Natural frequency and mode shapes are property of the system being analyzed and depend on how the mass and elasticity are distributed throughout. They describe the tendency of the system to vibrate when subjected to dynamic loading. Mode shapes can be considered relative displacement of a system which are not true displacements and that is why they are commonly normalized to 1 or mass normalized. The modal displacements and rotations describe the set of natural "shapes" or "patterns" of the system when vibrating with no external load.
Note: when watching an animated mode shape, remember that the animation is severely over exaggerated. With results grids opened, on Mode shape / frequency tab, place cursor in cell with value to see the true number (most often actually a very small number that cannot be correctly displayed in grids or output report).
Therefore, mode shape displacement has no units of length measurement because it has been mass normalized. Again, there are no actual displacements, it just a shape with any amplitude, positive or negative. Modal analysis is a requirement for all dynamic analysis in AutoPIPE, such as response spectrum, time history, force spectrum and harmonic loads. These analysis will have true displacements and true stresses. Once more, Mode shapes or Resonance shapes or Natural Vibration shapes are real and when a system is excited it most likely will vibrate in one or a combination of many mode shapes. It all depends on the loading pattern and loading frequency.
Number of frequencies and mode shapes with which a structure can vibrate depends on the number of mass degrees of freedom in the structure and frequency of the applied load.
The captured modal mass percentage tells how much of the response is attributed to a particular mode and also tells the mode orientation (X,Y or Z). The sum of modal masses should be 100% if all modes are counted. But since many modes are not counted, the sum is less than 100% and hence the importance of the ZPA and missing mass options for dynamic analysis.
Modal analysis advantage is that it can give good results by analyzing few modes. But that is not true for all systems.
Seismic / Earth quake:
Earthquake loading typically do not have frequencies above 50 Hz (most often between 0 and 33 Hz ) and so setting cut-off frequency to 33 or 50 Hz and including missing mass or ZPA will usually give satisfactory results. Some people like to make sure the captured modal mass is at least 75% or even 90%.
Response Spectrum analysis:
In some modal analysis 60% Participation is not bad for earthquake analysis so long as the cut-off frequency is above 33 Hz and both "Include missing mass" and "ZPA" options are enable.
Water hammer is usually a high frequency load, typical cut-off frequencies range from 150-400 Hz.
Machine vibrations, the analysis is often done up to 12 or 20 harmonics of the reciprocating equipment. For 300 rpm, first harmonic is 300/60=5Hz. So 20 harmonics would be 20x5= 100 Hz.
AutoPIPE provide the option to perform a modal analysis by a. Cut-off frequency or by Number of mode shapes.
Using Cut-off Frequencies, the frequency range of the modes should always cover the frequency content of the loading function and in some cases you need to analyze for all the modes by including a large cut-off frequency (e.g. 1.E20). This is usually done to capture support reactions in the absence of using ZPA or missing mass correction to correct for these unused higher modes.
Using Number of modes is just another way to limit the frequencies, recommend to set the number of modes to 999 to capture all modes up to a cut-off frequency. It is also important to look at the captured modal mass percentage in the frequency report and analysis summary. It is good to have about 75% or more of the mass captured if possible.
The more modes you have the more captured mass will be. Also recommend using missing mass and/or ZPA options to correct for all higher modes that are not included in the analysis.
The usual procedure for determining how many modes are sufficient is to extract a certain number of modes and review the results; then to repeat the analysis while extracting 5 to 10 additional modes, and comparing the new results to the old. If there is a significant change between the results, a new analysis is made, again extracting 5 to 10 more modes above those that were extracted for the second analysis. This iterative process continues until the results taper off, becoming asymptotic. The fact you are getting different results for different cut-off frequency indicates that your loading have higher frequencies.
Regardless of the mechanical cut off frequency used for span calculations, AutoPIPE modal analysis cut off frequency should always exceed the vibrational and forcing function frequencies expected. For example if you are interested in harmonic forcing functions at 88 Hz, Suggest using 1.5x88 or 2x88 Hz as the cut off frequency. In addition, never limit to certain mode number and suggest to set "Maximize number of modes" = 999.
For both static and dynamic analysis, many Users forget to use AutoPIPE’s mass discretization, but this feature is important to capture accurate modal mass in the piping system. Therefore, every time you set the cut off frequency for modal analysis, recommend to also set the cut off frequency for mass discretizations in Tools> Model Options> Edit. suggest that "Mass points per span" set to "A" (automatic) and enter a cutt off frequency. In general low compressor speeds requires forcing functions up to 100 Hz or 200 Hz cut off frequency. Higher compressor speeds requires higher harmonic frequencies and hence higher cut off frequencies. Furthermore, typically mass discretization cut off = 50Hz for Seismic Events and 150-200 Hz for Fluid Transient events.
AutoPIPE's "Extended Component" section of the "Model Listing" sub-report will indicate how many mass node points are used between each node point.
AutoPIPE lumps the mass of the pipe, components and contents, etc. at the associated node point. This assumption yields a diagonal mass matrix with no mass coupling terms. There are three mass degrees of freedom per node. Rotational mass is ignored, except for points with eccentric weights
AutoPIPE uses a subspace iteration method from SAP IV for solving mode shapes. SAP IV is a general purpose finite element analysis program for linear structural analysis originally developed at UC (Berkeley).
AutoPIPE does account for pressure stiffening of bends, but not tension stiffening in modal analysis. In addition, AutoPIPE finite element solver does consider transverse shear stiffness.
Again, a common rule of thumb is to capture at least 75% of the modal mass but the missing mass correction will capture the remaining modal mass for an accurate dynamic analysis.
The different standing-wave patterns, known as "normal modes of vibration", are shown in Figure below. The solid and dashed lines indicate the positions of the string at opposite phase positions in the cycle. You should be able to see that for each normal mode the string contains an integer number of half wavelengths. The first mode is known as the "fundamental mode" and, for this reason, the "first natural frequency" tends to be referred to as the "fundamental frequency". If the frequency at which the first mode occurs is denoted f1, then the frequencies at which the second, third and fourth modes occur are 2f1, 3f1 and 4f1 respectively. This set of frequencies and its indefinite continuation (5f1, 6f1, 7f1 … ) is known as a harmonic series.
Modal analysis are performed as a Dynamic analysis, as a results all dynamic analysis are analyzed using Linear Analysis, and all Linear Analysis ignore all Gaps, Friction, and Soil settings in the model. In addition, all Dynamic analysis results are (+)ve. Highly suggest reviewing the references in AutoPIPE help for more details on this comment.
Note: Consider the -ve direction of dynamic results by inserting / modifying code / non-code combinations, for indicated load case enter -1.0 scale factor, see reference #2 below for more information.
Please see the following AutoPIPE help section:
1. Help > Contents> Contents Tab> Reference Information> Analysis Considerations> Modal analysis> Modal Analysis> 3rd paragraph, starting with "It should be noted...."
2. Help > Contents> Contents Tab> Reference Information> Results Interpretation> Dynamic Support Forces Results
Dynamic "Modal" Analysis