MODELLING OF MECHANICAL SYSTEMS VOLUME 2 -- Contents -- Preface -- Introduction -- Chapter 1. Solid mechanics -- 1.1.Introduction -- 1.2. Equilibrium equations of a continuum -- 1.2.1. Displacements and strains -- 1.2.2. Indicial and symbolic notations -- 1.2.3. Stresses -- 1.2.4. Equations of dynamical equilibrium -- 1.2.5. Stress-strain relationships for an isotropic elastic material -- 1.2.6. Equations of elastic vibrations (Navier 's equations) -- 1.3. Hamilton's principle -- 1.3.1. General presentation of the formalism -- 1.3.2. Application to a three-dimensional solid -- 1.3.2.1. Hamilton 's principle -- 1.3.2.2. Hilbert functional vector space. -- 1.3.2.3. Variation of the kinetic energy -- 1.3.2.4. Variation of the strain energy. -- 1.3.2.5. Variation of the external load work -- 1.3.2.6. Equilibrium equations and boundary conditions -- 1.3.2.7. Stress tensor and Lagrange 's multipliers -- 1.3.2.8. Variation of the elastic strain energy -- 1.3.2.9. Equation of elastic vibrations -- 1.3.2.10. Conservation of mechanical energy -- 1.3.2.11. Uniqueness of solution of motion equations -- 1.4. Elastic waves in three-dimensional media -- 1.4.1. Material oscillations in a continuous medium interpreted as waves -- 1.4.2. Harmonic solutions of Navier 's equations -- 1.4.3. Dilatation and shear elastic waves -- 1.4.3.1. Irrotational,or potential motion -- 1.4.3.2. Equivoluminal,or shear motion. -- 1.4.3.3. Irrotational harmonic waves (dilatation or pressure waves) -- 1.4.3.4. Shear waves (equivoluminal or rotational waves) -- 1.4.4. Phase and group velocities . -- 1.4.5. Wave reflection at the boundary of a semi-infinite medium -- 1.4.5.1. Complex amplitude of harmonic and plane waves at oblique incidence -- 1.4.5.2. Reflection of (SH)waves on a free boundary -- 1.4.5.3. Reflection of (P)waves on a free boundary -- 1.4.6. Guided waves.

1.4.6.1. Guided (SH)waves in a plane layer -- 1.4.6.2. Physical interpretation -- 1.4.6.3. Waves in an infinite elastic rod of circular cross-section -- 1.4.7. Standing waves and natural modes of vibration -- 1.4.7.1. Dilatation plane modes of vibration -- 1.4.7.2. Dilatation modes of vibration in three dimensions -- 1.4.7.3. Shear plane modes of vibration -- 1.5. From solids to structural elements -- 1.5.1. Saint-Venant 's principle -- 1.5.2. Shape criterion to reduce the dimension of a problem -- 1.5.2.1. Compression of a solid body shaped as a slender parallelepiped -- 1.5.2.2. Shearing of a slender parallelepiped -- 1.5.2.3. Validity of the simplification for a dynamic loading -- 1.5.2.4. Structural elements in engineering -- Chapter 2. Straight beam models:Newtonian approach -- 2.1. Simplified representation of a 3D continuous medium by an equivalent 1D model -- 2.1.1. Beam geometry -- 2.1.2. Global and local displacements -- 2.1.3. Local and global strains -- 2.1.4. Local and global stresses -- 2.1.5. Elastic stresses -- 2.1.6. Equilibrium in terms of generalized stresses -- 2.1.6.1. Equilibrium of forces -- 2.1.6.2. Equilibrium of the moments. -- 2.2. Small elastic motion -- 2.2.1. Longitudinal mode of deformation -- 2.2.1.1. Local equilibrium -- 2.2.1.2. General solution of the static equilibrium without external loading -- 2.2.1.3. Elastic boundary conditions -- 2.2.1.4. Concentrated loads. -- 2.2.1.5. Intermediate supports -- 2.2.2. Shear mode of deformation -- 2.2.2.1. Local equilibrium -- 2.2.2.2. General solution without external loading -- 2.2.2.3. Elastic boundary conditions -- 2.2.2.4. Concentrated loads. -- 2.2.2.5. Intermediate supports -- 2.2.3. Torsion mode of deformation -- 2.2.3.1. Torsion without warping -- 2.2.3.2. Local equilibrium -- 2.2.3.3. General solution without loading -- 2.2.3.4. Elastic boundary conditions.

