smart machines with flexible rotors

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 373–382

0888-32

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Smart machines with flexible rotors

A.W. Lees

Swansea University, Singleton Park, Swansea SA2 8PP, UK

a r t i c l e i n f o

Article history:

Received 22 April 2010

Received in revised form

1 September 2010

Accepted 4 September 2010Available online 22 September 2010

Keywords:

Rotor

Vibration

Control

70/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ymssp.2010.09.006

ail address: [email protected]

a b s t r a c t

The concept of smart machinery is of current interest. Several technologies are relevant

in this quest including magnetic bearings, shape memory alloys (SMA) and piezo-

electric activation. Recently, a smart bearing pedestal was proposed based on SMAs and

elastomeric O-rings. However, such a device is clearly relevant only for the control of

rigid rotors, for flexible rotors there is a need for some modification on the rotor itself. In

this paper, rotor actuation by piezo-electric patches on the rotor is studied.

A methodology is presented for the calculation of rotor behaviour and appropriate

control strategies are discussed.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The concept of smart machinery is in its infancy and has received significantly less attention than has been devoted tothe related study of smart structures. A smart machine is one in which there is some facility for automatic diagnosis offaults coupled with a capacity to apply corrective loads to optimise machine duty until such time as correctivemaintenance can be undertaken. The precise form which these facilities take will depend significantly on the duty of themachine in question. The development of smart machines is inherently more complex than the associated structuralproblem owing to the rotor motion and problems associated with the bearings. However, in essence, the idealized machinehas three main features:

(a)
The facility to infer its own internal state. (b) A capability to diagnose faults. (c) The introduction of corrective forces in the event of faults.
The simplest case to consider is a machine with a rigid rotor, such as occurs in a range of small machines. This is aparticularly simple case to consider since the natural frequencies (or more particularly, critical speeds) are controlledentirely by the stiffness of the bearing pedestals. Lees et al. [1] have shown how a controllable bearing pedestal may bedesigned using shape memory alloys and elastomers, although it is appreciated that there are a number of possible routesto achieve this goal. Other approaches to the introduction of controllable support stiffness have been considered by Zaket al. [2] and Cartmell et al. [3]. Recently, Maslen [4] has given a brief review of progress on smart machinery, but this toofocuses of bearings, with particular emphasis on magnetic bearings: this is not surprising as this technology offers greatpromise.

However, looking further ahead one is naturally led to consider the possibility of controlling machines with flexiblerotors. This is important for two reasons: in machinery generally there is a trend towards the use of flexible rotors and

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A.W. Lees / Mechanical Systems and Signal Processing 25 (2011) 373–382374

secondly, adequate control would enable the operation of much lighter machines leading to higher efficiencies and bettermaterial utilization.

2. Piezo-electric patches

Piezo-electric devices have been used for the control of structures in aero-space applications for a number of years.There are numerous papers on this and an introductory account is given in Ref. [5]. Control may be exerted by either amodification of natural frequencies or, more usually, by changing the damping properties. The initial part of this studyconsiders the possible effect of such devices on a rotor. Signals may be input by means of slip-rings.

Consider a simple, single degree of freedom rotor system. We suppose there is a vibration amplitude w, and we mayassume that the deflection is of the form

wðxÞ ¼ 4x

L1�

x

L

� �ð1Þ

Using this deflection, the extension of the neutral axis is given by

DL¼1

2

Z L

0

dw

dx

� �2

dx ð2Þ

However, the situation under consideration here is the introduction of a piezo-electric patch, not on the neutral axis,but on the extremity of the rotor. Assuming this to be displaced by r from the neutral axis, the extension here will be ratherdifferent and will be given by

DS�1

2

Z L

0

dw

dxþr

d2w

dx2

� �2

dxþ rdw

dx

� �L

0

ð3Þ

Establishing a numerical methodology is a relatively straightforward task; what is rather more demanding is thedetermination of the most effective philosophy to use. The requirement is to establish a method to determine the dynamicbehaviour of a rotor with piezo-patches, which can be used to optimise design parameters. It is a fairly straightforwardmatter to supply the requisite electrical signals to the rotor and this can be achieved either using slip-rings or telemetrysystems. Whilst the voltages will be high, the currents and consequently the power will be negligible.

