Reporte Bomba Vertical Vibracion

CSI RBM University 2001 – ODS & Moda l 

Operational Deflection Shape and Modal Analysis Testing To Solve Resonance Problems  By Tony DeMatteo Consultant  Emerson Process Management / CSI Division

The objectives of this paper are to illustrate the typical steps required to solve resonance  problems and to emphasize the power and flexibility of CSI’s 2120-2 Analyzer and Vibrant Technology’s ME’scope ODS/Modal software. Every analyst has faced difficult difficult machinery vibration vibration problems. This paper describes the use of operational deflection shape (ODS) and Modal Analysis testing for problem solving. The ODS and Modal techniques are powerful tools that enhance an analyst's ability to understand the sources of vibration. The vertical pump case history, presented in this  paper, details the testing progression from problem identification in route vibration measurements to resonance testing, Operational Deflection Shape testing and modal analysis. Resonance problems are difficult to solve. ODS and Modal Analysis give a clear picture picture of the machine’s motion, however neither tool has the capability to solve resonance  problems. In the following example, Finite Element Analysis was used to model the pump structure and evaluate modifications that move the natural frequencies away from forced vibrations  – thus eliminating resonance.

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CSI RBM University 2001 – ODS & Moda l 

Resonance Testing/ODS/Modal and FEA Case History The Problem: Problem: Vibration on Pump #1 Background : Pumps 1-4 are sewage pumps. The pumps are Morris Morris two-vane, vertical, centrifugal pumps. The drive motors on pumps 1 & 2 are new, 800 HP, U.S. induction Motors, on a variable frequency drive. drive. The motors on pumps 3 & 4 are older Reliance motors with a liquid rheostat speed control. The operating speed range of the pumps is 700-890 rpm. The pump is bolted to the floor and the motor is supported on top of a tube covering a 15 foot long drive shaft. Figure 1 is a picture of the motors on pumps #1 and #3.

Figure 1  Left – New Motor (pumps 1 & 2), Right – Old Motor (pumps 3 & 4)

The new motor is a different design and weighs about 800 pounds more than the old motor. A mounting plate was fabricated to connect the new motor to the top of the tube. The vibration problem on Pump #1 began after its motor (an 800 HP, wound rotor,  Reliance motor with a liquid rheostat speed control) was control) was replaced with a new U.S. motor. The old motor was in service for many years. Since the new motor was installed, the vibration on the machine has been extremely rough. Some other characteristics of the  problem are listed below. • • • • •

The highest vibration is over .9 inches/second – peak (IPS) at twice rotational speed. The problem direction is in-line with the discharge piping. The worst vibration occurs when the pump speed is above 820 rpm. The pump cannot be operated at the desired speed. The VFD was programmed to exclude the 820-890 rpm speed range.

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Resonance Testing/ODS/Modal and FEA Case History The Problem: Problem: Vibration on Pump #1 Background : Pumps 1-4 are sewage pumps. The pumps are Morris Morris two-vane, vertical, centrifugal pumps. The drive motors on pumps 1 & 2 are new, 800 HP, U.S. induction Motors, on a variable frequency drive. drive. The motors on pumps 3 & 4 are older Reliance motors with a liquid rheostat speed control. The operating speed range of the pumps is 700-890 rpm. The pump is bolted to the floor and the motor is supported on top of a tube covering a 15 foot long drive shaft. Figure 1 is a picture of the motors on pumps #1 and #3.

Figure 1  Left – New Motor (pumps 1 & 2), Right – Old Motor (pumps 3 & 4)

The new motor is a different design and weighs about 800 pounds more than the old motor. A mounting plate was fabricated to connect the new motor to the top of the tube. The vibration problem on Pump #1 began after its motor (an 800 HP, wound rotor,  Reliance motor with a liquid rheostat speed control) was control) was replaced with a new U.S. motor. The old motor was in service for many years. Since the new motor was installed, the vibration on the machine has been extremely rough. Some other characteristics of the  problem are listed below. • • • • •

The highest vibration is over .9 inches/second – peak (IPS) at twice rotational speed. The problem direction is in-line with the discharge piping. The worst vibration occurs when the pump speed is above 820 rpm. The pump cannot be operated at the desired speed. The VFD was programmed to exclude the 820-890 rpm speed range.

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Baseline Data – Baseline vibration data was measured with the pump operating at 870 rpm (14.5 Hz.). The largest vibration was .7 IPS – peak in the discharge direction on the motor.

Vertical .7 IPS @ 2x

Discharge

Cross Discharge

.88 IPS @

Discharge

Cross Discharge .4 IPS @ 2x

.65 IPS @

Vertical Discharge .42 IPS @ Cross Discharge

Figure 2  Baseline Data © Copyright 2001, Computational Systems Incorporated. All rights reserved.

