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34.3 Rotordynamic Analysis

Back

Rotordynamic analysis is a part of current design procedures for rotating machinery that is carried out to

predict the vibration behavior during operation of the machine. Potential problems are identified and

eliminated by analytical means well before the manufacturing of components is begun. Furthermore,

when a machine in operation displays unusual vibration behavior, analytical means are employed to

study, identify, and help resolve the problem. In order for the analysis to be useful, it must be accurate

and cost-effective.

During the last 100 years, several analytical procedures have been developed to understand the

vibration behavior of rotating machinery. Some of these techniques are of historical interest only, and

their usefulness in practical systems is very limited. With the advent of computer technology and

advanced modeling techniques, several computer-based procedures have been developed to predict the

vibration behavior of rotating machinery quite precisely. Of the most commonly used procedures, two

are based on the lumped-parameter model where the distributed elastic and inertial properties are

* The two major categories of vibration phenomena that occur in rotating machinery are

forced vibration and self-excited instability.

* Parametric instability is a special case of self-excited instability where some of the normally

constant parameters vary, influencing rotor motion.

* Torsional vibrations are similar to lateral vibrations but occur in the planes perpendicular

to the shaft axis.

34-18 Vibration and Shock Handbook

represented as a collection of rigid bodies connected by massless elastic beam elements. These two

procedures are the transfer matrix formulations introduced by Myklestad (1944) and Prohl (1945), and

the direct stiffness matrix formulations proposed by Biezeno and Grammel (1954). Ruhl and Booker

(1972) introduced the third commonly used method, based on the finite element analysis (FEA) model in

which the rotor is represented as an assemblage of elements with distributed elastic and inertial

properties. Several of the well-known procedures are discussed next, some of them for historical

significance.

34.3.1 Analysis Methods

34.3.1.1 Rankine’s Numerical Method

Rankine (1869) proposed that, for a shaft of a given length, diameter, and material, there is a limit of

speed, and for a shaft of a given diameter and material, turning at a given speed, there is a limit of length,

below which centrifugal whirling is impossible. The limits of length and speed depend on the way the

shaft is supported. The critical speed of the shaft is given by the following equation:

v ¼

kðHgÞ1=2

b2 ð34:15Þ

where

v ¼ critical speed in rad/sec

k ¼ radius of gyration of the cross section of the shaft

g ¼ acceleration due to gravity

H ¼ modulus of elasticity expressed in units of height of itself ðH ¼ E=rÞ

E ¼ Young’s modulus

r ¼ density of the material

l ¼ shaft length

b ¼ l=p for a simply supported shaft

b ¼ l=0:595p for an overhanging shaft

34.3.1.2 Greenhill’s Formulae

Greenhill (1883) introduced the following differential equation of motion for a uniform shaft slightly

deformed from straightness by centrifugal whirling:

d4y

dx4 2

mv2

gEak2 y ¼ 0 ð34:16Þ

The general solution to Equation 34.16 is given by

y ¼ B cosh mx þ A cos mx ð34:17Þ where

m4 ¼ mv2=gEak2

m ¼ weight of the shaft per unit length

v ¼ rotational speed

a ¼ cross-sectional area of the shaft

The constants A and B depend on the boundary conditions at the support locations.

34.3.1.3 Reynolds’ Equations

Reynolds extended the differential equation of motion for a uniform rotating shaft (Equation 34.16) to

include shafts loaded with pulleys (disks) and for multispan rotors (Dunkerly, 1894).

Vibration in Rotating Machinery 34-19

At a bearing support, the difference in shear force must equal the bearing load P:

dMr

dMl

dx ¼ P ð34:18Þ

where Mr and Ml are bending moments in the right (r) and left (l) sides of the load.

At a load consisting of a revolving weight, W ; the above equation becomes

dMr

dMl

dx ¼

v2y ð34:19Þ

A further equation may be obtained by considering the centrifugal couple (gyroscopic moment) as

given by

Mr 2 Ml ¼ v2I0 dy

dx ð34:20Þ

where I0 ¼ moment of inertia of the pulley.

The solution to Equation 34.20 is given by

y ¼ A cosh mx þ B sinh mx þ C cos mx þ D sin mx ð34:21Þ

The values of A; B; C; and D will depend on the boundary conditions between any two singular points.

34.3.1.4 Dunkerley Method

When considering the effects of the pulleys and the shaft together, the formulae derived by Reynolds were

found to be limited for practical purposes. Dunkerly (1894) proposed an empirical method to consider

the effects of the shaft and each of the pulleys separately, and then combine them using the following

formula to obtain the critical speed of the rotor:

c ¼

v2s

i¼1

v2i

ð34:22Þ

where

vc ¼ critical speed of the rotor

vs ¼ critical speed of the shaft alone

vi ¼ critical speed of the ith disk on a weightless shaft

In the case of the unloaded shaft, the critical speed, vs; is given by the following formula

mv2s

gEI

􀁻 !1=4

l ¼ a ð34:23Þ

where

I ¼ sectional inertia of shaft

l ¼ length of the span

a ¼ a coefficient dependent on the manner of support of the shaft

The critical speed of the rotor vi with a single disk of weight Wi on a weightless shaft is given by

vi ¼ u

gEI

Wic3

􀀏 􀀐1=2

ð34:24Þ

where

c ¼ distance of disk from nearest support

u ¼ a coefficient dependant on the manner in which the shaft is supported, the position of the

disk within the span and the dimensions of the disk

34-20 Vibration and Shock Handbook

34.3.1.5 Rayleigh Method

The Rayleigh method is based on the premise that, when a system vibrates at its natural frequency, the

maximum potential energy stored in the elastic components is equal to the maximum kinetic energy

stored in the masses (Rayleigh, 1945).

The first natural frequency of a vibrating uniform beam is given by the following equation:

v2 ¼

ðl

d2y

dx2

􀁻 !2

ðl

y2 dx

ð34:25Þ

The Rayleigh formula for a lumped mass system is

v2 ¼

i¼1

miyi

i¼1

miy2

ð34:26Þ

where

v ¼ first natural frequency

y ¼ deflection of the beam

x ¼ distance along x-axis

l ¼ length of the beam

mi ¼ ith lumped mass

yi ¼ static deflection of ith mass

The accuracy of the Rayleigh method depends upon the selection of a suitable deflection curve that

approximates the fundamental mode shape. If the assumed curve represents the true mode shape, then

the correct fundamental natural frequency will result. All deviations from the true mode shape will yield

frequencies that are higher than the correct value.

