Strand7:  Web notes:  Shock modelling

Modelling shock problems in Strand7
Traditionally shock testing was carried out experimentally using a shock or shaker table. However in many cases it makes good sense to perform a shock analysis on the structure before testing to ensure that experimental shock tests will be passed on the first attempt.

Depending on the exact nature of the test required a shock analysis can be performed in Strand7 using either the transient or spectral solvers. The transient solver can provide a full time history of the reponse of the structure to the shock. The transient solver is suited to analysis of drop tests and tests where a pulse of load is applied to the structure, or to model the common base acceleration problems. The spectral solver is an aproximate method that is ideally suited to the majority of shock problems where a short pulse of load or acceleration is applied to the structure. The spectral solver will only yield the maximum response of the structure, it cannot calculate the full time history of the response.

The are basically three types of shock test:

  1. A drop test. The structure is dropped from a specified height.
  2. A base acceleration applied in accordance with a specified acceleration/time history (i.e. a sinusoidal acceleration pulse).
  3. A force is applied to the structure with a specified time history (i.e. a triangular pulse).
Tests 1 and 2 are by far the most common. Following is a discussion of the methods used to model each of these test types.

Use of the Transient Solver in Modelling Drop Tests
Often design specifications require a component to survive, without damage, a drop from a certain height. This sort of problem is best handled using the transient dynamics solver. If a structure is dropped from a given height we can readily calculate the velocity at the instant before impact using the equation:

sqrt.gif

where:
s = drop height (m)
a = gravitational acceleration = 9.81 m/s2.

Freedom conditions are applied to the model at the points where it will contact the ground when dropped. The transient solver is run using an initial velocity equal to the calculated impact velocity. Impact problems will require the use of a small time step due to the rapid rates of loading. It is important to use a time step less than or equal to approximately 1/20th of the period of the mode of the structure that will be excited during the impact. If the time step is too large, the analysis will not capture the full reponse of the structure. In particular the higher frequency components of the reponse will be missed.

Note that this method is valid only for modelling the response of the structure up to the point at which the structure begins to bounce off the impact surface. Past this point, a nonlinear transient dynamic analysis is required and this can be performed with Strand7 via the use of point contact elements.

Use of the Spectral Solver to Model Base Acceleration Problems
Spectral analysis is an approximate method that allows the calculation of the peak response of a specified loading. There are four different spectra types that can be used with the Strand7 spectral solver. These are the seismic acceleration, seismic velocity, seismic displacement and force spectrum methods. The first three spectra are assumed to excite the structure by movement of the base (i.e. the points where freedoms are applied). The force spectrum applies a more general spectral loading at any point on the structure. All the different spectra are used in a similar manner. The acceleration response spectra is used for most shock problems and will be considered here.

There are three factors which are combined to give the spectral acceleration applied to the base of the structure: these are the global acceleration applied to the model, spectral value and the direction factors in the solver panel. For the acceleration response spectrum the spectral acceleration applied to the model is:

Spectral acceleration = global acceleration x direction factor x spectral value

The spectral value is a function of the frequency of the structure. This is defined by a spectral curve.

The components of the direction vector are simply factors that multiply the applied loads. They define the direction of the seismic acceleration and may be either normalised or non-normalised.

In shock problems using the method outlined here the global acceleration x direction vector defines the maximum amplitude of the applied acceleration. The spectral value is a factor that defines the effect of the acceleration on the different modes of vibration of the structure.

The spectral value is also called the Dynamic Amplification Factor by many texts on dynamics. This factor defines a response ratio between the dynamic response and an equivalent static response when the structure is loaded with the peak acceleration. Basically the response of the structure to a loading will depend on the ratio of the frequency of load application and the natural frequencies of the structure. For a loading frequency much higher than the dominant frequency of the structure, the reponse of the structure will in general be less than the response from an equivalent static acceleration.

In all the impact analyses that we have been involved with the structure is loaded with a very short pulse of high acceleration. In these cases the response of the structure will be considerably less than that which would result if a steady acceleration were applied equal to the peak acceleration during the impact.

The calculation of the spectral value or amplification factor is covered in texts such as Clough and Penzien 'Dynamics of Structures'. The following graph shows the spectral value as a function of the frequency ratio for a number of common impulse loading waveforms. Equations for these curves are given in Clough and Penzien. The curves are graphed with the X axis as frequency ratio (frequency of structure divided by frequency of load). This is the normal way of presenting this sort of data as it is independent of the frequency of the load application and the frequency of the structure. Strand7 requires the spectral table to be input with an X axis of frequency of the structure. Thus the X axis of the gragh below needs to be converted to frequency of structure. This is readily done once the frequency of the load application is known.

WAF2.gif

For simple examples where one mode of the structure dominates the response, a constant value of spectral acceleration applicable to the dominant frequency can be used (i.e. a table with two points defining a constant spectral value).

For a more general problem with say 20 frequencies contributing to the response, a spectral curve is required that covers the entire frequency range and the spectral analysis should include all of these frequencies. A spectral value will then be calculated for each of the frequencies and used to excite the corresponding mode. The responses for each of the modes are then combined to get the total response using either the CQC or SRSS methods.

If the spectral curves shown in the above graph are not applicable to a particular problem, you can calculate the spectral curve using some simple single degree of freedom models in Strand7. Basically the procedure is as follows:

  1. Establish the loading input, ie. magnitude, period and shape of load vs time curve.
  2. Establish the frequency range of interest. ie. check which modes of the structure will contribute to the response when the loading is applied.
  3. For each of the frequencies of interest, f, construct a simple single degree of freedom model such that the simple model has a fundamental frequency equal to f (f=square root of k/m). Normally this model would be a mass on a spring (beam element). If the spectral curve for an acceleration response is being determined the loading will be applied as an acceleration.
  4. For each of the models carry out a full time history transient solution with the loading defined by a load vs time table. The loading in all cases should be the input established in step 1. From the transient response analysis determine the peak acceleration response. This is the spectral acceleration required by Strand7.
  5. The spectral value used in the above method is the ratio between this calculated spectral acceleration and the applied peak acceleration.
  6. Plot a graph of spectral acceleration or spectral value vs the frequency of the structure and fit a curve through this. This is the spectral curve.
Modelling Impact Loading Problems
When an impact loading is applied to a structure in some general manner such as the structure being hit at some point, both the transient and spectral solvers can be used. In both cases a time history of the impact force must be assumed.

As discussed above the transient solver can calculate the full time history of the reponse of the structure to the impact. In this case the maximum amplitude of the loading is applied to the structure. The time history of this load is defined by the input of a load vs time table. This table is linked to the appropriate load case in the transient solver panel. The transient solver is run, once again using a small time step to capture the response.

An approximate but faster method for this sort of analysis is to use the spectral solver with a load spectra. The procedure is very similar to that discussed above for base acceleration problems except that the solver assumes that the loading is applied at some point other than the base. The spectra is derived as described above and in most cases can use the standard spectra defined in Clough. The amplitude of load would be input as a point force in most cases at the point of application on the structure. The spectral curve will define the spectral value as a function of frequency.