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CSI RBM University 2001 – ODS & Moda l 

Coast-down Testing:

The pump was operated at maximum speed (899 rpm or 15 Hertz). A tachometer was used to measure motor speed and accelerometers were placed at the top of the motor in the discharge and cross discharge directions. The machine speed was decreased slowly using the VFD control. Vibration and phase data was recorded during coast-down using the Analyze | Monitor | Monitor Peak/Phase  function on the 2120 Analyzer.

Accelerometer Positions

Tachometer reading reflective tape on shaft

Figure 3 – Setup for Coast-down Testing

The Monitor Peak/Phase function was configured to measure coast-down vibration and  phase at 2x turning speed. The screen below shows the set-up.

Figure 4 – 2120 Setup for Coast-down Testing 

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The Bode plot below is a trace of 2x turning speed vibration and phase during coastdown. It indicates that at speeds below 830 rpm, the vibration on the motor was smooth (less than .1 IPS) in the discharge direction. Above 830 rpm, the 2x vibration increased and peaked at 0.9 IPS when the motor speed was 876 rpm. The 2x vibration frequency at o this point was 1752 cpm or 29.2 Hertz. A phase changed of about 180  was noted through the amplification area.

2x

Vibration Peaks at 876 rpm Below 830 rpm is smooth

Phase changes from 268 o to 94o over amplification area

Figure 5 2x Coast-down Bode Plot data (discharge direction)

The coast-down data for the perpendicular to discharge direction indicated increasing amplitudes as well. The Bode plot (below) shows the 2x turning speed vibration during coast-down. The vibration amplification peaks at a point above maximum speed. The response in this direction was less than half the amplitude of the discharge direction.

Resonance above maximum operating speed in perpendicular  to discharge direction

Figure 6 2x Coast-down Bode Plot data (perpendicular to discharge direction)

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Coast-down Testing Results: 1

The coast-down data identified a natural frequency  at 29.2 Hertz. Amplification due to resonance occurs when the pump speed is between 820 – 890 rpm. The resonance is coincident with 2x rotational speed. The 2x vibration level is amplified to .9 inches/second – peak at 29.2 Hertz when the pump speed is 876 rpm. The direction of the vibration is in line with the discharge pipe. Vibration from a mechanical defect at 2x rotational speed is exciting a natural frequency of something on the pump structure.

Discharge

Figure 7  Pump1

1

 A natural frequency is the frequency at which a part likes to vibrate. Resonant amplification results whenever forced vibrations, from mechanical defects, coincide with the natural frequencies in a system. At resonance, a small change in the excitation from mechanical defects can produce a significant change in vibration. The amount of amplification depends on the system damping characteristics. © Copyright 2001, Computational Systems Incorporated. All rights reserved.

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CSI RBM University 2001 – ODS & Moda l 

Impact Testing:

Once the pump was shut down, impact testing was performed on the motor. One sensor was used to measure response to an impact made with a three pound impact hammer. The impact and response were measured at the same position and direction on the motor. Both the discharge and cross discharge directions were tested. Impact & response direction

Figure 8 – Impact Test Setup

Impact testing identified natural frequencies in the discharge direction at 5, 30 and 39 Hertz. Hammer Impact TWF

Motor Response TWF

Cross Channel Coherence

Motor Response Spectrum 30

5

39

Figure 9 – Discharge Direction Impact Test © Copyright 2001, Computational Systems Incorporated. All rights reserved.

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Impact testing identified natural frequencies in the cross-discharge direction at 5.1, 31 and 40 Hertz.

Hammer Impact TWF

Motor Response TWF

Cross Channel Coherence

Motor Response Spectrum 31 40 5.1

Figure 10  – Cross-Discharge Direction Impact Test

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Impact Testing Results:

 Natural frequencies were found at 5, 30 and 39 Hertz in the discharge direction. When the pump is operating in the 830-890 rpm range, the second harmonic of turning speed is coincident with the 30 Hertz natural frequency resulting in resonance.

 Natural Frequencies at 5 and 30 Hertz

1x and 2x operating speed ranges Figure 11 Cross-Discharge Direction Impact Test

The pump has two vanes. Vane-pass frequency (pump speed x # vanes) and/or misalignment are potential sources exciting the resonance. The coast-down data indicates that the 2x vibration is smooth below 830 rpm. The amount of mechanical vibration related to vane pass or misalignment is, therefore, very small and probably can’t be reduced any further. The natural frequency at 5 Hertz is not coincident with any forced vibrations over the operating speed range of the pump. Vibration at this frequency is noticeable only during start-up or coast-down of the machine. The natural frequencies in the cross-discharge direction are very close to the frequencies in the discharge direction. The amount of 2x vibration in the cross-discharge direction was less than the discharge direction.

