EVALUATION OF MOTOR STRUCTURAL DYNAMIC RESPONSE BY

FORCED VIBRATION ANALYSIS

Copyright Material IEEE Paper No. PCIC-2013-43

Rafael Fialho, P.E. Suresh Pinapati, P.E. Renato Yabiku, P.E. GE Motors GE Motors GE Motors Campinas, Brazil Hyderabad, India Campinas, Brazil rafael.fialho@ge.com suresh.pinapati@ge.com Renato.yabiku@ge.com

Abstract – The manufacturer assessment during the design stage inception for potential resonances near any predicted operating force as well as the overall system amplitude response is essential to ensure the proper field operation. Using a model that takes into account the magnetic and mechanical loads it is possible to optimize the design and improve reliability, minimize resonances and reduce vibration amplitudes of the structure, shaft and bearing housing This paper discusses the forced vibration methodology to assess the system amplitude response when subject to forces coming from the motor itself – such as mechanical unbalances and magnetic unbalance pull. A brief design tutorial associated to the analysis is presented, such as the structural stiffness sensitivities, modeling of the fluid film bearing characteristics, mass lumping techniques and damping coefficient that allow the forced vibration analysis. To corroborate with the proposed model two studies case are discussed and used as validation for the proposed model. Index Terms — industrial motors, vibration, forced

vibration, modal analysis, vibration amplitude prediction

I. INTRODUCTION

Surveys performed by many industries list vibration as one

of the top issues associated to electric motor failure, therefore a continuous effort from many stakeholders to understand it sources, propose design methodologies, operation and maintenance control to mitigate them are continuously updated and reinforced. Vibration data is a critical motor quality indicator and must

be kept as low as possible. Predicting the vibration levels of the motor early during design stage helps to achieve an outstanding behavior at field, reducing down time and extending the operating life. To mitigate the field vibration issues, several techniques

can be employed such as: a. Dampening the vibration source to reduce the

amplitudes by using pads or isolation. b. Changing stiffness, mass or the ratio among them

to move the natural frequency; c. Fine tuning the balancing d. Fine tuning the alignment among components:

motor and load; e. Using different coupling constructions to isolate

sources of vibration excitation

Such approaches are sometimes required, but indeed demand significant effort, time and are usually costly when performed at site Nevertheless the vibration behavior measured in the field

may derive from components and geometry selection coming from the very beginning of the designing stage – definition of the bearing type, the geometry of the rotor and shaft, the frame and end shield geometry and therefore its stiffness and damping capabilities are examples of the manufacturer assessments and tradeoffs carried out during the process that significantly impact the performance and unbalance sensitivity. Among alternatives the following two methods are

frequently used during the designing: I. Modal analysis: design the motor structure to keep

any exciting frequencies away from resonances. A good practice comes from API 541 [1], which calls for a separation margin (SM); II. Force vibration: perform a forced vibration analysis

to calculate the amplitudes at desired voltage frequencies and rotational speed – usually the rated speed when the motor is a fixed speed type. Each case provides inputs to further work on the system –

frame, end shields, shaft, etc – in order to achieve the desired vibration levels. The modal analysis approach is focused on the separation

margin requirement and does not reveal the actual vibration amplitudes of the motor when subjected to the mechanical and magnetic forces, The forced vibration approach, also called frequency

response analysis, gives a vibration signature of the complete motor at various frequencies at a given exciting force. The designer may be interested in specific exciting frequencies like 1X, 1F, 2X, 2F (X - Rotational frequency and F – electrical line frequency). The method provides a significant data quality vibration prediction. The exciting forces and their influence on the resonating

frequencies are discussed in Section II. The assumptions and a brief tutorial about the modeling

are presented in Section III. Section IV will give some highlights regarding calibration

process to validate the numerical model. Section V gives a result overview. Finally at Section VI there will be the case studies where it

is possible to verify the correlation for one tested machine designed following this method.

The foundation stiff nesses are considered as infinitely rigid for the sake of the present analysis, as this is a traditional assumption when the actual data is not available.

