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Rotordynamic Design Optimization of aSteam Turbine Rotor Bearing System

Submitted: April 14, 2008

Udayraj SomashekarRamzi Bazzi

In this project, the dynamic properties of a rotor-bearing system of a single stage steam

turbine are studied, and the design is optimized for minimum weight and the placementof critical speeds. In the optimization for weight, the objective is to achieve a reduction in

the weight of the rotor while ensuring maximum fatigue life of the shaft. In the optimal

placement of critical speeds, the objective is to obtain an optimum design of the rotor and

bearings so as to yield the critical speeds as far from the operating speed range of the

turbine as possible.

The optimum design of rotor-bearing systems employed in turbo-machinery is an

iterative process involving several conflicting objectives and constraints. One attractive

approach is to look upon this design process as a multi-objective optimization problem,

where the design is optimized for different objectives individually, and the designs are

ultimately integrated to determine the overall optimum design. As a culmination of the

optimization, the individual sub-systems are integrated and optimized as a multi-

objective optimization to find the overall optimum design.

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ACKNOWLEDGEMENTS

It is our foremost obligation to thank Jarod Kelly, Doctoral Candidate, Department of

Mechanical Engineering, The University of Michigan, for taking keen interest in constantly

guiding us on this project and putting us on the right track throughout the semester.

No less are we indebted to Dr. Michael Kokkolaras, Associate Research Scientist, Department

of Mechanical Engineering, The University of Michigan, for his continual support and invaluable

advice throughout the course of this project.

We also acknowledge the assistance of all those who have lent their support directly or indirectly

to this project.

Students

Udayraj Somashekar

Ramzi Bazzi

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Table of Contents

1 Introduction 1

1.1 Constraints on Shaft Dimensions Based on Dynamic Fatigue Failure Criteria 4

1.2 Finite Element Model Development 9

2 Subsystem Design 13

2.1 Nomenclature 13

3 Optimization for Minimum Weight 15

3.1 Problem Statement 15

3.2 Mathematical Model 16

3.3 Design Variables and Parameters 18

3.4 Model Summary 19

3.5 Monotonicity Analysis 20

3.6 Optimization Study 21

3.7 Discussion of Results 23

3.8 Parametric Study 24

4 Optimal Placement of Critical Speeds 27

4.1 Problem Statement 27

4.2 Mathematical Model 31

4.3 Design Variables and Parameters 38

4.4 Model Summary 39

4.5 Monotonicity Analysis 40


4.7 Discussion of Results 44

5 System Integration 49


5.2 Results 51

6 References 52

7 Appendices 53

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ME555 - Design Optimization Winter 2008

1. Introduction

Considered in this project is the rotor-bearing system of a single stage steam turbine. The design

has been adapted from a previous project work on Rotordynamic Analysis of a Steam Turbine

Rotor carried out by Udayraj et al. during the course of their undergraduate degree. The shaftand disc have been modeled in CATIA V5, and is shown in the following page. The rotor-

bearing model comprises of a bladed disc with 72 shrouded blades arrayed around its periphery,

which is shrink fitted onto the shaft. The shaft comprises of 21 elements of different lengths and

diameters. Torque transmission is achieved through the interference contact. The rotor assembly

is mounted on two hydrodynamic oil film journal bearings that are fixed to a rigid foundation.

In the design of a modern steam turbine, there are increasing requirements for high efficiency,

reliability, operability and maintainability. These considerations usually lead to the use of more

flexible and more complex rotor systems. The trend towards greater flexibility results in criticalspeeds near the operational speed, which may cause severe vibration problems. The increasing

complexity of the system makes both system simulation and design much more complicated due

to the large number of parameters under consideration. Among these quantities, critical speeds,

unbalance response, deflection of shaft, and transmitted loads through bearings are the most

important ones to be taken into account in the design process. When designing rotating

machinery, the stability behavior and the resonance response can be obtained from the

calculation of complex eigenvalues.

Two kinds of optimization variables were widely used in the previous studies. One is thegeometry of the rotors, such as shaft element lengths and diameters, disk size, and the position of

the bearings and disks. The other is the system support parameters, such as the stiffness and

damping of the bearings on which the rotor is mounted. However, the dimensions of the rotor

system and the positions of the bearings and disks are constrained by other considerations such

as the overall machine structure, the performance envelope, and structural strength criteria,

besides just its dynamic performance. This makes the optimization problem and the imposition

of the appropriate constraints increasingly challenging.

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Image 1.1 The Rotor - Shrink Fit Assembly of Shaft and Bladed Disc

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The system has been divided into two subsystems by discipline decomposition, i.e., the rotor-

bearing system is optimized for two different objectives subject to their respective constraints,

and finally, the individual optimum designs are integrated and optimized simultaneously. First,

the rotor is optimized for minimum weight with constraints on the fatigue life of the rotor, shaft

geometry (elemental lengths and diameters), and bearing span (distance between the bearingmounts). In this case, the obvious tradeoff is between the weight and the fatigue life of the shaft.

Second, optimal placement of critical speeds is done where the critical speeds are moved as far

from the operating speed range of the turbine as possible, with constraints on the shaft geometry,

bearing properties (stiffness and damping), bearing span, and of course, the fatigue life of the

shaft. In this case, two predominant tradeoffs are observed - one, between the shaft geometry and

the critical speeds, and two, between the bearing properties and the critical speeds which are

harder to explain qualitatively. A more detailed explanation of these tradeoffs has been provided

in the subsystem design section. However, it is well established that the dynamic characteristicsof a rotor is substantially influenced both by the rotor geometry and the stiffness and damping

characteristics of the bearings, with the latter having a more significant impact.

The rotor-bearing system design is highly constrained by feasible regions for the damped natural

frequencies which are dictated by operating speed requirements, weight of the rotor assembly

which is dictated by cost and performance envelopes prescribed for the system, and the dynamic

response characteristics of the rotor which are dictated by the system geometry and bearing

parameters. Therefore one of the most important tasks for the designer is to determine feasible

designs and select the optimum design that fulfills all these constraints.

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1.1. Constraints on Shaft Dimensions Based on Dynamic Fatigue Failure Criteria

A rotating shaft loaded by stationary bending and torsional moments would be stressed by a

completely reversed or alternating bending stresses in every rotational cycle (tension and

compression), but the torsional stress would remain steady. In the case of designing a steppedshaft, it is necessary to realize that a stress analysis at a specific point on a shaft can be done

using only the shaft geometry in the vicinity of that point. The number of components on the

stepped shaft increases the complexity of analysis. Additionally, when the stress concentrations

due to notch effects are taken into account, the calculations become very involved and time

consuming when utilizing analytical methods. For each calculation, it is also difficult to read the

notch sensitivity and stress concentration factors which are typically presented graphically or

given in tabular form in machine design data handbooks.

When the shaft is subjected to completely reversed bending and steady-state torsion, the criticalbending stress is usually located at a point of stress concentration. The stress concentrations (Kt)

known as stress risers, occur at the limited zone, where a geometrical discontinuity, such as a

shoulder in a stepped shaft (known as a notch area) begins. The stress concentration effects are

hence indispensable in the design of a stepped shaft for maximum fatigue life. Based on these

considerations, the diameter of the a particular shaft element can be defined taking into

consideration the fatigue stresses in terms of the mean and alternating bending moments (Mm and

Ma), mean and alternating torsional moments (Tm and Ta), safety factor (n), and ultimate strength

of the shaft material (Su). We note that the shaft design equations to be presented assume an

infinite fatigue life design (a reliability of 100 %) of a material with an endurance limit (Se).

Using the maximum energy of distortion theory incorporated with the Soderberg failure

criterion, the minimum diameter of the a particular shaft element can be defined as:3/1

2/122

4

332

++

+ a

e

y

msta

e

y

msb

y

TS

STKM

S

SMK

S

nd

, where

n = factor of safety,

yS = yield strength of the shaft material,

uS = ultimate strength of the material,

'eS = endurance limit of the material,

'1

e

f

dcbe SK

KKKS

= , modified endurance limit of the material,

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dcb KandKK ,, are the size, notch, and surface factors, respectively,

( 11 += tf KqK ) , stress concentration factor for fatigue considerations,

tK = geometric stress concentration factor (ratio of maximum stress in the shaft element to the

nominal stress),

1

1

+=

r

aq , notch sensitivity (where a is the Neubers constant and r is the fillet radius

between the shaft sections under investigation), and finally,

stsb KandK are the shock factors in bending and torsion respectively. These are considered to

account for the shocks experienced by the shaft due to vibrations induced during critical speed

traversal of the rotor.

As mentioned above, during an iterative optimization procedure where the rotor dimensions, on

which the notch sensitivity and stress concentration factors are dependent, are changed at every

iteration, it is difficult for the subroutine to extract the requisite data that is typically presented

graphically or given in tabular form in design handbooks. Hence, for the purposes of

optimization, reasonable assumptions are made in the calculation of the modified endurance limit

by allowing for a liberal margin on the stress concentration factor, Kf.

For steels, Se = 0.504 Su .

Assume that ueef

dcbe SSS

K

KKKS 1.0'

5

1'

1

=

1.1.1. Steady Torsional Load

For rotational motion, Newton's second law can be adapted to describe the relation between

torque and angular acceleration:

pIT = , where,

T = total torque exerted on the body,

Ip

= polar mass moment of inertia,

= angular acceleration.

