Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 829-844, Warsaw 2008

ANALYSIS OF COUPLING MECHANISM IN
LATERAL/TORSIONAL ROTOR VIBRATIONS

Zdzisław Gosiewski

Aviation Institute, Warsaw, Poland

e-mail: gosiewski@post.pl

Coupling phenomenaof lateral-torsionalvibrations in rotatingmachinery can
lead to rotor unstable behaviour. In some types of rotating machinery, the
angular speed is subjected to wide ranges of change. The problem is to find
the range of angular speeds for which amachine exhibits stable behaviour in
thepresenceof the couplingofdynamicalmodes.Usually,unstable regionsare
in the vicinity of angular speeds where maps of natural frequencies for both
dynamical systems intersect with each other. Using the root locus method,
it is described which intersections lead to unstable vibrations and why.

Key words: rotor, coupled vibrations, stability, root locus

1. Introduction

Two dynamical systems influence each other, if connected. In the case of rota-
tingmachinery, lateral andtorsional vibrationsareusuallyanalysed separately.
Such analysis is allowed in the case when the coupling effect is weak. When
the coupling is strong, one should analyse coupled vibrations (Gosiewski and
Muszyńska, 1992; Sawicki et al., 2004). It appears that in the case of lightly
damped structures, there are ranges of unstable rotor speeds. The unstable
speed ranges are at frequencies for which the natural frequencies of both dy-
namical systems intersect with each other on the Campbell diagram. As far
as the author knows, the literature does not contain clear explanation of the
rotor instability at the intersections on natural frequencies maps.
In the paper, the coupling effect is analysed in the case of a simple three-

mode rotor model (Jeffcott’s model). In this case, vibrations are described
by three nonlinear equations. To obtain equations of motion, one can apply
Lagrange’s equations. After linearization in the inertial co-ordinate system,
rotor vibrations are described in the rotating co-ordinates system.



830 Z. Gosiewski

First, the couplingeffect and its influenceon the stabilitywill be considered
in the classical way (Muszyńska et al., 1992). Next, we define the feedback in
the coupled systemand carry out investigation of the system stabilitywith the
help of the Evans method known from the control theory (Takahashi et al.,
1972). Such an approach leads to simple explanationwhen a given intersection
on the map of natural frequencies is stable or not.

2. Mathematical model

We consider the physical model of a flexible rotor shown in Fig.1. Themodel
consists of a rigid disc and a massless flexible shaft. The unbalanced disc is
located at the midspan of the shaft. The shaft is driven by a motor which
rotates with a constant angular speed Ω. The disc has amass m andmoment
of inertia Io. We assume that the flexibility of the shaft in both directions ξ,
η are the same and the stiffness coefficients are k1 = k2 = k. The torsional
flexibility is kt.

Fig. 1. Physical model of the symmetric rotor

Kinetic Ek and potential Ep energies of such a dynamical system are as
follows

Ek =
m

2
(ẋ2S + ẏ

2
S)+

IO

2
γ̇2

Ep =Ep =
k

2
ξ2+
k

2
η2+

kt

2
(γ−Ωt)2−mgh



Analysis of coupling mechanism... 831

where

xs =x−ecos(γ+ δ) ys = y+esin(γ+ δ)

are coordinates of the disc mass centre S in the inertial coordinate system
XYZ, while

ξ=xcosγ+y sinγ η=−xsinγ+ycosγ

are co-ordinates of the disc geometrical centre W in the rotating co-ordinate
system ξηζ, which rotates with the rotor angular velocity Ω. Furthermore,
γ is the angle of the shaft twist, e – the eccentricity (distance) of the rotor
mass centre S from its geometrical centre W , while δ is the angle between
the unbalance vector and the axis ξ. We omit energy dissipation caused by
external and internal damping.

Lagrange’s equations were used to obtain the equations of motion of the
rotor. They have the form

ẍ−e[γ̇2cos(γ+ δ)+ γ̈ sin(γ+δ)]+ω2x=0

ÿ+e[γ̈ cos(γ+ δ)− γ̇2 sin(γ+ δ)]+ω2y=0 (2.1)

γ̈+Reγ̈+R[ÿcos(γ+ δ)− ẍsin(γ+ δ)]+µ(γ−Ωt)=T(t)

where

ω2 =
k

m
µ2 =

kt

I0
R=
me

I0

and T(t) is the difference between the driving and loading torque. It is seen
that the equations are nonlinear and coupled by rotor unbalance (e 6=0).