II. MAJOR EXCITING FORCES IN A INDUCTION

MOTOR

A typical horizontal induction motor frame is shown in Fig

1. The frame, shields, cooler, etc are subjected to the forces that are inherent of an electric motor. The magnetic force attraction between the stator core and the rotor can be balanced by a perfectly aligned and concentric stator bore and rotor outer diameter, however the attraction forces among them are still there and yield to deformations that ultimately reflect on the vibration measurement. The selection of dynamic forces considered while setting

the model for the analysis is based on their relevance to excite the motor as a whole. In this paper the following forces were considered: 1) mechanical unbalance; 2) magnetic attraction among stator and rotor both, fully balanced and unbalanced due to hypothetical rotor to stator misalignment in the air gap. It is also worth that other exciting forces from different

sources could be easily introduced in the analyses if required. The modeled frame system can be used as it is, demanding updates on the applied forces only that should be previously calculated to represent the expected phenomena.

Fig 1 Typical stator frame model

A. Mechanical Unbalance The mechanical unbalance is inherent to any rotating

system due to the non-homogenous mass distribution arising from unavoidable manufacturing tolerances. The rotor is made of multiple parts, including shaft, laminations, bars, short-circuit rings, retaining rings and fans. An unbalance mass on them will cause lateral vibration and also, according the bearing and end shield structure, excite axial displacement. Balancing is more critical and also more difficult to perform on rotors that operate above its first critical speed than the ones that operates below it. For the constructions operating above the first critical

speed, also known as flexible rotor constructions, a balancing

procedure performed at lower speed than the critical speed may not yield a good balancing at rated condition. For example, if a given rotor has 2200RPM as the critical

speed, any balancing correction performed at lower speeds than this may not represent a solution when the rotor is spinning at 3600 RPM. The mechanical unbalance force has a frequency of 1X,

where X is the rotor speed. The force magnitude is function of the residual unbalance and the speed. It is common to specify the maximum allowable residual

unbalance according industrial standards as ISO 1940 and API 541. For instance, API 541 – 4th Ed. defines the maximum allowable residual unbalance mass in correction plane as:

Nmc

WrUb 6350= (1)

where Ub residual unbalance [g*mm]; Wr journal static loading [Kg]; Nmc maximum continuous [RPM]. The mechanical unbalance is transmitted to the frame-

stator subsystem through the bearings. Fig 2 depicts the schematic of rotor unbalance in stator-rotor representation. Also, this picture has an infinite base – foundation stiffness

consideration.

Fig 2 Mechanical unbalance

B. Magnetic Force and Magnetic Unbalance The magnetic force is inherent to any induction motor and

it is caused by the core magnetization. The stator / rotor magnetic interaction through air the gap generates an attraction force between them. The current flowing through the winding creates a revolving

magnetic field that has a peak flux density. The speed of the revolving filed is associated with the power supply frequency. For instance, a 2 pole machine fed by a 60Hz power supply has a revolving field that has approximately a sine distribution through the air gap periphery with a peak positive and another negative value. In regards to the attraction among the rotor and stator surface the fundamental frequency amplitude is the only significant one, therefore for a 2 pole machine, there are 2 peak flux densities (one negative another positive), and they create 2 peak attraction force

locations. These peak force locations, apart 180 degrees in these machines, revolve at 60 Hz. This yields to a stationary observer a perception of a force pulsating at 120Hz. The relationship of magnetic force and flux density and air

gap is shown in eq. 2. The magnetic force attraction, distributed through the periphery of the stator bore in a certain moment is illustrated in Fig 3. The following equation represents the magnetic flux force:

a

BFmag

2

α→ (2)

where,

Fmag magnetic flux foce [N]; B flux density [Tesla];

a air gap distance [m].