Since torque transmission is effectuated between the disc and the shaft through an interference

fit, and the torque is determined by the force with which steam impinges on the blades on the

disc, let us assume that a constant torque is being applied to the shaft, which causes a constant

angular acceleration.

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As a test case, suppose that the shaft takes 60 seconds to attain a rotational speed of 10000 rpm

from rest. Hence,

( ) 2/4533.17sec60

sec/10472.010000srad

rad=

=

Let 17.4533 rad/s2

be the constant angular acceleration of the rotating shaft, and letT = Ip x 17.4533 N-m be the constant torque causing this angular acceleration.

Hence, 4533.1784

4533.178

4533.178

4533.17

2222

==== iiii

iii

sipi

dl

ddla

dmIT

( iii ldT44

103159.1 = ) (i = 1, 2, 5, 6, 9, 10, 13, and 14, with no summation on the index i)

1.1.2. Alternating Bending Load

For a shaft with varying diameters (or other causes of stress concentration), the section of worst

combination of moment and torque may not be obvious. Hence, the Soderberg fatigue failure

criterion equation is applied to each shaft element, whose diameter is a degree of freedom, to

determine the minimum diameter that the element can take for an infinite fatigue life of the shaft.

Thus, expressions for bending moments are required to be derived at each such section, which

are then substituted into the failure criterion equation to derive the lower bound constraints on

the diameters of those sections.

The bending moment at a section through a structural element may be defined as "the sum of themoments about that section of all external forces acting to one side of that section". The disc

exerts a vertically downward force of Fd due to its weight, the shaft a vertical force of Fs due to

its weight, and the bearings are approximated as linearized rigid supports for the purposes of this

derivation, which exert two vertically upward reaction forces, Rb1 and Rb2, due to the weights of

the shaft and disc which are mounted on them. For simplicity, let us assume that each bearing

bears an equal radial load, i.e., R b1 = Rb2 = Rb. This assumption is reasonable because the

individual reaction forces at the bearings do not change by a considerable degree with changes in

the elemental lengths and diameters of the shaft since this is a symmetric rotor.

Total load on the bearings is the sum of the disc and shaft forces,

( )

==

+

=+=+=+=14

1

2214

1 44 ii

i

d

io

i

siddsdsdv gld

gLDD

gmgLAgmgmFFF

( ) ( )

+=+= =

14

1

224 068.0109173.5i

iidosdv ldLDFFF

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Bearing reaction force,

2

4

22

14

1

214

1

14

1

===

+

=

+

=

+

= ii

id

i

iid

i

sid

b

gld

FglaFgmF

R

( ) ( )

+=

=

14

1

224 068.0109586.2i

iidob ldLDR

Hence, the bending moments on the shaft elements (whose diameters are degrees of freedom for

the optimization) are as follows:

( ) ( ) ( )123456789101112345671231 lllllllllllRlllllllFlllRM bvb +++++++++++++++++++=( ) ( ) ( )234567891011234567232 llllllllllRllllllFllRM bvb ++++++++++++++++=

( ) ( )5678910115675 lllllllRlllFM bv +++++++++=

( ) ( 67891011676 llllllRllFM bv )+++++++=

( ) ( 987654989 llllllRllFM bv )+++++++=

( ) ( 10987654109810 lllllllRlllFM bv )+++++++++=

( ) ( ) ( )131211109876541312111098131213 llllllllllRllllllFllRM bvb ++++++++++++++++=

( ) ( ) ( )141312111098765414131211109814131214 lllllllllllRlllllllFlllRM bvb +++++++++++++++++++=

1.1.3. Constraints on Elemental Shaft Diameters

The definitions of mean and alternating bending moments (Mm and Ma) and torques (Tm and Ta)

are as follows.

( )

( )minmax

minmax

2

1

2

1

TTT

MMM

m

m

+=

+=and

( )

( )minmax

minmax

2

1

2

1

TTT

MMM

a

a

=

=

As mentioned earlier, the operation of shafts under steady loads involves a completely reversed

alternating bending stress and a steady torsional mean stress. In the case of a rotating shaft,

constant moment M = Mmax = Mmin and torque T = Tmax = Tmin. Therefore,

( )[ ]

( ) TTTT

MMM

m

m

=+=

=+=

2

1

02

1

and

( )[ ]

( ) 02

1

2

1

==

==

TTT

MMMM

a

a

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Hence, the Soderberg failure criterion equation reduces to,3/1

2/1

2

2

4

332

+

isti

e

y

sb

y

i TKMS

SK

S

nd

This is essentially the ASME shaft design equation. Assumptions and the usage of hypothetical

data at certain phases, either due to the lack of availability of requisite data or due to imposed

system limitations, are a common practice because simulating real world conditions in an

idealized analytical environment is virtually impossible. A host of assumptions and

simplifications have been made in deriving the closed form expressions for the alternating

bending moment M and the steady torque T. For a more accurate design of real world systems,

the finite element method is generally employed to accomplish this.

Let us assume a lenient factor of safety, n = 3.

For structural ASTM A36 steel,

Yield strength, Sy = 250 MPa = 2.5 x 108

N/m2,

Ultimate strength, Su = 400 MPa = 4 x 108

N/m2,

Modified endurance limit, Se = 0.1 Su = 4 x 107

N/m2,

Shock factors, Ksb = Kst = 1.5 (for minor shocks).

Hence,

[{ } 3/12/1227 125.15938.58102223.1 iii TMd + ] , where i = 1, 2, 5, 6, 9, 10, 13, and 14.These are the constraints on the elemental diameters (diametral degrees of freedom) of the shaft.

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1.2. Finite Element Model Development

Rotordynamic analysis and optimal design of a rotor-bearing system require a suitable simulation

method to calculate critical speeds and unbalance responses. To perform the optimization, the

first step is to analyze the system dynamic behavior. The preliminary and most important phaseof the optimization is to determine the damped natural frequencies of the rotor-bearing system

for the initial design, which will be referred back to and recalculated at every stage of the

optimization process for modified designs to verify their feasibility in accordance with operating

speed stipulations.

Early dynamic models of the rotor-bearing system were formulated either analytically or using

the transfer matrix approach. The transfer matrix method solves dynamic problems in the

frequency domain, which makes itself reasonable to analyze the steady-state responses of the

rotor-bearing system. Usually, the rotor bearing system is modeled as an assemblage of thediscrete blades and bearings and the rotor segments with distributed mass and elasticity. To

perform an accurate analysis of the complex rotor-bearing system, the complex eigenvalues and

eigenvectors are calculated using general finite element procedures.

A typical configuration of a simple rotor-bearing system, which consists of the components of

rigid disk, flexible rotor, bearing, and foundation, is shown.

Figure 1.2.1 A Rotor-Bearing-Foundation system

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1.2.1. Shaft element

Initially, the shaft element is considered to be straight and modeled as having eight degree of

freedom elements - two translations and two rotations at each station of the element. The cross

section of the element is taken to be circular and uniform. The continuous shaft mass, forconstant density, is taken to be the equivalent lumped mass. The moment of inertia of each

element is divided into two and applied at both ends of each element.

The equation of motion, in a fixed frame, for a shaft element rotating with a constant speed is

given by,

(1)

Here, is the (81) displacement vector that corresponds to the two translational and two

rotational displacements at both ends of the shaft element. are the translational androtational mass matrices, is a gyroscopic matrix,

eq

e

R

e

T M,MeG

eK is a bending matrix, and is the force

vector acting on the shaft element.

eF

1.2.2. Bladed Disc Element

The turbine bladed disc elements are modeled as rigid disks. The rigid disk is required to be

located at a finite element station. If the rotating speed is assumed to be a constant then the

coordinates are governed by the following equation.dq

(2)

1.2.3. Bearing Elements

The nonlinear characteristics of the bearings can be linearized at the static equilibrium position

under the assumption of a small vibration. The dynamic characteristics of the bearings can be

represented by stiffness and damping coefficients. The forces acting on the shaft can be

expressed as

(3)

where, ,bCbK are the bearing damping and stiffness matrices, respectively. is the bearing

force acting on the shaft.

bF

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Figure 1.2.2 Modeling of a Fluid-Film Bearing

he bearing stiffness and damping have a significant impact on the vibration characteristics of a

.2.4. System Equation and Eigenvalue Analysis

T

rotor-bearing system. The addition of bearing flexibility to the rotor-bearing system computation

tends to lower the natural frequencies of the system, as determined by Rouch et al. (1991).

1

nce the element equations (1), (2), and (3) are established for a typical element, these equations

he assembled damped system equation of motion in the fixed frame is

O

are repeatedly used to generate equations recursively for the other elements. Then they are

assembled to find the global equation, which describes the behavior of the entire system.

T

where,

setting up the complex eigenvalue problem for the whirl frequencies of the system governedIn

by (4), it is convenient to write the system equation in the first-order statevector form

where the matrices A, B, and the vector x are defined as

(4)

(5)

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For an assumed harmonic solution of (5), the associated eigenvalue problem istexx 0=

For nontrivial solution, the determinant of this must be zero, |I + C| = 0,

y complex values and

te roots,

sing the Finite Element Method to analyze the eigenvalues of the rotor-bearing system will

where BAC 1= and is the eigenvalue. The eigenvalues are usuall

conjuga j = j I j, where j, j are the growth factor and the damped natural

frequency of the jth

mode, respectively.

