three dimensional resonant vibrations and stresses in
TRANSCRIPT
Rochester Institute of Technology Rochester Institute of Technology
RIT Scholar Works RIT Scholar Works
Theses
1981
Three dimensional resonant vibrations and stresses in turbine Three dimensional resonant vibrations and stresses in turbine
blade groups blade groups
Patrick J. Kline
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THREE DIMENSIONAL RESONANT VIBRATIONS AND STRESSES IN
TURBINE BLADE GROUPS
by
Patrick J. Kline
A Thesis Submitted
In
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
in
Mechanical Engineering
Approved by:
Prof. Neville F. Rieger(Thesi 5 Advisor)
Prof. Will iam Halbleib
Prof. Wayne Walter
Prof. P.M. Karlikan(Department Head)
DEPARTMENT OF MECHAN ICAL ENGINEERING
COLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
NO"","ber. 1981
ACKNOWLEDGEMENT
I gratefully acknowledge my debt to Dr. Neville F. Rieger for
his guidance and patience, and my wife for her patience.
ABSTRACT
This thesis describes an efficient procedure for calculating three
dimensional resonant vibrations and stresses in intermediate and
high pressure turbine blade groups. This procedure is capable of
calculating all the natural frequencies, mode shapes, and bending
stresses in the tangential, axial, and coupled modes of vibration.
Simple beam theory is applied to develop a dynamic stiffness matrix.
The solutions to this matrix give the natural frequencies and mode
shapes for the blade group. Prohl's energy method is used to deter
mine the amplitude of the forced vibrations and the dynamic stresses.
A Goodman diagram fatigue criterion is applied to evaluate the proba
bility of blade group failure. Comparing this procedure's numerical
results with experimental results for a rectangular beam structure,
the largest difference for the first five tangential natural frequencies
is 1.2 percent. This method of analysis is simple and can be applied
in twenty hours. Sample calculations and results are given for a ty
pical blade group, and the advantages and limitations of this method
are discussed.
TABLE OF CONTENTS
Page
LIST OF TABLES V
LIST OF FIGURES V!
LIST OF SYMBOLS X
I INTRODUCTION 1
II LITERATURE REVIEW 8
III THEORY 31
A. Basic Equations 31
B. Resonant Tangential Vibrations and
Stresses 40
C. Resonant Axial Vibrations and Stresses. 62
D. Coupled Resonant Vibrations and
Stresses 84
IV SAMPLE CALCULATIONS AND RESULTS ... 84
V DISCUSSION 136
VI CONCLUSIONS 143
VII RECOMMENDATIONS 145
VIII REFERENCES 147
IX APPENDIX 148
A. Computer Program for Tangential and
Axial Resonant Vibrations and Stresses 148
B. Computer Program for Coupled Resonant
Vibrations and Stresses 159
V
LIST OF TABLES
Table Description Page
1 Natural Frequencies of the Rectangular
Beam Structure 108
2 Natural Frequency of a Turbine Blade
Group1^3
3 Resonant Bending Stresses of a Turbine
Blade Group 134
VI
LIST OF FIGURES
Figure Description Page
1 Turbine Blade Showing Component
Terminology 2
2 Turbine Blade Group 3
3 Vibrations of a Six Blade Group 9
4 Schematic Representation of a Six Blade ....
Group. 12
5 Tangential Vibrations with Resonant Response
Factor 13
6 Axial Vibrations with Resonant-Response
Factors 14
7 Campbell Diagram Relating Fundamental
Tangential Frequencies to Coupling and
Pinion Tooth Frequencies. .20
8 General Analysis Procedure for Fatigue of
Steam Turbine Blades induced byNon-
Steady Steam Forces 26
VII
Figure Description Page
9 Beam with Positive Displacements, Slopes,
Moments, and Shears 32
10 Beam with Positive Angular Displacements
and Torques 38
11 Free Body Diagram of Blade Group for
Tangential Vibrations 41
12 Dynamic Stiffness Matrix for Tangential
Vibrations 46
13 Exciting Forces Acting on a Blade Group. . . 51
14 Tangential Force Spectrum for a Blade in
the IP Stage Test on a Water Table 53
15 Stations of the Blade and Cover 55
16 Heywood Strength Reduction Procedure with
Application to Blade Root Stresses 61
17 Free Body Diagram of Blade Group for
Axial Vibrations. 64
VIII
Figure Description Page
18 Dynamic Stiffness Matrix for Axial
Vibrations 68
19 Free Body Diagram of Blade Group
for Coupled Vibrations. 73
20 Dynamic Stiffness Matrix for Coupled
Vibrations. 79
21 Bar Structure. 85
22 Parameter Section of Computer Program
for Bar Structure, 86
23 Output of Computer Program for Bar
Structure - 10 modes 90
24 Mode Shapes for the Bar Structure 96
25 Cambered Turbine Blade. 109
26 Schematic of Cambered Blade Group 110
IX
Figure Description Page
27 Parameter Section of Computer Program
for Cambered Blade Group, Ill
28 Output of Computer Program for
Cambered Blade Group- 10 modes. 113
29 Plots of 10 Mode Shapes for the Cambered
Blade Group 122
X
NOTATIONS
2A a rea
,in .
A,B subscripts indicating ends of beam
A,B,C,D integration constants
b blade
c cover
E energy. Lb. -in.
2E modulus of elasticity. Lb. /in.
F driving force per blade, lb.
F. frequency functions
2G shear modulus of elasticity, lb. /in.
2
g gravitational constant, in. /sec.
4I second moment of area, in.
i,j subscripts
J torsional weight moment of inertia per station, lb.
K torsional moment of inertia for noncircular cross
4section, in.
K resonant response factor
K stiffness ratio, (EI)b/(EI)c
K strength reduction factors
k root stiffness factor, Ib.-in./rad.
k number of nozzles per360
L number of stations per blade
I length, in.
XI
M bending moment, in. -lb.
m number of blades per360
N number of covers or bays
norder of harmonic
p frequency variable
q intensity of the exciting force per unit length
of blade, lb. /in.
q notch sensitivity index
Rk
root stiffness ratio,pj-
R rotor speed,rev. /sec.
r relative
S shear force, lb.
S fractional value of stimulus
s number of stations per cover
T torque, in. -lb.
TGK
torsional stiffness ratio,-pry
c
t time, sec.
u deflection,in .
V,X,Y,Z displacement amplitude, in.
V angular displacement, rad.
w
3specific weight, lb. /in.
X length variable, in.
y displacement, in.
z3
section modulus, in.
displacement, in,
XII
a phase angle, rad.
a slope in y direction, rad.
6 slope in z direction, rad.
y rotation about the x axis, rad.
6 logarithmic decrement of damping
8 slope, rad.
0 angular rotation of a cambered blade, rad.
X frequency variable, in.
2a bending stress, lb. /in.
$. frequency function groups
I. INTRODUCTION
Blading problems accounted for 14.9 percent of the forced outages
in the fossil machines and 17.0 percent of the forced outages in
nuclear machines from the year 1964 to 1973 [1]. These outages
cost the utility and customer valuable time and money. A typical
outage may cost the utility and customer $ 60,000 per day for re
placement power and $250,000 for labor and materials for the repair.
The total costs for a four week repair may exceed $2,000,000. A high
number of these blading problems are due to fatigue failures. The
fatigue failures occur in the cover, tenon, vane, tie wire, base, or
root section. The blade sections are shown in Figure 1 and a blade
group is illustrated in Figure 2.
The objective of this thesis is to provide an efficient procedure for
calculating three dimensional resonant vibrations and stresses in inter
mediate and high pressure turbine blade groups. To accomplish this
objective a procedure is developed to determine the level of the fatigue
stress and the probability of blade group failure. Therefore, if blade
group failure is likely, corrective design actions may be taken.
The method described in this thesis is simple, easy to apply, inexpen
sive, and has been shown to give accurate results. This procedure is
capable of calculating three dimensional uncoupled or coupled resonant
vibrations and stresses. The results provide a tool for designing new
blade groups or analyzing existing blade groups. The analysis is for
Tenon
Cover
Vane
Platform
Blade Root
Disk Root
FIGURE 1. TURBINE BLADE SHOWING COMPONENT TERMINOLOGY,
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blade groups with covers but without lacing wires. Because of the
many variables in blade group design, manufacturing, assembly, and
operation, no method of analysis gives absolute results.
The method presented in this thesis applies simple beam theory and a
dynamic stiffness matrix to calculate the blade group natural frequencies
and mode shapes. This procedure includes force vibrations and damping
of the blade group in the calculations of the dynamic bending stresses.
The Heywood strength reduction factors and the Goodman diagram is
applied to determine the fatigue stress level and the probability of failure.
The method does require experimental data to determine the equivalent
blade root stiffness factor used in the calculations. The calculations
are performed by a computer program. The program outputs the natural
frequencies and mode shapes of the blade group and the dynamic bending
stresses at selected locations along the blade and cover. The input
parameters of the program include the mechanical properties and geom
etries of the blades and covers and the exciting force for the blade group.
Conventional structural design of steam turbine blade groups is accom
plished by two methods. The first method involves the calculation of static
stresses at given failure sites throughout the blade groups under combined
centrifugal and steam bending loads [2], The natural frequencies of the
blade group are calculated next including centrifugal stiffening. The
frequency results are plotted on a Campbell diagram. The blade group
design is then modified if necessary by tuning to avoid coincidence between
any integer multiple of the rotational speed and any of the first six or so
natural frequencies of the blade group. The principle applied is that
non-resonant blades have low dynamic stress and will not fail by fatigue.
This procedure has been mostly successful in the design of constant speed
steam turbine blades for more than twenty years. The second method
uses shorter, stiffer blades due to their statistical success. Present
design analysis consider blade bending in two planes plus torsion. The
root section is generally considered to be built into the disk at the first
hook. Suitable root stiffness values based on test experience are used
to fine tune the natural frequency calculations. Practical blade group
tuning is often less accurate than desired due to component tolerances
and assembly techniques. Consistent tuning to avoid resonance is
difficult to accomplish for these reasons.
In cases where blade resonance is a strong possibility, such as in variable
speed or marine turbines, a dynamic stress procedure for blade groups
has been developed by Prohl and Weaver [3] and others. Dynamic
stresses calculated by this method may be evaluated against some fatigue
criterion. This method has been used in the design of high pressure and
intermediate pressure blading of large turbines. Until recently, the
input technology for non-steady blade excitation, blade group damping
and material fatigue properties has been somewhat limited. With good
excitation and material properties the dynamic stress method gives the
potential for development of superior technology- This method cannot
yet directly account for machining and assembly tolerance effects nor for
certain three dimensional stress conditions, except through the use of
design factors.
The shortcomings of present calculation procedures for high output
blading may be summarized as follows:
1. Dynamic stresses are not usually calculated because of the
above difficulties. Instead,blade groups are tuned to
avoid resonance.
2. The amplitude of the exciting force is not considered in tuned
blade calculations.
3. Present fatigue stress design procedures are elementary.
mainly due to the lack of material test data. Multiple loading
effects, actual stress concentration effects, size effect, and
cycles to failure are often inadequately represented. The
problem of corrosion fatigue is not fully understood and solid
design data is lacking.
4. The blade-root interface is difficult to represent effectively.
5. Test data show blade group frequency scatter is commonly between
two percent and five percent. Precise tuning is frequently not
possible.
Present detuning procedures should be considered an inadequate design
technique for future blading for the above reasons. The design of
improved blading will required practical procedures which account for
dynamic stresses and for variability of material properties in a
more accurate manner.
II. LITERATURE REVIEW
One of the earliest papers on vibrations of turbine blade groups was
written by Smith [4]. Smith made a two dimensional free vibrational
analysis in the tangential direction using the dynamic stiffness matrix
method on a six and a twenty-blade group. Group frequencies and
mode shapes were determined. The blade and covers were separated
at their joints and equilibrium equations were written in terms of the
blade tip deflections and slopes. The results of the calculations were
shown on graphs in terms of three dimensionless parameters frequency
ratio, mass ratio, and rigidity ratio. The graph for the six blade group
is shown in Figure 3. This paper discussed the use of lacing wires
in turbine blade groups. Experience and mode shapes show lacing wires
have a significant effect on suppressing the second group of tangential
vibrations if the wires are inserted at the proper height. Rotor
speed has two effects on the blades; (1) the centrifugal force has a
tendency to straighten each blade along a radial line, (2) it tightens
the blade joints. No stress calculations were made. Smith's paper is
theoretical and does have design applications for the blade group
problem.
The first well-known blade group design paper was by Prohl [5].
Prohl presented a method of calculating natural frequencies, mode shapes,
and bending stresses for three dimensional free and forced vibrations in
VIBRATIONS OF PACKET OF 6 BLADES
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10
the tangential and axial directions. The analysis of this paper followed
the approach of Smith [4] and is extended to consider axial and torsion
al vibrations. Prohl used a series of concentrated masses and concentra
ted inertias to represent the blade group. The blade was broken into
n stations and the mass of one cover section was added to the tip blade
section. A modified Holzer technique was used to calculate the natural
frequencies and mode shapes. This method of analysis gives all of the
tangential and axial natural frequencies and mode shapes. The blades
were considered to be inextensional, but the covers were extensional.
Shearing deformation and rotary inertia were disregarded.
Vibrational amplitude and stress at resonance are calculated by equating
the input energy to the damping energy. The input energy of the blade
group is a function of the nozzle passing force, the deflection of the
blades, and a phase angle. The damping energy is equal to twice the
logarithmic decrement times the total vibrational energy of the blade
group. The vibrational amplitude is obtained by equating the input
energy to the damping energy and solving the equation for the maximum
deflection. The bending stress at the blade root due to resonant
vibrations is determined by the following equation.
a = K yS a
v 6 s
11
where
a - resonance stress
K = resonance response factor or the ability of the blade
group to accept input vibrational energy.
6 = logarithmic decrement for the given mode
S = stimulus or ratio of total exciting force per blade to driving
force
a = steam steady bending stress at the blade root
In a companion paper by Weaver and Prohl [3], Prohl's method was used
to calculate the natural frequencies, mode shapes, and stress levels for
a simple blade group. The results of the calculations for the blade group
shown in Figure 4 are illustrated in Figures 5 and 6. Figures 5 and 6
nkalso have a plot of the resonant response factor K vs where
n = order of harmonic
k = number of nozzles per360
m = number of blades per360
This paper discussed the design of turbine blade groups using vibra
tional stresses. Reliability of operation at all rotational speeds is the
main goal in the design of marine turbine blades. Operating records do
show that in a sample of 1000 ships the average number of problem rows
of blades was less than 0.05 percent of the total number of rows in
service. It is claimed by Weaver and Prohl that most of these problems
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15
associated with resonant vibrations. On constant speed turbines, it
is usually possible to avoid blades resonance by careful selection of
numbers of nozzle and blades in the respective rows. In general, this
cannot be done as easily for variable speed turbines. In these cases an
analysis of the resonant vibrational stress is required to show if the stress
is at an acceptable level.
The papers by Prohl and Weaver advanced the technology in the fields
of calculation and design of vibrational frequencies and stresses in blade
groups. In the discussion of Weaver and Prohl's paper, Wundt and Caruso
compared experimental natural frequencies with calculated natural
frequencies from Prohl's method. The correlation of 18 natural frequencies
for a six blade group was very encouraging with the largest discrepancies
being under ten percent. The tangential, axial, and torsional mode shapes
occur in groups of N where N is the number of blades. In the tangential
direction the first cantilever mode and the (N-1) fixed-supported modes
form the first group. This is shown in Figure 5 and labeled T through
Tfl . Within the fixed-supported modes the vibration patterns of the
blades alternate between odd and even symmetry. Even symmetry occurs
when the corresponding blades in either half of the group are in phase,
and odd symmetry occurs when the corresponding blades are out of
phase.
Prohl's method is easy to program and obtain results. These authors did
a very thorough job of covering the important factors in turbine blade
group design and their experience shows in their assumptions, calculations,
and computer output. This was an excellent paper!
16
Ellington and McCallion [6] wrote a paper on two dimensional tangential
vibrations of laced turbine blades. The special feature of this case is
the lacing wires located only at the tip of the blade group. This
method introduces a simplification of Smith's (N + 1) matrix method where
N is the number of blades, through the application of the calculus of
Finite Differences. By this technique the (N + 1) equations are re
duced to a single non-homogenous, second order linear difference equa
tion. This equation can be solved to obtain the general solutions in
the form of separate frequency equations for (a) in-phase vibrations
of the blade group (b)anti-phase vibrations of the group. The general
solution is first obtained and the appropriate boundary conditions are
then fitted to obtain the frequency equations and mode shapes. The
frequency equations for the symmetrical and asymmetrical modes of
vibrations of a blade group are derived and discussed in this paper. The
frequencies have been shown to consist of a series of modes corresponding
to the fundamental and overtones of an end loaded cantilever. These
frequencies are interspaced by bands of modes with little or no displace
ment of lacing wires referred to as the group modes.
This method by Ellington and McCallion applies only to the special case
of a blade group with a tie wire which joins the blade tips together to
form the group. Often the blade groups have tie wires at the center or
shrouds at the end or both. The technique of this paper is good for
design guidance, but not for absolute values because of the assumption
of the blade being rigidily attached at the root. A root stiffness factor
and experimental results are not included. The paper does not include
17
any forced vibrations or stress calculations. It also ignores blade
centrifugal stiffening.
A method of analysis for a laced group of rotating exhaust blades was
developed by Deak and Baird [7]. This analysis included three
dimensional coupled tangential and axial free vibrations with root
stiffness. Both flexural and torsional motions were considered. This
method gives all of the natural frequencies and mode shapes. An
important point made in Deak and Baird's paper was that the disk
effect and centrifugal stiffening effects are very important in long
exhaust blades. Blade to blade coupling does occur through the
lacing wires. The analysis did not include damping, forced vibrations,
or resonant stresses, and the blades did not have covers, but this
could be added.
The frequencies calculated by this method were compared to experimental
results with good correlation. Some judgement was required to define the
effective point of blade fixity in the disk rim and the effective lacing
wire constraints. The complicated root of the blade was replaced by an
equivalent beam encasement.
The vibrations of exhaust blades were tested statically, and also in a
rotating rig. In the static test the blade group was excited magnetically
while the frequencies are noted and mode shape identified. In the rotating
rig the frequencies were obtained by piezoelectric crystals attached to the
blades and the signals brought out through slip rings.
18
Rieger and McCallion [8] presented a paper on two dimensional free,
undamped tangential vibrations of frame structures. This work is on
single-story multi-bay frameworks which represents a simplified geometry
of a turbine blade group. The method employed by Rieger and McCallion
separated the portal frame at the intersection of the horizontal and
vertical members. Equilibrium force and moment equations are written
for the intersections. These equations are in terms of the deflections
and slopes at the end of the framework members. These equations were
used to develop a dynamic stiffness matrix which was solved for the
natural frequencies and mode shapes. This method is the basis of the
author's thesis.
Rieger and McCallion's paper did include important information on blade
groups. The effects of the blade/cover mass ratio, stiffness ratio, and
length ratio between the blades and covers were shown. The computer
output of the frequencies and mode shapes were compared to experimental
results with excellent agreement. The effect of root stiffness was included
in the computer program but stress calculations were not included in this
method. The method was applied to a simple geometry, but this paper
showed accurate results can be achieved if the input parameters are
accurately described. An advantage of this method is the parameters
are simple to input and the results are quickly calculated.
The paper by Fleeting and Coats [9] is an excellent example of using
theory to solve an existing problem. This analysis involved calculating
the three dimensional forced vibrations and stresses in the high pressure
turbines of the R.M.S. "Queen Elizabeth II", a luxury oceanliner. The
19
natural frequencies, mode shapes, and stresses were calculated by a
method first proposed by Smith [4] in 1937 and expanded by Prohl [5]
in 1958. This method as described earlier involves a dynamic stiffness
matrix developed from the equilibrium equations of the blades and covers.
The blades are divided into a number of stations, as described previously.
Both of the high pressure turbines on the R.M.S. "Queen ElizabethII"
had broken blades on the proving voyage. A thorough investigation
was initiated to discover the causes of the blade failures. Analysis was
done on the vibrational frequencies, mode shapes, and stresses of the
rotor and blade groups. Other sources of vibrations, such as primary
pinion gear teeth contact, axial shuttling, flexible coupling effects, and
rotor sag were investigated. Figure 7 shows a Campbell diagram. This
type of diagram was used repeatedly to show the relationship between the
frequency of a particular mode of vibration and the frequency of a possible
exciting force. The vibratory stress at resonance due to wake excitation
was expressed as:
S = eQSv
where
S =vibratory stress
e = excitation fraction of the steady steam force = 0.1
Q = dynamic magnification factor = 100
S =steady steam bending stress at resonant speed
20
6000
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4000
3000
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Speed Ratio
FIGURE 7. CAMPBELL DIAGRAM RELATING FUNDAMENTAL TANGENTIAL
FREQUENCIES TO COUPLING AND PINIONTOOTH FREQUENCIES.
21
This equation was derived from Prohl's stress equation.
The conclusions were ( 1) the primary cause of the blade failure was
due to the resonant vibrational stresses, (2) the level of the stress
was increased due to the stress concentration at the junction of the
blade aerofoil and root section, and (3) other resonant vibrations may
have contributed to the blade failures, but were not substantiated. The
blade corrections included a redesign of the blades to reduce the
resonant stresses, adding tiewires, and reducing the stress concentration.
The repeated and extensive trials over a period of service time have
shown the corrections to be successful. Evidence in the failures showed
a high probability of breakage due to the excessive amplitudes of the
fixed-supported tangential modes. This is a practical and very inform
ative paper because of the thorough job of defining, investigating, and
solving the particular blade failure problem involved.
Tuncel, Bueckner, and Koplik [10] applied diakoptics to determine blade
group frequencies. Diakoptics is a method of uncoupling weakly dependent
systems. This paper was a two-dimensional analysis on lower order free
tangential vibrations. Better results were achieved on long slender blades
with weak coupling than short rigid blades. This method is capable of
handling tiewires. Root stiffness, damping, forced vibrations, and
resonant stresses were not included. The calculated frequencies of this
method is compared to experimental data and to results from the Prohl-
Myklestad method. The blades and covers were represented by the lumped
parameter approach which is characteristic of the Myklestad method.
Diakoptics deals with subsystems independently in the first step and
22
includes the coupling in the second step. Diakoptics along with a
method of perturbations was used to find the natural frequencies
of the blade group. This method used the natural frequencies and
mode shapes of a single blade to obtain the frequency of the blade
group. Under such circumstances, the correct representation of the
single blade is of extreme importance.
Provenzale and Skok [11] wrote a paper on a method for curing steam
turbine-blade failures. The blade analysis used generalized beam
theory. This paper presented a computer program capable of calculating
three dimensional coupled tangential and axial forced vibrations and
stresses in blade groups. The program analyzes the complete blade
including the airfoil, platform, root and span constraints. The program
is capable of simulating elastic, damped constraints at the root, tip, and
mid-span locations. This allows the program to handle root fixity and
blade to blade coupling of shrouds and tiewires. This analysis permits
a direct solution of blades subjected to aerodynamic excitation and
damping. Therefore, with proper input, the authors claim that blade
reliability can be evaluated by determining the steady and alternating
stresses during operation. The main portion of the analysis is done
an a single blade with the effects of adjacent blades being introduced
by restraints on the single blade and a blade to blade phase angle. The
program outputs steady-state displacements, natural frequencies, mode
shapes, and dynamic stresses. This program is not totally suited for
calculating all of the blade group natural frequencies. The program is
best suited for exhaust blades with integral shrouds and lacing wires.
23
This paper cited several examples of the application of the program
with excellent correlation between calculated and experimental results.
