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

- 2016.09.12 (theory)
**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 - 2016.09.13 (theory)
**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 - 2016.09.13 (recitation)
**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 - 2016.09.13 (recitation)
**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 parameterExercises: [pdf]

Solution to exercises: [pdf]

- 2016.09.14 (theory)
**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 - 2016.09.14 (recitation)
**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 - 2016.09.15 (test)
**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 basisTest problem set: [pdf]

Solutions to the test: [pdf]

Supplementary exercises (i.e. after-test): [pdf]

- 2016.09.16 (test)
**Self-assessment test: hyperbolic trigonometry and ODEs**

Parametrization of a curve, integrals and derivatives of hyperbolic functions, simple ODEsTest problem set: [pdf]

Supplementary exercises (i.e. after-test): [pdf]

- 2016.09.19 (celebration)
**Five Years of Double Degree**

in Building Engineering and ArchitecturePoster: [jpg]

- 2016.09.20 (theory)
**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 - 2016.09.20 (recitation)
**Rotations in two dimensions**

Angle of rotation, matrix representation, eigenvalues and eigenspaces - 2016.09.21 (theory)
**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 - 2016.09.21 (recitation)
**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 - 2016.09.22 (theory)
**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 - 2016.09.22 (recitation)
**Rotations in three dimensions**

Angle of rotation, matrix representation, rotation axis, eigenvalues and eigenspaces, alternative representations of rotations - 2016.09.23 (theory)
**Determinant of a tensor, inverse tensor**

Triple product and the definition of determinant, properties of the determinant, inverse tensor, adjugate tensor - 2016.09.23 (recitation)
**Reflections**

Reflections (mirror, axial and central): how to write them with tensors, orthogonality, eigenvalues and eigenspaces. basic exercises on tensor algebra - 2016.09.23 (theory)
**Orthogonal tensors, isotropy**

Orthogonal tensors, special orthogonal group, rotations about an axis as subgroups, isotropic tensor, transversely isotropic tensor - 2016.09.26 (theory)
**Introduction to inertia**

Basic definition of inertia tensor, axial moment of inertia, fundamental properties - 2016.09.26 (recitation)
**Differentiation of vectors and tensors**

Basic principles, useful formulae - 2016.09.27 (theory)
**Inertia tensor**

Principal axes, principal moments, preliminary examples, matrix representation, geometrical and physical intuition - 2016.09.27 (recitation)
**Tensor algebra, homework correction**Exercises: [pdf]

- 2016.09.28 (theory)
**Properties of the inertia tensor**

Central tensor of inertia, Steiner's formula, composition theorem - 2016.09.28 (recitation)
**Inertia tensor**

Computing inertia tensor, using Steiner's formulaExercises: [pdf]

- 2016.09.29 (theory)
**Material symmetries**

Material symmetries of a body, symmetry group, symmetries and the inertia tensor - 2016.09.29 (recitation)
**Differentiation of vectors and tensors**

Application to the twist tensor and the twist vectorExercises: [pdf]

- 2016.09.30 (recitation)
**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 - 2016.09.30 (recitation)
**Homework correction**Exercises: [pdf]

- 2016.10.03 (theory)
**Introduction to curves**

Arc length parametrizaztion, the Frenet frame, curvature and torsion - 2016.10.03 (recitation)
**Rotations in 3D, homework correction** - 2016.10.04 (theory)
**Properties of curves**

Osculating plane and osculating circle, general properties - 2016.10.04 (recitation)
**Rotations in 3D, homework correction** - 2016.10.05 (theory)
**Two particular curves, generalisation**

Circle and planar curves, cylindrical helix and the meaning of torsion and its sign - 2016.10.05 (recitation)
**An example of curve**

Spherical curves parametrized by arc lengthNotes: [pdf]

- 2016.10.06 (theory)
**Introduction to Cosserat theory**

Special theory of Cosserat rods, unidimensional body, directors, geometric deformation, strain vectors - 2016.10.06 (recitation)
**More on the osculating circle**

Tangent lines to curves and the osculating circleNotes: [pdf]

- 2016.10.07 (theory)
**Mechanics of Cosserat rods**

Resultant contact forces and couples, stress vectors, constitutive equations, equilibrium, distributed forces and couples, reference configurations - 2016.10.07 (theory)
**Introduction to cables**

Specific conditions, tension, unlimited flexibility, fundamental properties - 2016.10.10 (theory)
**More on cables**

Distributed forces with constant direction, planar deformations, general setup, specific constraints, resolvent system, a first example: catenary, boundary conditions and solution - 2016.10.10 (theory)
**Cables from examples**

Suspended bridge, boundary conditions, solution, railway bridge (i.e. arc), boundary conditions, solution - 2016.10.11 (theory)
**Kirchoff Rods**

Distributed forces with constant direction, planar deformations, general setup, specific constraints, resolvent system. A first example: clamped cantilever, boundary conditions and solution - 2016.10.11 (recitation)
**Cables from examples**

Suspended cables, boundary conditions and solutions - 2016.10.12 (theory)
**Kirchoff Rods (contd.)**

Concentrated forces and jumps. A revised example: clamped cantilever with concetrated force, boundary conditions and solution - 2016.10.13 (recitation)
**Further configurations of cables**

Cable on a wheel, helicoidal cable on cylinder - 2016.10.13 (recitation)
**Beams with concentrated and distributed load**

Different types of constraints, reactive forces, boundary conditions and solutions - 2016.10.14 (recitation)
**Euler's Elastica**

Critical load for a vertical beam - 2016.10.14 (recitation)
**More on beams and equilibrium conditions**Exercises: [pdf]

- 2016.10.19 (final test)
Test problems: [pdf]

Solutions: [pdf]

- 2016.11.11 (excercises)
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)