2.2.3.5. Concentrated loads. -- 2.2.3.6. Intermediate supports -- 2.2.3.7. Torsion with warping:Saint Venant 's theory -- 2.2.4. Pure bending mode of deformation -- 2.2.4.1. Simplifying hypotheses of the Bernoulli -Euler model -- 2.2.4.2. Local equilibrium -- 2.2.4.3. Elastic boundary conditions -- 2.2.4.4. Intermediate supports -- 2.2.4.5. Concentrated loads -- 2.2.4.6. General solution of the static and homogeneous equation -- 2.2.4.7. Application to some problems of practical interest -- 2.2.5. Formulation of the boundary conditions -- 2.2.5.1. Elastic impedances. -- 2.2.5.2. Generalized mechanical impedances -- 2.2.5.3. Homogeneous and inhomogeneous conditions -- 2.2.6. More about transverse shear stresses and straight beam models -- 2.2.6.1. Asymmetrical cross-sections and shear (or twist) centre -- 2.2.6.2. Slenderness ratio and lateral deflection -- 2.3. Thermoelastic behaviour of a straight beam -- 2.3.1. 3D law of thermal expansion -- 2.3.2. Thermoelastic axial response -- 2.3.3. Thermoelastic bending of a beam -- 2.4. Elastic-plastic beam -- 2.4.1. Elastic-plastic behaviour under uniform traction -- 2.4.2. Elastic-plastic behaviour under bending -- 2.4.2.1. Skin stress -- 2.4.2.2. Moment-curvature law and failure load -- 2.4.2.3. Elastic-plastic bending:global constitutive law -- 2.4.2.4. Superposition of several modes of deformation -- Chapter 3. Straight beam models:Hamilton 's principle -- 3.1. Introduction -- 3.2. Variational formulation of the straight beam equations -- 3.2.1. Longitudinal motion -- 3.2.1.1. Model neglecting the Poisson effect -- 3.2.1.2. Model including the Poisson effect (Love -Rayleigh model) -- 3.2.2. Bending and transverse shear motion -- 3.2.2.1. Bending without shear:Bernoulli -Euler model -- 3.2.2.2. Bending including transverse shear:the Timoshenko model in statics.

3.2.2.3. The Rayleigh -Timoshenko dynamic model -- 3.2.3. Bending of a beam prestressed by an axial force -- 3.2.3.1. Strain energy and Lagrangian -- 3.2.3.2. Vibration equation and boundary conditions -- 3.2.3.3. Static response to a transverse force and buckling instability -- 3.2.3.4. Follower loads -- 3.3. Weighted integral formulations -- 3.3.1. Introduction -- 3.3.2. Weighted equations of motion -- 3.3.3. Concentrated loads expressed in terms of distributions -- 3.3.3.1. External loads -- 3.3.3.2. Intermediate supports -- 3.3.3.3. A comment on the use of distributions in mechanics -- 3.3.4. Adjoint and self-adjoint operators -- 3.3.5. Generic properties of conservative operators -- 3.4. Finite element discretization -- 3.4.1. Introduction -- 3.4.2. Beam in traction-compression -- 3.4.2.1. Mesh. -- 3.4.2.2. Shape functions -- 3.4.2.3. Element mass and stiffness matrices -- 3.4.2.4. Equivalent nodal external loading -- 3.4.2.5. Assembling the finite element model -- 3.4.2.6. Boundary conditions -- 3.4.2.7. Elastic supports and penalty method -- 3.4.3. Assembling non-coaxial beams -- 3.4.3.1. The stiffness and mass matrices of a beam element for bending -- 3.4.3.2. Stiffness matrix combining bending and axial modes of deformation. -- 3.4.3.3. Assembling the finite element model of the whole structure -- 3.4.3.4. Transverse load resisted by string and bending stresses in a roof truss. -- 3.4.4. Saving DOF when modelling deformable solids -- Chapter 4. Vibration modes of straight beams and modal analysis methods -- 4.1. Introduction -- 4.2. Natural modes of vibration of straight beams -- 4.2.1. Travelling waves of simplified models -- 4.2.1.1. Longitudinal waves -- 4.2.1.2. Flexure waves -- 4.2.2. Standing waves,or natural modes of vibration -- 4.2.2.1. Longitudinal modes. -- 4.2.2.2. Torsion modes. -- 4.2.2.3. Flexure (or bending)modes.

4.2.2.4. Bending coupled with shear modes -- 4.2.3. Rayleigh 's quotient -- 4.2.3.1. Bending of a beam with an attached concentrated mass. -- 4.2.3.2. Beam on elastic foundation -- 4.2.4. Finite element approximation. -- 4.2.4.1. Longitudinal modes. -- 4.2.4.2. Bending modes. -- 4.2.5. Bending modes of an axially preloaded beam -- 4.2.5.1. Natural modes of vibration -- 4.2.5.2. Static buckling analysis. -- 4.3. Modal projection methods -- 4.3.1. Equations of motion projected onto a modal basis -- 4.3.2. Deterministic excitations. -- 4.3.2.1. Separable space and time excitation -- 4.3.2.2. Non-separable space and time excitation -- 4.3.3. Truncation of the modal basis -- 4.3.3.1. Criterion based on the mode shapes -- 4.3.3.2. Spectral criterion -- 4.3.4. Stresses and convergence rate of modal series -- 4.4. Substructuring method -- 4.4.1. Additional stiffnesses -- 4.4.1.1. Beam in traction-compression with an end spring -- 4.4.1.2. Truncation stiffness for a free-free modal basis -- 4.4.1.3. Bending modes of an axially prestressed beam -- 4.4.2. Additional inertia -- 4.4.3. Substructures by using modal projection. -- 4.4.3.1. Basic ideas of the method -- 4.4.3.2. Vibration modes of an assembly of two beams linked by a spring -- 4.4.3.3. Multispan beams -- 4.4.4. Nonlinear connecting elements -- 4.4.4.1. Axial impact of a beam on a rigid wall -- 4.4.4.2. Beam motion initiated by a local impulse followed by an impact on a rigid wall -- 4.4.4.3. Elastic collision between two beams -- Chapter 5. Plates:in-plane motion -- 5.1. Introduction -- 5.1.1. Plate geometry -- 5.1.2. Incidence of plate geometry on the mechanical response -- 5.2. Kirchhoff -Love model -- 5.2.1. Love simplifications. -- 5.2.2. Degrees of freedom and global displacements -- 5.2.3. Membrane displacements,strains and stresses. -- 5.2.3.1. Global and local displacements.

5.2.3.2. Global and local strains.