We introduce coordinates of forward and backward motion

u=x+jy u=x− jy (2.2)

which allows us to represent equation of motion (2.1) in complex inertial co-
ordinates as follows

ü−eγ̇2ej(γ+δ)+jeγ̈ej(γ+δ)+ω2u= jg

ü−eγ̇2e−j(γ+δ)− jeγ̈e−j(γ+δ)+ω2u=−jg (2.3)

γ̈+Reγ̈+
R

2j

[

üe−j(γ+δ)− üej(γ+δ)
]

+µ2(γ−Ωt)=T(t)

Equation sets (2.1) and (2.3) are sets of nonlinear and coupled equations.



832 Z. Gosiewski

To start analysis of rotor vibrations, we linearize (2.3) assuming that the
torsional difference γ from the angle ΩT by φ(t)

γ =Ωt+φ(t) (2.4)

so φ(t) describes torsional vibrations of the shaft. During linearization, we
take into account only first linear components of the Taylor series

e±jφ =1± jφ+ . . .

The linearized set of equations in the inertial complex coordinates has the
following form

ü+e(−Ω2− jΩ2φ−2Ωφ̇+jφ̈)ej(Ωt+δ)+ω2u= jg

ü+e(−Ω2+jΩ2φ−2Ωφ̇− jφ̈)e−j(Ωt+δ)+ω2u=−jg (2.5)

φ̈+eRφ̈+
R

2j

[

üe−j(Ωt+δ)− üej(Ωt+δ)
]

+µ2φ=T(t)

Now, we introduce the transformation of the coordinates

u=wejΩt u=we−jΩt (2.6)

where
w= ξ+jn w= ξ− jn (2.7)

which allows us to transfer set (2.5) into a set of rotor dynamical equations in
the complex rotating co-ordinates

ẅ+2jΩẇ−Ω2w+e(jφ̈−2Ωφ̇− jΩ2φ)ejδ +ω2w= eΩ2ejδ +jge−jΩt

ẅ−2jΩẇ−Ω2w+e(−jφ̈−2Ωφ̇− jΩ2φ)e−jδ +ω2w= eΩ2e−jδ − jgejΩt(2.8)

φ̈+Reφ+µ2φ+
R

2j

[

(ẅ+2jΩẇ−Ω2w)e−jδ − (ẅ−2jΩẇ−Ω2w)ejδ
]

=T(t)

The first two equations are introduced to the third one with the external
and internal dampings neglected. By transforming (2.7), we have obtained the
equations of motion in real rotating co-ordinates

ξ̈−2Ωη̇−Ω2ξ+ω21ξ+eΩ
2φsinδ−2eΩφcosδ−eφsinδ=

= eΩ2cosδ+g sin(Ωt)

η̈+2Ωξ̇−Ω2η+ω22η+eΩ
2φcosδ−2eΩφsinδ+eφcosδ= (2.9)

= eΩ2 sinδ+gcos(Ωt)

φ̈+ReΩ2φ+µ2φ+Rω2 sin(δξ)−Rω2cos(δη) =T(t)−Rgcos(Ωt+ δ)



Analysis of coupling mechanism... 833

Linearized equations in inertial coordinates (2.5) have time dependent co-
efficients. It means that the rotor vibrations can be considered as parametric
vibrations. From the vibration theory we know that such vibrations for some
parameters can be unstable. The angular speed Ω is one of the main rotor
parameters. Equations in rotating coordinates (2.8) and (2.9) have constant
coefficients. Therefore, calculations of unstable ranges of the angular speed
will bemuch simpler by analysing equations (2.9) in the rotating coordinates.

