#74275  12/18/19 04:09 PM
DLF Spectrum

Member
Registered: 07/15/14
Posts: 42
Loc: Denver, Colorado, USA

Hello: I am using the Analysis Type: Relief Loads (spectrum) in the dynamic module. I feed the module a force vs time curve from my hydraulic model and out pops a frequency vs DLF curve (DLF spectrum). It doesn't seem that the piping is playing a role in creating this DLF spectrum. Any help on how this curve is formed is appreciated.
Thanks in advance,
_________________________
 Jonathan

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#74278  12/19/19 04:24 AM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 09/29/07
Posts: 700
Loc: Romania


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#74279  12/19/19 06:40 AM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 09/29/07
Posts: 700
Loc: Romania

You may see also a book Introduction to Structural Dynamics, J. M. Biggs, McGrawHill Book Co., 1964 which is referred by Figure II3.5.1.32 Dynamic Load Factors for Open Discharge System, Appendix II of B31.1 (search the book on internet).

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#74280  12/19/19 10:11 AM
Re: DLF Spectrum
[Re: mariog]

Member
Registered: 07/15/14
Posts: 42
Loc: Denver, Colorado, USA

Thank you very much! This answers my question. I appreciate the additional review material as well.
_________________________
 Jonathan

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#74281  12/19/19 11:21 AM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 07/15/14
Posts: 42
Loc: Denver, Colorado, USA

Hello mariog: After further review I have a couple questions pertaining to the Duhamel's Integral (attached) that CAESAR II uses to create the DLF spectrum.
First off, I am assuming that the equation only uses the force vs time curve that is provided by the user. The equation calculates the dynamic displacement of an arbitrary system based on the force vs time curve.
Based on those assumptions I have a few questions:
1) Does Duhamel's Integral consider the piping system?
2) How is the static displacement calculated that is used to create the DLF? Is it calculated using Duhamel's Integral?
3) What is the 1/M value in Duhamel's Integral?
Thanks,
Attachments
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 Jonathan

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#74282  12/19/19 02:40 PM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 09/29/07
Posts: 700
Loc: Romania

In fact in mathematics Duhamel's integral is presented rather as a technique to find a solution of a nonhomogeneous differential equation based on the known solution of homogeneous equation. Maybe in this abstract form is more easier to understand why is useful for a complex linear system. You have a good technique to calculate particular oscillations which are not specifically linked with a perturbation (let's say modal analyses) and you need a technique to calculate the system response under various perturbations as forces. The perturbations are the terms responsible for "nonhomogeneous". Duhamel's integral makes this step in solving the problem.
For the case of oscillations for one degree linear system subject to various forcing forces, Duhamel's integral can be presented with a physical meaning as in the book I've mentioned.
In the same book the Duhamel's solution has a more general form which is "a perfectly general expression for the response of an undamped, linearly elastic onedegree system subjected to any load function and/or initial conditions."
In the form you've written, M is the mass of the system, ω is the natural circular frequency and depends on both k and M (with k the "spring constant"). So k, M and ω are specific to the system.
The article you've downloaded informs that the displacements of a system due to any arbitrary load can be calculated through the application of Duhamel's principle and integral.
"CAESAR II provides the user with a Pulse Table/DLF Spectrum Generator, which performs the automatic integration of Duhamel's integral. This module takes a user supplied, segmented pulse, and creates the appropriate equation for each segment. Displacements are calculated at each terminus of the segment, and the equation is differentiated in order to locate any displacement minima or maxima occurring within the interior of the segment (this assures that the maximum displacement is found, without using a hitormiss approach). The absolute maximum dynamic displacement is then selected from the largest of the segmental values, and the DLF calculated from that. This process is repeated for the number of natural frequencies specified by the user, which, when plotted, create the response spectrum."

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#74292  12/22/19 09:42 AM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 09/29/07
Posts: 700
Loc: Romania

Dear Jonathan,
I don't know my explanations were enough to solve your questions. If I were you, I would try to not be so focused on the math beyond the Duhamel integral. It is just a tool to calculate what we need.
The DLF vs. Frequency plot is rather an indicator (of dynamics) sensitive to the frequency content of the applied load, much more useful in practice than a plot of "exact" DLF vs time, for example.
In case you downloaded the book I've mentioned or just seen Coade article, you may see that in the simple case of the "perturbation" as a force which is suddenly applied and remains constant indefinitely, DLF has expression 1cosωt.
However, in practice we are not interested to know the dynamic response vs time (or DLF vs time) but to see only the maximum value of the DLF which is really of interest. In the case considered, this maximum is 2. You may say that for each ω the DLF is 2 (because mathematically I can find the time when cosωt=1 for each ω considered but more important is that in this case DLF=1(1)=2 and this is the maximum possible since 1<=cos<=1) and this is told as "always the DLF is maximum 2"  for the case when a single instantaneously "hit" is applied ,I would add (my last "addendum" is too often forgotten!). Moreover, as you can see, one can realize that this "2" is not really related to a particular system but to any system with single DOF which is subject to this kind of "perturbation".
I hope this example gives you DLF spectrum feeling you needed or at least why is a "spectrum". If not, I guess Mr. Diehl or some other contributor can explain much better than me.

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#74437  02/12/20 04:36 PM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 07/15/14
Posts: 42
Loc: Denver, Colorado, USA

Thanks Mariog and I apologize for the late reply. I have been collaborating with a colleague of mine on this topic and we discovered that the DLF spectrum is related only to the FvsT curve that is input. It is independent of the mass of the piping system or any portion of the piping system for that matter. I believe this is what makes the Duhamel's integral so powerful. Thanks for all of your feedback!!
_________________________
 Jonathan

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#74443  02/14/20 03:50 PM
Re: DLF Spectrum
[Re: Jonathan_B]

Member
Registered: 09/29/07
Posts: 700
Loc: Romania

Dear Jonathan,
You're welcome. I would like to say that the "nondependence" you've mentioned is not directly by Duhamel's integral (which is dependent on ω), it is rather a result of our engineering approach to calculate the maximum of DLF for each natural frequency we specify and by repeating this procedure for "a range of natural frequencies specified by the user, which, when plotted, create the response spectrum."

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