R22 M.Tech. ENGG. DESIGN

#### JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD M. TECH. I Year I Sem. (ENGG. DESIGN)

#### MECHANICAL VIBRATIONS (Professional Elective - I)

L	T	P	C
3	0	0	3

Prerequisite: Basic concepts of Physics

###### Course Objectives:

- • understand the fundamentals of Vibration Theory • understand the fundamentals of Vibration Theory
- • be able to mathematically model real-world mechanical vibration problems • be able to mathematically model real-world mechanical vibration problems

Course Outcomes: Outcomes: At the end of the course the students will be able to

- • study the vibrations in machine elements and how to control them. • study the vibrations in machine elements and how to control them.
- • to analyze the mathematical model of linear vibratory system to determine its • to analyze the mathematical model of linear vibratory system to determine its response
- • linear mathematical models of real life engineering systems • linear mathematical models of real life engineering systems
- • vibratory responses of single and multi degree of freedom systems to harmonic, • vibratory responses of single and multi degree of freedom systems to harmonic, periodic and non-periodic excitation

UNIT-I: Free Vibration of Single Degree of Freedom Systems: Introduction, Free Vibration of an Free Vibration of Single Degree of Freedom Systems: Introduction, Free Vibration of an Undamped Translational System, Equation of Motion using Newton's second law of motion, Equation of motion using other methods, Equation of motion of a spring, mass system in vertical position, solution, Harmonic Motion Free Vibration of an Undamped Torsional System- Equation of motion. Free Vibration with Viscous Damping- Equation of motion.

UNIT-II: Forced Vibration of Single Degree of Freedom Systems: Introduction, Response of an Undamped Forced Vibration of Single Degree of Freedom Systems: Introduction, Response of an Undamped system under harmonic force, Total response, Beating Phenomenon. Response of a Damped System under Harmonic Force- Total Response, Quality Factor and Bandwidth, Response of a Damped system under the Harmonic Motion of the base, Fore Transmitted, Relative Motion.

UNIT- III: Two Degree of Freedom Systems: Introduction, Equations of Motion for forced Vibration, Free Two Degree of Freedom Systems: Introduction, Equations of Motion for forced Vibration, Free Vibration Analysis of and undamped system, Torsional system, Coordinate Coupling and Principal Coordinates, forded Vibration Analysis, Semi definite Systems, Self- Excitation and stability Analysis.

UNIT-IV: Multi-degree of Freedom Systems: Introduction Modeling of Continuous systems as Multi-degree of Multi-degree of Freedom Systems: Introduction Modeling of Continuous systems as Multi-degree of Freedom systems, Using Newton's second law to derive equations of motion, Influence Coefficients. Potential and kinetic energy expressions in matrix form, Generalized coordinates and generalized forces, Using Lagrange's equations to derive equations of motion, Equations of motion of undamped systems in matrix form, Eigen value problem, solution of the Eigen value problems - solution of the characteristic equation, orthogonality of normal modes, repeated Eigen values.

UNIT-V: Determination of Natural Frequencies and Mode Shapes : Introduction, Dunkerley's formula, Determination of Natural Frequencies and Mode Shapes : Introduction, Dunkerley's formula, Rayleigh's Method - Properties of Rayleigh's Quotient, Computation of the Fundamental Natural Frequency, Fundamental Frequency of Beams and Shafts. Holzer's Method -Torsional systems, Spring Mass Systems. Jacobi method, Standard Eigen value Problems.