ADVENTURES IN RESONANCE: AN INTRODUCTION TO NATURALFREQUENCIES AND MODE SHAPES

Robert J. Sayer, PEApplied Structural Dynamics

Medina, Ohio

Abstract:

Resonance is a common problem in industry that results in reduced reliability of mechanicalequipment and in some cases catastrophic failure. Resonance occurs when a naturalfrequency is excited. This paper presents an introduction to resonance with an emphasis onmodal participation. Common indicators of resonance are discussed. Several case studiesare presented that illustrate the problems that resonance can cause.

Key Words:

Modal participation, Mode shape, Natural frequency, Resonance

Introduction:

A natural frequency is a frequency at which a system tends to oscillate in the absence of adriving force. All structural-mechanical systems have at least one natural frequency. Mostsystems have numerous natural frequencies. Natural frequency is generally expressed incycles or revolutions per units of time, such as cycles/second (cps) or Hz, revolution perminute (rpm), and cycles per minute (cpm).

A natural frequency is not a problem unless a dynamic force is applied to the structural-mechanical system that has frequency content that is close to the natural frequency. Amachine does not necessarily have to rotate or translate at a speed equal to the natural toexcite it, but simply have a frequency component that is close to the natural frequency. Anexample is a structure excited by a rotating shaft in a loose bearing assembly that producesdynamic force not only at the rotational frequency of the shaft, but also at harmonic multiples(2x, 3x, 4x, etc.) thereof.

When a natural frequency of a machine or structure is excited, it is said to be in resonance. Resonance is a common problem in industry that results in reduced reliability of mechanicalequipment and in some cases catastrophic failure. Examples are numerous at technicalconferences dedicated to mechanical vibration, structural dynamics and rotor dynamics. Forexample, a paper may be presented on the vibration of centrifugal fans where sources ofvibration are discussed such as imbalance, mis-alignment, blade-pass pulsation, rotating stalland surge. Then a case study is presented as part of the presentation and it invariably dealswith some sort of resonant excitation of a fan. A review of the proceedings of a recentVibration Institute annual training conference [1] revealed that natural frequency, modalanalysis or resonance were mentioned in almost every technical paper.

Figure 1: Spring-Mass System

Single Degree-of-Freedom System:

All machines and structural supporting systems are made of an assemblage of stiffness andmass. The value of the natural frequencies are dependent upon the amount and distributionof stiffness and mass. To understand this, consider the simple single degree-of-freedom(sdof) vibrating system shown in Figure 1 consisting of a rigid single lumped mass (m)supported by two elastic springs. To further simplify this system, the mass is constrainedsuch that it can only travel in the vertical direction.

Since this sdof system has two springs,each having stiffness (k), the total stiffnessof the system in the vertical direction wouldbe 2k. Note that this example is presentedas an academic exercise in order toillustrate the difference in the response ofmachine structures to static loading versusdynamic loading. Most structural-mechanical systems are too complex to betreated as such a simple vibrating system.

staticIf the force (F ) produced by the machineis statically applied to the structure, the

resulting deflection is given by Equation (1).

staticDeflection = Force/Stiffness = F /2k Equation (1)

The response of the structure to a dynamic load is more complex. It depends upon the ratiodynamic nof the frequency of the dynamic force (F ) to the natural frequency (f ) of the structure.

The vibration amplitude that will occur as the result of dynamic loading is given in Equation(2).

dynamicAmplitude = MF(F /2k) Equation (2)

This is similar to the calculation for static deflection except that it contains a magnificationdfactor (MF) that accounts for the proximity of the frequency (f ) at which the dynamic force

is applied to the natural frequency of the structure. For a simple single degree-of-freedomsystem with damping (), the magnification factor is given by Equation (3).

d n d nMF = 1/[{1-(f /f ) } + {2(f /f )} ] Equation (3)2 2 2 1/2

Figure 2 is a plot of the magnification factor for different amounts of damping. For a sdofsystem with 2 percent of critical damping ( = 0.02), the magnification factor, when thefrequency of the dynamic force equals the natural frequency, would be 25:1! ! If thedamping were to be increased to 5 percent of critical damping ( = 0.02), the magnificationfactor at resonance would be reduced to 10:1.

