Mechanical Vibrations, 2nd Edition
Table of Contents
1. Fundamentals of Vibration.
Preliminary Remarks.
Brief History of Vibration.
Origins of vibration.
From Galileo to Rayleigh.
Recent contributions.
Importance of the Study of Vibration.
Basic Concepts of Vibration.
Vibration. Elementary parts of vibrating systems.
Degree of freedom.
Discrete and continuous systems.
Classification of Vibration.
Free and forced vibration.
Undamped and damped vibration.
Linear and nonlinear vibration.
Deterministic and random vibration.
Vibration Analysis Procedure.
Spring Elements.
Combination of springs.
Mass or Inertia Elements.
Combination of masses.
Damping Elements.
Construction of viscous dampers.
Complex number representation of harmonic motion.
Complex algebra.
Harmonic Motion.
Vectorial representation of harmonic motion.
Complex number representation of harmonic motion.
Complex algebra.
Operations on harmonic functions.
Definitions and terminology.
Harmonic Analysis.
Fourier serious expansion.
Complex Fourier series.
Frequency spectrum.
Time and frequency domain representations.
Even and odd functions.
Half range expansions.
Numerical computation of coefficients.
2. 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. Solution.
Stability Conditions.
Rayleigh's Energy Method.
Free Vibration with Viscous Damping.
Equation of motion. Solution.
Logarithmic decrement.
Energy dissipated in viscous damping.
Torsional systems with viscous damping.
Free Vibration with Coulomb Damping.
Equation of motion.
Solution.
Torsional systems with Coulomb damping.
Free Vibration with Hysteretic Damping.
3. Harmonically Excited Vibration.
Introduction.
Equation of Motion.
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 F(t) = F.
Response of a Damped System Under the Harmonic.
Motion of the Base.
Force transmitted.
Relative motion.
Response of a Damped System Under Rotating.
Unbalance.
Forced Vibration with Coulomb Damping.
Forced Vibration with Hysteresis Damping.
Forced Motion with Other Types of Damping.
Self-Excitation and Stability Analysis.
Dynamic stability analysis.
Dynamic instability caused by fluid flow.
4. Vibration Under General Forcing Conditions.
Introduction. Response Under a General Periodic Force.
Response Under a Periodic Force of Irregular Form.
Response Under a Nonperiodic Force.
Convolution Integral.
Response to an impulse.
Response to general forcing condition.
Response to base excitation.
Response Spectrum.
Response spectrum for base excitation.
Earthquake response spectra.
Design under shock environment.
Laplace Transformation.
Response to Irregular Forcing Conditions Using.
Numerical Methods. Computer Programs.
Response under an arbitrary periodic forcing function.
Response under arbitrary forcing function using the methods of section.
5. Two Degree of Freedom Systems.
Introduction.
Equations of Motion for Forced Vibration.
Free Vibration Analysis of an Undamped System.
Torsional System.
Coordinate Coupling and Principal Coordinates.
Forced Vibration Analysis.
Semidefinite Systems.
Self-Excitation and Stability Analysis.
6. Multidegree of Freedom Systems.
Introduction.
Modeling of Continuous Systems as Multidegree of Freedom Systems.
Using Newton's Second Law to Derive Equations of Motion.
Influence Coefficients. Stiffness influence coefficients.
Flexibility influence coefficients.
Inertia 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.
Eigenvalue Problem. Solution of the Eigenvalue Problem.
Solution of the characteristic (polynomial) equation.
Orthogonality of normal modes.
Repeated eigenvalues.
Expansion Theorem.
Unrestrained Systems.
Free Vibration of Undamped Systems.
Forced Vibration of Undamped Systems.
Forced Vibration of Viscously Damped Systems.
Self-Excitation and Stability Analysis.
Computer Programs.
Generating the characteristic polynomial from the matrix.
Roots of an nth order polynomial equation with complex coefficients.
Modal analysis of a multidegree of freedom system.
Solution of Simultaneous linear equations.