3. Free lateral/torsional vibrations

In this case, equations of motion (2.9) describe free undamped vibrations of
the rotor

ξ̈−2Ωη̇−Ω2ξ+ω2ξ+eΩ2φsinδ−2eΩφ̇cosδ−eφ̈sinδ=0

η̈+2Ωξ̇−Ω2η+ω2η−eΩφcosδ−2eΩφ̇sinδ+eφ̈cosδ=0 (3.1)

φ̈+eRΩ2φ+µ2φ+Rω2ξ sinδ−Rω2ηcosδ=0

After application of the Laplace transformation, the differential equations
change into the algebraic form as follows






s2−Ω2+ω2 −2Ωs eΩ2 sinδ−2eΩscosδ−es2 sinδ
2Ωs s2−Ω2+ω2 −eΩ2cosδ−2eΩssinδ+es2cosδ
Rω2 sinδ −Rω2cosδ s2+eRΩ2+µ2













ξ

η

φ






=







0
0
0







(3.2)
From the above matrix we can obtain the characteristic equation

s6+a1s
4+a2s

2+a3 =0 (3.3)

where

a1 =2ω
2+2Ω2+µ2z +pRh

a2 =(ω
2−Ω2)2+2(ω2+Ω2)µ2z +ω

4Rh+2Ω
2pRh

a3 =µ
2
z(ω
2−Ω2)2−ω4Rh+Ω

4pRh

p=ω2 Rh =Re=
me2

I0
µ2z =µ

2+RhΩ
2

In the above formulas for coefficients of the characteristic equation the
parameter δ, angle between the unbalance vector and the axis ξ, does not



834 Z. Gosiewski

influence the roots.This is only true in the case of an isotropic rotor.Therefore,
we can orient the rotating coordinate system in any way with respect to the
rotor. It can be seen from the last equation that the unbalance influences the
natural frequency of torsional vibration. This influence is shown in Fig.2.

Fig. 2. Effect of non-dimensional unbalance on the torsional natural frequency

For small values of the unbalance it can be assumed (see Fig.2) that
µz ≡µ≡= const.

The system is stable when all roots of the characteristic equation have
negative real parts. Using the Cardan solution of the 3-order for algebraic
equation (3.3), we have obtained the stability conditions

a3 > 0 a1 > 0 (3.4)

and

a21 < 3a2 or 2(a
2
1−3a2)

3 > [al(9a2−2a
2
1)−27a3]

2

when a21 > 3a2.

From first inequality (3.4), we can calculate one of the unstable ranges of
the rotor speed

ω

√

µ2z
Rhω

2+µ2z
<Ω<ω (3.5)

It is seen that the unstable range always exists when the rotor is unbalanced
(e 6=0).

Second inequality (3.4) is always satisfied. From third inequality (3.4) we
can extract one more range of the unstable rotor speed. The range is in the
vicinity of the value

Ω=ω+µ (3.6)

The analytical considerations are confirmed by the computer simulation
results.The changes of natural frequencies as a function of rotor angular speed



Analysis of coupling mechanism... 835

are shown inFigs.3-5. Inall simulationswehave obtain bothmentioned ranges
of unstable rotor speeds.

Because of symmetry, in Figs.3-5 and further figures, we show only one
quarter of the calculated natural frequencies. The unstable speeds are at the
vicinity of intersection of natural frequencies. For these frequencies, the real
part of roots has positive values (see plots (a) in mentioned Figs.3-5). We
denote by pi the ith pole of the transfer function (root of the characteristic
polynomial) and by zi the ith zero of transfer function.

Fig. 3. Real (a) and imaginary (b) parts of characteristic polynomial roots versus
rotor speed Ω for: ω=100s−1, µ=150s−1, Rh =0.1. Two ranges of unstable

rotor speed: 97s−1 <Ω< 100s−1, 256s−1 <Ω< 292s−1

Fig. 4. Real (a) and imaginary (b) parts of characteristic polynomial roots versus
rotor speed Ω for: ω=100s−1, µ=50s−1, Rh =0.1. Two ranges of unstable rotor

speed: 81s−1 <Ω< 100s−1, 148s−1 <Ω< 213s−1



836 Z. Gosiewski

Fig. 5. Real (a) and imaginary (b) parts of characteristic polynomial roots versus
rotor speed Ω for: ω=100s−1, µ=250s−1, Rh =0.1. Two ranges of unstable

rotor speed: 96s−1 <Ω< 100s−1, 545s−1 <Ω< 664s−1

4. Coupling of torsional/lateral vibrations as a feedback loop

In the above analysiswehave found that theunstable ranges of the rotor speed
appear in the vicinity of some angular speeds for which the lines of natural
frequencies intersect with each other. But not all the intersections are the
cause of unstable behaviour of free vibrations. So, the problem arises how to
recognize which of the crossings indicates the unstable ranges. To answer this
question, we reach for synthesis methods known from the theory of control.
We repeat equation (3.2) in a compact form