Mode Shapes:

The one thing that distinguishes one natural frequency from another is its mode shape. Themode shape is a unique deflection pattern of how a system responds when a naturalfrequency is excited. A structural-mechanical system can have multiple natural frequenciesthat are identical in value. However, each of the identical natural frequencies will have adistinct unique mode shape and, thus, will respond differently when excited. An exampleof this would be a rotor supported in bearings that has a stiffness in the horizontal plane thatis identical to that of the vertical plane.

Figures 3a and 3b show two distinct mode shapes of a fan rotor. Both are associated withflexure of the fan shaft. However, one is dominated by flexure in the vertical direction (Fig3a) and the other by flexure in the horizontal direction (Fig 3b). Since the mass is the samefor both modes and the stiffness is identical in all directions, the natural frequencies for thesemodes are identical at 20.88 Hz.

Figure 2: Magnification Factor

Figure 3b: Mode Shape (Horizontal) Figure 3a: Mode Shape (Vertical)

Excitation of these modes by unbalance would result in an orbital whirl of the rotor. However, if the stiffness of the bearings in the horizontal direction differed from the verticaldirection, a condition that is common for most bearings and support structures, then thenatural frequencies for these two modes would differ. In that case, the rotor could flex morewhen the unbalance force is in one direction than the other, resulting in an elliptical whirlingpattern (as measured at mid-span) instead of an orbital whirling pattern.

Multiple Degree-of-Freedom System:The number of natural frequencies thatexist is dependent upon the number ofdegrees-of-freedom of the vibratingsystem. Consider the sdof example givenin Figure 1 in which the response of themass was constrained to only allowmotion in the vertical direction. If theconstraints were removed, the mass couldhave a horizontal component that wouldresult in a rocking response as shown inFigure 4. This mode shape can bedescribed as a rocking mode wherein the

vertical motion at one spring support is out-of-phase with that of the other spring. Thisspring-mass system would also have a mode shape where the spring deformation is in-phaseresulting in the vertical translation of the mass without any rocking component.

In the example of Figure 4, the springs (or supporting structures) are constrained so as onlyto allow motion or travel in the vertical direction. Since there are 2 springs, each with 1degree-of-freedom, this is a 2 dof vibrating system. It will have 2 natural frequencies and2 mode shapes. The mode shapes are obviously unique and different from each other, onebeing associated with vertical translation and the other with rocking.

Modal Participation:

The mode shape dictates how sensitive a natural frequency may be to a dynamic force. If thedynamic force is applied at a rate close to the natural frequency and is applied at a locationand in a direction corresponding to maximum motion of the mode, then the vibrationresponse will be maximum. Conversely, if the dynamic force is applied at a location inwhich the mode shape does not have any motion, then there will not be any resonantamplification of response. This is referred to as modal participation.

For example, consider the modes for the fan rotor shown in Figure 3a and Figure 3b. If thedynamic force is applied at mid-span of the shaft and in the vertical or horizontal directions,resonant amplification will be maximized. If the dynamic force is applied at the bearings,then there will not be any resonant amplification.

Figure 4: Rocking Mode

The concept of modal participation is very important when modifying a design to change anatural frequency. Modifications should be made near locations of significant modalparticipation in order to maximize their beneficial effect. If a structural modification is madeto a mechanical system at a location of zero modal participation, then it will not have anyeffect on the natural frequency associated with that particular mode. Indicators of Resonance:

There are several indicators that a resonant condition might be present in a mechanicalsystem. Speed sensitivity and direction sensitivity are two of the most common. An exampleof speed sensitivity would be a pump, operating with variable-speed control, that vibratesexcessively at a lower speed and smooths out at a higher speed. Vibration levels, notassociated with flow-related phenomena, would normally be expected to increase as thespeed of the pump increases. However, if a natural frequency exists at the lower speed,resonant amplification of vibration could result in higher vibration.