Turbine design experience shows the reduction of the endurance limit
due to the effects of a corrosive environment may be substantial . A
debatable statement in this paper was pointed out in Sohre's [12]
discussion of the paper. The point in the paper was that some natural
frequencies of blades with integral tip shrouds decrease with increasing
turbine speed. Customarily, the natural frequencies increase with
increasing turbine speed. This increase is due to the straightening
of the blades and the tightening of the joints due to the increasing
centrifugal forces. Sohre also recommended using a titanium alloy over
the conventional ASTM 403 or 422 stainless steel in the stages of the
wet region of turbines. This paper comprehensively describes the
capabilities of the computer program, but little was written about the
theory or method of analysis.
Rao [13] made an analysis on turbine blade groups using Hamilton's
principle. Rao's paper is a two dimensional analysis of free vibrations
in the tangential direction . The first step is to develop the potential
and kinetic energies for the tangential motion of the blades and shrouds.
Second, Hamilton's principle is applied to derive the differential equation
of motion and the boundary conditions. Then, these equations are solved
to determine the natural frequencies. Rao compared his calculated frequencies
for lower order vibrations with frequencies calculated by Prohl's method.
The agreement was excellent. This paper did not include stresses.
24
A complete analysis covering all the variables of the blade group
problem was presented by Rieger [14]. This paper is a three-
dimensional coupled forced vibrational analysis of the blade group
with steady and dynamic stresses using finite elements. The
main thrust of the paper is the development of the finite element
procedures for turbine blade fatigue. Some of the many topics
included are damping, disk-blade coupling, transient, centrifugal
effects, and fatigue. Specific problems in the analysis of high,
medium, and low aspect ratio blades are presented including exper
imental results. The type of element used to investigate each type
of blade is discussed.
For certain blades like low pressure steam turbine blades, Rieger
points out that it is necessary to include the influence of the
support disk structure in the analysis. The blade length, width
and shape will determine the type of element required. Turbine
blades vary from thin three dimensional curved shell structures
to three dimensional thick shell sections. Three dimensional solid
elements are necessary near the blade-disk junction especially where
disk flexibility effects are important. Root flexibility effects need
to be included in all natural frequency and dynamic stress calcula
tions of blade groups. Most analyses are in the areas of static stress
and deflection or natural frequencies and modes. Very little work
has been done with fatigue stresses. Rieger presented a complete
fatigue stress procedure which may be summarized as follows:
25
1. Calculate steady stress details for a typical single
blade and root segment.
2. Calculate natural frequencies and mode shapes of the
blade group.
3. Calculate dynamic stresses throughout blade group at
resonant frequencies.
4. Apply the described fatigue stress procedure.
Errors are introduced into the stress calculations by the blade exci
tation modeling, blade damping, fatigue strength of the materials, and
root modeling. Rieger's method is accurate and applicable, but it
is costly and laborious.
Rieger and Nowak [2] wrote a second complete and thorough paper on
fatigue stress in blade groups. This paper is a three dimensional
finite element analysis of free, forced transient, and damped vibrations.
Recent advances in flow excitation technology and electrical excitation
technology are included in the stress procedure. A Goodman diagram
fatigue criterion is used which includes static mean stress, dynamic
alternating stress, notch effect, number of cycles, and size effect.
Sample calculations are given which demonstrate the use of the method
for typical blade group geometries. This method does include centrifugal
and steam loading of the blade and blade-rotor vibrations. A flow
chart of Rieger and Nowak's fatigue stress procedure is shown in Figure
8. Ideally the analysis would be one large computer program with water
26
1i"1
;-c UJ
35C-
o11
00
litCL
Ul3 /CCUI /
31/^
to
in <UI^
UJ
yr
c
sc /2
z>-
*
0
to
UJ
*=">-
IO (C <
0 <J 0
S**t-
vt
J .
V)1-
_l -i ZUito
zUJ
5
Z UJ Ul_J-I
UJ 0 O UJ UJ
*30
-^ 3 ODD T <r
u 2 5O0 UJ 0
-1
UJ sUJ 2 S
o 1- y-
UJ U_
ec 0
_J O1
Ul
ul
CC
Xt-
u.
0
3 < 0 0 l^-i UJ _l
HI
Ul
_l 0 0 S: Cj 1- UJ
zDEC
O O Oz
til
0
0
u. X1-
nnmn*
J >
111
0
Z
O u.
O
k1-
en
a
<
O-1
1-
1/1?- UJ
O
O t /X
/T \ /
01
zCO 0u 5=
to I"!CO <
UJ u
tr Q1- -<
CO J0 <0
S< tz trv 0
1-
<
A
CO
o zui o
x 1-
t- to
o UJ
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=
"<
UJ _l
ii
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O o
IIUJ
0z
z 2< tfcS zo 0X "1-
e>
1- t
r
o
to
Is
=:
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LL) ?
= P
1-
<u.
cc
ou.
UJ
cc
OUJ
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Occ
a.
in
>-I
<z
<
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UJ
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z <0
z1-
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CD z
OUJ
U
0z
z> in
a in
7 <a.
in UJUJ
_j
O N
< N
-1 Om z
111
<
_l
2 21-
D?U. H
>- = <O E -< t: 0UJ
to
ui 0
H z
<1-
z*
0(O
V d a-
0 2 0< < ^Ul Ul1- ?-
10 10
0
z
y 0
II
>'
Oz
tn
n
<
1-
zUJ
-1
z
0
in
1-
z
O
3cc1
a. 10
z
1-
cr
m lStc
spcr t
- tc 0
UJ (E a.<
UJ_J
X X a. ISI
0
1-
z
z
-I UJ
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$I cc
a.
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cc
o
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5 toc o z
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27
table data. Such calculations are presently time consuming and costly.
The following short-comings of current design practices are noted in
this paper:
1. Beam theory does not always apply.
2. Concentration factors are estimated.
3. Dynamic stresses are not usually calculated.
4. Excitation amplitude of tuned blades are rarely calculated.
5. Blade groups frequencies have a scatter of two - five percent.
6. Present natural frequency calculations are typically within
one percent- first mode, three percent
-
second mode, five
percent- third mode, and five - fifteen percent in higher
modes.
7. Transient stresses are not usually calculated .
8 Goodman diagram fatigue stress procedure is not always
adequate for a stress approach.
A comparison of various techniques used for measuring blade excitation
is presented. A method using the water table hydraulic analogy is
advocated. The water table measurements are inexpensive and versatile,
but care must be taken in modeling the flow parameters. Typical damping
ratio c/c values range from 1.0 percent at 200 hertz to 0.1 percent at
c
10 kilohertz.
Two strong points of this method are the accuracy and the elimination
28
of theoretical assumptions. The paper is very concise and well written
covering all important considerations of turbine blading design. The
stress oriented approach used is good!
A paper was presented by Salama and Petyt [15] on a blade group
represented by rectangular beams. A beam finite elements method is
used in a two dimensional forced vibrational analysis of blade groups.
Longitudinal and transverse displacements are included in the tangential
vibrations. A reduction technique is used to reduce the number of
degrees of freedom. The secondary degrees of freedom to be eliminated
are the longitudinal displacements and the slopes in the transverse direc
tion. This paper shows the relationship between the natural frequency
of the blade group and the stiffness ratio, mass ratio, number of blades
in the group, size and position of the lacing wire, and rotational speed.
The dynamic response of the blade group to periodic loadings and partial
admission loading are presented. The application of the technique of
periodic structures to reduce the number of degrees of freedom is restricted
to blade groups of six or more blades because of the effect of the cover
end conditions. When exciting the blade group in the range of lower natural
frequencies, the results of the paper showed the total contribution of the
modes above the seventh did not exceed one percent of the total response.
Salama and Petyt's paper does give good information. The paper did not
include any root stiffness factor, experimental data, or stress calculations.
Since the calculations are based on simple geometries, the results are
general and only good for guidance.
29
In conclusion there are many methods of analyzing the blade group
vibration problem. The methods range from simple beam theory to
finite elements. The accuracy, labor, cost, and capability of the
different methods vary considerably. The most important item
which controls the accuracy of the output of any given method is
the accuracy of the input parameters. The areas of exciting forces,
damping, fatigue strength, and root fixity are not well known and
need further investigation. Very few of the analyses include stress.
Areas which require technological development and research are:
1. Excitation of the blade group vibrations
2. Damping properties of the blade group
3. Material fatigue strength
The blades are subjected to one of two types of steam exciting forces.
The first is the exciting force of the passing nozzle. The second type
is due to unsymmetries in the medium flow path. Presently, the
magnitude of these exciting forces are approximated using a percentage
of the steam driving force. These approximations are from measure
ments made on similar turbines or from past experience. Studies have
been made on the damping of blade groups. Two types of damping
are present: (1) material damping due to internal friction and (2)
structural damping due to mechanical fits and joints. Measurements
on material damping are accurate, but the structural damping varies
over a large range because of manufacturing tolerances, assembly
tolerances, rotational speed, and corrosion. The value used for
30
blade group damping is approximated based on experience. Many
studies have also been made of material fatigue strength. Like
structural damping, material fatigue strength varies over a large
range. Fatigue strength is dependent on the dynamic load, static
load, number of cycles, corrosion, manufacturing tolerances,
assembly tolerances, stress concentration, elevated temperature,
three dimensional stress, material uniformity, and surface finish.
The value of the fatigue strength of a material is generally obtained
from a Goodman diagram with additional strength reduction factors
applied to this value. The three above areas have a direct influence
on the results of any stress analysis. Therefore, development and
research will result in more precise data to be inputted into the stress
analysis. More precise input will produce more accurate and reliable
stress calculations.
The development of engineering theory needs to stay in touch with
the needs of the turbine design industry. The turbine design
industry is looking for design tools which are accurate, but simple to
apply.
31
111. THEORY
The following analysis applies to groups of steam turbine blades
which do not have significant taper or twist along their length.
Three-dimensional vibrations and stresses of such groups are
considered. Simple beam theory is used to develop a dynamic
stiffness matrix which relates group structural displacements to
joint moments, shears, and axial forces. The solutions to the
structure dynamic stiffness matrix gives the blade group natural
frequencies and mode shapes. Prohl's energy method [5] is then
applied to determine the amplitude of the forced vibrations of the
blade group. The dynamic stress is calculated from the vibration
al amplitudes. Blade damping is assumed to be negligible when
calculating the natural frequencies, but is considered when calcu
lating the resonant vibration and resonant stresses. The blade
group is assumed to consist of a number of identical blades equally
spaced and identical covers. The blades and covers are considered
inextensional .
BASIC EQUATIONS
To develop the basic equations the end moment, shear, and torque
equations for a simple beam must be written in terms of the end
displacements and slopes of the beam. The positive direction of
end moments, shears, displacements, and slopes are shown in Figure
9.
32
y ,-
u. uB
FIGURE 9. CONVENTION OF SIGNS FOR POSITIVE DISPLACEMENT, ROTATION,
MOMENT AND SHEAR.
33
The equation of motion for free transverse vibrations of a uniform
beam in a conservative system is
^+
3c-rat2 (1)
This equation is for a beam free of axial forces. The shear and rotary
inertia effects are assumed to be negligible, and the assumption of the
Bernoulli-Euler beam apply -
By separation of variables, the following equation is a solution to
equation (1) for the normal modes of vibration.
/-
U(x) COS ait (2)
u(x) is independent of time and has the form
u(xV AcesV + B s,n\x"*- C cosK^x + D smhlx (3)
where
34
A,B,C, and D are constant of integration and are determined from the
boundary conditions of the beam. Successive derivatives allow
equations for slope, moment, and shear to be determined .
du& =
3 "-A^sinAx + B/Uos^y + OsmK^x + D> cosh'Ax
(5)
IV\=5I~|=
ll_-A"fcos Ax -
B>%iV\}y + C>2ccsk")x t D^a.nh^xJ(6)
5*-EI~5a
-l[A>3sinA< -fc)?ces}x +C>i5-,nV\>x +
D/^c^h>xJ*
(7)
The four constants in equation (3) may be determined in the general
case using the displacement and slope equations for end conditions A
and B, and the notation shown in Figure 9. The end conditions for
A and B are:
x = 0 x = I
y=
uA y=
uB
e =
eAe =
eB
This gives:
="
LeFTJ u******* *t
B z")*6* (8)
Fo 4.pi* Fa 1n _b_
F/0 A f01
^ uA+rerJ0A~2F3^ -?5^dB
(9)B =
2F3
35
6N-h&*+ -~^-Uo
-
-Ex & do)2^3
A
2F3B
ZW.3B
D =-_PL
2F3 UA-fe]^+
^a^4a^ (11)
in which [16]
(12a)
(12b)
F1= sin A*
sinh A
F2= cos XI'
cosh XI
F3= cos A- cosh XI -1 (12c)
Fy= cos A- cosh A +1 (12d)
F5= cos XI-
sinh XI -
sin XI-cosh A (I2e)
Fg= cos XI- sinh A + sin A cosh XI (12f)
F = sin XI + sinh A (12g)
F = sin Xi -
sinh A (12h)
Fg= cos A + cosh A (12i)
F = cos A -
cosh A (12j)
36
Substituting these expressions for A,B,C, and D into the
moment and shear equations gives the generalized end moment
and shear equations. These equations were previously developed by
McCallion and Rieger [8].
JVNii -.
44 eA-
-Ei e* +--^uk ? iu7vF3 7\F3
BF3 P3
Cosuif (13a)
If I a2 to3G* + fF3^ +
t^^ "& cos wt (13b)
3A z J5l a. -
e/o
LAFa^ 7\P,a B + -j* uA
- B"&|
COS uST (i3c)
EIa3
~[}f: *F36a +
-fe 6B ;es vwt (13d)CO
where
6 , 4*(t)dx
A general equation for displacement at any location along the blade
and cover can be derived by sustituting A,B,C, and D into equation
(3),
uW= [-(jb)uA. ^e,-|-uB+^0b]coS^
da)
37
A general equation for the moment may also be derived by sub
stituting the values of A,B,C, and D into equation (6).
The equation of motion for free torsional vibrations of a uniform
shaft is
o^fr _
_W_
3 ft (15)3x2
Ggat2
The material is assumed to follow Hooke's law and to be homogeneous
and isotropic. Displacements are considered sufficiently small so that
the response to the dynamic excitation is linearly elastic. The
positive angular displacement and torques are shown in Figure 10.
Again, by separation of variables the following equation is a solution
to equation (15) for the normal modes of vibrations.
v(x) is independent of time and has the form
(16)
(J) = C ccspx, + D Sin pX(17)
where
2_
0J~VJ (18)
38
FIGURE 10. BEAM WITH POSITIVE ANGULAR DISPLACEMENTS AND TORQUES.
39
C and D are constants of integration and are determined from the
boundary conditions. Applying the derivative, an equation for torque
is written.
T - OK-^
-
GKUcp sin px +Dp cos pxl (19)
where K is a function of cross section [17]. Using equation (17) the
end conditions for A and B, and the notation shown in Figure 10, the
two constants of integration in equation (17) may be determined for the
generalized case. The end conditions for A are x = 0 and v =y . .
The end conditions for B are x = and v = yb> By substituting the
values of C and D into the torque equation, the following general
equations may be written.
Ta = Tp co+qn (pfl) 6j\ f p sec (pi) *$ |o?s uif
(19a)
pp sec (p?) h*~
p tofen (p$) 3J cos i (19b)
OK L -
40
RESONANT TANGENTIAL VIBRATIONS AND STRESSES
To determine the resonant tangential vibrations and stresses, the
following steps must be taken.
1. Separate the blade group into simple beams.
2. Write the moment and force equations.
3. Load the dynamic stiffness matrix.
4. Determine natural frequencies and mode shapes.
5. Calculate the input and damping energies for each mode involved.
6. Solve for the maximum displacement.
7. Calculate the moments and stresses at selected locations
along the blades and covers.
Tangential vibrations have motion only in the plane of the blade group.
The principal axis of inertia of the blade is parallel to the rotor axis.
The turbine blade group is separated into individual blades and
covers so that, the general equations for moment and shear can be
applied at the joints. Figure 11 shows a blade group broken into
individual blades and covers. The vertical members are identical and
the horizontal members are identical. The blade-disk attachments are
assumed to permit slope rotation but no longitudinal or lateral movement.
The attachment is considered to have a constant slope dynamic stiffness
factor k. The joints between the covers and blades are assumed to be
41
Joint6 7
'///.
(a)
*/////////// ///>ysv
Joint 0
u0 M
ocA
rM
obB
obB
Joint i
M'CB ^"^TT1^'icA
S,ibB
Joint N
"ncB
u MNbB
*NbB
MobB
Ka,
(b)
Figure 11
(a) Typical multi-bay framework.
(b) Moment and shears at joints..
42
rigid, and the angle between any two members remains at right
angles during any displacement. The length, cross-sectional area,
weight density, second moment of area, and the modulus of elasticity
may be different for the cover and blade. Therefore, the subscript
b will be used when referring to the blade parameters and c will be
used when referring to the cover parameters.
Before the moment and shear equations can be applied, it is necessary
to write the general equations with the proper end conditions for the
blade and cover. For the blade:
uA=
o
M/^ k'
6rt COS uit
x- X
<V dB
(20a)
(20b)
(20c)
(20d)
(20e)
(20f)
where A is at the blade attachment and B is at the blade tip. k is
a root stiffness factor in pound-inch per radian which is determined
experimentally. Setting equation (13a) equal to equation (20c) and
43
solving for 8. in terms of 9B gives
ALrf, +PsJ & ^
L RF3 + A5F3Jb (21)
3.'
where
R*
1
After substituting the blade end conditions and equation (21) into
equations (13b) and (13d), the following blade equations are obtained
[8]:
M6~
ETM-^dB r feaBJ C6S oit (22a)
Sb=
Ei^Lit o& +
$*a*lcos ^ (22b)
where
*1
2
-2/^SF, + RF$
;ufs*-
F5
;uFfc + RF,3 F5 + RF3
2n Fa + RF,^XFs- + /?F3
44
For the cover end conditions
(23a)
uA = 0 (23b)
(23c)
(23d)
uB= 0 (23e)
B UB(23f)
Substituting these conditions into equations (13a) and (13b), the
cover equations become:
ma-
-Eu[^eA?" is^&Jeos^t (24a)
A
M & ei^'s eA* iH eB] co, *t
<2b>
where
3"
= _E-
?*.
45
Using the notation illustrated in Figure 9, the free body diagram
is drawn and shown in Figure 11. Writing the equilibrium equations,
we have the following (N + 2) equations where N is the number of bays.
MocA +MobB
- 0 (25o)
"M,tA~
MitB f M,c,j= 0 (251)
-MNtB+
MNbB^
0 (25N>
S.bB* 9-bB +
SMbB--[NJwy^/]c{25(N + 1)}
where the subscripts have the following meanings:
first = joint number
second = blade or cover
third = end of member
i = 1 through N-l
Substituting equations (22) and (24) into the equilibrium equations
(25) and combining terms, it is now possible to write the dynamic
stiffness matrix. This matrix is illustrated in Figure 12. The
matrix includes the mass and spring stiffness according to the
following equation .
[-MJ-
f w]y = o (26)
II
z
46
r4
csl
rei
i
rf1
T2.
t/l
Z
o
I-
<DC
CO
0
O
o
in
KMo
rH
0
he*
J
o
-0
in
u
KM
U4 UJ
K*
it
(V
0)
j=
3
_l
<
z
UJ
a
z
<
DC
oLL
><
DCh-
<
1/1
to
Ui
u_
H
u
<z>-
a
LU
a:
D
O
LL
C
to
3
rrOJ
Eo
r0
mto
to
47
By obtaining the roots of the determinant of the coefficients for this
set of simultaneous transcendental equations in Figure 13, the natural
frequencies and mode shapes are calculated. When the determinant of
the coefficient matrix goes to zero, a natural frequency is found. The
natural frequency or eigenvalue is calculated by a predicted method
of trial and error. The brut force method is used to find when the
sign of the determinant of the coefficient matrix changes. Once this
occurs, the method of false position is used to refine the value of the
natural frequency.
Assuming a value for y, the mode shape or eigenvector can be deter
mined. This is done by deleting one of the (N + 2) equations and
solving the remaining (N + 1) nonhomogeneous equations for the re
maining (N + 1) coordinates. The equation which is dropped gives the
largest absolute value for the determinant of the coefficient matrix of
the remaining (N + 1) equations. The geometry of the mode shape is
correct, but the magnitude of the coordinates are relative values.
Applying equations (3), (6), and the relative coordinates, relative values
of displacements and moments are calculated for selected locations along
the blades and covers. A relative maximum displacement Y is deter-mr
mined. The relative displacements and moments are normalized with
Y Mrespect to Y giving r and r .
^mr
3 a
yV~~
mr mr
An energy method is applied to calculate the magnitude of the forced
vibrations and is based on Prohl's [5] energy method. The following
assumptions are required in addition to the assumptions made for the
48
frequency calculations.
1. The blade groups move at a constant speed behind a full row
of equally spaced identical nozzles.
2. Because of the non-uniform flow from a nozzle and the asymmetries
in the flow path, the blade group is subjected to an exciting
force which varies harmonically with time. The forces can be separ
ated into tangential, axial, and torsional components.
3. A condition of resonance is assumed to occur between the given
forcing function and a natural frequency of vibration of the blade
group. Under this condition of resonance only the given harmonic
of the exciting force will supply a net energy to the vibrating
system.
4. The amplitude and phase angle is assumed to be constant along the
length of the blade.
5. At resonance the input energy is completely dissipated by the
damping energy.
6. The damping of the blade group is considered to be small, so that,
the mode shape based on no damping is valid.
49
7. The total damping of the blade group can be expressed as a
function of the logarithmic decrement.
Assuming the blade group vibrates at some frequency oj, the displacements
for the blades are
Yj = Y,(>0 sin <*f (27i)
where
i = 0 through N
The blades are subjected to one of two types of steam exciting forces.
The first is the exciting force of the passing nozzles. As the blades
pass the nozzles they are subjected to a cyclic force of frequency p.
p- Z 7f n k R (28)
where
n = order of the harmonic
k = the number of nozzles per360
R = rotor speed (rps)
50
This is illustrated graphically in Figure 13. The exciting force
acting on the blades from the passing nozzles in differential form is
dF, = ^dx sin [p(t-^p)+f>] C29D
- c\jdx. sinTp't - ,'oc 4 (p]
where
q= intensity of the exciting force over length dx
= phase angle
a = 2tt nk/m
m = number of blades per360
The second type of exciting force is due to un symmetries in the
medium flow path. In this case, the frequency of the exciting force
is also represented by p. The force acting on the blades from the
unsymmetries in differential form is
cjFv= c^ck sin [pt -*
+-
<p\(30i)
where
a =j*~
51
t + <j>
FIGURE 13. EXCITING FORCES ACTING ON A BLADE GROUP.
52
The tangential force spectrum for a blade in an intermediate pressure
stage test on a water table [2] is shown in Figure 14.
The energy dE supplied by the force dF. acting on a blade element of
length dx for a unit of time dt is
dEs = dF; M df(J))
The force and displacement is in the same direction. By substituting
equations (29i) for (30i) and the derivative of equation (27i) into
equation (31), the total energy is derived for one cycle of vibration
of the ith blade.
5 21Y
(Eg);" f f^ S',ttLV~,"' +^J*X'CX) COS ult d(u>t) (32)
Letting p= u<
,the condition for resonance, and integrating gives
(Es>, KqJ,Y;6ft sin(<p-i<)ix ()
The integration with respect to x is performed numerically. Summing
over all the blades, the total energy for the blade group is equal to
N'
Lf(34)
. -. "J""
M'-"
'0J1
N L
L i-o .H
53
i_ oo
(J r>
afN
*~
00O
ino
oo
i
IIII
<U
0.>
ra
***. BM*
0 (U.2
ro4-> N
DC ra
rr
N
O
c
Vc U)
L. >, l_
3ifl
ifl
cuJ.
0.
*-> U) ra
8c
h-
00
>
ra
a(n
co (A
*-> 0)aj 4) _ <jrn n ro (1)
N
ra roX
i_O
in CO < 5 z
Ifl
<u
a
GO
>
u
UJ U
So
El
o
N
X
>>
u
cQ)
O"
cui_
LL
LU
a
<_i
CO
Qi
oUL
2 LU_)
_l
a: COh-
<u r-
LU
0. t*
(/) 111
LU
uDC
<
o ZLL o
<r-
t/1
r-
Z
LUr-
LU LU
o oz << r-
r-
l/>
LU
DC
o
LL
54
where L represents the number of stations per blade. Figure 15
shows the stations of a blade and cover.