There are four possible approaches to use the forces arising in the piezo-electric MFC patches, which may be outlined asfollows:

(a)
Use the forces to directly change the natural frequencies. This may be achieved by using opposite sides of the rotor tomodify shaft properties. A simple application of perturbation techniques shows that this approach appears to belimited to small machines.
(b)
Apply an axial force which will modify the rotor’s properties—this is akin to the common changes of natural frequencyassociated with axial forces.
(c)
By suitable scaling and timing use the force to control damping rather than the stiffness. (d) Use the patches to induce rotor bends, which can counteract the unbalance.
Of these possible approaches, it seems that (d) is the most promising and some simulations have been studied toinvestigate the operation of such a device. The calculations presented here study a machine with a shaft of diameter12 mm and a span between the bearings of 1 m. Each bearing has symmetrical bearing properties with a stiffnessparameter of 100,000 N/m and associated damping parameter of 800 Ns/m. The single central mass gives a resonantfrequency of 124 rad/s and the system damping coefficient is 1.8%.

3. Problem formulation

Fig. 1 shows the basic layout for a simple rotor. Piezo-electric multi-fibre composite patches are positioned at the centreof the rotor span, but for a system involving higher modes, further patches would be required. Depending on the phasing ofthe applied voltages, either forces or bending moments may be applied. Using two pairs of patches around the periphery ofthe rotor, resultant forces/moments can be applied at any phase angle with respect to the rotor.

It is well known that unbalance and rotor bends both give synchronous vibration, but the two dependencies with rotorspeed differ [6]. A typical comparison of the vibration resulting from a bend and unbalance is shown in Fig. 2, which hasbeen arranged to yield identical excitation at a rotor speed of around 1200 rev/min (the minor differences being due todiscretisation). From these two different variations with frequency it is clear that a bend may be compensated by a balance

Fig. 1. Basic layout showing central piezo-electric patches.

0 500 1000 1500 2000 2500 3000 3500 40000

0.005

0.01

0.015

0.02

0.025Comparison of shaft bend and imbalance

Rotor speed (rpm)

Dis

plac

emen

t (m

)

Imbalancebend

Fig. 2.

A.W. Lees / Mechanical Systems and Signal Processing 25 (2011) 373–382 375

mass, provided that the rotor operates at a single speed. Indeed such a compensation has been used in many industrialapplications. The proposal in this work is precisely the reverse of this: an unbalance is to be compensated by inducing abend in the shaft through the piezo-activation.

For the unbalance case, the response for a steady shaft rotation rate O is given by

KyþC _yþM €y ¼MeO2expðjOtÞ ð4Þ

The response due to the bend is given by

KyþC _yþM €y ¼ Kb0expðjOtÞ ð5Þ

For the light machine considered, Fig. 2 shows the response as a function of speed for two distinct cases. Bothexcitations have been given zero phase for the sake of convenience. Note that although the response is synchronous in bothcases, the response curves, away from resonance, are rather different.

Fig. 3 shows an example in which a bend counteracts the unbalance at a specific frequency. It can be shown that thevibration can be limited to the magnitude of the bend. However, since in the case of this study the force exerted bythe patch (P) can be time dependent, it is clear that the bend formation cannot occur instantaneously and there is inertia inthe system.

The objective is now to solve the equations simultaneously during a machine run up or run down. The forcing function P

may be chosen. The equations are integrated using a Runge–Kutta scheme in Matlab. To do this, the equations aretransformed to a state-space formulation using the variables ½yb _y _b�T . In this formulation, the equation of motion becomes

AqþB _q ¼ F ð6Þ

However, in the device under consideration, the extent of the bend xb may be changed at will, but this does mean thatthe ancilliary equations must be considered in some detail. This is considered as a Lagrangian formulation.