Fig 3 Pattern of magnetic flux pressure profile for 2 pole

motor Stator core vibrations due to the alternating magnetic force

can be transferred to the frame to a certain extent. A method to isolate the link and therefore the frame response to the core excitation is by using springs or other isolating techniques that enable the stator to vibrate freely, not transmitting force to the frame. Also another source of excitation would be unbalanced

magnetic pull (Fig 4). This is indeed a force that is supposed to be reduced by ensuring a tight concentricity among the rotor and stator. Nevertheless it is recognized that a normal capacity –machining and assembly – may yield approx. 10% air gap difference throughout the periphery of the air gap. For the sake of engineering analysis it is a fair assumption to estimate the frame performance considering that such lack of concentricity will happen. As a worst case scenario however the manufacturing process shall always be set to have it minimized during the production. The force estimate is accurate as the flux density levels

derivate from the electromagnetic design. Parts variations, assembly tolerances, run out, geometrical

imperfections of the shaft, thermal bow etc lead to non-uniform air gap, that yield to unbalanced magnetic forces. The magnitude of magnetic pull is higher in the direction of

minimum air gap. The results presented at this paper consider a machine

operating at rated voltage. To take into account any overvoltage, due to drive usage, the magnetic flux density shall be adjusted to provide the proper force magnitude.

Fig 4 Magnetic unbalance

III. FORCED VIBRATION ANALYSIS

As mention in Section I, the motor frequencies can be

obtained by using modal analysis. The equations of motion for free vibration may be expressed by the following equation and Eigen values, Eigen vectors can be calculated using this:

0*2

/2

* =+ xkdtxdm (3)

where, m mass [kg]; k stiffness [N/m]; x displacement [m]; d

2x/dt

2 acceleration [m/s

2].

Modal analysis can calculate the natural frequencies of the

motor; however it provides limited information about the amplitudes as the forces are not set to reflect the actual sources from the machine in this analysis. In other words, the analysis is carried using a generic excitation that has no correlation with the actual forces and excitations arising from the mechanical or electromagnetic phenomena that occur. Hence, the common output of the modal analysis is the

resonance frequency only, with little or no information that enable checking the vibration amplitudes when the system faces the expected forces inherent to the electromagnetic design and expected manufacturing variability that yields to potential mechanical unbalance. The modal analysis is extremely useful to identify the

vibration modes, enabling a qualitative evaluation about the critical ones for specific criteria – vibration and noise mainly. For the sake of the vibration analysis, focus of this discussion, the low order modes are the most relevant. Using the modal analysis results, to investigate the location of the highest displacement amplitudes in the model it is very important, and shall be used to define the model for the forced vibration analysis. A more effective deformation prediction result is expected when the model resonances are within a good control level. The forced vibration approach is a design technique that

gives the vibration amplitudes at any desired location and at any specific frequency, but as highlighted previously it is necessary to carry out pre analysis to define the relevant forces and locations to be observed in the model. The forced vibration provides additional information that

the modal analysis does not offer. With the forced analysis

addition during design stage it is possible to access also how should the machine vibrate regarding its amplitude. This allow an extra design point where it is possible to compare these values with the applied standard and other machines with test data to inquiry the predicted numbers. Ultimately, to ensure a reliable operation, it is imperative to

have a system comprised by the frame, end shields, rotor, cooler, etc that will have a reduced response to the mechanical and electrical unbalance, despite how close or how far the natural frequencies are.

Fig 5 Flow chart for forced vibration analysis

The steps involved in performing the forced vibration

analysis using finite element analysis are listed in Fig 5. Finite Element Analysis includes geometric modeling,

meshing, applying loads, boundary conditions and post processing the results. Performing the analysis for the particular frequency range of interest and capturing the amplitudes and stresses at the right frequencies (could be 1X, 2X) is vital. The equation of motion for forced vibration is expressed

as:

)(*/*2

/2

* wFxkdtdxcdtxdm =++ (4)

where, c damping [N-sec/m]; dx/dt velocity [m/s]; F exciting force [N];

w frequency of the exciting force [Hz]. The forcing function F(w) in equation 4 is discussed in

Section II. The mode superposition method is applied to analyze the

case discussed in the following section. Mode superposition is a method of using the natural frequencies and mode shapes from the modal analysis to characterize the dynamic response of a structure to steady harmonic excitations.