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Natural Frequency and Stiffness Comparison – pumps 1-4:

Impact tests were completed on three other pumps. Both the discharge and perpendicular to discharge directions were measured. The natural frequencies and stiffness values are listed in table 1 below. Only data for the natural frequency near 2x rotational speed are included in the table. The pumps were not operating during impact testing.

Table 1  Natural Frequencies and Stiffness Values for Pumps 1-4 P1

P2

P3

P4

 Natural Frequency (Discharge Dir) - cpm

30.0

22.5

33.7

28.6

Stiffness (Discharge Dir)  – mils/lb.

.011

.024

.008

.017

 Natural Frequency (X-Discharge Dir) - cpm

31.2

22.7

36.8

28.5

Stiffness (X-Discharge Dir) – mils/lb.

.011

.017

.007

.021

 Note: All machines have the same pump configuration. Pumps 1 and 2 have US  Motors. Pumps 3 and 4 have the old style Reliance Motors.

The natural frequencies of pumps 1 and 2 should be similar since both pumps have U.S. Motors. They are not. The natural frequencies of pumps 3 and 4 should be similar  because both pumps have Reliance motors. They are not. The variability in frequencies and stiffness values of the machines is probably due to differences in the boundary conditions at the machine base (i.e. the connection to the floor is different). All of the natural frequencies are lightly damped as indicated by tall peaks with narrow skirts. This means that a small amount of 2x vibration energy from misalignment or vane  pass frequency can cause large amplification in the 2x operating speed range. The graphic below shows the proximity between natural frequency and 2x operating speed range for each pump. Operating Speed

2x O perating Speed Range 2X 2D

Pump 1 Pump 2

4 D 4 X 1D 1X

3D

3X

Pump 3 Pump 4 D = Disch Dir  Direction D = Discharge X = Prpendic X = Cross-Discharge

   7   1 .   1

   3    3 .   1

   0    5 .   1

   7    6 .   1

   3    8 .   1

   0    7    0 .   1 .    2    2

   3    0    7    3 .    5 .    6 .    2    2    2

   3    8 .    2

   0    0 .    3

   7   1 .    3

   3    3 .    3

   0    5 .    3

   7    6 .    3

   3    8 .    3

Hertz

Figure 12  Discharge and Cross-discharge Direction Natural Frequencies near 2x Operating Speed Range for Pumps 1-4

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CSI RBM University 2001 – ODS & Moda l 

The data indicates that Pump 1 (discharge direction) and Pump 4 (both directions) have natural frequencies in the 2x operating speed range. Pump 1 has a vibration problem related to resonance at 2x operating speed. Based on the impact data measured on Pump 1, amplification factors of 10-15 can be expected at the 30 Hertz natural frequency. Pump 4’s natural frequencies are below the upper end of the 2x operating speed range. Pump 4 usually operates at full speed and 2x operating speed is far enough away from the natural frequency that resonance will not occur. Pumps 2 and 3 do not resonate because the natural frequencies fall outside of the 2x operating speed range.

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CSI RBM University 2001 – ODS & Moda l 

Operational Deflection Shape Test: 2

An Operational Deflection Shape Test  (ODS) was completed on the #1 Pump. The  purpose of ODS testing was to determine the shape of the structure when the 2x operating speed vibration is at high levels. ME’scope Visual ODS-pro software was used for the test. In order to generate operational deflection shapes, the ODS program requires: 1) Phase and magnitude data for each measured position 2) A structure drawing of the machine Data were collected on the #1 pump using the 2120-2 analyzer and the Advanced 2-Channel DLP. The Advanced 2-channel DLP provides the 2120 with additional cross channel capability and allows storage of cross channel data to analyzer memory. The Advanced 2channel DLP facilitates ODS data collection.

Motor 

Motor  baseplate

The pump was operated normally at 870 rpm during the ODS testing. One accelerometer was positioned at the top of the motor as the reference accelerometer. A second accelerometer was moved to each  position/direction where cross-channel phase and magnitude measurements were made. A total of 250 measurements were made over the entire length of the machine. All bolted or welded joints were measured to show any looseness, bending, soft joints or weakness in the machine structure.

Steel Tube

Pump

The figure to the right is the structure file that was created in the ME’scope ODS software representing the #1 Pump. The black dots indicate the measurement points. After collecting the Advanced 2-Channel DLP data, the measurement file was downloaded from the analyzer to a computer using VibPro software.