A(s) −B(s) −D(s)
B(s) A(s) −F(s)
−K(s) −H(s) C(s)













ξ

η

φ






=0 (4.1)

where

A(s)= s2−Ω2+ω2 B(s)= 2Ωs

C(s)= s2+eRΩ2+µ2 D(s)=−e(Ω2 sinδ−2Ωscosδ−s2 sinδ)

H(s)=−Rω22 sinδ F(s)= e(Ω
2cosδ+2Ωssinδ−s2cosδ)

K(s)=Rω21 cosδ

We consider now separately lateral and torsional vibrations. The coupling
forces are assumed as external excitations

A(s)ξ(s)−B(s)η(s)=D(s)ϕ(s) B(s)ξ(s)+A(s)η(s)=F(s)ϕ(s)
(4.2)

C(s)ϕ(s)=H(s)ξ(s)+K(s)η(s)



Analysis of coupling mechanism... 837

Thefirst two equations describe lateral vibrationswhile the thirdonedescribes
torsional vibrations. Themathematical model can be presented in the form of
a block diagram, given in Fig.6.

Fig. 6. Block scheme of torsional/lateral vibrations of the flexible rotor

It is a dynamical systemwith a feedback loop verywell known from theory
of control, where particular transfer functions have the form

G1(s)=
ξ(s)

ϕ(s)
=
B(s)F(s)+A(s)D(s)

A2(s)+B2(s)

G2(s)=
η(s)

ϕ(s)
=
A(s)F(s)−B(s)D(s)

A2(s)+B2(s)
(4.3)

G3(s)=
ϕ(s)

ξ(s)
=
H(s)

C(s)
G4(s)=

ϕ(s)

η(s)
=
K(s)

C(s)

We break the feedback loop in the place indicated by tildes to obtain an open-
loop system. The open-loop transfer function is obtained bymultiplication of
the matrices

Go(s)= [G3(s),G4(s)]

[

G1(s)
G2(s)

]

=G3(s)G1(s)+G4(s)G2(s) (4.4)

Taking into account transfer functions (4.3), we have

Go(s)=
H(s)B(s)F(s)+H(s)A(s)D(s)+K(s)A(s)F(s)−K(s)B(s)D(s)

[A2(s)+B2(s)]C(s)

When we introduce rotor parameters (4.1), the open loop transfer function
has the form

Go(s)=
−Rhω

2[s4+s2(2Ω2+ω2)− (ω2−Ω2)Ω2]

[s4+s22(ω2+Ω2)+(ω2−Ω2)](s2+µ2z)
=RhGr(s) (4.5)

In theEvansmethod,we consider the rotor unbalance parameter Rh as a gain
which varies from zero to infinity.



838 Z. Gosiewski

5. Evans method

Now we analyse the influence of the unbalance on the system dynamics. The
closed loop system from Fig.6 have the following characteristic equation

1+RhGr(s)= 0 or Gr(s)=−
1

Rh
for Rh ­ 0 (5.1)

In our case, the characteristic equation of the closed-loop system has the
form

1+
−Rhω

2[s4+s2(2Ω2+ω2)− (ω2−Ω2)Ω2]

[s4+s22(ω2+Ω2)+(ω2−Ω2)](s2+µ2z)
= 0

or

1

[s4+s22(ω2+Ω2)+(ω2−Ω2)](s2+µ2z)

{

[s4+s22(ω2+Ω2)+(ω2−Ω2)]·

(5.2)

