[Free vibration of 1D cantilever beam] Difference between Newmark and Modal

[Enginit]

Dear all,

I have just compared the results (displacement, velocity, and acceleration at the mid-beam) between Newmark and Modal analysis of the free vibration of a 1D cantilever beam (a pulse at the end of beam).
However, they are different from each other. I try to reduce the step of time (dt = 0.1 and 0.001)  but it just doesn't work.
Please leave me any suggestion or mistake in the input file which I could make.

Best regards

[Prof. R.L. Taylor]

Set the pulse to be multiple time steps and see if the agreement is better.  The modal does an exact integration of the equations of motion for piecewise inputs.  Newmark does an approximate integration.

[Prof. R.L. Taylor]

Also check the periods of the modes.  Newmark should be doing many time steps for the periods you want to integrate.  Your periods are very very small!!!!

[Enginit]

Dear Professor,

Thank you very much for your response. Based on your suggestion, I increase the length of the pulse so the agreement gets better. However, the Newmark method seems not to equal the Modal method even though I try to increase the number of elements to catch higher frequencies for Newmark model. Regarding the periods, I didn't notice that since I just want to compare these 2 methods.
In the following models, I use a longer step pulse and increase the mass density to raise the natural periods. It turns out that there are 2 points:
1. There is a considerable difference between the 2 models. (Probably due to the number of natural frequencies (waves) involve in each model are different, I think)
2. The modal method seems to be better as you suggested. (In this example, It shows a clear step pulse propagates along the beam).
Once again, thank you very much for your kind suggestion.

[Prof. R.L. Taylor]

If you want the Newmark to agree with the modal solution you need to use a large number of time steps for the shortest period (highest frequency) in the model.   Say a dt such that there are about 20-30 increments for this period.  However, this solution is that of the 'discrete' problem, not that of an exact solution to the PDE for this problem.  Also, there will be numerical errors from the solution.  One better way to compare solutions is to use a smooth input, say a sine-squared pulse.  This should excite a fewer number of modes in the solution (with significant magnitude) and give a better comparison between the methods.

[Enginit]

It is true that we can eliminate the high frequency via using a pulse dominated by a prescribed frequency so that the partition of this wave is so high which could overlay the other higher frequencies. It is also interesting to know your first comment on increasing the number of timestep for the lowest natural period.
Thank you very much for your really helpful comment, Professor.