Legs Steel block Concrete Pad Floor 

Figure 13 ODS Structure

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An O perational Deflection Shape (ODS) is a non-intrusive test used to analyze the motions of rotating equipment and structures under normal operating conditions. An ODS is an extension of phase analysis. In an ODS, points on a computer generated model of the machine are animated using phase and magnitude data measured during normal operation. If Frequency Response functions were measured using the Advanced 2-channel DLP, the structure can be animated at any frequency of interest. © Copyright 2001, Computational Systems Incorporated. All rights reserved.

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VibPro software is a CSI product used to download and analyze and print data collected with the Advanced 2-Channel or Advanced Transient DLP’s. VibPro is also used to export data to spreadsheet format so it can be imported into the ME’scope ODS software and used to animate the structure drawing. The graphic below shows the animation shape at 2x turning speed (29 Hertz). It indicates that the pump tube is bending. The shape approximates the second bending mode of a cantilevered structure. Click on the link to play the AVI file. ODS 29 Hz -2x.Avi

Figure 14 ODS Animation of Natural Frequency at 29 Hertz

In addition to bending of the tube, the ODS showed looseness on one leg of the pump  base. Another ODS study was made on the leg. Many additional points were measured to get a better view of the problem. A new structure drawing of just the one leg was made. It is shown below.

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CSI RBM University 2001 – ODS & Moda l 

Tripod Leg (discharge side)

Tripod Foot Cast Steel Spacer Block

Concrete pad Floor 

Figure 15a  Picture of one Pump Foot

Figure 15b ODS Structure of the Pump Foot

The results of the new ODS test confirmed looseness in the base. Relative motion was observed between the cast steel spacer block and the concrete pad. The motion was largest on the leg closest to the discharge. The predominant motion was in the vertical direction. Click on the link to play the AVI file. ODS 29 Hz Foot only REPEAT TEST.Avi

It was thought the looseness could lower the natural frequency of the pump structure causing its natural frequency to shift into the 2x operating speed range. The bolts holding the steel shim block to the concrete were checked and tightened, however no change in vibration was noted after the adjustments were made. Figure 16 ODS Animation of Discharge Side Leg

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ODS Test Results:

The Operational Deflection Shape test showed that at 2x turning speed, the shape of the  pump structure approximates the second bending mode of a vertical cantilevered structure. The largest motion was in the discharge direction. The ODS test identified relative motion between the cast steel block and the concrete pier on the discharge side foot. The motion on this leg was noticeably larger than the other two legs. All bolts were tightened and the base was inspected. No obvious problems were found and it was determined that the loose foot was a separate issue and not contributing to the bending of the tube.

About analyzing ODS data…

After viewing the animations, the ODS results must to be interpreted. The ODS analysis involves studying the views, analyzing the motions and determining what is wrong with the machine. Some mechanical faults will be more obvious than others. For example, misalignment or looseness between bolted joints is easily spotted. Some analysis tips are listed below: •

• • • • •

Look for global motions -- where the entire machine is moving together with no relative motion between components. This could be a result of a machine mounted on isolators or floor and building vibrations Look for relative motions between bearing housings or shafts (if shaft data was taken) -- an indication of misalignment Look for phase lag and relative motion between bolted or welded joints -- indicating looseness. Looseness problems will show similar motion at different frequencies. Look for twisting of the machine base -- indicating torsional bending modes or structural weakness Look for bending of structural components – an indication of resonance (note: ODS does not prove resonance) Look for localized motion on machine feet or bases – an indication of soft-foot

Being a good vibration analyst means being a good detective. Study the ODS animations and look for clues that will help solve the problem.

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Attempt to Stiffen the Structure:

Resonance problems are often difficult to solve and usually require a modal survey and Finite Element Analysis to identify a solution. Whenever possible, try to temporarily shift the natural frequency by adding mass or stiffness. Doing so will give you an idea of how much change is required to avoid the effects of resonance. The technique doesn’t always work, however it is a quick and inexpensive test. Change the mass by adding sand or shot bags to the top of a machine or structure. Another method of changing the mass is to stand on the machine and see if your body mass reduces the amount of vibration. Adding mass lowers the natural frequency. Use caution when climbing on rotating machinery. Wedging steel or lumber between the machine and a solid object such as a column or wall changes the stiffness. Increased stiffness raises the natural frequency. In the case of the #1 pump, a 8” x 8” timber was wedged between the concrete wall of the  building and the tube. The timber was placed about half way down the tube in the discharge direction where the bending was greatest.