For ease of calculations it is desirable to have the values of the dis
placement in dimensions quantities. Also in this step a stimulus
factor, S, is added.
Eor ^Ym ^y3 (A*)Z] H %*S-A (<p-i*) (35)
mo j3i Ym
where
Y = maximum tangential displacement of the blade group
S = ratio of the total tangential exciting force per blade
to the tangential driving force per blade
Values of ij are calculated in the mode shape step. The ratio
y .Ym V .
'
\j is equal to ~L
Ym Ymr
The E is maximized by the following steps. The trigonometric function
in equation (35) is expanded.
Lu;, ui 'm Ji"o
'<*
(36)
Blade
Stations
Cover Stations
4 5 12 3
55
Tenon
Cover
Vane
Platform
Blade Root
Disk Root
FIGURE 15. STATIONS OF THE BLADE AND COVER.
56
Letting w L
a = 7] T] s,rv io< (37a)
M. _L y..
T"- COS IOC (37b)4-i 4-* Y^1=0 i-O
equation for the total energy supplied to the blade group becomes
Es= 7rYmSyc^(Ax)[Bsinp- k
cosp]( 38)
Using the first and second derivatives of E with respect to p, the
maximum value of E is calculated.s
E, = +e.2
Sm**1*13^(39)
where
q-
T
F = tanqential driving force per blade
y
Ax = r-
Simplifying, equation (39) becomes
Ec,=n\ 'ro -7 j_
(40)
57
The energy dissipated by damping is
where ^ = logarithmic decrement
Ej= total vibration energy of the blade group
The logarithmic decrement used in this equation represents the total
damping of the blade group. Two types of damping are present:
1. material damping due to internal friction and
2. structural damping due to mechanical fit and joints.
Aerodynamic damping is neglible for this type of blading.
The total vibrational energy for the blade group in the tangential
direction is written as the summation of the kinetic energies of all the
stations of the blades and covers. In general form
Ed =
-^fYm C (>
58
where
N L v5_ N-l
<-z:E[^]b*.E[;,[^g]*]..
i~0j-
1 D i'Oj'i C
N-1 = number of covers
s = number of stations per cover
Yij = tangential displacements of the blades
Xij = tangential displacements of the covers
Setting the input energy equal to the damping energy and solving for
Ym, an equation for Y,^ is developed
s~
- [n5)
There are two main differences between the author's method and
Prohl's energy method. The author assumes a constant driving force
and phase angle along the length of the blade, whereas Prohl's energy
method allows the driving force and phase angle to be varied.
59
M
Relative values of f^r ]. are calculated for selected locations alongmr
''
the blades and covers in the mode shape step. The actual values
of the moments are calculated with the following equation :
^- l>" (ti (46)
where
M. = tangential moment
Y = maximum tanqential displacementm
ai-
Y = relative maximum tangential displacementmr
r-
M = relative tangential moment
Using the maxmimum bending moment at the root of the blades, Mmbr
a resonant response factor is calculated [5].
u - M"^ ?-g Va*+b*J (47)*
Ymy L^ C
To qive the stress a. at selected locations along the blades and covers
3J
the moment is divided by the tangential section modulus.
>- [*] :
(48)
60
The possibility of fatigue stress failure at a selected location [2] may
be evaluated by the Goodman diagram approach shown in figure 16.
The figure plots alternating stress vs. mean stress. The failure
enveloped shown is for unnotched fatigue test data obtained under
similar mean and alternating stress conditions. The effective
notched stress failure envelope is obtained using a strength reduction
factor method by Heywood [18]. The strength reduction factors are:
Ka=
^=
Ks'OvKsH (50)
where
K = mean strength reduction factorm
K = alternating strength reduction factora
am
mn
maximum value of mean stress
nominal value of mean stress
maximum value of alternating stress
nominal value of alternating stress
K = static stress concentration factor
s
%
aan
JK = alternating
fatigue strength factor (> 10 cycles)
tnnominal tensile strength of material (U.T.S.)
a = notch sensitivity index
^a
61
Unnotched fatigue curve107
cycles
-Notched fatigue curve107
cycles
an
Altern
Stress
Nominal Stress
Point
MEAN STRESS
(A) SCHEMATIC OF PROCEDURE
Unnotched strength
from test results
20 40 60
(B) SAMPLE FATIGUE STRENGTH CALCULATION
Material: STAINLESS STEEL
Location: Blade root upper
notch
K_= 2.43
mri
= 7298 psimn
r
Mode 1Qanl
= 8078 psi (345 Hz)
Ktm= 2-47'
Nominal mean a
Mode 2(Tn= 2918 psi (539 Hz)
FIGURE 16. HEYWOOD STRENGTH REDUCTION PROCEDURE WITH
APPLICATION TO BLADE ROOT STRESSES.
62
These factors include all the major fatigue parameters. Nominal mean
stress and nominal alternating stress values at the selected location are
plotted on the Goodman diagram. The relationship between this stress
point and the failure envelope defines the possibility of fatigue failure.
If sufficient data is available, this possibility can be defined statistically.
RESONANT AXIAL VIBRATIONS AND STRESSES
The resonant axial vibrations and stresses are calculated by steps similar
to the tangential vibrations.
1. Separate the blade group into simple beams
2. Write the moment, shear and torque equations
3. Load the dynamic stiffness matrix.
4. Determine natural frequencies and mode shapes.
5. Calculate the input and damping energies for each mode involved.
6. Solve for the maximum displacement.
7. Calculate the moments and stresses at selected locations
along the blades and covers.
Axial vibrations have motion only in a plane perpendicular to the blade
group. The principal axis of inertia of the blade is perpendicular to the
rotor axis.
The explanations of some of the above steps in the following paragraphs
have been shortened due to their similarities with the tangential case. The
reader must be familiar with the tangential section before reading this section!
63
Figure 17 shows the blade group separated into simple beams. The
moments, shears, and torques are also shown in this figure. The
same assumptions are made for the blades and covers in the axial
direction as for the tangential case. The equations for moment and
shear of the blade is the same as equations (22) for the tangential
vibrations. In the axial case, the attachment is also allowed to twist
about the x-axis.
X= 0 (51a)
(51b)
(51c)
Setting equation (19a) equal to equation (57b) and solving forya
gives
*AZ sgc p}
fe + CoTc
GK"an
$ e>(52)
Substituting equation (52) into equation (19b), an equation for Tg of
the blade is derived.
TB-- [GKpi^J cos ui (53)
64
in
<DC
CO
X
<
DC
oLL
Q.
ODC
o
LU
Q
<_J
CO
LL
O
<DC
o
<
>-
Q
O
CO
LU
LU
DC
LU
DC
O
LL
X -<
65
where
if~ ^-!
4 CoRm p%
The cover has the following end conditions:
x =0 (54a) x = jt (54e)
UA:=
UA (54b)UB
=
UB (54f)
9A:=
6A (54c) 0B:=
9B (54g)
YA:=
YA (54d) ^B:=
YB (54h)
With these end conditions the general equations (13) and (19a),
and (19b) for moment, shears, and torque apply directly.
The next step is to write the equilibrium equations for the moments,
shears, and torque using the sign convection shown in figure 17.
The number of equation is 3(N +1) where N is the number of bays.
Joint 0
Mxoca"
T<ob&-= O (55 O)
T/ocA f M/cbB~"
& (56o)
SzccA- SzotBl * 0 (57 O)
66
Joint i:
MxicA~
Mxl'tfc-
Tx,'^ * 0 (55 i)
TytcA -
lyxcb + M/lbB ^ 0 (56 i)
SZKA"
SjrjtB-
SZl't,B'
0 (57 i)
Joint N
,VVNcB + ^xNbB "
O (55 N)
~TyNcfe"*
MvNbJ3"*
O (56 N)
^Nck+
^iNbB - O (57 N)
The subscripts have the following meanings:
First = axis
Second = joint number
Third = blade or cover
Fourth = end of member
i = 1 through N-1
67
Now, equations (13), ( 19a) , ( 19b) , (22) .and (53) are substituted into
the above equations. With considerable work combining terms a series
of equations are developed which are used to load the dynamic stiffness
matrix. The final equations are in terms of the coordinates and are
shown in figure 18.
With the dynamic stiffness matrix developed, the natural frequencies,
mode shapes, relative displacement, relative angular displacements, and
relative moments are calculated. The procedure used to calculate the
above items is the same as for the tangential vibrations.
The input energy is summed in the same manner as for the tangential
case, but for the axial case torque is also included.
fr.=2!^ _J
A*
+ft*
'
(58)
where
W L
a
=L[v^+ ^%]^- (59a)
B =
^ ^JSzk^i + S*T*f^|COS,OC (59 b)
i--o \-.\
Z.. = axial displacements of the blades
'J
$.. = angular displacements of the blades
68
r n
0
l I
0 2 z
N
z
Nh-u
ifliT
u. |n
i
U-|u-
t-| roU.U.
ro u
lc|Ll
"a?
*:
i
urir
u. u.
n
LO
I-
<DC
CO
*:
Ulu"
i u
r<:
4-
tf| ro
I
m
I
rt:
u.|u_
<
02
oLL
><
DC
^
rc
10 CVJi +u
r< iei
h^
*
t&
J in
+ o
f<
0-
,J)
h
2
li u.21 ro
u.|u-
2 mU- U.
r?
in
l/i
LU
zLL
to
u
X
X
Q-
I
<C 0-
I
c
a.
+
Q-
+
I/)
rd
eJ
:*:
LU
02
Z)
o
LL
<tf
r<
&v5r
o-
rj
i
69
Z = maximum axial displacement in the blade group
F = axial driving force per blade
S = ratio of the total axial exciting force per blade to
the axial force per blade
S = ratio of the total exciting torque per blade to the
torque per blade
T = torque per blade
The equation for the damping energy is:
C - ij=2. 7 f (60)
where2.
c =t[^tet-^feL
(61)
C
r-C ^1
AZ *= differential axial displacement of the covers
A? = differential angular displacement of the covers
'J
70
Differential displacement is the difference between the displacement of a
cover station and the average displacement of the complete cover.
By setting E = E . and solving for Z,the following equation is derived:
7 - 2LSL V A
^ Wl C(62)
Knowing the value of Z,the bending moment at selected locations can be
calculated using the following equation:
Mj=
m VZvm-/(63)
M
The values are calculated in the mode shape step.
Zmr
Using the maximum bending moment at the root of the blade, Mmfc)r, a
resonant response factor may be determined.
v. M.mfcr eg
+
(64)
The stress a. may be calculated by dividing M. by the section modulus
Z..J
(f -
jM
Z (65)
71
To determine if fatigue stress at a selected location may cause failure,
the Goodman diagram and Heywood's strength reduction factors are
applied. This is explained in the tangential section.
COUPLED RESONANT VIBRATIONS AND STRESSES
Coupled resonant vibrations and stresses are a third type of vibrations.
The tangential and axial motions are not independent but are coupled.
This coupling of the motions occurs when the principal axes of inertia
of the blades are rotated with respect to the principal axes of the rotor.
This analysis is performed by calculating the moments, shears, and torques
at the end of the blades with respect to the principal axes of the blades.
These moments, shears, and torques are then transferred to the rotor
coordinate system. The steps of the calculations are very similar to the
tangential vibrations.
1. Separate the blade group into simple beams.
2. Write the moment, shear, and torque equations.
3. Load the dynamic stiffness matrix.
4. Determine natural frequencies and mode shapes.
5. Calculate the input and damping energies for each mode
involved.
6. Solve for the maximum displacement.
7. Calculate the moments and stresses at selected locations
along the blades and covers.
The explanation of some of the above steps in the following paragraphs
has been shortened due to their similarities with the tangential case.
72
The reader must be familiar with the tangential and axial sections before
reading this section!
The blades and covers are separated as shown in figure 19. The
moments, shears, and torques are also shown in this figure. The prime
coordinate system is the blade coordinate system, and the unprimed is the
cover or rotor coordinate system.
The blade members are identical and the cover members are identical.
The blade attachment is assumed to permit rotation about all three axes
but no longitudinal or lateral motion. The attachment is considered to have
an independent dynamic stiffness factor k in each direction of rotation.
The same assumptions are made for the coupled vibrations as for the
tangential vibrations.
The equations of equilibrium for the joints are
Joint 0
Ma -T .
' 0 (66 O)
MXccA '*obB
M/otacosd "^cbBs'nd f 7/-a- o (670>
Mz<obBccs6 + 'VbB> MG *
MzecA "- * (68 >
^cbB sm0 "Sz'ofcBcc^ + SzocA * (690)
73
%r^ S
in
<DC
00
QLU
_J
0.
o
u
02
OLL
0.
o02
O
LU
Q
<_l
CO
LL
O
<02
a
<
>
Q
O
00
LU
LU
02
LL
LU
02
a
LL
74
where 6 =clockwise rotation of the cambered blade -
top view.
Joint
"T*ibB+
Mx,'cA "MxcB ~' (66i)
M/ibB co^-
Mz,ibB sin +T/;cA
~T/(cE>-
0 (67 i)
",Vlz',bBCebG"M/'bB &'n0
-
M7lJV+ Mzic8""0 <68i>
""Sz'.bB Cc'^ + V'iB sk& +Sxl-cfc
-
SZ(cBL-6 (69 i)
Joint N :
T*Nb&+
MxNeB " (66 N)
^/Kibft Cc'-G"
^I'lMbfe 5,n6 "TyNcB'
(67N)
Mz'NbB Ccs6 VMv'Nb& Sm(5
'
Mz.Nc& "O (68 N)
^/NhB Sm~
^z'N-bB CcbQ
~"5iKJtB ^0 (69 N)
The subscripts have the following meanings:
First = axis
Second = joint number
Third = blade or cover
Fourth = end of member
i = 1 through N-1
75
For three-dimensional coupled vibrations it is necessary to have
(4N + 5) equations where N is the number of bays. Presently,
(4N + 4) equations have been determined and the last equation is
obtained by considering the shears in the y-direction for the cover as
a complete unit.
S/obBcc56 +
SrobB Si^e + S/a>B cosG fSZ'.bi3 ^^
+
%'NbB Co&0 +^VNbB &m6r N w A J?
<*fy
^ Jc(70)
In order to simplfy the substitution into the above equations, a summary
of the moment, shear, and torque equations is presented with the appro
priate end conditions.
Blade -
<
y and direction
X = 0 (20a)
uA-
0 (20b)
M.=k6A cos M)
X= *
uB
B
*B
JB
(20c)
(20d)
(20e)
(2Of)
YB=
<B(51b)
TA = ky. cos (tot)T A
(51c)
76
MB EI 7>|-tj-d|5 + Ij"B C<. W+ (22a)
SB EnH^-fi",]"^ (2*)
where
*1
3
R
asrfe-*-
pf
/UF5 + RF3
'B [GKp$fc #BJ COS ojt (53)
where
_li+ Co+cin pV.
_Jil_ f cc+dVA pv
GKp
Cover x direction
77
X = 0 (23a) x = 1 (23d)
UA= 0 (23b)
UB= 0 (23e)
9A=
9A (23c) 9B=
9B (23c)
M,
M B
~E I *[$+ 0A + $s 0&j CCS uit
Ei7v[f5 eA + $H eB] cos ool
(24a)
(24b)
where
$*~~
Fa x5 r
Cover y direction
x = 0
u uA
9A
A
'A
(54a)
(54b)
(54c)
(54d)
Ma
ETTf-[-^-^"fc^
Y
UB
0B
YB
4 -EfS-UB
(54e)
(54f)
(54g)
(54h)
Ml , fOT L
F*GA f If 0B
*r/l^
4- ^,C,1 A + f"
I1-___ LA A
+ ij
F3 4 8-i
CCS ^t (13a)
Ccs wT(13b)
&n3 >F,dA 06 f Js.ufl
--~i-i cl
B3 J
S wtCtii W
(13c)
78
ElX̂ ["fe^+ V* * %"*- iuB]c.,+(13d)
>t (19a)JA. - [Ip coiin (f>$) &A 4 p SCC (pi) S&lces "S
-2k . [-p sec Cp) *a -rpcoita (pO *B"1 cos WT
<19b>
By substituting these specialized equations into the equilibrium equations
for the appropriate moments, shears, and torques, new equations are
generated which are in terms of their own coordinates. The coordinates
for the blades are in their prime coordinate system and must be con
verted to the unprimed cover or rotor coordinate system. This is
accomplished with the following matrix equation.
y cos 6
sir\$
SU\6
CO sfc
(71a)
(71b)
With a tremendous amount of manipulation and simplification, the equil
ibrium equations are converted into a form used to fill the dynamic
stiffness matrix shown in figure 20. The final equations are in terms
of the rotor coordinate system.
By obtaining the roots of the determinant of the coefficients for
this set of simultaneous trancendental equations, the natural
frequencies, relative deflections, relative angular displacements,
relative moments, and mode shapes are determined. The same procedure
o79
f
z
N
1 ',J MW
aJ>
s: ^ 8i . * 7
J! w S
<5 :
Li i V
J fi
km *ej
+
XI
u ttlu?
(^ "t \JI o
ie< < u. u.
rt f< A
s: x.
M
r
I at
^>
o-a -a
N
*
l!j +
I- rtfl
H7. * 3
--to ^ -"
u
2 *
u.|u.
i
u.|u-
o
*:
21 nu-|a
u
2
a -ifliTn *u
i i i
rz rt re
> i i
Itri x
I-+
0 H o
z
Q1X1
0.
o
u
_
n11
JO J3 '*
* * s. I .
* o
1? rs 8 44
wc 8"
^1IJ
^5
LU
#x
+
* m h u
kll + .
cs0-
J)
"n
+
i+
i i a> "r "n c
3Si
0
i>-
Q-
caj
a
01
2* t
S-5
~8 -S
"5'
x:
a).
"3 -
z
i
a
-a
CD
<
U
CC
HUJ
SS>to
"E
.
>-
* ai ra
E ra
\. r v" ra-C
I- >h-
02
oUL
><
02
10
tn
LU
Z
LL
LL
Hin
<z
Q
o
(SI
LU
02
Z)
a
80
as described for the tangential case is used to calculate these values.
Next, the relative displacements calculated in the previous step is used
to determine the input energy. With coupled vibrations the input energy
is summed numerically in the y and z direction using the following
equation :
e5=xy*Ljr\z
+ (72)
where
: -^ .>t-
Stn K(73a)
B
m
=
L l[s'F>t + +
S'T'%]" '* l73b'
= maximum displacement in the blade group
F = tanqential driving force per blade
y
F = axial driving force per blade
T = torque per blade
ratio of the total tangential exciting force per
blade to the tangential force per blade
81
Sz= ratio of the total axial exciting force per blade to the
axial force per blade
Sx= ratio of the total exciting torque per blade to the torque
per blade
Y.. = tangential displacement of the blades
Z.. = axial displacements of the blade
$.. = angular displacements of the blades
The damping energy is also summed in the axial and tangential
directions for the blades and covers.
Z..ZE.-^V.C
where
c--LL\*P$2i+"Ptot)^m1*0 j=0
b
.c
i--0 jSl
X.. = tanqential displacements of covers
'J
AZ.. = differential axial displacements of covers
A$.. = differential angular displacements of covers
82
By setting E = E. and solving for V, the following equation is
derived.
TT<3+b*
V" "
S^L C(76)
Knowinq the value of V,the values of the bendinq moment at selected
am
3
locations may be calculated using the following equation.
M "' [V" Umt; (77)
M
The values of r^ are calculated in the mode shape step.
mr
Using the maximum bending moment at the root of the blade,Mmbr. a
resonant response factor may be calculated.
KM^r 2
aJh? +
' mrT
(78)
The stress a., may be calculated by dividing M. by the appropriate
section modulus Zj:
OjhA
-Z-J
(79)
83
To determine if fatigue stress at a selected location may cause
failure, the Goodman diagram and Heywood's strength reduction
factors are applied. This is explained in the tangential section.
84
IV. SAMPLE CALCULATIONS AND RESULTS
To simplify and demonstrate the theory of the three types of
vibration; tangential, axial, and coupled, a set of sample cal
culations will be performed for each case. The tangential and
axial vibrations and stresses will be calculated on a simple
rectangular beam structure. The coupled vibrations and stresses
will be calculated on an actual turbine blade group.
The tangential and axial vibrations are calculated on the bar
structure shown in figure 21. The vibrations of the same case
was performed in a paper by Rieger and McCallion [8]. The
parameter section for this structure of the author's computer
program is shown in figure 22. This computer program is for
uncoupled resonant vibrations and stresses in the tangential and
axial directions.
Looking down the list of parameters, most parameters areself-
explanatory. Some are not and shall be explained. The logar
ithmic decrement for stainless steel type 403 is .00063 to .035 per
Lazan [19]. For this structure .02 is used. The stimulus is the
ratio of the exciting force to the driving force of a blade. This
parameter is directional.
The next parameter to be discussed is the sectional modulus about
85
86
UNCOUPLED VIBRATIONS AND STRESSES
D IMEMS I ON Z < 37 , 37 ) , Ui ( 3 ) , X ( 4 / , WA < 3 > , XA < 3 ) ,
I ZA < 37 , 37 ) , COR ( 37 ) , ZB < 37 , 37 ) , YBLA ( 200 ) , XEuA ( 200 ; ,
1 ZBLA ( 200 ) , XEL ( 200 ) , YBL ( 200 ) , YMOM (12), XMGM (12).
1 ZBL ( 200 ) , XMO ( 200 ) , YMO ( 200 ) , ZhO ( 200 ) , SECZB (11),
lZM0M<2t>> ,STRI<25) ,ANG<200> ,SECYB<11) , 5TRXU2) , S7PY< ,2\
1 XBLt. ( 200 ) , Y3L3 ( 200 ) , ZBLB ( 200 ) , ANGB ( 200 ) , SECZC < 22 ) ,
15ECXC(22) , XBLAA<200> , YBLAA(ZOO)
DIMENSIONS ARE IN LBS, INCHES, SECONDS, RADIANS
GRAVITATIONAL CONSTANT
G=38G.33E
PI=3. i41532B5
NUMBER OF NOZZLES PER 380 DEGREES
UN=92-
NUMBER CF BLADES PER 360 DEGREES
UB=1S2.
LGGAR I THM I C DECREMENT
DC=.02
ST I MULL 3 Y-.DI RECT I ON
STI^.V=. i
STIMULUS Z-DIRECTION
STTMZ=. 04
5TI-;UlLS A-DIRECTIONST.IMA--
. 04
AMPLITUDE CF DRIVING FORCE - L3/BLD Y-DIRECTIGU
AMY- 10 .
AMPLITUDE OF DRIVING FORCE -
lB/ELD Z-3 IREOTIG.1-.
AMZ=5.
AMPLITUDE OF DRIVING FORCE - LE/BlD A-D.7PEC4 ION
AMA-E ,
ROTATIONAL. SPEED - RAD/SEC
ROTS = GO .*2.*PI
BLADE PARAMETERS
SPECIFIC WEIGHT
D3-0, 23103
AREA
AB =,059057
MGDU,_US 3F ELASTICITY
EE-2S..3E-0C
3 -iEAR MODULUS CF ELASTICITY
SHRB-i .1 . 25E+0B
INERTIA ABCUT Y'-A;<13
c YB^.4S04E -3
INERTIA A30UT Z'-AXIS
BZB=. :"'4S7E-3
-EPTTuN MODULUS A3CUT Z-AXIS
St 223 t .1 ) = OO I bo (J
DO '73 1 = 2,10
SECZ3' I '< ~
. 001E30
5ECZ3< 1 ) ~
.001 EGO
SEC7Iu"-i MODUlUS A3GL7 Y-AXIS
FIGURE 22
PARAMETER SECTION OF COMPUTER PROCRAM FOR BAR STRUCTURE
l . 000
.000
c
c
3 . 000 c
A . 000
3 . 000
G u 000
7.
