Università degli Studi di Pavia

Facoltà di Ingegneria

Analytical Mechanics

Code: 500397, 2016-2017

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


  • Reports of errors in the resources below are always welcome
    1. 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

    2. 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

    3. 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

    4. 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 parameter

      Exercises: [pdf]

      Solution to exercises: [pdf]

    5. 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

    6. 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

    7. 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 basis

      Test problem set: [pdf]

      Solutions to the test: [pdf]

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

    8. 2016.09.16 (test)

      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]

    9. 2016.09.19 (celebration)

      Five Years of Double Degree
      in Building Engineering and Architecture

      Poster: [jpg]

    10. 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

    11. 2016.09.20 (recitation)

      Rotations in two dimensions
      Angle of rotation, matrix representation, eigenvalues and eigenspaces

    12. 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

    13. 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

    14. 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

    15. 2016.09.22 (recitation)

      Rotations in three dimensions
      Angle of rotation, matrix representation, rotation axis, eigenvalues and eigenspaces, alternative representations of rotations

    16. 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

    17. 2016.09.23 (recitation)

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

    18. 2016.09.23 (theory)

      Orthogonal tensors, isotropy
      Orthogonal tensors, special orthogonal group, rotations about an axis as subgroups, isotropic tensor, transversely isotropic tensor

    19. 2016.09.26 (theory)

      Introduction to inertia
      Basic definition of inertia tensor, axial moment of inertia, fundamental properties

    20. 2016.09.26 (recitation)

      Differentiation of vectors and tensors
      Basic principles, useful formulae

    21. 2016.09.27 (theory)

      Inertia tensor
      Principal axes, principal moments, preliminary examples, matrix representation, geometrical and physical intuition

    22. 2016.09.27 (recitation)

      Tensor algebra, homework correction

      Exercises: [pdf]

    23. 2016.09.28 (theory)

      Properties of the inertia tensor
      Central tensor of inertia, Steiner's formula, composition theorem

    24. 2016.09.28 (recitation)

      Inertia tensor
      Computing inertia tensor, using Steiner's formula

      Exercises: [pdf]

    25. 2016.09.29 (theory)

      Material symmetries
      Material symmetries of a body, symmetry group, symmetries and the inertia tensor

    26. 2016.09.29 (recitation)

      Differentiation of vectors and tensors
      Application to the twist tensor and the twist vector

      Exercises: [pdf]

    27. 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

    28. 2016.09.30 (recitation)

      Homework correction

      Exercises: [pdf]

    29. 2016.10.03 (theory)

      Introduction to curves
      Arc length parametrizaztion, the Frenet frame, curvature and torsion

    30. 2016.10.03 (recitation)

      Rotations in 3D, homework correction

    31. 2016.10.04 (theory)

      Properties of curves
      Osculating plane and osculating circle, general properties

    32. 2016.10.04 (recitation)

      Rotations in 3D, homework correction

    33. 2016.10.05 (theory)

      Two particular curves, generalisation
      Circle and planar curves, cylindrical helix and the meaning of torsion and its sign

    34. 2016.10.05 (recitation)

      An example of curve
      Spherical curves parametrized by arc length

      Notes: [pdf]

    35. 2016.10.06 (theory)

      Introduction to Cosserat theory
      Special theory of Cosserat rods, unidimensional body, directors, geometric deformation, strain vectors

    36. 2016.10.06 (recitation)

      More on the osculating circle
      Tangent lines to curves and the osculating circle

      Notes: [pdf]

    37. 2016.10.07 (theory)

      Mechanics of Cosserat rods
      Resultant contact forces and couples, stress vectors, constitutive equations, equilibrium, distributed forces and couples, reference configurations

    38. 2016.10.07 (theory)

      Introduction to cables
      Specific conditions, tension, unlimited flexibility, fundamental properties

    39. 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

    40. 2016.10.10 (theory)

      Cables from examples
      Suspended bridge, boundary conditions, solution, railway bridge (i.e. arc), boundary conditions, solution

    41. 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

    42. 2016.10.11 (recitation)

      Cables from examples
      Suspended cables, boundary conditions and solutions

    43. 2016.10.12 (theory)

      Kirchoff Rods (contd.)
      Concentrated forces and jumps. A revised example: clamped cantilever with concetrated force, boundary conditions and solution

    44. 2016.10.13 (recitation)

      Further configurations of cables
      Cable on a wheel, helicoidal cable on cylinder

    45. 2016.10.13 (recitation)

      Beams with concentrated and distributed load
      Different types of constraints, reactive forces, boundary conditions and solutions

    46. 2016.10.14 (recitation)

      Euler's Elastica
      Critical load for a vertical beam

    47. 2016.10.14 (recitation)

      More on beams and equilibrium conditions

      Exercises: [pdf]

    48. 2016.10.19 (final test)

      Test problems: [pdf]

      Solutions: [pdf]

    49. 2016.11.11 (excercises)

      Further problems & solutions: [pdf]


    1. Marco Piastra


    Assistant Instructors

    1. Andrea Pedrini


    2. Andrea Seppi


    Final Test

    1. Wednesday, October 19th, h 9:15 am, Room AT1


    (for Tensor Algebra, Inertia and Curves):

    1. Biscari, C. Poggi, E.G. Virga, Mechanics Notebook, Liguori Editore (Napoli), Serie di Matematica e Fisica 3 (1999)