Wall

Timber 

Figure 17 –Picture showing 8” x 8” Timber used to stiffen tube (left) and  Sketch showing location of the tim ber in the discharge direction (right)

Once the timber was wedged tightly in place, the pump was operated and another coastdown test was performed. The data indicated that the natural frequency of the structure (near 2x operating speed) didn’t change much, however the response at resonance was cut in half. The Bode plot below shows that the vibration was reduced from .89 IPS to .45 IPS. The vibration in the perpendicular to discharge direction also decreased. The timber was left in place after the test and secured with a chain-fall. The speed restriction was removed from the drive control and the Operating Department began to use the pump in this speed range. Note: The timber should not be left in place as a permanent solution to the resonance problem. © Copyright 2001, Computational Systems Incorporated. All rights reserved.

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Without Timber .9 ips With Timber .46 ips

Figure 18 2x Coast-down Bode Plot of #1 Pump with a 8” x 8” Timber Wedged Between the Building Wall and the Pump Tube

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Conclusions based on ODS, Coast-down and Impact Testing:

The ODS animation showed that the structure was bending at 2x turning speed. Bending is an indication of resonance. The source of excitation causing resonance at 29 Hertz is 2x turning speed vibration from misalignment or vane pass frequency. The coast-down and impact tests confirmed a natural frequency at about 29 Hertz. Outside of the amplification curve, the amount of 2x vibration was very slight. Reducing the amount of 2x vibration is, therefore not an option. Changing the speed of the pump is also not desirable. Correcting the resonance problem must be accomplished by changing the mass or stiffness of the pump structure.  Natural frequencies cannot be eliminated. Changing the mass or stiffness of the structure changes all the natural frequencies. To correct the resonance problem, the natural frequency at 29 Hertz must be moved out of the 2x turning speed range. The question is how to change the mass or stiffness to get the desired results? Sometimes, the modifications needed are obvious, simple and inexpensive to implement. In many cases, they are not. Making structural modifications by trial and error is a roll of the dice resulting in one of the following: 1) Vibration is reduced or eliminated at the intended frequency 2) Vibration is not reduced 3) Vibration is reduced or eliminated at the intended frequency, however changing mass or stiffness results in excitation of a different natural frequency. 4) Vibration on the machine is reduced, however the modification element itself is resonating. 5) The structural modification results in mechanical or structural failure causing machine down time, expensive repairs or safety issues. After identifying a resonance problem, the steps towards correction are a Modal survey and Finite Element Analysis.

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Modal Survey: 3

Modal survey  data were collected on the #1 pump using the 2120-2 analyzer and the Advanced 2-Channel DLP. The Advanced 2-channel DLP is required for modal data collection. The modal survey on the #1 Pump consisted of Frequency Response Function (FRF) measurements at 194  positions/directions. At each measured point: a) Natural frequencies were excited by striking a 12  pound impact hammer against the top of the motor (discharge (X) direction)  b) An accelerometer was used to measure the 0-100 Hertz response at each measurement point and direction. Four averages were acquired at each measurement point. Figure 19  Modal Structure

The figure below is a Modal Peaks plot. It is a summation of all 194 FRF’s and makes it easy to see the response of all points and directions in one view. Pump #1 had natural frequencies at 4.8, 29.81 and 39.3 Hertz. Additional natural frequencies were found above 50 Hertz. Each natural frequency has a unique shape called a mode shape. After curve fitting the FRF data, the mode shapes were animated and analyzed.

29.81

56.3 68.8

4.817

39.3

39.9 51.4

24.6

Figure 20  Modal Peaks Plot is a Summation of all FRF Measurements

3

 Modal Analysis is an experimental method of determining the natural frequencies, damping values and mode shapes of a structure. A modal analysis is done with the machine off-line. © Copyright 2001, Computational Systems Incorporated. All rights reserved.

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Modal Survey Results:

The modal survey confirmed that the mode shape at this frequency is the second bending mode of a cantilevered structure. A frames view of the deformation for the second bending mode (29.8 Hertz) is shown  below. Click on the links to play the AVI file. Modal 2nd bending mode - In Phase - 29.8 Hz Quad View.Avi Modal 2nd bending mode - In Phase - 29.8 Hz 3D.Avi Modal 2nd bending mode - In Phase - 29.8 Hz 3D Closeup.Avi

Figure 21 Frames View of 29.8 Hertz Natural Frequency

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The first bending mode was 4.82 Hertz. No mechanical vibrations are exciting this mode into resonance except during start-up and coast-down. The frames view of the 4.82 Hertz mode shape is shown below. Click on the link to play the AVI file. Modal 1st bending mode - 4.82 Hz Quad View.Avi

Figure 22 Frames View of 4.82 Hertz Natural Frequency

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Table 2 lists the Modal Survey results and characterizes each mode shape observed. There were seven modes between 0-60 Hertz.

