torsional modal analysis of single shaft systems

Torsional modal analysis of single shaft systems containing permanent magnet synchronous motors using novel broadband torque excitation

T. Kimpián1, F. Augusztinovicz2 1 ThyssenKrupp Presta AG, Competence Center Acoustics

Essanestrasse 10, FL-9492 Eschen, Liechtenstein

e-mail: [email protected]

2 Budapest University of Technology and Economics, Department of Telecommunications

Magyar Tudósok körútja 2., H-1117, Budapest, Hungary

e-mail: [email protected]

Abstract The growing demand in the automotive industry for high performance DC motor drives is more and more

satisfied using permanent magnet synchronous motors. Naturally the development and testing of these

motors require – beside others – sophisticated experimental torsional modal analysis techniques. In this

paper a novel torsional modal analysis method is presented, which is based on a unique broadband torque

excitation that is provided by a permanent magnet synchronous motor. As in tipical cases the synchronous

motor is also part of the investigated mechanical system no additional external excitation is required.

Another great advantage of this excitation method is that the investigated system does not have to rotate

during the measurements, therefore no special torsional vibration instrumentation is needed, so standard

accelerometers and modal analysis software can be used for the data aquisition and for the mode shape

extration too.

1 Introduction

The continuous pressure on the automotive industry to reduce fuel consumption and CO2 emission in

conjunction with the latest requirements towards the steering systems such as automatic parking, easily

adjustable steering feel while driving, lack of hydraulic oil, etc. created a high demand of Electric Power

Assisted Steering (EPAS) Systems on the automotive supplier market. To fulfill this demand with a

growing market share ThyssenKrupp Presta develops and produces various EPAS systems for different

vehicle classes from mid sized through SUVs to luxury cars.

To follow the continuously increasing rack-force requirements and to challenge the shrinking space for

packaging high power density electric drives are used incorporating Permanent Magnet Synchronous

Motors (PMSMs). The reduced packaging space also means more challenging magnetic design for the

motors; therefore the NVH (Noise Vibration and Harshness) engineers also face usually higher torque-

ripple and higher level of housing vibration on this EPAS component.

For the supplier of a complete EPAS system the above mentioned trends mean more sophisticated

development processes that include special care on NVH issues. At ThyssenKrupp Presta one pillar of the

development process is the detailed testing of the EPAS components against strict NVH requirements. In

case of the electric drive – for the PMSM alone and also with the dedicated drive electronics together – the

signature analysis of the vibration as well as the torque signal is a key element. To get representative

component level test data, the testrig conditions (motor speed, load, mounting, etc.) have to be similar to

the real operating conditions, therefore the appropriate load has to be applied on the motor.

2081

Figure 1: Typical test setup for motor NVH tests. 1.: Tested motor, 2.: Accelerometer, 3.: Adapter, 4.:

Clutch, 5.: Torque sensor, 6.: Clutch, 7.: Load motor

A typical testrig setup for motor tests can be seen on figure 1. To apply the necessary load on the EPAS

motor an asynchronous load motor (7.) is attached directly to the motor shaft via clutches (3., 4., 6.) and a

torque sensor (5.). In case of this test setup the appropriate measurement of the motor vibration can be

easily achieved by means of a stiff fixation of the motor (for example in the middle of the housing like on

figure 1., or on the front shield), but the measurement of the torque-ripple in a wider speed and analysis

frequency range is more challenging. The most difficulties arise from the fact that the spectrum of the

measured torque signal is modified by the torsional resonances of the shaft assembly, thus making the

spectral and especially the order analysis of the torque signal cumbersome.

In this article a new torsional modal analysis method is presented that was developed to approach the

problem of the torsional resonances in a new way, without the use of the standard equipment of torsional

vibration analysis. To achieve this goal first of all the physical analogy between torsional and rectilinear

systems were studied, and to develop the proper torsional excitation method, this analogy was extended to

electrodynamic transducers, to take the advantage of the presence of a PMSM in the torsional system.

After the theoretical explanation the experimental setup and the results of the modal analysis will be

presented. The article is closed by the conclusions and the generalization of the excitation method is also

briefly discussed.

