SDOF system response to Impulsive Loads
Intro to Step by Step methods
Today we started with the construction of a spreadsheet to integrate the equation of motion of an undamped system using the Duhamel integral. A spreadsheet that does the same for a damped sytem is available (see below).
Next, we have seen two exercises, one about aliasing and the other about solution in the frequency domain. Printouts of the two exercises are available (see below).
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.
Intro to SbS 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 volocity 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 simple step-by-step method is the so called piecewise-linear method: the loading is assumed to have a linear behaviour during a time step, identified by its initial and final values, so that the state of the system at the end of the current step can be computed in terms of these two values and of the initial state.
Materials available for this class:
- the slides i used for my lesson
- a ready to print version of the slides,
- the complete spreadsheet with also the responses of damped systems.
- the printout of the exercise on aliasing,
- the printout of the exercise on DFT,
- a simple implementation of FFT,
- just for fun, a one liner fft function...