An Analytical Approach to Solving Motor Vibration Problems
Copyright material IEEE
Paper No. PCIC-99-20
William R. Finley Mark M. Hodowanec Warren G. Holter
Senior Member Member
Large Motors & Pumps Industrial Products Division Industrial Products Division Siemens Energy & Automation, Inc. Siemens Energy & Automation, Inc. Siemens Energy & Automation, Inc. 4620 Forest Ave. 4620 Forest Ave. 4620 Forest Ave.
Norwood, OH 45212 Norwood, OH 45212 Norwood, OH 45212
Abstract: Vibration problems in induction motors can be extremely frustrating and may lead to greatly reduced reliability. It is imperative, in all operations and manufacturing processes that down time is avoided or minimized.  If a problem does occur the source of the problem is quickly identified and corrected.  With proper knowledge and diagnostic procedures, it is normally possible to quickly pinpoint
the cause of the vibration.  All too often erroneous conclusions are reached as a consequence of not understanding the root cause of the vibration.  This may result in trying to fix an incorrectly diagnosed problem, spending a significant amount of time and money in the process.  By utilizing the proper data collection and analysis techniques, the true source of the vibration can be discovered.  This includes, but is not limited to:
Electrical imbalance
Mechanical unbalance – motor, coupling, or driven equipment Mechanical effects – looseness, rubbing, bearings, etc. External effects - base, driven equipment, misalignment, etc. Resonance, critical speeds, reed critical etc.
Once the electrical and mechanical interactions in a motor are understood, and the influence external components have on the apparent motor vibration, identification of the offending component is usually straightforward.  This paper provides an analytical approach for expeditiously understanding and solving these types of problems.
Index Terms:  Induction Motors, Troubleshooting Vibrations, Cause of Vibration.
I.    INTRODUCTION
Much has been written about vibration over the years.  This includes many papers and books on vibration in general and a number of papers on vibration in induction motors in particular.  This is an ongoing subject, continually extended by advances in analytical and diagnostic tools and methods.  For this reason, and because this is an important and complex subject, it is worthwhile periodically to both present any new knowledge and experience as well as to review prior knowledge and concepts.
Vibration problems can occur at anytime in the installation or operation of a motor. When they occur it is normally critical that one reacts quickly to solve the problem. If not solved quickly, one could either expect long term damage to the motor or immediate failure, which would result in immediate loss of production.  The loss of production is oftentimes the most critical concern. To solve a vibration problem one must differentiate between cause and effect. For this to happen, one must first understand the root cause of the vibration. In other words: where does the force come from?  Is the vibratory force the cause of the high levels of vibration or is there a resonance that amplifies the vibratory response. Perhaps the support structure is just not stiff enough to minimize the displacement.  In this paper the various sources of electrical and mechanical forces will be explained. Additionally, how the motor reacts or transmits this force and how this force can be amplified or minimized will be explained as well.  When a vibration problem occurs it is important that
one use a good systematic, analytical approach in resolving the problem. This includes performing the proper diagnostic tests. The process starts by listing all the possible causes for the particular identified frequency of vibration and any variations under different operating conditions. Then eliminate the incorrect causes one by one until all that remains is the true source of the problem, and now this can be efficiently eliminated.
II.    SOURCES OF VIBRATION
There are many electrical and mechanical forces present in induction motors that can cause vibrations.  Additionally, interaction of these various forces make identification of the root cause elusive.  In subsequent sections, the major mechanisms are discussed.  For a more comprehensive list of electrically and mechanically induced vibrations Table I should be referenced.
FIG. 1.  Stator and Rotor
Twice Line Frequency Vibration:
There are many different forces and interactions as a result of the power source and the interactions between the stator and rotor as seen in Fig. 1. The power source is a sinusoidal voltage that varies from positive to negative peak voltage in each cycle.  Many different problems either electrical or mechanical in nature can cause vibration at the same or similar frequencies. One must look closely to differentiate between the true sources of vibration.
A power supply produces an electromagnetic attracting force between the stator and rotor which is at a maximum when the magnetizing current flowing in the stator is at a maximum either positive or negative at that instant in time. As a result there will be 2 peak forces during each cycle of the voltage or current wave reducing to zero at the point in time when the current and fundamental flux wave pass through zero as demonstrated in Fig. 2. This will result in a frequency of vibration equal to 2 times the frequency of the power source (twice line frequency vibration). This particular vibration is extremely sensitive to the motor's foot flatness, frame and base stiffness and how consistent the air gap is between the stator and rotor, around the stator bore.  It is also influenced by the eccentricity of
the rotor.
