Natural frequency analysis of a functionally graded rotor-bearing system with a slant crack subjected to thermal gradients

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Arnab Bose ◽  
Prabhakar Sathujoda ◽  
Giacomo Canale
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Power Law ◽  
Beam Theory ◽  
Functionally Graded ◽  
Thermal Gradients ◽  
Element Analysis ◽  
Rotor Bearing ◽  
Natural Frequency Analysis ◽  
Bearing System

Abstract The present work aims to analyze the natural and whirl frequencies of a slant-cracked functionally graded rotor-bearing system using finite element analysis for the flexural vibrations. The functionally graded shaft is modelled using two nodded beam elements formulated using the Timoshenko beam theory. The flexibility matrix of a slant-cracked functionally graded shaft element has been derived using fracture mechanics concepts, which is further used to develop the stiffness matrix of a cracked element. Material properties are temperature and position-dependent and graded in a radial direction following power-law gradation. A Python code has been developed to carry out the complete finite element analysis to determine the Eigenvalues and Eigenvectors of a slant-cracked rotor subjected to different thermal gradients. The analysis investigates and further reveals significant effect of the power-law index and thermal gradients on the local flexibility coefficients of slant-cracked element and whirl natural frequencies of the cracked functionally graded rotor system.

Natural frequency analysis of a functionally graded rotor-bearing system with a slant crack subjected to thermal gradients

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Arnab Bose ◽  
Prabhakar Sathujoda ◽  
Giacomo Canale
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Power Law ◽  
Beam Theory ◽  
Functionally Graded ◽  
Thermal Gradients ◽  
Element Analysis ◽  
Rotor Bearing ◽  
Natural Frequency Analysis ◽  
Bearing System

Abstract The present work aims to analyze the natural and whirl frequencies of a slant-cracked functionally graded rotor-bearing system using finite element analysis for the flexural vibrations. The functionally graded shaft is modelled using two nodded beam elements formulated using the Timoshenko beam theory. The flexibility matrix of a slant-cracked functionally graded shaft element has been derived using fracture mechanics concepts, which is further used to develop the stiffness matrix of a cracked element. Material properties are temperature and position-dependent and graded in a radial direction following power-law gradation. A Python code has been developed to carry out the complete finite element analysis to determine the Eigenvalues and Eigenvectors of a slant-cracked rotor subjected to different thermal gradients. The analysis investigates and further reveals significant effect of the power-law index and thermal gradients on the local flexibility coefficients of slant-cracked element and whirl natural frequencies of the cracked functionally graded rotor system.

Finite Element Analysis for Dynamic Response of Rotor-Bearing System With Cracked Functionally Graded Turbine Shaft

2017 ◽  
Author(s):  
Debabrata Gayen ◽  
Debabrata Chakraborty ◽  
Rajiv Tiwari
Keyword(s):  
Finite Element ◽  
Power Law ◽  
Natural Frequencies ◽  
Crack Size ◽  
Fe Analysis ◽  
Functionally Graded ◽  
Gradient Index ◽  
Crack Orientation ◽  
Rotor Bearing ◽  
Bearing System

This paper describes finite element (FE) analysis of a rotor-bearing system having a functionally graded (FG) shaft with a transverse breathing crack. Two nodded Timoshenko beam element with four degrees of freedom (DOFs) per node has been considered and effects of translational and rotary inertia, transverse shear deformations, and gyroscopic moments are also considered. The FG shaft is considered to be composed of zirconia (ZrO2) and stainless steel (SS) with the volume fraction of SS increasing towards the inner radius of the shaft. Thermo-elastic material properties are considered in the radial direction of the FG shaft following power law gradation. 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 with crack orientation using the Castigliano’s theorem and Paris’s equations which are used to compute the stiffness matrix in the FE analysis. The FE formulation has been validated with the analytical and FE solutions reported in the literatures, and then natural frequencies and whirling (forward and backward) frequencies are determined. Influences of crack size, power law gradient index, slenderness ratio, and temperature gradient with crack orientation, on the dynamic responses of the rotor-bearing system with an FG shaft are studied. Results show that the power law gradient index has significant influence on the natural frequencies and whirling frequencies for the rotor-bearing system with the breathing cracked FG shaft and the choice of power law index could play an important role in design of FG shafts under thermo-mechanical environment from the view point of damage tolerant design.

Analysis and Optimization of Bellows With General Shape

1998 ◽  
Vol 120 (4) ◽  
pp. 325-333 ◽  
Author(s):  
B. K. Koh ◽  
G. J. Park
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Beam Theory ◽  
Inequality Constraints ◽  
Shell Element ◽  
Scalar Function ◽  
Programming Algorithm ◽  
Multiple Objective ◽  
Axially Symmetric ◽  
Element Analysis

A bellows is a component in piping systems which absorbs mechanical deformation with flexibility. Its geometry is an axially symmetric shell which consists of two toroidal shells and one annular plate or conical shell. In order to analyze the bellows, this study presents the finite element analysis using a conical frustum shell element. A finite element analysis program is developed to analyze various bellows. The formula for calculating the natural frequency of bellows is made by the simple beam theory. The formula for fatigue life is also derived by experiments. A shape optimal design problem is formulated using multiple objective optimization. The multiple objective functions are transformed to a scalar function with weighting factors. The stiffness, strength, and specified stiffness are considered as the multiple objective function. The formulation has inequality constraints imposed on the natural frequencies, the fatigue limit, and the manufacturing conditions. Geometric parameters of bellows are the design variables. The recursive quadratic programming algorithm is utilized to solve the problem.

Finite element analysis of flexible functionally graded beams with variable Poisson’s ratio

2016 ◽  
Vol 33 (8) ◽  
pp. 2421-2447 ◽  
Author(s):  
João Paulo Pascon
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Poisson’S Ratio ◽  
Scientific Literature ◽  
Functionally Graded ◽  
Poisson's Ratio ◽  
Transverse Strain ◽  
Thickness Direction ◽  
Element Analysis ◽  
Content Type

Purpose The purpose of this paper is to deal with large deformation analysis of plane beams composed of functionally graded (FG) elastic material with a variable Poisson’s ratio. Design/methodology/approach The material is assumed to be linear elastic, with a Poisson’s ratio varying according to a power law along the thickness direction. The finite element used is a plane beam of any-order of approximation along the axis, and with four transverse enrichment schemes, which can describe constant, linear, quadratic and cubic variation of the strain along the thickness direction. Regarding the constitutive law, five materials are adopted: two homogeneous limiting cases, and three intermediate FG cases. The effect of both finite element kinematics and distribution of Poisson’s ratio on the mechanical response of a cantilever is investigated. Findings In accordance with the scientific literature, the second scheme, in which the transverse strain is linearly variable, is sufficient for homogeneous long (or thin) beams under bending. However, for FG short (or moderate thick) beams, the third scheme, in which the transverse strain variation is quadratic, is needed for a reliable strain or stress distribution. Originality/value In the scientific literature, there are several studies regarding nonlinear analysis of functionally graded materials (FGMs) via finite elements, analysis of FGMs with constant Poisson’s ratio, and geometrically linear problems with gradually variable Poisson’s ratio. However, very few deal with finite element analysis of flexible beams with gradually variable Poisson’s ratio. In the present study, a reliable formulation for such beams is presented.

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