2 Analogy between torsional and rectilinear mechanical systems

2.1 Analogy between the basic variables

Table 1. shows the well known analogy between the rectilinear and torsional systems [1]. As the equation

of motion is the same differential equation for both types of mechanical systems, the methods used in

experimental modal analysis of rectilinear systems can be easily adapted to torsional systems. The key

issue is to find a way to excite and measure the corresponding variables in the torsional systems. In the

past hundred years plenty of ideas, solutions and established methods were developed and used for

torsional modal analysis, however nowadays the spread of high power density permanent magnet

synchronous motors enabled the use of a method that is based on the analogy between the operation of

electrodynamic shakers and PMSMs.

1.

2.

3. 4.

5.

6.

7.

2082 PROCEEDINGS OF ISMA2012-USD2012

Physical quantity Rectilinear vibration Torsional vibration

Symbol Unit Symbol Unit

Time t s t s

Displacement x m ϕ rad

Velocity x& , v m/s ϕ& , θ rad/s

Acceleration x&& , a m/s2 ϕ&& , β rad/s

2

Force / Torque F N M Nm

Mass / Inertia m kg J Nms2/rad

Spring constant k N/m K Nm/rad

Damping

coefficient c Ns/m C Nms/rad

Equation of

motion Fkxxcxm =++ &&& MKCJ =++ ϕϕϕ &&&

Natural frequency m

k=0ω rad/sec

J

K=0ω rad/sec

Damping factor ξ dimensionless ξ dimensionless

Table 1: Analogy between rectilinear and torsional mechanical systems

2.2 Analogy between conventional electrodynamic transducers and permanent magnet synchronous motors

On figure 2. a typical cross-section of an electrodynamic shaker is shown [2]. The connection between the

electrical and mechanical variables is very simple [3].

Figure 2: Cross-sectional view of an electrodynamic shaker

The force F acting on the winding is proportional to the current i flowing through the coil, and the

induced voltage u on the winding is proportional to the linear velocity v of the coil. The proportionality

MODAL TESTING: METHODS AND CASE STUDIES 2083

coefficient T is the length of the wire l in the homogenous field multiplied by the magnetic flux density

B . Therefore the equations describing the electrodynamic shaker as a transducer:

)()()( tTvtBlvtu == (1)

)()()( tTitBlitF == (2)

Figure 3: Cross-sectional view of a PMSM

A typical simplified cross-section of a PMSM can be seen on figure 3. In EPAS applications the use of

motors with sinusoidal magnetic field distribution is relatively common, therefore

)sin(ˆ)( ϕϕ pΨ=Ψ , (3)

where Ψ̂ is the magnetic flux density, p is the number of pole-pairs and ϕ is the angular position of the

rotor. To calculate the induced voltage, the flux density has to be expressed as a function of time, as the

induced voltage )(tu is the time derivative of the flux density [4].

))(sin(ˆ)( tpt ϕΨ=Ψ (4)

dt

tdptp

dt

tdtu

)())(cos(ˆ)(

)(ϕ

ϕΨ=Ψ

= (5)

To verify the correctness of this formula, let us calculate the induced voltage in case of constt =Θ )(

constant rotor speed:

00

0 260

2)( fN

constt ππ ==Θ==Θ , (6)

where 0N and 0f are the rotor speed in [ ]rpm and [ ]Hz respectively. To insert the above result in

equation (5) )(tϕ has to be calculated first.

[ ] 000000

0

00

0

)()( ϕϕτϕτϕττϕ +Θ=+Θ=+Θ=+Θ= ∫∫ tdddtt

tt

(7)

Inserting the result of equation (7) in (5) gives the following result

))2(cos(ˆ))(cos(ˆ)( 000000 ϕπϕ +ΨΘ=Θ+ΘΨ= tfppptptu (8)

As expected in one phase the induced voltage will be a sine wave with 0pf frequency and 0ϕp initial

phase. Of course in case of a three phase two pole motor the windings are dividing the circumference by

120° so that the induced voltages will also be shifted by 120°.


In case of vibroacoustic investigation, it will be shown, that it is meaningful to assume a single phase

sinusoidal excitation, hence sinusoidal angular speed with ω frequency and Θ̂ amplitude.