Let b represent the bend is stationary co-ordinates and b0 is the bend in the rotating frame. Normally, a bend can beconsidered as a constant, but in the present problem it is allowed to vary with time.

Then the Lagrangian may be written as

L¼U�T ð7Þ

where

U ¼1

2fy�bgT K y�b

� þaPT b ð8Þ

0 500 1000 1500 2000 2500 3000 3500 40000

0.2

0.4

0.6

0.8

1x 10−3

Dis

plac

emen

t (m

)

Rotor speed (rpm)

Combined response

Fig. 3.


where a is a geometric factor (which is the radius at which the patch is fixed multiplied by a vector of length 52. This vectorcontains zeros and +/�1) and

T ¼1

2f _yþ _b� _eg

TM _yþ _b� _en o

ð9Þ

Note that in steady running at speed O, y¼ y0cosOt

But the bend is time dependent because of the applied force, hence

b¼ RðtÞb0ðtÞ

‘ _b ¼ _Rb0þR _b0

‘ €b ¼ €Rb0þ2 _R _b0þR €b ð10Þ

where the rotation matrix is given by

R¼

r 0 ^ 0

0 r ^ 0

. . . . . . & ^

0 0 0 r

26664

37775 ð11Þ

and

r¼

cosOt sinOt 0 0

�sinOt cosO 0 0

0 0 cosOt �sinOt

0 0 sinOt cosOt

26664

37775 ð12Þ

Lagrange’s equation

@L

@q�

d

dt

@L

@ _q

� �¼ 0 ð13Þ

yields the relationship

�K y�b�

þaPþd

dtM _yþ _b� _eh i

¼ 0 ð14Þ

But we can now eliminate the terms in y given by combining the equations to give

KbþaPþM €b ¼ 0 ð15Þ


Hence, using the relations (10), the equation governing bend development becomes

KRb0�aRPþM €Rb0þCr_b0þ2M _R _b0þMR €b0 ¼ 0 ð16Þ

This is readily transformed to the rotating frame by multiplying by RT to give (assuming axial symmetry)

Kb0�MO2b0þ2OGMb0þCr_b0þM €b0 ¼ aP ð17Þ

where G is the skew matrix given by

G¼

0 �1 0 0

1 0 0 0

0 0 0 �1

0 0 1 0

&

&

&

&

0 �1 0 0

1 0 0 0

0 0 0 �1

0 0 1 0

26666666666666666666666664

37777777777777777777777775

ð18Þ

A damping term is included in Eq. (17): the magnitude of this can be modified by the application of appropriatevoltages.

The complete system is described by the equations

Kb0�O2Mb0þ2OGMb0þCr

_b0þM €b0 ¼ aP ð19Þ

KyþC _yþM €y ¼MueiOtþKRðtÞb0 ð20Þ

Note that in the current model some damping has been assumed on the rotor: damping can be readily introduced byappropriate phasing of the applied voltages. The equations of motion for the complete system, whose speed varies withtime, in the presence of both imbalance and activation producing a bend, can now be written as

Kb0�O2Mb0þ2OGMb0þCr

_b0þM €b0 ¼ aP

KyþC _yþM €y ¼KRðtÞb0þmaO2ejOt�jmadOdt

ejOt ð21Þ

These equations assume shaft symmetry. They may be written in state-space form as

0 0 �K 0

0 0 0 �K

K �KR C 0

0 K�O2Mþ2OGM 0 Cr

26664

37775

y

b0

_y_b0

8>>><>>>:

9>>>=>>>;þ

K 0 0 0

0 K 0 0

0 0 M 0

0 0 0 M

26664

37775

d

dt

y

b0

_y_b0

8>>><>>>:

9>>>=>>>;¼

0

0

maO2ejOt�jma _OejOt

aPðtÞ

8>>>><>>>>:

9>>>>=>>>>;

ð22Þ

Of course, the effective phases of the deflection arising from an unbalance and a bend is somewhat involved and duerecognition must be made of this. It can be shown that the combined deflection due to the two may be limited to the extentof the bend and Fig. 3 shows the case in which the magnitude and phase of the bend have been chosen to exactly balance ata set rotation speed.

