The familiar concept described by the word "vibrations" suggests the rapid alternating motion of a system about and in the neighbourhood of its equilibrium position, under the action of random or deliberate disturbing forces. It falls within the province of mechanics, the science which deals with the laws of equilibrium, and of motion, and their applications to the theory of machines, to calculate these vibrations and predict their effects. While it is certainly true that the physical systems which can be the seat of vibrations are many and varied, it appears that they can be studied by methods which are largely indifferent to the nature of the underlying phenomena. It is to the development of such methods that we devote this book which deals with free or induced vibrations in discrete or continuous mechanical structures. The mathematical analysis of ordinary or partial differential equations describing the way in which the values of mechanical variables change over the course of time allows us to develop various theories, linearised or non-linearised, and very often of an asymptotic nature, which take account of conditions governing the stability of the motion, the effects of resonance, and the mechanism of wave interactions or vibratory modes in non-linear systems.

Table of Contents:
I. Forced Vibrations in Systems Having One Degree or Two Degrees of Freedom.- Elastic Suspension with a Single Degree of Freedom.- Torsional Oscillations.- Natural Oscillations.- Forced Vibrations.- Vibration Transmission Factor.- Elastic Suspension with Two Degrees of Freedom. Vibration Absorber.- Response Curve of an Elastic System with Two Degrees of Freedom.- Vehicle Suspension.- Whirling Motion of a Rotor-Stator System with Clearance Bearings.- Effect of Friction on the Whirling Motion of a Shaft in Rotation; Synchronous Precession, Self-sustained Precession.- Synchronous Motion.- Self-maintained Precession.- II. Vibrations in Lattices.- A Simple Mechanical Model.- The Alternating Lattice Model.- Vibrations in a One-Dimensional Lattice with Interactive Forces Derived from a Potential.- Vibrations in a System of Coupled Pendulums.- Vibrations in Three-Dimensional Lattices.- Non-Linear Problems.- III. Gyroscopic Coupling and Its Applications.- 1. The Gyroscopic Pendulum.- Discussion of the Linearised System.- Appraisal of the Linearisation Process in the Case of Strong Coupling.- Gyroscopic Stabilisation.- 2. Lagrange's Equations and Their Application to Gyroscopic Systems.- Example: The Gyroscopic Pendulum.- 3. Applications.- The Gyrocompass.- Influence of Relative Motion on the Behaviour of the Gyrocompass.- Gyroscopic Stabilisation of the Monorail Car.- 4. Routh's Stability Criterion.- 5. The Tuned Gyroscope as Part of an Inertial System for Measuring the Rate of Turn.- Kinematics of the Multigimbal Suspension.- a) Orientation of the Rotor.- b) Co-ordinates of an Intermediate Gimbal.- c) Relations Between the Parameters ? and ?.- The Equations of Motion.- Inclusion of Damping Terms in the Equations of Motion.- Dynamic Stability. Undamped System.- Frequencies of Vibrations of the Free Rotor.- Motion of the Free Rotor.- Case of a Multigimbal System Without Damping. The Tune Condition.- Examination of the Two-Gimbal System.- IV. Stability of Systems Governed by the Linear Approximation.- Discussion of the Equation Aq? + ??q? = 0.- Discussion of the Equation Aq? + ??q?+ Kq = 0.- Systems Comprising Both Gyroscopic Forces and Dissipative Forces..- 1. Case E = 0.- A Modified Approach in the Case of Instability.- 2. Case E ? 0.- Eigenmodes.- Rayleigh's Method.- Effect on the Eigenvalues of Changes in Structure.- An Example.- V. The Stability of Operation of Non-Conservative Mechanical Systems.- 1. Rolling Motion and Drift Effect.- 2. Yawing of Road Trailers.- 3. Lifting by Air-Cushion.- The Stationary Regime.- Case of an Isentropic Expansion.- Dynamic Stability.- VI. Vibrations of Elastic Solids.- I. Flexible Vibrations of Beams.- 1. Equations of Beam Theory.- 2. A Simple Example.- 3. The Energy Equation.- 4. The Modified Equations of Beam Theory; Timoshenko's Model.- 5. Timoshenko's Discretised Model of the Beam.- 6. Rayleigh's Method.- 6.1. Some Elementary Properties of the Spaces H1 (0, l), H2(0, l)..- 6.2. Existence of the Lowest Eigenfrequency.- 6.3. Case of a Beam Supporting Additional Concentrated Loads.- 6.4 Intermediate Conditions Imposed on the Beam.- 6.5 Investigation of Higher Frequencies.- 7. Examples of Applications.