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University of Central Florida University of Central Florida
STARS STARS
Retrospective Theses and Dissertations
1975
Simulation of a Spring Constrained Hypocyclic Roller Mechanism Simulation of a Spring Constrained Hypocyclic Roller Mechanism
Wayne R. Bomstad University of Central Florida, [email protected]
Part of the Engineering Commons
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STARS Citation STARS Citation Bomstad, Wayne R., "Simulation of a Spring Constrained Hypocyclic Roller Mechanism" (1975). Retrospective Theses and Dissertations. 139. https://stars.library.ucf.edu/rtd/139
SIMULATION OF A SPRING CONSTRAINED HYPOCYCLIC ROLLER MECHANISM
WAPNE R. BOBSTAD B.S.E., Florida Technological University, 1973
THESIS
Submitted in partial fulf21llment of the requirements for the degree of Master of Science in Engineering
in the Graduate Studies Program of Florida Technological University
Orlaado, Florida 1995
TABLE OF CONTENTS
Page
TABLEOFCONTENTS . . . . . . . . . . . . . . . . . . . . . . . i
LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . ii
. . . . . . . . . . . . . . . . . . . . . LIST OF ILLUSTRATIONS iii \
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . iv
CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . . .
v.
APPENDIX
. . . . . . . DEVELOPMENT OF THE MATHEMATICAL MODEL
Spring A n a l y s i s . . . . . . . . . . . . . . . . 6
Roller Analysis. 11 . . . . . . . . . . .
DISCUSSION OF RESULTS. . . . . . . . . . . . . . . 23
CONCLUSIONS AND RECOMMENDATIONS 35
COMPUTER PROGRAM 37
Computer N o m e n c l a t u r e . . . . . . . . . . . . . 38
Computer Source L i s t i n g . . . . . . . . . . 42
COMPUTER PROGRAM SABPLE OUTPUT DATA LISTING SO
FOOTNOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
BIBLIOGRAPHY... . . m e . . . . . . . . m . . . . . . . . . . 58
LXST OF TABLES
Table Page
1. , Computer Program Basic Input Data. . . . . . . . . . 15
2. Computer Program Basic Output Data . . . . . . . . 16
iii
LIST OF ILLUSTRATIONS
Figure Page
1 . Schematic of the Spring Ccmetrained Hypocyclic Roller Mechanism . . . . . . . . . . . . . . . . . . . 2
2 . MechanismGeometry . . . . . . . . . . . . . . . . . . . . 5
3 . Spring Set-Up for Fini te Difference Solution . . . . . . . 7
. . . . . 4 . Set-Up Used i n Solving f o r the Binding Moment. Mi 8
5 . Correction t o the Last F in i t e Difference Step . . O m . . . 10
6 . Roller Free Body Diagram . . . . . . . . . . . . . . . . . 12
7 . Computer Program Generalized Block Diagram . . . . . . 17
8 . Programstructure . . . . . . . . . . . . . . . . . . . . . 2 1
9 . Geometry Defining Angle $ . . . . 25
10 . Maxi- Bending St ress i n t h e Spring fo r One Camplete Cycle of Operation . . . . . . . . . . . . . . 26
11 . Tangential Spring Force Acting a t the Spring's FreeEnd . . . . . . . . . . . . . . . . . . . . . . . 28
12 . Radial Spring Force Acting a t the Spring's Free End . . . . . . . . . . . . . . . . . . . . . . . . . . 29
13 . Tangential Companent of the Roller Mass Center Acceleration . . . . . . . . . . . . . . . . . . . . . 30
14 . Roller Tangential Force Arising From the Roller Tangential Acceleration . . . . . . . . . . . . . . . . 32
15 . Roller Fr ic t ion Force Arising From the Angular Acceleration of the Roller . . . . . . . . . . . . . . . 33
Because of the extensive itse of mathematical nomnenclature i n this
work, the sydiols are defined as they are introduced into the various
analyses. Therefore, an extensive nomenclature list i s not presented.
Instead, a brief summary of the more frequently used sgmbols is
included here.
'Definition
Bending moment, in-lb . 4
Moment of inertia, i n . 2
Modulus o f elast ic i ty , lb/in . Mass, lb.
The author wishes to express his gratitude to his advisor,
Dr. D. B. Wall, for hie able guidance throughout the study. Thanks
are also given to Dr. R. C. Rapson, Jr., and Dr. A. H. Hagedoorn for
their ideas and criticism.
Special thanks are given to Dr. .C. E. Nuckolls for his time
m d interest throughout this project . m
Appreciation is also extended to Dr. T. C. Edwards for his
ideae, wise advice and encouragement.
Deepest appreciation is expressed to the author's wife,
Henrietta, for her interest and encouragement throughout this work.
