Links to lessons

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

  • 2017-03-07:
    Introduction - Free Vibrations
    • An introduction to dynamics of structures,
    • the single degree of freedom (SDOF) linear oscillator,
    • the free vibrations of SDOF oscillators.
  • 2017-03-09:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2017-03-14:
    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.
  • 2017-03-16:
    Impulsive loads, Step by Step methods
    • Impulsive Loads
    • definition, usual assumptions,
    • analytical solutions,
    • approximate analysis.
      •
    • Step by Step Methods
    • general discussion, pros & cons,
      • the piecewise linear method,
      • the central differences method,
      • the constant acceleration method,
      • the linear acceleration method.
  • 2017-03-21:
    Generalized SDOF
    • Assemblages of Rigid Bodies, Principle of Virtual Displacements
    • Deformable Bodies, Separation of Variables
    • Rayleigh Quotient Method
    • Refinements of Rayleigh Quotient Estimates
  • 2017-03-28:
    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
  • 2017-03-31:
    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
  • 2017-04-04
    Truncation, Matrix Iteration
    • modal responses as coefficients in eigenvector expansion of response
    • truncated eigenvector expansion
    • matrix iteration
      • is about refining equilibrium
      • converges to the first eigenvector
      • can be forced to converge to each eigenvector in sequence
    • Ritz Coordinates - reduction of dimensionality
    • Rayleigh Quotient as a function of Ritz coordinates - reduced eigenvalue problem
    • Choice of base - Subspace Iteration procedure
  • 2017-04-06
    Modal Contribution Factors, Numerical Integration, Multiple Support Excitation.
    • Modal Partecipation Factors $\Gamma_i$
    • Modal Contributions to Load Vector
    • Modal Partecipation Factors
    • Static Correction
    • Wilson's $\theta$ method
    • Multiple Support Excitation
    • Influence Matrix
      •
  • 2017-04-11
    Continuous Systems
    • Continuous Systems
    • Beams
    • EoM
    • Separation of Variables
    • Free Vibrations, Eigenvalue Problem, Orthogonality
    • Forced Response
      •
  • 2017-04-20
    • Earthquakes
    • Response Spectra
    • Tripartite Plots
    • Idealized Response Spectra
    • Elastic Design Spectra
    • Inelastic Design
    • Inelastic Eartquake Response
    • Inelastic Response Spectra
    • Inelastic Design Spectra
      •

Written Tests

Positive completion of a written test is a requirement for the admission to the oral exam. The texts of the assignments, complete with reference solutions, are here.

Homeworks

The homeworks are strictly optional and have no direct influence on your final marks. On the other hand, working on them lets you test your understanding of the subject and prepare for the written test. You can find the homeworks and the solutions here.

Organization

This year we'll have 11 classes and 3 tutorial sessions.

The classes will be every Tuesday and Thursday, except for Thursday March 23rd, when you'll enjoy a tutorial session. The other two tutorials are scheduled for Friday April 7th and Friday April 28th.

Following each of the first two tutorials you will receive a set of homework problems: if you return your homework (detailed instructions will follow) it will be corrected so that you can check your preparation. The homeworks are strictly optional.

The slides I use will be posted here after each class, as well as solved problems and other classroom materials [1], but please understand that the slides, etc, are not a substitute for a real textbook (see below).
If you want to have a look at the slides before a class, last year's slides are a click away.

[1] Other classroom materials means, mostly, small computer programs, written in the Python programming language rather than in other languages with wich you might be most familiar. OTOH, in the tutorials we will use the programming languages you're possibly most familiar with.

Exam

The exam consists in a classroom written test and an oral test.

The written test, mostly an admission test for the oral, consists in a set of problems that require no software for their solution. The problems will be similar to the homeworks' ones.

The oral test consists in a Q&A session, you are expected to be able to discuss all the different topics of the course.

Recommended text books

  • Anil K.Chopra, Dynamics of Structures (Theory and Applications to Earthquake Engineering), 4th ed.
  • Ray W. Clough, Joseph Penzien, Dynamics of Structures.
    This classic text is solely sold by the software house linked above.
    You can find an used copy with a bit of luck.

The course is mostly inspired by Clough and Penzien's book, but for many topics it follows Chopra's approach and examples. Should you prefer to buy a single book, my advice is "Buy Chopra's".

Note that

  • you can find copies of former editions of Chopra's book, that are OK for my course, at much lower prices,
  • a fair number of copies of Chopra's book (3rd ed.) are available from the Campus Library in Lecco.

Homeworks & Software

The homework problems sometimes require you to plot a time series or perform some matrix algebra, but every half-decent spreadsheet (say Excel or Calc) can do these things.

Of course you can do better and write a computer program using, e.g., Mathematica, Matlab or Matlab's free clone Octave to perform all the computations and generate the plots.

All the proprietery software I mentioned, as well as many other applications and development tools, can be freely downloaded from a specificPolitecnico site.

In the classes I sometimes will use, to demonstrate live coding, the Jupyter notebook. To have an idea of what I'm talking of, here it is the solution of a recently assigned problem presented as a notebook.































Giacomo Boffi