Mechanical Vibrations

Citation

Rao, S. S. (2018). Mechanical Vibrations (6th ed.). Pearson Education India.

Chapter Summary

The sixth edition has fourteen chapters and six appendices. In the preface the author describes the book as an introduction to vibration engineering at the undergraduate level, presenting theory, computational aspects and applications as simply as possible, with computer techniques emphasised. The reader is assumed to know basic statics, dynamics, strength of materials and differential equations; matrix theory and Laplace transforms are desirable, and appendices review them. In the Global Edition the print book contains Chapters 1–12, with Chapters 13 and 14 provided as downloadable files on the companion website.

Chapter 1: Fundamentals of Vibration

A short history of vibration, why vibration matters (including how the human ear converts vibration to sound), basic concepts and classification, the vibration analysis procedure, and the three elementary parts of a vibrating system: spring elements, mass or inertia elements and damping elements. The chapter ends with harmonic motion and harmonic analysis using Fourier series, with a first set of MATLAB examples.

Chapter 2: Free Vibration of Single-Degree-of-Freedom Systems

Free vibration of undamped translational and torsional systems, response of first-order systems and the time constant, Rayleigh's energy method, free vibration with viscous, Coulomb and hysteretic damping, logarithmic decrement, graphical representation of characteristic roots, root-locus representation of parameter variation, and stability of systems.

Chapter 3: Harmonically Excited Vibration

Response of undamped and damped systems to harmonic force, beating, quality factor and bandwidth, harmonic motion of the base with force transmitted and relative motion, rotating unbalance, forced vibration with Coulomb, hysteretic and other damping, self-excitation and stability analysis, and the transfer-function, Laplace-transform and frequency transfer-function approaches.

Chapter 4: Vibration Under General Forcing Conditions

Response under periodic and nonperiodic forces, the convolution integral, response spectrum (including earthquake response spectra and shock design), Laplace transforms with the description of transient response, and numerical methods including Runge–Kutta.

Chapter 5: Two-Degree-of-Freedom Systems

Equations of motion, free-vibration analysis of undamped and torsional systems, coordinate coupling and principal coordinates, forced-vibration analysis, semidefinite systems, self-excitation, and transfer-function, Laplace and frequency-response solutions.

Chapter 6: Multidegree-of-Freedom Systems

Modelling continuous systems as multidegree-of-freedom systems, influence coefficients, energy expressions in matrix form, Lagrange's equations, the eigenvalue problem and orthogonality of normal modes, the expansion theorem, unrestrained systems, and modal analysis of forced vibration of undamped and viscously damped systems.

Chapter 7: Determination of Natural Frequencies and Mode Shapes

Numerical methods for natural frequencies and modes: Dunkerley's formula, Rayleigh's method, Holzer's method, matrix iteration, Jacobi's method and the standard eigenvalue problem.

Chapter 8: Continuous Systems

Transverse vibration of strings, longitudinal vibration of bars, torsional vibration of shafts, lateral vibration of beams (including axial force, rotary inertia, shear deformation and beams on elastic foundation), vibration of membranes, and the Rayleigh and Rayleigh–Ritz methods.

Chapter 9: Vibration Control

The vibration nomograph and criteria, reduction at the source, balancing of rotating machines and reciprocating engines, whirling of shafts, control of natural frequencies, introduction of damping, vibration isolation (including shock isolation and active control) and vibration absorbers.

Chapter 10: Vibration Measurement and Applications

Transducers, vibration pickups, frequency-measuring instruments, vibration exciters, signal analysis, dynamic testing of machines and structures, experimental modal analysis, and machine-condition monitoring and diagnosis.

Chapters 11–12: Numerical Integration Methods in Vibration Analysis; Finite Element Method

Chapter 11 covers finite difference, central difference, Runge–Kutta, Houbolt, Wilson and Newmark methods for single- and multidegree-of-freedom and continuous systems. Chapter 12 introduces finite elements for bars, torsion elements and beams, element transformation, assembly, boundary conditions and consistent and lumped mass matrices.

Chapters 13–14: Nonlinear Vibration; Random Vibration

Nonlinear vibration covers exact and approximate analytical methods, subharmonic and superharmonic oscillations, the Mathieu equation, phase-plane methods, stability, limit cycles and chaos. Random vibration covers random processes, correlation functions, stationarity, power spectral density and the response of single- and multidegree-of-freedom systems.

