ME555!08!01 Final Report
-
Upload
tarkan-karagoz -
Category
Documents
-
view
220 -
download
0
Transcript of ME555!08!01 Final Report
-
8/4/2019 ME555!08!01 Final Report
1/72
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.
-
8/4/2019 ME555!08!01 Final Report
2/72
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
-
8/4/2019 ME555!08!01 Final Report
3/72
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
4.6 Optimization Study 41
4.7 Discussion of Results 44
5 System Integration 49
5.1 Optimization Study 49
5.2 Results 51
6 References 52
7 Appendices 53
-
8/4/2019 ME555!08!01 Final Report
4/72
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.
1
-
8/4/2019 ME555!08!01 Final Report
5/72
ME555 - Design Optimization Winter 2008
Image 1.1 The Rotor - Shrink Fit Assembly of Shaft and Bladed Disc
2
-
8/4/2019 ME555!08!01 Final Report
6/72
ME555 - Design Optimization Winter 2008
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.
3
-
8/4/2019 ME555!08!01 Final Report
7/72
ME555 - Design Optimization Winter 2008
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,
4
-
8/4/2019 ME555!08!01 Final Report
8/72
ME555 - Design Optimization Winter 2008
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.
5
-
8/4/2019 ME555!08!01 Final Report
9/72
ME555 - Design Optimization Winter 2008
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
6
-
8/4/2019 ME555!08!01 Final Report
10/72
ME555 - Design Optimization Winter 2008
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
7
-
8/4/2019 ME555!08!01 Final Report
11/72
ME555 - Design Optimization Winter 2008
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.
8
-
8/4/2019 ME555!08!01 Final Report
12/72
ME555 - Design Optimization Winter 2008
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
9
-
8/4/2019 ME555!08!01 Final Report
13/72
ME555 - Design Optimization Winter 2008
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
10
-
8/4/2019 ME555!08!01 Final Report
14/72
ME555 - Design Optimization Winter 2008
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)
11
-
8/4/2019 ME555!08!01 Final Report
15/72
ME555 - Design Optimization Winter 2008
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.
12
-
8/4/2019 ME555!08!01 Final Report
16/72
ME555 - Design Optimization Winter 2008
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
13
-
8/4/2019 ME555!08!01 Final Report
17/72
ME555 - Design Optimization Winter 2008
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
14
-
8/4/2019 ME555!08!01 Final Report
18/72
ME555 - Design Optimization Winter 2008
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.
15
-
8/4/2019 ME555!08!01 Final Report
19/72
ME555 - Design Optimization Winter 2008
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 .
16
-
8/4/2019 ME555!08!01 Final Report
20/72
ME555 - Design Optimization Winter 2008
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
17
-
8/4/2019 ME555!08!01 Final Report
21/72
ME555 - Design Optimization Winter 2008
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.
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,
and to afford more flexibility in the other elemental degrees of freedom.
18
-
8/4/2019 ME555!08!01 Final Report
22/72
ME555 - Design Optimization Winter 2008
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
19
-
8/4/2019 ME555!08!01 Final Report
23/72
ME555 - Design Optimization Winter 2008
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.
20
-
8/4/2019 ME555!08!01 Final Report
24/72
ME555 - Design Optimization Winter 2008
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
21
-
8/4/2019 ME555!08!01 Final Report
25/72
ME555 - Design Optimization Winter 2008
The optimization results from Optimus are shown below.
Image 3.1 Optimus Results for Weight Minimization
22
-
8/4/2019 ME555!08!01 Final Report
26/72
ME555 - Design Optimization Winter 2008
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
23
-
8/4/2019 ME555!08!01 Final Report
27/72
ME555 - Design Optimization Winter 2008
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
24
-
8/4/2019 ME555!08!01 Final Report
28/72
ME555 - Design Optimization Winter 2008
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
25
-
8/4/2019 ME555!08!01 Final Report
29/72
ME555 - Design Optimization Winter 2008
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
26
-
8/4/2019 ME555!08!01 Final Report
30/72
ME555 - Design Optimization Winter 2008
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.