ABSTRACT
Hypocyclic mechanisms arc the .basic building blocks in the design
of many widely used mechanical systems such as gear differentials,
computing devices'and other useful instruments. This paper presents
a unique variation to the conventional hypocyclic system configuration
in that the rotating elements are spring-constrained instead of rigid
arm constrained. A mathematical model was developed to simulate the
operational characteristics of the mechanism. The model was coded in
Fortran IV conputer language and a simulation survey was conducted for
a set of geometrical and system constraints. The results of this survey
indicate that the rathematical mdel could be a useful tool in the
parametric study and possible design and application of a spring
cone trained hypocyciic roller mechanism.
The objective of this work wab to Bimulate the operation of a
spring constrained hypocyclic roller mechanism. It is the purpose
of this work to supply basic analytical information which could aid
future investigation and possible design and application of such a
mechanical ays tem.
Hypocyclic mechanisms, as stated in Reference 111, describe in
general a large f&ly of mechanical systems in which one or more
concentric elements are rotating about a moving axis while at the
same time are rolling along the inner contour of a concentric cavity.
These types of mechanisms have many unique applications in gear
differentials, computing devices, and other useful instruments.
The hypacyclic mechanism considered here ie shown schematically
in Figure 1. This mechanism is unique with respect to other hypocyclic
mechanisms in that the rotating elements are spring constrained instead
of rigid arm constrained, which adds a high level of complexity to the
problem.
The mechanism consists of a stationary housing with circular
cavity, a rotor, a roller and 'a flexible spring connecting the roller
to the rotor.
The rotor is located eccentrically within the housing and is
assumed to rotate at a constant angular velocity. The roller element,
pulled along by the spring and.puehed radially outward by the centrifugal
and apring forces, rolls along the houbipg inner.wal.1. The roller
rotates at a varying angular velocity due t o . thh changing geometrical
configuration of the system.
Fig. 1 - Schematic of the Spring Constrained .. Hypocyclic Roller Mechanism
The mathematical model is developed to simulate the operating
characteristics of the spring constrained hypocyclic roller mechanism
which is subject to a given set of getmetrical and system constraints.
The geometrical constraints include physical parameters such as housing
size, rotor eccentricity, spring size, rotor diameter and roller size.
The system constraint is rotor speed.
The ro l l e r kinetics are related direct ly t o the changing . . geometrical
configuratfcm of the spring, L,e,, its deflection characteristics.
However, the geametrical configuration of the epring is also, i n part,
a functian'of the ro l l e r kinet%cs which contribute t o the spring force
system. This complex' interiiependence is further complicated by f i n i t e
spring (thin bean) deflections caused.by both l a t e r a l and axia l forces.
Only steady s t a t e conditions, i.e., constant rotor angular velocity,
are considered f o r ' a speed regime where the ~ c c e l e r a t i m forces within
the spring are negligible. That is, the spring is aesumed mass-less.
The jus t i f ica t ion fo r t h i s is that a t low rotor speed regimes which
could be useful fo r some machinery, the centrifugal.forcee and a l l other
acceleration forcee within the spring e re -11 compared to the
acceleration forcee of the ro l l e r and the spring forces that a r i se due
t o flexure.
CHAPTER XI
DEVELOl;mENT THE MBlXEMATICLIL MODEL
The pert inent geametrical parameters of the spring constrained
hypocyclic r o l l e r mechanismare b h m ' i n Figure 2. The direct ion of
rotor ro ta t ion is assumed counter clocbcwise.
The namenclatore f o r Figure 2 is as follows:
mle fran housing horieuntal axis t o l i n e QO,
measured counter-clockwise as shawn
angle from houeing horizontal axis t o l i n e PC
meamred count er-.clockwise a s ahawn
center of rotor rotation
center of housing
ecc ro tor eccentricity; off set
point at which the spring ie fixed t o the rotor;
epring'e r-y coordinate system origin
length of unbent spring; measured along spring x-axis
spring thickness
free end of spring; point a t which r o l l e r is attached
t o the spring
L' X Coordinate of point P when the spring is i n the bent
posit ion
Y coordinate of point P ( = o when spring is unbent)
angle l i n e PC.makes with the spring x-axis
Y angle apringl s =-aria is rotated f ram line QO
W xqo
angular velocity of rotor ' element - (constant)
radius of the' e tat ionary housing
R radius ,of the rotor qo
R r radius of the roller
%c length of line PC %
Fig . 2 - Mechanism Geometry
6
The following paragraphe c~mAder the analysis of'the spring and
roller' elmenta,
.... ..-....... . . . . . . . . .
'Sprhg 'Analysis
The spring element in the mechanism can be considered as an end-
loaded thin rectangular beam with uniform cross-section. Since large
deflections are involved, the general curvature equation [2]
must be used in.the spring bending analysis.
