### Tangential Displacement and Shear Stress Distribution in Non-Uniform Rotating Disk under Angular Acceleration by Semi-Exact Methods

Document Type : Research Paper

Author

Department of Mechanical Engineering, University of Bojnord, Bojnord, Iran

10.22034/jsm.2019.1879783.1504

Abstract

In this paper semi-exact methods are introduced for estimating the distribution of tangential displacement and shear stress in non-uniform rotating disks. At high variable angular velocities, the effect of shear stress on Von Mises stress is important and must be considered in calculations. Therefore, He’s homotopy perturbation method (HPM) and Adomian’s decomposition method (ADM) is implemented for solving equilibrium equation of rotating disk in tangential direction under variable mechanical loading. The results obtained by these methods are then verified by the exact solution and finite difference method. The comparison among HPM and ADM results shows that although the numerical results are the same approximately but HPM is much easier, straighter and efficient than ADM. Numerical calculations for different ranges of thickness parameters, boundary conditions and angular accelerations are carried out. It is shown that with considering disk profile variable, level of displacement and stress in tangential direction are not always reduced and type of changing the thickness along the radius of disk and boundary condition are an important factor in this case. Finally, the optimum disk profile is selected based on the tangential displacement-shear stress distribution. The presented algorithm is useful for the analysis of rotating disk with any arbitrary function form of thickness and density that it is impossible to find exact solutions.

Keywords

[1] Gamer U., 1984, Elastic–plastic deformation of the rotating solid disk, Ingenieur-Archiv 54: 345-54.
[2] Guven U.,1992, Elastic–plastic stresses in a rotating annular disk of variable thickness and variable density, International Journal of Mechanical Sciences 43: 1137-1153.
[3] Guven U.,1995, On the applicability of Tresca’s yield condition to the linear hardening rotating solid disk of variable thickness, ZAMM 75: 397-398.
[4] Guven U., 1994, The fully plastic rotating disk of variable thickness, ZAMM 74: 61-65.
[5] Eraslan A.N., Orcan Y., 2002, Elastic–plastic deformation of a rotating solid disk of exponentially varying thickness, Mechanics and Materials 34: 423-432.
[6] Eraslan A.N., Orcan Y., 2002, On the rotating elastic–plastic solid disks of variable thickness having concave profiles, International Journal of Mechanical Sciences 44: 1445-1466.
[7] Eraslan A.N., 2002, Inelastic deformations of rotating variable thickness solid disks by Tresca’s and von Mises criteria, International Journal of Computational Engineering Science 3: 89-101.
[8] Eraslan A.N., Orcan Y., 2002, Von Mises yield criterion and nonlinearly hardening variable thickness rotating annular disks with rigid inclusion, Mechanics Research Communications 29: 339-350.
[9] Eraslan A.N., Orcan Y., 2003, Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions, International Journal of Mechanical Sciences 45: 643-667.
[10] Hojjati M.H., Jafari S., 2007, Variational iteration solution of elastic non uniform thickness and density rotating disks, Far East Journal of Applied Mathematics 29: 185-200.
[11] Hojjati M.H., Hassani A., 2008, Theoretical and numerical analyses of rotating discs of non-uniform thickness and density, International Journal of Pressure Vessels and Piping 85: 694-700.
[12] Hojjati M.H., Jafari S., 2008, Semi exact solution of elastic non uniform thickness and density rotating disks by homotopy perturbation and Adomian’s decomposition methods Part I: Elastic Solution, International Journal of Pressure Vessels and Piping 85: 871-878.
[13] Hojjati M.H., Jafari S., 2009, Semi-exact solution of non-uniform thickness and density rotating disks. Part II: Elastic strain hardening solution, International Journal of Pressure Vessels and Piping 86: 307-318.
[14] HassaniA., Hojjati M.H., Mahdavi E. , Alashti R.A., Farrahi G., 2012, Thermo-mechanical analysis of rotating disks with non-uniform thickness and material properties, International Journal of Pressure Vessels and Piping 98: 95-101.
[15] Jafari S., Hojjati M.H., Fathi A., 2012, Classical and modern optimization methods in minimum weight design of elastic rotating disk with variable thickness and density, International Journal of Pressure Vessels and Piping 92: 41-47.
[16] Alashti R.A., Jafari S., 2016, The effect of ductile damage on plastic behavior of a rotating disk with variable thickness subjected to mechanical loading, Scientia Iranica B 23: 174-193.
[17] Zheng Y., Bahaloo H., Mousanezhad D., Mahdi E., Vaziri A., Nayeb-Hashemi H., 2016, Stress analysis in functionally graded rotating disks with non-uniform thickness and variable angular velocity, International Journal of Mechanical Sciences 119: 283-293.
[18] Salehian M., Shahriari B., Yousefi M., 2018, Thermo-elastic analysis of a functionally graded rotating hollow circular disk with variable thickness and angular speed, Journal of the Brazilian Society of Mechanical Science and Engineering 2018: 41.
[19] Shlyannikov V. N., Ishtyryakov I. S. , 2019, Crack growth rate and lifetime prediction for aviation gas turbine engine compressor disk based on nonlinear fracture mechanics parameters, Theoretical and Applied Fracture Mechanics 103: 102313.
[20] Nayak P., Bhowmick S., NathSaha K., 2019, Elasto-plastic analysis of thermo-mechanically loaded functionally graded disks by an iterative variational method, Engineering Science and Technology, an International Journal 23: 42-64.
[21] Calladine C.R., 1969, Engineering Plasticity, Oxford, Pergamon Press.
[22] Timoshenko S., Goodier J. N., 1970, Theory of Elasticity, New York, McGraw-Hill.
[23] Johnson W., Mellor P.B., 1983, Engineering Plasticity, Chichester, UK, Ellis Horwood.
[24] Guraland U., Fenster S.k., 1995, Advanced Strength and Applied Elasticity, London, Prentice-Hall International.
[25] He J.H., 1999, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178: 257-262.
[26] He J.H., 2003, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135: 73-80.
[27] He J.H., 2004, Asymptotology by homotopy perturbation method, Applied Mathematics and Computation 6: 156-591.
[28] He J.H., 2005, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26: 827-833.
[29] He J.H., 2005, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation 6: 207-208.
[31] Adomian G., 1994, Solving Frontier Problems of Physics: The Decomposition Method, Boston, Kluwer Academic.
[32] Wazwaz A.M., 2002, Differential Equations: Methods and Applications, Rotterdam, Balkema.
[33] El-Wakil S.A., Abdou M.A., 2007, New applications of adomian decomposition method, Chaos Solitons Fractals 33: 513-522.
[34] Nakmura S., 1991, Applied Numerical Methods with Software, Prentice-Hall International Inc.
[35] Ashok K., Singh K., Bhadauria B.S., 2009, Finite difference formulae for unequal sub-intervals using lagrange’s interpolation formula, International Journal of Mathematical Analysis 3: 815-827.
[36] Tang S., 1970, Note on acceleration stress in a rotating disk, International Journal of Mechanical Science 12: 205-207.