U

sometimes cause serious errors if the rotational effects are neglected. The rotational effects that

influence the structural frequencies come from two major sources. One is the centrifugal forces,

which are proportional to the square of the spinning speed. These centrifugal forces tend to

increase the stiffness of some mechanical components on the rotating shaft. Therefore, the

natural frequencies are actually found to be higher than expected. The effects due to centrifugal

forces can be accounted for by incorporating the geometric stiffness matrix into the finite

element model. The other rotational effects are caused by the gyroscopic forces. This force

couples motion in one plane with the motion in another plane. This depends on the spinning

speed as well; the greater the spinning speed, the greater the coupling effect. The finite element

model formulated in this section inherently considers these effects.

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2. Subsystem Design

ubsystem 1

S : Optimization for Minimum Weight, by Ramzi Bazzi

Subsystem 2: Optimal Placement of Critical Speeds, by Udayraj Somashekar

.1. Nomenclature2

Parameter / Variable Description Units

Youngs Modulus N/m2

E

G Shear Modulus N/m2

Poissons Ratio -

Mass Density kg/m3

g Acceleration due to Gravity m/s2

il , i = 1 .. 14 Elemental Length of Shaft m

id , i = 1 .. 14 Elemental Diameter of Shaft m

ia , i = 1 .. 14 Elemental Cross Sectional Area of Shaft m2

sL Total Length of Shaft m

sm Mass of Shaft kg

oD Outer Diameter of Disc m

iD Inner Diameter of Disc m

dL Length of Disc m

dA Cross Sectional Area of Disc m2

dm Mass of Disc kg

bL Length of Bearing m

sB Bearing Span (Distance Between Bearings) m

Eccentricity Ratio of Bearing -

L

yy

L

xx kk , aring NPrincipal Stiffnesses of Left Be /mL

yx

L

xy kk , Cross Coupling Stiffnesses of Left Bearing N/m

R

yy

R

xx kk , Principal Stiffnesses of Right Bearing N/m

R

yx

R

xy kk , Cross Coupling Stiffnesses of Right Bearing N/m

L

yy

L

xx cc , Principal Damping of Left Bearing N-s/m

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L

yx

L

xy cc , Cross Coupling Damping of Left Bearing N-s/m

R

yy

R

xx cc , Principal Damping of Right Bearing N-s/m

R

yx

R

xy cc , Cross Coupling Damping of Right Bearing N-s/m

N Rotational Speed of the Rotor rpm

lowN d RangeLow End of the Operating Spee rpm

highN High End of the Operating Speed Range rpm

i ith

Critical Speed of the Rotor rpm

if ith

Natural Frequency of the Rotor Hz

21 ,aa Separation Margins -

yS Yield Strength N/m2

u

SUltimate Strength N/m

2

'eS Endurance Limit N

2

eS Modified Endurance Limit N/m2

n Factor of Safety in Fatigue -

stsb KK , d TorsionShock Factors in Bending an -

fK Stress Concentration Factor in Fatigue -

M N-mBending Moment

T Torque N-m

am MM , d Alternating Bending MomentsMean an N-m

am TT , Mean and Alternating Torques N-m

pI Polar Mass Moment of Inertia kg-m2

Rotational Angular Velocity of the Rotor rad/s

Rotational Angular Acceleration of the Rotor rad/s2

vF Total Load on the Bearings N

dF Load due to Disc Weight on the Bearings N

sF Load due to Shaft Weight on the Bearings N

bR Reaction Force at the Bearings N

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3. Optimization for Minimum Weight (by Ramzi Bazzi)

.1. Problem Statement

3

the design of modern turbo-machinery, it is necessary to increase the dynamic performance of

educing the weight of the shaft affects not only the amplitude response of the rotor at the

is always important to achieve a certain minimum fatigue life of the shaft and reduce its weight

Inrotor-bearing systems. This necessitates the design of increasingly compact and light weight

designs, which greatly increase fuel economy during their service life. Furthermore, the

longevity and durability of rotor-bearing systems can be significantly increased by minimizing

the weight of the rotor, thus reducing the forces transmitted to the bearings, and those transmitted

through the bearings to the foundation, consequently also improving the performance and

maximizing the fatigue life of the bearings.

R

critical speeds, and hence the stresses induced in it as a result of vibrations, but also the fatiguelife of the shaft in constant rotation. As the weight is reduced, the amplitude of vibrations tends

to increase which increases the stresses in the shaft, which consequently decreases its fatigue life.

Hence, there is a clear tradeoff between the weight and the fatigue life of the shaft, since the

weight cannot be reduced indiscriminately without regard to the reliability of the shaft in fatigue.

It

up until a point where the limitations imposed by the fatigue life considerations on the elemental

diameters and lengths of the shaft are satisfied. Hence, constraints are imposed on the elemental

diameters and lengths of the shaft, and the bearing span. The derivation of the constraints onshaft diameters based on dynamic fatigue failure criteria as well as the theory behind it has been

presented in section 1.1.

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3.2. Mathematical Model

bjective Function:

O

he objective is to minimize the weight of the shaft. Since the shaft has 14 elements, the

1i

ii

onstraints:

Tobjective function becomes,

Minimize =14

laW =

C

Constraints on Shaft Dimensions Based on Dynamic Fatigue Failure Criteria

s derived earlier,

} 3/12/1227 125.15938.58102223.1 iii TM + , where i = 1, 2, 5, 6, 9, 10, 13, and 14,where,

) ld4410 (i = 1, 2, 5, 6, 9, 10, 13, and 14, with no summation on the index i),

A

[ ]{d

(iT 3159.1= ii( ) ( ) ( )11231 lFlllRM vb 2345678910111234567 llllllllllRlllllll b +++++++++++++++++++=( ) ( ) ( )234567891011234567232 llllllllllRllllllFllRM bvb ++++++++++++++++=

( ) ( )5678910115675 lllllllRlllFM bv +++++++++=

( ) ( 67891011676llllllRllFM

bv )+++++++=

( ) ( 987654989 llllllRllFM bv )+++++++=

( ) ( 10987654109810 lllllllRlllFM bv )+++++++++=

( ) ( ) ( )131211109876541312111098131213 llllllllllRllllllFllRM bvb ++++++++ + + ++++++=

( ) ( ) ( )141312111098765414131211109814131214 lllllllllllRlllllllFlllRM bvb ++ + + + + ++++++=

, and

+ + + +++ +

( ) ( )

+=+= =

14

1

224 068.0109173.5i

iidosdv ldlDFFF

( )( )

+=

=

14

1

224 068.0109586.2i iidob

ldlDR .

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Constraint on Total Shaft Length

constraint on the total length of the shaft has been imposed based on operating limitations.

1i

is

A

=14

lL

=

910870 sL

( ) 910870 1413121110987654321 +++++++++++++ llllllllllllll

Constraint on Bearing Span

constraint on the bearing span (distance between the two bearings) has been imposed, again

4i

is

A

based on operating limitations. The bearing span has to be greater than 450mm to facilitate

mounting of the disc housing, and has to be less than 530mm to allow for an adequate shaft

overhang for rotor balancing, and to accommodate other assemblies.

=11

lB =

540440 sB

( ) 540440 1110987654 +++++++ llllllll

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3.3. Design Variables and Parameters

esign Variables:

D

Shaft element diameters, d1, d2, d5, d6, d9, d10, d13, d14.

esign Parameters:

Shaft element lengths, l1, l2, l5, l6, l9, l10, l13, l14.

D

Shaft element diameters d3 = d4 = 50mm, d7 = d8 = 58mm, d11 = d12 = 50mm - fixed at the

Disc dimensions, Di = 58mm, Do = 300mm, Ld = 90mm - fixed at the indicated values based

Shaft element lengths l3 = l4 = 35mm, l7 = l8 = 45mm, l11 = l12 = 35mm - fixed at the

Bearing length, Lb = 70mm.

indicated values since these are the stations of bearing and disc mounting, and are required to

be constant for the particular configuration of bearing and the dimensions of the disc used.

on operating limitations.


be constant for the particular configuration of bearing and the dimensions of the disc used,

and to afford more flexibility in the other elemental degrees of freedom.

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3.4. Model Summary

bjective function: Minimize

Subject to:

=

=14

1i

iilaW O

{ [ ] } 0125.15938.5810 13/12/12

1

2

1

7 + dTM

2223.1

[ ]{ } 0125.15938.58102223.1 23/12/12

2

2

2

7 + dTM

[ ]{ } 0125.15938.58102223.1 53/12/12

5

2

5

7 + dTM

[ ]{ } 0125.15938.58102223.1 63/12/12

6

2

6

7 + dTM

[ ]{ } 0125.15938.58102223.1 93/12/12

9

2

9

7 + dTM

[ ]{ } 0125.15938.58102223.1 103/12/12

10

2

10

7 + dTM

[ ]{ } 0125.15938.58102223.1 133/12/12

13

2

13

7 + dTM

[ ]{ } 0125.15938.58102223.1 143/12/12

14

2

14

7 + dTM

where M1, T1, M2, T2, M5, T5, M6, T6, M9, T9, M10 10 3, T13, M14, T14 are as described in

)

, T , M1

the constraints section.