)cos(ˆ)( tt ωΘ=Θ (9)

Again let us first calculate the angle from the angular speed:

000

0

0

0

)sin(ˆ)sin(ˆ)sin(ˆ

)cos(ˆ)( ϕωϕϕωω

ϕω

ωτϕτωτϕ +=+

Θ=+

Θ=+Θ= ∫ ttdt

tt

(10)

Naturally sinusoidal angular speed means sinusoidal displacement with a peak amplitude ϕ̂ and an initial

phase 0ϕ . Inserting the above result into (5):

)cos(ˆ)))sin(ˆ(cos(ˆ)( 0 tptptu ωϕωϕ Θ+Ψ= (11)

Assuming that 00ˆ ϕϕϕ ≅+ meaning that the amplitude of the resulting torsional vibration is very small,

(11) can be linearized:

)()()()cos(ˆ)cos(ˆ)cos(ˆ)( 000 tTtpptpptu Θ=ΘΨ=ΘΨ= ϕϕωϕ (12)

The practical meaning of (12) is very substantial. This equation means that around an equilibrium position

of the rotor, the vibration angular velocity of a small amplitude torsional vibration is proportional to the

induced voltage. This is analog to the behavior of a rectilinear electromagnetic transducer, except that the

electromagnetic transmission ratio depends on the actual angular position of the rotor.

Using the same assumption, that the amplitude of the torsional vibration is small and that the flux density

has a sinusoidal distribution (3), the connection between the generated torque )(tM and the phase current

)(ti provides similar results with much less calculation steps.

)()()()cos(ˆ)sin(ˆ)()()( 00 tiTtipp

d

pdti

d

dtitM ϕϕ

ϕ

ϕ

ϕ=Ψ=

Ψ=

Ψ= (13)

Equation (12) and (13) describe the behavior of a PMSM in case of single phase broadband excitation and

small amplitude torsional vibration of the rotor around an equilibrium position. The absolute value of the

angular position determines the coupling between the electric and the torsional subsystems.

3 The proposed method for experimental torsional modal analysis

3.1 The experimental setup

As shown in the previous section, broadband torque excitation can be produced by a PMSM by using

single phase current excitation while the shaft system is standing still. This extremely useful feature of a

PMSM can be utilized in experimental modal analysis, as this is an easy way of providing the necessary

known torsional excitation of the investigated shaft assembly.

The biggest benefit of this method is that the measurement of the angular velocity can be extremely

simplified. At a certain measurement point the angular velocity is given by the tangential channel of a

standard triaxial accelerometer multiplied by the radius of the measurement point.


Figure 4: The experimental setup. The measurement points on the shaft assembly and the accelerometer

placement. The motor was driven on a single phase by broadband white noise current.

Figure 4. shows the measurement points as a wire frame on the investigated shaft assembly, and the

accelerometer placement is also well visible. The motor was excited by white noise current supplied by a

dedicated current-output analog linear power amplifier. The signals (the phase current and the

acceleration) were recorded with an LMS Scadas III data acquisition system, and Frequency Response

Functions (FRFs) were calculated using H1 estimator. The only additional step in case of torsional modal

analysis is that the FRFs have to be multiplied by the radius of the measurement point to have a properly

scaled animation afterwards. The measurement time can be shortened by measuring only one point in a

cross-section and projecting the FRFs with an appropriate coordinate transformation to the other three

points on the wire frame model.

3.2 The obtained mode shapes

After having all the necessary FRFs available a standard modal analysis were performed using the LSCE

(Least Squares Complex Exponential) method and the natural frequencies and the mode shapes were

obtained. Figure 5-7. show the three mode shapes found in the 0 – 1000 Hz frequency band.

The first mode (fig. 5.) is the rigid body mode of the system, because there is hardly any deformation

present, the complete assembly moves together. The relatively high natural frequency of this mode shape

can be explained by the presence of the cogging torque. The cogging torque is the torque on the shaft

needed to turn the rotor at a low (typically 10 rpm) speed without any current in the phases. The reason of

this non-zero fluctuating torque is the interaction of the permanent magnets with the slotted stator steel.

The cogging torque is a periodic function with zero-crossings, so that one of these zero-crossings will give

the equilibrium of the motion. This means that if the rotor is moved from this equilibrium point, a counter

force will be present acting as a spring pulling the rotor back to the equilibrium point.

Figure 5: The 1st mode of the shaft assembly @ 47 Hz


Figure 6: The 2nd

mode of the shaft assembly @ 298 Hz

Figure 7: The 3rd

mode of the shaft assembly @ 695 Hz

This spring and the inertia of the system give the natural frequency and the shape of the first mode.

