An out-of-balance compressor is inducing an oscillating vibration in the attached piping system. This causes concern with the forces, which are produced at the compressor nozzle, and pipe deflections near the compressor. Therefore, it is desired to model the harmonic loads in order to quantify the impact of the vibration on the system. Typically, the pipe/compressor connection point is instrumented, resulting in measurements of the amplitude of motion (A_{m}) and frequency of vibration (f_{m}). In order to accurately solve for piping displacements and the forces exerted on the compressor, it is necessary to determine the harmonic forces, which are induced at the compressor. Then, a harmonic analysis can be performed which produces the desired displacement and pipe force results.
AutoPIPE's Solutions to Piping Vibrations:
AutoPIPE V8i 09.04.xx.xx. and higher have harmonic displacement and velocity option which should be used instead of the previous methods used by the program in earlier versions of the software (see bellow for procedure using 09.03.xx.xx and lower) . Recommend the following suggestions when modeling with the newer version of AutoPIPE:
Apply the harmonic frequency to the source by using the new harmonic impose displacement, Impose Force, etc...
If your nozzle has some flexibility, the imposed displacement will be that of the pump and because of the flexibility only part of that displacement is applied to the pipe side.
Recommend using the ZPA option when using harmonic loads as this will give you better nozzle reactions.
The cut-off frequency should be at least 1.5 times the highest harmonic load frequency.
Select Tools> Model options> Edit> "Mass points per span", to add intermediate mass points based on the cut-off frequency.
The procedure refereed to below is meant for earlier versions of AutoPIPE (09.03.xx.xx and Lower). NOTE: If there is more than one harmonic vibration source, at other points in the piping system, Steps 1-8 must be applied at each source independently in order to calculate all harmonic forces to be defined in a harmonic load case (HI - H3). However, once the stiffness values have been adjusted at an anchor they do not need to be temporarily set to "rigid" while the other sources are evaluated
The procedure refereed to below is meant for earlier versions of AutoPIPE (09.03.xx.xx and Lower).
NOTE: If there is more than one harmonic vibration source, at other points in the piping system, Steps 1-8 must be applied at each source independently in order to calculate all harmonic forces to be defined in a harmonic load case (HI - H3). However, once the stiffness values have been adjusted at an anchor they do not need to be temporarily set to "rigid" while the other sources are evaluated
1. At the anchor, which has been defined to represent the compressor, modify the translational stiffnesses by replacing "rigid" with "0" (zero) for each global coordinate direction, which is a component of Δ_{m}. For example, if Δ_{m} is observed to act along global X only, then only the "X" DOF is set to zero. However, if Δ_{m} is observed to act in a skewed, 3D direction the "X", "Y", and "Z" DOF's should be zeroed.
2. Apply a unit "concentrated force" (f_{A}) at the anchored point for each released DOF. For example, if only the "X" DOF was set to zero, apply the force in the global X direction (f_{AX}). If all three DOF's were zeroed, apply forces in each global direction (f_{AX}, f_{AY}, and f_{AZ}). The applied forces should be placed into separate, isolated load cases (such as f_{AX} -> U1, f_{AY} -> U2, and f_{AZ} -> U3; where no other load effects are considered in Ul - U3).
NOTE: Any applied force value can be used for the unit force since its magnitude scales the corresponding displacement calculated by AutoPIPE. However, care should be taken when considering the magnitude of the resulting displacement value because of the significant figures limitations in AutoPIPE reports.
3. Perform a static linear analysis in order to determine the translational displacements (Δ_{AX}, Δ_{AY}, and Δ_{AZ}) at the anchored point.
NOTE: The applied force magnitude (f_{A} ) can influence a nonlinear analysis result set (lift-off and gap closure), and AutoPIPE is only capable of linear supports in a dynamic analysis.
4. Calculate the stiffnesses (k) of the anchor for each released DOF as follows:
K = ƒ_{A} / Δ_{A} · 10^{3}
this will result in a corresponding k_{X}, k_{Y}, and k_{Z} value. The value "10E3" is somewhat arbitrary, but it is a reasonable assumption based on the stiffness of a "rigid" anchor DOF. Then change the anchor stiffnesses, which were set to zero in Step 1, to the values calculated in this step.NOTE: this 10^{3 }factor used in the above equation is to ensure that the anchor has a rigid stiffness. However this factor could be even larger higher thus making the anchor a more rigid in an attempt to better match the resultant anchor displacement.
Again, by iteratively increasing this factor the user may be able to better match with model results with the field measured amplitude at the resultant anchor displacement.
5. Calculate the (piping system) mass of the anchor point (manc). This is done by summing the weights of any components defined at the anchor point (i.e. flange, or weight) and the weight of the pipe (and contents) based on the half length to the next point (piping, soil, or mass). If the length of pipe between the anchored point and the next adjacent piping point is buried or if "automatic mass discretization" has been employed, the MODEL "list" report must be consulted to determine the number of "transparent" points added by AutoPIPE in order to calculate the correct half length. Once this total weight has been summed, divide by "g" (using appropriate units) to obtain the mass.
6. Calculate the cut-off frequency (f_{c}) to be specified for the modal analysis from the following equation:
f_{c} = √(K_{max}/m_{anc})
where K_{max} is the maximum of k_{X}, k_{Y}, and k_{Z} calculated in Step 4.
7. Perform a modal analysis and specify the cut-off frequency calculated in Step 6. Make sure that the last mode calculated by AutoPIPE reaches the cut-off frequency. If it does not, rerun the modal analysis with a greater number of modes specified (along with the calculated f_{c}). Iterate until the last mode reaches f_{c}.
NOTE: This requirement can result in a large number of modes being captured. Missing mass and ZPA corrections do not impact the requirement of capturing the appropriate number of modes.
8. The harmonic force (F_{H}) loads are calculated from the following equation:
F_{H} = k · Δ_{m}
where "k" corresponds to the appropriate global direction, (k_{x}, k_{v}, and k_{z}). Thus, there may be F_{HX}, F_{HY}, and F_{HZ} (these are the values entered in the HARMONIC LOAD form). The frequency value associated with these forces is the measured frequency (f_{m}).
9. Now, move on to the next (vibration source) anchor and repeat Steps 1 - 8 in order to calculate the harmonic forces (FH) to be applied at this location.
NOTE: Step 9 can be ignored for a single vibration source and 0 phase angle used. Where multiple vibration sources exist in a piping system, each set of harmonic forces can be applied as an individual load case (one-to-one correspondence of source and load case), or they may be grouped into a single load case. If the latter scenario is desired, the phase angle relationship can be evaluated.
10. Once all harmonic forces are modeled and defined as desired, perform a final Modal analysis where the cut-off frequency is the maximum calculated for all vibration sources. Then, perform the harmonic analysis. The desired results can be obtained from the "Displacement" and "Restraint" reports, created using the Result / Output report, for the appropriate load cases (Hi - H3) or combinations of load cases.
If all relevant modes have been captured, the resultant anchor displacements should be within 2% of the measured amplitude Δ_{m}). Piping forces and displacements for the remainder of the system will also be correct.
However, if fewer modes are extracted, the anchor displacements reported by AutoPIPE will be near zero and the corresponding forces will be incorrect. Piping forces and displacements for the remainder of the system will be close to the correct values.
Dynamic "Harmonic" Analysis
Bentley AutoPIPE