Functionally Graded Shaft Research Articles

Steady-state and transient vibration analysis of a thermally loaded power law-based functionally graded rotor system with induced porosities

The dynamic analysis of a power law-based functionally graded (FG) rotor-bearing system with induced porosities for non-uniform porosity distributions has been performed using the finite element method. A novel porous rotor element considering the effects of transverse shear deformation, gyroscopic moments, translational and rotational inertial has been developed for non-uniform porosity distributions based on the Timoshenko beam theory. Material properties are graded based on the power law along the radial direction of the FG shaft. The nonlinear temperature distribution law has been considered in conjunction with the power law to vary the temperature across the cross-section of the FG shaft. Material is graded in such a way that the stainless steel is comprised at the core of the FG shaft and the outer periphery of the FG shaft is made of Zirconia. The steady-state and transient time and frequency responses of a thermally loaded porous FG rotor-bearing system have been computed for both types of uneven porosity distribution using the Houbolt time marching technique and Fast Fourier Transform, respectively. The free vibration analysis has also been performed on the porous FG rotor-bearing system for various parameters such as power law index, the volume fraction of porosity and porosity distribution under different thermal gradients. The reduction of critical speed with increased amplitude is observed from the steady-state and transient frequency responses with an increase in volume fraction of porosity and temperature gradients

A Study on Dynamic Behaviour of Thermally Distributed Exponentially Graded Rotor System with Induced Porosities

The present work deals with the dynamic analysis of exponential law-based functionally graded (FG) rotor-bearing systems. The effect of thermal gradation and porosity on dynamic characteristics of FG rotor shafts has been studied first time, using exponential law with a novel two-nodded FG rotor element based on Timoshenko beam theory (TBT). Porous material properties are assorted using exponential law and thermal gradation across the cross section of the FG shaft using exponential temperature distribution (ETD). The effects of temperature and porosity on natural frequencies and whirl frequencies are studied. It has been observed that there is a significant reduction in natural frequencies and whirl frequencies with an increase in volume fraction of porosity and temperature. Attempts have been made to obtain suitable reasons for the behaviours based on the material properties. Furthermore, the steady-state and transient vibration responses have been simulated using the Houbolt time marching technique for the ceramic-based FG rotor shaft system. The result shows the maximum amplitude of the steady-state and transient vibration responses is increased, and the critical speed of the FG rotor system shifts towards the left with the increase in volume fraction of porosity and temperature.

Analysis of Temperature, Displacement, and Stress in Shafts Made of Functionally Graded Materials with Various Grading Laws

An exact solution for thermo‐elastic analyses of radially graded circular hollow shafts made of functionally graded materials (FGMs) is studied. The material properties of functionally graded (FG) shafts are assumed to be temperature‐dependent (TD) and graded in the radial direction according to various grading patterns such as linear law of material gradation (LLMG), power law of material gradation (PLMG), and exponential law of material gradation (ELMG). Temperature fields of radially graded FG shafts are obtained analytically for the different grading patterns, using a steady‐state Fourier equation of heat conduction. Subsequently, based on linear strain‐displacement relations, thermo‐elastic equations are solved analytically to obtain displacement and stress fields as functions of radial distances, material gradient indices, and temperature gradients. The present mathematical derivations for thermo‐elastic analyses and solution methods are successfully validated by comparing with the results of published literature. Numerical results are presented to highlight the importance of material parameters, temperature distributions, radial distances, and aspect ratios on the temperature profiles, displacement, and stress fields of the FG shafts under thermal/mechanical loads. The numerical results have explicitly shown that the thermo‐elastic analyses of FG shafts are significantly influenced by material gradient indices, temperature gradients, and temperature‐dependent properties in high‐temperature environmental applications.

Fundamental frequency analysis of a thermally loaded corroded exponentially graded rotor bearing system

ABSTRACT Corrosion is an unintentional degradation of a material, which transpires due to harsh environmental conditions. It is a surface phenomenon that results in loss of material, which affects the dynamic characteristics and structural integrity of any structure. Fundamental frequency analysis of a corroded functionally graded (FG) rotor-bearing system using finite element method (FEM) for flexural vibrations is presented in work. A functionally graded (FG) shaft, Stainless Steel as the inner-metal core and Zirconia as the outer-ceramic layer is considered. Exponential gradation law is followed to assign the material properties along the radial direction of the shaft. The exponential temperature distribution (ETD) method based on Fourier law of heat conduction has been used for the temperature distribution across the cross-section of the FG shaft. Finite element formulations of a corroded FG shaft element have been developed using Timoshenko beam theory. An FE code is developed to compute the natural and whirl frequencies of an FG rotor-bearing system for various parameters such as corrosion length, position, depth and thermal gradients to investigate the influence of corrosion on fundamental frequencies. It has been investigated that there is a significant influence of corrosion parameters on the natural and whirl frequencies of an FG rotor-bearing system.

