ME663A

Metal Forming

Credits:

  

 

3L-0T-0L-0D (9 Credits)
 

Course Content:


Fundamentals of plasticity, yield and flow, anisotropy, instability, limit analysis, slipline field theory. Applications to forging, wire and tube drawing, deep drawing, extrusion and rolling. High velocity forming. 

Lecturewise Breakup (based on 75min per lecture)


I. Introduction (1 Lecture)

  • Introduction to the course: different metal forming processes, importance of plasticity in the course. [1 Lecture]

II. Fundamentals of Plasticity (17 Lectures)

  • Review-Analysis of stress: transformation relations, principal stresses and directions, maximum normal and shear stresses, invariants, hydrostatic and deviatoric parts; Analysis of (infinitesimal) strain: transformation relations, principal strains, invariants, hydrostatic and deviatoric parts; (Infinitesimal) rotation, Stress-strain relations for isotropic, linearly elastic material. [4.5 Lectures]

  • Experimental observations on plasticity: yielding, strain-hardening, viscoplasticity, temperature softening, Baushinger effect, hysteresis, incompressibility of plastic deformation, anisotropy, plastic instability. [1 Lecture]

  • Yield criterion for isotropic materials: von Mises and Tresca yield criterion, their geometric interpretation, convexity of the yield surfaces, experimental validation. [2 Lectures]

  • Incremental and rate forms of the measures of plastic deformation: linear incremental strain tensor, strain rate (i.e. the rate of deformation) tensor and their relation, incremental rotation tensor and spin tensor. [2 Lectures]

  • Change in yield criteria due to isotropic hardening: strain hardening and work hardening hypotheses, experimental validation of the hypotheses. [2 Lectures]

  • Plastic stress-strain relations for isotropic materials: plastic potential and associated flow rule, incremental and rate forms of elasto-plastic stress-strain relations, simplifications for non-hardening and rigid-plastic materials (Prandtl- Reuss and Levy-Mises relations),Objective measures of stress rate and incremental stress. [3.5 Lectures]

  • Incremental and flow formulations of plasticity: updated Lagrangian and Eulerian formulations, boundary and initial conditions, examples. [2 Lectures]

  • Anisotropy: strain rate ratio, normal and planer anisotropies, Hill’s anisotropic yield criterion, example. [2 Lectures]

III. Approximate Methods of Plastic Analysis (6 Lectures)

  • Approximate methods of solving plasticity problems: upper and lower bound theorems, upper and lower bound methods, slip line field equations, different boundary value problems of slip line method, one example of all 3 methods.

IV. Metal Forming Processes (14 Lectures)

  • Slab method for sheet and wire drawing processes for predicting drawing force and die pressure, comparison with Wistreich experimental results, optimum die angle, maximum reduction for non-hardening material, upper bound method for sheet/wire drawing, slip-line method for sheet drawing, correction for hardening effect in upper bound and slip line  methods. [4 Lectures]

  • Slab method for extrusion of rod and sheets for predicting extrusion pressure, upper bound method for sheet/wire extrusion, dead-metal zone for square die, slip-line method for sheet extrusion. [2 Lectures]

  • Slab method for plane strain rolling for predicting roll force and roll torque, limiting reduction, roll diameter to sheet thickness ratio and friction coefficient. [2 Lectures]

  • Slab method for sheet and disc forging processes for predicting forging force, sticking radius, slab method for hollow disc forging, neutral radius, upper bound method for sheet and disc forging with and without bulge, slip-line method for sheet forging. [4 Lectures]

  • Slab method for the flange analysis in deep drawing, limiting drawing ratio for non-hardening materials. [2 Lectures]

Term Paper:


It should be based on the analysis of a different forming process (other than drawing, extrusion, rolling, forging and deep drawing) using either the upper bound method or the lower bound method or the slip line method. .