34.3.1.6 Ritz Method

The Ritz method is an improvement on the Rayleigh method (Timoshenko et al., 1974) where the mode

shape is represented by several orthogonal functions with unknown coefficients that satisfy the boundary

conditions. The orthogonal functions are represented by a series of functions, FiðxÞ; where i varies from

1 to n: The mode shape is represented by the following expression:

y ¼

i¼1

aiFiðxÞ ð34:27Þ

In order for the coefficients ai in the above equation to yield minimum values when substituted in the

energy balance equation proposed by Rayleigh, the following expression needs to be satisfied:

›

›ai

ðl

d2y

dx2

􀁻 !2

ðl

y2 dx

¼ 0 ð34:28Þ

From Equation 34.25 and Equation 34.28, we find

›

›ai

ðl

d2y

dx2

􀁻 !2

v2m

" #

dx ¼ 0 ð34:29Þ

Vibration in Rotating Machinery 34-21

Substituting Equation 34.27 for y in Equation 34.29 and performing the mathematical operations, a system

of linear equations in ai is obtained. The number of such equations will be equal to n: These equations will

yield solutions different from zero only if the determinant of the coefficients of ai is equal to zero. This

condition yields the frequency equation, from which the frequency of each mode can be derived.

34.3.1.7 Stodola – Vianello Method

The Stodola – Vianello method is a numerical iterative process (Timoshenko et al., 1974) that can be used

to calculate the natural frequencies and mode shapes of vibrating systems. An approximate mode shape is

first assumed and by successive iterations it is refined until convergence is obtained to the desired level of

accuracy. This method can be used to refine the assumed mode shape when using the Rayleigh formulae

or in the more general case of the matrix iteration process illustrated below.

Using Newton’s Second law, the equations of motion for a multi-DoF system in matrix notation are

{Y } ¼ v2i

½A􀀉½m􀀉{Y } ð34:30Þ

½A􀀉 ¼ ½K􀀉21 ð34:31 where Þ

½A􀀉 is the flexibility matrix

½m􀀉 is the mass matrix

½K􀀉 is the stiffness matrix

To start the iterative process a trial vector, {Y }1; representing the mode shape is substituted to both

sides of Equation 34.30 and solve for the natural frequency, vi: For this reason, let the product of ½A􀀉;

½m􀀉; and {Y }1 be {Y }02

: The first approximation for vi may be obtained by dividing any one of the

elements on {Y }1 by {Y }02

(Note that, if {Y }1 was the true mode shape, then the ratio for all such elements

will be equal.) The vector {Y }02

is then normalized by dividing all the elements by the first element to

produce {Y }2: The vector {Y }2 is premultiplied by ½m􀀉 and ½A􀀉 to produce {Y }03

: Once again the ratio of

corresponding elements of {Y }03

and {Y }2 are compared for equality.

This procedure is repeated until the mode shape and the associated frequency is determined to the

desired level of accuracy. In the above iteration procedure, the mode shape converges to the one

corresponding to the lowest natural frequency. If the stiffness matrix had been used instead of the

flexibility matrix, then convergence at the highest natural frequency is obtained. After the first mode of

vibration is determined, it is removed from the system matrices by the use of a sweeping matrix so that

higher modes can be obtained. This procedure is repeated until all the desired mode shapes and natural

frequencies are determined.

34.3.1.8 Myklestad – Prohl Method (Transfer Matrix Method)

The Myklestad – Prohl transfer matrix formulation (Myklestad, 1944; Prohl, 1945) is commonly used to

analyze lumped parameter models of rotating machinery. The distributed elastic and inertial properties

of the rotor are represented as a collection of rigid bodies connected by massless elastic beam elements as

illustrated in Figure 34.10. This method is best suited to calculate critical speeds and mode shapes of

rotors neglecting the effects of viscous damping. The Myklestad – Prohl procedure can also be adopted to

perform synchronous response and stability analysis, including for the effects of damping.

In order to demonstrate the transfer matrix procedure, an axisymmetric rotor is analyzed to determine

its undamped critical speeds and mode shapes. Refering to Figure 34.10, the rotor is divided into n nodes,

and each node is connected to the adjacent node by a massless elastic beam with uniform cross-sectional

properties. The mass of components such as disks, impellers, and so on, together with the mass of the

adjacent portion of the shaft, is lumped at the nodes. The Myklestad – Prohl method is based on the

solution of the Bernoulli – Euler equation and the variables of interest are displacement ðyÞ; slope ðuÞ;

moment ðMÞ; and shear ðV Þ: The development of the following procedure follows Childs (1993).

34-22 Vibration and Shock Handbook

At a typical nodal point ðnÞ; the variables on the left-hand side (l) are related to the variables on the

right-hand side (r) by the following relationship:

V r

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

1 0 0 0

0 1 0 0

0 2Jnv2 1 0

2mnv2 0 0 1

6666664

7777775

u l

V l

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

ð34:32Þ

For the purpose of abbreviation:

ðQÞTn

¼ ð yn un Mn VnÞ ð34:33Þ

and Equation 34.32 can be written in a more compact form as follows:

ðQÞr

n ¼ ½Tmn􀀉ðQÞl

n ð34:34Þ

where ½Tmn􀀉 represents the transfer mass matrix at node n:

At a massless beam section, connecting node n to node n þ 1 the transfer matrix is given by

nþ1

u l

nþ1

M l

nþ1

V l

nþ1

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

1 ln

2EIn

2l3

6EIn

0 1

EIn

2EIn

0 0 1 2ln

0 0 0 1

666666666664

777777777775

u r

V r

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

ð34:35Þ

Equation 34.35 may be written in a more abbreviated form as

ðQÞl

nþ1 ¼ ½Tbn􀀉ðQÞr

n ð34:36Þ

where ½Tbn􀀉 represents the beam element transfer matrix connecting node n to node n þ 1:

From Equation 34.34 and Equation 34.36, we obtain the combined transfer matrix for nodes

n and n þ 1:

ðQÞl

nþ1 ¼ ½Tbn􀀉½Tmn􀀉ðQÞl

n ¼ ½Tn􀀉ðQÞl

n ð34:37Þ

Massless

elastic beam

elements

Rotors (rigid

bodies)

2 3 4

N-1

n-2

n-1

FIGURE 34.10 Lumped-parameter model of rotor.