It is easily shown that a bend and a discrete unbalance can exactly compensate at a single rotational speed, and in thiscase the natural frequency has been chosen as the speed at which the response is minimized. However, in the case understudy here, because actuators are on the rotor, the response may, in principle be minimized at all speed simply by varyingthe voltages to the piezo-patches. As this involves not only magnitude but phase, the control of this is non-trivial and a fulldiscussion is beyond the scope of the present paper.

4. An example

The simple rotor system shown in Fig. 1 was modelled with Euler beam finite element and the piezo-patch wasmodelled to exert equal and opposite bending moments on neighbouring nodes close to the shaft centre. The rotor has12 elements, the model has a total of 52 degrees of freedom. For the calculation of the forced response the model wasreduced to eight degrees of freedom using standard node condensation methods. The degrees of freedom used were thoseof the two nodes between which the two piezo-patches (in two orthogonal planes) were connected. With this reduced set,the state-space system model (Eq. (23)) has 32 degrees of freedom.

02

46

810

12

0

100

200

300

4000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

x 10−3

Applied moments (Nm)

Variation of vibration with patch force

Relative phase (Degrees)

Dis

plac

emen

t (m

)

Fig. 4.


We consider now taking a fixed unbalance by varying the amplitude of rP, the applied torque between 0 and 10 N m (forcesof the corresponding magnitudes may realistically be obtained from commercially available patches, albeit with the applicationof considerable voltages. By varying two orthogonal components of P, any desired phase can be achieved. The phase anglebetween the unbalance and the bend is varied between 01 and 3601 in steps of 101 degrees. The response is shown in Fig. 4.

The effect of changing the actuation voltage was examined by simulating the operation of the rotor at a steady speed.The speed used is not particularly important, but that use was 8 Hz being below the first critical speed. The numericalintegration was carried out using a Runge–Kutta scheme, and the pre-determined voltage was applied after 2 s, a pointchosen to allow for the decay of the initial transients. The result is shown in Fig. 5. The effect of the correction is veryobvious and this is an impetus to take the study further.

The next obvious step must be to develop a means of feedback control to correct errors, but the best way to achieve thisgoal is far from obvious as parameters have cyclic variation.

The next step must be to automate this process and here too there are several possible approaches. The simple idea ofusing a feedback loop has some difficulties owing to the oscillatory nature of the problem and so for this initial study analternative procedure is adopted. After allowing the system to settle at the chosen running speed, an arbitrary force ischosen in the piezo-patch (which may be easily related to the applied voltage). Following a further settling period, the newvibration amplitude and phase is monitored and used in a calculation very similar to a standard single place balancingprocedure, which yields the required voltage to be applied. The results of this procedure are shown in Fig. 6. In thiscalculation, the initial settling time allowed was 1.5 s, with a further 6 s following the trial change; the voltage change wasapplied steadily over 1 s to avoid transient disturbances.

Note that after applying the calculated voltage, the vibration level is greatly reduced, but it is not zero. There are severalreasons for this: first, the deflected shape of the shaft due to the application of moments is not precisely the same as thedeflection resulting from a single unbalance force (a problem easily overcome be considering modal contributions). Moreimportantly is the non-stationarity of the response: whereas in Fig. 6 a settling time of 6 s is used, Fig. 7 shows thesituations with a settling time of 3 s. Comparing the two figures it is seen that the longer settling time leads to reducedvibration levels, but clearly this is dependent on the level of damping. Of course in generalising this conclusion, it is thenumber of shaft revolutions, which is the parameter of physical relevance rather than the time itself and system dampingwill have an important influence.

This highlights a problem: accuracy requires that the machine is in steady-state condition, but provided there is a goodmodel of the system available, the unbalance forces may be evaluated using the state space equation of motion. This isillustrated in Fig. 8, which shows the first second of the simulation. Although the vibration trace is far from settled anddisplays transient features, the derived force on the lower trace shows an accurate value very quickly and hence anappropriate bending moment may be calculated. This approach would lead to a faster control strategy, but of course it isdependent on an accurate system model. A suitably designed observer is likely to be appropriate, but this is notinvestigated in the present work.