( 0680.014131096521 +++++++ llllll ll

( ) 0640.0 14131096521+++++++

llllllll ( ) 0380.010965 +++ llll

( ) 0280.0 10965 +++ llll

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3.5. Monotonicity Analysis

he Monotonicity Table is a convenient tool to determine whether or not a model is wellT

constrained and to identify active constraints a priori. The columns are the design variables and

the rows are the objective and constraint functions, the entries in the table being themonotonicities of each function with respect to each variable. Positive (negative) sign indicates

an increasing (decreasing) function, and (u) indicates an undetermined or unknown

monotonicity. An empty entry indicates that the function does not depend on the respective

variable or that it is non-monotonic with respect to that variable.

Variable d1 d2 d5 d6 d9 d10 d13 d14 l1 l2 l5 l6 l9 l10 l13 l14

F + + + + + + + + + + + + + + + +

g1 + + + + +

g2 + + + + +

g3 + + + + + + + + + + + + + +

g4 + + + + + + + + + + + + + + +

g5 + + + + + + + + + + + + + + +

g6 + + + + + + + + + + + + + +

g7 + + + + +

g8 + + + + +

g9 + + + + + + + +

g10

g11 + + + +

g12

Table 3.1 Monotonicity Table for Weight Minimization

rom the monotonicity table, it can be observed that g1, g2, g3, g4, g5, g6, g7, and g8 are all activeF

and bound the diameter variables d1, d2, d5, d6, d8, d10, d13, and d14 from below. It can also be

inferred that g10 and g12 are active and bound the length variables l2, l5, l6, l9, l10, l13, and l14 from

below. The activity of these constraints can be verified from the optimization results from

Optimus that are presented in the following section. Hence, from the above observations, it can

be concluded that the model is well constrained.

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3.6. Optimization Study

he optimization was performed in Noesis Optimus 5.2 SP1 using the Sequential QuadraticT

Programming algorithm. The workflow for the optimization routine is shown below.

Figure 3.1 Optimus Workflow for Weight Minimization

he original and optimum values of the elemental lengths and diameters of the shaft are shown

ShaftE

OriginalLe

OptimumL

OriginalDia

OptimumDi

Mounting

T

in the table below. The original and optimum weights of the shaft, and the percentage reduction

are also shown in a table.

lement ngth (mm) ength (mm) meter (mm) ameter (mm)

1 75 155.10 40 22.58

2 100 97.40 45 22.72

3 35 35 50 50 Bearing

4 35 35 50 50 Bearing

5 35 3 37.37 65 7.98

6 130 105.81 50 37.85

7 45 45 58 58 Disc

8 45 45 58 58 Disc

9 45 3 40.00 70 1.92

10 120 106.81 50 38.04

11 35 35 50 50 Bearing

12 35 35 50 50 Bearing

13 55 4 23.79 55 2.79

14 100 63.70 50 22.64

Original Weight (Kg) Optimum Weight (Kg) Percentage Reduction

14.39 7.54 47.60 %

Table 3.2 Results of Weight Minimization

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The optimization results from Optimus are shown below.

Image 3.1 Optimus Results for Weight Minimization

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3.7. Discussion of Results

he inferences from the monotonicity analysis are verified by the results of the optimization.T

Constraints g1, g2, g3, g6, g7, and g8 are found to be active which bound d1, d2, d5, d10, d13, and d14

from below, and g4 and g5 are found to be semi-active. Furthermore, constraints g10 and g12 arefound to be active which bound the elemental lengths of the shaft from below. Hence, design of

the rotor for maximum fatigue life has yielded a design with a significantly reduced weight. The

original and optimum configurations of the rotor are shown below.

Image 3.2 Original Configuration of the Rotor

Image 3.3 Optimum Configuration of the Rotor for Minimum Weight

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3.8. Parametric Study

parametric study of the system was performed to explore the effects of changes in each

d3, d4 Optimum Weight (Kg) l3, l4 Optimum Weight (Kg)

A

parameter on the optimum while keeping the other parameters constant, and also the effects of

changes in a combination of all the parameters (referred to as a configuration of the rotor in thissection) on the optimum. The parametric tables are presented along with the curves indicating

the variations of the optimum as functions of parameter changes within a certain range.

40 7.168885 25 7.609909

45 7.349161 30 7.576593

50 7.54212 35 7.54212

55 7.77165 40 7.529503

60 8.014716 45 7.517134

d11, d12 Optimum Weight (Kg) l11, l12 Optimum Weight (Kg)

40 7.168885 25 7.613381

45 7.349161 30 7.576074

50 7.54212 35 7.54212

55 7.77165 40 7.528433

60 8.014716 45 7.514916

d7, d8, Di Optimum Weight (Kg) l7, l8, Ld Optimum Weight (Kg)

38 6.515355 25 7.927695

48 6.978311 35 7.69645158 7.54212 45 7.54212

68 8.23065 55 7.510896

78 9.02094 65 7.6916

Table 3.3 Param tric Table - Set 1e

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Optimum Weight - vs - Parameters d3, d4

7

7.2

7.4

7.6

7.8

8

8.2

35 40 45 50 55 60 65

d3, d4

Optimum

Weight

Optimum Weight - vs - Parameters l3, l4

7.5

7.52

7.54

7.56

7.58

7.6

7.62

20 25 30 35 40 45 50

l3, l4

Optimum

Weight

Optimum Weight - vs - Parameters d11, d12

7

7.2

7.4

7.6

7.8

8

8.2

35 40 45 50 55 60 65

d11, d12

OptimumWeigh

t

Optimum Weight - vs - Parameters l11, l12

7.5

7.52

7.54

7.56

7.58

7.6

7.62

20 25 30 35 40 45 50

l11, l12

OptimumWeigh

t

Optimum Weight - vs - Parameters d7, d8, Di

6

6.5

7

7.5

8

8.5

9

9.5

10

30 40 50 60 70 80 90

d7, d8, Di

OptimumWeight

Optimum Weight - vs - Parameters l7, l8, Ld

7.4

7.5

7.6

7.7

7.8

7.9

8

20 30 40 50 60 70

l7, l8, Ld

OptimumWeight

Graph 3.1 Optimum Weight vs Different Parameters

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Configuration 1 Configuration 2

l3, l4 25 l3, l4 30

d3, d4 40 d3, d4 45

l11, l12 25 l11, l12 30

d11, d12 40 d11, d12 45

l7, l8, Ld 25 l7, l8, Ld 35

d7, d8, Di 38 d7, d8, Di 48

Optimum Weight (Kg) 5.227168 Optimum Weight (Kg) 6.265812

Configuration 4 Configuration 5

l3, l4 40 l3, l4 45

d3, d4 55 d3, d4 60

l11, l12 40 l11, l12 45

d11, d12 55 d11, d12 60

l7, l8, Ld 55 l7, l8, Ld 65

d7, d8, Di 68 d7, d8, Di 78Optimum Weight (Kg) 8.824083 Optimum Weight (Kg) 11.007087

Table 3.4 Parametric Table Set 2

Rotor Configuration - vs - Optimal Weight

4

6

8

10

12

0 1 2 3 4 5

Rotor Configuration

OptimalWeig

ht

6

Graph 3.2 Optimum Weight vs Rotor Configuration

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4. Optimal Placement of Critical Speeds (by Udayraj Somashekar)

4.1. Problem Statement

In the optimum placement of critical speeds, the objective is to position the critical speeds awayfrom certain regions in the operating range of speeds of the rotor system. When the spin speed of

the rotor coincides with one of the natural frequencies of whirl of the rotor system, the spin speed

is referred to as a critical speed. The separation margin of the critical speed of a rotor-bearing

system under the constraints of the dimensional variables is investigated. Usually, three major

procedures have to be accomplished for an optimum design problem of this nature. The first is to

set up the objective function, which is the separations of the critical speeds and the operating

range of the rotor-bearing system. The second is to choose the appropriate design variables, the

changes of whose values the critical speeds are most sensitive to. The last is to decide the major

constraints of this problem, which may be the most challenging since these constraints aretypically based on experience, analytical study of the system dynamic properties (which is

performed in ANSYS and explained in the sections that follow), and possibly experimentation.

The objective of this study is to find the optimum dimensions of the rotor and bearing dynamic

properties such that the optimized rotor system can yield the critical speeds as far from the

operating speed as possible. The diameters and lengths of the shaft elements and the stiffnesses

of the bearings are the primary design variables since they play a very important role in the

determination and movement of critical speeds. In practice, the diameters of the shaft elements

cannot be sharply changed from the original values due to strength considerations; hence, it is achallenging problem to obtain the optimum combination of the shaft dimensions and the bearing

properties such that the objective of moving the critical speeds away from the operating speed

range is accomplished.

It is important to note that the third mode is a flexible rotor and relatively rigid support

mode, as opposed to the first two modes, which are relatively rigid rotor and flexible support

modes. Hence, the dimensions of the shaft play a predominant role in effectuating the upward

movement of the third mode, whereas, the bearing dynamic coefficients play a predominant role

in shifting the second mode downward. Hence, the objective here is to move the damped natural

frequencies in the operating range away from each other by reducing the value of the second

rigid body mode frequency (second critical speed) and increasing the value of the first bending

mode frequency (third critical speed), and avoiding the proximity of the 2 modal frequencies that

are moved for the choice of operating speed.