Table 2  Pump #1 Modal Frequencies Frequency (Hz) 4.82 25 29.81 39.3/39.9 51.4 56.3

Mode Description

First bending mode First torsional mode – not sufficiently excited in the modal survey Second bending mode, Shaft/Support Pipe in Phase Second bending mode, Shaft/Support Pipe out of Phase Vertical mode Third bending mode

About analyzing Modal data…

The products of a modal survey include a list of natural frequencies, damping values and mode shape of each natural frequency. The existence of natural frequencies is not a  problem. A bell, for example, does not ring until it is struck. Natural frequencies exist in machinery and structures and only become a problem when excited by forced vibrations such as unbalance, misalignment, looseness, gearmesh, vane pass, and other mechanical defects. Modal analysis does not provide a solution for resonance problems. A Modal Survey is the first step towards correcting resonance problems. When the modal analysis is complete, the mode shapes must be evaluated. Things to look for when evaluating modal results: • • • •

•

Evaluate modal frequencies -- What is the source of the excitation? Is it mechanical, electrical or impactive type energy? Evaluate the mode shape -- Is the motion a global mode or a local mode? Is the structure bending or twisting? What component is deforming? What are the closest adjacent modes? How will corrections affect other modes? Structural changes affect all natural frequencies. Speed changes may coincide with other modes. What is the damping?

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 Natural frequencies can not be eliminated. The effect of resonance may be diminished or natural frequencies may be shifted up or down in the frequency range. Some methods of correcting resonance are discussed below. Reduce the exciting force  – Nothing resonates without an excitation force. Forcing frequencies (mechanical vibrations) are most often the excitation for resonance. Reducing the exciting force, by any amount, diminishes the effect of the resonance. It’s usually less expensive to reduce or eliminate the exciting force than it is to modify the structure. Some examples of reducing the excitation include • • • •

Balance to precision levels Precision alignment of shafts and belts Use precision parts Replace worn or broken isolators

Change the speed  -- Move the exciting force away from the natural frequency. The rule of thumb is to change the speed 10%-15% on either side of the natural frequency. Depending on the damping value for a given bending mode, more or less speed change will be required.

system natural frequency     e      d     u      t      i      l     p     m     a

move shaft speed above or   below natural frequency  Rule of thumb: Move at least  15% away for forcing frequency frequency

Figure 23 Changing the Speed to Move a Forced Vibration Away from a Natural Frequency

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Change the Mass  -- Increasing the mass of a structure lowers its natural frequencies. Consider this simplified natural frequency formula:

 fn = 1/2  k / m

Where: Fn = natural Frequency K = Stiffness M = Mass ½ Π = Constant

If “K” remains constant and “M” is increased then “Fn”, decreases If “K” remains constant and “M” is decreased then “Fn”, increases Examples of changing a structure’s mass are filling a steel base frame with concrete or adding a steel plate to the top of a steel base frame.

Change the Stiffness  -- Increasing the stiffness of a structure raises its natural frequencies. Consider the simplified natural frequency formula below:

n = 1/2   k / m If “K” is increased and “M” remains constant then “Fn”, increases If “K” is decreased and “M” remains constant then “Fn”, decreases Examples of changing a structure’s stiffness include adding bracing or gussets to a base frame and changing the thickness of components. A modal survey alone does not indicate how to correct resonance. The SDM version of ME’scope software has some Finite Element Tools that can be used to estimate the required structural modifications. The alternative is to have a Structural Engineer complete a Finite Element Analysis.

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Finite Element Analysis (FEA): 4

A Finite Element Analysis  was completed on the #1 Pump Structure. The purpose of doing a FEA is to identify potential structural modifications to move natural frequencies away from forced vibrations. A finite element model of Pump 1 was created. The model is built after gathering all the available information about the structure. The information in the model includes all dimensions, material properties, discrete stiffnesses, component masses and boundary conditions (how the pieces are attached). Some of the information is easy to obtain. Other information, like the motor’s stiffness and inertia, ma y end up being an educated  guess. The structural components in the model include the concrete pads, metal blocks, pump  base, pump volute, support pipe, motor shaft, motor adapter plate, and the motor. The motor is included in the model using a combination of rigid and concentrated mass and inertia elements. Given the relatively large dimension s of the motor and the cantilevered configuration, the mass moments of inertia are important in the dynamic behavior of the structure. However, the only mass property data available on the motor was its total weight of 9200 pounds. Therefore, estimates of the moments of inertia are required to accurately predict the natural frequencies of the pump. As stated in the assumptions above, the mass moments of inertia were calculated by approximating the motor geometry as a cylinder with uniformly distributed mass. The actual values used in the model are shown in Table 3. Table 3  Motor Mass Properties and Weight

Mass Moments of Inertia ^ Weight

Mass

IXX

IYY 2

IZZ

(lbf)

(lbf-sec /in)

(lbf-sec -in)

(lbf-sec -in)

(lbf-sec2-in)

9200

23.8

21950

21950

13750

^

2

2

The principal mass moments of inertia are calculated at the motor center of gravity.

