyze non-stationary signals, considered wavelet transform tools to mitigate this situation.In order to analyze the monitored vibration signals, two popular wavelet methods, namely continuous wavelet and wavelet packet transforms were used.Moreover, a novel algorithm, which incorporates a combination of both these and the concept of windowing a particular pulse into several revolutions of shafts to emphasize faults, was proposed and implemented in this paper.Koizumi [12] discussed and introduced functionally graded material.Rao et al. [13] came up with the idea of a mathematical model using power law and TMBT for FE modelling.Boukhalfa et al. [14] analyzed FG rotor shaft by using P-version of FEM with trigonometric shape functions and studied whirling frequencies, natural frequency response, and the whirling speeds of the rotating FG shaft.Further they also employ hpversion of FEM in to study vibrations of composite shaft by using kinetic and strain energy equations to formulate the equation of motion [15].In the available literature very few research work reported on the modal analysis of FGM rotor shaft.The present work intends to demonstrate the superiority of FG shaft over steel shaft via Finite Element Analysis on ANSYS.It also provides an insight on influence of material properties, radial thickness and power law index on vibration response of FG shaft.

MODELLING AND ANALYSIS
In the present work the FG shaft is modelled using ANSYS considering rotation in x direction (axis of rotation of shaft) is free and is zero in the y and z directions.The shaft is modelled as a shell having mean diameter 129.5 mm as shown in Fig. 1 and other parameters as mentioned in table 1 and 2.

Figure 1. Rotor Shaft Model
In order to achieve the best performance of Functionally Graded rotating shafts, which are heterogeneous, precisely estimated material property and optimal volume fraction selection are necessary.An FG shaft as shown in Figure 2 has been considered, having finite length L, inner radius (r a ) and outer radius (r b ).And it has taken Aluminium and Alumina as top and bottom surfaces respectively.

Modelling of shaft
After taking into consideration the inertia and gyroscopic couple effects the FG shaft is modelled using ANSYS solver.The shaft is modelled as a shell having mean diameter 129.5 mm and it undergoes rotation for a range of angular velocity about its geometrical axis.The x axis is assumed to be the shaft geometrical axis.The displacement fields are given as follows: By utilizing the kinetic energy and the elastic potential energy of the FG shaft, external loads and bearings performing virtual work done, combined with Hamilton's principle for rotating shaft gives rise to the following quations: In the above equation [A] refers to mass matrix, [B] refers to gyroscopic matrix, [C] refers to damping matrix, and [D] refers to the stiffness matrix.Also, in the above equation, {E} is the external force vector and {x} is the displacement vector.From the rotor dynamic Lagrangian equation of motion, including both internal viscous and hysteretic damping of a shaft disk element, the equation can be modified to obtain: The variation of properties of Young's modulus, density and Poisson's ratio of the FG shaft to be modelled was obtained using power law, with properties varying along the radial direction and value of k=0.5.The effective material properties P can be written as, The relationship between alumina's and aluminum's volume proportions is given by: Where V A and V a are the volume proportions of alumina and aluminium respectively at any point z throughout the radius r; r A and r a are the corresponding radii, where the former is the external radius and the latter is the internal radius of the shaft.According to power law, V A can be expressed as ( ) Here, z = r.This particular material distribution was incorporated in the designed shaft of given dimensions.This was achieved by layer wise stacking of different plies of different material composition (as per power law), with the plies stacked in inward direction.Furthermore, the fiber orientation between all the plies was the same.Through the above plot, if observed carefully, it can be seen that the critical speed for the second modal frequency lies in the range of 1100-1200 rad/s.The model of the functionally graded rotor shaft was solved to obtain Campbell Diagram and frequency response: In the above plot, the critical speed for the second modal frequency occurs at almost 5500 rad/s, which is almost five times the value of the critical speed obtained in the case of conventional steel shaft.Based on the modal analysis comparison of various parameters of steel shaft and FG shaft are presented in table 5 and 6.The results shown in the avbove three tables are the comparison between FG shaft and steel shaft in terms of frequency, maximum deformation and minimum deformation.It is evident from the results that FG shaft results are significantly better than the steel shaft.

CONCLUSION
It can be observed from the above analysis that Functionally Graded shaft is superior to the one made from conventional steel as FGM shaft allows for higher critical speeds, thereby increasing its capability and effectiveness.Also in FG shafts deformations are smaller than those in conventional steel shaft for corresponding modes.The results obtained show the benefits of functionally graded shaft over conventional steel shaft.Finally it can be concluded that by using rotor shaft made of FG materials shows better dynamic behaviour than conventional steel shaft.The proposed shaft system will be very useful for numerous applications such as driveshaft in automobiles, jet engines and helicopter drive applications, turbines shafts and other rotating machineries.