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The tradeoffs, or more appropriately, the somewhat abstract relationships between the shaft

geometry and the critical speeds, and those between the bearing properties and the critical speeds

can be explained with a preliminary understanding of and experience in rotor dynamics, and by

the following study of the modal properties of the rotor bearing system performed in ANSYS.

From the analysis, it was also observed that each mode shape typically occurs twice - once as apredominantly horizontal mode, and once as a predominantly vertical mode.

Image 4.1 First Lateral Bouncing Mode

The first mode, distinctively referred to as the bouncing mode, peaks near the center of the

rotor. The mode shape shows the rotor displacement in-phase, with most of the strain energy in

the rotor, rather than in the bearings. This yields a relatively high amplification factor as this

mode is traversed because of the low damping contribution from the supports. Consequently, the

first mode must be well balanced if it has to be traversed during startup and coastdown. It is

difficult to shift the first mode frequency very much with bearing or support changes because of

the minimal participation from the supports in the dynamic response. However, this mode is

located well below the operating speed, and therefore, it is usually unnecessary to move this

mode.

Image 4.2 Second Lateral Rocking Mode

The second mode, distinctively referred to as the rocking mode, crosses the axis near the

center of the rotor, or at the point on the shaft where the disc is mounted, and may sometimes

have two bending peaks. This mode is frequently located in the operating range and may be

moved by either a bearing modification or by altering the stiffness of the rotor in the regions

where the strain energy is high.

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Unlike the first mode, where the energy is concentrated in the shaft, the second mode has

significant support motion in the mode shape. This makes the second mode much better

attenuated and more sensitive to bearing stiffness and damping changes. Because of the better

modal damping, this mode could very likely be safely traversed as long as it is reasonably well

balanced. Furthermore, the greater participation from the bearings and supports in the modeshape makes it possible to shift its frequency, if necessary, through bearing or support stiffness

changes.

Image 4.3 Third Lateral Bending Mode

The third mode is distinctively referred to as the first bending mode. The third mode is

fundamentally different from the first two modes because this is the first "flexible-rotor" mode,

where the bearing and support properties have less effect on the frequency or response

amplitude. This is because of very high strain energy in the shaft, caused by shaft bending, that

biases the energy distribution toward the shaft and away from the bearings. Like the first mode, it

is difficult to shift the frequency very much with bearing or support changes because of the

minimal percentage of strain energy in the bearings relative to the shaft.

From the results of vibration analysis and from experience, it is observed that the rotor has the

largest resonance amplitude when the speed of the rotor traverses the third critical speed.

Moreover, it is usually an unstable mode with a very high amplification factor and an

exponentially increasing response. Traversal of this frequency, or operating the turbine in the

vicinity of this critical speed, causes severe vibrations, and possibly catastrophic failure of the

rotor if this critical speed is not well damped by the bearings. Therefore, it is necessary to move

this frequency as far from the operating speed range of the turbine as can be achieved with

allowable changes in bearing dynamic properties and shaft dimensions.

The effect of changes in bearing stiffnesses on the natural frequencies of the system has been

investigated to understand the behavior of the system and to constitute meaningful constraints on

the bearing stiffnesses for the optimization, and has been presented in the following.

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Critical Speed Map

100

1000

10000

100000

1000000

1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 1.00E+12

Bearing Stiffness (N/m)

CriticalSpeed(rpm)

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Graph 4.1 Critical Speed Map

From the analysis, it was also observed that, in addition to modifying the bearing span and rotor

shaft dimensions to move the critical speeds, the stiffness of the bearings has a significant impact

on both the location of the natural frequencies and the shape of the modes. From the CriticalSpeed Map shown above, it can be inferred that the modal frequencies increase with an increase

in support stiffness, and beyond a certain value characteristic to each mode, the modal

frequencies remain constant. It is also observed that, as the support stiffness is increased, the

amount of bending in the rotor increases, as shown below.

Mode Shapes for (a) Bearing Stiffness = 1x105

N/m, and (b) Bearing Stiffness = 1x1012

N/m

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4.2. Mathematical Model

Objective Function:

Suppose that the operating speed of the turbine has the range, highlow NNN , and that it isdesirable to achieve a rotor design where the low end of the operating speed range, Nlow, at least

a1 times higher than the second critical speed, 2 , and at the high end of the speed range, a

desirable design is to have the bending critical speed, 3 , at least a2 times higher than the high

end of the operating speed range, Nhigh. The objective is to maximize the separation between the

second critical speed and the low end of the operating speed range, and the separation between

the high end of the speed range and the third critical speed.

Hence, the objective function becomes,

)()( 32 highlow NNMaxf +=

A common way to identify the critical speeds of a rotor is the Campbell Diagram, which is a key

plot in the dynamic design process of rotating machinery. It is a plot of Frequency - vs. -

Rotational Speed, and is essential in estimating the critical speeds that could be encountered

during operation. The diagram features crossings of frequency lines of the running speed

harmonic (and typically other ambient sources of excitation as well such as the nozzle passing

frequencies, vane passing frequencies, engine order excitations, etc.) with the natural frequencies

of the rotor. At these crossings, the risk of resonant excitation of the structure exists. The criticalspeeds of the rotor can be estimated by extrapolating the points of intersections of the running

speed harmonic line with the natural frequency lines onto the rotational speed axis.

The Campbell Diagram for the present rotor design is shown on the following page. It can be

observed that the second critical speed occurs at approximately 13,850 rpm, and the third critical

speed at approximately 15,250 rpm. According to the operating specifications of the turbine, the

low end of the operating speed range, Nlow, is 12,000 rpm, and the high end of the operating

speed range, Nhigh, is 18,000 rpm. Hence, there is clearly a risk of running the turbine at a

dangerous proximity to the second or third critical speed, or worse, of consecutively traversing

both of these critical speeds due to fluctuations in torque. In order for the turbine to operate

safely within the specified operating speed range, it is important that the second critical speed be

moved below the low end of the operating speed range, and the third critical speed be moved

above the high end of the operating speed range.

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Campbell Diagram

0

50

100

150

200

250

300

350

400

2000 6000 10000 14000 18000 22000 26000

Rotational Speed (rpm)

Frequency

(Hz)

Mode 1

Mode 2

Mode 3

Mode 4

Running Speed Harmonic

Mode 5

Graph 4.2 Campbell Diagram for the Present Design

Constraints:

It may not always be possible to achieve a very wide separation between the critical speeds andthe limits of the operating range due to complex dynamic considerations as explained earlier, and

of course, the dimensional limitations on the system, but it is nevertheless important to stipulate

that a certain design is not acceptable unless a certain separation margin is achieved between the

critical speeds and the limits of the operating range. Furthermore, one of the critical speeds may

be easier to move with changes in shaft dimensions and bearing properties than the other due to

the inherent nature of the respective modes as explained earlier. Hence, it is important to choose

appropriate separation margins for the respective critical speeds such that the accomplishment of

these margins during the optimization procedure and obtaining a feasible design is likely.

Therefore, to make the design more practicable and the optimization more amenable, inequality

constraints are imposed on the separation between the second critical speed and the low end of

the operating speed range, and the separation between the high end of the speed range and the

third critical speed, and are of the form shown in the following.

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Separation Margin between the upper bound on the second critical speed (second rigid body

mode) and the lower limit of the operating speed range,1

2a

Nlow .

This constraint stipulates that the design is unacceptable if the low end of the operating speed

range is not at least times higher than the upper bound on the second critical speed.1a

Since, as explained earlier, this mode is more easily shifted with changes in the bearing dynamic

coefficients and shaft dimensions than the third mode, let us postulate that a separation margin of

at least 10% is required between Nlow and the upper bound on 2.

Low end of the operating speed range, rpmNlow 000,11=

Second critical speed, ( ) rpmf 6022 = , where f2 = second modal frequency of the rotor in Hz

Separation Margin, 10.11001011 =+=a

Therefore, the constraint becomes,

( )601.1000,11 2 f

Separation Margin between the lower bound on the third critical speed (first bending mode)

and the upper limit of the operating speed range, highNa23 .

This constraint stipulates that the design is unacceptable if the lower bound on the third critical

speed is not at least times higher than thehigh end of the operating speed range.2a

Since, as explained earlier, this mode is not as easily shifted with changes in either the bearing

dynamic coefficients or shaft dimensions as the second mode, and also since it is a dangerous

mode with a very high amplification factor which is desired to be moved as far from the

operating speed as possible, let us postulate that a separation margin of not less than 10% is

required between Nhigh and the lower bound on 3.

High end of the operating speed range, rpmNhigh 000,18=

Third critical speed, ( ) rpmf 6033 = , where f3 = third natural frequency of the rotor in Hz

Separation margin, 10.1100

1012 =+=a


( ) 000,181.1603 f

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Separation Margin between the lower bound on the second critical speed (second rigid body

mode) and the lower limit of the operating speed range,3

2a

Nlow .

This constraint stipulates that the design is unacceptable if the low end of the operating speed

range is not at least times higher than the lower bound on the second critical speed.3a

Separation Margin,

50.1100

5013 =+=a


( )5.1

000,12602 f

Separation Margin between the upper bound on the third critical speed (first bending mode)

and the upper limit of the operating speed range, highNa43 .

