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Grao en Enxeñaría Mecánica
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  Mechanics
   Contents
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NOTE. The following blocks or themes develop the contents established in the Verification Report card => Kinematics of the rigid solid. Static Dynamic of systems.
1. Introduction to kinematics. 1.1. Change of orthonormal reference. Transformation of the components of a vector.
1.2. Matrix form of a rotation.
1.3. Second order Cartesian Tensors.
1.4. Rate of change of a vector.
1.5. Frenet frame. Frenet formulas.
1.6. Speed and acceleration. Intrinsic components
2. Kinematics of the rigid body. 2.1. Rigid body definition.
2.2. Translation and rotation movements.
2.3. Helical speed distribution. Chasles Theorem.
2.4. Kinematic group. Invariants.
2.5. Instantaneous axis of rotation. Minimum sliding speed.
2.6. Axoid Surfaces.
2.7. Acceleration distribution.
2.8. Angles and rotations of Euler.
3. Relative Motion 3.1. Relative velocity.
3.2. Addition theorem for angular velocity.
3.3. Relative acceleration.
3.4. Addition theorem for angular acceleration.
3.5. Inverse movements.
3.6. Movement of two solids in contact.
4. Plane Motion 4.1. Instantaneous centre of rotation. Base and rolling curve.
4.2. Speed of succession of the instantaneous centre of rotation.
4.3. Distribution of accelerations in the plane movement
5. Distributed forces. 5.1. Centre of mass.
5.2. Inertia tensor.
5.3. Steiner's theorem or parallel axes.
5.4. Diagonalization of the inertial tensor.
5.5 Symmetries in mass distributions.
5.6. Inertia Ellipsoid
6. Rigid body equilibrium 6.1 Rigid body equilibrium. Free-Body Diagrams
6.2. Principle of virtual work.
6.3. Potential energy and equilibrium conditions. Stability
7. Cable equilibrium 7.1. Equilibrium of the ideal cable.
7.2. Equilibrium under a system of parallel forces.
7.3. Cable under the action of its own weight. Catenary
8. Principles of dynamics. 8.1. Principles and laws of Newtonian mechanics.
8.2. D'Alembert's principle.
8.3. Hamilton's principle
9. Basic elements of Analytical Mechanics. 9.1. Constraints in physical systems. Definition, properties and classification.
9.2. Equilibrium conditions and equations of movement in generalized coordinates.
9.3. D'Alembert's principle.
9.4. General equation of the dynamics for a system with constraints without friction.
9.5. Forces, work and energy in generalized coordinates.
10. Lagrange formulation. 10.1. Lagrange equations.
10.2. Generalized potentials and dissipation function.
10.3. Simple applications of the Lagrange formulation.
10.4. Constants of movement. Conservation theorems.
10.5. Variational principle of Hamilton. Application to the derivation of the Lagrange equations.
10.6. Hamiltonian function.
10.7. Elimination of cyclical coordinates. Routh function.
11. Dynamics of Rotational Motion about a Fixed Axis
11.1. Equations of movement
11.2. Reactions in the supports. Static and dynamic equilibrium
12. Dynamic of the rigid body with one fixed point 12.1. Equations of motion of a rigid body with one fixed point. Linear moment, angular moment and kinetic energy.
12.2. Application of the angular moment theorem. Euler equations.
12.3. Integration of Euler equations in the absence of pairs. Cases of ellipsoid of revolution and asymmetric ellipsoid.
12.4. Stability of the rotation around the principal axes.
12.5. Movement of a heavy solid around a fixed point. The Lagrange top.
13. Small oscillations about equilibrium 13.1. Small oscillations around stable equilibrium.
13.2. Determination of natural frequencies and normal modes.
13.3. Characterization of the movement according to the different modes of oscillation. Stability of motion.
13.4. Temporal response of the system to applied forces. Machine vibrations like forced oscillations.
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