Consider the time waveformshown in Figure 5, acquiredduring the start-up of a fan. Itactually contains a bit ofinformation acquired near theend of a coast down (0 - 16seconds). The start-up rampbegins at the obvious jog in thedata at around 16 seconds intothe record. Maximum speed isachieved at around 64 secondsinto the record. However,vibrations are the highest,approaching 3.0 ips, at around

50 seconds into the record, as the speed of the equipment passes through a natural frequency.

When a natural frequency is exited, the responding deformation of the structural-mechanicalsystem will be the mode shape associated with the natural frequency. Many modes aredirectional in nature, resulting in vibration in one direction far exceeding other directions.Whenever this occurs, resonance should be suspected. For example, if the horizontalvibration of a motor is 5x the vertical vibration, it could indicate that the structural supportor foundation is resonant.

Figure 6 shows the mode shape for a pedestal supporting an air-handling fan. This modeconsists of the horizontal flexure of the fan shaft combined with horizontal rocking of thesupport pedestal. If the natural frequency for this mode were excited, vibration levels of thebearings would be much larger in the horizontal direction then in the vertical direction.

Figure 5: Start-Up Data

These indicators should be considered as general rules. It is important to note that they maynot be applicable to all situations.

Case Study 1: Indicators are not always a foolproof method of identifying resonance.

An annealing furnace at an aluminum processing plant contained several plug fans locatedin the side wall of the furnace. The fans were direct-driven by an induction motor and hadan impeller which was overhung from the bearings. They were variable speed units, with anormal operating speed of around 960 rpm (16 Hz) for most products loaded into the furnace.

These type of fans are called plug fans since they are supplied with a support panel whichplugs into the side wall of the furnace. Vibration levels, measured from the motor and fanbearings, were excessive. The horizontal vibrations were nearly the same amplitude as thevertical vibrations. The phase between maximum vertical and maximum horizontal vibrationwas almost exactly 90 degrees. The facts of the data (H/V ratio ~ 1.0 and Phase = 90 )osupported a diagnosis of excessive vibration caused by imbalance. The data did not indicateany directional sensitivity. Resonance was not originally suspected. However, afternumerous unsuccessful attempts at balancing the fans, a study was performed to investigateresonance as the probable root-cause of the vibration problem.

Figure 7a is a result of a numerical finite element analysis (FEA) that shows a mode shapefor a natural frequency calculated at 15 Hz (900 rpm). It is dominated by horizontal swayof the fan caused by the flexural deformation of the furnace side wall. Figure 7b shows amode shape for a natural frequency calculated at 17 Hz (1020 rpm). It is dominated byvertical sway of the fan caused by the flexural deformation of the furnace side wall. Thenormal operating speed of 16 Hz (960 rpm) fell directly between both natural frequencies. The proximity of the frequency of the dynamic force created by the rotating fan resulted inalmost equal amounts of resonant excitation of both modes. Since one mode was dominatedby vertical response, the other mode by horizontal response, and both were equally excited,vibrations were excessive and nearly equal in both directions.

Figure 6: Mode Shape of Fan Pedestal

Often, resonant issues involving equipment operating with variable-frequency drives can beresolved by changing the speed and placing an exclusion zone that prohibits operating neara natural frequency. This was not practical in this case. Decreasing the speed would haveput it closer to the lower natural frequency. Increasing the speed would have placed it closerto the higher frequency. Prohibiting the operation of the fan over the entire excitation rangeof both natural frequencies would have adversely affected its capacity and productivity. Thefinal solution was to increase the flexural stiffness of the side wall, increasing both naturalfrequencies above the excitation range of the fan.

This case shows that indicators of resonance should be used only as a general guide and notbe considered as conclusive evidence as to whether a resonant condition exists or not. Thefollowing case studies provide other indicators of when to suspect resonance.

Case Study 2: Suspect Resonance when Equipment decides to leave the confines inwhich it was intended to operate and visit other areas of the plant.

A overhung single-wide, single-inlet (SWSI) fan experienced two catastrophic failures. Figure 8 is a picture of the second failure, in which all the blades and the side plate separatedfrom the backplate of the wheel.