A novel corrosion detection method using wavelet transformed mode shapes of a functionally graded rotor-bearing system

Development of online fault detection based on vibration characteristics of a rotating system has been the subject of interest for the last few decades. A novel corrosion detection method is suggested in the present work, based on the wavelet transformed modal analysis of a functionally graded (FG) rotor system. Material properties of an FG shaft are position and temperature dependant and graded in a radial direction following the power-law and non-linear temperature distribution (NLTD) respectively. An FG rotor consisting of a mixture of Zirconium Dioxide (ZrO2) and Stainless Steel (SS) is considered, where the volume fraction of metal decreases toward the outer radius and ceramic volume fraction increases. The finite element (FE) formulations have been developed using Timoshenko beam theory (TBT) for modal analysis of a corroded FG rotor bearing system. A spatial variation of a wavelet transform is used to transform the computed FG rotor mode shapes into the wavelet domain to identify and locate the local corrosion defect. A sensitivity analysis and a parametric study has been carried out to further verify the validity of the suggested novel method.

Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor–Bearing System

The present work deals with natural and whirl frequency analysis of a porous functionally graded (FG) rotor–bearing system using the finite element method (FEM). Stiffness, mass and gyroscopic matrices are derived for porous and non-porous FG shafts by developing a novel two-noded porous FG shaft element using Timoshenko beam theory (TBT), considering the effects of translational inertia, rotatory inertia, gyroscopic moments and shear deformation. A functionally graded shaft whose inner core is comprised of stainless steel (SS) and an outer layer made of ceramic (ZrO2) is considered. The effects of porosity on the volume fractions and the material properties are modelled using a porosity index. The non-linear temperature distribution (NLTD) method based on the Fourier law of heat conduction is used for the temperature distribution in the radial direction. The natural and whirl frequencies of the porous and non-porous FG rotor systems have been computed for different power law indices, volume fractions of porosity and thermal gradients to investigate the influence of porosity on fundamental frequencies. It has been found that the power law index, volume fraction of porosity and thermal gradient have a significant influence on the natural and whirl frequencies of the FG rotor–bearing system.

Stability behavior of two-crack functionally graded shaft in a rotor-disc system: finite element approach

Stability behavior of a rotor-disc system having a functionally graded (FG) shaft with fully open transverse cracks are studied through a finite element (FE) approach, considering the rigid end bearing conditions, various crack parameters, shaft internal damping and geometrical parameters. FG shaft is modeled with considering the effect of internal damping using Timoshenko beam theory (TBT)using two nodded with four degrees of freedom (DOFs) per node are taken into the formulation. Zirconia (ZrO2) and stainless steel (SS) are considered as FG material where material properties are considered temperature dependent and radially graded in the cracked FG shaft. Local flexibility coefficients (LFCs) are derived analytically with depth of crack, gradient index and temperature using Castigliano's theorem and Paris’s equation to compute the modified stiffness matrix at each instant of shaft rotation. Carrying out several numerical examples, the present formulation is validated with availableresults. Influences of shaft internal damping coefficients, gradient index, temperature gradient, shaft’s slenderness and relative locations of crack on the dynamic responses of the FG cracked shaft system are computed and discussed in detail.

Natural frequency analysis of a functionally graded rotor system using three-dimensional finite element method

Three-dimensional Finite Element (FE) analysis has been carried out using ANSYS software to study the natural frequencies of functionally graded (FG) rotor system. Temperature and position-dependent material properties of the FG shaft system are considered to be graded in the radial direction. Power-law with the nonlinear temperature distribution (NLTD) and exponential law with exponential temperature distribution (ETD) have been used to model the material gradation and temperature distribution. Rotor systems of two different FG materials, namely Stainless Steel-ZrO2 and Stainless Steel-Al2O3 have been studied. Python codes have been developed to generate ANSYS macros that apply the material properties. Simply supported FG shaft and FG rotor system supported on bearings have been analyzed in the current work. The results show the influence of different power-law coefficients, different material and material laws on the natural frequencies of the rotor system. The effect of these parameters on whirl frequencies has also been studied for FG rotor systems.