0 5 10 15−8

−6

−4

−2

0

2

4

6

8x 10−4 Response change with gradual transition

Res

pons

e (m

)

Time (secs)

Fig. 6.

0 1 2 3 4 5 6 7 8 9 10−3

−2

−1

0

1

2

3

4

5

6x 10−4

Time (secs)

Res

pons

e (m

)

Response with pre−determined patch force after 2 seconds

Fig. 5.


5. Scaling issues and practical problems

The example used to illustrate the principles of this paper is a very slender rotor, with an aspect ratio of 80:1. Thelength between bearings is 1 m whilst the diameter is 10 mm. This slender example was chosen as a bend is easily induced

0 5 10 15−8

−6

−4

−2

0

2

4

6

8x 10−4

Time (secs)

Res

pons

e (m

)

Response with 3 second settling time

Fig. 7.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

0

2

x 10−4 Vibration levels

Res

pons

e (m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5Derived Forces

Imba

lanc

e fo

rce

Time (secs)

Fig. 8.


in such a rotor. However the real issue is the magnitude of the force available on the piezo-patch. The appendix considersthe bending of a simple beam of length L, simply supported at either end with a patch of total length lp at the centre. Thispatch is assumed to transmit force only at the end points (although other modelling representations are easily achieved).


If the patch exerts moments M=Pr, then the maximum bending introduced is

yL

2

� �¼

ML3

12EIlp¼

PrL3

12EIlpð23Þ

where r is the shaft radius. Equating this to an unbalance deflection gives

PrL3

12EIlp¼ Fu

L3

48EIð24Þ

where Fu is the unbalance force. This yields the overall constraint on this concept

Fu ¼ 4Pr

lpð25Þ

Hence the limitation is set on the unbalance force, which can be counteracted by the force capacity of the patch. Notethat in principle, r can be enhanced by mounting a patch between discs.

In practice, voltage would be supplied to the rotor by means of slip-rings: although voltages used may be high, theimpedance is virtually infinite and currents minimal. An issue to be resolved is as to whether voltages should be amplifiedon the rotor or transmitted. In any event no major problems are envisaged.

6. Discussion

Interest in mechatronics and the concept of ‘smart machines’ have developed in the past few years. But it has so farfocused on the properties of bearings and support structures. Whilst this is appropriate for machines with rigid rotors, theoverall trend in machinery is the enhanced use of designs involving flexible rotors. Such rotors require some controllingelement to be mounted on the shaft. The voltage applied to piezo-patches can be very considerable, of order hundreds ofvolts (dc) albeit at negligible current levels. Wilkie et al. [7] give an overview of the force levels which may be obtainedfrom multi-fibre composite piezo-patches. One might envisage that the signal would be fed to the rotor via a slip-ringarrangement: the question as to whether the signal would be at full voltage or amplified on the rotor is a question forfuture development.

In Section 5 a discussion is given on scaling and it is shown that the limitation is on the magnitude of the unbalanceforces, but there are some far reaching implications of this work. If any critical speed can be effectively negated by someautomated balancing procedure, this immediately removes some important design constraint from the designer of rotatingmachine.

To illustrate this point consider a 500 MW. At the axial position with maximum torque the shaft diameter will typicallybe of the order of 0.5 m. A simple calculation reveals that the torsional stress will be of the order of 8 MPa. This is farlower than the self-weight bending stress, brought about in part by designing the structure to have an adequate naturalfrequency. Removing this constraint may tend towards lighter, more agile machinery. However, there are a number of issues tobe resolved.

The work presented in this paper is very exploratory. Nevertheless a potential means of controlling flexible rotors hasbeen outlined. In essence the control is affected by imposing a bend to compensate for the imposed imbalance. It is wellknown that a bend can only be balanced out at a single rotational speed, but in the present case the extent of the bend canbe varied by the voltage applied to the piezo-electric patch.