Vibration in Rotating Machinery 34-23

Starting with node one, successive matrix multiplications are carried out until node n þ 1 is reached. The

last node ðn þ 1Þ is a dummy node with the beam length, l; equal to zero, and the mass and inertias also

equal to zero. This makes the nodal parameters on the left-hand side of node n þ 1 equal to those on the

right-hand side of node n: The result is as follows:

ðQÞr

n ¼ ½Tn􀀉½Tn21􀀉· · ·½T1􀀉ðQÞl1

or ðQÞr

n ¼ ½T􀀉ðQÞl1

ð34:38Þ

The matrix ½T􀀉 is a function of the rotational speed; v: The Myklestad – Prohl method uses a trial and

error solution to determine the values of v which satisfy the boundary conditions and Equation 34.38

simultaneously. It is not necessary to store and multiply all the matrices together. The transfer matrix

procedure is used to proceed from one end to the other without having to store all the nodal matrices. In

all cases, two boundary conditions each are known at the two ends of the shaft, and the frequencies that

satisfy these boundary conditions are the critical speeds of the rotor. Once the critical speeds are

calculated, the corresponding mode shapes can also be determined using the transfer matrix procedure. It

should be noted that other types of elements, such as elastic supports, flexible couplings, and so on, could

also be introduced very conveniently.

34.3.1.9 Direct Stiffness Method

The direct stiffness method uses a lumped-parameter formulation to evaluate the dynamic characteristics

of a flexible rotor. The general differential equation of motion that characterizes its behavior (less the

damping and gyroscopic forces) is as follows:

½m􀀉 0

0 ½ J 􀀉

" #

ðY€ Þ

ð u€Þ

( )

þ ½K􀀉

ðY Þ

ðuÞ

( )

ðFÞ

ðTÞ

( )

ð34:39Þ

where ½m􀀉 and ½ J 􀀉 are diagonal matrices which contains the nodal masses, mi; and nodal moments of

inertia, Ji; respectively. The stiffness matrix, ½K􀀉; contains the internal stiffness terms of the beam

elements as well as any external spring stiffness at the supports. The vectors ðFÞ and ðTÞ represent external

forces and moments acting on the system, respectively.

The stiffness matrix for a typical beam element based on the Bernoulli – Euler equations is as follows

(Childs, 1993):

½Ki􀀉 ¼

2EIi

l3i

6 3li 26 3li

3li 2l2i

23li l2i

26 23li 6 23li

3li l2i

23li 2l2i

6666664

7777775

ð34:40Þ

The overall stiffness matrix, ½K􀀉; has to be assembled by combining the individual component matrices in

a systematic manner. The following procedure illustrates the process.

The stiffness matrix of the ith beam element in matrix notation is

½Ki􀀉 ¼ ½kij

;k􀀉 ð34:41Þ

where j and k vary from ð2i 2 1Þ to ð2i þ 2Þ:

To form the overall stiffness matrix, the elements with the same subscripts of adjacent beam elements

are added over n beam elements as given by the following equation:

½K􀀉 ¼ ½Kj;k􀀉 ¼

i¼1

2Xiþ2

j¼2i21

2Xiþ2

k¼2i21

kij

;k ð34:42Þ

Once the inertia matrix and the stiffness matrix for the entire system are assembled, the eigenvalues

and eigenvectors can be evaluated by solving the following homogeneous equation derived

34-24 Vibration and Shock Handbook

from Equation 34.39:

½M􀀉ðY€ Þ þ ½K􀀉ðYÞ ¼ 0 ð34:43Þ

There are numerous analysis procedures (Meirovitch, 1986) for the solution of Equation 34.43 that

yield the eigenvalues and eigenvectors of the system. The method of choice will depend on the

complexity and nature of the inertia and stiffness matrices. Perhaps the most widely known is the

matrix iteration using the power method in conjunction with the sweeping technique. However, this

method is not necessarily the most efficient, particularly for higher-order systems. The Jacobi’s

method, which uses matrix iteration to diagonalize a matrix by successive rotations, is more

commonly used owing to its higher efficiency. Details of these techniques are given in the text by

Meirovitch (1986).

When the damping matrix and the gyroscopic matrix is also included in Equation 34.39, the direct

stiffness method can be used to calculate damped critical speeds, forced rotor response, and instability of

the rotor in addition to the eigenvalues using similar methods of solution.

34.3.1.10 The Finite Element Analysis Method

The basis of the FEA method is to provide formulation for complex and irregular systems that can utilize

the automation capabilities of computers (also see Chapter 9). The FEA method considers a

rotordynamic system as an assemblage of discreet elements, where every such element has distributed

and continuous properties, namely, the consistent representation of both mass and stiffness as distributed

parameters. As illustrated in Section 34.3.1.9, the lumped-parameter method uses a consistent stiffness

matrix equation (Equation 34.40) in its formulation, and therefore, the identical procedure can be

adopted for the finite element method as well. For the distributed mass representation of an element,

Archer (1963) procedure, which is based on the assumption that the mass distribution is proportional to

the elastic distribution similar to the Rayleigh – Ritz formulation, is utilized. The resulting mass matrix is

as follows:

½mi􀀉 ¼

mili

420

156 22li 54 213li

22li 4l2i

13li 23l2i

54 13li 156 222li

213li 23l2i222li 4l2i 2

6666664

7777775

ð34:44Þ

The overall stiffness matrix, ½K􀀉; for the entire system is assembled by combining the individual

component matrices in a systematic manner according to Equation 34.41 and Equation 34.42. The

overall mass matrix can also be assembled in precisely the same manner, as given by Equation 34.45 and

Equation 34.46.

The mass matrix of the ith beam element in matrix notation can be represented as

½mi􀀉 ¼ ½mij

;k􀀉 ð34:45Þ

where j and k varies from ð2i 2 1Þ to ð2i þ 2Þ:

½M􀀉 ¼ ½Mj;k􀀉 ¼

i¼1

2Xiþ2

j¼2i21

2Xiþ2

k¼2i21

mij

;k ð34:46Þ

Once the mass matrix and the stiffness matrix for the entire system are assembled, Equation 34.43

that describes the free vibration of the complete system can be solved. The solution methods of

the eigenvalue problem, which can be utilized, are the same as those used for the direct stiffness

method illustrated in Section 34.3.1.9 above. Details of the FEA methods are given in Ruhl and

Booker (1972).