FE approach for dynamic response of a functionally graded spinning shaft system containing a transverse fully open crack

Free vibration and stability analysis are studied for a rotor-disc-bearing system having a radially functionally graded (FG) shaft with a transversely fully open crack, based on finite element (FE) approach. Both internal viscous and hysteretic damping are incorporated in the FE model of FG cracked shaft using two nodded Timoshenko beam element having four degrees of freedom at each node. Material properties of the FG shaft are assumed temperature dependent and graded with different material law of gradation. Aluminium Oxide (Al2O3) and stainless steel (SS) are composed as FG material. Local flexibility coefficients (LFCs) are derived analytically as a function of crack size, power-law gradient index and temperature using Paris’s equation and Castigliano’s theorem to compute the stiffness matrix at each instant in the FE analysis. Using the developed MATLAB code, the FE formulation and the cracked model are verified with the published results. Parametric studies are conducted to study the influences of different material gradient index, temperature gradient, crack size, internal damping, slenderness ratio and boundary conditions on the vibration responses of the FG cracked shaft system.

Finite element based stability analysis of a rotor-bearing system having a functionally graded shaft with transverse breathing cracks

In spite of gaining practical importance of functionally graded (FG) shafts, stability analysis of such FG rotor-bearing system has not been reported yet. This paper presents the development of a finite element (FE) procedure and code for stability analysis of cracked FG rotor-bearing system under thermal environment. Two-noded Timoshenko beam elements are used to model the FG shaft considering the effects of gyroscopic moments, translational and rotary inertia, bending and shear deformation and material damping. Zirconia (ZrO2) and stainless steel (SS) are considered as constituents of the radially graded FG shaft with temperature dependent material properties. Considering breathing crack behaviour, local flexibility coefficients (LFCs) are derived using the Paris's equation and the Castigliano's theorem. Results show that while the depth, orientation and locations of cracks, thermal gradient and material damping affect the stability threshold speed, it is important to choose the material gradient index judiciously. Thus, even when surface cracks appear on the FG shaft, threshold speed could still be in the desired range, which is important for damage tolerant design in high temperature applications.

Free Vibration Analysis of Functionally Graded Shaft System with a Surface Crack

Free vibration analysis of functionally graded (FG) non-spinning simply supported shaft having a transverse surface crack has been carried out. Material properties of the FG shaft are assumed to be graded radially following the power law of material gradation. Local flexibility coefficients as a function of depth of crack are computed with the help of Paris’ equation along with Castigliano’s theorem. Finite element (FE) formulation has been done using Timoshenko beam elements with two nodes and having four degrees of freedom per node. Based on the formulation, FE analysis has been performed to understand the effects of shaft’s slenderness, crack’s depth and location, gradient parameter and thermal gradient on the free vibration response of the cracked FG shaft. FE analysis show that for a cracked FG shaft, both gradation parameter and crack parameters have significant influence on the free vibration response. Gradation parameter could be suitably chosen in the design of FG shafts so as to minimize the increase in flexibility due to appearance of cracks.

Parametric study on free vibration and instability of a functionally graded cracked shaft in a rotor-disc-bearing system: finite element approach

Free vibration and stability analysis are studied for a rotor-disk-bearing system having a radially functionally graded (FG) shaft with a transversely fully open crack, based on finite element (FE) approach. Both viscous and hysteretic internal damping are incorporated in the FE model of FG cracked shaft using two nodded Timoshenko beam element having four degrees of freedom (DOFs) at each node. Material properties of the FG cracked shaft are assumed temperature dependent and graded along radial direction following different material gradation law. FG shaft is made of two constituents material namely zirconia (ZrO2) and stainless steel (SS) where metallic (SS) contain is decreasing towards the outer diameter of the shaft. Extended Hamilton’s principle is employed to derive the system equations of motion (EOMs) of the FG cracked shaft system. A complete code is developed in MATLAB for correcting the formulation of modeling of crack and verified with existing published results. Influences of different material gradient index, temperature gradients, size and location of crack, viscous and hysteretic internal damping, slenderness ratio, and boundary condition on dynamic responses of the FG cracked shaft system are studied.

Finite element analysis for a functionally graded rotating shaft with multiple breathing cracks

Finite element (FE) formulation of a functionally graded (FG) shaft having multiple cracks has been presented to study the transverse vibration of such shafts in a rotor bearing system. Two nodded Timoshenko beam elements with four degrees of freedom (DOFs) per node are used to model cracked FG shaft considering the translational and rotary inertia, transverse shear deformations and gyroscopic moments. Local flexibility coefficients (LFCs) for the cracked FG shaft are derived using Paris's equation and Castigliano's theorem. Breathing effect of cracks has been considered taking into account the stiffness variation during rotation of the FG shaft. Zirconia (ZrO2) and stainless steel (SS) with temperature dependent material properties have been considered as the constituent materials of a radially graded FG shaft. Using the FE code developed, the forward and backward whirl frequencies and critical speeds of the cracked FG shaft are obtained to study the effects of important parameters and power law gradient index. Results show that besides being affected by the crack locations, orientations and size, the extent of percentage reductions in fundamental frequencies and critical speeds are also influenced by the power-law gradient index of the FG shaft. Therefore, from the view point of damage tolerant design of FG shafts, the choice of power-law gradient index has significant importance where multiple transverse cracks appear in the shaft during service.