Flux - Flux Around a Stator on a 2 Pole Motor
Force - Force Between a Stator & Rotor on a 2 Pole Motor 0180360
0180360
Fig. 2.  One Period Flux Wave & Magnetic Force Wave Some people are inaccurately under the premise that twice line frequency vibration varies with load. This misconception comes from the belief that twice line frequency vibration excitation is due to a magnetic field generated by the current in the stator coil which varies with load and creates a magnetic force which varies with the load current squared. In reality the ampere-turns of the stator and rotor tend to balance one another except for the excitation ampere-turns.  To explain this to those not familiar with motor electrical theory, the excitation ampere-turns are created by the motor no load current. This establishes the magnetic field in the motor necessary to generate a back EMF approximately equal to the applied voltage. As load is applied to the motor, both stator and rotor currents increase together and balance one another, therefore, there are no significant changes in flux.  This means that the basic magnetic forces are independent of load current and are nearly the same at no load or full load. Therefore the
main component of twice line frequency vibration which is created by an unbalanced magnetic pull due to air gap dissymmetry and does not change with load.
On 2 pole motors, the twice line frequency vibration level will appear to modulate over time due to it’s close relationship with 2 times rotational vibration. Problems in a motor such as a rub, loose parts, a bent shaft extension or elliptical bearing journals can cause vibration at 2 times rotational frequency. Due to it’s closeness in frequency to twice line frequency vibration the two levels will add when they are in phase and subtract when they are out of phase and then add again when they return to being in phase. This modulation will repeat at a frequency of 2 times the slip on 2 pole motors. Even at no-load, twice rotation vibration on 2 pole motors will vary from 7200 cpm (120Hz) due to slip. Since there is some slip on Induction motors, although small at no load, it may take 5 to 15 minutes to slip one rotation. For those of you not familiar with the term slip, there is a rotating field around the stator that the rotor is trying to stay in phase with, but the rotor will fall behind the stator field a certain number of revolutions per minute depending upon the load. The greater the load the greater the slip. Slip is typically 1% of rated speed at full load, and decreases to near 0 slip at no-load. Since vibration levels are not constant, to measure vibration, many times it is necessary to perform what is referred to as a modulation test.  In a modulation vibration test the motor is allowed to run for a perio
d of typically 10 or 15 minutes, and vibration is recorded continuously to allow the maximum and minimum to be established.
Elliptical Stator due to Fundamental Flux:
As can be seen in Fig. 3, for 2-pole motors the electromechanical force will attempt to deflect the stator into an elliptical shape. The primary resistance to movement is the strength of the core back iron and the stiffness of the housing around the stator core, which is restraining the core's movement.  On 4 pole motors the distance between the nodes is only 45 mechanical degrees, ½ that seen on 2 pole motors, thereby making the 4 pole stator core much stiffer to movement resulting in much lower twice line frequency vibration.  Calculations on a typical 1000 HP two pole motor at 60 Hz show 120 Hz vibration at the stator core OD of about .12 inches per second, peak, while values for a four pole motor of the same size are only about .02 to .03 inches per second, one sixth to one quarter of this value.  This twice line frequency vibration is transmitted through the motor frame to the bearing brackets where it is reduced somewhat in amplitude.
Fig. 3. Electromechanical Force on 2  & 4 Pole Motors
Non Symmetrical Air-gap:
Twice line frequency vibration levels can significantly increase when the air gap is not symmetrical between the stator and rotor as shown in Fig. 4.
Fig. 4. Unsymmetrical Air Gap Around Rotor
This particular condition will result in the force being greater in the direction of the smaller air gap.  That is, an unbalanced magnetic pull will exist in the direction of the minimum air gap. Force ≈ B2/d
Where B= Flux density
And d= distance across air gap
Of interest here, not only is the stator pulled in one direction, but also the rotor is pulled in the opposite direction, to the side that has the minimum air gap. This causes higher shaft vibration, which is more detrimental to bearing life. Note that in Fig. 4 the rotor OD is concentric with the axis of rotation thereby causing the force to remain a maximum in the direction of minimum air gap.