7. 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.
Matrix Iteration Method.
Convergence to the highest natural frequency.
Computation of intermediate natural frequencies.
Jacobi's Method. Standard Eigenvalue Problem.
Choleski decomposition.
Other solution methods.
Computer Programs.
Jacobi's method.
Matrix iteration method.
Choleski decomposition.
Eigenvalue solution using Choleski decomposition.
8. Continuous Systems.
Introduction.
Transverse Vibration of a String or Cable.
Equation of motion.
Initial and boundary conditions.
Free vibration of a uniform string.
Free vibration of a string with both ends fixed.
Traveling-wave solution.
Longitudinal Vibration of a Bar or Rod.
Equation of motion and solution.
Orthogonality of normal functions.
Torsional Vibration of a Shaft or Rod.
Lateral Vibration of Beams.
Equation of motion.
Initial conditions.
Free vibration.
Boundary conditions.
Orthogonality of normal functions.
Forced vibration.
Effect of axial force.
Effects of rotary inertia and shear deformation. Other effects.
Vibration of Membranes.
Equation of motion.
Initial and boundary conditions.
Rayleigh's Method.
The Rayleigh-Ritz Method.
9. Vibration Control.
Introduction.
Reduction of Vibration at the Source.
Balancing of Rotating Machines.
Single-plane balancing.
Two-plane balancing.
Whirling of Rotating Shafts.
Equations of motion.
Critical speeds.
Response of the system.
Stability analysis.
Balancing of Reciprocating Engines.
Unbalanced forces due to fluctuations in gas pressure.
Unbalanced forces due to inertia of the moving parts.
Balancing of reciprocating engines.
Control of Vibration.
Control of Natural Frequencies.
Introduction of Damping.
Vibration Isolation.
Vibration isolation system with rigid foundation.
Vibration isolation system with flexible foundation.
Vibration isolation system with partially flexible foundation.
Shock isolation.
Active vibration control.
Vibration Absorbers.
Undamped dynamic vibration absorber.
Damped dynamic vibration absorber.
Computer Program
10. Vibration Measurement and Applications.
Introduction.
Transducers.
Variable resistance transducers.
Peizoelectric transducers.
Electrodynamic transducers.
Linear variable differential transformer (LVDT) transducer.
Vibration pickups.
Vibrometer.
Accelerometer.
Velometer.
Phase distortion.
Frequency Measuring Instruments.
Vibration Exciters.
Mechanical exciters.
Electrodynamic shaker.
Signal Analysis.
Spectrum analyzers.
Bandpass filter.
Constant percent bandwidth and constant bandwidth analyzers.
Dynamic Testing of Machines and Structures.
Using operational deflection shape measurements.
Using modal testing.
Experimental Modal Analysis.
Representation of the frequency response of a system.
Testing and analysis.
Test preparation and setup.
Measurement of frequency response functions.
Identification of modal parameters.
Computer-aided modal testing.
Machine condition monitoring and diagnosis.
Vibration severity criteria.
Machine maintenance techniques.
Machine condition monitoring techniques.
Vibration monitoring techniques.
Instrumentation systems.
Choice of monitoring parameter.
11. Numerical Integration Methods in Vibration Analysis.
Introduction.
Finite Difference Method.
Central Difference Method for Single Degree of Freedom Systems.
Runge-Kutta Method for Single Degree of Freedom Systems.
Central Difference Method for Multidegree of Freedom Systems.
Finite Difference Method for Continuous Systems.
Longitudinal vibration of bars.
Transverse vibration of beams.
Runge-Kutta Method for Multidegree of Freedom Systems.
Houbold Method. Wilson Method.
Newmark Method.
Computer Programs.
Fourth order Runge-Kutta method.
Central difference method. Houbold method.
12. Finite Element Method.
Introduction.
Equations of Motion of an Element. Mass Matrix, Stiffness Matrix, and Force Vector.
Bar element.
Torsion element.
Beam element.
