From Sept. 12th to Oct. 14th
Theory: 9:15 a.m. - 11 a.m., Room AT1
Recitations: 11:15 a.m. - 1 p.m., Room AT1
Course presentation Syllabus, useful info, self-assesment tests, final test
A quick recap of linear algebra, Part 1 Vectors, matrices, matrix multiplication, scalar product, norm of a vector, orthogonality, linear vector space, basis, matrices and linear transformations, change of basis
A quick recap of linear algebra, Part 2 Determinant and trace of a matrix, image, kernel and rank of a matrix, basis for a vector sub-space, eigenvectors, eigenvalues and eigenspaces
Curve parametrization in 2D, trigonometry, hyperbolic trigonometry Parametrization of a curve, motivation and techniques, tangent as derivative, arc length, unit speed parametrization, trigonometric functions as parametrization of the unit circle, parametrization of an hyperbola, functions cosh, sinh and tanh, derivatives and Taylor approximation
Linear vector spaces, eigenvectors and eigenspaces Finding a basis for a vector sub-space, orthonormal basis, verifying linear independence, eigenvalues and characteristic polynomial, eigenvectors, eigenspaces, finding a basis for an eigenspace, eigenvalues and eigenspaces are invariants under changes of basis, matrix inversion, matrix depending on a real parameter
Exercises: [pdf]
Solution to exercises: [pdf]
Introduction to tensor algebra Tensors and matrices, Euclideans space, points, vectors, linear transformations as second-order tensors, linear space of tensors, diadic product, parallel and perpendicular projectors
Derivation and integration hyperbolic functions, ODEs Derivatives and integrals of hyperbolic functions and their inverses, Taylor polynomials, simple ordinary differential equations involving trigonometric and hyperbolic functions
Self-assessment test: linear algebra Basic operations with matrices, matrix inversion, linear independence and orthorgonality of vectors, basis of a linear space, orthonormal basis, image and kernel of a matrix, eigenvalues and eigenspaces, change of basis
Test problem set: [pdf]
Solutions to the test: [pdf]
Supplementary exercises (i.e. after-test): [pdf]
Self-assessment test: hyperbolic trigonometry and ODEs Parametrization of a curve, integrals and derivatives of hyperbolic functions, simple ODEs
Test problem set: [pdf]
Supplementary exercises (i.e. after-test): [pdf]
Five Years of Double Degree in Building Engineering and Architecture
Poster: [jpg]
Diads, linear space of tensors Linear space of tensors, diadic product, bilinearity, parallel and perpendicular projectors, geometrical interpretation, a basis for the linear space of tensors, transpose of a ternsor, matrix representation of diadic product
Rotations in two dimensions Angle of rotation, matrix representation, eigenvalues and eigenspaces
Symmetric and skew tensors, trace, inner product Decomposition, skew and symmetric component of a tensor, linear subspaces of skew and symmetric tensors, trace of a tensor, inner product of tensors and fundamental properties, norm of a tensor
Diads and skew tensors Diads and geometrical interpretation, symmetry with respect to a point, a plane and an axis, eigenspaces of a diad, skew tensors
Skew tensors and cross product Matrix representation of a skew tensor in three dimensions, rank of a skew tensor, nullspace, linearity, norm of a skew tensor, axis, orthogonality
Rotations in three dimensions Angle of rotation, matrix representation, rotation axis, eigenvalues and eigenspaces, alternative representations of rotations
Determinant of a tensor, inverse tensor Triple product and the definition of determinant, properties of the determinant, inverse tensor, adjugate tensor
Reflections Reflections (mirror, axial and central): how to write them with tensors, orthogonality, eigenvalues and eigenspaces. basic exercises on tensor algebra
Orthogonal tensors, isotropy Orthogonal tensors, special orthogonal group, rotations about an axis as subgroups, isotropic tensor, transversely isotropic tensor
Introduction to inertia Basic definition of inertia tensor, axial moment of inertia, fundamental properties
Differentiation of vectors and tensors Basic principles, useful formulae
Inertia tensor Principal axes, principal moments, preliminary examples, matrix representation, geometrical and physical intuition
Tensor algebra, homework correction
Exercises: [pdf]
Properties of the inertia tensor Central tensor of inertia, Steiner's formula, composition theorem
Inertia tensor Computing inertia tensor, using Steiner's formula
Exercises: [pdf]
Material symmetries Material symmetries of a body, symmetry group, symmetries and the inertia tensor
Differentiation of vectors and tensors Application to the twist tensor and the twist vector
Exercises: [pdf]
Mirror symmetries Using mirror symmetries for the computation of inertia tensors, usage of the composition theorem, two different ways for finding the principal axes of inertia
Homework correction
Exercises: [pdf]
Introduction to curves Arc length parametrizaztion, the Frenet frame, curvature and torsion
Rotations in 3D, homework correction
Properties of curves Osculating plane and osculating circle, general properties
Rotations in 3D, homework correction
Two particular curves, generalisation Circle and planar curves, cylindrical helix and the meaning of torsion and its sign
An example of curve Spherical curves parametrized by arc length
Notes: [pdf]
Introduction to Cosserat theory Special theory of Cosserat rods, unidimensional body, directors, geometric deformation, strain vectors
More on the osculating circle Tangent lines to curves and the osculating circle
Notes: [pdf]
Mechanics of Cosserat rods Resultant contact forces and couples, stress vectors, constitutive equations, equilibrium, distributed forces and couples, reference configurations
Introduction to cables Specific conditions, tension, unlimited flexibility, fundamental properties
More on cables Distributed forces with constant direction, planar deformations, general setup, specific constraints, resolvent system, a first example: catenary, boundary conditions and solution
Cables from examples Suspended bridge, boundary conditions, solution, railway bridge (i.e. arc), boundary conditions, solution
Kirchoff Rods Distributed forces with constant direction, planar deformations, general setup, specific constraints, resolvent system. A first example: clamped cantilever, boundary conditions and solution
Cables from examples Suspended cables, boundary conditions and solutions
Kirchoff Rods (contd.) Concentrated forces and jumps. A revised example: clamped cantilever with concetrated force, boundary conditions and solution
Further configurations of cables Cable on a wheel, helicoidal cable on cylinder
Beams with concentrated and distributed load Different types of constraints, reactive forces, boundary conditions and solutions
Euler's Elastica Critical load for a vertical beam
More on beams and equilibrium conditions
Exercises: [pdf]
Test problems: [pdf]
Solutions: [pdf]
Further problems & solutions: [pdf]
Marco Piastra
Andrea Pedrini
Andrea Seppi
Wednesday, October 19th, h 9:15 am, Room AT1
Biscari, C. Poggi, E.G. Virga, Mechanics Notebook, Liguori Editore (Napoli), Serie di Matematica e Fisica 3 (1999)
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