References:

  1. The Mathematical Theory of Plasticity by R. Hill, Oxford University Press, 1950

  2. Engineering Plasticity by W. Johnson and P.B. Mellor, von Nostrand Co. Ltd, 1972

  3. Theory of Plasticity by J. Chakrabarty, McGraw-Hill Book Co., International Edition, 1987

  4. Metal Forming: Processes and Analysis by B. Avitzur, McGraw-Hill Book Co., 1968

  5. Continuum Theory of Plasticity by A.S. Khan and S. Huang, John Wiley and Sons Inc., 1995.
 

ME681A

Mathematics For Engineers

Credits:

  

 

3L-0T-0L-0D (9 Credits)

 

 

Course Content:


Linear Algebra: Vector space and its basis; Matrices as coordinate-dependent linear transformation; null and range spaces; Solution of linear algebraic equations: Gauss elimination and Gauss-Jordon methods, LU Decomposition and Cholesky method, Gauss-Seidel/ Jacobi iterative methods; Condition number; Minimum norm and least square error solutions; Eigenvalues and eigenvectors of matrices and their properties; Similarity transformation; Jordon canonical form and orthogonal diagonalization; Mises power method for finding eigenvalues/eigenvectors of symmetric matrices. Tensor Algebra and Index Notation. Vector and Tensor Calculus: Curves and surfaces; Gradient, divergence and curl, Line, surface and volume integrals; Gauss (divergence), Stokes and Green’s theorems. Topics in Numerical Methods: Solution of a non-linear algebraic equation and system of equations; Interpolation methods, Regression; Numerical Integration. Ordinary Differential Equations (ODEs): Techniques of the separation of variable and the integrating factor for 1st order ODEs; Solutions of linear, 2nd order ODEs with constant coefficients and Euler-Cauchy ODEs; System of 1st order ODEs; Numerical methods for solving ODEs, Homogeneous, linear, 2nd order ODEs with variable coefficients: power series and Frobenius methods; Sturm-Louville problem; Laplace transform method for non-homogeneous, linear, 2nd order ODEs: discontinuous right-hand sides

Lecturewise Breakup (based on 50 min per lecture)

 

I. Introduction (1 Lecture)

  • Introduction to the course. [1 Lecture]

II. Linear Algebra (12 Lectures)

  • Vector spaces: definition, linear independence of vectors, basis, inner product and inner product space, orthogonality, Gram-Schmidt procedure, subspaces. [2 Lectures]

  • Matrices: coordinate-dependent linear transformations, null and range spaces. [1 Lectures]

  • Linear algebraic equations: existence and uniqueness of solution, elementary row/column operations, Gauss elimination and Gauss Jordon methods, Echelon form, pivoting, LU decomposition and Cholesky method, Gauss-Seidel and Jacobi iterative methods, condition number, minimum norm and least square error solutions. [4.5 Lectures]

  • Eigenvalues and eigenvectors of matrices: properties like multiplicity, eigenspace, spectrum and linear independence of eigenvectors, similarity transformation and Jordon canonical form, eigenvalues/eigenvectors of symmetric matrices: orthogonal diagonalization. [3 Lectures]

  • Iterative methods to find eigenvalues/eigenvectors of symmetric matrices: forward iteration and Mises power method, inverse iteration. [1.5 Lectures]

III. Tensor Algebra (4 Lectures)

  • Index Notation and Summation Convention. [1 Lectures]

  • Tensor algebra: tensor as a linear vector transformation, dyadic representation, transformation of components, product of tensors, transpose, decomposition into symmetric and antisymmetric parts, invariants, decomposition into isotropic and deviatoric parts, inner product and norm, inverse, orthogonal tensors, eigenvalues and eigenvectors, square-root, positive definite symmetric tensor, polar decomposition, tensors of higher order. [3 Lectures]

IV. Vector and Tensor Calculus (4 Lectures)

  • Review of multi-variable calculus. [0.5Lecture]

  • Curves: parametric representation, tangent vector, arc length, curvature, principal normal vector, osculating plane, bi-normal vector; Surfaces: parametric representation, tangent vector and tangent plane. [1.5Lecture]

  • Scalar fields: gradient, directional derivative, potential; Vector fields: divergence, curl, solenoidal and irrotational vector fields; Line integral and path independence; Surface and volume integrals; Gauss (divergence), Stokes and Green’s theorems (without proof). [1.5Lecture]

  • Tensor calculus: tensor derivative of a scalar field, gradient of a vector field, divergence of a tensor field. [0.5Lecture]