Transformation of Element Matrices and Vectors.
Equations of Motion of the Complete System of Finite Elements.
Incorporation of Boundary Conditions.
Consistent and Lumped Mass Matrices.
Lumped mass matrix for a bar element.
Lumped mass matrix for a beam element.
Lumped mass versus consistent mass matrices.
13. Nonlinear Vibration.
Introduction.
Examples of Nonlinear Vibration Problems.
Simple pendulum.
Mechanical chatter, belt friction system.
Variable mass system.
Exact Methods.
Approximate Analytical Methods.
Basic philosophy.
Lindstedt's peturbation method.
Iterative method.
Ritz-Galerkin method.
Subharmonic and Superharmonic Oscillations.
Subharmonic oscillations.
Superharmonic solution.
Systems with Time-Dependent Coefficients (Mathieu Equation).
Graphical Methods.
Phase Plane Representation.
Phase velocity.
Method of constructing trajectories.
Obtaining time solution from phase plane trajectories.
Stability of Equilibrium States.
Stability analysis.
Classification of singular points.
Limit Cycles. Chaos.
Functions with stable orbits.
Functions with unstable orbits.
Chaotic behavior of Duffing's equation without the forcing term.
Chaotic behavior of Duffing's equation with the forcing term.
Numerical Methods.
14. Random Vibration.
Introduction.
Random Variables and Random Processes.
Probability Distribution.
Mean Value and Standard Deviation.
Joint Probability Distribution of Several Random Variables.
Correlation Functions of a Random Process.
Stationary Random Process.
Gaussian Random Process.
Fourier Analysis. Fourier series.
Fourier integral.
Power Spectral Density.
Wide-Band and Narrow-Band Processes.
Response of a Single Degree of Freedom System.
Impulse response approach.
Frequency response approach.
Characteristics of the response function.
Response Due to Stationary Random Excitations.
Impulse response approach.
Frequency response approach.
Response of a Multidegree of Freedom System
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Table of Contents
1. Fundamentals of Vibration.
Preliminary Remarks.
Brief History of Vibration.
Origins of vibration.
From Galileo to Rayleigh.
Recent contributions.
Importance of the Study of Vibration.
Basic Concepts of Vibration.
Vibration. Elementary parts of vibrating systems.
Degree of freedom.
Discrete and continuous systems.
Classification of Vibration.
Free and forced vibration.
Undamped and damped vibration.
Linear and nonlinear vibration.
Deterministic and random vibration.
Vibration Analysis Procedure.
Spring Elements.
Combination of springs.
Mass or Inertia Elements.
Combination of masses.
Damping Elements.
Construction of viscous dampers.
Complex number representation of harmonic motion.
Complex algebra.
Harmonic Motion.
Vectorial representation of harmonic motion.
Complex number representation of harmonic motion.
Complex algebra.
Operations on harmonic functions.
Definitions and terminology.
Harmonic Analysis.
Fourier serious expansion.
Complex Fourier series.
Frequency spectrum.
Time and frequency domain representations.
Even and odd functions.
Half range expansions.
Numerical computation of coefficients.
2. 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. Solution.
Stability Conditions.
Rayleigh's Energy Method.
Free Vibration with Viscous Damping.
Equation of motion. Solution.
Logarithmic decrement.
Energy dissipated in viscous damping.
Torsional systems with viscous damping.
Free Vibration with Coulomb Damping.
Equation of motion.
Solution.
Torsional systems with Coulomb damping.
Free Vibration with Hysteretic Damping.
3. Harmonically Excited Vibration.
Introduction.
Equation of Motion.
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 F(t) = F.
Response of a Damped System Under the Harmonic.
Motion of the Base.
Force transmitted.
Relative motion.
Response of a Damped System Under Rotating.
Unbalance.
Forced Vibration with Coulomb Damping.
Forced Vibration with Hysteresis Damping.
Forced Motion with Other Types of Damping.
Self-Excitation and Stability Analysis.
