Links to lessons

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

  • 2011-06-12, Earthquake excitation, elastic response spectra and elastic design spectra, inelastic response, ductility and inelastic design spectra.
  • 2012-06-07, Continuous systems with an infinite number of degrees of freedom.
  • 2012-05-29, Derived Ritz Vectors, Direct Numerical Integration, Multiple Support Excitation.
  • 2012-05-22, Modal Contributions and Truncation Errors analysis. Static Correction Procedure.
  • 2012-05-15:
    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
  • 2012-05-08:
    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
  • 2012-05-03:
    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
  • 2011-04-17:
    Generalized SDOF
    • articulated rigid bodies, generalised properties
    • separation of variables, deformable bodies
    • Rayleigh method
    • Rayleigh method's refinement
  • 2011-04-12:
    Step by step integration of the EOM
    • Piecewise exact integration
    • Explicit methods (central differences)
    • Implicit methods (Newmark Beta family)
    • Modified Newton-Raphson algorithm
  • 2012-04-03:
    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
      •
  • 2012-03-27:
    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, relationship between the two solutions.
  • 2012-03-20:
    Response to Harmonic Loading
    • undamped response, dynamic amplification factor, resonant response
    • damped response
    • accelerometer
    • vibration isolation
    • measuring equivalent damping ratio
  • 2012-03-13:
    Introduction - Free Vibrations
    • An introduction to dynamics of structures,
    • the single degree of freedom (SDOF) linear oscillator,
    • the free vibrations of SDOF oscillators.
ditto

The Extra Homework

Some students have submitted neither the first nor the third homework. To help these people, so that they can try their Dynamics exam in February, here it is the Extra Homework (<---- hey, that's a link, click on it!).

Of course you may have different exams in your agenda, in such case you could wait for the assignments of the 2013 class and try my exam in July/September.

If your submission gets at least a 60/100 mark , the Extra Homework fulfills all my requests for the admission to my part of the exam, in other terms no need for the second homework.

On the other hand, with respect to HW 1 or 3 you have to do one more exercise and some of the exercises are a little more tedious to be worked out completely.

The term for submission is February 5.

The Third Homework

.
2012-08-29
The solutions of the third homework are online.
You can browse the solutions or you can download the notebooks that implement the solutions.
BrowseDownload
Dynamical Testing Dynamical Testing
Vibration Isolation Vibration Isolation
Numerical Integration Numerical Integration
Generalized Coordinates Generalized Coordinates
Rayleigh Quotient Rayleigh Quotient
Three DOF System Three DOF System

If you follow the Browse column links, beware that some browsers (in my experience, chrome) introduce small errors in the rendering of fractions.

If you download the notebook files, you'll need a recent release of ipython, that you could obtain

  • for windows, downloading the academic version or the free version of the Entought Python Distribution,
  • for linux, using the package manager of your distribution,
  • for macos, probably you can use one of the Entought distributions but i have no experience at it.
2012-07-20
The text of the third homework is now available.

The third homework is due by Friday 24 August, must be submitted by email and concerns only who has not submitted the first one.

The Second Homework

2012-06-12
The text of the second homework is now available.

The First Homework

2012-05-08
The text of the first homework is now available.
2012-05-23
The solutions of the first homework are now available.
I will post the programs that I used to do the math as soon as I can put in them some comments and a general clean-up...
2012-05-26
Errata Corrige: the generalized mass in exercise no.3 is
m* = 8 m /3
I posted a new version of the solutions, with the above correction applied and, bonus!, a free comparison between the analytical results of exercise no. 6 and a numerical solution.

Homeworks explained

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

  • the FIRST or the THIRD, one of them, and
  • the SECOND one.
The FIRST assigment is about SDOF systems and a touch of MDOF, and is due on May 22. The SECOND will be released at the end of the course, will be about MDOF systems and is due for the day of your oral examination.

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

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, to the numerical methods for the eigen-analysis of multiple degrees of freedom systems and to earthquake engineering applications.

Organization

The course comprises about 14 weekly classes; the classes include tutorials and computational exercises.

The slides i use in classes will be made available on this page in a short time, but older versions of the slides are already available, as i taught the same course, more or less the same course, in the previous academic years.

Would you like to peek at the material i posted during the previous years, here it is...

Your degree for my module will depend on two home assignments and on an oral exam.

The two home assignments will be assigned in may, due two weeks later, and at the end of the lessons, due for your oral exam.

Recommended books

  • Ray W,Clough, Joseph Penzien, Dynamics of Structures.
    Please note that the standard 2nd edition is out of press; the link points to a software house that has bought the rights and issued a revised edition of the classic book by Clough and Penzien.
  • Anil K.Chopra, Dynamics of Structures (Theory and Applications to Earthquake Engineering), 3rd ed.

The first part of the course is inspired from Clough and Penzien's book, the second part from Chopra's.

Should you prefer to buy a single book, i'd suggest to buy Chopra's.

A number of copies of Chopra's book are available from the campus library in Lecco.

Should you find a hugely discounted used copy of the 2nd edition of prof. Chopra's book, don't worry and buy it, the 2nd ed. is good enough for our purposes.

Home assignments

Each home assignment comprises a set of exercises. Most of these exercises require no more than pencil, paper and a hand-held calculator.

The remaining exercises need something that is capable of a bit of elementary matrix algebra, a thing that every decent spreadsheet can do.

Of course specialized programs, let's say Mathematica, Matlab or Matlab's free clone Octave can be helpful.

Just for the record, when i work preparing and solving the assignments, my setup comprises:

  • Calc, an Emacs' plugin that gives you symbolic manipulation of matrices, symbolic derivation and integration and much, much more.
    Calc is not Mathematica, it's very different from Mathematica but it's useful and free.
  • Scipy, a library of numerical methods for the programming language Python.
  • Gnuplot, a command-line driven interactive data and function plotting utility, for 2- and 3-D graphs. Gnuplot does also parametric plots, model fitting, etc.

Giacomo Boffi