A. Modeling the fluid film bearing characteristics Critical speeds of the rotor are highly dependent on oil film

bearing stiffness and damping coefficients in addition to shaft and support stiffness. Due to its complex geometry of sleeve bearings, it is difficult to build a precise model with exact stiffness levels. Fluid film stiffness and damping in plain cylindrical journal

bearings are anisotropic in nature. Usually the horizontal stiffness is lower than the vertical stiffness. A simplified modeling approach used to represent the actual stiffness of the bearings is shown in Fig 6. In the model the base stiffness is not presented as it was

assumed as a massive one. This assumption may not represent the actual construction, and if necessary another spring element in each direction shall be considered, or alternatively, the frame stiffness spring can be adjusted by reflect the combined values. Both approaches yield the same results, and very few questions arises while deciding among one or another alternative. Although the way to represent it in the model is not critical, the base stiffness value per se is extremely relevant and can impair severely the dynamic performance. The rigidity of a foundation is a relative quantity. It must be

compared with the rigidity of the machine bearing system. The ratio of bearing-housing vibration to foundation vibration is a characteristic quantity for the evaluation of foundation flexibility influences. One indication that a foundation is massive is if the vibration amplitudes of the foundation (in any direction) near the machine feet or base frame are less than 30% of the amplitudes that could be measured at the adjacent bearing housing in any direction [1].

B. Lumped mass for cooler Natural frequencies of the motor are highly dependent on

stiffness and mass of sub systems. Ventilation is one such sub system which contributes less in stiffness and more in mass. A lumped mass approach is used to address the contribution of a cooler to the motor. Cooler and enclosing covers are represented as a lumped mass positioned at the center of gravity.

Fig 6 Fluid film and bearing equivalent model

IV. CALIBRATION

A. Damping ratio The severity of the vibration depends on the amount of

damping within the motor. Damping for the complete motor is the sum of many separate system responses. In general, increased damping will decrease the amplitude of vibration at resonance. The measurement permit the calculation of the damping-ratio associated at system level. It is possible to estimate the amount of damping present at a particular mode from the amplitude response curve at the resonance. Damping is determined from the sharpness of the peak. Damping ratio is calculated in the analysis for the complete motor by matching the impulse response of the test, assuming the FE Model is accurately modeled.

B. Natural Frequencies As stated previously, the forced vibration analysis is

usually performed afterward the modal analysis evaluation. As the machine model is the very same for both analysis and only the boundary conditions of the original modal analysis are required to be updated to consider the excitation forces of interest prior running the forced analysis. Nevertheless, it is beneficial when test data is available, to

perform a model calibration and validation between the two analyses. Comparing the natural frequency measurement is an

important step towards the validation of the FE model. The calibration of the model using test frequencies imparts the real behavior of motor to simulation. The bump test is a test procedure that enables the

measurement of the frequencies. A series of accelerometers are attached to the structure and have their readings recorded during a strike promoted by a hammer provided with another sensor, used to synchronize the vibration reading with the strike moment. The bump test data, post processed can be used to generate Figures that can reveal the system response and to be assessed against the analysis result. Fig 7 and Fig 8 put side by side the test data and the model for the horizontal and axial modes, which are highly significant for the purpose of the vibration studies in discussion. Bump test measurement techniques and model calibration

suggestions can be found in Ewins [2]. Provided all information coming from the modal analysis

and bump test data, it is possible to design a machine with all natural frequencies with enough separation margin to any

forcing frequency foreseen under operation. The range of frequencies that are undesired is usually referred as exclusion zone.

Fig 7 Bump test result 1 – lateral mode at 75.2 Hz

Fig 8 FEA calculation – lateral mode at 81.3 Hz

Provided all information coming from the modal analysis

and bump test data, it is possible to design a machine with all natural frequencies with enough separation margin to any forcing frequency foreseen under operation. The range of frequencies that are undesired is usually referred as exclusion zone.