Appendices

Mathematical relations and material properties, deflection of beams and plates, matrices, Laplace transform, units and an introduction to MATLAB.

Key Concepts

Chapters 1–5: Single and two degrees of freedom

• Natural frequency, viscous damping and logarithmic decrement: the parameters that characterise free vibration.
• Transmissibility and rotating unbalance: forced-response phenomena for harmonic excitation.
• Convolution integral and response spectrum: the response to general forcing.
• Principal coordinates and coordinate coupling: how two-degree-of-freedom systems decouple.

Chapters 6–8: Many degrees of freedom and continua

• Eigenvalue problem and modal analysis: natural frequencies and modes from the equations of motion in matrix form.
• Rayleigh quotient and related methods: approximate natural frequencies.
• Separation of variables for continuous systems: strings, bars, shafts, beams and membranes.

Chapters 9–14: Applications and extensions

• Vibration isolation and absorbers: control of vibration transmitted from or to a machine.
• Experimental modal analysis: determining modes from measurements.
• Finite element method for vibration problems.
• Nonlinear and random vibration: limit cycles and chaos; power spectral density.

Critical Analysis

Strengths

• Computation is built in. The preface emphasises computational aspects, with MATLAB-based examples in the last section of every chapter and 55 MATLAB programs in total.
• A large amount of practice material. The preface counts 252 illustrative examples, 988 review questions, 1,214 problems and 34 design-project problems.
• Topics grouped by purpose. The author deliberately collects all vibration control topics in Chapter 9, measurement in Chapter 10 and numerical integration in Chapter 11, rather than spreading isolators, absorbers and balancing across chapters.
• Wide coverage with a flexible syllabus. The preface suggests Chapters 1–5, Chapter 9 and parts of Chapter 6 for a basic course, with Chapters 7 and 11, 8, 10, 12, 13 or 14 added for numerical, continuous-system, experimental, finite element, nonlinear or random emphases, or the whole book for a two-course sequence.
• Historical touches. Short biographical notes on 22 scientists and engineers appear on the opening pages of chapters and appendices.

Limitations

• A large book. The fourteen chapters and about 1,100 printed pages cannot be covered in one course, and a reader has to choose a route.
• Parts online. In the Global Edition the last two chapters, on nonlinear and random vibration, are available only as downloads on the companion website, which also holds MATLAB programs and answers.
• Prior knowledge needed. The preface assumes statics, dynamics, strength of materials and differential equations; matrix theory and Laplace transforms are only reviewed in appendices.
• Edition differences. The sixth edition has a US version and a Global Edition in SI units, with the US text dated 2017 and the Global Edition 2018, so chapter and problem numbers should be matched against the version a course uses.

Where It Fits in the Course

In this statics and dynamics course, Rao is the specialised vibration text. Hibbeler's Engineering Mechanics: Statics and Meriam and Kraige's Engineering Mechanics: Dynamics supply the mechanics Rao assumes: force and moment balance, and the motion of particles and rigid bodies using energy and momentum. Williams' Fundamentals of Applied Dynamics also covers vibration of lumped-parameter and continuous systems (in its last two chapters), but from the viewpoint of formulating equations of motion; Rao is a more detailed vibration treatment with measurement, control and numerical methods.

Bird and Ross's Mechanical Engineering Principles and the OpenStax University Physics Volume 1 are gentler foundations for readers who need to build up mechanics first. Readers heading toward buildings and earthquakes can continue with Chopra's Dynamics of Structures, which begins with single-degree-of-freedom systems and builds up to multi-degree-of-freedom and continuous systems.

How to Use This Book

The book is for mechanical, aerospace and civil engineering students who have studied dynamics, differential equations and some linear algebra, and for practising engineers who need a reference on vibration.

A practical plan for a first course:

• Core: Chapters 1–5, then the multidegree-of-freedom theory of Chapter 6 as far as the eigenvalue problem and modal analysis.
• Next: Chapter 9 on vibration control, with Chapter 3 first for transmissibility and isolation.
• By emphasis: Chapters 7 and 11 for numerical solution, Chapter 8 for continuous systems, Chapter 10 for measurement, Chapter 12 for finite elements, and Chapters 13 and 14 for nonlinear and random vibration.

Work through the MATLAB examples at the end of each chapter and use the review questions to check understanding before attempting the problems. Review the matrix and Laplace transform appendices before Chapters 4 and 6.