Vibration in Rotating Machinery 34-25

34.3.1.11 Torsional Analysis (Holzer Method)

The development of torsional analysis methods have gone through a similar evolutionary process to

lateral vibration methods. Holzer (1921) first introduced the lumped-parameter numerical method to

calculate torsional natural frequencies of a multi-DoF system. Even to-date, this is the most commonly

used method because of its simplicity and reasonable degree of accuracy. The Holzer method is a transfer

matrix formulation that uses a lumped parameter model similar to that used in the Myklestad – Phrol

method described in Section 34.3.1.8. The only difference is that the transfer matrices represented by

Equation 34.32 and Equation 34.35 are replaced by the equations

( )r

n¼

1 0

2v2J 1

" #

( )l

n ð34:47Þ

( )l

nþ1¼

0 1

nþ1

( )r

n ð34:48Þ

Starting with node one, successive matrix multiplications are carried out until node n þ 1 is reached. The

result can be represented by Equation 34.38. The matrix ½T􀀉 is a function of the rotational speed, v: In all

cases, one boundary condition at each end of the rotor is known. A trial-and-error solution to determine

the values of v which satisfy the boundary conditions and Equation 34.38 are simultaneously determined.

These values are the torsional critical speeds of the rotor. Once the critical speeds are calculated, the

corresponding torsional mode shapes can also be determined using the transfer matrix procedure.

In the case of branched systems and geared systems, particular attention has to be paid to the relative

rotational speeds of the components. The rule is quite simple: multiply all stiffness and inertias of the

geared shaft by N2; where N is the speed ratio of the geared shaft to the reference shaft.

Other methods such as the distributed mass matrix method, direct stiffness method, and finite element

method can also be used to determine torsional critical speeds of rotors. These procedures are very

similar to those for lateral critical speed analysis.

34.3.2 Modeling

The design and analysis of rotordynamic systems require the development of models that simulate the

behavior of the physical system. In the past, the critical speed of the rotor was considered to be the main

criterion for stable operation. Today, stable, well-damped rotordynamic response to the exciting forces

within a machine is considered to be a necessary condition for high reliability. The accuracy and

reliability of the results greatly depends on the credibility of the system model and its adaptability to the

analytical procedure. Even the most accurate and efficient analytical method cannot produce good results

from a bad model. The methods that are commonly used to model shaft sections and disks and other

such elements attached to shafts have been discussed in the previous sections. Useful formulae for

calculating critical speeds of simple systems are given in Table 34.2. Models to represent bearings, rotor

dampers, seals, and rotor– stator interactions are discussed in the following sections.

34.3.2.1 Journal Bearings

Journal bearings were used in rotating machinery for a long time before their dynamic characteristics

were fully understood. Considerable effort has been expended in the last few decades to understand and

develop techniques for their accurate representation in rotordynamic analysis. A variety of bearing types

with improved characteristics have been developed over the years. Figure 34.11 shows the most

commonly used types in rotating machinery. Hagg and Sankey (1958) were amongst the first to provide

dynamic stiffness and damping coefficients for a number of these bearing types. However, these

coefficients are considered incomplete as cross-coupling terms were not considered. Soon after, there was

a flurry of activity related to the analysis of journal bearings; Sternlicht (1959), Warner (1963),

34-26 Vibration and Shock Handbook

TABLE 34.2 Useful Formulas in Vibration Analysis and Design

Rankine formula

v ¼

kðHgÞ1=2

Note: This formula is of

historical interest only

and has limited practical value

Greenhill formula d4 y

dx4 2

mv2

gEak2 y ¼ 0

Dunkerly equation 1

v2c

v2s

i¼1

v2i

The above equation reduces to

v2c

g P

ystat

Formulas for natural

frequency calculation

(Blevins, 2001;

Gorman, 1975)

a b

vc ¼

ffiffiffiffiffiffiffi

3EIL

vc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

6EI

W ð3L 2 4aÞ

vc ¼

ffiffiffiffiffiffi

3EI

vc ¼

12EIL3

Wa3 b2ð3L þ bÞ

􀁻 !1=2

a b vc ¼

3L3 EI

Wa3 b3

􀁻 !1=2

vc ¼

3EI

WL3

􀀏 􀀐1=2

vc ¼

98EI

mL4

􀀏 􀀐1=2

m a vc ¼

64a

􀀏 􀀐2

􀀏 􀀏 􀀐

3 Þ2

mL4

􀀏 􀀐1=2

;

$ 0:25

(continued on next page)

Vibration in Rotating Machinery 34-27

TABLE 34.2 (continued)

vc ¼

237EI

mL4

􀀏 􀀐1=2

vc ¼ 274:7

􀀏 􀀐2

þ22:1

􀀏 􀀏 􀀐

þ 3:14Þ2

mL4

􀀏 􀀐1=2

;

, 0:25

m vc ¼

502EI

mL4

􀀏 􀀐1=2

m vc ¼

12:4EI

mL4

􀀏 􀀐1=2

Formulas for torsional

natural frequency

calculation

k J

vc ¼

􀀏 􀀐1=2

k1 J k2 1 J2

vc ¼

1ffiffi

2 p

k1 þ k2

J1 þ

􀀒

k1 þ k2

J1 þ

􀀏 􀀐2

4k1 k2

J1 J2

􀀙1=2

#1=2 (

k1 k2

vc ¼

k1 þ k2

􀀏 􀀐1=2

k1 J1

k2 J2

vc ¼

1ffiffi

2 p

k1 þ k2

J1 þ

k2 þ k3

􀀒

k1 þ k2

􀀘􀀏

k2 þ k3

􀀐2

4ðk1 k2 þ k2 k3 þ k1 k3 Þ

J1J2

􀀙1=2

#1=2

J1 k J2

vc ¼

J1 þ

􀀏 􀀐1=2

J1 J2 J3

k2 k1

vc ¼

1ffiffi

2 p

J1 þ

k1 þ k2

J2 þ

􀀒

J1 þ

k1 þ k2

J2 þ

􀀏 􀀐2

(

4k1 k2 ðJ1 þ J2 þ J3 Þ

J1 J2 J3

􀀙1=2

#1=2

Rayleigh equations

v2 ¼

ðl

d2 y

dx2

􀁻 !2

ðl

y2 dx

34-28 Vibration and Shock Handbook

TABLE 34.2 (continued)