The pump assembly includes numerous bolted interfaces between the structural components. Considerable effort was made to accurately model these connections. While the bolts themselves are considered to be rigid, only assumed preloaded areas, which are local to the bolts, are used to connect the parts. These preloaded areas are modeled using special constraint elements that do not add stiffness to the bolted components themselves. This technique was used to model the connections between the concrete and metal  blocks, the top of the metal blocks and the base plates of each leg of the pu mp base, and the top flange of the pump base and the bottom flange of the support pipe. 4

 A Finite Element Analysis (FEA) is an analytical technique that utilizes a mathematical model of a structure to predict its natural frequencies and mode shapes. A resonance problem is corrected by using the FEA model to evaluate the effectiveness of mass, stiffness and damping modifications. Sigmadyne, Inc. (http://www.sigmadyne.com) completed the FEA on Pump 1.

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CSI RBM University 2001 – ODS & Moda l 

The finite element model was created for use in MSC/NASTRAN, a commercial finite element code. The model includes a total of 6575 beam, shell, and solid elements. Solid  base, support pipe and adapter plates were modeled using shell elements. The thickness and internal design of the volute is not known. It is modeled as a uniform shell structure with a thickness of 5/8 inch. The motor shaft was modeled using beam elements connected to the model at the base of the motor, the top of the volute, and the top of the conical structure mounted on the volute. The conical structure inside the legs of the pump  bases was modeled using a rigid element. The entire model is constrained at the bottom surface of each of the three concrete blocks. Physically this represents the interface between the concrete blocks and the floor beneath the pump. The complete model is shown below.

Figure 24 FEA Structure

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CSI RBM University 2001 – ODS & Moda l 

The FEA model is only as accurate at the information used in its construction. The FEA  prediction of the natural frequencies will not be accurate if any of the information is incorrect. In addition, the estimates for structural modification may not be valid. Examples of things that can lead to an inaccurate FEA include: • • • • •

Incomplete or inaccurate information about the structure Weakened structural members due to rusting, corrosion and cracks Loose or weakened concrete base Stretched or loose bolts Incorrect estimates of material properties, masses, stiffness and boundary conditions

The assumptions used in the FEA model of the #1 Pump are listed below. 1. The weight of the motor, specified by the customer, was 9200 lbf. Since the mass moments of inertia are unavailable, they are calculated from this weight based on a diameter of 67.5 inches and a height of 92.5 inches. The mass moments of inertia are calculated using the equations for a solid cylinder with uniformly distributed mass. 2. The motor is assumed to be rigid over the frequency range of interest. 3. Concrete is assumed to be a homogeneous and isotropic material and to behave in a linear elastic manner. The following mechanical properties are used in the analysis: Modulus of elasticity:

2.0 MSI.

Poisson’s Ratio:

0.2

Weight density:

145 lbf/ft

3

4. The rebar used in concrete does not significantly alter its mechanical properties. 5. Steel is assumed to be a homogeneous and isotropic material and to behave in a linear elastic manner. The following mechanical properties are used in the analysis: Modulus of elasticity:

30 MSI.

Poisson’s Ratio:

0.3

Weight density:

481 lbf/ft

3

6. The pump is fixed at the bottom surface of the concrete block. Any additional compliance due to the surrounding structure is ignored. 7. All dimensions used in the creation of the model were measured on Pump 1 as  part of the modal survey or were supplied by Plant personnel.

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CSI RBM University 2001 – ODS & Moda l 

 Natural frequency or modal analysis is used to analytically predict the natural frequencies and mode shapes of vibration of a structure. As previously stated a modal survey is an experimental procedure used to measure both the natural frequencies and mode shapes of a structure. The accuracy and validity of the finite element can be evaluated and enhanced by correlating the predicted natural frequencies with the measured values. This correlation is based on comparison of the mode shapes of vibration rather than a simple comparison of the frequency values. This ensures that similar modes (i.e. the first  bending mode of a cantilevered structure) are being appropriately compared. Once the finite element model was correlated to the modal survey, a design optimization analysis was performed using the model. The objective of this analysis was to determine the changes in the natural frequencies due to alterations of the pump geometry.