V
.000
''!00
w. , 000
10 .000
11 . 000 c
i. v . 000 w
13 . 000
14 .000
15,. 000r-
IB.. 000
17 . 000 c
18 . 000
15 .000 c
20.. 000
2i ,. 000 c
'/'?'
,. 000
.000 c
24 .000
~? '~-l.doo c
T,
"-1
000'"< 'j
. 000 c
7^ ,. ooo
2J\,. 000 c
30 .. 000
O t. ,. 000 0
3'-000
33,. 000 r
34,. 0003:'
,000 c
\.Zo ,. 000 c
/ ,. ooo r
wJ
C'
000
L'w 1. 000
40,. 000
r *
. 000 l~J
42 . 00 0
43,. 000.
'
* i i= 000/<
'-}. 000 c
n ~'
= 000
4/
48 i. 000
, 000
~
50,
000
. 000
,-
5.". , . 000
52,. 000"
'5;-;v, 000
34., 000 r
(continued)
87
53.000 3 E C Y B ( 1 ) =. 0 0 3 0S 0
53.000 DO i 12 1=2,10
57, 000 1 1 2 SECYE ( I ) - .. 0030E0
50 . 000 SECYE ( 1 1 )-
. 00 3 0 8 0
59.000 ,C TORSIONAL MOMENT OF INERTIA
BO . 000 POLE*. 43 1OE-3
SI. 000 0 LENGTH
G2.000 CB=S.0
S3. 000 C TORSIONAL HEIGHT MOMENT OF INERTIA
b'- 000TQB~
POLE#CB/ 1 0 . #DB
03.000 C STEAM FOMENT AT THE ROOT OF THE 8LADE-
SG . 0 00 S ;vi0Z-
AMY #C 3/'
2 .
G7.000 C
G8.000 C COVER PARAMETERS
GS.000 C SPECIFIC NEIO.-T
70.000 DC=0.2S1S3
7:. .000 C AREA
72.000 AC-. 055037
73.000 C MODULUS OF ELASTICITY
74.000 EC=23.3E+0B
7D.000 C SHEAR MODULUD OF ELASTICITY
7B.000 S.l-;RC~.l i .25Ei-0G
7 7.000 C INERTIA A POUT X-AXIS
7S.000 BXC=. 4804E-3
7S.000 C INERTIA ABOUT Z-AXTS
80.000 BZC"-. 17487E--3
81.000 C SEC7IGN MODUlLS ABOUT Z-AXI3
82.000 SECZC< 12)=. 00136
33.000 DO IZS 1-13,21
S''i .OOC 123 SECZC \ I ) =.00 1SG
85 . 000 SECZC ( 22 )-
. 00 1 EG
83.000 C SECTION MODULUS ABOUT X-AXIS
87 . 000 SECX C ( 1 2 ) = . 0030 3
88.000 DO 130 1=13,21
89.000 130 3ECXCC I > -.00708
90.000 SECXC( 22) =.00308
9'..000 3 TORSIONAL MOMENT CF INERTIA
SP.,000 P0LC-.4310E-3
93.000 0 LENGTH
04.000 CC-S-0
S5.000 0 TORSIONAL HEIGHT "lOMENT CF INERTIA
SB. 000 70C-PGLC#CC/i0.*DC
97.000 C
B8.000 0
S3 000
100 . 000
10 i .000
102.000
103. 000
104.000 0
105.000 3 NU^EER OF "?HE HARMONIC
10S . 0 00 HAP~
1
107. 000 SA - ( DB tGrRC/SHRG/DC ; *-*0
108 . OOC CO"
' 33/SHRC/3 ) >" 0 .
?
POGT ATTACHMENT;-
y\ i-
HY-3.EE10
> J"~
rz
~
CjC . : ;
HT -3.SE..0
RY =HY-!CB/ ;E3-sSYEi
RZ =!-!Z*C?./ !EO-*BZB
\
(Figure 22 Continued)
88
109.000 C NUMBER OF BAYS110.000 DO 80 l-3,3
111. 000 X<1 )-.05
1 i 2.000 X (3) =.05
.1 13.000 V= ( DC*AC/EC/G ) *#0 . 25
114.000 S= ( DB-K-AB/EB/G ) **0 . 25
1 1 5.000 W< 1 >=0.0
1 i. 3 . 000 A3==L
117.000 DO 90 IA =5, 1100,5
(Figure 22 Continued)
89
Y and Z axis. Generally, these values are input according to the
location shown in figure 15. For this case, the values of the section
moduli are the same for all locations about their respective axis.
The torsional moment of inertia, K, is calculated according to Roark
[17] with the following formula.
K ^ -L-2,iL
3c
o i ? ^ i [80]
where a = long side of the bar
b = short side of the bar
10The blade attachment factors, HY,HZ, and HA, are equal to 1. x 10
inch-pounc! oer radian which represents infinity .This means the root
rotation in all directions is zero and the root is supported rigidly.
X(1),X(3), and W(1) are constants which do not change. IA is the
frequency range in cycles per second over which the calculations are
to be performed.
The output of the computer program for the first 10 modes is shown
in figure 23. The first ten mode shapes are plotted in three dimensions
and are shown in figure 21.
The mode shapes are labeled according to the following scheme.
90
RUN
FREQUENCY (CPS) =.13405E 03
AXIAL COORDINATES.23878E 00 .32248E 00 ,33014E 00 .25233E 00
Mim-n -Ammi -Innml \\mmlMAXIMUM DEFLECTION s .31044
LOC AXIAL STRESS
1 tU708E 062 ,10378E 06
I -Mhil 815 .64232E OS
6 ,51395E 05
,38fQ5E OS;2689iE OS
,16884E OS.10342E OS
11 ,48543E 0412 ,54030E 0413 .B8478E 0414 *U?37S OS
n imiiUi17 .1610SE OS18 .1S811E 05
18 :|M8H8S21 .82964E 0422 ,47472E 04
PLOT NUMBER 1 COMPLETED
xl
FREQUENCY (CPS) =.13597E 03
TANGENTIAL COORDINATES,104736 00 .51478E-01 .51478E-01 .10473E00 .1QO00E01
MAXIMUM DEFLECTION = 1,17049
LOC TANG, STRESS
1 ,48648E 06
2 ,39006E 063 .29385E 06
i -Mimit6 ,41386E OS7 .75647E 05
8 ,i559E 069 ,23823E 06
10 3i074E 06
11 .37652E 06
12 .26816E 0613 ,22112E 06
14 >17376E 06
15 ,I2590E 06
1* :llttfttt18 f77S32E OS
19 ,I2590E 06
ii mm a22 .26816E 06
PLOT NUMBER 2 COMPLETED
FREQUENCY (CPS) * ,17S21E 03
nl%iWlthlOQm 00 ,10031E 00 .23025E 00
FIGURE >* OUTPUT OF COMPUTER PROGRAM FOR BAR STRUCTURE. 10 MODFS
91
t537j9K0i .13289E 00 ,133Q7E 00*,99966E 00 -,44240E 00 ,44100E 00
MAXIMUM DEFLECTION =,01369
LOC AXIAL STRESS,S6101E 0449148E 04
mm n26554E 04
- -
04
,54044E01
,iooooe oi
68505772979SIDE 8110695E 04104UE 04943961 0377581E
1921n
0303
636E 03PLOT NuABER 3 COMPLETED
FREQUENCY (CPS) .2715SE 03
AXIAL COORDINATES
.-Aim n -'Anm-n
,99808E 00 -.38303E 00
MAXIMUM DEFLECTION s ,01380
LOC AXjJSl|p6SS
-.38126E 00.mm n,iooooe oi
4
5
6789
I?1213
ll9
20
*45t09E 04.35621E 04,2643lE 04.17737E 04,97677E 03.27746E 03.S4724E 03
'.mm n.29550E 04.2886SE 04,33125E 04;36277 04
,38209E 04
38854E 04,38189E 04I36237E 04
,33067E 04;28792E 04
29621E 04
PLOT NuABER 4 COMPLETED
FREQUENCY (CPS) s ,46632E 03
AXIAL COORDINATES^ taatp nn...aas. rtrt
.15879E 00 .11188E OO *tlH88E 00
:48145t 00 I94148*01 -,9399tE-01
1999281 00 ,864431 OO -I864S4E 00
MAXIMUM DEFLECTION ,00308
.15891E
,481526
,10000E
LOC AXIAL STRESS
fPigure 23 Continued)
5Z
1 ::U!!H813 ,11369E 044 ,7700SE 03
1 ::!3iUE8J.20865E 0339027E 03
it 'Atmiii
li 'Aiimn
15 .21373E 0416 ,1S69E 04
H --a:llSI "8119 .21374E 0420 .23178E 04
21 .23853E 0422 .24110E 04
PLOT NUMBER 5 COMPLETED
FREQUENCY (CPS) =.57138E 03
TANGENTIAL COORDINATES
.13820E 00 -,16351E-01 -,16332E-0i -.13819E 00 .10000E 01
MAXIMUM DEFLECTION a ,05468
LOC TANG. STRESS
1 .S5048E 05
5 -.mm n4 ,34077E 04
5 .10928E 056 ,21218E 05
7 ,26809E 05
8 ,26940E 059 .21226E 05
10 .96620E 04
U ,12976E 05
12 ,U019E OS
13 .80980E 0414 .83692E 04
15 .80276E 04
16 .69407E 04
17 .50912E 04
a -.mm 8t20 .83678E 04
,80962E 04
pl8I nuAb^r21E6completed
FREQUENCY (CPS) a .65840E 03
TAM?f!IHl! 8SR2;?}SS8e 02 -,10983E 02 .19673E 02 .10000E 01
MAXIMUM DEFLECTION a ,00415
LOC TANG, STRESS
1 f3861E 04
5 *2397|l 04
% ,,3 3*0 lb WJ
5 ,14353E 04
6 J22374E 04
7 .26775E 04
8 .27345E 04
9 ,24376E 04
10 ,18678E 04
(Figure 23 Continued)
il.11522E 0733783E 04
ii24261E 04,19503E 04
IE 'AlUll liMUil 82
IE.29732E 04
J19461E 04
il :imn nPLOT NUMBER 7 COMPLETED
FREQUENCY (CPS) a .80697E 03
TA2?IS?ltrl 0R2^SIflE oo ..mrn m ..w,o .iooooeoi
MAXIMUM DEFLECTION a ,02881
LOC TANG. STRESS
T ,44745E 05
2 .26793E 05
I ,94668E 044 59335E 042 H7826E 05
24783E 05
)E 05
E 05i 05
,56744|04
I2356SE 05
H :lsli$fo1
it 'Miinti
18 .?7277E 04
19 ,U061E 05
20 ,1288SE 05
21 ,15485E OS
22 .20486E OS
PLOT NUABER 8 COMPLETED
FREQUENCY (CPS) a .90991E 03
AXiAi8JS6ED00ATE?95417E-03 .93352E-03
.if|0$g00
Milt II -:?I4iJI88 ::! 88 :W8tt 8!
MAXIMUM DEFLECTION a ,00140
LOC AXIAL STRESS
1 ,14104E 04
2 .95455E 03
i
ifSSSlE 8!'Amft.nI35700E 02.36277E 0378792E 03,13145E 04
-:itittE 8J*19119E 04^18929E 04
(Figure 23 Continued)
94
,16966E 04,13146E 04.78808E 03.36255E 03
NUABER 9 COMPLETED
FREQUENCY (CPS) a,98682E 03
TANGENTIAL COORDINATES12745E 01 .13633E 01 ,13660E 01 .12762E 01 ,10000E 01
MAXIMUM DEFLECTION a,01676
3 ,39627E 04
i :!MM1 U6 .21001E 05
7 .19633E 05
1.13336E 05.32948E 04
10 ,87558E 04
lA20950E 05.16884E 05
Uv1414t>E 05
.10842E 05
i,93243 04,68Q23E 04,29370E 04
*s 67646E 04
I,92941E 04
.mm 8i22 16959E 05
PLOT NUMBER 10 COMPLETED
?STOP* 0
(Figure 23 Continued)
95
811 511 Sil RIT RII HI KIT RIT R* RIT RIT RIT RIT RIT RIT RIT R1T RIRIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIT RIi
PARTITION NUMBER00*09*33
3
TOTAL CPU TIME 1,5774
vmm nmiMiW*":!1K
USER EXECUTION TIMEUSER SERVICE TIME
,5197
,0258
tm\ 'ttBst8SHoMH IUSER PAGES 5
COREI PEAK CORE(PAGES)
i/o,EihflHlliH
65
93.26^CALS 3660
TOTAL JOB COST)
FILE SPACE
PEAK DISK TEMPORARY 120
uftisEiLoP1itiA&iR,,,'ENI
mRESOURCES ALLOCATED
CQa 64(PAGES)
ACCOUNT STATUS
BUDGET APPROVED = $I N F O R
593^2SIGMA 9 BATCH = $SIGMA 9 TIMESHARING a $ 397,79CREDITS a $TOTAL EXPENDITURES a $
.00
991,01BALANCE REMAINING a $ -991,01
RATE
i 108,000
in Ml#**
***
**
NONENONE
:881.030
NONE
88!NONE
/HOUR***
***
***
**
tfftgg/PAGb***
COST
2,84
:88
:Ji.15
,00
l:H,00
5,84
M A T I 0 N
(Figure 23 Continued)
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106
T.. = tangential
A.. = axial
Rj= = torsional or rotational
i = number of nodes in the blade
j = number of nodes in the blade group cover
0 thru N -1
N = number of blades in the blade group
The tangential, axial, and torsional mode shapes occur in groups of
N. This is shown in figure 5 and figure 6. In the tangential direc
tion, the first cantilever mode and the (N-1) fixed-supported modes
form the first group .
Looking at the output of the computer program for the first mode,
the first line is the natural frequency. The second line of
printout tells if the vibration is axial or tangential. The axial
coordinates are listed in the following format.
V* 3i""
3N
y0 ri N
107
7 7 7
0 i CH
Figure 17 illustrates the coordinate system. The tangential
coordinates are listed in this format.
a0 aj aN
The maximum deflection is calculated by forced vibrations. The
last printout is the maximum dynamic bending stress for a particular
station or location on the blades or covers. For this example,
the blades and covers are divided into 10 sections giving 11 stations
per blade and cover and a total of 22 stations. Figure 15 demonstrates
the stations. Station 11 is at the tip of the blade. For the first
mode, the tip of the blade has a maximum dynamic bending stress of
4850 pounds per square inch. The computer looks at the stress at
the tip of all the blades to determine the maximum stress.
Table 1 compares the calculated tangential natural frequencies of the
author's method to the calculated and measured tangential natural
frequencies of Rieger and McCallion [8]. Only the tangential natural
frequencies are compared because Rieger and McCallion did not cal
culate axial natural frequencies. The agreement of the results is
excellent. The percent differences between the two calculated
frequencies range from 0.7 to 0.9. The percent differences between
the measured frequencies range from 0.1 to 1.2.
108
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! C COOFLtU VXtiRAIiUwS Aul) STKLSbBb
3. C
4. OIMENSIufti ZC5o,50J,w(3J,Xl4J,wAl3),AAt3J,= . i ZA(50,bu) ,CUl5u) , Z,b C 30 , bO ) , tftiLiA U65) , XtfLAC-265) ,b. A bLA I2bb) , XbLUbbJ . l bLl^bb) , XiiUis ( 1 2 J , Xi*iUMl,l2) ,/. i Z c. I- 1 2 b b J , \MuC/.hb j , 4>:0(.2obJ , ^iuUbb ) .SfcXZFHll) ,8. 1ZOmU5) r6i'K'/,C'/'.b) , Aw(.iC^obJ ,oELSbtl U ,bTKXC12) .iJ'i'Ki C A 2 j ,9. 1 XbLb ( 2 bb J , ialtiUa-o) , ZbLbUbb) , AnGb ( 2bb) ,
CURrJ ( bO ,) ,
19- iStCZ^^J ,SKCXCc2*l,XbLAA^2b5J,ibL<AA(2bb)r , A c
U. i- UlMt.JiiurJfa AHu ,i(\l LbS, IiMCHfcS, oKCUftDS, RALUAwS
12. C GHAVI'I A'ilO'MALCUHbTAMl'
H. c;=.3ttb. iciy14. fc>l = J,14iby2b!>
15. C i^UribKhur-
^'JZ^Lt.S e\:.H 3bO uhUKKtb
16. U.x = 15b.17,- C iMUMbfcV. UK HLAUKS t^brt jhu DKURt-bfa
i.8. Ub=^1.19. c buij/Hi rii.ij.c L>bCi<Kiih>rr
20. UKC=.U2
21. C SlLiiJijUii Y-OiRcC'J'lON
22.,
Si i/,V =.i
23. C SJIi'iUL'ib /.-i).l.K'fcXi:i.in\
24. t>iJi-iZ=.l
^1C SHfW" A-OlKfcCi-IUK
27. C AnPun Out. OF IJRIVifJt, FOKCii - bt./HbO t-UlRKCTiOw
'28. AwY=238,
29. C AhPljUiii>t Ur uRiVliJG FOhCb - uu/bLu Z-DlkfcCTiON
JO. A.'iZ = 3b.,
,.
31. C AnFjjJ'ii.nJHJ OK UHiVltoc, FOKCb - bb/ubl,' A-UlKhdiUiM
32. A(.A =0.
33. C RU1AI J.UnAL bPKKU - KAO/bMJ
34. KUTS =M).*2.*P1
35. C C". KO'iAilu.J OF bLAUcS - i'OP Vlbw
36,RU'l'
= -ft2.b/lt,0.*r'I
37. CO =LUu(-'Mi. j
3. i>i =6lu0.uj )
39. C
40. C BijAL'K PMKAMbTbHo
'il. CSfcUlFlL'
f.bKJHL
'VI. ba=0.2 33
43. C AkKa
44, At> =A.725
4b. C muDUbuS OF tlLASl ICll'V
tb.c..-* = 3U . c+0 n
i7. C Sutw.R huUuLUS Of KbAbi JC1.U
<*8, isn.^nsl 1 . K+Ob
49. C i.i.KK'i LAAoujl.'
'/'-AXIS
50, fstf=,bc5ol
ol. C l^bHTiA AyOul Z'-AXlti
52. 4ZB=.3 2u7
i>3. C SoClMu.'i fiuUULUS AbOUT Z-AXlb
*:>4, vSr..Cz,b(.l)=. 3ob4
55. a,j bb l-2,ly
56. bb c.hCZ.H(l j = . .1931
i>7. SLCZHtil) = ,u/vj4
58 ^ C Si^C'l'lu-'i "ii.ii;uLUS AaUUT Y-AXlo
59. 5eX"xbUJ = .10<Jb
oO. Uu U * 1 = ^ # 10
6 1. 112 SbCi'uU ) = .iloib2. .ShCxbl 1. i ) = .ubo2
,
yj, C 'I. Urffclu-'.Al.- l-HJilFi'l'l UP litK'UA
o4.I'ULr-tJb
t>5. C luM-'y fi"!
b^; c Tt^'slu";'u wbluHi fiu,-ih,iJT'li- Wert iLn
68 1 XL.r. =^Obr^Cb/lt'. *Db ,
59; c S'it-.HM rtu^-fciMX AT THt RUOT UF 'l'hfi. dLAUt Z-AKl.b
/o!
'
Si-iiiZ= A;i i*c'b/2.
7 1 1 C
72. C CJVe:<< FAKAf-li-.lLH^
73. C 5jJKc.it it rth'lGrii
74. uc=0.2bj
75. C AkKa
76. AU = 1 .u2.
.,..
77, c i-;uL;liLjuS UK tLASi ILIA 1
^U = J'^ . ''^ + Ub
^ Snt'/K .-itiD JJUufjOr'
buAS'i ic. .1 j, 1
ai)'.S:iro=i 1 ,C+i)b
CURE 27: PARAMETER SECTION OF COMPUTER PROGRAM FOR CAMBERED BLADE GROUP
112
1. C livifclKTiA AoUJT A-AAlo
82. saC=.30ci/
83, C lnbliTl j A Ar.OJT /.,-AXlS
4. Bi.C=.0 3b'D
85. C SfcC'i'JUi-' KuiJJLUSAbULil'
Z-AA.lt>
8b. SFCZCU 2) = ,0b94j 7. uu 129 1 = 1 3. 21Hd, U9 J>cCiCU) = .0b94
89. ScC.ZCl2/,i = .0b94
90. C bcCl'lOM i-'iuDuljUb AbOor /-AaIS
91. SfcCXCi U) = ./bl4
92, L'U iiU l =13,/l
93. 130 ScCXCUJ = .2bl4
94. bfcCXCl22?=.2bl4,
9b. C IuK^UmhLi '"l.ji-it.iV
i'
Or Inert') 1 A
9b. P.jr.,C=. u /4/.
^7. C l.jt.r-'UTd
98, 00=2.^*2
99, C TuR6Uii-'"jwc.ll.Hi'
:<iUwtiNT Ulr i n r rt T 1 A
loo. ,ruc=pubt*cc/i.o.*i)C
i u i . c
102. t KuOl ATI ACiK'ifc.wT F'aClUKS - AoUU L Tt-ib AXlb
lo3, tU=b.co
104, H^, =b.r, /
105. h.L=b.0b4
10b. rt Y = .i t*Ct/ ctb*oj.'B j
107. KZ =hZ*i:b/vE*bZb)
lob. C
juy. C mi-ibKK UF 'Ltib HAhl-iuwlC
ui.~ "",
112.
113. C
114, ...-
US. X ( 1 j = . u b
lib. V = U>C":*AC/rC/G)**0.2b
14,7. S=(uB*Ab/r_d/GJ**0.2b
118. w (,1) =o.o
1)9. a J = ij
1 20 1 >.u 90 lA=/bU,7oii,lu
ia, 5 *
122. C
123.U4. C ih-itcA
1 ii5 . *,( 2 J =U*b, 28 3185 398
l^b. C ijA.;bUh uK CUuKrt
U7. >AC=.'*'-'K2j**0,5/bA(>*U./Jb
1^8. Y iii:= V * vJ U J * *G . b / LUC * * u . 2 5
129. luhC = *-'(2 J*Sb
13U." "
131. .- -
.
132, It ia(2J*aU J Jo4,*y,9
!W,K=i .
SA=u^*;>rti.C/JL'.HRr./UCJ**0.b
bi>=lUc,/^HKC/GJ **U.5
'4Uf-c>KK HP hAYb
i)i.t r>0 i.i =5, b
X ( 1 j = . u 5;iL'f:jfAC/rC/G)**0.'2t
;(uB*Ab/r_d/GJ**0.2^
H'( 1 )=0.0
i 90 lA=/bU, 7ou, lc
fuH.iA'l (17,4t,l /.8)CiCubS l- Krt oKCO^i;
vJ= 1m
133. b
134,13b,13b. , . . .
137. i)u oi i-i= l,b
C A - u u
X(.2j=uK.tKCZ, T..O
iJ A C 2 ) =l"
t 2 Jv /' ( J. ) = -f I 1,1
K w ( 2 ) = a (. 2 J
Xrt( U = X. c 1 ).li 1.-1 u.
(Figure 27 - Continued)
113
the same as for the tangential and axial vibrations will not be
discussed again in this case. The logarithmic decrement has
been selected to be .02. "Clockwise rotation of the blade
from the topview"
is the amount of clockwise rotation of the
principal axes of the blade with respect to the rotor axes. The
blade airfoil areas and moments of inertia are calculated by a
numerical integration technique. The torsional moment of inertia
is calculated using a formula 15 from Roark [17].
The blade support stiffness factors; HY, HZ, and HT, are tuned
using experimental data. The frequencies of the three basic modes
of vibration; tangential, axial, and torsional, are determined by
measuring the actual blade group. The root attachment factors are
adjusted to tune the computer program to match the frequencies of
these three basic modes. This allows the computer model to be
adjusted to match the actual blade group.