v2 ¼

i¼1

mi yi

i¼1

mi y2

Ritz method y ¼

Pni

¼1 aiFi ðxÞ

›

›ai

ðl

d2 y

dx2

􀁻 !2

ðl

y2 dx

¼ 0

›

›ai

ðl

d2 y

dx2

􀁻 !2

v2m

" #

dx ¼ 0

Stodola – Vianello method {Y } ¼ v2i

½A􀀉½m􀀉{Y }; ½A􀀉 ¼ ½K􀀉21

Transfer matrix —

Myklestad – Phrol

method

nþ1

u ln

þ1

M ln

þ1

V ln

þ1

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

1 ln

l2n

2EIn

2l3n

6EIn

0 1

EIn

l2n

2EIn

0 0 1 2ln

0 0 0 1

666666666664

777777777775

u rn

M rn

V rn

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

Stiffness matrix for a

beam element

½Ki􀀉 ¼

2EIi

l3i

6 3li 26 3li

3li 2l2i

23li l2i

26 23li 6 23li

3li l2i

23li 2l2i

6666664

7777775

Mass matrix for a beam

element

½mi􀀉 ¼

mi li

420

156 22li 54 213li

22li 4l2i

13li 23l2i

54 13li 156 222li

213li 23l2i

222li 4l2i

6666664

7777775

Squeeze-film damper

coefficients

k ¼

24R3 Lmv1

r ð2 þ 12 Þð1 2 12 Þ

c ¼

12pR3 Lm

r ð2 þ 12 Þð1 2 12 Þ1=2

k ¼

2RL3mv1

r ð1 2 12 Þ2

c ¼

pRL3m

2C3

r ð1 2 12 Þ32

Unbalance sensitivity SF ¼

Rolling element bearing

defect frequencies

fbor ¼

60d

1 2

cos u

􀀏 􀀐2

" #

N¼rotational speed (rpm),

D ¼ rolling element pitch diameter

(continued on next page)

Vibration in Rotating Machinery 34-29

Lund (1964), Lund (1965), Glienicke (1966), Orcutt (1967), Lund (1968), Someya et al. (1988), and

several others provided complete bearing coefficients, including cross-coupling terms, for several bearing

types. This information is considered to be a valuable resource for those engaged in rotordynamic

analysis. The general form of the rotordynamic model for a journal bearing resulting from the above

contributions is given by the following equation:

( )

¼ 2

k11 k12

k21 k22

" #

( )

c11 c12

c21 c22

" #

( )

ð34:49Þ

Since the dawn of the digital computer era, several computer codes have been developed to analyze all

aspects of journal bearings, including stiffness and damping coefficients. Many of these codes have been

developed by equipment manufactures and research centers for their exclusive use. Several commercially

available software codes popularized in North America are given in Table 34.3. Although bearing

coefficients given in the form of charts and tables from the earlier studies are still in use, computer-based

codes are growing in popularity.

34.3.2.2 Rolling Element Bearings

Rolling element bearings are used in numerous types of rotating machinery which are required to be

compact, manage high loads, and have low heat rejection and simple lubrication systems. Unlike journal

bearings, their load-carrying capacity is not speed-dependent and as a result is capable of full load

capacity down to zero speed. Some of these salient features make rolling element bearings very attractive

to many industries.

From a rotordynamic standpoint, rolling element bearings are modeled as linear spring elements with

direct spring coefficients only. The damping terms are insignificant and as a result do not attenuate rotor

deflections at critical speeds. A typical rolling element bearing is represented by the following equation:

( )

¼ 2

k 0

0 k

" #

( )

ð34:50Þ

The absence of cross-coupling stiffness and damping terms signifies that bearing induced rotor instability

will not occur. Although, for convenience of analysis, the spring stiffness is considered linear, its

TABLE 34.2 (continued)

fir ¼

120

1 þ

cos u

􀀏 􀀐

d ¼ rolling element diameter,

N ¼ number of rolling elements

for ¼

120

1 2

cos u

􀀏 􀀐

u ¼ contact angle with respect

to axis, bor ¼ ball or roller defect

fc ¼

120

1 2

cos u

􀀏 􀀐

ir ¼ inner race defect,

or ¼ outer race defect,

c ¼ cage defect

Lomakin formula for

radial stiffness for a

close clearance bushing

k ¼

8 ð1 þ 6Þlm4 l

􀀏 􀀐2

DpD

m2 ¼

1 þ 6 þ ðll=2bm Þ

bm ¼ radial clearance, D ¼ diameter,

l ¼ length of bushing, Dp ¼ differential pressure

across bushing,

z ¼ inlet loss coefficient, l ¼ friction coefficient

34-30 Vibration and Shock Handbook

true behavior can be quite the opposite, leading to some calculation inaccuracies. The nonlinearities are

most significant where the bearings have no preload and some internal clearance in the bearings exists.

Preloaded bearings with little or no internal clearance behave quite linearly. Jones (1960), Harris (1991),

and Kramer (1993) have analyzed the bearing stiffness coefficients for the common types of rolling

element bearings, and this data can be utilized for rotordynamic study of rotating machinery.

34.3.2.3 Squeeze-Film Dampers

Squeeze-film dampers are used to introduce damping capacity to a rolling element bearing or, in the case

of journal bearings, to provide additional damping and stiffness to eliminate rotor instability problems.

Squeeze-film dampers have come into prominence through the modern aircraft gas turbine industry

where the bearing of choice is the rolling element bearing. In the mid-1970s, several designs were

introduced to add damping capacity and predictable stiffness to the rolling element bearings. In its basic

form, a squeeze-film damper is very similar to a nonrotating cylindrical journal bearing where the outer

race of the rolling element bearing forms the journal as illustrated in Figure 34.12. The addition of end

seals (to control leakage) and centering springs are modifications that have been introduced to enhance

its performance. The interactive force at a bearing using a squeeze-film damper can be represented by the

following equation:

( )

¼ 2

k 0

0 k

" #

( )

c 0

0 c

" #

( )

ð34:51Þ

The stiffness and damping coefficients for the squeeze-film dampers have been derived (Ehrich, 1999),

from the solution of the Reynolds’ equation for the case of a nonrotating journal bearing. For dampers

with end seals, the long journal bearing theory is used to generate the following stiffness and damping

20∞

R+C

W W W

R+C

(a) Plain Cylindrical (b) Two groove Cylindrical (c) Partial-arc Cylindrical (d) Offset Cylindrical

(e) Two groove Elliptical (f) Three lobe (g) Four lobe (h) Tilting pad

FIGURE 34.11 Common types of journal bearings.