This ia a highly melinear ordinary differential ;equation which,
fortunately, can be solved quite easily and accurately using a finite
difference numerical method. [3] The finite difference method breaks
the differential equation up into a number of sfnnr1taneous algebraic
equations which may then be solved by the digital computer or any other
well-known method. The resulting solution is an approximate solution
to the original differential equation.
The finite difference approximations for the first and second
derivatives of y with respect to x at i + 2
are substituted into equation 2.1 to get
Solving for the spring def lec t ioa a t 1+2 yields
Figure 3 i l l u s t r a t e s how the epring is set-up fo r the f i n i t e
difference solution. The length of the spring is divided up i n to s m a l l
Ax increments the f i r s t of whahich starts on the "-r" s ide of point Q,
the origin of the spring coordinate system Where the spring is
cantilevered t o the rotor.
The reason for s t a r t i ng here is t o equate the f i r s t two spring
deflections t o zero which simplifies the analysis.
Y
X - - -X
Fig. 3 - Spring Set-Up fo r F in i t e Difference Solution
The force system a t point P is resolved in to two components; a force
vector i n .the r d i r e c t i o n and a- force vector .in the y-direction. Since
these two forcee act simultaneously to cause bending of the spring, the
t o t a l deflection is no longer aelinear function of the forces and cannot
be found by superposition methods. Due to large deflections an i t e ra t ive
solution is required.
Using equation 2.5 the f i n i t e difference eolution w i l l begin a t
4 and etep i n the +x direction u n t i l the end of the spring is reached.
Rewriting equatfon 2.5 fo r the deflection at any xi yields
However, before t h i s equation can be solved the bending moment a t xi, Mi,
amst be known. Using the method of sections s l i c e the spring a t xi a s
shown i n Figure 4. S w m i n g moments about x yields i
- 'Y
,Y
P Fx ----- -T yi 1
X a -
I X i , (i-2)AX do
Fig. 4 - Set-up Used i n Solving for the Binding Moment, Mi
Asaume tha t F and Fx are hum, This leaves three amknowns i n the Y
1 moment equation: .L , 6, and yi, By assw4ng s ta r t ing valueo f o r these
three tmknarvns the beam equation can be'solved by an i t e ra t ion procees.
The required i t e r a t i on process is:
1. Assume in i t i a l . values f o r L', 6 and y3.
2. Solve equation 2.7.
3. Solve equation 2.5.
4. Repeat oteps two and three f o r the en t i r e spring length at which
time new values for L', 6 and y are obtained. 3
5. Repeat s teps two through f ive u n t i l the nth i t e ra t ion produces
values f o r L' and 6 which d i f f e r insignif icantly with those
produced by the n-1 i te ra t ion .
The numerical solution (equation 2.5) s teps i n the "+x" direction
u n t i l
C A S > L - (2.8)
where ZAS is an apprdmat ion f o r the curved spring length which fo r
small &x's is a close approximation ( refer t o Figure 3).
However, a correction must be made t o the l a s t s tep because the
f i n i t e difference solution, except i n a rare case has stepped past point
P. Figure 5 shows how t h i s correction ie made. After the correction,
L' and d become respectively the x and y coordinates of point P.
Assuming Hooke's Law is followed, the pure bending s t ress , 8 , i n
each Ax-element of the th in rectangular beam can be expressed as,
'Mh ab =-I-
The lost finita d i f f l L \
step went to here 7,' -i' %.< \ as\
e- Correction
I // bock . to /hm
7
Fig. 5 -- Correction to the Last Finite Difference Step
where :
h = spring thicknees/2
I = spring moment of inertia
M = bending moment
In the foregoing analysis it was assumed that the force system at
the end of the spring required to bend the top of the spring to the
unique position P was.knuwn. However, this is not the case. The
procedure to f ind the required force system (Fx and F ) at P is Y
discwsed in a subsequent section.
R 6 l Z e t h a l y a i s
To analyze the forces' exerted on' the r o l l e r element i t i s
considered as a f ree body existing i n e? general position with a l l
known forces indicated, as shown i n Figure 6, where
F centrifugal force due t o the centrifugal acceleration C
of the ro l l e r
Ft tangential force arising from the tangential acceleration
of the ro l l e r
Ff = ro l l e r f r i c t ion f o ~ c e .
F = normal reaction force a t the housing wall. n
F - radia l component of the apring force sr
F = tangential component of the spring force st
are the s ix ' fo rces influencing the motion of the rol ler . The rol ler-
to-spring connection at P is assumed fr ic t ionless , otherwise there
would be another tangential force arising from f r ic t ion a t the pinned
joint.
