Vibrations in Mechanical Systems

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. Padé'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.