Vibration in Rotating Machinery 34-31

TABLE 34.3 Rotordynamic Analysis Software

Name of Software Type of Analysis Supplier

CAD20 Lateral critical speeds of flexible rotors CADENSE Programs,

Foster Miller Technologies

Inc., Albany, NY, USA

CAD21 Unbalance response of flexible rotors

CAD21a Response of flexible rotors to

nonsynchronous sinusoidal excitation

CAD22 Torsional critical speeds and response

of geared systems

CAD24 Transient torsional critical speeds of geared system

CAD25 Dynamic stability of flexible rotors

CAD25a Transient response of flexible rotors

CAD26 Lateral critical speeds of multilevel rotors

CAD27 Unbalance response of multilevel rotors

CAD30 Dynamic coefficients of liquid lubricated journal bearings

CAD30a Dynamic coefficients of ball bearings

CAD31 Dynamic coefficients of liquid lubricated tilting

pad journal bearings

CAD32 Dynamic coefficients of liquid lubricated

axial-groove and single pad journal bearings

CAD34a Performance of tilting pad thrust bearings

CAD34b Performance of tapered-land thrust bearings

CAD36 Dynamic coefficients of liquid lubricated pressure

dam journal bearings

CAD38 Dynamic coefficients of liquid lubricated

deep-pocket hydrostatic journal bearings

CAD40 Dynamic coefficients of gas lubricated journal bearings

CAD41 Dynamic coefficients of gas lubricated tilting pad

journal bearings

CAD42 Dynamic coefficients of gas lubricated spiral

groove journal bearings

CAD42i Dynamic coefficients of liquid lubricated spiral groove

journal bearings

FEATURE Rotor bearing system analysis

COJOUR Analysis of journal bearings

DYNROT A program designed to perform a complete study of

the rotordynamic behavior of rotors. It is capable

of linear, nonlinear and torsional analysis of rotors

Dipartimento di Meccanica,

Politecnico di Torino,

Torino, Italy

DyRoBeS Comprehensive rotordynamic analysis software for

lateral and torsional analysis, including bearing

analysis of rotor-bearing systems

AGILE SOFTWARE

CONCEPTS NREC White

River Junction,

RotorLab A software package for agile modeling of rotor systems, VT, USA

bearings, and seals. It combines the tasks of design,

modeling, analysis, post processing, and data

management into a consistent user interface

DAMBRG2 Coefficients and rigid rotor stability information for

two-lobe isoviscous bearings with a pressure dam in

only one pad

ROMAC—Rotating Machinery

and Controls Laboratory,

University of Virginia,

HYDROB Predicts the steady state and dynamic operating Charlottesville, VA, USA

characteristics of hybrid journal bearings

PDAM2D This program can analyze stiffness and damping

coefficients, and the rigid rotor stability threshold

of multipad pressure dam bearings

SQFDAMP Determines stiffness and damping coefficients for short and long

squeeze-film bearings with and without fluid film cavitation

THBRG Dynamic coefficients of multilobe journal bearings with

incompressible fluid

34-32 Vibration and Shock Handbook

TABLE 34.3 (continued)

Name of Software Type of Analysis Supplier

THPAD Dynamic coefficients of tilting pad journal bearings with

incompressible fluid

THRUST Predicts the steady-state operating characteristics of

tilting-pad and fixed geometry fluid-film thrust bearings

CRTSP2 Undamped lateral critical speeds of dual-level rotor systems

MODFR2 Undamped lateral critical speeds of single or

dual-level rotor systems

TWIST2 Undamped torsional critical speeds and mode shapes

of rotor systems

FRESP2 Predicts the modal frequency forced response of

dual rotor systems with a flexible substructure

RESP2V3 Nonplanar synchronous unbalance response of

dual-level multimass flexible rotors

HCOMB Dynamic coefficients of straight-through honeycomb

seals with a compressible gas

LABY3 Dynamic coefficients for straight-through and uniform

interlocking type labyrinth seals with a compressible fluid

SEAL2 Stiffness and damping coefficients for plain and grooved

seals with incompressible turbulent axial flow

SEAL3 Stiffness, damping and mass coefficients of both plain

and circumferentially grooved seals

TURSEAL Stiffness and damping coefficients of turbulent flow annular

seals or water lubricated bearings

FSTB3 Stability, damped critical speeds, and whirl mode shapes

of multispool rotor systems

ROTSTB Stability, damped critical speeds, and whirl mode shapes

of single spool rotor systems

COTRAN Nonlinear time transient analysis of multilevel rotors

with substructure

TORTRAN3 Transient torsional rotor response

hydrosealt Stiffness and damping coefficients, and threshold speed

of instability of cylindrical-pad journal bearings and

pad-hydrostatic

bearings of arbitrary arc lengths and preloads

Rotordynamics Laboratory,

Texas A&M University,

College Station, TX, USA

hydroflext Stiffness and damping coefficients, and threshold speed of

instability of a variety of bearing and seal types

hydrotran Predicts the transient force response of a rigid rotor

supported on fluid film bearings

hydrojet Force coefficients for a variety of hybrid bearing and seal

types handling process fluids

hydroTRC Stiffness and damping coefficients for a variety of bearing

and seal types and for different types of fluids

hseal2p Stiffness and damping coefficients of seals that operate

under two-phase flow conditions

fembear Stiffness and damping coefficients of cylindrical and fixed

arc pad hydrostatic and hydrodynamic bearings for

laminar and isothermal flow conditions

sfdfem Damping force coefficients of finite length squeeze-film

dampers executing circular centered motion

sfdflexs Instantaneous fluid film forces for arbitrary journal motions

and circular centered orbits in multiple pad integral

squeeze-film dampers

hsealm Stiffness and damping coefficients of cylindrical annular

pressure seals

(continued on next page)

Vibration in Rotating Machinery 34-33

TABLE 34.3 (continued)

Name of Software Type of Analysis Supplier

lubsealn Stiffness and damping coefficients of single-land

and multiple-land high pressure oil seal rings and

cylindrical journal bearings

ROTECH Lateral rotordynamic analysis for critical speeds;

unbalance response, linear stability and nonlinear

transient response of rotors. Also includes a torsional

rotordynamic analysis program

ROTECH Engineering

Services, Delmont,

PA, USA

ROTOR-E Acomprehensive software package for lateral rotordynamic

analysis of rotating equipment

Engineering Dynamics Inc.,

San Antonio, TX, USA

ROTORINSA A software package devoted to the prediction of the

steady-state lateral dynamic behavior of rotors

Laboratoire de Mecanique des

Structures, LMST INSA Lyon,

Lyon, France

TURBINEPAK

A software package for rotordynamic analysis of nonlinear

multibearing rotor-bearing-foundation systems

Scientific Engineering Research,

Mt Best, Vic., Australia.