One Times  Line Frequency Vibration:
Although not nearly as prevalent as twice line frequency vibration, one times line frequency vibration can exist. Unbalanced magnetic pull may result in vibration at line frequency (one times line frequency) as well as the usual twice line frequency vibration.  If the rotor or stator moves from side to side, the point of minimum air gap may move from one side of the motor to the other.  When the frequency of this motion corresponds to the frequency of the traveling flux wave, the unbalanced magnetic pull will shift from side to side with the point of minimum gap, resulting in vibration at line frequency.  This line frequency vibration is normally very small or non-existent, but if the stator or rotor system has a resonance at, or near, line frequency, the vibration may be large. One Times Rota
tion Vibration - Electrical
Eccentric Rotor:
An eccentric rotor, which means the rotor core OD is not concentric with the bearing journals, creates a point of minimum air gap which rotates with the rotor at one times rotational frequency.  Associated with this there will be a net balanced magnetic force acting at the point of minimum air gap, since the force acting at the minimum gap is greater than the force at the maximum gap, as illustrated in Figure 5.  This net unbalance force will rotate at rotational frequency, with the minimum air gap, causing vibration at one time rotational frequency.
The flux causing the magnetic force is the fundamental flux wave, which rotates around the stator at the synchronous speed of the motor.  The rotor attempts to keep up with the rotating flux wave of the stator, but the rotor slips behind the stator field as needed to create the necessary torque for the load.  When the high point of the rotor (point of minimum air gap) aligns with the high point (maximum) of the stator flux, the force will be a maximum, and then it will decrease, becoming small under a point of minimum flux.  Thus, an unbalance force is created which rotates at rotational speed and changes in magnitude with slip.  The end result is a one times rotational speed vibration, which
modulates in amplitude with slip.  This condition occurs at no load or full load.  At no load, the frequency approaches synchronous speed and could have a modulation period of 5 to 15 minutes.  At full load the frequency of modulation in CPM will equal the slip in rpm times the number of poles.  The slip is equal to the synchronous speed minus the full load speed, typically 1% of rated rpm.  For example, a 2-pole motor with a full load speed of 3564 rpm at 60 Hz will have a slip of 3600-3564 = 36 cycles per minute (1% slip) and will result in a modulation frequency of 2*36 = 72 cycles per minute.
Fig. 5.  Eccentric  Rotor
Broken Rotor Bar:
If a broken rotor bar or open braze joint exists, no current will flow in the rotor bar as shown in Fig. 6.  As a result the field in
the rotor around that particular bar will not exist. Therefore the force applied to that side of the rotor would be different from that on the other side of the rotor again creating an unbalanced magnetic force that rotates at one times rotational speed and modulates at a frequency equal to slip frequency times the number of poles.
Fig. 6.  Rotor with Broken Rotor Bar
If one of the rotor bars has a different resistivity a similar phenomenon (as in the case of a broken rotor bar) can exist. It should be noted that this is one of the few conditions that can not be seen at no-load. But there is an additional phenomenon associated with this condition that can be seen at no load after the motor is heated to full load temperature by any method that creates rotor current. These methods would include, coupled full load test, dual frequency heat run, multiple accelerations or heating by locking rotor and applying voltage.  In addition, broken rotor bars or a variation in bar resistivity will cause a variation in heating around the rotor. This in turn can bow the rotor, creating an eccentric rotor, causing basic rotor unbalance and a greater unbalanced magnetic pull, thereby creating a high one times and some minimal twice line frequency vibration.
Rotor Bar Passing Frequency Vibration:
High frequency, load-related magnetic vibration at or near rotor slot passing frequency is generated in the motor stator when current is induced into the rotor bars under load.  The magnitude of this vibration varies with load, increasing as load increases. The electrical current in the bars creates a magnetic field around the bars that applies an attracting force to the stator teeth.  These radial and tangential forces which are applied to the stator teeth, as seen in Fig. 7, create vibration of the stator core and teeth.