The output of the computer program for the first ten modes of the
coupled blade group is presented in figure 28. These ten mode
shapes are plotted in three dimensions in figure 29 .Modes6 through
10 have two types of mode shapes in their plots because of the coupled
vibrations.
The system coordinates of the computer output are listed in the
following format:
I It
RUN
1 ,3lo21E 04 . 1 3 b 2 4 c.
2 .49634E 04 ,3^9lbfc
3 ,40bOOE 0<* .2932bb
4 .31670E 04 ,25814b
5 .22952E
,14577b
04 ,22422b
.19199fcb 0 4
7 ,1356K 03 .lblSbb
8 .24104E 03 .134b9b
. 9 .72378E
.12529E
03 .1U55E
10 04 ,93bblE
11 .45802E 04 .44049fc
12 .36097E 04 ,55250b
13 .309406; 04 ,613Bbb
14 ,2377bb 04 ,6 7203c,
15 .16b05E 04 ,72073b
16 ,94^b3 03 ,74993fc
17 .22408E 03 ,75963c
18 .94296E 03 .74982E
19 .lbbl2E 04 , 72051b
20 .237 86E 04 .67171b
21 .30952E 04 .60828b
22 .38112b 04 ,54636b
PLOT NUMBER 3i COMPLEX ED
FREQUENCY (CPS) =,7bl00E 03
SYSTEM COORDINATES.44694E-01 .30bb9E-01 .32b08b-01 ,i2blOE-01 .30b79h-01
.44713E-01
121017c! 00 .22007E 00 .2il/9fco< .'/3189E 00 ,22033b
00
.21053E UO
.33638E-01 .25707E-01 .953j8c-02-. 92522E-0/'. -.2b513fc-01
~
-.33501E-01
.99B41E 00 ,1092b 01 .11427b 01 ,ll43l 01 .10903E 01
.10000E 01
MAXIMUM DEFLECTION =.00341
LOC STKESS-TANb'. SIRbSS-AXlAL
05
04
04
04
0404
04
0 4
04
03
04
03
03
0 30 3
03
0 3
0303
03
03
0 3
FREQUENCY (CPS) = ,88bl5E 03
S^!fS2SlE^{'A^tob93E-0, -.18347fc-02 .l29,b-02 .b0blE-u2
.10275E-01
-lioilyb'So -.13291E 00 -.4b037fc-0l U599bE-01 .13288b 00
U1589E 00 .13175E 00 ,14710b 00 .147UE Ou .13176b 00
-liiSolb 01 -,b3b0/t 00 -,2<!02bb UO ,220l4b 00 .63594b 00
.lOOuOb 01
MAXIMUM DEFLECTION = .U0133
LOC STRESS-TAMG, STRESS-AXIAL
1 ,95852b 03 ,67293b 04
2 ,15456b 0 .15748b 04
3 .12/88K Oh .134/3E 04
4 ,101b5E 04 .11250b04
5 ,7b33bb 03 ,91092b 03
6 52512b 03 .pMh ){\
8 ,12080b 03 ,35137b 03
9 ,30291b Oi ,20441b 03
10,13688b-Oi ,10247b 03
11 :b2441b 03 .2o02b 03
12 .36392K Oi ,3923/b 03
13 .298H^E Oi ,37912b ui
14 ,2336*b 03 .359b7b y3
15 .16833E 03 .33415b 03
16 .1029UE 03 .30327b 03
17 ,3734bE Ol ,292fcbc, 03
F|CU"P 58r OUTPUT OF COMPUTER PROGRAM FOR CAMBERED BLADE GROUP
115
18 .10284E
,16&22E
03 ,30285b 03
19 03 .33343b 03
20 ,2334bE Oj .35857b 03
21 ,29861b 0 3 . 3 7 7 2 c, 03
22 .36366E 03 . 39078t 03
PLOT NUMBER 4I COMPLETE!,
FREQUENCY (CPS) =. 10b82b 04
SYSTEM COORDINATES
-.92451E 00 -,b2159E 00 -,64914b 00 -.6491BE 00 -.62160b 00
-.92452E 00
-.67809E 01.42872E 00 ,67l8bE 00 .70665E 00 .70663E Oo .67185E 00.42868E 00
.12978E 00 .10540E 00 .4u99uE-01 -.4101-iE-Ol -.10541E 00
-.12980E 00
.99994E 00 .I359lh 01 ,15b46b Ul ,15a46E 01 ,13592b 01
.10000E 01
MAXIMUM DEFLEO10.M = .00104
LOC STRESS-TAWG, S1RESS-AX1AL
1 ,11667b 04 ,66974b 04
2 .17179E 04 .14702b 0 4
1.12306E
,77325b %%.11477b
.83255b
0 40 3
5 -40036b 03 ,56509b 03
6 .15691E 03 .32892E 03
.
.56750E 03 .U142E 03
,937 7 2E Oi ,31098b 0 3
9 12599E 04 ,52555b 0 3
10 ,152bOE 04 .70644E 03
11 Ami0404
46714b
.13643b
04
03
13 .32927E 04 ,14227b 03
14 .25349E 0% ,15211b 03
15 .17 754E 04 ,15916b 03
1? mm US Aim U18 ,10141b 04 ,16347b 03
19 ,17754b 04 .15928b 03
20 .25349E 04 ,15229b 421 ,32927b 0 4 ,14251b 03
22 .40493E 04 ,13674b 03
PLOT 1MUMbEK 3 CO /ilr'LETEU
FREQUENCY (CPS) " 1462E 04
SYSTEM COOkDlWATEb
-.55581E-02 ,B390yb-03 ,10543b-O2 ,i0b42E-02 .83893fc-03
-.55577E-02
-.99608b-02
,17899b 00 90 7 5/E-02"-.11^ 52b 0 0 -
. 1 1 2 5 0 fc 00 .9Ub7b-u2
,17904b 00
-, 336 3 4b 00 -,^9*49E 00 -
.1243bE 00 .12<*41E 00 .299blfc 00
. 33634E Oo
.99976E 00 ,/i9262b-01
-6 32b4b oo -.63275b 00 .29477E-01
.1000OE 01
MAXIMUM1 DEFLECT 10a ;= .00049
LOC STREkS-'IAi i'-i G , STRESS-AXXAb
1 ,*5l97E 03 ,26968b04
2 ,o42bE 03 ,59252b03
3 ,51519b 03 ,46452b 03
4 ,35189b03 .34235b 0 3
5
6
,2000iE
.bb3bob
0 301
.22935b
,l*!906fc
03
0 3
7 .41563b 01 .45168b02
g ,11605k. 03 ,18562b0 2
J.14951b
,13469b
0 303
.58342b
.7u391b
0202
11 ,17784b 03 ,2b02BE 03
sXFiaure 28 - Continued)
116
.53001E 02 ,14699b 04,45967E 02 ,14480b 04
,38881b 0 2 .14752b 04
.31710b 0 2 .14948b 0 4,24438b 04. .lb066E
.15105E
04,17064F 0^ 04.24436E 02 .15066E 04
.31707E 02 .14947E 04
.3B878E 02 .14751b 04,45964b Oi ,14480b 0 4,62998b 02 .14700b 04
NUMBER <Sk COMt-LP.lED
121314
15
If18
19
202122
PLOT
FREQUENCY (CPS) =.2H&72E 04
SYSTEM COORDINATES'i^$fc 0Q, -./Ub93b-01 -.5O810E-01 .50613E-01 .7U892E-01
-, 11407b 0 0-.88109E-06
-.12933E-01 .28707E-01 .3o449b-Ol -.30450E-01 -.2M706E-01
12936E-01
?fli?fr W, .35394E 00 -.395b5b 00 -.39564E 00 ,3b395b 00/ol ibb 00
-.99993E 00 .B4434F. Oo .724H6E 00 -./2489E 00 -.84432E 00. 1O000E 01
MAXIMUM DEFLECTION =.00020
LOC STRESS-TANG. S j. KbSS-AXlAL1 .50279E 03 .14065b 04
2 .64172E 03 ,24242b 033 .33941E 03 ,11293b 034 .58876E 02 ,11944b u2
i -Atim a -.mat n7 .46479E 03 ,23413b 038 .477B6E Oi ,24901c 039 ,39337b 03 ,22465k 03
10 ,2096^E 03 ,l56b9b 0311 .29304E Oi ,33939b 03
12 ,39035b 03 ,lb923b 04
13 .31885E 03 ,19096b 04
14 .26824E 03 .19121E 04
15 ,25109b Oi ,lb985b o4
16 ,23025b 03 .lb6blb o4
20535E 03 ,18207b 04
23026b 03 ,lb6blb 04
u nam u :isH8g n21 , 31885b 03 .1909
22 .39036E 03 .1892
PLOT NUMriER s CUFFLbTEu
FREQUENCY (CPS) = .3964/E 04
SYSTEM COORDINATbi,.,
-.23325E 01 .26u/7E 01 -,13133b 01 -.A3i34b 01 .26076E 01
.23325E 01
oo
__.,16E 01 ,24927b 01 -,12969b ul -. 12970b 01 ,24926b 01
-.20/46E 01
-,24003b 01 .46B90E 00 ,16511b ol -.lbbllb 01 -,46895b 00
.24002E Ol
,99999b 00 -.29985E 01 ,2o231E 01 ./G233E 01 -,299d4t ul
,10000b 01
MAXIMUM UbKLEC'ilUN = .00046
LOC STkESS-T^iK-.. STKbSo-AxlAL
1 .3038ob 0<* ,2l040fc 04
2 ,34310b Ot ,25471b 03
3 :il952E 04 :39800E 02
4 .80284b 03 ,30327b 03
5 ,23839b 04 ,5l9H9t 03
jFigure 28 - Continued)
117
04 ,67508b 03
04 ,75641b 03
04 ,75425b 03
04 ,6b2C5b 0303 .47639E 0 304 ,10812b 04
04 ,39212b 04
04 .36659E 04
0 4 .33791k. 0404 ,3o616b 0404 ,2713h 0404 .27351b 04
04 .2 /183b 04
04 ,30b 16b:33791b
0404 04
04 .36659b 04
0 4 .39212b 04
6 .33820E
7 .36865E
9 ..SlbbV.b
.69670E
11 .41397E
12 .26937E
13 .22350E
14 .21429E
15 .20083E
16 ,2004bE
17 .19826E
18 .20049E
19 .20082E
20 .21428E
21 .22349E
22 ,2693E
PLOT NUMBER 6 COMPLEX Kb
FREQUENCY (CPS) s . 41616b 04
SYSTEM COORDINATES-.29042E 01 .40316E 01 -.59213b 01 .59193E 01 -,40326b. 01,29061b 01
-.10986E-03
-.25210E 01 ,3588oE 01 -,5ol68b 01 .56152E 01 -,35899b 015 H 9 'J Q
fr'
0 1
I44801E 00 UOiObE 01 -,467b9b 00 -.46660E 00 ,10313c Ol
-.44878E 00
-.99956E 00 -,/4/0bE 00 .192B5E 01 -,!9i0bE 01 .74823E 00
,10000b 01
MAXIMUM DEFLECTION =.OOOob
LOC STRESS-TANG. STRESS-AXIAL1 .37654E 03 .990UE 02
2 .41600E 03 .10926E 0*2
i :iim i -.mm a5 ,320lbE Oi ,28007c. 02
6 .44048b: 03 ,36713b 02
7 ,47044E Oi ,39686b 02
8 ,40799b 03 ,39489fc 029 ,26292b 03 ,34865b 02
10 .55313E 02 ,25738t 02
H -.urn 81 :iUill a13 .26967E 03 .21190b 03
14 .27537E oj .17939k. 03
15 ,27434k. u3 .I4555t 03
16 .28591E 03 .U052fc 03
17 .29172E 03 ,74487b 02
18 ,28586b 03 .11052b 03
19 .27444k". 03 ,14553b 03
20 ,27550b Oi ,17935b 03
21 .26985E 03 .21184b 03
22 . 260 39E 03 ,24298b 03
PLOT NUMBER 7 COMPLETED
FREQUENCY (CPS) =.45749E 04
Y-198271bR00N
-Uo5b 00 ,2ubl6b Oo -,2073/t 00 ,108o3b 00
i 98474b 00
-i96174E~00 -.22716E 00 .lblbbb 00 -.lWil9E Oo .227J4E 00
;lo523b 00 .232UE 00 -,B26ilE-Ul -.b2628b-0l ,232b6b 00
-i99807hi 00 -.22009F-02 ,15474b 00 -.lbi>6ib 00 .l9644b-02
UOOOOE 01
MAXIMUM DEFLECTION = .0001/
(Figure 28 - Continued)
118
LOC STRESS-TANG. STRESS-AXIAL1 .12442E 04 .97813b 932 .13126E 04 .11233b 033 .31525E 03 ,3b7b5b 02
?/iifc 0i .14986b 035 12163E 04 .?4251E 036 .1860E 04 .29741E 037 .16301E 04 .30701E 038 .13595E 04 ,26535b 039 .83508E 03 .19316E 03
10 .15929E 03 ,1962E 0311 .14813E 04 ,11181b 0412 .18467E 04 .46978b 0313 .16281E 04 ,45319b 03
14 .13935E 04 .43545E 03
15 .113292 04 .41558E 0316 .84243E 03 .392b3E 0317 .52394E 03 :3b725b 03
l ?M,?3&Pi .39351E 03
19 .U353E 04 :41634E 03
20 ,13963b 04 .43626b 0321 ,16314b 04 .45405E 0322 .18503E
04*.47069E 03
PLOT NUMBER a COMPLETED
FREQUENCY (CPS) = -45944b 04
SYSTEM COORD IWAXES
, 96560c. 00 .23157E-01 -,634b2E 00 -.b4396E 00 .28580E-01
.1012 3 E 01
,65554b-01
.93451E 00 ,/b499E-01 -,77637b oO -.7642/E Og ,b/672b-Ol
,980/OE 00
-.4J453E 00 -,^707UE 00 -.454b4E-0l ,4l/14b-01 ,2bl84E 00
.43 3 59b 00
I95217E 00 -,92644E-01 -. 47230b 00 -,479b4E 00 -.91838E-01
.10000b 01
MAXIMUM DEFLECTIOh =.00055
LOC STRESS-TAWc. STKtSo-AXlAL1 .40800b 04 .30079E 04
2 .42952E 04 .34179E 03
3 .10166E 04 ,10937b 03
4 ,18516E 04 .47080E 03
5 ,40illb 04 ,75799b 03
6 .52190E 04 ,92832b 03
7 .63546E 04 .95854E oi
8 .44576E 04 .83047b 03
J illlffl ii Miti 8111 ,73658b 04 ,60510b 04
12 .59741E
04'.12742b u4
13 ,49517b 04 ,12296b 04
14 .43081E 04 ,U745t 04
15 ,35793b 04 .U048E 04
16 .27507E 04 .10170E 04
17 .19025E 04 ,96406b 03
18 ,28799b 04 ,10/b/b 04
19 .37533E 04 ,11678b 04
20 -45223b 04 ,12404b 04
21 ,52021b 04 . 1290b 04
22 ,59919b 04 . 13450E 04
PLOT NijftbER 9 COMPLEX ED
FREQUENCY(CPS)'
=-47029E 04
SYSTEM COORDINATES
--70424E 00 .b744bE Ol .36869b Oi. -,i694yb Ol -.5/b31b 01
."70051E 00
-.32442E-02
.21058E 00 .05524E 01 ,420o3E 01 -.421bb 01 -.65697E 01
-,2lo.->5b uO
,32760b 01 ,9695ob 00 -.22023b 01 -,/!20bOb 01 ,9o8oWfc 00
119
.32872E 01
"*.1^00E 8Jbb6b4E 0i .35121b 01 -.35141E 01 -.5S828E 01
MAXIMUM DEFLECTION s,0004b
LOC STRESS-TANG. STRbSS-AXlAL1 .37497E 04 .14571E 0 4
I ,3928E
.90320E
P4 ,20/43E 0303 . 12938c 03
4 .17280E 04 ,3148 4b 0 3
5 .36794E 04 -46704b 036 .47195E 04 .b//14b 03
I .47339E
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04 ,63897b 0304 ,64976b 03
9 .19876E 04 ,61045b 0310 27602E
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H.56030E 04 ,12214b 04,46711b 04 ,13893b 04
14 ,37031b 04 ,15103b 0 4
15 .26820E 04 ,15832b 04
16 .21008E 04 .16076b 0 4
17 .17543E 04 ,1586bb 04
18 .20982E 04 .16U2E 04
19 ,2b860E 04 .15878b 0 4
20 ,37090b 04 .15158b 04
21 .46788E 04 ,13955b 04
>ihi nuA&B2^04
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FREQUENCY (CPS) = ,51034b 04
SYSTEM COORD 1* AT bs-.43252E 01 -.38544E 02 .13710b 01 .l3bl/E 01 -.38532E 02
-.43240E 01
-.15125E 02
".14266E 02 -,38b44b 02 -. 10068b 0/i -,10o59 02 -.38633E 02-.14263E 02
-.40394E 02 . /72b2E 01 .22963b u2 -.229b4b 02 -.77307b ul
.40385E 02,?9718E 00.10000E 01
-.d2o22E 0<! .lolbVb 02 .J0204K 02 -.52601E 02
MAXIMUM DEFLbCTXO;J =.00003
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,2bl6E 03
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,31640b 02
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iPLblED
FREQUENCE (CPS) =,5 1763c 04
SYSTEh COOkDJNAlbo
( Figure 28 Continued)
LOC STRESS-TANu.1 . 2H28 5E Oi
2 .28437E Oi
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PLC)T NUMbEK XX CO
120
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MAXIMUM DEFLECT LO.\i =.00010
LOC STRESS-TANG. STHbSS-AXlALh 4^49fe ^ .U726E 042 .10620E 04 ,67701b 02
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SYSTEM COORDluATbS
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MAXIMUM DEFLECTION = ,000 0 0
LOC STKESS-TANG. STRESS-AXIAL
1 .363UE 01 . 13060b 0 2
2 .35131E Ot .83665E 0 0
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\l Mm 81 .mm u19 .44302b 01 ,17588b
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21 .51212E 01 .1807 9b 0 2
22 ,52923E 01 .18195b 0 2
PLOT NUMBER 13i CO^HLbTEi;
(Fiqure 28 - Continued)
121
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A comparison of the natural frequencies calculated by Stress
Technology Incorporated, Westinghouse, and the author are
listed in table 2. Stress Technology Incorporated and Westinghouse
applied a finite element method to calculate the natural frequencies.
The percentage differences range from 0 to 12 percent. The
correlation of the calculated natural frequencies are excellent
considering the calculations were performed independently by
three different people using two different concepts of analysis and
three different models on a very complicated blade group.
The variable which accounts for the largest part of the differences
is the root modeling. The blanks in table 2 indicate that the
data was not calculated.
Table 3 compares the resonance bending stresses of four modes.
The calculations are performed by Stress Technology Incorporated
133
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135
and the author. As an overview, the resonance dynamic stresses
are in the same ranges for the two different methods. This is
especially true when a person considers the variables of damping,
exciting forces, root modeling, and blade modeling. The stresses
of two of the modes agree closely, but two of the modes do not.
When the author's stresses are prorated to match the maximum dis
placement of Stress Technology Incorporated, the percentage
differences between three of the modes range from 2 to 13 percent.
This demonstrates that the modeling of the blade group by the
author and Stress Technology Incorporated is in close agreement.
The differences in the stresses is mainly related to the modeling
of the forced vibrations. A large disagreement exists for the
stress of mode 8. The mode shape from the calculations of Stress
Technology Incorporated was unexplainable.
136
V. DISCUSSION
The problem of forced outages in turbomachinery has been addressed
from the standpoint of fatigue failure due to three dimensional resonant
vibrations and stresses. The method described in this thesis is simple,
easy to apply, inexpensive, and gives accurate results. An average
cost to apply the procedure is about eighteen hours of labor and $150
of computer time for the vibrations and stresses of a blade group with
chambered blades. This excludes any experimentation time and cost.
The output of the computer programs gives the fatigue bending stress
level for a given blade group exposed to a given excitation force.
The computer programs are capable of handling uncoupled or coupled
vibrations in intermediate and high pressure turbine blade groups.
The programs may be used for new designs or for failure analysis.
If the stress is above an acceptable limit for a given blade group, one
of the following solutions may be applied to reduce the stress:
1. Increase the cross section of the blades or covers at the
location of high stress.
2. The number of blades or nozzles may be changed to reduce
the blade groups acceptance of excitation energy.
3. The length or geometry of the blades or covers may be
changed to cause a high stress mode of vibration to occur at a
137
different frequency where the stress is lower.
1. Stress concentration factors may be reduced.
5. Tie wires may be incorporated to reduce the stress in the
fixed supported tangential modes or the fixed supported
tangential coupled modes.
Some of the above solutions are similar to the methods used in the de
tuning of blade groups. Unlike the detuning method, the above
solutions are based on fatigue bending stress instead of avoiding the
coincidence of the natural frequency with the frequency of the exciting
force .
Many variables affect the accuracy of the results from the procedure
described in this thesis sir from any procedure. The variables which
have the most influence on the accuracy of the calculations for a typical
blade group are:
o Exciting force
o Material properties
o Damping
o Root modeling
o Centrifugal loading
o Blade geometry
138
Naturally, the accuracy of any method is only as good as the input.
For the variables of exciting force, material properties, and damping,
the values used in the analysis are approximated. For some cases
experimental data can be obtained to improve the accuracy of the
approximations. Root modeling is very complicated. The effect of
the root is included in a root stiffness factor. This root stiffness factor
is varied to correlate the analytical model to the actual blade group
through the use of experimental data on the natural frequencies of the
three basic modes. The three basic modes are the first tangential, first
axial, and the first torsional. Centrifugal loading is usually handled
by increasing the calculated natural frequencies by a small percentage
factor. The centrifugal stress is combined with the steam stress and
vibrational bending stress. These are applied to the Goodman Diagram
to give the complete stress picture. Often, because of the complicated
geometries of the blade aerofoil and root sections, assumptions are made
to simplify these geometries.
A second set of variables which have a lesser effect on the accuracies
of the stress calculations are:
o Manufacturing tolerances
Assembly tolerances
e Stress concentrations
Elevated temperatures
Corrosion
139
Manufacturing and assembly tolerances cause a spread in the occurence
of the natural frequencies and result in frictional damping in the blade
group. Two examples of manufacturing tolerances are the clearances
and the radii in the blade root section. Two examples of assembly
tolerances are the tightness of the swedged tenon and nicks caused
by hammering when assembling the blade to the disk. Stress concen
tration, elevated temperatures, and corrosion are variables which are
handled by approximation factors.
A third set of variables which have the least effect on the accuracy of
the procedure are:
Transient vibrations
Material uniformity
Interacting groups
Disk-blade interaction
External sources of vibrations
Operation of the blade group outside of the design specifications
Surface finish
Arced covers
Partial admission
Torsional constant approximations
Deviation of the theory from the real world
Normally, each of the above variables are relatively unimportant, but
must be considered in special cases.
140
Many assumptions have also been made which affect the accuracy of
this analysis. For the equations of motion for transverse and torsional
vibrations of the blades and covers the following assumptions are made:
Conservative force field
Uniform beam
Linear elastic displacement
No axial loads
Negligible shear
Negligible rotary inertia
Bernoulli-Euler beam theory applies
The above assumptions are valid and have only a small influence on the
outcome of the stress calculations in most cases. The correction factor
for shear and rotary inertia effects for a rectangular beam, where
G=3E/8, and where the wave length is ten times larger that the depth,
is -1.7 percent. See reference [20]. At very high frequencies the
Bernoulli-Euler beam theory is not valid. The blades and covers are
under axial loads. These axial loads cause small changes in the natural
frequencies and mode shapes of the blade group. The changes in the
natural frequencies are handled by adjusting the blade root stiffness
factor.
The following are assumptions made which are directly related to the
blade group:
o Right angles between the blades and covers remain at right
angles.
141
The mode shapes are based on undamped vibrations.
Longitudinal vibrations of the blades and covers are
negligible.