Dynamic stability analysis.
Dynamic instability caused by fluid flow.
4. Vibration Under General Forcing Conditions.
Introduction. Response Under a General Periodic Force.
Response Under a Periodic Force of Irregular Form.
Response Under a Nonperiodic Force.
Convolution Integral.
Response to an impulse.
Response to general forcing condition.
Response to base excitation.
Response Spectrum.
Response spectrum for base excitation.
Earthquake response spectra.
Design under shock environment.
Laplace Transformation.
Response to Irregular Forcing Conditions Using.
Numerical Methods. Computer Programs.
Response under an arbitrary periodic forcing function.
Response under arbitrary forcing function using the methods of section.
5. Two Degree of Freedom Systems.
Introduction.
Equations of Motion for Forced Vibration.
Free Vibration Analysis of an Undamped System.
Torsional System.
Coordinate Coupling and Principal Coordinates.
Forced Vibration Analysis.
Semidefinite Systems.
Self-Excitation and Stability Analysis.
6. Multidegree of Freedom Systems.
Introduction.
Modeling of Continuous Systems as Multidegree of Freedom Systems.
Using Newton's Second Law to Derive Equations of Motion.
Influence Coefficients. Stiffness influence coefficients.
Flexibility influence coefficients.
Inertia 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.
Eigenvalue Problem. Solution of the Eigenvalue Problem.
Solution of the characteristic (polynomial) equation.
Orthogonality of normal modes.
Repeated eigenvalues.
Expansion Theorem.
Unrestrained Systems.
Free Vibration of Undamped Systems.
Forced Vibration of Undamped Systems.
Forced Vibration of Viscously Damped Systems.
Self-Excitation and Stability Analysis.
Computer Programs.
Generating the characteristic polynomial from the matrix.
Roots of an nth order polynomial equation with complex coefficients.
Modal analysis of a multidegree of freedom system.
Solution of Simultaneous linear equations.
7. 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.
Matrix Iteration Method.
Convergence to the highest natural frequency.
Computation of intermediate natural frequencies.
Jacobi's Method. Standard Eigenvalue Problem.
Choleski decomposition.
Other solution methods.
Computer Programs.
Jacobi's method.
Matrix iteration method.
Choleski decomposition.
Eigenvalue solution using Choleski decomposition.
8. Continuous Systems.
Introduction.
Transverse Vibration of a String or Cable.
Equation of motion.
Initial and boundary conditions.
Free vibration of a uniform string.
Free vibration of a string with both ends fixed.
Traveling-wave solution.
Longitudinal Vibration of a Bar or Rod.
Equation of motion and solution.
Orthogonality of normal functions.
Torsional Vibration of a Shaft or Rod.
Lateral Vibration of Beams.
Equation of motion.
Initial conditions.
Free vibration.
Boundary conditions.
Orthogonality of normal functions.
Forced vibration.
Effect of axial force.
Effects of rotary inertia and shear deformation. Other effects.
Vibration of Membranes.
Equation of motion.
Initial and boundary conditions.
Rayleigh's Method.
The Rayleigh-Ritz Method.
9. Vibration Control.
Introduction.
Reduction of Vibration at the Source.
Balancing of Rotating Machines.
Single-plane balancing.
Two-plane balancing.
Whirling of Rotating Shafts.
Equations of motion.
Critical speeds.
Response of the system.
Stability analysis.
Balancing of Reciprocating Engines.
Unbalanced forces due to fluctuations in gas pressure.
Unbalanced forces due to inertia of the moving parts.
Balancing of reciprocating engines.
Control of Vibration.
Control of Natural Frequencies.
Introduction of Damping.
Vibration Isolation.
Vibration isolation system with rigid foundation.
Vibration isolation system with flexible foundation.
Vibration isolation system with partially flexible foundation.
Shock isolation.
Active vibration control.
Vibration Absorbers.
Undamped dynamic vibration absorber.
Damped dynamic vibration absorber.