This source of vibration is at a frequency which is much greater than frequencies normally measured during normal vibration tests. Due to the extremely high frequencies, even very low displacements can cause high velocities if the frequency range under test is opened up to include these frequencies. Though these levels and frequencies can be picked up on the motor frame and bearing housings, significant levels of vibration at these higher frequencies will not be seen between shaft and bearing housing where they could be damaging.  For this reason vibration specification requirements normally do not require that these frequencies be included in overall vibration.
Fig. 7. Magnetic Field around Rotor Bar and Resulting Force
on Stator Teeth
Since vibration at rotor bar passing frequency occurs at a high frequency, the vibration velocity level may be significant, but the effect on motor reliability is insignificant.  Considering the stress that results in the motor as a consequence of the vibration makes this determination.    For example, suppose a two pole motor exhibiting a vibration at 2800 Hz due to rotor bar passing frequency plus a 120 Hz side band:
Velocity, (IPS) 0.1 0.5
Displacement (mils) 0.011 .057
Stress in Stator Core Iron 30 psi 150 psi
Stress in Stator Tooth Iron 50 psi 250 psi
The typical fatigue strength of the core iron is 35,000 psi. Similar low stress levels can be calculated for all parts of the motor (including the stator windings).  In addition, the typical minimum oil film thickness ranges from 1.0 mils to 1.5 mils. Since only a small displacement such as .011 to .057 mils as mentioned above could be seen, this vibration will not have an adverse affect on bearing performance.
The rotor slot and side band frequencies are in the frequency range normally related to noise rather than vibration performance, and are taken into account in noise predictions during motor design.  In fact, these force components are the principal sources of high frequency noise in electrical machines, which has been for some time subject to noise regulations and limits.  Experience has shown that where noise has been within normal or even high ranges, there has been no associated structural damage.  The significance of these high frequency vibrations is distorted by taking measurements in velocity and then applying limits based on experience with lower frequency vibratio
n.
Load Related Magnetic Force Frequencies and Mode Shapes The frequencies of the load related magnetic forces applied to the stator teeth and core equal the passing frequency of the
rotor bars plus side bands at + or – 2f, 4f, 6f and 8f Hz, where f is the line frequency.  A magnetic force is generated at the passing frequency of the rotor slot (FQR), which is motor speed in revolution per second times the number of rotor slots as shown in (3).
FQR = RPM*Nr / 60, Hz (3)
where
Nr = number of rotor slots
For the typical two pole 3570 rpm motor with 45 rotor slots in the example above, FQR = 2680 Hz.
The side bands are created when the amplitude of this force is modulated at two times the frequency of the power source. On a 60 Hz system the 120 Hz modulation produces the side bands, giving excitation frequencies of FQR, FQR + 120, FQR – 120, FQR + 240, FQR – 240 Hz, etc.
The forces applied to the stator teeth are not evenly distributed to every tooth at any instant in time; they are applied with different magnitudes at different teeth, depending upon the relative rotor- and stator-tooth location.  This results in force waves over the stator circumference.  The mode shape of these magnetic force waves is a result of the difference between the number of rotor and stator slots as shown in (4).
M = (N s – N r) +/-KP (4)
Where
N s  =  number of stator slots
N r  =  number of rotor slots
P    =  number of poles
K  =  all integers 0, 1, 2, 3, etc.
Mode Shapes and Natural Frequencies of Core Vibration: Under the applied magnetic forces the stator core is set into vibration in the same manner that a ring of steel would respond if struck.  Depending upon the modal pattern and frequencies of the exciting force, as described above, the stator would vibrate in one or more of its flexural modes m of vibration, as shown in Figure 8.  Each of the mode shapes has its associated natural frequency.  The core may be somewhat influenced by the stator frame in actuality, but in analysis the frame is usually neglected, both due to complexity an
d because the effect on higher frequency modes is minimal.
To understand the resonant frequency of the core at a given mode of vibration, the core can be represented as a beam, which is simply supported on both ends and flexes between the ends due to forces applied on the beam.  The length of the beam is equal to the circumferential length of the mean diameter of the stator core for one-half the mode wave length (see Fig. 9) [8].
M
D
L s
2
Π
=
If the resonant frequency of the core is close to the forcing frequency, a high level of vibration will result.  The lower modes of vibration may produce resonant frequencies that are
close to the primary forcing frequencies.
modulateFig.9.  a) Fourth Mode of Vibration
b) Linear Representation of Core
for one-half Wavelength of Force

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