SDOF system response to Impulsive Loads
An impulsive load is characterized by a short duration and is usually analyzed under the assumption of negligible effects of damping.
Simple analytical solutions can be given for an half-sine impulse, a rectangular impulse and a triangular impulse.
The results of our analyses can be summarized in a shock-spectrum, a graph of the peak values of the response ratio.
For really short impulse a very good approximation to the peak response is given by a simple formula, based on an approximation to the value of the system's momentum.
Step by Step methods
We are able to compute numerically the response of a SDOF using either the Duhamel integral or the frequency response function plus the DFT, but these methods hold only for linear systems, being based on the superposition of effects. Many applications require taking into account different sources of non-linear behaviour. A reasonable solution is given by the use of linear methods to compute the solution in a short time step, during which the characteristics of the system can be held constant.
It is recognized that if we want to restart our computations in a generical moment, we have to know the position AND the velocity of our linear system. These two quantities are said to constitute a state vector of the linear system. For a non-linear system, the state vector must be augmented with some other info, e.g., plastic deformation, cumulated plastic deformation, etc.
A number of different algorithms have been exposed, namely the piecewise-linear method, the central differences method, the constant acceleration method and the linear acceleration method.
A brief summary of the equation of motion of a SDOF elasto-plastic oscillator will be presented next Tuesday.
Materials available for this class:
- the slides i used for my lesson
- a ready to print version of the slides,
- a brief note, that anticipates and expands on what I'll expose in class, about the integration of the EoM for an elasto-plastic SDOF system,
- two notebooks, an example of use of the piecewise linear approximation method and an example of use of the constant acceleration method.