V. RESULTS Designing the system to have no natural frequencies within

the exclusion zone is a strong recommended practice. However, this criterion may not be enough to ensure a stable and reliable performance. If the excitation force is applied to the system, even away from any natural frequency, it may have energy enough to make the whole machine vibrate at undesirable displacement levels. By using forced vibration method, the system response

amplitudes in any frequency, included the one that are listed in the exclusion zone, can be evaluated. Exclusion zones for 2 pole motors fed by a 60Hz power

supply, as per API 541 [1], are the ones comprised in the range from 51Hz to 69Hz and from 111Hz to 129Hz. (Fig 9)

Fig 9 Dynamic response with exclusion zone highlighted The dynamic response of the motor at locations of interest

(e.g. bearing housing) are captured, with the input of various forcing functions discussed in section II, at appropriate frequencies. Dynamic response at bearing housing in frequency domain is shown Fig 9. It depicts the natural frequencies of the motor are away from exclusion zone and the displacements are below the design criteria. Using FEA, stresses at maximum amplitude due to dynamic forces are calculated. Structural stresses on frame and brackets due to the mechanical unbalance load are shown in Fig 10.

Fig 10 Deformation plot

The output of forced response analysis gives a clear

evaluation of the expected vibration on the machine, as it takes into account the forces foreseen during the operation. In addition, it is possible to correlate modeling simulation

output to actual test data (velocity and/or displacement) that are usually part of the routine testing. The correlation and data evaluation is helpful for model assumptions validation that ultimately can help to improve the accuracy of updated models. The fatigue life of the machine can be calculated for the

stresses due to dynamic response at various locations especially at welds and foot mounting plates. This is an important consideration, as the static loads may be surpassed by the dynamics in some components.

VI. CASE STUDIES

A. Case study I The motors tested in this case study are 2 pole motors, 3.8

or 4kV, 1006 HP, with WPI and TEAAC enclosures. The vibrations amplitudes are taken at the bearing housing

at the frequency of 60Hz. Tests were carried to measure the amplitudes at the same bearing locations and compared the FEA output. Understanding the forces acting on the motor while

operating and corresponding forcing frequencies is critical as the motor unbalance will act at operating frequency and the magnetic loads at line frequency. Amplitudes of the motor are different at different load cases and the presented vibration amplitudes in Fig 12 are maximum amplitudes among all load cases.

Fig 11 Setting the model – including pull and unbalance

magnetic forces Fig 12 presents the vibration amplitudes from tested and

calculated data at bearing: • Unit A: 2 pole – 1000 HP – 4.0 kV – TEAAC • Unit B: 2 pole – 1100 HP – 3.8 kV – WPI

Fig 12 Filtered vibration amplitude at 60 Hz measurement The vibration amplitudes indicate that all motors are

compliant with the requirements – amplitudes lower than 0.40 mil pk-pk for 60 Hz filtered vibration.

Considering several key critical functionalities of components like fluid film characteristics, complex core construction, ventilation influence, foundation stiffness, etc makes the system very complex. It is recommended to calibrate the FE model using the bump test mode shapes to ensure model accuracy and thus to achieve accurate data.

B. Case study II The second case presented discusses the benefits of

using the forced vibration methodology along with the modal analysis. The design, although featuring the peak displacement

under the modal analysis with a decent separation margin had to be modified to meet the requirements of maximum vibration displacement at rated speed. The modification need only became evident after the

mechanical unbalance and magnetic forces were applied over the model following the steps previously presented. From the modal analysis, the natural frequency of the

motor is 134Hz (Fig 13) and is meeting the API 541 separation criteria i.e natural frequency should out of the range comprised from 111Hz up to 129Hz, as this is a 60Hz fed machine. Using this Figure to perform the forced vibration analysis it

is identified that the vibration amplitude is more than the design limit established. The system response is presented in Fig 13, and at 120Hz

the forecasted displacement is 0.75 mil pk-pk. This amplitude is not related to shaft displacement but to bearing housing vibration displacement. Further design changes were made to stiffer the frame.

With this change the natural frequency of the motor is barely modified, but the amplitude is limited to meet 0.10 mil pk-pk the amplitudes requirements.