V. Topics in Numerical Methods (5 Lectures)

  • Solution of non-linear algebraic equations, Newton-Raphson method for a system of non-linear algebraic equations. [1.5Lectures]

  • Lagrange and Hermite interpolation methods. [1Lectures]

  • Regression: linear least-square method. [1Lectures]

  • Numerical integration: trapezoidal and Simpson’s rules, Gauss quadrature. [1.5Lectures]

VI. Ordinary Differential Equations (ODEs) (14 Lectures)

  • Initial value problem (IVP) of a 1st order ODE: existence, uniqueness and continuity with initial conditions. [0.5 Lectures]

  • Methods of solving 1st order ODEs: separation of variable technique, change of variable to make ODE separable; exact ODEs, integrating factor to make ODE exact, linear 1st order ODEs. [1.5 Lectures]

  • Homogeneous, linear, 2nd order ODEs: existence and uniqueness, 2 fundamental (linearly independent) solutions and Wronskian, superposition for obtaining general solution, fundamental solutions of ODEs with constant coefficients, method of reduction of order to find 2nd linearly independent solution, fundamental solutions of Euler-Cauchy ODEs. [2 Lectures]

  • Non-homogeneous, linear, 2nd order ODEs: existence and uniqueness, methods of undermined coefficients and variation of parameters, introduction to higher order ODEs. [1.5 Lectures]

  • System of 1st order ODEs: existence and uniqueness of IVP, solution of the homogeneous system with constant coefficients, generalized eigenvector to find other fundamental solutions, method of variation of parameters [1.5 Lectures]

  • Numerical methods for solving IVP of ODEs: Euler and Runge-Kutta methods, stability of numerical methods. [1.5 Lectures]

  • Homogeneous, linear, 2nd order ODEs with variable coefficients: power series method, solution of Legendre equation; Frobenius method, solution of Bessel equation; Sturm-Louville problem with regular, periodic and singular (homogeneous) boundary conditions and use of its eigenfunctions as an orthogonal basis for the representation of functions. [3.5 Lectures]

  • Laplace transform method for IV problem involving non-homogeneous, linear, 2nd order ODEs, properties of transform, inverse transform using tables, discontinuous right-hand sides involving unit step, impulse and Dirac-delta functions, t-shifting theorem. [2 Lectures]

    References:

    1. Advanced Engineering Mathematics by E. Kreyszig, John Wiley and Sons, International 8th Revised Edition, 1999.

    2. Applied Mathematical Methods by B. Dasgupta, Pearson Education, 2006.
 

ME401

Energy Systems-II

Credits:

  

 

3L-0T-1P-0A (10 Credits)
 

Pre-requisite:


ESO201, ME231.

Course Content:


Introduction: General Theory and Classification of Turbomachines; Similarity and Dimensional Analysis; Two-dimensional Cascade Theory; Axial and Radial Flow Machines: Turbines, Compressors and Fans; Gas Turbine Power Plant Cycles; Thermal Power plant: Flow through Nozzle and Steam Turbines; Hydraulic Machines: Pelton, Francis and Kaplan Turbines; Pump and Cavitation.

Lecturewise Breakup:


I. Introduction: General Theory and Classification: (3 Lectures)

  • Definition of turbomachines, Classification: mixed, axial and radial flow impellers and their applications, Equation of motion and energy in rotating frame of reference, Effect of Coriolis forces, Euler equation for torque, Concept of velocity triangles.

II. Similarity and Dimensional Analysis: (2 Lectures)

  • Geometric and kinematic similarity, Similarity rules, Non-dimensional characteristics, Specific Speed and Specific Diameter, Similarity analysis for compressible flow.

III. Two-Dimensional Cascade Theory: (4 Lectures)

  • Introduction to turbine and compressor cascades, Two-dimensional analysis of inviscid and incompressible flows through cascade,   Kutta-Joukowski theorem, Circulation and lift, Cascade tunnel, Blade efficiency and losses, Cascade nomenclature.

IV. Axial flow machines: Compressors, Turbines and Fans: (9 Lectures)

  • Two-dimensional pitch line analysis and design, Work done factor, Degree of Reaction, Losses, Compressor/Turbine Blade efficiency,   Off-design performance, Multi-stage machines, Compressible flow analysis, Preheat and reheat, Overall-pressure ratio, Turbine   blade cooling.