Manufacturing and assembly tolerances play a major role in the first
assumption. Since damping is very small, assumption two is correct.
Unless the vibrations are of a very high frequency, the third assumption
is valid.
The last set of assumptions pertains to the energy method used to
determine the amplitude of the forced vibrations. The assumptions
are:
Constant rotor speed
Identical blades and covers
Harmonic exciting force
Constant amplitude and phase angle of the exciting force along
the length of the blade
The input energy is completely dissipated by damping
Damping is a function of the lagarithmic decrement.
These assumptions have the biggest impact on the outcome of the calcu
lated stresses. The assumptions simplify an area which is normally
complicated and difficult to describe analytically- The assumptions are
112
valid and the parameters related to the assumptions must be input with
experimentation, experience, and common sense.
Two simplifications of the blades have been incorporated which have
a small effect on the accuracy of the results.
1 . The length of the blade is selected to be from the top of the root
to the middle of the tenon .
2. The nonuniform blade is converted into a uniform beam. This
is accomplished by taking an average of the cross sections at the
different stations for the vibrational analysis. For the stress
calculations the actual cross section at the station is used.
From this discussion it is easy to see that the blade group problem is
complicated and has many variables. The errors are additive andsub-
tractive. Due to the law of averages the total error is considerably less
than the sum of the individual errors.
143
VI. CONCLUSIONS
1. A comprehensive analysis has been presented for three dimen
sional resonant vibrations and stresses in intermediate and
high pressure turbine blade groups. This procedure is inexpen
sive and is easy to apply.
The analysis is a fatigue stress approach using simple beam
theory and a dynamic stiffness matrix. This procedure is
capable of handling uncoupled and coupled vibrations of the
blade group.
2. Areas requiring technological development have been identified.
These areas are:
o> Excitation of the blade group vibrations
Damping properties of the blade group
Material fatigue strength
Root modeling
3. Fatigue stresses are compared with fatigue test data on a Goodman
diagram. The relationship between the stress point and the failure
envelope defines the probability of fatigue failure.
144
4. For applications where the assumptions of the Bernoulli-Euler
beam theory are not validated, the calculated natural frequen
cies are very precise. Comparing this procedure's numerical
results with Rieger and McCallions [8] experimental results,
the largest percent difference for the first five tangential
natural frequencies is 1.2 percent.
5. The root stiffness factor is extremely important in correlating
the analytical model to the actual blade group.
6. At higher natural frequencies shear and rotary inertia effects
begin to decrease the accuracy of the described procedure.
7. The time objective of twenty hours to apply this fatigue stress
procedure has been accomplished.
8. This new procedure could be used in the design of new blade
groups, or for the analysis of failures of existing blade groups.
145
VII RECOMMENDATIONS
1. The new design procedure described in this paper should
be used for the design and failure analysis of intermediate
and high pressure turbine blade groups. This procedure
should be used because of the following reasons:
o Fatigue stress approach
Easy to apply in twenty hours
e Inexpensive
Handles uncoupled and coupled vibrations
2. Technological development needs to be initiated in each of
the following areas to improve blade group reliability through
improved design parameters from precise experimental data.
e Excitation of the blade group vibrations
Damping properties of the blade group
e Material fatigue strength
Root modeling
3. This method of analysis needs to be applied to a variety of
blade groups so that confidence in this procedure may be
achieved through experience.
146
4. An accurate record should be kept of the reliability of blade
groups to which this procedure has been applied.
5. This procedure needs to be compared to other methods of
analysis to determine which method is best for a particular
application.
6. This procedure should be expanded to include the capability
of handling exhaust blade groups and blade groups with tie
wires. Also transient vibrations should be included.
147
REFERENCES
1. EEI Data for 1964 - 1973.
2. Rieger, Neville, F. and Nowak, William J., "Analysis
of Fatigue Stresses in Steam Turbine BladeGroups,"
Rochester Institute of Technology, Wehle Research Lab.,
Report 77 WRL Ml, March 1977.
3. Weaver, F.L. and Prohl, M.A., "High Frequency Vibration
of Steam TurbineBuckets,"
Trans, of the A.S.M.E., pp 181-189,
January 1958.
4. Smith, D.M., "Vibration of Turbine Blades inPackets,"
Proc.
of the Seventh Inter. Cong, for Appl. Mech.,Vol. 3, pp 178-192,
1948.
5. Prohl, M.A., "A Method for Calculating Vibration Frequency and
Stress of a Banded Group of TurbineBuckets,"
Trans, of the
A.S.M.E., pp 169-180, January 1958.
6. Ellingon, J. P. and McCallion, H., "The Vibrations of Laced
TurbineBlades," Journal of the Royal Aeronautical Society,
Vol. 61, pp 563-567, August 1957.
148
7. Deak, A.L.and Baird, R.D.,"A Procedure for Calculating the
Packet Frequencies of Steam Turbine Exhaust Blades," J. of
Engr. for Power, pp 324-330, October 1963.
8. Rieger, N.F. and McCallion, H., "The Natural Frequencies of
Portal FramesII,"
Inter. J. of Mech. Sci., Vol. 7, pp 263-267.
1965.
9. Fleeting, R., and Coats, R., "Blade Failure in the H.P. Turbine
the R.M.S. 'Queen Elizabeth2'and Their
Rectification,"The
Inst, of Marines Engr., Advance Copy, October 1969.
10. Tuncel, 0., Bueckner, H.F. and Koplik, B., "An Application
of Diakoptics in the Determination of Turbine Bucket Frequencies
by the Use ofPerturbations,"
J. of Engr. for Ind., pp 1029-1034,
November 1969.
11. Provenzale, G.E. and Skok, M.W., "Cure for Steam-Turbine-
BladeFailure,"
A.S.M.E. Publications, 73-PET-17, 1973.
12. Sohre, J.S., "Discussion of A.S.M.E. Publication73-PET-17,"
September 1973.
13. Rao, J.S., "Application of Hamilton's Principle to Shrouded
TurbineBlades," Congress of I.S.T.A.M., Indian Inst, of Tech.
149
14. Rieger, N.F., "Finite Element Analysis of Turbomachine Blade
Problems", Paper No. 5, pp 93-120, A.S.M.E. Monograph
"Finite Element Applications in Vibration Problems", Editors
M.M. Kamal and J.A. Wolf, Jr.
15. Salama, A.L. and Petyt, M., "Dynamic Response of Packets of
Blades by the Finite Element Method,"A.S.M.E. Publications,
77-DET-70, 1977.
16. Rieger, N.F., "Vibrations ofFrameworks,"
Ph.D. Thesis,
University of Nottingham, 1959.
17. Roark, R.J., Formulas for Stress and Strain, 4th Edition,
McGraw Hill, 1965.
18. Heywood, R.B., Designing Against Fatigue of Materials,
Reinhold Publishing Corp., New York, 1962.
19. Lazan, B.J., Damping of Materials and Members in Structural
Mechanics, Pergamon Press, 1968.
20. Timoshenko, S., Young, D.H., and Weaver, W., Vibration
Problems in Engineering, Fourth Edition, John Wiley S Sons
Incorporated, 1974
148
IX. APPENDIX A
COMPUTER PROGRAM
FOR
TANGENTIAL AND AXIAL RESONANT VIBRATIONS
AND STRESSES
20.
31:23,24.26.2b.
27.
28,2?.
30,
31.32.
li:
35.
3b.
37.
38,39.*40.
41.42,H*44,
45,46,47,48.
49,bOjbl.
52.
53,
36,
B:39.bO.ol .
62,63,t>4.
rcb,
67.&8.
69./O,71.72,73,74.75./6,77.78.79.
80.
C
C
c
149
UNCOUPLED VIBRATIONS ANU SThESSES
CC
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
Sb
c
112
C
C
c
c
c
c
c
c
c
c
c
c
DIMEI ZAC31 ZBLAi Zbli1ZM0MC2
iHAVG=3
R
UMBER6=192
LOGARI
DEC=,USXIMULiSX1MY=
STImULSTlhZ=
SllMUJjSiIftA=
AWPL1T
AMYalO
AMPL1XAMZ=5;AMPLll
AMA=b,KUTAT1R0X5S6
NS1GN Zt37,37j,W(3),Xt4j .wA(31 .XAC3J .
7*l7i#fiOR(3/j.ZB(37.37J;yibJjAC200)>X6L.AC200),
2^O}tXMUC2Ou),XMOUU0>,ZMu(2OO)/SEC*6Ui),6) .STRZC25).ANGU00>.SECY6(lt) .6TKXC123 ;5TR*(l2) ,
Pf;lfeSi^45^I3E^hi6r<iBC26o,'5,sczc^^''
SiONS ARE IN;LBSv ANCHt-S, SECONDS, RADIANS
ftWswmsm
ITAU0NAL-
CONSTANT6 ,3 39 '\
-..or*
lOf Bfcft'DgS PER 360 DECREES
tHMiC DECREMENT2US--- Y-DIRECTION.1
US*-* Z-DIRECTION
04
5fi A-D1RECTIQN.04
QUE DF DRIVING FORCE
UDE DF DRIVING FORCE
UDE OF DRIVING FORCE
ONAL SPEEU - RAD/SEC0,*2.*P1
- LB/bLD
- LB/BLO
- Lb/BLb
i-UlKEC'TlOw
-DlKECTJ.On
A-U1KECT10N
BLAOE PARAMETERS
SPECIFIC WEIGHT
DB=u, 28183
AKEA
AB=, 059057HUDULUS OF ELASTICITY
E6=^8,3E+06-
SttEAR MODULUS Of ELASTICITY
SttRb=ll,2bE+Q6
INEKT1A ABOUT X'-AXIS
BYBS.4804E-3
INERTIA ABOUT Z'-AXIS
BZB=.l /487E-3
I?W?8!MoABO!JI z-4"6
OU 551=2, 10~
Sfc.CZBCD = .001B60
SECZBCUJa, 001860lg'fI?S!)^8B^o4aoul'
V-AXIS
UO 112 1=2,10
EHIHW88SII80TuRSIQNAL MOMENT UF INEKTIA
PuLB=.4310E-3
LENGTH'"~"'CtJ=6'
0"r~"'
""""T" '
TORSIONAL MOMENT OF INERTIA
STEAM^UMEN'^A^TME ROOT OF THE BLADE Z-AXIS
SrtUZ=AM**CB/2,
COVER PARAMETERS
SFECIHC WEIGHT
DC=0, 28183
AREA
AC=. 069057 . ,
MODULUS.OK ELASTICITY
ShEAk'mODULUD OF ELASTICITY
SHKC =U,25Et0b
AwErtTlA ABOUT X-AAlo
B4C=.17487E*3
81.
82.d3,
84,
.85,
86,87,38,
.139,
90.
9lt92i
s:"9b
97,98,99,
100,101,JOT."
103,
104,105,10b,
182:109,
lit:
lii:lib.
116,
118.119.120,
Hi:123,124,12b.
126.
127.
128,129,130,
lsh132.
Ill:135.
136.
137.138,139,140,
129
C
130
C
C
C
C
c
58C
C
C
48
1615
64
14
13
5*126
56
bO
150
SECTION MODULUS ABOUT Z-AXlS
Sfc.CZCU2) = . 0018b
OU 129 1=13.21
SECZCCIJ=. 00186
SEcflO^MoSuLUS^ABOUT X-AXISStCXC 112) =,00308
DO 130 1=13/21-
SECXCCip=. 00308-
SECXC(22), 00308
TORSIONAL MOMENT OF INERTIA
POLC=,4310E"3
;ngth
T^J*|lfoc *CC/ 10 . *DC &, -'-a
ROOT "ATTACHMENT FACTORS
HY=9.9E10
H5S=9,9E10
hl=9.9E10
RY=HY*CB/(EB*BYB)
RZ=HZ*CB/(,EB*BZB)
NUMBER OF THE .HARMONIC
H AR"""
1
SA=CDB*SHRC/SHRB/DCJ**0.5
SB=CDC/SHKC/CJ**0,5
NUMBER OF BAYSIt&AV-3
Wd)=0,0
A3 = L
DO 90 IA=5, 1100,5x
F0RMAT(17;4E17.8)
CxCLES PER SECOND
G=IA
St2)=U*6, 28318530b*xis?sst2Vts:^gxc**o.2b
yzC=vwu)**0.5/Bz.C**o,ab
TGRC=*(2)*SB
CALL U
DO 48 1=1, L+2
DO 4 8 J=l,L+2
m. INERTIA
DO 4 o=aut*
ZBCArUO^ZU/JpXl4)=DlElZB,L+2)
x(2J=LETEtZ,lBJ
iKAascxm'sitV*
X K.I 1=Uc.TEj.<, xdj
CAaSUC2)J-I.0E+2021
WA(2J=Wt2J"
WAU)=W(U
XAC2J=Xt2;
XAU2Sm4 5
WKITEUU8;58) "M'
GO TO"
1 8
15, lb, 16
13,14
SSc3) =i2AC2}*XAaj-WAUHXAC2))/(XAtU-XAC2))
yXC=V*WAC3)**p.6/BXC*J0.2b
W^ifaFfteuUENCY'ttPS) =,EU.b,/JTEUOo,
IMA'i C///, iWRITE IFOKMA.L
lKXM3f*XACl))bO,60,bl
KllEUJ08'58)M,UA,XAC3)AA12)=XAC3J
229,230.
61
62
89
9080
52
127
28
41
33
98
39
29
43
42
34
:A)1-B*SANCTA)-83*C0SH
151
GO TO 62
WACl)=WAC3)H*KliEUo8>58)M,UA,XA(3)XA(1)=XA(3)-
CONTINUE
GO TO 53WU) =W(2)
!XtlJaXt2)
S8N?5NV&rpRMATU5fSE13.5)
*0'^|i
IBL+2
WKITEUQ8.127)FORMATC TANGENTIAL COORDINATES')CALL DE
- '
WRITE (108, 76) (CORU), 1=1, L+2)M=l
'
P....
K= lC=2.0*(RZ*F3ZB+XZ6L*F5ZB)
DO 39 J=l;L+l%
Bj=CRZ*F8ZB/YZB*CORCJ)-RZ*FlOZB*CORtLt2))/C
B=U2:*ZBL*SINH(*ZBL)-RZ*FlOZB)/XZ6*COK(J)
1 +(-2,0*YZBL*COSHC>:ZBL)-RZ*f 7ZB)*CORCLt2))/C
Bl3 =U-2.*YZBL*SlNUZbL)+RZ*Fl0ZB)/XZl3*CORCd)
l+C2,*XZBL*COSCYZBL)+RZ*F7ZB)*COKCL+2))/C'
DO 41 1=0,10" --" ~
UA=A
XbLACH-K)=UA/l0.0*C6
TA=XBLAII+K)*1ZB
TB=tXBLA(I+K)+CB/20,)*YZBYBLA(A+K)=CB3*COSCT
1 CTA)+B13*SINH(TA))YBLBU+M)=(B3*COSCT6)+B*S1NITB)-B3*COSH
ZMUUtK)=EB*BZB*YZB**2,*C-B3*C0SCTA)-b*SINClA)-B3*CoSH
HTA)tBl3*SlNHCTA) J
If (VA-ABSUBLA(H-K)))28,41,41
VA=A6SUBLAUtKJJ" "
IEl+fcCONTINUE
v rt
IF (VBABSCZM0CK)))33,98,98
VBBABSCZM0CK3)IF=K
"
K=K+11
T=T+CCM=M+10
CONTINUE
M=(L"H)*10 + 1
Bl=2,0*XZC*F3ZC
DO 42J=17L"
B2=*F5ZC/Bl*CORCJ)-F8ZC/61*COKCU+l)
Al=-UlZC?F3ZC);Bi*COK(U)+Fl0zC/Bl*COKCU+l)
A2=lFlZC-fr3ZC)/bl*COR(,J)-M0ZC/Bl*CoRU + l)
DO 43 1=0,-10* '
UA = 1T=YZC*CC*UA/10,
TB=T+CC/20.*YZC
XBLACI+K)=(B2*COSCT)+Al*SlNtT)-B2*CoSHlT)+A^*SlNH(T))
XBLB(l+M) = CB2*C0S(TB)+Al*SlW('J;B)-b2*C0SHCTB)^A2
1*SINH CTB) )ZMOCI+K)=EC*BZC*YZC**2,*C-B^*C0S(,i)-Al*SliMCi)
1-B2*C0SH(T)+A2*SHMHI.T);'
lli?CVA-ABStxBLACi+K)n29,43,43VA=ABSCXBLAU+KJ)
' '
1E=A+K
CONTINUE
K=K+11M=M+10
CON'UNUE
DO 34 I=1,(L+1)*11
YBLAC1)=YBLA(I)/VA
CONTINUE
DO lib l = l,tL+l')*10
152
2^1 . lib YBLBU) =YBLBC1)/VA, ^ % t
2^2, DO y xsiLtl)*il+l,C2*L+l)*ll
243, XBLAtl)=XBLACD/VA- -
--
244. 9 CuwjUiMUt
245.__D0
117 I=IL+1)*1Q+1,(2*L+1)*10
24b. 117 XBLBCI)sXBLB(i;/VA
247, r. CALCULATING 'MAXIMUM DEFLECTION
248. ADA=0,249. ,ADB=0,^
250, AE=0,251, K=0
232, AF=2,*PI*WA(3)/R0TS/UbHi:- *8.-8i-ttl{iSK
-
!2b5, :una*i.".''"::
2*6,, Ai5AsSY6LBCI*K)*C0SCUA*AF)+ADA297. :ADBaYBLB(I+l\5*SINCUA*AF;+ADB
258, AE=tYbLfaU+j\))**2,*06*AB*CB/lO.+AE259. 85 CONTINUE
_ . .
2b0, K=K+1U
.201,84 CONTINUE
262. DO 83 I=(L+l)*10+l,C2*Ltl)*lO
203, AE=tXBLBU))**2,*DC*AC*CC/l0.+AE
264, W3 CONTINUt----
205, DMAX=PI*G*ST1MY*AMY/1U.*CADA**2.+ADB**2. )**,5/DEC/
2bb. 1WA(3)**2,/AE. . .
207. '^KIIEUUB, liilJDMAA
268. 121 FORMAi'(/,21HMAXiMUM DEFLECTlOl* =,F/,5,/)
209, DO 86I=1,22*L+11- "
7.'
270. ZMOU)=ZMOU)/VA*DMAX
271, 86 CONTINUE
2 72, RRFY=VB/VA*2.*G/Cb/10,*lADA**2.+ADB**2.)**.5/
273. HfcAC3)**2,/AE. - -
274, Do 91 0=1,11
275, ZMOn(U)=0,2/b*
00 97 I=l!CL+l)*ll,ll
27 7, IF(ZMOMCU)-ABSCZMOCI+U-l)))88,97,97
278, 88ZMOMCd)=ABStZMO(ItJ-l))- ~
279, 97 CONlINUt
280, 91 CONTINUt
281, #0.94 Us12'^2
liil SoUS54=CL+l)*ll + l,22*L +n.ll
284. It (ZMQM(U)-ABSlZMO(H-U-l2j))9b,y5,9b
2B5. 9b ZMOM(U) =ABSIZMOU+J-12)J- - - -
28b, 95 CONTINUE
287, 94 CONTINUE
288. WRITEC108,122)
289, 122 FORMAT ( LOC TANG, STKESS')
290, DO 92 1=1,11
2*i; STRZ(I)=ZMOMCI)/SECZB(I)
wRIIEtl08,59)l,STKZl.I)*
92 CHBT55U4-12 ?2SlRz?i)=zAOMTI)/SECZCCI)
WKriEUU8,59)I,STKZU)*
297. 93 CONTINUE
298. T=0.
299. K=l
300, DO 133 0=1, L+l
301, DO 134 1=0,10
W2 XBLtl+K)=XBLACI+K)303*
YBL(I+K)=T
304, ZBLCI+K)=0,
305. XBLAA(I+K)=XBLA(I+K)
J8J: fgWi{!^)=YBLACI +K)+T308, 134 CONTlwUE
309, K=K+11
3to! T=T+CC
311 133 CONTINUE
312, K=CL+1)*U + 1
3lSl DO 135 0 = 1, b
3li. DO 13b 1=0,10
315, UA=1
{^ci^J=CC*UA/10. + iBL(K-L*ll-lj
(if fSLA/lu +K)=XBLACI+K)tC6
jJo! xaLAAU +K)sCCUA/10.-
+-BLAA(N-L*U-l)
153
321. ZBLAC1+K)=0,322. 13b continue323, K=K+11324, 135 CONTINUE
325.'
I=C2*Ltimi
327: S=YBLAAti)*0,5+XBLAA(0)328, X6LAA(U)=YBLAA(U)*,86&025-ZBLA(0)
329, YBLAA(0)=T'
'
330. T=YBL(U)*0.5+XBLCU)
331, XBLCO)=YBLU)*,86b025-ZBLCJ)332, YBUJJal
iH V-4
2JiE-A3 *c$fT.Q+i , o336.
? IC*LL WINDOW (3 ,9., 9,)&37,... ;>iXaLAAUtl)=*3.. :_;..
338, XBLAA(I+2)=3,339, YBLAACI +U=0.340, YBLAAtI+2)=3.
,,,,,
1 1 4 * -
T^E^Aiiii)il34'2,5'8HYi}u^H{'mB'slzt>*''*BuhAU+1)'
343^ CALL AX1S(2,,3.,8HX (INCH) ,8 ,T,90, , YBLAAU + 1 ) ,
344. lYbLAAUt2))'
. , ,,
345, "CALL AXIS(3.,3,,8HZ (INCH) , 8 , 1 , , 180, , YBLAAU + 1 ) ,34b. lYoLAAU + 2))
- - - - -
347. CALu PLUT(2.,3.,-3)
348. CALL NEWPEN13)
-388:1iM!HJiflM'I'1'-!'U)
3bl. XbLUt2)=XBLAA(H"2$
352. YSL(I+1)=YBLAA(I+1)353. YbL(I+2)=YBLAA(I+2)
354, CALL NEWPENU)
355. CALL LINE(XBL,YBL,1,1,-1,3)
35b. CALL UNPT" "
357. GO TO 89
358; C AA1AL VIBRATION
359. 27 Ib=3*L+3pp360, DO 44 1=1, IB
361, DO 44 0=1, IB
302^ 44 Z(I>0)=Z(I+L+2,0+L+2)
303 ^KliEC108.128)
304, 128 FORMAT( 'AXIAL COORDlNATtS )^j. l /* A I 1 I }
U*
366! WKITEU08,76)(COR(I),I = 1,L+1)
367, WRITEC108,76)(COR(I),I=L+2,2*L+2)
368 1 WKlTE(108,76)(COR(IJ,l=2*L+3,3*L+3)
369, K=f"
:' '---
llii A^I^S^R^^
313. A/=i:/SIN(TOHBL)-A6*l,/TAW(TOKBL)
ill'B3=?Ri*F6YB/YYB*COR(0)-RY*F10YB*CORt2*L+2+0))/C
376 b2f(2,0*YYBL*SINHCYYBL)*RY*FlOYb;i/YXB*COR(>U;
iU' ?|)13a(fi2,*i5BL*SlN(*yBL)*KYFiOYB)/Y5tb*COKCj)-
379 ; 1;t2.*YYB*COS(Y'iBL)fRY*F7YB)*COK(2*L +2+J))/C
3S0! -A8=A6*C0R(0+L+1)
38l! A9=A7*CUKC0+L+1)182P"'
'DG_8 1 =0,10
XBLA(1+K)=UA/10.0*CB
TA=XBLA(ltK)*lYB
I1 mititii ?<^/20.)*TORBfgZ* 1C-iitAU +K) = (B3*C0S(iA5tB*SiNCTA)-B3*C0SH
389 1 CTA)+B13*SINH(TA)J
390.'" ZBLBd+M)sCB3*COSCTB)+Bt[SiN(Tb)-b3*COi>H
|Jj|*-ytfu'tXfK)=iB*B^^
Uh UJiG?i?KjJA8*cis(TjRBXBLAlI+K))+A9*SJNlTOKBXBLA(I+N))|JJ'
ANGB(lW)=A8*COS(XC)+A9*SlN(TC)
(II'i?(VA-ABS(ZBLACX+K)))31>8,b
~
397; 31 VA=ABS(ZBLA(1+K))
398/ Ic=I + Kt ^Sag*
8 CONTINUE
Hoi IKVB-AbS(YhO(K)))2b,10o,l0b
401. '26 Vb =ABS(YMO(M)
402. lt=*403. 106 K=K+11404, TaT+cc405, M=M+104ub, TORUB=SHRB*PQLB*TORB*(-A6SlNtTORBL)407, 1+1,/TAN(TORBL))*COR(L+It0)408. 6
"
^CONTINUE:*
'
409. 76 F0RMAT(5E13,5)410. B4a^,0*F3XC
till "Bfa2,0*$XC*F3XC412, Ka(L+l)*U + l
413. Ma(L+l)*10+l
414, M*(Fixc-F3*a#s>i;&15* .A|F5XC/B
16 r 82F10XC/PI17.; BbapexC/Bli4J8, B6=F6XC/B4
419. B7=(F1XC+F3XC)/B1420. B8=F7XC/B4
421., B9=F10XC/B1
422, DO 17 0=1, L423, IB=U+2*L+2
424, IC=0+L+1
m: tiiiuwiwmffliiv-mmmtv-tiiimi&tv427, Cl=(COR(0+U-C0KC0)*COStTORCL^)/SiN(TORCL)
428, B16=B/*YXC*C0KCIB)-A2*C0RClC)+B2*COh(lB+l)-B5*COR(IC+l)
429, Bl7=-Bb*COR(IB)-Al/YXC*COR(IC)+B8*CoKUB+l)+B9*COR(lC+l)
43ll UA=1
432, T=YXC*CC*UA/10.