The second mode (fig. 6.) is the first dynamic mode of the system showing real deformation in the shafts

and in the clutches. Similarly to the first mode this motion can also be explained by a simple mass-spring

system, where the mass is the inertia of the rotor and the stiffness is given by the shafts and the clutches.

This mode shape is very important from practical point of view as its natural frequency (~300 Hz) lies in

the middle of the analysis frequency range, therefore analyzing the mode shape may give an indication

how to move this mode shape out from the frequency range of interest.

Figure 7. shows the third mode, that is the resonance of the torque sensor. This is also a vital peace of

information meaning that this natural frequency can only be changed significantly by changing the

complete torque sensor unit.

3.3 Validation of the natural frequencies by means of signature analysis

To make sure that the obtained mode shapes are relevant under normal rotating operating conditions as

well, signature analysis of a speed run-up was performed on the same test setup. The motor speed was

increased from 0 – 2000 rpm in 80 s, and the torque signal was recorded with the same data acquisition

system. Figure 8. shows the Campbell diagram of the torque signal. The well distinguishable horizontal

lines indicate the torsional resonances of the system, and as expected the orders coming from the motor

are amplified significantly while crossing these resonances.

To determine the natural frequencies more precisely, the averaged spectrum of the complete run-up were

also calculated (fig. 9.). Comparing the natural frequency values from the modal analysis and from the

signature analysis, the relative error for the second and the third mode shape are 5.7% and 0.7%

respectively. As in this case the complete shaft assembly is rotating the first mode is not present in the

system, the harmonics of the cogging torque are contributing as pulsating torque excitation.


Figure 8: Campbell diagram of an 80 s speed run-up from 0 – 2000 rpm

Figure 9: Averaged spectrum of the torque signal over the complete 80 s speed run-up from 0 – 2000 rpm

4 Conclusion

In this paper a novel torsional modal analysis method was presented. The continuous development of the

permanent magnet synchronous motor market – especially in the automotive industry for electric power

assisted steering systems – created the demand of low noise high power density electric drives. The

acoustic testing of PMSMs under load is an especially difficult task due to the torsional resonances of the

shaft system that connects the tested motor to the load machine. To identify the mode shapes and the

natural frequencies of the disturbing resonances pulsating torque excitation was generated using the tested

PMSM fed by broadband white noise current. The response – the angular acceleration – of the shaft

system was measured by a standard triaxial accelerometer, and the angular acceleration was calculated

from the tangential acceleration.

To provide a correct understanding of the behavior of the PMSM in such a non-rotating operating mode,

the theoretical connection was determined between the electrical (phase voltage and phase current) and the

mechanical (torque and angular velocity) variables of the motor. This theoretical analysis provides the


electrodynamic transducer model of a permanent magnet synchronous motor that is in complete analogy

with the well-known model of a rectilinear electromagnetic transducer (shaker, loudspeaker). However the

rotor position dependence of the electromagnetic transmission ratio has to be bear in mind in case of

theoretical calculations and practical applications as well.

Using the proposed method experimental torsional modal analysis was carried out on an EPAS motor

testbench to determine the torsional natural frequencies and mode shapes. To verify the obtained modal

model signature analysis was performed to see that the measured mode shapes are also relevant in normal

operation as well. The coherence of the results is more than satisfying, therefore future work is planned to

generalize the outlined concept and to widen the field of applications.

Acknowledgements

The results presented in this paper were achieved with financial and technical support of ThyssenKrupp

Presta Hungary Ltd and ThyssenKrupp Presta AG which is gratefully acknowledged. The theoretical

background and the research framework were provided by Budapest University of Technology and

Economics, Department of Telecommunications, Laboratory of Acoustics for which the authors are

exceptionally thankful.

References

[1] J. C. Wachel, F. R. Szenasi, Analysis of torsional vibrations in rotating machinery, Proceedings of

22nd

Turbomachinery Symposium, Texas (1993), pp. 127-151.

[2] http://www.labworks-inc.com/engineering_info/shaker_eng.htm

[3] L. L. Beranek, Acoustics, McGraw-Hill Book Company, Inc., New York / Toronto / London (1954)

[4] A. Kapun, M. Curkovic, A. Hace, K. Jezernik, Identifying dynamic model parameters of a BLDC

motor, Simulation Modelling Practice and Theory, 2008, pp. 1254-1265.


torsional modal analysis of single shaft systems

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