The plant at which this fan operated hadanother unit of identical design directlyadjacent thereto. This other fan operatedwithout incident. Although the rotor(shaft and wheel) design of the two fanswere identical, there were severalinteresting differences in the installation. The fan that experienced the failures wasdriven by an 1,800 rpm motor, while thefan that did not have any problems wasdriven by a turbine at 1,600 rpm. Thedynamic forces produced by a fan at 1,800rpm are generally greater than those

Figure 7a: Mode Shape @ 15 Hz Figure 7b: Mode Shape @ 17 Hz

Figure 8: Fan Wheel after Failure

produced at 1,600 rpm, yet the lower speed fan failed twice and the higher speed did not. This fact pointed to possible speed sensitivity that supported resonance as a probable root-cause.

The design of the fan wheel was changed between the first and second failures. This designchange was implemented on both fans, even though the second fan did not experience afailure with the original design. Natural frequency testing was not done after the initialfailure, even though it should have been. Because the original fan design was not availablefor testing on either fan, a numerical finite element analysis (FEA) was done to investigatethe possibility of resonance as the root-cause of the failure.

Figure 9 shows the mode shape for anatural frequency calculated as 18.2 Hz. This mode is best described as theprincipal flexural mode of the rotor. ForSWSI fans, this mode always consists ofa combination of the flexure of the shaftwith the flexure of the backplate of thewheel [2][3]. Fan rotors of this type willbe subjected to large centrifugal stressesand gyroscopic effects that will increasethe effective stiffness of the wheel when itrotates. The natural frequency calculatedby the FEA in Figure 9 did not includethese stiffening effects and, thus, predicts

only the at rest natural frequency of the rotor.

Figure 10 is a curve that reports the change innatural frequency (blue line) with an increasein rotational speed (red line). If thefrequency of the dynamic force is directlyrelated to rotational speed (i.e. unbalance),the FEA predicted that it would be coincidentwith the natural frequency at an operatingspeed of 1,600 rpm. At a higher speed of1800, there is a small separation marginbetween natural frequency and dynamicforce. The FEA provided insight into thereason that the lower speed fan failed and thehigher speed fan did not. The questionremained as to why the fan failed a secondtime, even after the design was changed.

Figure 9: Mode Shape of Fan Rotor

Figure 10: Natural Freq vs. Speed

After the second failure of the turbine-driven fan, the replacement fan on the motor-drivenunit was still available to perform a natural frequency test. Figure 11 is the result of this testand indicated that the at rest natural frequency of the replacement fan was around 22.5 Hz. The at rest natural frequency of the replacement fan was around 4.3 Hz or nearly 24%higher than the original design. Assuming a similar stress stiffening effect, this designshould have provided a separation margin sufficient enough to preclude the resonantexcitation of the fan wheel and prevent the second failure.

The FEA model was revised to reflect thechanges for this modified replacement designto estimate the effects of centrifugalstiffening during operation. Figure 12 is acurve that reports the change in naturalfrequency (blue line) with an increase inrotational speed (red line) for the replacementdesign. At the 1,600 rpm operating speed ofthe turbine-driven fan, the blue line (naturalfrequency) is separated from the red line(dynamic force) by a sufficient margin. Thissupported a conclusion that resonance wasnot the probable root-cause of the secondfailure. In fact, the separation margin at 1,800rpm was less than at 1,600 rpm. If any of thereplacement fans should have experiencedelevated vibration due to resonance, it should have been the motor-driven unit.

After presenting this data to the operation, along with the conclusion that the original fanfailure was caused by resonance but the replacement fan failure was most probably not,maintenance personnel admitted that the replacement unit had momentarily operated at 2,400rpm after the speed governer on the turbine malfunctioned. Fan wheels experience highcentrifugal stresses during operation and are typically designed for over-speeds of around20%. Even though the turbine-driven fan was operating at 1,600 rpm, it was designed for1,800 rpm. The momentary over-speed was 33% higher than design speed.

Figure 11: Natural Freq Test Results

Figure 12: Natural Frequency vs. Speed(Replacement Fans)

Figure 13 shows a contour plot of centrifugal stress at a rotational speed of 2,400 rpm. Thered contours indicate areas where the stress levels exceed the yield strength (50,000 psi) ofthe material. What made this situation worse was the fact that, per Figure 12, the fan wheelwas also resonant at 2,400 rpm resulting in stresses even greater than reported in Figure 13.

