Links to lessons

You can access the material that i used in my classes following the links (in white) that I will post below.

  • 2014-06-10 Applications to Earthquake Engineering: Response Spectra, Design Spectra, Ductility Spectra.
  • 2014-06-03 Continuous systems with an infinite number of degrees of freedom.
  • 2014-05-27 Derived Ritz Vectors, Direct Numerical Integration, Multiple Support Excitation.
  • 2014-05-20:
    Matrix Iteration Repeat, Modal Contributions and Static Correction,
    • Recall on Ritz-Rayleigh and Subspace Iteration
    • Modal Partecipation Factor
    • Modal contributions to the load vector
    • Modal contributions to pseudo-static response
    • Pseudo-displacements
    • Peak factors
    • Static correction
  • 2014-05-13:
    Superposition repeat, Matrix Iteration
    • a first approach to the issue with modal truncation errors
    • the idea of Matrix Iteration
    • convergence to 1st eigenvector/value
    • forcing convergence to higher mode, sweeping matrix
    • a first optimization, inverse iteration
    • the shifted eigenvalue problem
    • Rayleigh-Ritz procedure and subspace iteration
  • 2014-05-06:
    Structural Matrices in multi DOF systems
    • further relationships of orthogonality
    • flexibility and stiffness
    • strain energy, symmetry
    • mass matrix, consistent or lumped>
    • damping matrix by linear combination
    • static condensation procedure
  • 2014-04-29:
    Multiple DOF systems
    • equations of dynamic equilibrium
    • vector equation of equilibrium, matrix formulation
    • homogeneous equation, eigenvalues and eigenvectors
    • vector orthogonality
    • modal expansion, uncoupled equations of motion
  • 2014-04-15:
    Generalized SDOF
    • separation of variables, deformable bodies
    • Rayleigh method
    • Rayleigh method's refinement
  • 2014-04-08
    Step by Step methods, SDOF rigid bodies assemblages
  • 2014-04-01:
    Impulsive loads, intro to Step by Step methods
    • Impulsive Loads
    • definition, usual assumptions,
    • analytical solutions,
    • approximate analysis.
      •
    • Step by Step Methods
    • general discussion, pro"cons,
    • e.g., the piecewise linear method
      •
  • 2014-03-25:
    Response to Periodic and Non-Periodic Loading
    • Analysis in the frequency domain, Fourier Series and Fourier Transform, Discrete Fourier Transform and Aliasing, fast algorithm for DFT (FFT).
    • Analysis in the time domain, Duhamel integral.
  • 2014-03-18:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2014-03-11:
    Introduction - Free Vibrations
    • An introduction to dynamics of structures,
    • the single degree of freedom (SDOF) linear oscillator,
    • the free vibrations of SDOF oscillators.

Homework #3 clarification

In exercise 3, "Numerical Integration", I reference $T_1$ without a previous proper, explicit definition. $T_1$ is the period of vibration associated with the first mode of the system. Further, the load is described as $p(t) = \frac{kL}{50}\cdots$ while it is, of course, a vector: $$\boldsymbol{p}(t) = \begin{Bmatrix}1\\0\end{Bmatrix}\frac{kL}{50}\cdots$$

Homework #3

The text of the new first homework is online. This homework is due for September 1st.

Homework #2

The text of the second homework is online. This homework is due for the day of your oral exam.

Solutions for the HW #1

Here are the solutions, first in PDF format

and also as IPython notebooks

Homework #1

The first homework assignment has been published. The assignment is due by Tuesday May 20th 2014.

Objectives

The course deals with the dynamical response of mechanical systems, linear and non-linear, under the assumption of small displacements.

Focus is given to

  • analytical and numerical methods for the integration of the equations of motion, both in time and in frequency domain,
  • the numerical methods for the eigen-analysis of multiple degrees of freedom systems and
  • earthquake engineering applications.

Organization

This year we'll have 14 or 15 weekly classes, inclusive of tutorials and computational exercises.

All the slides used in classes will be (almost) immediately made available on this page after each class.

If you want to have a look at the slides before a class, you can: last year's slides are still available. Remember, however, that I'm still slowly changing the content and the organization of my lessons and that the reference material is only what I post here.

Exam

You have to submit, generally speaking, two positive homeworks to be admitted to the final oral test in July or September. My evaluation will be mainly based on the oral test.
Would you be unable to pass before the start of the 2014-15 academic year (October 2014, that is) I reserve the right to ask you to submit an additional/alternative homework.

Recommended books

The first part of the course follows Clough and Penzien's book, the second part follows Chopra's. Should you prefer to buy a single book, my advice is buy Chopra's.

Note that a fair number of copies of Chopra's book (3rd ed.) are available from the campus library in Lecco, note also that Chopra's 2nd and 3rd editions are widely available at lower prices and are perfectly appropriate for my course.

Homeworks

Each homework comprises a set of exercises. While many of these exercises won't require more than paper, pencil and a hand-held calculator, the remaining ones will require plotting a time series or a bit of really elementary matrix algebra, i.e., things that every half-decent spreadsheet can do.
Of course specialized programs, let's say Mathematica, Matlab or Matlab's free clone Octave can be helpful.

A pleasant alternative to proprietary stuff is the IPython notebook, an example of which is the solution of an exercise I've assigned in 2012.

Homeworks explained

You have to present TWO homeworks, on different subjects, out of THREE homework assignments, either

  • the FIRST and the SECOND, or
  • the THIRD and the SECOND.
The FIRST assigment is about SDOF systems and a touch of MDOF, and is due at the end of May. The THIRD assignment will be about SDOF systems and a touch of MDOF's (that is, exactly the same subjects of the 1st one), will be released at the end of July and will be due at the end of August.
The SECOND will be released at the end of the classes, will be about MDOF systems and is due for the day of your oral test.

If you aren't ready for the first homework, if you plan to do the exam in September, if you have other things to do in May, then you can do the THIRD assignment instead of the FIRST one.
 
Giacomo Boffi