43

### Chapter 4

## 4.

## ROTORDYNAMICS

Reliability and accuracy of turbomachines are influenced by noise and vibration that may occur. The most important source of vibration is set from the dynamic of the shaft and other components associated with it as bearings and impellers. Hence vibrations should be always minimized and critical speeds should occur at nonoperating expected values. Although rotordynamic instability is a principal concern in the dynamic of the shaft, oscillating radial forces can be caused by unsteady flows by means of flow oscillations and can reach 20% of the axial force.

### 4.1.

### ROTORDYNAMIC INSTABILITY

In real situations the shaft of the turbomachine is not in the ideal position due to nonsymmetrical
forces or for uncertainties on bearings, seals, stator and rotor components that lead to a deflection of
*the shaft. This deflection may create a secondary motion of the axis of rotation known as whirl. *

The deflection is usually acceptable until it reaches a critical configuration for the clearance
between stator and rotor elements. This limiting value of deflection is a function of the ratio between
the rotational speed and the critical velocity. The latter can be defined by means of the physical system
*of Figure 4.1 where an impeller with mass m concentrated in its center of gravity G is rotated from a *
*shaft without mass. In this configuration d is the distance between G and the axis of rotation O of the *
*shaft. The axis of rotation is finally supposed to rotate with a precession motion at a constant speed ω. *

Hence the rotor will be in equilibrium condition if the centrifugal force and the elastic force are equal to each other:

2
3
( )
*c* *EJ*
*F* *k* *m* *d*
*l*
ε ε ω ε
= → + = (4.1)

*where E is the Young's modulus, J is the cross sectional moment of inertia and l is the length of the *
shaft. From equation 4.1 it is possible to obtain the following expression for the lateral deflection of
the axis of rotation

2
2
*md*
*k m*
ω
ε
ω
=
− (4.2)

44

*c* _{m}k

ω = (4.3)

Two situations can arise from previous considerations:

•* For ω > ωc the modulus of ε will decrease for increasing ω and its direction will be inward. *

•* For ω < ωc* the centrifugal force is in phase with the deflection and ε will be in outward

direction.

**Figure 4.1 Ideal system of a rotor with center of mass displaced with respect to axis of rotation. Jery [2] **

*If an external damper is considered, a phase displacement, β, will be present between the *
centrifugal force and the vector position of the center of mass. This displacement can be indicated as
*in Figure 4.2 and it is equal to 90° when ω = ωc*. Hence a damper can actually reduce the peak of the

vibration but it does not affect its frequency.

Although it is possible to find the critical velocities for these ideal situations, in real turbomachines the issue becomes much more complex and there will be an infinite number of critical velocities. Since it is impossible to know all of them, only the first and more important speeds are found by means of an equivalent idealized and discrete system composed by a set of masses, dampers and springs.

Appropriate margins will be set to operate at a speed far enough from critical values and if the rotor will go through these speeds, it has to be a rapid process.

Moreover the whirl motion can be forced or self-excited. The former case is a synchronous whirl and therefore the modulus presents a maxima at resonance condition. On the other hand, the latter case exhibit a nonsynchronous frequency close to the critical speed of the rotor but its intensity does not assume significant values until the so called Onset Speed of Instability is reached.

The forced whirl motion is a classical situation of forced vibration and therefore it is easy to predict. This allows to avoid excessive noise and vibration in the turbomachine, limitating its

45

consequences. On the other hand the self-excited whirl is still not well understood and should be treated carefully because of its intrinsic instability and for being nonsynchronous, forcing continuous reversals of force direction on the rotor.

**Figure 4.2 Ideal system of a rotor with center of mass displaced with respect to the axis of rotation, in the presence of a damper. **
**Jery [2] **

### 4.2.

### ROTORDYNAMIC THEORY

The instantaneous resulting force 𝐹⃗(𝑡) that the fluid exerts on the impeller can be expressed as the sum of two different components:

1. The steady forces 𝐹⃗0. They are also termed as radial forces or radial thrust. It is the only

component in case of centered rotor (ε* = 0) and it is independent from the eccentricity. *

They are obtained as the time-averaged value of 𝐹⃗(𝑡).

2. The fluid induced rotordynamic forces 𝐹⃗𝑅𝐷 which arise as a consequence of the

displacement of the axis of rotation.