- 7.1. Beam Fixed at x = 0, Free at x = l.- 7.2. Beam Fixed at Both Ends.- 7.3. Beam Free at Both Ends.- 7.4. Beam Hinged at x = 0, Free at x = l.- 7.5. Beam Fixed at x= 0 and Bearing a Point Load at the Other End.- 7.6. Beam Supported at Three Points.- 7.7. Vibration of a Wedge Clamped at x = 0. Ritz's Method.- 7.8. Vibrations of a Supported Pipeline.- 7.9. Effect of Longitudinal Stress on the Flexural Vibrations of a Beam and Application to Blade Vibrations in Turbomachinery.- 7.10. Vibrations of Interactive Systems.- 8. Forced Vibrations of Beams Under Flexure.- 9. The Comparison Method.- 9.1. The Functional Operator Associated with the Model of a Beam Under Flexure.- 9.2. The Min-Max Principle.- 9.3. Application to Comparison Theorems.- 10. Forced Excitation of a Beam.- 10.1. Fourier's Method.- 10.2. Boundary Conditions with Elasticity Terms.- 10.3. Forced Vibrations of a Beam Clamped at One End, Bearing a Point Load at the Other End, and Excited at the Clamped End by an Imposed Transverse Motion of Frequency ?.- II. Longitudinal Vibrations of Bars. Torsional Vibrations.- 1. Equations of the Problem and the Calculation of Eigenvalues.- 2. The Associated Functional Operator.- 3. The Method of Moments.- 3.1. Introduction.- 3.2. Lanczos's Orthogonalisation Method.- 3.3. Eigenvalues of An.- 3.4. Pade's Method.- 3.5. Approximation of the A Operator.- III. Vibrations of Elastic Solids.- 1. Statement of Problem and General Assumptions.- 2. The Energy Theorem.- 3. Free Vibrations of Elastic Solids.- 3.1. Existence of the Lowest Eigenfrequency.- 3.2. Higher Eigenfrequencies.- 3.3. Case Where There Are No Kinematic Conditions.- 3.4. Properties of Eigenmodes and Eigenfrequencies.- 4. Forced Vibrations of Elastic Solids.- 4.1. Excitation by Periodic Forces Acting on Part of the Boundary.- 4.2. Excitation by Periodic Displacements Imposed on Some Part of the Boundary.- 4.3. Excitation by Periodic Volume Forces.- 5. Vibrations of Non-Linear Elastic Media.- IV. Vibrations of Plane Elastic Plates.- 1. Description of Stresses; Equations of Motion.- 2. Potential Energy of a Plate.- 3. Determination of the Law of Behaviour.- 4. Eigenfrequencies and Eigenmodes.- 5. Forced Vibrations.- 6. Eigenfrequencies and Eigenmodes of Vibration of Complex Systems.- 6.1. Free Vibrations of a Plate Supported Elastically over a Part U of Its Area, U Open and ? ? ?.- 6.2. Eigenfrequencies and Eigenmodes of a Rectangular Plate Reinforced by Regularly Spaced Stiffeners.- V. Vibrations in Periodic Media.- 1. Formulation of the Problem and Some Consequences of Korn's Inequality.- 2. Bloch Waves.- VII. Modal Analysis and Vibrations of Structures.- I. Vibrations of Structures.- Free Vibrations.- Forced Vibrations.- Random Excitation of Structures.- II. Vibrations in Suspension Bridges.- The Equilibrium Configuration.- The Flexure Equation Assuming Small Disturbances.- Free Flexural Vibrations in the Absence of Stiffness.- a) Symmetric Modes: ? (x) = ?(? x).- b) Skew-Symmetric Modes: ? (x)= ? ? (? x).- Torsional Vibrations of a Suspension Bridge.- Symmetric Modes.- a) Flexure.- b) Torsion.- Vibrations Induced by Wind.- Aerodynamic Forces Exerted on the Deck of the Bridge.- Discussion Based on a Simplified Model.- A More Realistic Approach.- VIII. Synchronisation Theory.- 1. Non-Linear Interactions in Vibrating Systems.- 2. Non-Linear Oscillations of a System with One Degree of Freedom.- 2.1. Reduction to Standard Form.- 2.2. The Associated Functions.- 2.3. Choice of the Numbers m and N.- 2.4. Case of an Autonomous System.- 3. Synchronisation of a Non-Linear Oscillator Sustained by a Periodic Couple. Response Curve. Stability.- 4. Oscillations Sustained by Friction.- 5. Parametric Excitation of a Non-Linear System.- 6. Subharmonic Synchronisation.- 7. Non-Linear Excitation of Vibrating Systems. Some Model Equations.- 8 On a Class of Strongly Non-Linear Systems.- 8.1. Periodic Regimes and Stability.- 8.2. Van der Pol's Equation with Amplitude Delay Effect.- 9. Non-Linear Coupling Between the Excitation Forces and the Elastic Reactions of the Structure on Which They Are Exerted.- Application to Bouasse and Sarda's Regulator.- 10. Stability of Rotation of a Machine Mounted on an Elastic Base and Driven by a Motor with a Steep Characteristic Curve.- 11. Periodic Differential Equations with Singular Perturbation.- 11.1. Study of a Linear System with Singular Perturbation.- ?(dx/dt) = A(t)x + h(t).- 11.2. The Non-Linear System.