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CSI RBM University 2001 – ODS & Moda l 

FEA Results:

Both predicted and measured values for the first eight natural frequencies of Pump 1 are summarized in Table 4. Natural frequencies of a symmetric structure occur in orthogonal  pairs. The physical significance is that the pump can actually vibrate (bend) in any direction based on the direction of the applied excitation. It is difficult to exp erimentally measure multiple modes with nearly identical frequencies. During the modal survey of Pump 1, the frequency resolution was insufficient to individually measure the two first  bending modes and the first set of second bending modes. The frequenc y separation of the next mode pair (motor shaft and support pipe oscillating out of phase) is sufficient and both were identified in the test. Table 4 Correlation of Measured and Predicted Natural Frequencies of Pump 1 Measured Value Mode Shape Description

(Hz)

Analytical Prediction (Hz)

st

1 Bending

4.8

st

1 Bending

4.4 4.4

Torsion

24-25

23.5

2nd Bending, Shaft/Support Pipe in Phase

29.8

29.2

2nd Bending, Shaft/Support Pipe in Phase

29.5

2nd Bending, Shaft/Support Pipe out of Phase

39.2

41.8

2nd Bending, Shaft/Support Pipe out of Phase

39.9

42.2

Vertical Extension

51.4

60.3

The correlation between the analytical prediction and the measured frequency values was very good. A picture of the mode shape at 29.2 Hertz is shown below. Click on the link  below to play the FEA animations. FEA 2nd bending mode 29.2 Hz.mpg FEA 1st Bending mode 4.3 Hz.mpg

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CSI RBM University 2001 – ODS & Moda l 

Figure 25  Second bending mode of the Pump1 structure showing the drive shaft vibrating in-phase with the support pipe at 29.2 Hz.

Since the natural frequency at 29 Hertz is at the upper end of the 2x operating speed range, the structural engineer recommended increasing the machine stiffness. Increasing stiffness results in a higher natural frequency. To accomplish the change, several  potential solutions were evaluated using the FEA software including: • • • • •

Shortening the length of the pump tube Increasing the wall thickness of the tube A different motor Improving the attachment to the floor Bracing the tube to the building wall

The only option that was feasible was to reduce the length of the pump tube. The customer agreed that changing the tube length was an acceptable solution. Tube length reductions of 48, 62 and 72 inches were evaluated. All three models produced acceptable results. Table 5 shows the change at each natural frequency based on pipe length reductions. The stiffness of the support pipe (T  p ) was considered by treating the support  pipe thickness as a design variable. It was found that this parameter has little effect on the modes near 29 Hz.

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CSI RBM University 2001 – ODS & Moda l 

Table 5  Effect of Support Pipe Length on the Predicted Natural Frequencies of Pump 1 Length Reductions Baseline

Tp=0.75”

48”

62”

72”

(Hz)

(Hz)

(Hz)

(Hz)

(Hz)

1st Bending

4.4

4.8

5.4

5.8

6.2

1st Bending

4.4

4.9

5.4

5.9

6.3

Torsion

23.5

25.5

24.6

25.1

25.5

2nd Bending, Shaft/Support Pipe in Phase

29.2

30.0

37.1

39.9

41.8

2nd Bending, Shaft/Support Pipe in Phase

29.5

30.2

37.3

40.2

42.2

2nd Bending, Shaft/Support Pipe out of Phase

41.8

39.2

49.6

54.1

58.5

2nd Bending, Shaft/Support Pipe out of Phase

42.2

39.6

49.8

54.2

58.6

Vertical Extension

60.3

64.9

64.4

66.0

67.3

Mode Shape Description

Decreasing the length of the support pipe and correspondingly, the overall height alters the natural frequencies of the pump. A length reduction of 48 inches increases the frequency of the first pair of second bending modes (shaft and support pipe bending in  phase) to approximately 37 hertz. An important assumption in this prediction is that all other stiffnesses remain unchanged. This is especially important given the number of  bolted connections in the pump assembly. Operating Speed

2x Operating Speed Range

  n   n    i  o   n    i  o    t    t   c    i  o   c    t   u   u   c   d   u   d   d   r  e   r  e   r  e    h    h    t    t    h    t   g    g    g    e  n   e  n   e  n    ”   l    ”   l    ”   l    2    8    2    6   4    7

  n    i  o    t    i   d   n   o    t  c   n   e   r   u  r   C

   0  .   1  1

   0    3 .   1

   0    5 .   1

   0    7 .   1

   0    9 .   1

   0  .    2  1

   0    3 .    2

   0    5 .    2

   0    7 .    2

   0    9 .    2

   0  .    3  1

   0    3 .    3

   0    5 .    3

   0    7 .    3

   0    9 .    3

   0  .   4  1

   0    3 .   4

Figure 26   Pump 1 Second bending mode Frequency Shift With Pipe Length Reduction

The vibration measurements, made during operation of the pump, did not indicate any 3x vibration that might excite the (new) second bending mode natural frequency.

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