Assuming a linear perturbation model (small displacement ε), the relation between the
displacement and the rotordynamic force can be considered linear by means of a time independent
*rotordynamic force matrix [A]: *

### [ ]

0 0

( ) * _{RD}* ( )

*F t* =*F* +*F* =*F* + *A* ε *t* (4.4)

In a spatial fixed reference frame, according to Figure 4.3, the hydrodynamic force can be expressed also as:

46
0
0
( ) ( )
( ) ( )
*X*
*X* *XX* *XY*
*Y*
*Y* *YX* *YY*
*F*
*F t* *A* *A* *x t*
*F*
*F t* *A* *A* *y t*
= +
(4.5)

*It is also important to note that [A] is a function not only of the geometry but also of the whirl *
*frequency ω [rad/s]. *

**Figure 4.3 Forces acting on a displaced rotor center with whirl motion. Jery [2] **

In addition to the forces 𝐹⃗(𝑡), it is also possible to obtain the fluid induced bending moments 𝑀��⃗(𝑡). With the same notation of equation 4.5 the following relation can be obtained as the sum of two different analogous components:

### ( )

### ( )

00### [ ]

### ( )

### ( )

*X*

*X*

*Y*

*Y*

*x t*

*M t*

*M*

*B*

*y t*

*M t*

*M*

_{=}

_{+} (4.6) 0 0 ( ) ( ) ( ) ( )

*X*

*X*

*XX*

*XY*

*Y*

*Y*

*YX*

*YY*

*M*

*M t*

*B*

*B*

*x t*

*M*

*M t*

*B*

*B*

*y t* = + (4.7)

*where B is the rotordynamic moment matrix. *

*An important property of the matrices [A] and [B] is that the geometry is invariant to a rotation of *
*the x,y axis since they transform the eccentricity vector into a coplanar vector. Hence their components *
will be related to each other as follows:

;
*xx* *yy* *xy* *yx*
*A* =*A* *A* = −*A* (4.8)
;
*xx* *yy* *xy* *yx*
*B* =*B* *B* = −*B* (4.9)

Setting a displacement modulus, ε*, with a whirl motion at a frequency ω on a circular orbit (see *
*Figure 4.4), the displacements x(t) and y(t) can be expressed as *

### (

### )

### (

### )

0 0 0 0 cos sin*X*

*X*

*XX*

*XY*

*Y*

*Y*

*YX*

*YY*

*F*

*t*

*F*

*A*

*A*

*F*

*t*

*F*

*A*

*A*ε ω ω ε ω ω + = +

_{+} (4.10)

*where ω0* is the angle between the absolute reference frame and the eccentricity vector at the initial

47

**Figure 4.4 Representation of the rotordynamic forces in absolute and rotating reference frames. Torre et al. [1] **

Another notation for rotordynamic forces and rotordynamic moments is to define the normal and
*tangential components as in Figure 4.4. The component FN is then defined positive as outward and FT*

*is defined positive if directed in the same direction of the impeller rotation Ω. This alternative notation *
is important because from these components it is easy to determine the local stability of the whirl
motion. From the general definition of the normal and tangential components, a dependence from the
rotordynamic force matrix is found in case of circular whirl orbit (see also following equations 4.17
and 4.18):

### [

0 0### ]

0 0 XX YY 1 (t) (t) 1_{cos( t}

_{)}

_{sin( t}

_{)}(t) 1 ( ) 2

*T*

*T*

*N*

*X*

*Y*

*N*

*XX*

*YY*

*F*

*F*

*dt*

*F*

*F*

*dt*

*T*

*T*

*F*

*A*

*A*

*A*

*A*ε

_{ω ω}

_{ω ω}ε ε ε ε ⋅ = = + + + = + = =

### ∫

### ∫

(4.11)### [

0 0### ]

0 0 1 (t) (t) 1_{sin( t}

_{)}

_{cos( t}

_{)}(t) 1 ( ) 2

*T*

*T*

*T*

*X*

*Y*

*T*

*YX*

*XY*

*XY*

*YX*

*F*

*F*

*dt*

*F*

*F*

*dt*

*T*

*T*

*F*

*A*

*A*

*A*

*A*ε

_{ω ω}

_{ω ω}ε ε ε ε ⋅ = = − + + + = − = − =

### ∫

### ∫

_{(4.12) }

• The normal force is considered positive in the outward direction. If so it is destabilizing as it tends to increase ε.

• The tangential force is considered positive when directed in the same direction of the
*impeller rotation Ω. In this case it sustains the rotation of the eccentricity and is therefore *
destabilizing.