·(s2+µ2z)−Rhω
2[s4+s2(2Ω2+ω2)− (ω2−Ω2)Ω2

}

=0

We can simply check that the numerator of transfer function (5.2) is a cha-
racteristic polynomial of closed-loop system (5.2), and the denominator is a
characteristic polynomial of the open-loop system.
Gr(s) is a complex number and can be represented by a vector on the

complex plane. So, according to equation (5.1), the transfer function Gr(s)
should satisfy two conditions imposed on the angular location of the vector

argGr(s)=−180
◦±360◦N (5.3)

where N – integer number and on its absolute value

|Gr(s)|=
1

Rh
(5.4)

From the above conditions one can conclude that for Rh increasing from zero
to infinity, the open-loop poles (roots of the numerator polynomial in the
transfer function Gr(s)) move towards zeros of the same open-loop transfer
function Gr(s) (roots of the nominator in equation (4.6)). If the number of
poles is bigger than the number of zeros, other poles escape to infinity along
asymptotes which start from the central point on the complex plane.
It means that zeros of the open-loop transfer function should play an im-

portant role in the analysis of the unbalance effect on dynamical behaviour of
coupled torsional/lateral vibrations of the flexible rotor. The real and imagi-
naryparts of the transfer function zeros for the rotor parameters: ω=100s−1,



Analysis of coupling mechanism... 839

µ= 150s−1, Rh = 0.1 are presented in Fig.7. It is a rotor, whose poles (na-
tural frequencies) are given in Fig.3. Two ranges of unstable rotor speeds are:
97s−1 <Ω < 100s−1 and 256s−1 <Ω < 292s−1. From now, all calculations
will be carried out for this case.

Fig. 7. Real (a) and imaginary (b) parts of zeros in the transfer function Gr(s)
versus rotor angular speed Ω drown as bold lines. Slim lines in (b) show natural
frequencies of uncoupled torsional/lateral vibrations (poles of the transfer

function Gr(s)

Aswe can see inFig.7, thenonzero (positive andnegative) real parts of the
zeros exist only for the rotor speed Ω ranging from0s−1 to 100s−1. Imaginary
parts (frequencies) of the zeros change togetherwith the rotor angular speed Ω
and increase parallel to real parts of the poles (natural frequencies) of the
lateral vibrations.

Fig. 8. Root locus of unbalanced torsional/lateral vibrations for rotor angular speed
60s−1 when the unbalance Rh changes from 0 to infinity

For a smaller angular speed of the rotor (60s−1 in Fig.8), the system can
be unstable for very big values of Rh because two of the zeros in the transfer



840 Z. Gosiewski

function Gr(s) are real and positive. But the value of unbalance for which it
works is unrealistic.

The root locus for the first range of unstable speed (99s−1) is shown in
Fig.9. It is seen that for small unbalances (Rh > 0.045), motion of the rotor
becomes unstable because of the positive zeros in the transfer function.

Fig. 9. Root locus of unbalanced torsional/lateral vibrations for rotor angular speed
99s−1 when the unbalance Rh changes from 0 to infinity

Fig. 10. Root locus of unbalanced torsional/lateral vibrations for rotor angular
speed 105s−1 when the unbalance Rh changes from 0 to infinity. All poles, zeros
and root locus lines lie on the imaginary axis. The 10−14-order shift of the zeros and

poles from the imaginary axis is caused by numerical errors

When all zeros and poles of the transfer function are purely imaginary,
plot of the the root locus shows their location with well seen errors. They are
very small computation errors. In this case the polesmove along the imaginary
axis. In this case, each pole has its associated zero which is a target for the
locus line. Two biggest poles move to infinity along the imaginary axis. Such
a rotor is completely stable for all values of the unbalance Rh.



Analysis of coupling mechanism... 841

Fig. 11. Root locus of unbalanced torsional/lateral vibrations for rotor angular
speed 250s−1 when the unbalance Rh changes from 0 to infinity

Such harmony is destroyed when the rotor angular speed crosses 245s−1

(see Fig.11). For angular speed Ω = 250s−1, the alteration of poles and
zeros is replaced by close neighbourhood of two poles. To reach the zeros, the
locus lines are forced tomake circles. The circles enter the positive side of the
complex plane. It means thatmotion becomes unstable for some values of the
unbalance. On amagnified picture (Fig.12), the gain Rh > 0.21 for the rotor
speed Ω=250s−1 is sufficient to destabilize motion. The dynamic behaviour
of the rotor is similar for Ω =270s−1 (Figs.13 and 14) but motion becomes
unstable for much smaller value of the unbalance (Rh > 0.001) than in the
case of the rotor speed Ω=250s−1.