The side-plate (or inlet shroud) of the fan failed at each blade, resulting in twelve individualpieces cantilevered off the center-plate. These pieces then failed at the intersection of theblades and the back-plate, leaving the rotor shown in Figure 8. This case demonstrates thatresonance, although the root-cause of many rotating equipment failures, is not always toblame. Case Study 3:Suspect Resonance when Equipment vibrates excessively and it is noteven operating.

The stator cooling water pumps at a nuclear power plant had experienced several prematurefailures of couplings. There were two identical single-stage centrifugal pumps supported ona common equipment skid. Figure 14 is a picture of the pump skid. The pumps, which wereidentical, were identified as Pump A and Pump B. Only one pump was required to providecooling water, the other was a back-up. Both pumps were direct-driven by a 3,600 rpminduction motor.

Maintenance records indicated that Pump B was the bad actor and had experienced mostof the premature failures. Because of this, Pump A was designated for operation much ofthe time and Pump B became the standby back-up. Although these pumps serviced thegenerator and not the reactor and, as such, were not designated as safety-related equipment,they were definitely commercial critical equipment. If Pump A failed and Pump B were notable to operate, then the generator would not be able to operate.

Figure 13: Centrifugal Stress Contour PlotRotational Speed = 2,400 rpm

A vibration level of 0.65 ips was measured from the drive-side bearing of the motor drivingPump B. This vibration is considered excessive for the equipment. What made this evenmore critical, was the fact that, when this data was acquired, Pump B was not operating. Thevibration levels of the pump were excessive and it was not even operating. The vibrationsof the idle pump were the result of vibration transmitted from the adjacent pump. Anotherinteresting fact was the vibration levels of the operating Pump A were less than the idlePump B.

Based upon the unusual facts of the case, resonance was suspected as the root-cause of theproblem. An impact test was performed to identify the natural frequencies of the Pump Bmotor. Because of the large amount of transmission from Pump A, the impact test had to bedone during an outage when both pumps could be idled.

Figure 15 contains the results of the impact test. A significant natural frequency wasapparent at 60.5 Hz that was very close to the operating speed of the pumps. Note that othernatural frequencies (38.0 Hz and 49.0 Hz), although not as sensitive, were evident in the testresults.

Figure 14: Pump Skid (Motor for Pump B on Left)

Figure 15: Natural Frequency Test Results

In order to efficiently change the natural frequencies, the mode shapes must be known. If theobjective of a modification was to increase the natural frequency, stiffness must be added insuch a way as to address the modal participation of the mode of natural frequency inquestion. Figures 16a and 16b show the mode shapes calculated by finite element analysis(FEA) for two of the natural frequencies.

The mode for the natural frequency at 60.5 Hz consists of axial rocking of the motor forPump B. The mode for the natural frequency at 49.0 Hz consists of axial rocking of themotor for Pump A. Excitation of either mode would result in elevated vibration levels inboth the vertical and axial direction. Since the operating speed of the pumps were close tothe natural frequency at 60.5 Hz and not the natural frequency at 49.0 Hz, it was only themode of Figure 16a that was an issue. Note that when the mode is excited, it results insignificant relative motion between the motor and pump and would create secondarydynamic forces at the coupling.

The mode shapes are dominated by flexural deformation of the plate to which the motors aremounted, meaning the plate is a major participator in the modes. It was decided to increasethe natural frequency that was being excited. Stiffening the upper plate of the equipmentskid should have a significant effect on the natural frequency at 60.5 Hz because of its modalparticipation. However, flexural deformation of the plate also dominates modal participationof the natural frequency at 49.0 Hz. Thus, increasing the stiffness of the plate could alsoincrease the lower natural frequency. This was a concern since the operating speed of thepumps were currently greater than the lower natural frequency.

The modification consisted of adding two steel bars (6" x 6") across the front and back sidesof the motor that were welded to the upper surface of the skid plate as shown (in red) on thesketch in Figure 17. Because the equipment skid was just inches off the main floor platform,it was not possible to add structural reinforcement underneath the plate. The steel beams thatwere underneath the plate in the original design are shown as dotted lines. Note that fromleft-to-right, there were not any beams under the front nor back of the motors. The beamunder the middle of the motors served as a pivot point about which the motors rocked.