*Assuming a reference frame x,y fixed with the impeller and therefore rotating at a speed Ω, it is *
possible to obtain the following relations for the two reference frames:

0 0
0 0
(t) (t)cos( t ) (t)sin( t )
(t) (t)sin( t ) (t)cos( t )
*X* *x* *y*
*Y* *x* *y*
*F* *F* *F*
*F* *F* *F*
= Ω + Ω − Ω + Ω
= Ω + Ω + Ω + Ω (4.13)

48 Moreover combining the equations 4.13 with 4.10:

0 0 0 0 0

0 0 0 0 0

(t)cos( t ) (t)sin( t ) F cos( t ) sin( t )

(t)sin( t ) (t)cos( t ) F cos( t ) sin( t )

*x* *y* *X* *XX* *XY*
*x* *y* *Y* *YX* *YY*
*F* *F* *A* *A*
*F* *F* *A* *A*
ε ω ω ε ω ω
ε ω ω ε ω ω
Ω + Ω − Ω + Ω = + + + +
Ω + Ω + Ω + Ω = + + + + (4.14)

and by their integration the steady forces can be found

0 0 0
0
0 0
0
0 0
1 _{(t)cos( t} _{)} _{(t)sin( t} _{)}

1 cos( t) (t)cos( ) (t)sin( ) sin( t) (t)sin( ) (t)cos( )

*T*
*X* *x* *y*
*T*
*x* *y*
*x* *y*
*F* *F* *F* *dt*
*T*
*F* *F*
*T*
*F* *F* *dt*
= _{} Ω + Ω − Ω + Ω _{} =
= _{} Ω _{} Ω − Ω _{}+
− Ω _{} Ω + Ω _{}

### ∫

### ∫

(4.15) 0 0 0 0 0 0 0 0 0 1_{(t)sin( t}

_{)}

_{(t)cos( t}

_{)}

1 cos( t) (t)sin( ) (t)cos( ) sin( t) (t)cos( ) (t)sin( )

*T*
*Y* *x* *y*
*T*
*x* *y*
*x* *y*
*F* *F* *F* *dt*
*T*
*F* *F*
*T*
*F* *F* *dt*
= _{} Ω + Ω + Ω + Ω _{} =
= _{} Ω _{} Ω + Ω _{}+
+ Ω _{} Ω − Ω _{}

### ∫

### ∫

(4.16)*The components AXX and AYX* can be obtained also integrating the first and second equation in 4.14

respectively, as follows:

### [ ]

00

2*T* _{cos( t} _{)dt}

*T*

### ∫

ω ω+ (4.17)*whereas the components AXY and AYY* are found by means of the integration of the same equations

### [ ]

00

2*T* _{sin( t} _{)dt}

*T*

### ∫

ω ω+ (4.18)*In general it is quite common to decompose the matrix [A] into different components: added mass, *
damping and stiffness matrices. Hence

### [ ]

*A*

*x*

*M*

*m x*

*C c x*

*K k*

*x*

*y*

*m M y*

*c C y*

*k K y*

_{= −}

_{−}

_{−}

_{−}

_{−}

_{−} (4.19)

*where the dot indicates the differentiation with respect to time. The matrix [A] is therefore linked to *
the acceleration, velocity and displacement of the rotating reference frame by means of the three
*matrices [M],[C] and [K]. *

*In this notation M and m are respectively termed direct and cross-coupled added mass, C and c are *
*the direct and cross-coupled damping, K and k are the direct and cross-coupled stiffness. From *
previous relations the following expression for normal and tangential forces can be obtained

* * 2 * *
* * 2 * *
( ) ( / ) ( / )
( ) ( / ) ( / )
*N*
*T*
*F t* *M* *c* *K*
*F t* *m* *C* *k*
ω ω
ω ω
= Ω − Ω −
_{= −} _{Ω −} _{Ω +}
(4.20)

where the asterisks indicate the corresponding dimensionless parameters.

*It is therefore clear that FN, FT* and the elements of the rotordynamic force matrix present a

49

common situation in some kind of pumps and therefore the aim of studying the rotordynamic instability is to understand the rotordynamic mechanism that give rise to cross sectional forces and to understand how the rotordynamic coefficients affect the shaft motion.

*In practice, it is commonplace to observe a small value of m so it can be neglected and the relation *
between tangential force and whirl frequency ratio becomes linear. In this situation the stability of the
*tangential force is set by the value of k*_{Ω / ωC}*_{ where k}* _{/ C}*_{ = k / ΩC is recognized as the whirl ratio }*

(not to be confused with the whirl frequency ratio). Larger values of whirl ratio means a major possibility of rotordynamic instability, due to a wider region of destabilized tangential force.