TURBINEPAK

NONLINEAR

Designed to study transient responses of rotor-bearingfoundation

systems, including the loss of stability of

the system

XLrotor Acomplete suite of analysis tools for rotating machinery

dynamics. Handles both lateral and torsional analysis of

rotors. Also includes codes for calculating coefficients for

fluid film and antifriction bearings

Rotating Machinery Analysis

Inc., Austin, TX, USA

XLTRC Asuite of codes for executing a complete lateral

rotordynamic analysis of rotating machinery

The Turbomachinery

Laboratory, Texas A&M

University, College Station,

TX, USA

XLAnSeal Force and moment coefficients for annular turbulent seals

in the laminar, turbulent, and transition flow regimes

XLCGrv Coefficients for centered grooved-stator, turbulent flow,

annular pump seals

XLLaby Stiffness and damping coefficients for tooth-on-rotor or

tooth-on-stator gas labyrinth seals

XLIsotSL Coefficients for smooth rotor/honeycomb stator annular seals

XLLubGT Coefficients for high-pressure oil bushing seals of compressors

or smooth pump seals in the laminar flow regime

XLJrnl Stiffness and damping coefficients for fixed-arc and tilting-pad

bearings

HLHydPad Stiffness and damping coefficients for hydrostatic and hybrid

journal-pad bearings in the laminar flow regime

XLTFPBrg Stiffness and damping coefficients for fixed-arc, tilting-pad

and flexure-pivot hydrostatic bearings

XLPresDm Stiffness and damping coefficients for multilobed, rigid-pad

arc bearings with preload and pressure-dam bearings with

relief tracks

XLBalBrg Stiffness coefficients for ball bearings

XLLSFD Damping and mass coefficients for locally sealed squeeze-film

dampers

XLOSFD Damping and mass coefficients for open ended squeeze-film

dampers

XLSFDFEM Damping coefficients for squeeze-film dampers with various

types of end seals

XLPIMPLR Stiffness, damping, and mass matrices for centrifugal pump

impellers

XLWachel Destabilizing cross-coupled force coefficients for impellers of

centrifugal compressors

XLClrEx Destabilizing cross-coupled stiffness coefficients for

unshrouded turbines

34-34 Vibration and Shock Handbook

coefficients:

k ¼

24R3Lmv1

r ð2 þ 12Þð1 2 12Þ ð34:52Þ

c ¼

12pR3Lm

r ð2 þ 12Þð1 2 12Þ1=2 ð34:53Þ

where

R ¼ the damper radius

v ¼ whirl speed

L ¼ length of damper

m ¼ viscosity of oil

Cr ¼ the radial clearance

1 ¼ eccentricity ratio (orbit radius/Cr)

Similarly, for a damper without end seals, the

short journal bearing theory yields the following

stiffness and damping coefficients:

k ¼

2RL3mv1

r ð1 2 12Þ2 ð34:54Þ

c ¼

pRL3m

2C3

r ð1 2 12Þ

2 ð34:55Þ

Although the above equations, based on the

Reynolds’ equation, have been proposed to predict

damper characteristics, the experimental evidence

does not validate these equations. Therefore, these

equations should be used with caution for

practical purposes.

34.3.2.4 Annular Seals

Annular seals are primarily used in pumps, compressors, gas turbines, and steam turbines to minimize

leakage and thereby improve the volumetric efficiency of the machine. In addition to their basic function,

they also play a vital role in the rotordynamics of the machine, especially in multistage machines,

providing stiffness and damping and thereby enhancing high-speed operational capability. In fact, in the

last few decades, most of the development work on seals has focused on understanding and improving

their dynamic vibration characteristics rather than improving their efficiency in sealing.

Lomakin (1958) was the first to publish on the restoring forces in smooth annular clearances in

pumps. However, it was more than a decade later that Black (1968) provided the major initial impetus for

the understanding and development of seals. Childs (1993) provided an excellent compendium of the

research work conducted in the area of seals. His book also provides the most comprehensive coverage of

the subject of seal dynamics.

In the present context, seals are handled in the same manner as the stiffness and damping

characteristics of journal bearings with some degree of modifications. In particular, fluid inertia effects

are included, and it is assumed that the center of the shaft orbit is the same as the center of the stationary

seal ring. Assuming rotational symmetry the reaction force-seal motion model can be represented by the

following equation:

( )

¼ 2

k kc

2kc k

" #

( )

c cc

2cc c

" #

( )

m 0

0 m

" #

X€

Y€

( )

ð34:56Þ

Oil Inlet

'O' Rings

Rolling

element

bearing

Spring

element

FIGURE 34.12 A squeeze-film damper.

Vibration in Rotating Machinery 34-35

An added complexity is the predominance of turbulent flow in annular seals. This invalidates the use of

Reynolds’ equation for the derivation of seal coefficients. The highest degree of accuracy can be obtained

by the direct solution of the Navier– Stokes and continuity equations. However, at the present moment,

such methods are considered to be excessively costly and impractical. As a result, two practical

semiempirical methods have been developed to derive seal coefficients. In the first approach, the

semiempirical turbulent model is directly substituted in the Navier– Stokes equation and a numerical

technique is used for its solution. The second, most commonly used technique uses a bulk flow model

together with control volume formulations, namely, the continuity equation and momentum equation,

to obtain the desired results. For a detailed discussion of these methods, solution techniques, the

influence of various physical parameters on the coefficients, and an excellent compilation of

computational and experimental results, the publication by Childs (1993) is recommended.

34.3.2.5 Impeller – Diffuser/Volute Interface

It is widely known that the flow fields within certain types of rotating machinery can significantly

influence its vibration behavior. Thomas (1958) recognized and explained the presence of destabilizing

clearance excitation forces in axial flow steam turbines. Black (1974) was the first to suggest that

centrifugal pump impellers could also develop destabilizing forces. The nature of these forces and their

influence on rotor instability has been explained in Section 34.2.2.2 and Section 34.2.2.3 of this chapter.