Torsion of a functionally graded material

Within the framework of the nonlocal strain gradient theory, a size-dependent shaft model, which can account for the through-radius power-law variation of two-constituent functionally graded (FG) materials, is derived to investigate the small-scaled effects on the static and dynamic torsion behaviors. The equations of torsional motion and corresponding boundary conditions of the size-dependent FG shaft are derived in terms of the Hamilton’s principle. The shaft models can account for the small-scaled effects of the inter-atomic long-range force and the microstructure deformation mechanism by introducing material length scale and nonlocal parameters. An analysis on the harmonic propagation with time torsional waves in a nonlocal strain gradient FG shaft is carried out. In the case of clamped-clamped boundary conditions, analytical solutions are obtained for the free vibration and static torsion problems of nonlocal strain gradient FG shafts. The effects of small-scaled parameters and the through-radius power-law variation of a two-constituent FG material on wave propagation, free vibration and static torsion are investigated in details.

Whirl frequencies and critical speeds of a rotor-bearing system with a cracked functionally graded shaft – Finite element analysis

Abstract Finite element (FE) formulations have been developed for dynamic analysis of a functionally graded (FG) shaft having a transverse crack. Two nodded Timoshenko beam element with four degrees of freedom (DOFs) per node has been used where effects of translational and rotary inertia, transverse shear deformations, and gyroscopic moments are considered. Local flexibility coefficients (LFCs) of the cracked FG shaft are determined analytically as a function of crack size, power-law gradient index (k), and temperature using the Castigliano's theorem and Paris's equations which are used to compute the stiffness matrix in the FE analysis. Temperature dependent thermo-elastic material properties of the FG shaft are considered graded in the radial direction following power-law gradation. Using the present formulations, an FG shaft composed of zirconia (ZrO2) and stainless steel (SS) is considered and the forward and backward whirl frequencies and critical speeds are determined. Influences of crack size, power-law gradient index, slenderness ratio, and temperature gradient on the dynamic characteristics of the rotor-bearing system with an FG shaft have been studied. Results show that the power-law gradient index has significant influence on the whirl frequencies and critical speed both for cracked and un-cracked FG shaft and could be judiciously chosen in designing FG shafts.

Vibration Analysis of Functionally Graded Rotating Shaft System

This paper mainly deals with the dynamic analysis of functionally graded (FG) rotor shaft system. Power law gradation is adopted for mathematical material modelling of FG rotor shaft. Timoshenko beam theory (TMBT) is also been used for the finite element (FE) modelling of the FG shaft. The FG shaft model has been verified by comparing critical speeds with the available literature. Different analyses have been carried out (such as Campbell diagram, Stability speed limit (SSL) and damping ratio) by considering shear deformation, rotary inertia, gyroscopic effects, strain, the kinetic energy of shaft, and internal damping.

Variation of Local Flexibility Coefficients of Functionally Graded Cracked Shaft

The present paper deals with the analytical determination of local flexibility coefficients (LFCs) for a functionally graded (FG) cracked shaft. Finite element (FE) analysis has been performed using two nodded beam element having six degrees of freedom (DOF) at each node and Timoshenko beam theory (TBT) has been used. Material properties are assumed to be graded in radial direction following linear, power law and exponential law gradation respectively. In the present analysis, the mixture of Aluminum Oxide (Al2O3) and Stainless Steel (SUS304) is considered as FG material where metal content is considered decreasing towards the outer diameter of shaft. The LFCs of a cracked shaft is derived using Castigliano's theorem and Paris's equations in conjunction with the expression for stress intensity factors. A complete code has been developed using MATLAB program and validated with the existing results available in literatures. The LFCs are calculated for different crack depths and for different crack closer lines for a cracked shaft and the effect of crack depth and crack closer line position on the LFCs has been studied. Numerical results reveal that for a given crack configuration, LFCs values are maximum for metallic, minimum for ceramic shaft and the LFCs for FG shaft in between, and also the LFCs values increase with the increasing power law gradient indices.