Computer Program
10. Vibration Measurement and Applications.
Introduction.
Transducers.
Variable resistance transducers.
Peizoelectric transducers.
Electrodynamic transducers.
Linear variable differential transformer (LVDT) transducer.
Vibration pickups.
Vibrometer.
Accelerometer.
Velometer.
Phase distortion.
Frequency Measuring Instruments.
Vibration Exciters.
Mechanical exciters.
Electrodynamic shaker.
Signal Analysis.
Spectrum analyzers.
Bandpass filter.
Constant percent bandwidth and constant bandwidth analyzers.
Dynamic Testing of Machines and Structures.
Using operational deflection shape measurements.
Using modal testing.
Experimental Modal Analysis.
Representation of the frequency response of a system.
Testing and analysis.
Test preparation and setup.
Measurement of frequency response functions.
Identification of modal parameters.
Computer-aided modal testing.
Machine condition monitoring and diagnosis.
Vibration severity criteria.
Machine maintenance techniques.
Machine condition monitoring techniques.
Vibration monitoring techniques.
Instrumentation systems.
Choice of monitoring parameter.
11. Numerical Integration Methods in Vibration Analysis.
Introduction.
Finite Difference Method.
Central Difference Method for Single Degree of Freedom Systems.
Runge-Kutta Method for Single Degree of Freedom Systems.
Central Difference Method for Multidegree of Freedom Systems.
Finite Difference Method for Continuous Systems.
Longitudinal vibration of bars.
Transverse vibration of beams.
Runge-Kutta Method for Multidegree of Freedom Systems.
Houbold Method. Wilson Method.
Newmark Method.
Computer Programs.
Fourth order Runge-Kutta method.
Central difference method. Houbold method.
12. Finite Element Method.
Introduction.
Equations of Motion of an Element. Mass Matrix, Stiffness Matrix, and Force Vector.
Bar element.
Torsion element.
Beam element.
Transformation of Element Matrices and Vectors.
Equations of Motion of the Complete System of Finite Elements.
Incorporation of Boundary Conditions.
Consistent and Lumped Mass Matrices.
Lumped mass matrix for a bar element.
Lumped mass matrix for a beam element.
Lumped mass versus consistent mass matrices.
13. Nonlinear Vibration.
Introduction.
Examples of Nonlinear Vibration Problems.
Simple pendulum.
Mechanical chatter, belt friction system.
Variable mass system.
Exact Methods.
Approximate Analytical Methods.
Basic philosophy.
Lindstedt's peturbation method.
Iterative method.
Ritz-Galerkin method.
Subharmonic and Superharmonic Oscillations.
Subharmonic oscillations.
Superharmonic solution.
Systems with Time-Dependent Coefficients (Mathieu Equation).
Graphical Methods.
Phase Plane Representation.
Phase velocity.
Method of constructing trajectories.
Obtaining time solution from phase plane trajectories.
Stability of Equilibrium States.
Stability analysis.
Classification of singular points.
Limit Cycles. Chaos.
Functions with stable orbits.
Functions with unstable orbits.
Chaotic behavior of Duffing's equation without the forcing term.
Chaotic behavior of Duffing's equation with the forcing term.
Numerical Methods.
14. Random Vibration.
Introduction.
Random Variables and Random Processes.
Probability Distribution.
Mean Value and Standard Deviation.
Joint Probability Distribution of Several Random Variables.
Correlation Functions of a Random Process.
Stationary Random Process.
Gaussian Random Process.
Fourier Analysis. Fourier series.
Fourier integral.
Power Spectral Density.
Wide-Band and Narrow-Band Processes.
Response of a Single Degree of Freedom System.
Impulse response approach.
Frequency response approach.
Characteristics of the response function.
Response Due to Stationary Random Excitations.
Impulse response approach.
Frequency response approach.
Response of a Multidegree of Freedom System
link
http://ifile.it/2wf064o/mechanical_vibrations_-_s.s.rao-2ed.djvu