The sample machine taken here is by no means typical: it is extremely flexible, but if a machine can be controlled insuch a manner, it becomes realistic to operate these devices. This paper has outlined some basic considerations andformulated a method of analysis. Furthermore the possibility of actively controlling a flexible rotor in this way has beenestablished. To develop this idea further the following steps need to be undertaken:

(a)
A review of the range of machine dimensions, which would be feasible for this type of technology. (b) Derivation of an appropriate control algorithm. (c) The construction and testing of a device.
7. Conclusions

1.
It is more realistic to control the forcing on a rotor rather than the dynamic properties. Calculations show that it isdifficult to introduce substantial changes in natural frequency by the use of piezo-electric patches. It has been shownthat it is feasible to counteract an unbalance with an induced bend.
2.
Some simple cases have been examined and shown to be feasible. Whilst this is primarily of interest in small machines,the full range of applicability has yet to be established. Note, however, that if the constraint of avoiding critical speedscan be removed then machines can be made much smaller and lighter.
3.
An analysis approach has been presented.


Appendix 1. Rotor deflection

As there are no shear forces acting (except at the bearings) the static displacement is given by the expression

yðxÞ ¼ ax3þbx2þgxþd ðA1:1Þ

We can conclude immediately that d¼ 0The equal and opposite moments are applied equidistant about the centreline, so that at x¼ ððL�aÞ=2Þ

�M

EI¼ 6a L�a

2

� �þ2b ðA1:2Þ

and at x¼ ððLþaÞ=2Þ

M

EI¼ 6a Lþa

2

� �þ2b ðA1:3Þ

Therefore, subtraction gives

2M

EI¼ 6aa ðA1:4Þ

‘0¼6aðL�aÞ

2þ3aaþ2b

‘b¼�3aa

2ðA1:5Þ

So that

a¼ M

3EIaðA1:6Þ

To determine g consider the displacement at L

0¼ aL3�3aL3

2þgL ) g¼ aL2

2ðA1:7Þ

The deflection at the midpoint is given by

yL

2

� �¼

M

3EIa

L

2

� �3

�M

2EIa

L3

4þ

M

3EIa

L3

2¼

ML3

12EIaðA1:7Þ

References

[1] A.W. Lees, S. Jana, D.J. Inman, M.P. Cartmell, The Control of Bearing Stiffness Using Shape Memory, ISCORMA, Calgary, Alberta, August 2007.[2] A.J. Zak, M.P. Cartmell, W.M. Ostachowicz, Dynamics and control of a rotor using and integrated sma/composite active bearing actuator, in:

Proceedings of the Fifth International Conference on Damage Assessment of Structures, University of Southampton, UK, 2003, pp 233–240.[3] M.P..Cartmell, Th. Leize, L. Atepor, D.J. Inman, A.W. Lees, Controlling flexible rotor vibrations by means of an antagonistic SMA/composite smart

bearing, in: Proceedings of the IoP Conference, Modern Practice in Stress and Vibration Analysis, Bath, UK, July 2006.[4] E.H. Maslan, Smart machine advances in rotating machinery, in: Proceedings of the Ninth International Conference on Vibrations in Rotating

Machinery, I.Mech.E., Exeter, UK, 8–10 September 2008.[5] M. Brennan, Experimental investigation of different actuator technologies for active vibration control, Smart Material Structures, Institute of Physics

Publishing Ltd., 1998.[6] M.I. Friswell, J.E.T. Penny, S.D. Garvey, A.W. Lees, Dynamics of Rotating Machines, Cambridge University Press, 2010.[7] W.K. Wilkie, R.G. Bryant, J.E. High, R.L. Fox, R.F. Hellbaum, A. Jalink, B.D. Little, P.H. Mirick, Low cost piezocomposite actuator for structural control

applications, in: Proceedings of the SPIE, Seventh International Symposium on Smart Structures and Materials, Newport Beach, CA, March5–9th, 2000.

smart machines with flexible rotors

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