Fig 13 Vibration amplitudes on bearing housing

The changes were significant for the stiffness although

minor to the mass, not impacting significantly the total motor weight. For this specific case 12 tubes in the upper portion of the frame were added to stiffen the system at axial direction (Fig 14). Also the bracket thickness was slightly increased, by approximately half an inch.

Fig 14 Reinforced frame after analysis

If the method was not used a common recipe of just

improving the overall plate thickness may be applied and solve the issue, but probably with more weight and more costly than the final solution. Another advantage was that the existing frame was not discarded after issue due to localized rework done. After the modifications, the motor was tested and the

measured vibration was within limits.

VII. CONCLUSION

The method here presented was capable of proving the

required information regarding a machine vibration with load cases discussed – mechanical unbalance, magnetic excitation forces and magnetic unbalance. The requirement to have a good separation margin may

not be enough to ensure a successful operation. To foresee this issue it is a good practice to perform forced vibration analysis. The model shall be developed with care to ensure correct

correlations with tested machine to avoid miss decisions in design stage. With the presented approach and calibration it is possible to ensure a good relationship between design and test. According to case studies, the technique was able to

predict the machine vibration with a reasonable degree of accuracy to meet expected vibration levels. Also the approach can be used as a firewall against excessive displacement although operation far from any resonance. In addition the method can be used to provide special

designs with vibration amplitudes meeting requirements and with reduced prototyping time. This is very important in cases were the total machine weight is very limited. The forced vibration analysis procedure was applied in

both cases presented and the vibration amplitudes results were compliant with API 541. So, the method aligned with a good database can predict and provide reliable designs very earlier in the design stage. The error between the test and calculation is not negligible

but it was within an acceptable variance when new test data were added to the comparison. Even so more units shall be analyzed in future to update model. In the end, with a model accurate and information about

load forces – i.e. pump, compressor or fan – and coupling type it is possible to simulate motor behavior at full load condition. This is subject for a future planned presentation.

VIII. REFERENCES

[1] Form-Wound Squirrel-Cage Induction Motors-500

Horsepower and Larger, ANSI/API 4th ed., American Petroleum Institute, Standard 541,Jun. 2004. [2] D. J. Ewins, Modal Testing: Theory and Practice,

London, England: Research Studies Press LTD, 1983. [3] William R. Finley, An Analytical Approach to Solving

Motor Vibration Problems, IEEE Transactions of Industry Applications, Paper No. PCIC-99-20. [4] Sumit Singhal and Rajendra Mistry., “Influence of

design, manufacturing & installation practices on rotor vibrations of large induction motors used in petrochemical Industry” 2011, PCIC Europe conference proceedings, Page Nos. 1-10. [5] Ishibashi, F; Noda, S; Mochizuki, M., “Numerical

simulation of electromagnetic vibration of small induction motors” 1998, Ower, Electronics, IEEE Transaction, Vol No 145, Page Nos 528-534.

IX. VITA

Rafael de Lima Fialho graduated from State University of

Campinas (UNICAMP), Brazil, in Control and Automation Engineering. He joined GE Energy Motors in 2006 as an intern. In 2008 he became effective as Engineering Analyst. Currently, he is the leader for material productivity projects for motors and generators. He is co-author of previous PCIC papers. Renato Yabiku graduated from Universidade de São Paulo

- USP in 2003. Worked as a design engineer in Equacional Company till 2005. Since 2005 he joined GE and worked on designing and product management since then. He is author of previous papers in IEEE and MOMAG about special topology motors, temperature rise estimate on induction machines, and discussions about squirrel cage surface temperature for starting and conceptual trade off among different materials for squirrel cage constructions. He worked as a teacher on energy conversion lectures for Universidade Paulista - UNIP. Suresh Pinapati is post graduated from the Indian Institute

of Technology-Chennai, India in Applied mechanics. He joined GE in 2000, since then he has been involved in development of several new products for both motor controls and motors design. Now he works as technical leader engineer in motors material productivity, in the GE Technology Center in Hyderabad, India.