The impeller– diffuser/volute forces assuming rotational symmetry can generally be modeled by an

equation of the following form:

( )

¼ 2

k kc

2kc k

" #

( )

c cc

2cc c

" #

( )

m mc

2mc m

" #

X€

Y€

( )

ð34:57Þ

For analytical procedures for the derivation of impeller interaction coefficients and a comparison of

experimental data, the work by Childs (1993) is recommended. It is well recognized that a considerable

amount of work still needs to be done towards understanding the complex nature of impeller – diffuser/

volute interactive forces, especially at off-design conditions.

34.3.3 Design

Since the real machine is not available for tests, at the preliminary design stage it is a common

practice to develop an accurate mathematical model of the machine to predict its dynamic behavior

in operation. It is also prudent to understand and estimate how the machine will interact with its

operating environment and how the environment could influence the operation of the machine. A

suitable model of the rotor can be developed using the techniques described in Section 34.3.2, and

the rotordynamic characteristics of the machine can be analyzed using one of the methods described

in Section 34.3.1, above. Based on these methods, numerous computer-based rotordynamic analysis

programs have been developed. A listing of the most widely known computer programs in North

America is given in Table 34.3. The objectives of the analysis are to predict the critical speeds,

excitation frequencies, the amplitudes of deflection, and the magnitude of the forces of the rotor

within its full operating range. In certain situations, evaluation of the energy content of the

excitation may also be required.

Once the mathematical model is developed, the eigenvalues of the rotor and the mode shapes can be

determined. The results can then be presented in the form of a Campbell diagram, where the eigenvalues

along with the excitation frequencies are plotted as a function of rotor speed. Critical speeds occur at the

speeds corresponding to the points of intersection of the excitation frequency lines and the eigenvalue

lines. The Campbell diagram presentation (Figure 34.13) of the results is very useful since the influence of

key parameters such as stiffness, damping, clearances (new and worn conditions), and so on can all be

shown on the same diagram. A critical speed, although present, may be of little consequence if it is

associated with sufficient damping. As illustrated in Figure 34.4, when the damping ratio z $ 0:707; the

system is critically damped and above this level of damping there is no amplification of the rotor

deflection. At or near a critical speed the amplification factor is < 1=2z: Using this estimated value,

34-36 Vibration and Shock Handbook

depending on the internal clearances of the machine, it is possible to assess if the rotor can pass through a

critical speed without causing damage to the components. An amplification factor of 2.5 or below is a

typical acceptance limit for centrifugal pumps, even for continuous operation at or near it. However, if

the amplification factor exceeds the acceptable limits or the critical speeds are too close to the continuous

operating speed, then design modifications have to be made to change the critical speed values. At the

design stage, it is considered good practice to ensure that the critical speeds are not within ^10% of the

continuous operating speed; these limits are sometimes referred to as separation margins. The mode

shapes of the rotor are important from the standpoint of identifying where the maximum deflections

occur. It also provides a good guide for assessing design modifications to improve damping or reduce

sensitivity to unbalance forces.

0 40 80 120 160 200

1000 1520 2040 2560 3080 3600

---- Synchronous Excitation Rotor Speed [rpm]

Frequency [Hz]

1520 2040 2560 3080 3600

Rotor Speed [rpm]

List of Symbols: Mode 1.

Mode 2.

Mode 3.

Pump Stat: New

Analysis...... 807281012. FEED PUMP

Damping [%]

0 15 30 45 60 75

1000

nmin

nmax

FIGURE 34.13 Campbell diagram for a multistage pump.

Vibration in Rotating Machinery 34-37

The eigen analysis only provides relative deflections of the rotor. In order to estimate true

deflections, a forced response analysis has to be made. Forcing functions of estimated magnitude are

applied at selected locations to determine the resulting deflections at specific points on the rotor. This

type of analysis is typically carried out for synchronous excitation forces only. The nature of the

forcing function depends on the type of the machine; mechanical unbalance is common to all types

of machines, whereas hydraulic unbalance is relevant to centrifugal pumps and electrical unbalance to

electric motors. The challenge, of course, is to determine the magnitudes, directions, and locations of

the forces to apply and how the resulting rotor response should be judged. Of course, these criteria

are machine type-dependent and not necessarily applicable to all types of rotating machinery.

An example of how forced response analysis on centrifugal pumps is evaluated is given below

(Bolleter et al., 1992):

1. Maximum amplification factors and required separation margins are defined by specifications;

example as shown in API 610, 8th edn., 1995.

2. Excitation forces are defined and the response is judged relative to admissible shaft vibration

limits, and relative to clearances.

3. Apply unbalance forces of such a magnitude that maximum permissible vibration limits at the

vibration probe locations are reached, and then evaluate if the deflections exceed the minimum

clearances in the machine.

4. Apply an unbalance force of arbitrary magnitude and determine the resulting response at the

same or another location, and calculate the sensitivity factor (SF) using the following

formula:

SF ¼

M ð34:58Þ

where

a ¼ rotor deflection

U ¼ unbalance force

M ¼ rotor mass

The sensitivity factor should then be compared with experimental base values of similar machines for

acceptance. The rotor responses to the applied forces can be further analyzed to extract other parameters

of interest, such as phase angles and force magnitudes at the bearings, in order to evaluate the design.

In order to optimize the rotating machine design in terms of placement of critical speeds and control

of deflections and forces, a parameter sensitivity coefficients analysis (Lund, 1979; Rajan et al., 1986;

Rajan et al. 1987) may be carried out. For speed and convenience of analysis, the optimization routine

can be automated.

* Rotordynamic analysis is a part of the current rotating machinery design practice used to

predict their vibration behavior.

* The most current rotordynamic analytical procedures are computer-based and are derived

from the lump-parameter model or the transfer matrix method.

* In the lumped-parameter model method, the distributed elastic and inertial properties of

the rotor are represented as a collection of rigid bodies connected by massless elastic beams.

* In the transfer matrix method, commonly called the FEA method, the rotor is represented

as an assembly of elements with distributed elastic and inertial properties.

* Accurate modeling and representation of rotor components is vital to the accuracy

and reliability of analysis results. As a result, significant advancement in modeling shaft

sections, disks, impellers, bearings, seals, rotor dampers, and rotor– stator interactions have

been made.

34-38 Vibration and Shock Handbook

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