Numerical methods for non linear systems
The Rayleigh Quotient method
Under the assumption of separation of variables, the frequency of vibration can be computed equating the maximum values of the kinetic and strain energies.
Using an assigned shape is equivalent to imposing an additional constraint, the natural frequency is hence a lower bound of the Rayleigh Quotient.
Using the deflections produced by the application of the postulated inertial forces, it is possible to refine the first estimate.
MDOF Systems
Multiple degrees of freedom linear systems are very simple to study if we adopt a matrix notation for the equation of motion...
- Vectorial equation of dynamic equilibrium.
- Linear relationship between displacement, acceleration vector and elastic, inertial force vector are written in terms of a matrix product.
- Homogeneous equation of motion, equation of frequencies, eigenvalues and eigenvectors, eigenvectors' properties.
- Modal superposition.
- A complete example of solution.
- the Newton-Raphson slides,
- the Rayleigh Quotient slides,
- the MDOF systems slides used during most of the lesson, lesson
- a ready to print version of the last slides,
- a notebook with the solution of the 2 DOF problem discussed in class.