This constraint stipulates that the design is unacceptable if the upper bound on the third critical

speed is not at least times higher than thehigh end of the operating speed range.2a

Separation margin,

40.1

100

3014 =+=a


( ) 000,184.1603 f

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The above four constraints essentially mean that, the objective of this optimization is to move the

second critical speed below the low end of the operating speed range and the third critical speed

above the high end of the operating speed range such that the specified separation margins are

achieved for each critical speed. The Campbell Diagram is shown below indicating the operating

speed range and the allowable ranges for

2 and

3 as shaded regions on either side of theoperating speed range.

The constraints stipulate that the second critical speed should lie in the range1a

Nlow and3a

Nlow

which is represented by the bluish region on the left side of the operating speed range, and the

third critical speed should lie in the range and which is represented by the

greenish region on the right side of the operating speed range.

highNa2 highNa4

Graph 4.3 Campbell Diagram for the Present Design Indicating the Operating Speed Range

and the Allowable Ranges for the Second and Third Critical Speeds

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Constraints on Bearing Dynamic Coefficients

From the analytical study of the dynamic properties of the rotor bearing system in ANSYS, it

was observed that the natural frequencies of the rotor are more sensitive to changes in bearing

stiffness coefficients than to those in bearing damping. It was found that the latter contributes primarily to the attenuation of the response amplitude of the rotor at the critical speeds, thus

facilitating safe traversal of those critical speeds, and does not contribute predominantly to the

location or movement of those critical speeds.

Hence, the stiffness coefficients of the bearings are selected as the most important variables in

this optimization study while keeping the bearing damping constant at a value characteristic to

the running speed and the geometry of the rotor, and constraints are imposed on the stiffnesses

based on the minimum and maximum stiffness values that the bearing can assume for the

particular configuration used in this application, and also based on the range of stiffness valuesthat the modal frequencies of the rotor are most sensitive within as observed from the Critical

Speed Map. Also, only the principal stiffness and damping coefficients are considered, and the

effects of cross coupling stiffness and damping are ignored for simplicity of modeling.

Stiffness coefficients of left bearing,

mNkmN Lxx /102/10286

mNkmN Lyy /102/10286

Stiffness coefficients of right bearing,

mNkmN Rxx /102/10286

mNkmN Ryy /102/10286

Again, due to stipulations on the configuration of the bearing required for this application, the

x-direction principal stiffness of each bearing is assumed to be equal to its y-direction principal

stiffness, since these values are typically within a very close proximity of each other due to the

inherent dynamic nature of oil film hydrodynamic journal bearings.

L

L

yy

L

xx kkk == , .RR

yy

R

xx kkk ==

Hence, the constraints on the bearing stiffnesses become,

mNkmN L /102/10286

mNkmN R /102/10286

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Constraints on Shaft Dimensions

Of course, constraints are required to be imposed on the shaft dimensions based on operating

limitations, assembly requirements, and most importantly, fatigue life considerations. The theory

and the rationale behind each of these constraints have been explained in the earlier sections, andwill be mentioned below for completeness of the model.

(a) Constraints on Shaft Dimensions Based on Dynamic Fatigue Failure Criteria

[{ } 3/12/1227 125.15938.58102223.1 iii TMd + ] , where i = 1, 2, 5, 6, 9, 10, 13, and 14.

(b) Constraint on Total Shaft Length

( ) 910870 1413121110987654321 +++++++++++++ llllllllllllll

(c) Constraint on Bearing Span

( ) 540440 1110987654 +++++++ llllllll

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4.3. Design Variables and Parameters

Design Variables:

Shaft element diameters, d1, d2, d5, d6, d9, d10, d13, d14. Shaft element lengths, l1, l2, l5, l6, l9, l10, l13, l14.

Left and right bearing stiffnesses, kxxL, kyy

L, kxx

R, kyy

R.

Design Parameters:

Shaft element diameters d3 = d4 = 50mm, d7 = d8 = 58mm, d11 = d12 = 50mm - fixed at the


be constant for the particular configuration of bearing and the dimensions of the disc used.

Disc dimensions, Di = 58mm, Do = 300mm, Ld = 90mm - fixed at the indicated values based

on operating limitations.

Shaft element lengths l3 = l4 = 35mm, l7 = l8 = 45mm, l11 = l12 = 35mm - fixed at the


be constant for the particular configuration of bearing and the dimensions of the disc used,

and to afford more flexibility in the other elemental degrees of freedom.

Bearing length, Lb = 70mm, bearing eccentricity ratio (ratio of eccentricity at equilibrium tothe radial clearance), = 0.5.

Left and right bearing damping coefficients, cxxL, cyy

L, cxx

R, cyy

R= 3 x 10

10N-s/m.

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4.4. Model Summary

Objective function: { })000,1860()000,1260( 32 = ffMinf

Subject to:

01 006.0 2 f

01330

3

f

012222.122

2

f

010024.0

3 f

0102 8 Lk

0102 6 Lk

0102 8 Rk

0102 6 Rk

[ ]{ } 0125.15938.58102223.1 13/12/12

1

2

1

7 + dTM

[ ]{ } 0125.15938.58102223.1 23/12/12

2

2

2

7 + dTM

[ ]{ } 0125.15938.58102223.1 53/12/12

5

2

5

7 + dTM

[ ]{ } 0125.15938.58102223.1 63/12/1

26

26

7+ dTM

[ ]{ } 0125.15938.58102223.1 93/12/12

9

2

9

7 + dTM

[ ]{ } 0125.15938.58102223.1 103/12/12

10

2

10

7 + dTM

[ ]{ } 0125.15938.58102223.1 133/12/12

13

2

13

7 + dTM

[ ]{ } 0125.15938.58102223.1 143/12/12

14

2

14

7 + dTM

where M1, T1, M2, T2, M5, T5, M6, T6, M9, T9, M10, T10, M13, T13, M14, T14 are as described in

the constraints section of the previous subsystem.

( ) 0680.014131096521 +++++++ llllllll

( ) 0640.0 14131096521 +++++++ llllllll

( ) 0380.010965 +++ llll

( ) 0280.0 10965 +++ llll

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4.5. Monotonicity Analysis

Variable d1 d2 d5 d6 d9 d10 d13 d14 l1 l2 l5 l6 l9 l10 l13 l14 kl kr

F + + + + + + + + + + + + + + + + (u) (u)

g1 + + + + +

g2 + + + + +

g3 + + + + + + + + + + + + + +

g4 + + + + + + + + + + + + + + +

g5 + + + + + + + + + + + + + + +

g6 + + + + + + + + + + + + + +

g7 + + + + +

g8 + + + + +

g9 + + + + + + + +

g10

g11 + + + +

g12

g13 (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u)




g17 +

g18

g19 +

g20

Since a finite element model was used to determine the frequencies of the rotor system and the

analysis was implemented in ANSYS, it was difficult to identify the monotonicities that occur in

the frequency constraints g13, g14, g15, and g16. However, from the monotonicity table, it can be

observed that g1, g2, g3, g4, g5, g6, g7, and g8 bound the diameter variables d1, d2, d5, d6, d8, d10,

d13, and d14 respectively from below, and that g10 and g12 bound the length variables l1, l2, l5, l6,

l9, l10, l13, and l14 respectively from below. Also, constraints g17 and g18 bound kl from above and

below respectively, and g19 and g20 bound krfrom above and below respectively. Hence, it can be

concluded that the model is well constrained.

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The optimization was performed in Noesis Optimus 5.2 SP1 using the Sequential Quadratic

Programming algorithm. The workflow for the optimization routine is shown below.

Figure 4.1 Optimus Workflow for Optimal Placement of Critical Speeds

From the results of the optimization, the original and optimal positions of the critical speeds are

represented pictorially in the Campbell Diagram, with the downward and leftward arrows

indicating the downward movement of the second critical speed, and the upward and rightward

arrows indicating the upward movement of the third critical speed to their optimum positions.

Graph 4.4 Campbell Diagram for the Initial and Optimum Designs

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The original and optimum values of the critical speeds are shown in the table below.

Variable Original Value (RPM) Optimum Value (RPM) Percentage Difference

Critical Speed 2 13745.67 8000.04 41.80 %

Critical Speed 3 15188.37 23759.98 56.44 %

The original and optimum values of the bearing variables are shown in the table below.

Variable Original Value Optimum Value

Left Bearing 1.50E+07 8.94E+06Stiffness (N/m)

Right Bearing 2.50E+07 1.23E+08

Bearing Span (mm) 490 517.59

The original and optimum values of the elemental lengths and diameters of the shaft are shown

in the table below.

ShaftElement

OriginalLength (mm)

OptimumLength (mm)

OriginalDiameter (mm)

OptimumDiameter (mm)

Mounting

1 75 54.37 40 24.16

2 100 68.96 45 31.65

3 35 35 50 50 Bearing

4 35 35 50 50 Bearing

5 35 45.55 65 85.35

6 130 110.04 50 135.46

7 45 45 58 58 Disc

8 45 45 58 58 Disc

9 45 65.95 70 95.34

10 120 136.05 50 89.31

11 35 35 50 50 Bearing

12 35 35 50 50 Bearing

13 55 63.21 55 46.07

14 100 100.54 50 43.05

Table 4.2 Results of Optimal Placement of Critical Speeds

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4.6. Discussion of Results

First, it is important to understand that the use of the optimal design procedure requires that a

suitable objective function be chosen for the particular configuration of the rotor system that is

under investigation and the application that the rotor is used for. For instance, if the same rotorwere to be designed as a supercritical system where the operating speed range is well above the

bending mode, then it may be required to minimize the bending critical speed by moving it as far

below the operating speed range as possible, instead of maximizing it.