Figure 16a: Mode Shape @ 60.5 Hz Figure 16b: Mode Shape @ 49.0 Hz

The proposed modification was added to the numerical finite element analysis to predict theeffect that it would have on the two natural frequencies of concern. Figure 18 shows theresults of the FEA that predicted the lowest mode of natural frequency associated with themotors would be around 83.9 Hz.

The modification increased both the 49.0 Hz natural frequency associated with Pump A andthe 60.5 Hz natural frequency associated with Pump B.

Figure 17: Pump Skid Modification

Figure 18: Mode Shape at 83.9 Hz (Modified Base)

Case Study 4:Suspect Resonance when Equipment vibrates at unusual frequencies.

A used dual cylinder reciprocating compressor was purchased by a steel production facilityand placed on the second floor of the power house. This compressor had operated for severalyears at another facility without any major vibration issues. However, at the other facilityit was supported on ground level upon a massive, rigid concrete foundation. The foundationwas designed to comply with mass ratios provided by the manufacturer. At the steel mill,it was placed upon a structural steel framing system, although made of large steel wide-flange beams, was much more flexible than a rigid concrete foundation.

Figure 19 contains a picture of thecompressor. It had 2 cylinders and wasdriven by a synchronous motor. Theoperating speed of the compressor was330 rpm (5.5 Hz). It produced dynamicforces at the primary frequency and at 2xoperating speed (11 Hz).

Because of its mass and the flexibility ofthe steel framework it was going to beplaced upon, and recognizing thatreciprocating compressors producesignificant dynamic loads as part ofnormal operation, an inertia base withisolation springs was designed to support

the compressor and minimize transfer of dynamic loads to the building.

Immediately upon start-up, vibration levels of the entire inertia base were extreme to thepoint that the unit was considered inoperable. Operators at the steel mill were dismayed atthe fact that significant amount of money was spent to purchase the compressor and retrofitthe building, and the equipment was not useable.

Figure 20 is the frequency spectrum of vibration data acquired from the inertia base. A smallindication was evident at the operating speed of the compressor (5.5 Hz). However, thespectrum was dominated by a response at 1.7 Hz, a frequency not related to any dynamicforce produced by the equipment. The vibration at this low frequency was measured to bearound 0.60 ips or around 450 mils peak-to-peak.

Figure 21 contains the results of an impact test. A natural frequency was identified at 1.7 Hz,which was the same frequency of the dominant vibration in the inertia base. Everything innature has natural frequencies. They are only a problem if excited. Natural frequenciescannot be excited unless a forcing function exists. Yet, the equipment did not produce aforce at the frequency in question.

Figure 19: Reciprocating Compressor

An operating deflection shape (ODS) test was performed to investigate the dynamic responseof the inertia base in an attempt to locate the source of excitation.

The ODS animation for the response at 1.7 Hz indicated that the inertia base was shortcircuiting at one of the corners. An inspection of the springs in the corner confirmed thatall were good and none were bottoming out. However, after a more detailed inspection, asmall piece of steel bar stock, used as temporary support during the installation, was foundwedged between the bottom of the springs and the top of the floor steel. This caused arepetitive impact that excited the natural frequency of the inertia base. The small steel barwas removed and the vibration response at 1.7 Hz totally disappeared.

Figure 20: Spectrum of Inertia Block Vibration

Figure 21: Impact Test Result (Inertia Base)

References:

[1] Proceedings of the Vibration Institute Training Conference and 37 Annual Meeting,thThe Vibration Institute, Jacksonville, FL., June, 2013.

[2] R. J. Sayer, Flexural Critical Speed of Single-Wide Fan Rotors, Proceedings of the the 27 Annual Meeting of the Vibration Institute, New Orleans, LA, July, 2003.th

[3] R. J. Sayer, Modal Characteristics of Single-Wide Fan Wheels, Engineering Paper5207-03, AMCA International Engineering Conference, Schaumburg, IL, December 7-9,2003.