- 11.3. Stability of the Periodic Solution.- 12. Application to the Study of the Stability of a Rotating Machine Mounted on an Elastic Suspension and Driven by a Motor with a Steep Characteristic Curve.- 13. Analysis of Stability.- 14. Rotation of an Unbalanced Shaft Sustained by Alternating Vertical Displacements.- 15. Stability of Rotation of the Shaft.- 16. Synchronisation of the Rotation of an Unbalanced Shaft Sustained by Alternating Vertical Forces.- 16.1. The Non-Resonant Case.- 16.2. Analysis of Stability.- 17. Synchronisation of the Rotation of an Unbalanced Shaft Sustained by Alternating Forces in the Case of Resonance.- 17.1. The Modified Standard System.- 17.2. Synchronisation of Non-Linear System.- 17.3. Stability Criterion for Periodic Solution.- 17.4. Application.- IX. Stability of a Column Under Compression - Mathieu's Equation.- Buckling of a Column.- Analysis of Stability.- A Discretised Model of the Loaded Column.- The Discretised Model with Slave Load.- Description of the Asymptotic Nature of the Zones of Instability for the Mathieu Equation.- Normal Form of Infinite Determinant. Analysis of Convergence.- Hill's Equation.- X. The Method of Amplitude Variation and Its Application to Coupled Oscillators.- Posing the Problem.- Cases Where Certain Oscillations Have the Same Frequency 353 Coupled Oscillators; Non-Autonomous System and Resonance. A Modified Approach.- Case of Resonance.- Case Where Certain Eigenmodes Decay (Degeneracy).- Case of Oscillators Coupled Through Linear Terms.- Non-Autonomous Non-Linear System in the General Case; Examination of the Case When Certain Eigenmodes Are Evanescent.- Gyroscopic Stabiliser with Non-Linear Servomechanism.- XI. Rotating Machinery.- I. The Simplified Model with Frictionless Bearings.- Preliminary Study of the Static Bending of a Shaft with Circular Cross-Section.- Steady Motion of a Disc Rotating on a Flexible Shaft.- Flexural Vibrations When Shaft Is in Rotation.- Forced Vibrations.- II. Effects of Flexibility of the Bearings.- Hydrodynamics of Thin Films and Reynold's Equation.- Application to Circular Bearings.- Unsteady Regime.- Gas Lubricated Bearings.- Effects of Bearing Flexibility on the Stability of Rotation of a Disc.- 1. Case of an Isotropic Shaft: $${b_2} = {\tilde b_2},{c_2} = {\tilde c_2}$$.- 2. Case Where Shaft and Bearings Are Both Anisotropic.- Periodic Linear Differential Equation with Reciprocity Property 394 Stability of Rotation of Disc Where the System Has Anisotropic Flexibilities.- An Alternative Approach to the Stability Problem.- Application to the Problem of the Stability of a Rotating Shaft.- III. Stability of Motion of a Rigid Rotor on Flexible Bearings. Gyroscopic Effects and Stability.- Notation and Equations of Motion.- Analysis of Stability in the Isotropic Case.- Calculating the Critical Speeds of the Rotor.- Resonant Instability Near ? = (?1 + ?2)/2.- Instability Near the Resonance ? = ?1.- Ground Resonance of the Helicopter Blade Rotor System.- IV. Whirling Motion of a Shaft in Rotation with Non-Linear Law of Physical Behaviour.- Calculation of Ty, Tz.- The Equations of Motion.- Effect of Hysteresis on Whirling.- Stability of the Regime ? < ?0.- Analysis of the Rotatory Regime When ? > ?0.- V. Suspension of Rotating Machinery in Magnetic Bearings.- Principle of Magnetic Suspension.- Quadratic Functional and Optimal Control.- Application to the Model with One Degree of Freedom.- Characteristics and Applications of Magnetic Bearings.- XII. Non-Linear Waves and Solitons.- 1. Waves in Dispersive or Dissipative Media.- The Non-Linear Perturbation Equations.- An Example: Gravity Waves in Shallow Water.- 2. The Inverse Scattering Method.- The Method of Solution.- 3. The Direct Problem.- 3.1. The Eigenvalue Problem.- On Some Estimates.- The Finiteness of the Set of Eigenvalues.- 3.2. Transmission and Reflection Coefficients.- Eigenvalues (Continued).- 4. The Inverse Problem.- The Kernel K(x,y) (Continued).- The Gelfand-Levitan Integral Equation.- An Alternative Definition of the Kernel K(x,y).- Solving Gelfand-Levitan's Equation.- 5. The Inverse Scattering Method.- The Evolution Equation.- Integral Invariants.- Another Approach to the Evolution Equation.- 6. Solution of the Inverse Problem in the Case Where the Reflection Coefficient is Zero.- 7. The Korteweg-de Vries Equation. Interaction of Solitary Waves...- Investigation of Asymptotic Behaviour for t ? + ?.- Asymptotic Behaviour for t ? ? ?.- References.