Hence, a preliminary study of the natural frequencies and mode shapes of the system and an

understanding of its vibrational characteristics are indispensable in order to guide the

computations to achieve the desired design objectives. As explained in Section 2.2.1, due to the

inherent nature of the modes, it was hypothesized the dimensions of the shaft play a predominant

role in effectuating the upward movement of the third mode, whereas, the bearing dynamiccoefficients play a predominant role in shifting the second mode downward. The results of the

optimization confirm these hypotheses.

The original and optimum configurations of the rotor are shown below.

Image 4.4 Original Configuration of the Rotor

Image 4.5 Optimum Configuration of the Rotor

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The fact that the optimal design procedure yielded a thin section at the location of disc mounting

and a thick section close to either bearing is not surprising when one looks at the bending mode

shape. Clearly, in order to maximize the bending critical speed and consequently also decrease

the vibration amplitude of this mode under the constraints prescribed, one should minimize the

mass at the center section and at the shaft extremities where there is maximum displacement, andat the same time increase the bending stiffness of the shaft by increasing the cross sectional area

close to the sections of bearing mounting.

Image 4.6 Mode 3: Bending Mode

Again, the fact that the optimal design suggested an increase in the stiffness of the right bearing

is consistent with the relative normalized deflection level indicated at that bearing. Similar

conclusions about the decrease in the stiffness of the left bearing become obvious on examining

the mode shape for the second critical speed which is required to be minimized. Hence, the

optimal configuration of the rotor is consistent with these observations.

Image 4.7 Mode 2: Rocking Mode

A parametric study could not be performed on this system since the optimization routine was

taking several hours to converge at an optimum solution. Moreover, with a basic understanding

of rotordynamics and modal analysis, valuable insights can be obtained and discernible

inferences drawn about the nature of the system and the effect of changes of different parameters

on the vibrational characteristics of the system by studying the results of the optimization.

Hence, a formal parametric study is generally unnecessary in this case.

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The effects of certain important variables on the outputs of the optimization such as the

frequencies to be moved (which constitute the objective function) and the objective function

itself (and the constraints if desired) can be studied using the Model Editor in Optimus and

plotting 3D Plots representing a selected output variable as a function of any two selected input

variables on the X and Y axes, while the values of the other input variables are set as parameters.A Least Squares fit for a Taylor Polynomial was used to compute the surrogate model.

Graph 4.5 3D Plot of f2 vs kl and kr

From the above plot of the left bearing and right bearing stiffnesses versus the second natural

frequency of the rotor, and it can be observed that a decrease in the left bearing stiffness and an

increase in the right bearing stiffness from their nominal values would effectuate a downward

movement of the second critical speed.

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Graph 4.6 3D Plot of f3 vs kl and kr

From the above plot of the left bearing and right bearing stiffnesses versus the third natural

frequency of the rotor, and it can be observed that a decrease in the left bearing stiffness and an

increase in the right bearing stiffness from their nominal values would effectuate an upward

movement of the second critical speed.

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Graph 4.5 3D Plot of Objective Function vs kl and kr

From the above plot of the left bearing and right bearing stiffnesses versus the objective function,

and it can be observed that a decrease in the left bearing stiffness and an increase in the right

bearing stiffness from their nominal values would minimize the objective, thus effectuating a

downward movement of the second critical speed and an upward movement of the third critical

speed.

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5. System Integration

Minimization of rotor weight and placing the critical speeds are conflicting objectives with

significant tradeoffs. Minimizing the rotor mass tends to decrease the stiffness of the rotor and

increase the natural frequencies of all rotor modes. In the optimal placement of critical speeds,moving the third critical up tends to increase the bending stiffness of the rotor, and hence

increase the mass of the rotor, as was observed from the results of the optimization. Moving the

second critical down seems to affect predominantly the bearing dynamic coefficients than the

rotor geometry. Hence, there is a clear tradeoff between the two subsystems, which is the weight

of the rotor.


The optimization was implemented as a multi-objective optimization in Noesis Optimus 5.2using the Weighted Objective Method Multi-Objective Optimization Solver. The workflow for

the optimization routine is shown below.

Figure 5.1 Optimus Workflow for the Multi-Objective Optimization

The results of the optimization are shown in the following page. As expected, with weights of 1

and 0 respectively for the critical speed objective and the weight objective, the results obtained

from the multi-objective optimization were same as those obtained from the optimal placement

of critical speeds. Using weights of 0.75 and 0.25 respectively for the critical speed objective and

the weight objective, which is reasonable since in traditional dynamic design of a rotor, more

emphasis is usually placed on the placement of critical speeds than on the minimization of rotor

weight, it is observed that a reduction of nearly 8.5 kg (which is a percentage reduction of 20.94

%) is obtained. Also, interesting but expected changes are observed in the values of the modal

frequencies and the bearing stiffness values.

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Image 5.2 Optimus Results for the Multi-Objective Optimization

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5.2. Results

The original and optimum values of the critical speeds are shown in the table below.

Variable Original Value (RPM) Optimum Value (RPM) Percentage Difference

Critical Speed 2 13745.67 7321.254 46.74 %

Critical Speed 3 15188.37 25110.765 65.33 %

The original and optimum values of the bearing variables are shown in the table below.

Variable Original Value Optimum Value

Left Bearing 1.50E+07 6.74E+06Stiffness (N/m)

Right Bearing 2.50E+07 1.54E+08

Bearing Span (mm) 490 517.59

The Pareto plot for the optimization is shown below.

Graph 5.1 Pareto 2D Plot for the Multi-Objective Optimization

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6. References

[01] Krish Ramesh, Introduction to Rotor Dynamics: A Physical Interpretation of thePrinciples and Applications of Rotor Dynamics, Dresser-Rand, Houston, TX.

[02] Hagg, A. C., and Sankey, G. O., Elastic and Damping Properties of Oil-Film JournalBearings for Application to Unbalance Vibration Calculations, ASME J. Appl. Mech.,

80, 1958, p. 141.

[03] Anders Angantyr, Jan Olov Aidanpaa, A Pareto-Based Genetic Algorithm SearchApproach to Handle Damped Natural Frequency Constraints in Turbo Generator RotorSystem Design, ASME Journal of Engineering for Gas Turbines and Power, July 2004,

Volume 126, Issue 3, pp. 619-625.

[04] B.S. Yang, S.P. Choi, Y.C. Kim, Vibration reduction optimum design of a steam-turbinerotor-bearing system using a hybrid genetic algorithm, Springer Berlin/Heidelberg,

Industrial Applications, Volume 30, Number 1, July, 2005, pp. 43-53.

[05] Hamit Saruhan, Modeling and Simulation of Rotor-Bearing Systems, Proceedings ofthe 5th International Symposium on Intelligent Manufacturing Systems, May 29-31,

2006, pp. 292-303.

[06] Yih-Hwang Lin, Sheng-Cheng Lin, Optimal weight design of rotor systems with oil-film bearings subjected to frequency constraints, Finite Elements in Analysis andDesign, Volume 37, Number 10, September 2001 , pp. 777-798.

[07] H.D. Nelson, J.M. McVaugh, The dynamics of rotor-bearing systems using finite

elements, Trans. ASME J. Eng. Ind. 93 (2) (1976), pp. 593-600.

[08] Rajan, M., Rajan, S. D., Nelson, H. D., and Chen, W. J., Optimal Placement of CriticalSpeeds in Rotor-Bearing Systems, ASME J. Vibr. Acoust., 109 (1987), pp. 152157.

[09] A. C. Ugural, Mechanical Design: An Integrated Approach, McGraw-Hill Professional,2003.

[10] Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, MechanicalEngineering Design, McGraw-Hill Professional, 2004.

[11] G.K. Grover, S.P. Nigam, Mechanical Vibrations, Published by Nem Chand & Bros,India, Seventh Edition, 2001.

[12] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, SecondEdition, Cambridge University Press, 2000.

[13] Panos Y. Papalambros, Model Reduction and Verification Techniques, in Advances inDesign Optimization, H. Adeli (ed.), Chapman and Hall, 1994.

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Appendix I Optimization for Minimum Weight

1. MATLAB Code for Optimus-MATLAB Interface

clear all;close all;clc;

% Design Variables% Shaft element diameters, d1, d2, d5, d6, d9, d10, d13, d14.% Shaft element lengths, l1, l2, l5, l6, l9, l10, l13, l14.