433. IA=T0RC*UA*CC/10.434, 'j.b=T +CC/20.*YXC
illl il5LAcI +S{aBl4l8SSlT)+Bl&*SlNCl,)+B16*COSH(lO+Bl7*SINH(T3437. ZbLb(l>M)=B14*C0SCTB)+Bj:5*SiN(TB)+B16*C05H(TBJ+Bl7
439: ^XM0U+K)=EC*BXC*YXC**2f*(-Bl4*C0S(T)-bl5*6lN(T)+B16*440. 1C0SH(T)+B17*SINH(T))
- - - -
Ml: JS8iWTS8BMiTS88iIMTSHiSHM)443.
lF(VA-ABS(ZBLA(l+K)))32,-7b,yb-
444, 32 VA =ABSCZBLAU+K))~
445, IE=1+K
446, 75 CONTINUE
447, K=K+i!
449:TORaA=SHRC*P0LC*T0RC* ( 1 . /TAn (TORCL ) *COR (0 )
4b0, l-lt/SIN(TORCL)*CQK(U+lU
4b ll TOR0B=SHRC*P0LC*TORC*tl,-/SlN(l0KCL)*COR(0)
4b2j 1-1./,TAN(T0RCL)*C0K(0 +1))-
453. 17-
Continue- - - -
454. DO 101 I=1,(2*L>1)*11
455, ANG(IjaANG(I)/VA" "
456, 101 ZBLAU)=ZBLA(I)/VA
457. DO UB 1=1,(2L+1)*10
468. A1MGB(1)=ANGBU)/VA-
459. 118 ZBLb(l)=Z6LBU)VVAir,
.
,r
40o: C CALCULATING MAXIMUM DEFLECTION
401, ADA=0.
462: ADO=0,
403, AE=0.
464. K=0
4651 Ar=2.*PI*WA(3)/R0TS/UB
406: DO 102 0=1, L+l
467, DO 103 1=1,10
AaDB*AB*CB/10,*ZBLBU+K)**2.+T0B*ANGb(l + l\)**2,+AE
472^ 103 CONTINUE
473. K =K-U0
474, lo2 COivTIwUE
4/5, 1=0,476: TAaO.
4 77 =
Do'llb I=(Ltl)*l0+l,(2*L+l)*lU
j=ZBLb(l)+T
155
TA=ANGBU)+TAlib CONTINUE
UA=L*10
T=T/UA
Bfi=|0^4aCL + l)*lQ + l,(TBa(ZbLB(I)-T)**2,-iB
1Q+1,(2*L+1)*10
107
TC=(ANGBU)-TA)**2.+TC104 CONTINUE s
AEaDC*AC*CC/10,*TB+TOC*TC+AEDMAXapI*G/10,*(ADA**2,+ADB**2,)**,5/DEC/1WA(3)**2./AE
- . . .
-WKITEU08.12DDMAXD0 1 0 5
-
1= 17 2 2*L~* 1TYMOU)al#OU)/'VA*DMAX
105 WNTINilill- -
- ftR?Z=Vg?TA*2,*G/CB/l0,*(ADA**2,+ADB'l'*2.)**.5/AMY/ltIA(3)**2,/AE.......
DO, 113 0=1,11YMOM(d)=0,DO 108 1 = 1, (L+1)*U,U1F(YMOM(05-ABSUMOU+0-I)))l07,108,10bYMOM(0)=ABSUM0Ut0-l))- ' - - - -
504. 108 CONTINUE
505, 113 CONTINUE506, DO 114 1 = CL + 1)*U + 1,22*L +U
507. XMOU)=XMO(l)/VA*DMAX
508. U4 CONTINUE
509. DO 109 J=12,22
510. XMOMCO)=0,511. DO 110 I = (L + l)*ll + l,22*L+ll,U
512. IF(XMOM(0)-ABS(XMO(I+0-12))) 111, 110,110
513. Ul XMUMCO)=ABS(XMQU*0-12);- - -
514. U0 CuiVUwClt
515, 109 CONTINUE
516, WHITE(108,123)
517, 123 FORMAT (' LOC AXIAL STRESS')
518, DO y9 1=1,11
519. STRUI) =YMOM(I)/SfcCYB(I)520. WK1TEU08,59)I,STKYU)
'
521. 99 CoNllNUb* - - -
522. DO 100 1=12,22
523. SlRX(l)=XMOMU)/SECACU)
524. AK1TEU08,59)A,STKXU)'
526, 100 CONTINUt,
5 26. K=l
527. 1=0,528. DO 137 0=1, L+l
529. DO 1381=0,10"
530. XBLUtK)=XBLA(I+K)
531. YBL(ItK)=T
532. ZBLU+K)=0.
533, XbLAACItK)=XBLAUtK)
534: YBLAAU +K)=T-
535, 138 CONTINUE
536, K=K+U
537, T=T+CC
538, 137 CONTINUE
539J K=(L+1)*U + 1
540: DO 139 Ual,L
54i: PO 140 1=0,10
542: UA=I
543, XBL(I+K)=CB
544) yblu+k)=cc*ua/io, + j:bl(k-l*ii-1)
545, ZBL(I+K)=0."
54&: XBLAA(I +KJ=XBLU +N)
547. YBLAA(I +K) =YBLUtN)
548, 140 COn'1 J. ft UE
549. K=K+11
550: 139 CONTINUE
551.'
I = (2*L + 1)*U
jj7 XBL(U)=*BL(U)*.8bb025-ZbLC0)
SIS-,31 &5f51
56?: CUsizt=A3*CC/2. 0+1.0
156
561, TaCB+1,
HI: S8tt?HSil:,!564, XbLAAU +2) =3,565, YBLAA(I+1J30-566, YBLAACI+2)a3!
HI* CALL AXIS(2ii34, 2,5, 8HY (INCH) ,-8 ,SIZE, 30, ,XBLAAC1) ,boo, lXbLAA(I+2))
-.-. ....
1W: iWfckAt{+fJJ*'3,'8HX <*NcH),8,T,9o.,YBLAA(m),b,i"
CALL AX1S(3.,3.,8HZ ( INCH) , 8 , 1 , , 180 , , XBLAACI+1 ) ,572, AYoLAAU +2})
_...'-'-. -
573, CALD PLGT(2,.3,,-3)574, CALL NEwPENU)5'5. CALL LlNEUBLAA.YBLAA,I,l,-l,U)576, XBL(I+l)aXBLAA(A+I)|'K XBLU+2)3XBLAA(I+25578, YbL(H-l)=YBLAAU+ l)579, YbL(I +2) =YBiJAAU +2)580, CALL NEwPENU)581*- CALL HnE(XBL, YBL, I , 1 # -1 , 3)582, CALL FINPT
'
- ~
583. GO 10 69
584, 78 CONTINUE
621. SuBKOUTiNb U
622, C LAMBDA OF BLATJET
623, YYB=YXC/y*(bXC/BYB)*0.25*S624,
YZB3YXB*(BYB/SZBj**0,25- -
625, YXCLaYXC*CC
6267YZCljaifZe*eC-
627, YYBL=YYB*CB
628. YZBLaYZB*Cfl
629, TORB3SA*TOHC
6 30-7 T0RCL=ToRt:*ee
631, TORBLaTORBCB
632. FlYBaSINUYBL)*5INH(YYBL)
till r3H5SSg4I!*ifSSa4AfW4ij-*-.- -
637, t /YBa5lN(YYBL)+SlNH(YYBL)
636:F8YaaSt(YYBLi-SlNH(YYBto5-
639: FtO*B8COS(YYBL)-COSH(YBL)
640, F1ZB3SINUZBLJ*5INHCYZBL)*
157
541, f 2ZB=C0S(YZBL)*C0SHCYZBL)
642. F3ZB=C0SUZBL)*C0SHCYZBL)-1,, _ c ,uWut ,
643. f 5Zb=C0S(YZBL)*SlNH(YZBL)-SlNCYZBD)*C0SH(YZBLl
644, 5bZB=C0S(YZBL)*SlNHUZBL)+SN(YZBiJ)1'C0SHUZBL)
645, F7ZB=SltMCYZBL)+SINH(YZBD)'
646. F8Z6=SIN(YZBLJ-5INH(YZBL)
647. F10ZB=COS(YZBL)-COSII(YZBL)648, ;F1XC=SIN(YXCL)*SINH(YXCL)
649, :f3XC=C0S(YXCL)*C0SH(YXCL)-l.t , ,
650, F5XC=COS(YXCL)*SINHCYXCL)-sIN(YXCL)*COSH(YXCL)
661, F6XCaCOSCYXCL5*SlNH(YXCL)+StN(YXCL)*COSH(YXeL)
652, F7XeBSlN(YXCL)+SINH(YXCL)' "
m-. iimmmiiiWA^h.165, |FiZCSiN(rZCL;*SINHtYZCL)
gT3ZCC0StYZCL)*C0SH(YZCL)-l. '-
,*,,
hl\. iC52C=COSCYZqL5*SINH(YZCL)-SlNCYZqL)*COSH YZCL)
,58:F6ZC=C0b(YZCL)*5lNH(YiiCL)+SiNUZCL)*C0SH(YZLL)
659, F7ZC=SIN(YZCL)+SINH(YZCL)-
' " " ' '
660, F8ZC=S1NCYZCL)-SINH(YZCD
igi: ii^-S^Hi^^663j P2Y=(YYBL*F6Yb+FlY6*RY)/(RY*F3YB+YYBL*F5YB)
664, P3Y=C2,0*YYBL*F2YB+RY*FbYB)/(RY*F3YB+YYBL*FbYB)
665. PlZ=(*2,0*YZBL*FlZB+RZ*F5ZB)/(RZ*F3ZB*YZBL*FbZB)
6b&: P2Z=aZBL*FbZb+HzB*RZ)/(HZ*Fhb +YZbL*FbZBP
607. P3Z=(2,0*YZBL*F2ZB+RZ*FbZt))/(RZ*F3Zb+YZBL*tbZB)
608. P4=-l,/(HT/fc>HRB/P0LB/T0KBTl./TANCT0RBL))
bill KOT=0./!80.*PI
672. C0=C0S(R0T)
673. Sl =SIfiCRGT)67.4 lti~ifMb, tu
675, DO 10 0=1, IB
676, DO 10 K=1.IB
67?:
10Z(i;i}aiiB*BZB*YZB*PlZ*C0**2, +EB*BYB*YYB*PlY
67^: l*Sl**2.+EC*BZCYZC*FbZ,C/F3ZC-
680! ZU,2)=EC*BZC*YZC*F8ZC/F3ZC, u
68UZ(.ljL+2)=EBBZB*YZB**2,0*P2Z*C0**2.+EB*BYB*nB
Hi', ^Jtl |L+3?sfcB*C0*SI*(-BZBYZBPlZ+BYB*YYB*PlY
IS!1 '|J^J^5p^CO*SI*C-B.B*YZB^2.*P2Z+BYB
68?: p K=i; ',
till Z(0^Ktl)=ZCifl)+EC*BZC*YZC*F5ZC/F3ZC
C0>LtK+3)=Z(l,L+3)690.
692: Z(0^3*L + K +5)=ZU,3*L +b)
tUl2Q
-P':Z(Ltl,L+l)sZU,l)
6951ZCL+l,2*L+3)=ZU,L+3)
6961 ZCL+1 4*L+5)=Ztl,3*Lfb)
692.- ,698. 30 tjW'M*? i?yiim + l .*EB*LBZB*YZB**3.*P3Z*CO**2,
70l!ZtLt2,L+3)=ZU.3*L +b)
702"
DO 45 U=L+4, 2*L+3
Ul45 |j^;^K^Bi;CO^inBZBnZB^3.*P3,-BY
m: AS5^rl=?^b,4*L,5s*
70?: 4bZ(L+2,U)=ZCL+2,3*L+b)
?S- J?Ic.lC)afcBBZB*YZB*PlZ*SA*2,+EBBYB*YYB*PiY
7t2!K=Lt4'
7U,lb=2*L+2
714. ^911i^7tVr.f(n+SHKC*P0LC*T0RC/lAlHTOKCL)^jJJ)?ztic}iq+JHKC*tOLC*TORC/Ti
ZCD,Ktl)=Z(lt,lC+l)
K=Ktl'
Ib=2*L+;
zcibub;K=3*Ltb
717 U K=K + 1
718!Ib=2*L+3
4t5 ZCI6/1B)=ZUC,1C)
720^--a/-K
158
721. UO 12 0=IC,1B
7 22. ZC0,K)=EB*BZ6*YZB**2.*P2Z*SA**2.+EB*BYB*nB
723. i**2,*P2Y*CO**2....
724, 12 'K=K+1
7 27 , ZCI8,IB)=SHR8*P0LB*T0HB*P4+EC*BXC*YXC*F5XC
728,1/F3XC- ---- -
729, -Z(lB,IB+l)aEC*6XC*YXC*FBXC/F3XC730, ZUB iC)a-EC*BXC*YXC**2j*FlXC/F3XC731 J Z(lB,IC+l)=-EC*BXC*YXC**2i*F10XC/F3XC732, Ka2*Lt5
.-.-......_ . _ ..
731, |C=2*L+4
ll&l :D013 JalCtl/IB
716% 35CO;K)aZ(IC;iC)tEC*BXC*YXC*F5XC/F3XC737^ Z(0,Ktl)aZ(IC,irZ(0,Ktl)aZ(IC,IC+l)
Z(0,L+K)a-Z(IC,lB+3)Z(U L+K+2)=Z(lC,I6+3)?39
740! 13 K=Ki-l
741,-
lB=3*L+4
742, IO=4*Lt4
743. ZUB,IB)=ZUC,IC)744. ZUB,lD)=-ZUC,lB+2)745. ZUb,AD+1)=-ZUC,1B + 1)74b. Ib=3*L+5
7 47, ZU6,AB)=-EB*BZB*lZB**3.*P3Z*.bl**2,-EB*BYB*YYB
748, I**3.*p3Y*CO**2,-EC*BXC*YXC**3.*frbAC/F3XC
749. -ZUjg,i[m)=EC*BXC*YXC**3.*FyXC/F3XC
75ll J.C=3*L+ 6
752. Ab = <**Li + <J
754: Z(J^fK)=2UC-l,lC-l)-EC*BXC*YXL**3.*(rbXC/l-3XC
75b. Zlo,Ntl)=Z(AC-l,IC)
7bfa. 23 is = K + l
75 7. AB=4*L+b
758. Z(Ab,lB)=Z(IC-l,IC-l)
7b9. Du 48 0 = 1, lb
760. UO 48 K=l,lb
761. 48 Z(U;is)=Z(J.K)/Eb/BYB
7o2. UJ 24 0=2, lb
7b3. UO 25 i\=l,AB
764. Z(d,K)=ZU,U)
765. If C0-K-l)24,24,25
7 66. 2 5 CoN'i'lNUfc
707. 24 CONTINUE
768. RETURN, ,
7b9. C SUbKUuTANE TO F1NU COORDINATE
770. BUBHUUTlNb DE
771, COC=0.
772. DO /9 M=1,IB
773. DO 61 I=l,lb-1
774. 00 81 0=1, lb
775. 81 Zti(A,0) =ZU,0)
77b. uO 82 J = i ; At*
it (ABa(COEF)-ABS(COE) ) /y, /9,bb
7<J0. bb COEsCoEf
7*1. IC3M
782. 7y conxinUE
783. COK(lo)=l,0
7H4. n=ic
78o. UU 11 I=1,1B-1
78 /, uo / / 0 = 1 , lb
788, II Zu(I,0)=ZU,J)
7 89. LO 50 0 = 1 PB
lW. 5u ZdlM,0)=Z[ib.O)
7yl. uo 2 JV-1,AB-1
iril wo 22 l = l,Ib-l
7^3. uo 22 o=l,ib-l
lA. 22 ZA(l,0)=ZrJU,O)
Ul'2i ^(Aj^ = izMltlB)CUKU|
ill 2CoKlK)=DMKZA(iB-lJ/Co&*
jU*
59 FUR.4AiUb,5t.l3.5)
EivD799
800
159
APPENDIX B
COMPUTER PROGRAM
FOR
COUPLED RESONANT VIBRATIONS AND STRESSES
160
1. C COUPLED VIBRATIONS AND STRESSES2 , C
- -
3 C4*
DIMENSION Z(50 , 50) , W ( 3) , X (4) , WA ( 3 ) , AA (3 ) .
7, 1 ZBL(265),XMOC265);YMO\265j,ZMOC265);SECZBUl),6f lZMpMC2b),STRZrp),ANGr2657/aEOYB(U),iTKXtl2);STRY(l2),V. 1XBLBU65),YBLBP266),ZISLB(2&^
"
JO,lSECZC(v22),SECXC(22),XBLAA(265).YBLAA(265)-
XI, Cp*
DIMENSIONS ARE AN LBS; ANCHtS, SECONDS, RADAANS
12, C GRAVITATIONAL CONSTANT--'-
T-*l m, !$B3~-,14T52-6#r..,.
15,1 CI DUMBER OF'NQMLES PER '360 DEGREES
16, ,;i
$N=156,''Pp-F''P':; -*- -* - -
-p*p.,
17* ;CC SUHBKR OF BLADESVPER 360 DECREES
J9j c lSgarIthmic DECREMENT20,
DEC=,02-- -
21. C STIMULUS - Y-DARECTION22,
"-
ST1MY=,1"
23, C STIMULUS - Z-DARECTION
24, ST1MZ=,1
25, C A-DIRECTION
27: C ANPLAXUUE OF DRAVING FORCE - Lb/BLD X-DlKECTAON
28, AMY'=238,~ - - - - -
29, C AMPLITUDE OF DRIVING FORCE - LB/bLu Z-DlRECTIOlM
30, AfiZ=36,- -- ----- -
...........
31, C AMPLITUDE OF DRIVING FORCE - LB/BLD A-DlRECTAON
U: c8Slsii8IW
ME u " KAU/-C
35. C CV* KQTATAONOF BLADES - TOP VAEW
36.ROT=-42,5/180^*Pl-
37,C0=C05(RGT)-
38, SI=SIB(R0T)j-
Q-, .....
40: C BLAUE PARAMETERS
41. C SPECIFICWEIGHT-
2, DB=6,283--
43, C AREA -'
44 A&= 1 7 2 5
45: C MO00LUS OF ELASTICITY
46J Eb=30fE+06- - - - -
*!, CSnEAR'
MODULUS OF ELASTICITY
48,SrtRB=lA,E*U6- -----
49, C INERTIA -ABOUT'-Y' -AXIS
50, BYB=,6801
51, C INERTIA ABOUT Z'-AXA5
52. BZB=,T207~
b3, C SECTION MODULUS ABOUT Z-AAAS
54j SECZBU) ="
55, DO 55132,10"
50, 55 SECZBU) = .193l
57. Se.CZBUl)i0704
58: C SECTION -MODULUS ABOUT Y-AXIS59.*
ScCYBtl) =U006- - -
60. uu U21=2*10-
61, U2 StCXBU) = ,3751tri.- Sfi,C*BUDi068 2
C3, C TORSIONAL MOMENT OF INERTAA
64, POLB=,16
65, CLENGTH-
7"
J C TORSIONAL WEIGHT MOMENT OF ANtRXlA
68. Xo63POL6*CB7lOf*D6 >
6u r STEAM MOMENT AT 'THE ROOT OF TriE BLADE Z-AX1S
7UJ SMOZ=AMY*CB/2,..----
72! C'
COVER PARAMETERS
13. C SPECIFICWEIGH!"
74, DC=0,283
75, C ftRA
"ill C ACMqi5uLUS OF ELASTICITY
r ISeAr'MODULUD OF ELASTICITY
80: SHHC=U,EtOb-
tr,
161
81, C
82.83, C
84,
85,86,87,
88,s 12989 7
'
90j'X'
91.
92..9.3,.
130
';S>6%> C
!!:99, c
100,101. c
i o27 c
103.
104,
105,
106,107,
108, c
109,no.
c
IU.
112,113. c
114,
115,110.
HI:119,UO.121. 58
122,c-
123,124. c
: c
127,
128,i*t*130,131,132.
133, 64
134,""
136.136,137,138,139, 14
140,"
ttl: 13
1M:1
C
145,146,147,
148,149,150.
,r-
151. b4
152, 15
ill: 56
156.156.
60
lb/.156, -61
159,160, 62
ABOUT X-AXIS
INERTIA ABOUT X-AAIS
BAC=.3007~
INERTIA ABOUT Z-AXASBZC=,03b5
-
.
SECTION-MODULUS ABOUT Z-AXlS
SECZCU2) = ,0694LO
129- '
A=13>21St.CZCU) =
SHJiHii?ttit828iStCXCU2) = ,*2514DO'X30^A=13;2l*
.S|CXC,(I7a,2
6l^
TORSIONAL WEIGHT MOMENT Ot INERTIA
ToC=POLC*CC/10,*DC' - - -
ROOT ATTACHMENT FACTORS - ABOUT THE AXIS
HY =b.E6 ,..
" ' - -
HZ5,E7 fp-
HT=6.0E4 gRJ=ttY*CB/(EB*BYB)
RZ=HZ*CB/(EB*BZB)
DUMBER .GFPTHEHARMONICU A D iff 'i -
.- -
"':
* * '
SA=(DBSHRC/SHRB/DC)**0,5SB=tDC/SHRC/Gl*0^5' '
-NUMBER OF BAYS
DO 80 L=5,b~
XU7=,Ob'
V = (DC*AC/,EC/G)**0.2b
S=CDB*AB/EB/G)**0.25.... .
w CI 3*0.0A33L iso^co
DO90"
IA6-,-5*66-, 10
FORMATU7,4E17,8) _
CYCLES PER SECOND
QslA--*---
OMEGA
YXC=V*W(2)**0;S/BXC**0,25
rZC=V*W(2)**0,5/BZC**0,25
TORC3ft(2)*SB:P pp:p:_ :.' "
caLlu
XU)=DETEIZ.IB)1FCX(2)*XC1))64,89,89
WA(2)=W(2) p_P... .".P..
wA(i)=wU)
XA(2)=X(2)XA(i)=xiir *
-IF(ABS(Xa(2U-U0E+'6;113,13,14
WA(3) = (wA(2) +WAU))/2;0- -
WRITEU08;58)M- '
*% C 3 ) = tlAC2 ) *AATp-*AVT) *XA ( 2 ) ) / ( XAU ) -XA ( 2 ) )
0AaiA(3)/b. 283135308
LAMBDA-
OF-
CU V ER- - ~
YXC=V*rtA(3)*0,5/BXC**0,25
}ZC=V*WA(!5**0:5/BZC**0,25XORC=WA(3)*SB
-
CALL CI" "
WRIiEtl08jl5)UA
FORMAT(///,l7HFREOUENCY (CPS) =,EU.5,/)
fF(3EK(3j*XACI))b0,6O,6l
XAU)=XA(3)
GO TO 62 pWAU) =WA(3)AAU)=XAUJ
"CONUNUE"
R
by
90
80
59
53
52
82
65
.7 9
7 7
.&0_
22
212
16
7 6
C -.