V. Centrifugal Machines: (4 Lectures)

  • Pressure rise in Centrifugal compressors, work done factor, relative eddy and slip factor, effect of compressibility, diffuser system, inward flow radial turbine, basic design of the rotor, losses and efficiency.

VI. Power Plant Cycles: (2 Lectures)

  • Gas turbine cycles, Reheat and Regenerative cycles, Inter-cooling, Gas turbine cycles for propulsion, Turboprop and turbojet engines.

VII. Thermal Power plant: Flow through Nozzle and Steam Turbine: (9 Lectures)

  • Introduction, Boiler, thermodynamic relations of adiabatic flow through convergent and divergent nozzle, shock, over- and under-   expansion, effect of friction, nozzle efficiency, super-saturated expansion of steam in a nozzle, degree of super-saturation and under-   cooling, Wilson line, steam turbine, simple impulse turbine, pressure and velocity compounding, blade efficiency, stage efficiency,   optimum blade speed, reaction turbine, degree of reaction, performance analysis, governing of turbines.

VIII. Hydraulic Machines: (5 Lectures)

  • Pelton turbine, Francis turbine, Kaplan turbine, cavitation, draft tube, Specific speed and efficiency.

IX. Pump and Cavitation: (2 Lectures)

  • Centrifugal pump, types of impeller, performance analysis of a centrifugal pump, diffuser, Net Positive Suction Head (NPSH) and cavitation.

Laboratory sessions:


I. Study and performance characteristics of a two-stage axial flow fan.


II. Evaluation of performance characteristics for a multi-stage compressor.


III. Performance characteristics of centrifugal pumps operating in series & parallel mode.


IV. Performance characteristics of an Impulse Turbine for different loads.


V. Performance analysis of a miniaturised Power Plant.


VI. Performance, emission characterization and heat balance of an IC Engine.


VII. COP and Performance analysis of vapour compression system, operating in refrigeration and heat pump mode.

References:

  1. S. L. Dixon and C. A. Hall, Fluid Mechanics and Thermodynamics of Turbomachinery, Elsevier, Sixth Edition, 2010.

  2. H. Cohen, G. F. C. Rogers and H. I. H. Saravanamuttoo, Gas Turbine Theory, Addison Wesley Longman Ltd, Fourth Edition, 1996.

  3. S.M.Yahya, Turbines, Compressor and Fan, Tata McGraw-Hill, Second Edition, 2003.

  4. B.Lakshminarayana, Fluid Dynamics and Heat Transfer of Turbomachinery, John Wiley and Sons, Inc, 1996.
 

ME352

Theory of Mechanisms and Machines

Credits:

  

 

2L-0T-1P-0A (7 Credits)
 

Pre-requisite:


ESO 209.

Course content:


Kinematic pairs, diagrams and inversion.  Mobility and range of movement.
Displacement, velocity and acceleration analysis of planar linkages.  Dimensional synthesis for motion, path and function generation. Dynamic force analysis, flywheels.  Inertia forces and balancing for rotating and reciprocating machines.  Cam mechanisms, Cam profile synthesis. Gears and gear trains.

Lecture-wise break-up:


I. Kinematic pairs, diagrams, Mobility and range of movement: (5 Lectures)


II. Kinematic analysis of planar linkages: (5 Lectures)


III. Dimensional synthesis: (4 Lectures)


IV. Dynamic force analysis, flywheels: (2 Lectures)


V. Inertia forces and balancing: (4 Lectures)


VI. Cam mechanism: (4 Lectures)


VII. Gears and gear trains: (3 Lectures)


Laboratory sessions:


I. Assembly of mechanisms from links and joints, study of mobility.


II. Linkage design and experimental verification.


III. Development of turning moment diagram for an engine.


IV. Balancing of thin discs and/or rotors.


V. Balancing of IC engines.


VI. Analysis of cams.


VII. Study of gears. Design, assembly and operation of gear trains.


Suggested text and reference material:

  1. Theory of Mechanisms and Machines by Ghosh and Mallik (EWP).