Fig. 12. Root locus of the system for rotor angular speed 250s−1. Amagnified part
of interest of Fig.11

For higher angular speeds of the rotor, we have the same dynamical be-
haviour of the rotor but the unstable motion is excited by bigger values of
unbalance (see Figs.15 and 16).



842 Z. Gosiewski

Fig. 13. Root locus of unbalanced torsional/lateral vibrations for rotor angular
speed 270s−1 when the unbalance Rh changes from 0 to infinity

Fig. 14. Root locus of the system for rotor angular speed 270s−1. Amagnified part
of interest of Fig.13

Fig. 15. Root locus of the system for rotor angular speed 350s−1



Analysis of coupling mechanism... 843

Fig. 16. 16. Root locus of the system for rotor angular speed 350s−1. Amagnified
part of interest of Fig.15

Fig. 17. Ranges of the rotor speed with unbalances which can destabilize rotor
vibrations

After all considerations, we conclude that we are able to show the ranges
of unstable vibrations (Fig.17). In the first range (up to 100s−1), unstable
torsional/lateral vibrations are connected with positive value of one zero of
the open-loop transfer function. In the second range of ustable rotor angular
speeds, the alternating pole-zero pattern is destroyed (Preumont, 2002). The
neighbourhood of two zeros or two poles is a simple indicator for detecting
unstable behaviour of the system. The closer to the crossing point the smaller
value of the unbalance is needed to trigger the instability in both ranges.

6. Conclusions

The theory of control tools has been employed in the analysis of coupled vi-
brations of the rotor. Using Evans’method, the unstable regions of vibrations



844 Z. Gosiewski

for both coupled systems have been demonstrated. Analysis of pole and ze-
ro locations over a wide range of the rotor angular speed provides complete
insight into the rotor dynamics. The generated results can be applied for dia-
gnosing the technical state of rotors and also can be used in the control of
rotor vibrations.

References

1. Gosiewski Z., Muszyńska A., 1992,Dynamika maszyn wirnikowych (Dyna-
mics of rotating machinery), Publishing Office ofWSInż. Koszalin

2. Muszyńska A., Goldman P., Bently D.E., 1992, Torsional/lateral vibra-
tion cross-coupled responses due to shaft anisotropy: a new tool in shaft crack
detection,Conference of Institution of Mechanical Engineers, Bath, Great Bri-
tain

3. Sawicki J.T., BaakliniG.Y.,Gyekenyesi A.L., 2004,Dynamic analysis of
accelerating cracked flexible rotor, Paper GT2004-54095, Turbo ASME Turbo
Expo Conference, 14-17 June, Vienna, Austria

4. Takahashi Y., Rabbins M.J., Auslander D.M., 1972,Control and Dyna-
mic Systems, Addison-Wesley, Reading,MA.

5. PreumoutG., 2002,Vibration Control of Active Structures: An Introduction,
Kluwer Academic Publishers, Dordrecht

Analiza mechanizmu sprzęgającego drgania giętno-skrętne wirnika

Streszczenie

Zjawisko sprzęgania się drgańgiętnych i skrętnychwmaszynachwirnikowychmo-
że prowadzić do niestabilnego ruchu wirnika. W niektórych maszynach wirnikowych
prędkość obrotowa wirnika zmienia się w szerokim zakresie. Zadaniem jest znalezie-
nie takich zakresów prędkości obrotowej, dla których ruch wirnika pozostaje stabilny
mimo sprzęgania się jego postaci drgań. Zwykle niestabilne zakresywystępują w oto-
czeniu prędkości kątowych, w których przecinają sięmapy częstości własnychwyzna-
czonych oddzielnie dla drgań giętnych i drgań skrętnych. W artykule wykorzystano
metodę linii pierwiastkowych, abywykazać, które przecięcia są niestabilne i dlaczego.

Manuscript received March 19, 2008; accepted for print May 19, 2008