UC"
)
41
33
98
162
GO XO 53
WU)=WC2)X(1)=X(2)CONUNUECONTINUE
GO TO 78
fuRMAlCl5,5E13,j)CALL U
Ito=4*L+5
COEaO,"
DO 79 MSI, IBDO 81 I=1,AB-1DO, 81. 0 = 1,10
"
ZB(H,0)ZCIB,0)COEF=DETE(ZB,Ib-l)
lFCABaCC0EF)-ABS(,CUE))7y, /9,6bCOEaCOEt
' -
-...'--'..
CoEratOE
ZBU,0) =ZU,0)DO bO-0=l;lB
ZB(w,O)aZ(lB,0"
DU-2 lUl,I-i-
DO 22 A=1,IB-1DO 22 0=1,AB-1
Za(A,U)=ZB(A,0)DO 21 I=1,IB-1ZAU. K ) = -ZBU, IB) *COR( lb)CORvK)=DEXEtZA,IB-l;/COEFWRATEC1U8.16)
-
FORnAX ( 'SYSTEM COORDINATES 'J
WRITE(108,7 6MCORCIJ,1 = I,L-U)WRXXEUU8,76JC0R(L +2J
' * "' "
WRITE (108,76) (COR t I ),A=L+3,2*L+3)WRITE (108, 76 )( CORU),A = -2*L +4,-3*D +4)WRITE (108,70) (COR ( I ),I = 3*lj +b,tB7
-
FORMAT(5El3i;5)..CONVERTING COORDINATES ANXO PRAME COORDANATESDO illal/Ltl
- - - - - - - .
CORP(A)=COR(I)*CO-COR(I+L+2)*SICORPU+2*Lt2)=COH(U*SItCORU+L+2)*COCORP(A+L+ l)=CORTLl-2J*Cg*CURU + 3*L+4)*SICORPU +3*Lt3)=CORtLt2i*SltCORUt3*L+ 4^*CODETERMINING-MODE SHAPEVA=0."
VB=0,Kal ,
rtsi
C=2,0*(RZ*F3ZB+YZBL*F5ZB)f
DO 39 OaUL+lB3 = tR2*F8ZB/YZB*C0RP(0)-RZ*H0Zb):CORP(0+L+l)l/CB=((2,0*YZBL*SlNHtYZBL)-HZ*iU0ZB)/YZB*CORP(U)
I +l-2.0*YZBL*COSHUZBL>-RZ*FfZB)*CORir'(d+jj-t'iU/C
-B13=((-2,*YZBL*SIN(YZBLJ+RZ*FI0ZB)/XZB*CGRP(U)lt(2,*KZbL*COStYZBL)tHZ*f 7ZB?*tORPtO +
Ltl))/C" -
-
-DO 41 1 =0,10- - - ~ ' '
XBLA(A+lU=yA/lO.O*CB
TA=XBLA(A1'K)*iZB
TB=(XbLA(A+iW+CB/20,)*YZB
YBLAU+K) = (63*CGS(TA):t:6*SAN(TA)-B3*COSH
1 CTA)+B13*SINH(TA))YBLB(l+M)=(B3*CGS(TB)+B*SlN(TB)-B3*COSH
1 (XB)+B13*SINH(TB)J
-ZMOU +K)=EB*BZB*YZB**2,* (-B3*C0SUA)-B*olNCTA)-B3*C0SHKTA)+B13*SlNHtTA)-J------ -
- -
continue
IF 1.Va-ABS(ZM0(K) U33,9tt,*8
VB=ABS(ZMO(R)7-
- - -
IF =K"
KSK+11
MSM+IO
ShbB=EB*BZb*YZB**3.*(CORP(U)/YZB*F2Z-OORP(0+Ltl)*P3Z)
IbJ
3y
123
124
28
119
31
120
8;
26
106
29
43
42
CONTINUEK=l"
Mai-
C=2.0*(RY*F3YB+UBL*F5YB)
*D0 6 0 = 1, L+l' " ' '" * '"
B-3atRY*F8YB/YYB*qORP(0+2*L+2)-RY*F
1,/TAN(T0RBL))
lOYB*COHP(U+3*Lt3))/C
Y*B*CORP(0+2*L+2)'
RP(0+3*L+3) T/CYB*C0RPt0l-2*L+2)3*L+3))/C- '
i$mm,"DO "'8
BifWWM
TA=XBCA*$a/io,ocb
.iffcymB
TBTAfCB/^0i*YYB-
TC=(XBLA(l+K):fCB/20,)TORBZbLA(l+K)=(B3*COS(TA)tB*SAN(TA)-B3
r^'(TA7+B13*SINHCTA)7" "'
"
'ZBLB(ltM)aXb3*COS(TB)+B*SlN(TB)-B3
1 (TB)+B13SINH(TB))" '
-YM0U+K)=EB*BYB*YXB**2,*(-B3*C0S(TA)
HTAJ+Bl3*SlNHtTA)T~ " ~
-ANG(I+K3=A8*COS(ToRB*XBLA(I+K))+Ay*S
ANGB(A+M)sA8*COS(TC)+A9*SAN(TC)' "
YBLA(T+K)=Ar" ---
IF(A*T0)123,124,124, a
AiaYBLBtA+M)*CQt"ZBLB(l+M)*SlZBLB(A+M)=-YBL6(ItM)'tai+ZBLBU+M)*CU
YbLB(A+M)=Al-
" '
CONTINUEIF(VA-ABS(YBLA(A+t<J))28,U9,U9
VAaA8S(YBLAU+RT)* ' '"
IE=A+K P
CONTINUE
AF ( VA-AbS(ZBLAU+M) ) 3 1,1 20, 120
VA=ABS(ZBLAU +KU"'
lt=i+hCONTINUE
CONTINUE
IF(VB-ABS(YM0(K)))26,
VB=ABS(YMG(M) v------
ilSn'
M3M+10
T0RQB=SHRB*P0LB*T0RB*(-A6*SAN(T0RBL)
l + I./TAN(T0RBLU*COR(2*L +3+07'
-SflfiB=EB*BYB*YYB**3,*(CORP(0+2*L+2)/YYB*P2Y
1-CGRP10+3*L*3)*P3X)"
-
-Continue* - - ---- '
Ks(D*l)*U + l
"WSJJ!fi.r3*cDO 42
Jal'iL'
B2=*F5ZC/B1*C0R (0 ) -F8ZC/ 6 1*C0R (0+1
Al = -tMZC+F3ZC)/Bl*CORtU)tFttOZC/Bl
A2=(FIZCF3ZC7/Bl*CORC0 7-F10ZC/bl*
DO 43-1 =0710- - -
UA=1' * '
T=YZC*CC*UA/10,
XbXBLA?A+Nj=?b2*C0S(T)+Al*SlN(T) -B2*XBLB(X+M)=(B2*C0S(TB)i'Al*SllHTB)-B
?COSH
?COSH
-B*SIN(TA)-B3*C0SH
IN(1URB*XBLA(1 +M)
106,106
)*C0R(U+1)
COR(OU)
C0SH(T)+A2*SINH(T))
2*C05H(T6)+A2
1*S1NH(TB))
1-B2*C0SH(T)+A2*SINHUJJ
+K!=EC*BZCYZC**2,*(-b'2*00S(X)-Al'l'blN(T)
1F(VA*ABS(XBLA(1+K)3)29,43,43
VA=SBS(XBLA(X+K7)*
1C=A+K
-CONTINUE
KsK+11
5 SONUNUE*T*r-
b4 =2,0*lr 3XC
~B1=2.0*YXC*F3XC
164
K=(L+l)*lltl
M3(L + l")*10tS; yLJAA=AFiXC-F3XC)/B4
A2=F5XC7B1B2=F10XC/B4
B5=F8XC/BAB6=rbAC/B4
B7=tFlXC+F3XC)/BlB8=F7XC7'B4-
-p
B9=FlOXC/Bl
-DO I701,LIB=U+3*L*4
Mk4a-Al^iRaB)+A2*COR(AC)^B2*COR(IB+l)+B5*COR(K
Bi5=Bo*COR(AB)+B7*COR(lC)*B*COR(AB+i)-B9*COR(ICtl)
Cia(CoR(0*L*3)-C0KC0tL+a)*CdSfTORCL))/SlN(ToRCLfBl6aB7*YXC*COR(lB)*A2*CORtlC)+B2*COR(iB*l)-B5*COR(ICM
Bl7 = -Bb*COHUB)-A!/XXC*C0R(iC)+b8C0RaB-U)+B9*COKUC+ l)
*DO 75 1=0,10UA= 1
'
T=YXC*CC*UA/10,TA3TOKC*UA*CC/10,TB=T+CC/20,*YXC
TC =TAfCC/20.tT0RC
BLA(A+R)3614*C0S(T)+B15*5IN(T)+B16*C0SH
bLBU+M =Bi4*e0S(TB)+Sl5?SANtiB)i-Bl6*C0
CT)tB17*SANH(T)
SH(TB)+B17
1*S1nH1TB j
-XMO(I+K)=EC*BXC*YAC**2,*(-Bl4*COS(T)-Blb*SlN(X)+Bi6*
1C0SH(I)+B17*S1NH(T) 3' '
-ANG"i+R)=CpRtO+LV2)*COS(TA)+ClfSlN(TA^ANGB(ltM)aCOR(0tLt2)*COS(TC)+El*SlN(TC)lF(VA*ABS(ZBLA(l+A)))32;7b,75-
32 VA=ABS(ZBLA(I+K7)'
".p" "
1E=1+K
75 CONTINUE
K=K+11
MsM+10TORQA=SHRC*P0LC*T0RC*(l,/TAN(T0RCL)*COR(0+L+2)
l-l,/SIN(TORCL)*COR(0+L+3)3
TORGB=SHRC*POLC*TGRC*(l;/SlN(TORCL)*COR(U+L+2)
--aSftA2fia*S88SMSS8SJ*tit*A4*COKCIB>+riXC*a.^Bl*CORClC3
xgS&S2gSSpiiStJSi*5 ;SfSjSggiSgSili , -2 . *b9*cor c xc j- ^ .
1*B6*C0R(IB+1?+F1XC*2,/B1*C0R(IC-U))
17 CONTINUEr
DO 34 I=1,CL+1)*U
YBLA(1)=YBLA(I)/VA
34 . CONTINUE
DO 116 I=1.(L+1)*10
116 YBLB(I)=YBLB(I)/VA
DO 9 A=(L+1)*U +1.(2*L+1)*U
9 g8lVJlfr!=(L+l)*10+ l,(2*L+l)*10in XBLB(1) =XBLB(I)/VA
C CALCULATING MAXIMUM DEFLECTION
ADA=0.ADB=0.
AE=0.
AF=2,*PX*WA(3)/R0TS/UB
DO 84 Jal,L+l
UA=0-1
K8AaSliSi*AMVnBLB(I+K)*C0SCUA*AF)+ADAADB=STIMYAMY*YBLB I+K)*SlN(yA*AF)tADB
AE=(YBLB(I+K))**2.*DB*AB*CB/10,+AE
85- RywHUEp
84 l5n,3t1=CL+i)*iou,(2*L+i)noAE=(X6LB(I))**2,*DC*AC*CC/10,+AE
83 S8Nt0lUI-
= l,(2*Ltl)*U
ANG(I)aANG(I)/yA
101 ZBLA(I)=ZBLA(I)/yA
DO 118 1=1, (2*L+!)*10ANGB(1)=ARGB(I)/VA
UB :ZBLb(I)=ZBLB(I)/VA
165
KeO
DO 102 0=1, L+lUA=U-1
ADAa(STIMZ*AMZ*ZBLBCI*^ADB3(STIMZ*AMZ*ZBLB(I+K)+ST1MA*AMA*ANGB(I+K))*SIN(UA*A^ 3+AOB
AE=DB*AB*CB/1G,*ZBLB(U-K3**2#+T0B*ANGB(1+K)**2,+AE
103 CONTINUE... ,, _K=K+ 10102 CONTINUE
TAbO1,
'58"?U I3(L+1)*1G+1,(2*L+1)*10
tBZBLBU)+T_IAANGBtl)+W
115 CONTINUEUAL*10
TaT/UA
IA=TA/UADO 104 I = (L + 1)*1Q+ 1. (2*L +U*10TBs(ZBLB(I)-T)**2,+TB
TC= ( ANGB ( I ) -TA ) **2 . tTC
WRITE(108,121)DMAA^^
121 FQRMAT(/,21HMAX1MUM DEFLECTION a ,F7,5,/)
DO 86 I =1,22L+UZMO(l)=ZMO(l)/VA*DMAX
Aft CONTINUERRF=VB/VA*2.*G/CB/10.*(ADA**2.+ADB**2.)**.5/AMY/
1WA(3)**2./AE
DO 91 0=1,11
ZMOd(0)=0.
IF(ZM0S(05^AB^
68ZMOMC0)=ABS(ZM0(I+0-l))-
97 CONTINUE
91 CONTINUE-.
DO 94 0=12,22
ZMOM(O)=0,n^r
DO 95 I=(L + l)*U + le22*L+ ll.Ue nc
IF(ZM0M(0J-ABS(ZM0Ut0-12)5)9b,95,95
9b ZMOM(0)=A6S(ZMQ(I+0-12))
95 CONTINUE
94 CONTINUE
Sm'fiJaiA&Jun/sEczBCi)92 CONTINUE
STRZU)izJoA(I)/SECZC(I)
DO 105 I = 1,22*L +UYMO(1)=YMO(I)/VA*DMAX
105 CONTINUE
DO 113 0=1,11
YMOM(0)=0.
??(JSSM^^sHSohii-l)))107,108,10b
107 YMOM(0)=ABS(YMO(I+0-1))
'
108 CONTINUE
113 CONTINUE,rjLl^11it1 r.I+11
DO 114 l=(L+l)*lltl,22*L+U
XM0(I)=AMO(I)/VA*DMAX
114 CONTINUE
DO 109 0=12,22
93
XMM(0)-0,2 + n
IF(UoM(03-ABSCXMO(i|0-l23 3 5 1U,U0,1 10
111 j?MdM(0)=ABS(XMO(l+0-l2))
0 CONTINUE
109 CONTINUE-
122 FORMAtJ'^LOC STRESS-TANG. STRESS-AXIAL')
STRYCX)=YA0M(1)/SECY8(I)
WRITEU08,59)I,STRZ(I),STRY(I)
99 .CONTINUt
166
481. DO 100 1=12.22462, STRX(I)aXMOMU)/SECXC(I)48 3. WR1TE(108,59)1,STRZ(I),STRX(I)
484, 100 CONTINUE
.485,. K=l
486, 1=0.
487, DO 137 J1,L41
488. DO 138 lao.10
489. XBL(I*K)axBLACIfK)490. YBL(I+K)al491. ZBL(I+K)=0,492. XBLAA(I+K)=XBLA(I+K1
493. YBLAA(I+K)=YBLA(I+K)+T
494. 138 CONTINUE
495, KaK+U
49o. T=T+CC
^49 7, 137 CONTINUE
498. Kb(L+1)*11+1
499, DO 139 Oal, L500, DO 140 130,10
501. . UAaA502. XBL(I+K)aCB
# ^ . .
503. YBL(I +K)sCCUA/10.*YBL(K-L*U-l)
504. ZBL(I+K)aO,605, XBLAA(ItK)aXBLA(I+KjtCB _
-^ . ,
506, YBLAA(I+K)aCC*UA/10,+YBLAA(K-L*U-l)
507, 140 CONTINUE
50S. KaKUl
iw. mm\mi509. 139
510. _.._
511. DO 131 J=lbll. UO Ui 0= i,A, %
512. TaYBLAA(0)*0.5*XBLAACJ)
bit: ..XBLAA(U3=YBlJIaXJJ*,866025^BLACJ)
51^: YBLAA(0)3T
51b. TsYBL(0)*0.5+XBL(0)B , ,
bl&: XBL(0)3YBLtO)*,866025-ZBL(J)
517. YBL(J)al
518. 131 CONTINUE ~ ,^
'
n
519. SlZEsA3*CC/2, 0*1,0
520, TaCB+1,
521. CALL *IND0*(3,13*,1U>
522. XbLAA(I + l)3*;3.
523. XBLAA(I+2)33,
524. YBLAA(IU)aO,
till CALLAAxii(2h34,2.5,8HY (INCH) ,-8 , 8. ,30. ,XBLAA(1*1)
526: ^AL^AXIS^.^.^HX (INCH) ,6 ,5. ,90, , YBLAA (1 + 1 ) ,
530*: 1CALLAAX1S?3.,3,,8HZ (INCH) , 8 , 1 . , 180, , YBLAAU + 1 ) ,
53i: lYBLAA(I+2))
532, CALL PLOT(2,.3.,-3)
Il CAfci(L!!lE|lSfeB[XBLAAfYBLAA,I,l,-l,n)535,
ill; -iH-iiji!:ffiSim:
IIS: Eftfct 2BSKS1Jimi.i..,j)541. CALL FINPT
542: GO TO 89
543, 78
167
so ;
ill
SUBROUTINE U
^,YXC7vUtc'/B)YB>**012SSflT2BBYYB*(BYB/BZB)*0.2S
3fXCLYXC*CC
YZBLYZB*CBTORBaSATORC
f8lft:i8!EIiFlYBaSlN(YYBL)*SlNH(YYBL)
F2YBaC0S(YYBL)*C0SH(YYBL)
i,3YB3C0S(YYBL3C0SH(YYBL)-l,F5YB3C0S(YYBL)*SInH(YYBL)-S1N(YYBL)*C0SH(YYBL)
R?g:i?gH!Sl;jaiSI! ??? ??"(m,'J*CMHCnBL)
f!nBftUj&BIIHHU) -
F2 ZB=CO,<mim&F3ZBaCOS(YZBL3*COSH(YZBLtT5ZB3C0S(YZBL)*SINH(YZBl5F6ZB3C0S(YZBL)*SINH(YZBL)F7ZB3SIN(YZBL)+SINH(YZBL3F8ZB3SIN(YZBL)-SINH(YZBL)
F10ZBsC05(YZBL)-COSH(YZBL)F1XC3S1N(YXCL)*SINH(YXCL)F3XCsCOS(YXCL)*COSH(YXCL)-l
R5g:HS Ha Ki1 aF10XCsCOS(YXCL)-COSH(YXCL)TlZCaSlN(YZCL)*SlNH(YZCL)F3ZCaCOS(YZCL5*COSH(YZCL)-l
-1*-SINCYZBL)*COSH(YZBL)
?S1N(YZBL)*C0SH(YZBL)
IsiHajsgaiiHJffij
YZCL)*SINH(YZCL?S1N(YZCL)*CQSH(YZCL)
+SIN(YZCL)*COSH(YZCL)us- JSiNH(YZCL;i+SlNH(YZCL
' Y&CL'
F8ZCsS1N(YZCL5-SINH(YZCl:
F10ZC=COS(YZCL)-COSH(YZCL)
P1Y=(-2,0*YYBL*F1YB+RY*F5YB)/(RYF3YB+YYBL*F5YB)
MiZ5KWKttIJ?HttH6fKa>
76
10
TORBL)1+1, /TAN.FORMAT(SE13,5
TB=4*L+5
DO 10 0=1, IBDO 10 K=1,IB
5ZB)
6'
hEB*BYB*YYB*PlY"S[l|lJs|6*bZB*YZB"*PlZ*CO**2,*I
1*SI*2,+EC*BZC*YZC*F5ZC/F3ZC
Z(l,2)aECBZC*YZC*F8ZC/F3ZC
Z(l,L+2)at;B*BZB*YZB**2,0*P2Z*CO**2, +EB*BYB*YYB
CljLtSjaEB'
i)
*C( |*SI*(-BZB*YZB*P1Z+BYB*YYB*P1Y
168
Z(l,3*L+5 3=EB*CO*SI*(-BZB*YZB**2.*P2Z+BYB
L*YYB**2,*P2Y)Kal
Z(0,L+K+3)=Z(i,L+3);Z(0,3*L+K+5)aZ(i,3*L +5)..
.:K
=KU ...
t K..P._...
Z(L+l,L+l)aZ(l,l)_Z(L+l,2*L+35aZ(l,L+3)
Z(L+l,4*L+5)aZ(l,3*L+5)
DO 3 0 ..0 f 2 aL_J
l
l*BYB*YYB**3,*P3Y*SI**2,>EC*B*ZC*YZC**4.*A3*CC
22 (L+2l, L+ 3 3 =Z ( 1 . 3*L+5 ) . . P. :...:., j: .:.
DO 45 0=L+4,2*L+3
45 Z(L+2,0)aZ(L+2,L+3)ZCL+2 3*Lt5)aEB*CO*SI*(BZB*YZB**3.*P3Z-BYB
1*YYB**3,*P3Y)^r B
---
DO 46 0=3*L+6,4*L+5
46 Z(L+2,0)=Z(L+2,3*L+5)
ZCIctlC)aEB*BZB*YZB*PlZ*SI**2,EB*BYB*YYB*PlY
l?CO**2i+SHRC*POLC*TORC/TAN(TOftCL3
Z(IC,1CU3=-SHRC*P0LC*T0RC/SIN(T0RCL)
KaL+4
lBa2*L+2
DO U 0=IC+1,IBZ(0,K)=Z(XC,IC)+SHRC*P0LC*T0RC/TAN(TORCL)
Z(0,K+1)=Z(IC,IC+1)
11 K=K+1Ib=2*L+3^^Xg)=Z(IC,IC)
8?j}8)abt$6iB**ZBl**2.*P2Y*CO**2,
**2,*P2Z*S1**2,+B*BYB*YYB
12 K=K+1
Z(IB,IB)=SHRB*P0LB*T0RB*P4+EC*BXC*YXC*F5XC
1/F3XC
Z(IBUB + 1)=EC*BXC*YXC*F8XC/F3XC
K=2*L+5
- mitu1
-
z'(oiK)=Z(IC:iC)+EC*BXC*YXC*F5XC/t3XC
Z(0,K+1)=Z(1C,IC"U)
Z(0,L+K)=-Z(IC,lB+3)
Z(0,L +K+2)=ZdC,IB+ 3)
13 K=K+1
IB=3*L+4
lD=4*L+4
Z(IB,IB)=Z(IC,IC)
00:ZCIblxDJs-ZCI^IB+gJ
r.QiJZ(Ib,IDtlJ=-Z(IC,IB +U
lof* i(IB,i!6)a-EB*BZB*YZBUB?i!6ja-EB*BZB*YZB**3,*P3Z*SI**2.-EB*BYB*YYb
i**3.*P3Y*CO**2.-EC*BXC*YXC**3,*F6XC/F3XC
Z(IB.ABtl)=EC*BXC*YXC**3.*F7XC/F3XC
K=3*L+6
IC=3*L+6
lB=4*L+4
2Vo2^)=Z(Sfi-l^C-l)-EC*BXC*YXC**3,*F6XC/F3XC
Z(0)k+1)=Z(IC-1,IO
DO 48 J=1,IB
DO 48 Kal, lb , ,
DO 25 K=1,1B
Z(0,K)=Z(K,0)
7 ,>.l, ' 10-u-l J21,24,25
7 p . /. _' Co.i'i I u lie7 ; 3 . 2 4 Cl.iu UuUt.7 >*. Rfc I UKim
7 >- 6 . c , , 0
169