Syllabus (MATHEMATICS)

Course Type: MAJ-17

Semester: 8

Course Code: BMTMMAJ17T

Course Title: Advanced Mechanics

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: Upon successful completion, the students will be able to CO1: determine resultant of various force systems. CO2: determine centroid, moment of inertia and solve problems related to t

Syllabus:

Unit -1: Dynamics of Rigid Bodies [Credit: 2, 30L]

• Moments and Products of inertia (in three-dimensional rectangular co-ordinates), Inertia matrix, Principal values and principal axes of inertia matrix, Principal moments and principal axes of inertia for (i) a rod, (ii) a rectangular plate, (iii) a circular plate, (iv) an elliptic plate, (v) a sphere, (vi) a right circular cone, (vii) a rectangular parallelepiped and (viii) a circular cylinder, Momental Ellipsoid.
• Equation of motion of a rigid body about a fixed axis. Expression for kinetic energy and moment of momentum of a rigid body moving about fixed axis.
• Equations of motion of a rigid body moving in two-dimension, Expression for kinetic energy and angular momentum about the origin of rigid body moving in two dimensions, Necessary and sufficient condition for pure rolling, Two-dimensional motion of a solid of revolution moving on a rough horizontal plane, the following examples of the two-dimensional motion of a rigid body to be studied:

(i) Motion of a uniform heavy sphere (solid and hollow) along a perfectly rough inclined plane;

(ii) Motion of a uniform heavy circular cylinder (solid and hollow) along a perfectly rough inclined plane:

(iii) Motion of a rod when released from a vertical position with one end resting upon a perfectly rough table or smooth table.

(iv) Motion of a uniform heavy solid sphere along an imperfectly rough inclined plane;

(v) Motion of a uniform circular disc, projected with its plane vertical along an imperfectly rough horizontal plane with a velocity of translation and angular velocity about the centre.

Unit-II: Statics [Credit: 1, 15L]

• Forces in three dimensions: Forces, concurrent forces, Parallel forces, Moment of a force, Couple, Resultant of a force and a couple (Fundamental concept only), Reduction of forces in three-dimensions, Pointsot’s central axis, conditions of single resultant, conditions of equilibrium.
• Virtual work: Principle of Virtual work, Deduction of the conditions of equilibrium of a particle under coplanar forces from the principle of virtual work, Simple examples of finding tension or thrust in a two-dimensional structure in equilibrium by the principle of virtual work.
• Stable and unstable equilibrium, Coordinates of a body and of a system of bodies, Field of forces, Conservative field, Potential energy of a system, Dirichlet’s Energy test of stability, stability of a heavy body resting on a fixed body with smooth surfaces- simple examples.

Unit -3: Elements of Continuum Mechanics & Hydrostatics [Credit: 2, 30L]

• Deformable body, Idea of a continuum (continuous medium), Surface forces or contact forces, Stress at point in a continuous medium, stress vector, components of stress (normal stress and shear stress) in rectangular Cartesian co-ordinate system; stress matrix.
• Definition of ideal fluid and viscous fluid.
• Pressure (pressure at a point in a fluid in equilibrium is same in every direction), Incompressible and compressible fluid, Homogeneous and non-homogeneous fluids.
• Equilibrium of fluids in a given field of force; pressure gradient, Equipressure surfaces.
• Pressure in a heavy homogeneous liquid, Thrust on plane surfaces, Center of pressure, effect of increasing the depth without rotation, Centre of pressure of a triangular & rectangular area and of a circular area immersed in any manner in a heavy homogeneous liquid, Simple problems.
• Thrust on curved surfaces: Archemedes’ principle, Equilibrium of freely floating bodies under constraints. (Consideration of stability not required).
• Gas : Equation of state of a ‘perfect gas’, Isothermal and adiabatic processes in an isothermal atmosphere, Pressure and temperature in atmosphere in convective equilibrium

Reading References:

1. I.H. Shames and G. Krishna Mohan Rao, Engineering Mechanics: Statics and Dynamics, (4th Ed.), Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi, 2009.
2. R.C. Hibbeler and Ashok Gupta, Engineering Mechanics: Statics and Dynamics, 11th Ed., Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi.
3. Chorlton, F., Textbook of Dynamics.
4. Loney, S. L., An Elementary Treatise on the Dynamics of particle and of Rigid Bodies
5. Loney, S. L., Elements of Statics and Dynamics I and II.
6. Ramsey, A. S., Dynamics (Part I).
7. I.H. Shames and G. Krishna Mohan Rao, Engineering Mechanics: Statics and Dynamics, (4th Ed.), Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi, 2009.
8. R.C. Hibbeler and Ashok Gupta, Engineering Mechanics: Statics and Dynamics, 11th Ed., Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi.
9. Chorlton, F., Textbook of Dynamics.
10. Loney, S. L., An Elementary Treatise on the Dynamics of particle and of Rigid Bodies
11. Loney, S. L., Elements of Statics and Dynamics I and II.
12. Ghosh, M. C, Analytical Statics.
13. G. C. Gorain, Mathematical Hydrostatics, New Central Book Agency, Kolkata
14. Verma, R. S., A Textbook on Statics, Pothishala, 1962.
15. Matiur Rahman, Md., Statics.
16. Ramsey, A. S., Dynamics (Part I).
17. Chatterjee, R.N.-Elements of Continuum Mechanics.
18. Love, A.E.H.-Conitnuum Mechanics.