For centrifugal pumps, typical results for the dimensionless normal and tangential forces can be
seen in Figure 4.5. From the two diagrams it is possible to verify the previous statements. It is also
*clear that the stability of the two forces is easily obtained from the two graphics. In this case FN* is

positive for almost all of the range of whirl frequency ratios and therefore it is mainly destabilizing. On the other hand the tangential force becomes whirl destabilizing for the range 0 < ω/Ω < 0.35 but it is stabilizing in the rest of the range.

**Figure 4.5 Typical dimensionless normal and tangential forces for the Impeller X/Volute A configuration at 1000 rpm and ϕ = 0.092. ****Jery et al. [6] **

The rotordynamic coefficients of equation 4.20 can be therefore obtained from the two representations and it can be observed that for centrifugal pumps:

•* The direct stiffness, K, is always negative due to the Bernoulli effect. Indeed at high *
Reynolds number the Bernoulli equation is valid and, as an eccentricity is introduced, the
flow increases its velocity in the region in which the clearance is reduced, resulting in a
lower local pressure. The opposite thing will happen elsewhere in the rotor circumferential
region and a net force in the direction of the displacement will be present (negative
stiffness).

•* The cross-coupled stiffness, k, is always positive and it is related with the unstable region *
*at low positive whirl frequency ratios for the tangential force. Thus k indicates the degree *
of destabilization imposed by the fluid.

50

•* The direct damping, C, is positive and usually it is about half of the cross-coupled *
*damping, c. *

•* The cross-coupled added mass, m, is much smaller than the direct added mass, M, and, *
thus, it can be neglected.

On the other hand, in case of unshrouded axial flow pump or axial inducers prior to centrifugal pumps, different less understood behavior of rotordynamic forces are encountered due to the dependence on the dynamic response of the tip clearance flow.

Another important parameter is the nondimensional rotordynamic force 𝐹⃗𝑅∗ that can be expressed as

modulus and phase of the normal and tangential forces as follows:

### ( ) ( )

2 2 *_{*}

_{*}

*R*

_{N}

_{T}*F*=

*F*+

*F* (4.21) arctan

*T*

*N*

*F*

*F*φ=

_{}

_{} (4.22)

The conditions of stability previously stated for tangential and normal forces can then be summarized as in Figure 4.6, where the stability regions are shown as colored areas for positive whirl frequency ratios (top) and for negative whirl frequency ratios (bottom).

In previous considerations the attention has been focused on rotordynamic and radial forces, though relevant observations can also be obtained from rotordynamic moments. Similarly to normal and tangential forces, normal and tangential moments can be defined and, in the particular case in which the whirl orbit is circular with eccentricity ε and whirl frequency ω, their expressions will be analogous to equations 4.11 and 4.12:

1 ( )
2
*N* *XX* *YY* *XX* *YY*
*M* = *B* +*B* ε =*B* ε =*B* ε (4.23)
1 ( )
2
*T* *YX* *XY* *YX* *XY*
*M* = *B* −*B* ε =*B* ε = −*B* ε (4.24)

It is also important to understand that each component of a turbomachine (seals, bearings, rotors, etc) will present a set of rotordynamic coefficients different with respect to the others and a complete analysis of the issue would require to find these coefficients for any element that composes the turbomachine.

### 4.3.

### NONDIMENSIONAL PARAMETERS

Nondimensionalization of forces and moments is useful as it allows for easy generalization to and comparison of pumps operating under geometric and fluid dynamic similarity conditions (shape, working fluid, speed of rotation, eccentricity, flow turbulence, thermal/inertial cavitation).

The common nondimensional form, 𝐹0∗, used for radial forces is

* 0
0 2 3
2
*L* *T*
*F*
*F*
*R L*
πρ
=
Ω (4.25)

*where RT2 is the discharge radius and for centrifugal impellers L corresponds to b2* (width of discharge)

51

**Figure 4.6 Conditions of stability for normal and tangential unsteady forces in case of positive (top) or negative (bottom) whirl **
**frequency ratios. Torre et al. [1] **

Analogous expressions can be found for other parameters, as follows:
*
2 2
2
*N*
*N*
*L* *T*
*F*
*F*
*R L*
πρ ε
=
Ω (4.26)
*
2 2
2
*T*
*T*
*L* *T*
*F*
*F*
*R L*
πρ ε
=
Ω (4.27)
*
2 2
2
[ ]
*L* *T*
*A*
*A*
*R L*
πρ
=
_{Ω} (4.28)

52

In an analogous manner, fluid induced bending moments and rotordynamic moment matrix are nondimensionalized by means of following relations:

* 0
0 2 4
2
*L* *T*
*M*
*M*
*R L*
πρ
=
Ω (4.29)
*
2 3
2
*N*
*N*
*L* *T*
*M*
*M*
*R L*
πρ ε
=
Ω (4.30)
*
2 3
2
*T*
*T*
*L* *T*
*M*
*M*
*R L*
πρ ε
=
Ω (4.31)
*
2 3
2
[ ]
*L* *T*
*B*
*B*
*R L*
πρ
=
_{Ω} (4.32)

Lastly, the relations between the rotordynamic coefficients and their corresponding dimensionless
values are
* * * * * *
2 2 2 2
2 2 2
, , ,
, ; , ; ,
*L* *T* *L* *T* *L* *T*
*M m* *C c* *K k*
*M m* *C c* *K k*
*R L* *R L* *R L*
ρ π ρ π ρ π
= = =
Ω Ω (4.33)

### 4.4.

### FURTHER CONSIDERATIONS ON RADIAL FORCES

In section 4.3 the attention was focused on rotordynamic forces although radial forces may adversely affect the performance in a meaningful way. Indeed these forces, that are created by the nonaxisymmetric flows through a pump, are capable of leading the bearings to deflection and thus failures. Their deflection can displace the axis of rotation of the rotor and therefore rotordynamic forces can arise, degrading the hydraulic performance of the turbomachine.

Radial forces depends on the flow coefficient and on the geometry of the diffuser/volute.
*Measurements of these forces have been studied since 1960 by Agostinelli et al.,Iverson et al., *
*Biheller, Grabow and Chamieh et al, taking into account different impellers, diffusers and *
configurations. Stepanoff (1957) stated also an empirical equation to evaluate the modulus of the
nondimensional radial force in a centrifugal pump with spiral volute:

### (

_{2}

_{2}

### )

### {

### (

### )

2### }

0 0*X* 0*Y* 0.229 1 / *D*

*F* = *F* +*F* = ψ − *Q Q* (4.34)

and for collectors with uniform cross-sectional area:

0 0.229 / *D*

*F* = ψ*Q Q* (4.35)

Small changes in relative positions between impeller and volute can lead to large variations of
*radial forces and it can be seen through the stiffness matrix [K]. This means that at a particular *
location the radial force can be zero. This particular property can be seen in Figure 4.7, where the
locus of zero for radial forces, obtained by Chamieh et al. and Domm and Hergt, is plotted.

53

**Figure 4.7 Locus of zero of radial forces for the Impeller X/Volute A combination. Brennen [3] **

If a centrifugal pump is considered, the possible contributions to the radial force can be identified with:

• Circumferential pressure variation at the discharge.

• Leakage flow from discharge to inlet. This contribution can be affected from nonuniformity of the pressure at the discharge, resulting in a circumferential nonuniformity pressure in the leakage flow.

• Circumferential nonuniformity of the inlet flow rate may lead to a circumferential nonuniformity of the momentum flux. This third contribution can usually be considered as negligible with respect to the others.

From a 1-D theoretical model developed by Adkins and Brennen, based on the integration of the circumferential pressure gradient, it has been demonstrated that the circumferential nonuniformity pressure at the discharge is the main contribution due to the agreement between theoretical model and experimental data.

Clearly, bending pipes at the inlet may produce radial forces due to the consequent inlet nonuniformities of the flow.

Moreover radial forces are affected from cavitation phenomenon as the head rise of the turbomachine starts to be influenced from it.

54

### 4.5.

### REFERENCES

[1] L. Torre, A. Pasini, A. Cervone, L. Pecorari, A. Milani e L. d’Agostino, Rotordynamic
*Forces on a Three Bladed Inducer, ALTA S.p.A., 2010. *

[2]

### B.Jery, Experimental Study of Unsteady Hydrodynamic Force Matrices on

*Whirling Centrifugal Pump Impellers, Ph.D. Thesis, California Institute of *

*Technology, 1987 *

*[3] C.E. Brennen, Hydrodynamics of Pumps, Oxford University Press, 1994. *

[4] Chamieh, D.S., Acosta, A.J., Brennen, C.E., and Caughey, T.K., Experimental
measurements of hydrodynamic radial forces and stiffness matrices for a centrifugal
*pump-impeller, ASME J. Fluids Eng.,107, No. 3, 307-315, 1985. *

[5] Adkins, D.R. and Brennen, C.E., Analyses of hydrodynamic radial forces on centrifugal
*pump impellers, ASME J. Fluids Eng.,110, No. 1, 20-28, 1988. *

[6] Jery, B., Acosta, A.J., Brennen, C.E., and Caughey, T.K., Forces on centrifugal pump
*impellers, Proc. Second Int. Pump Symp., Houston, Texas, 21-32, 1985. *