27
-
8/4/2019 ME555!08!01 Final Report
31/72
ME555 - Design Optimization Winter 2008
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.
28
-
8/4/2019 ME555!08!01 Final Report
32/72
ME555 - Design Optimization Winter 2008
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.
29
-
8/4/2019 ME555!08!01 Final Report
33/72
ME555 - Design Optimization Winter 2008
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
30
-
8/4/2019 ME555!08!01 Final Report
34/72
ME555 - Design Optimization Winter 2008
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.
31
-
8/4/2019 ME555!08!01 Final Report
35/72
ME555 - Design Optimization Winter 2008
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.
32
-
8/4/2019 ME555!08!01 Final Report
36/72
ME555 - Design Optimization Winter 2008
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
Therefore, the constraint becomes,
( ) 000,181.1603 f
33
-
8/4/2019 ME555!08!01 Final Report
37/72
ME555 - Design Optimization Winter 2008
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
Therefore, the constraint becomes,
( )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
Therefore, the constraint becomes,
( ) 000,184.1603 f
34
-
8/4/2019 ME555!08!01 Final Report
38/72
ME555 - Design Optimization Winter 2008
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
35
-
8/4/2019 ME555!08!01 Final Report
39/72
ME555 - Design Optimization Winter 2008
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
36
-
8/4/2019 ME555!08!01 Final Report
40/72
ME555 - Design Optimization Winter 2008
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
37
-
8/4/2019 ME555!08!01 Final Report
41/72
ME555 - Design Optimization Winter 2008
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
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.
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
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,
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.
38
-
8/4/2019 ME555!08!01 Final Report
42/72
ME555 - Design Optimization Winter 2008
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
39
-
8/4/2019 ME555!08!01 Final Report
43/72
ME555 - Design Optimization Winter 2008
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)
g14 (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u)
g15 (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u) (u)
g16 (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.
40
-
8/4/2019 ME555!08!01 Final Report
44/72
ME555 - Design Optimization Winter 2008
4.5. Optimization Study
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
41
-
8/4/2019 ME555!08!01 Final Report
45/72
ME555 - Design Optimization Winter 2008
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
42
-
8/4/2019 ME555!08!01 Final Report
46/72
ME555 - Design Optimization Winter 2008
The optimization results from Optimus are shown below.
43
-
8/4/2019 ME555!08!01 Final Report
47/72
ME555 - Design Optimization Winter 2008
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
44
-
8/4/2019 ME555!08!01 Final Report
48/72
ME555 - Design Optimization Winter 2008
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.
45
-
8/4/2019 ME555!08!01 Final Report
49/72
ME555 - Design Optimization Winter 2008
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.
46
-
8/4/2019 ME555!08!01 Final Report
50/72
ME555 - Design Optimization Winter 2008
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.
47
-
8/4/2019 ME555!08!01 Final Report
51/72
ME555 - Design Optimization Winter 2008
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.
48
-
8/4/2019 ME555!08!01 Final Report
52/72
ME555 - Design Optimization Winter 2008
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.
5.1. Optimization Study
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.
49
-
8/4/2019 ME555!08!01 Final Report
53/72
ME555 - Design Optimization Winter 2008
The optimization results from Optimus are shown below.
Image 5.2 Optimus Results for the Multi-Objective Optimization
50
-
8/4/2019 ME555!08!01 Final Report
54/72
ME555 - Design Optimization Winter 2008
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
51
-
8/4/2019 ME555!08!01 Final Report
55/72
ME555 - Design Optimization Winter 2008
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.
52
-
8/4/2019 ME555!08!01 Final Report
56/72
ME555 - Design Optimization Winter 2008
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;
53
-
8/4/2019 ME555!08!01 Final Report
57/72
ME555 - Design Optimization Winter 2008
% 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);
54
-
8/4/2019 ME555!08!01 Final Report
58/72
ME555 - Design Optimization Winter 2008
% 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);
55
-
8/4/2019 ME555!08!01 Final Report
59/72
ME555 - Design Optimization Winter 2008
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
56
-
8/4/2019 ME555!08!01 Final Report
60/72
ME555 - Design Optimization Winter 2008
!*
/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