% Parameters% Shaft element diameters d3 = d4 = 50mm, d7 = d8 = 58mm, d11 = d12 = 50mm

% Disc dimensions, Di = 58mm, Do = 300mm, Ld = 90mm

% Shaft element lengths l3 = l4 = 35mm, l7 = l8 = 45mm, l11 = l12 = 35mm

% Initialization - Shaft Element Diameters - Variablesd1 = 40e-3; d2 = 45e-3; d5 = 65e-3;

d6 = 50e-3; d9 = 70e-3; d10 = 50e-3;

d13 = 55e-3; d14 = 50e-3;

% Shaft Element Diameters - Parametersd3 = 50e-3; d4 = 50e-3;

d7 = 58e-3; d8 = 58e-3;

d11 = 50e-3; d12 = 50e-3;

% Initialization - Shaft Element Lengthsl1 = 75e-3; l2 = 100e-3; l5 = 35e-3;

l6 = 130e-3; l9 = 45e-3; l10 = 120e-3;

l13 = 55e-3; l14 = 100e-3;

% Shaft Element Lengths - Parametersl3 = 35e-3; l4 = 35e-3;

l7 = 45e-3; l8 = 45e-3;

l11 = 35e-3; l12 = 35e-3;

% Disc Data - ParametersDo = 300e-3; % Outer DiameterDi = 58e-3; % Inner DiameterLd = 90e-3; % Length

% Calculations

% Objective Function (Weight of the Shaft)W = (7680*(1/4)*pi*d1^2)*l1 + (7680*(1/4)*pi*d2^2)*l2 +

(7680*(1/4)*pi*d3^2)*l3 + (7680*(1/4)*pi*d4^2)*l4 + (7680*(1/4)*pi*d5^2)*l5 +

(7680*(1/4)*pi*d6^2)*l6 + (7680*(1/4)*pi*d7^2)*l7 + (7680*(1/4)*pi*d8^2)*l8 +

(7680*(1/4)*pi*d9^2)*l9 + (7680*(1/4)*pi*d10^2)*l10 +

(7680*(1/4)*pi*d11^2)*l11 + (7680*(1/4)*pi*d12^2)*l12 +

(7680*(1/4)*pi*d13^2)*l13 + (7680*(1/4)*pi*d14^2)*l14;

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% Elemental TorquesT1 = ((1.3159e4)*(d1^4)*l1)*1e8;T2 = ((1.3159e4)*(d2^4)*l2)*1e8;T5 = ((1.3159e4)*(d5^4)*l5)*1e8;T6 = ((1.3159e4)*(d6^4)*l6)*1e8;T9 = ((1.3159e4)*(d9^4)*l9)*1e8;T10 = ((1.3159e4)*(d10^4)*l10)*1e8;T13 = ((1.3159e4)*(d13^4)*l13)*1e8;T14 = ((1.3159e4)*(d14^4)*l14)*1e8;

% Disc ForceFd = (5.9173e4)*(((Do-Di)^2)*Ld + (d1^2)*l1 + (d2^2)*l2 + (d3^2)*l3 +

(d4^2)*l4 + (d5^2)*l5 + (d6^2)*l6 + (d7^2)*l7 + (d8^2)*l8 + (d9^2)*l9 +

(d10^2)*l10 + (d11^2)*l11 + (d12^2)*l12 + (d13^2)*l13 + (d14^2)*l14);

% Bearing Reaction ForceRb = (2.9586e4)*(((Do-Di)^2)*Ld + (d1^2)*l1 + (d2^2)*l2 + (d3^2)*l3 +

(d4^2)*l4 + (d5^2)*l5 + (d6^2)*l6 + (d7^2)*l7 + (d8^2)*l8 + (d9^2)*l9 +

(d10^2)*l10 + (d11^2)*l11 + (d12^2)*l12 + (d13^2)*l13 + (d14^2)*l14);

% Moments on Shaft ElementsM1 = (Rb*(l3+l2+l1) - Fd*(l7+l6+l5+l4+l3+l2+l1) +

Rb*(l11+l10+l9+l8+l7+l6+l5+l4+l3+l2+l1))*1e3;M2 = (Rb*(l3+l2) - Fd*(l7+l6+l5+l4+l3+l2) +

Rb*(l11+l10+l9+l8+l7+l6+l5+l4+l3+l2))*1e3;M5 = (-Fd*(l7+l6+l5) + Rb*(l11+l10+l9+l8+l7+l6+l5))*1e3;M6 = (-Fd*(l7+l6) + Rb*(l11+l10+l9+l8+l7+l6))*1e3;M9 = (-Fd*(l8+l9) + Rb*(l4+l5+l6+l7+l8+l9))*1e3;M10 = (-Fd*(l8+l9+l10) + Rb*(l4+l5+l6+l7+l8+l9+l10))*1e3;M13 = (Rb*(l12+l13) - Fd*(l8+l9+l10+l11+l12+l13) +

Rb*(l4+l5+l6+l7+l8+l9+l10+l11+l12+l13))*1e3;M14 = (Rb*(l12+l13+l14) - Fd*(l8+l9+l10+l11+l12+l13+l14) +

Rb*(l4+l5+l6+l7+l8+l9+l10+l11+l12+l13+l14))*1e3;

% Constraintsg1 = (((1.2223e-7)*(((58.5938*(M1^2))+(1.125*(T1^2)))^0.5))^1/3) - d1;g2 = (((1.2223e-7)*(((58.5938*(M2^2))+(1.125*(T2^2)))^0.5))^1/3) - d2;g3 = (((1.2223e-7)*(((58.5938*(M5^2))+(1.125*(T5^2)))^0.5))^1/3) - d5;g4 = (((1.2223e-7)*(((58.5938*(M6^2))+(1.125*(T6^2)))^0.5))^1/3) - d6;g5 = (((1.2223e-7)*(((58.5938*(M9^2))+(1.125*(T9^2)))^0.5))^1/3) - d9;g6 = (((1.2223e-7)*(((58.5938*(M10^2))+(1.125*(T10^2)))^0.5))^1/3) - d10;g7 = (((1.2223e-7)*(((58.5938*(M13^2))+(1.125*(T13^2)))^0.5))^1/3) - d13;g8 = (((1.2223e-7)*(((58.5938*(M14^2))+(1.125*(T14^2)))^0.5))^1/3) - d14;g9 = (l1 + l2 + l5 + l6 + l9 + l10 + l13 + l14) - 0.680;g10 = 0.640 - (l1 + l2 + l5 + l6 + l9 + l10 + l13 + l14);g11 = (l5 + l6 + l9 + l10) - 0.380;

g12 = 0.280 - (l5 + l6 + l9 + l10);

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% Write Values of Objective Function and Constraints to Filefid = fopen('shaft_optimization.txt','w');fprintf(fid,'W = %f \n\n',W);fprintf(fid,'g1 = %f \n',g1);fprintf(fid,'g2 = %f \n',g2);fprintf(fid,'g3 = %f \n',g3);fprintf(fid,'g4 = %f \n',g4);fprintf(fid,'g5 = %f \n',g5);fprintf(fid,'g6 = %f \n',g6);fprintf(fid,'g7 = %f \n',g7);fprintf(fid,'g8 = %f \n',g8);fprintf(fid,'g9 = %f \n',g9);fprintf(fid,'g10 = %f \n',g10);fprintf(fid,'g11 = %f \n',g11);fprintf(fid,'g12 = %f',g12);fclose(fid);

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Appendix II Optimal Placement of Critical Speeds

1. ANSYS Macro for Optimus-ANSYS Interface

*SET,d1 , 40/1000

*SET,d2 , 45/1000

*SET,d3 , 50/1000*SET,d4 , 50/1000

*SET,d5 , 65/1000

*SET,d6 , 50/1000

*SET,d7 , 58/1000

*SET,d8 , 58/1000

*SET,d9 , 70/1000

*SET,d10 , 50/1000

*SET,d11 , 50/1000

*SET,d12 , 50/1000

*SET,d13 , 55/1000

*SET,d14 , 50/1000

*SET,l1 , 75/1000*SET,l2 , 100/1000

*SET,l3 , 35/1000

*SET,l4 , 35/1000

*SET,l5 , 35/1000

*SET,l6 , 130/1000

*SET,l7 , 45/1000

*SET,l8 , 45/1000

*SET,l9 , 45/1000

*SET,l10 , 120/1000

*SET,l11 , 35/1000

*SET,l12 , 35/1000

*SET,l13 , 55/1000

*SET,l14 , 100/1000

*SET,c1 , -(l1 + l2 + l3 + l4 + l5 + l6 + l7)

*SET,c2 , -(l2 + l3 + l4 + l5 + l6 + l7)

*SET,c3 , -(l3 + l4 + l5 + l6 + l7)

*SET,c4 , -(l4 + l5 + l6 + l7)

*SET,c5 , -(l5 + l6 + l7)

*SET,c6 , -(l6 + l7)

*SET,c7 , -(l7)

*SET,c8 , 0

*SET,c9 , (l8)

*SET,c10 , (l8 + l9)

*SET,c11 , (l8 + l9 + l10)

*SET,c12 , (l8 + l9 + l10 + l11)

*SET,c13 , (l8 + l9 + l10 + l11 + l12)*SET,c14 , (l8 + l9 + l10 + l11 + l12 + l13)

*SET,c15 , (l8 + l9 + l10 + l11 + l12 + l13 + l14)

*SET,kl , 1.5e7

*SET,kr , 2.5e7

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!*

/NOPR

/PMETH,OFF,0

KEYW,PR_SET,1

KEYW,PR_STRUC,1

KEYW,PR_THERM,0

KEYW,PR_FLUID,0

KEYW,PR_ELMAG,0

KEYW,MAGNOD,0

KEYW,MAGEDG,0

KEYW,MAGHFE,0

KEYW,MAGELC,0

KEYW,PR_MULTI,0

KEYW,PR_CFD,0

/GO

!*

/COM,

/COM,Preferences for GUI filtering have been set to display:

/COM, Structural

!*

/PREP7!*

MPTEMP,,,,,,,,

MPTEMP,1,0

MP

me555!08!01 final report

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