The centrifugal force, Fc, due t o rotor rotation is
where m is the ro l l e r mass and a is the normal component of the ro l le r n
mass center acceleration defined as,
where w is the angular velocity of l i ne PC and R is the radial . rpc PC
distance t o the center of rotation. Fc is always directed radial ly
outward.
Fig. 6 -- Roller Free Body Diagram
The tangential force, Ft, arising from the tangential
acceleration of the roller is,
where m is the roller maas and at is the tangential component of the
rolaer mass center acceleration defined as,
a = A t
R (2.13) SPC PC dw
where A is the angular acceleration (e) of line PC. A is =PC rpc
calculated through a roller-to-spring iteration process which will be
discussed subsequently.
Assuming the r o l l e r rolls without slippipg, the r o l l e r f r i c t i on
force, Ff , is found by' summing maaknte ' about point P, which yields
2 where I i e the r o l l e r mass moment of i n e r t i a (1/2mr ), a is the r o l l e r -
angular acceleration and Rr is the r o l l e r radius. Again, assuming the
r o l l e r r o l l s without slipping the r o l l e r angular acceleration, a, is
given by
a r a / R t r
(2.15)
where a is the tangential acceleration of the r o l l e r and R is the t r
r o l l e r radius. I f the r o l l e r s l i p s the f r i c t i on force is
Fr = IJ Fn (2.16)
where is the coefficient of f r i c t i on between housing and r o l l e r and
Fn is the no-1 reaction force which is discussed i n the next paragraph.
The normal reaction force, Fn, at the housing w a l l and the spring
force components, *sr and Fgts can be solved f o r by employing the two
force equilibrium conditions. That is, the summation of all forces i n
the r ad i a l and tangential directione are zero. Thus
f Z F r = O = F s r + F c - F n
+ + e F t = o = F f - F t - F st (2.18)
Fram equation 2.17, the normal force a t the housing wall is equal t o
the algebraic sum of the radia l spring force and the centrifugal force,
1.e. ,
P I F + F c n sr
Fn is always directed radia l ly inward, as sham i n Figure 6.
The spring.foforce components, and F are the forces resulting st'
from the spring's bent ccmdi,tione and from accelerating .the roller along
the inner contour of the- housing. F is the radial spring force sr
component and is always directed radially outward. F r m equation 2.18
the tangential conpanant of the spring force is
Ff - Ft
Assuming that the radial and tangential roller accelerations are
known, the only remaining unknowns in the roller force analysie are
Fn and Fsr. As can be seen by equation 2.17, these two factors are
dependent. Hence, one force must be known- before the other can be
solved for.
In the foregoing, the spring and roller analyses were conducted
assuming that the forces causing spring flexure and the accelerations
causing the roller forces were known. However, due to the interdependence
between spring flexure and roller kinetics these forces and acceleration8
are not known initially. They can be found through an iteration process
which couples the spring and roller analyses together. This iterative
procedure is discussed in the nact chapter.
.CHAPTER 111
COMP?J!I'XR ' PROCEDURE
The computer procedures for s i m l a t i n g the mechanical system were
developed using Fortran computer'language. The program was designed
t o calculate the syatemts r o l l e r and spring dynamica a t selected rotat ional
internale vhen supplied with the b a s k input data l i s t e d in Table 1.
TABLE 1
Canput- Program Basic Input Data
Input Parameter
Rotor rotational speed
Rotor eccentr ici ty
Rotor radius
Housing inside diameter
Roller radius
Roller mass
Spring thickness
Spring length
Coefficient of f r i c t i on between
r o l l e r and housing wall
Angular increment between each
station
Computer Svmhnl RPM
ECC
RSS
RR
R M .
TSPR
LSPR
-
I Units
rev/min.
inches
inches
inches
inches
inches
inches
dimensionless
degrees
Baaic.computer output data.is listed in Table 2. A ccmputer program
TABLE 2
. . . . . . . . . . . . . . . . . . . . Cos~puter:.Progrctn-Baeic Output -Data
. . .
+
Output -Parameter . . . . . .
.
Rotor rotationel angle
Line PC rotatianal angle
Roller centrifugal acceleration
Roller tangential acceleration
Roller angular acceleration
Roller rotational speed
Roller friction force
Roller tangential force
Roller centrifugal force
Housing wall normal force
Tangential spring force
Radial spring force
Spring merimurn bending stress
. . Units
degrees
degrees
in/sec 2.
in/sec 2
radlsec 2
rev/min
lb f
lb f
lb f
lbf
lb f
lbf
lb/in2
Hathematical
. . . . . . . . .SyPbol
8
$
a n
a t
a
-
Ff
Ft
F C
F n
F st
F sr
CJ
C-uter
. . . Sgrbol
THETA
BETA
ANRMC
ATRMC
AR
SPDR
FF
FT
FC
FN
FST
FSR
BsHAX
Pigwe 7 a slapllified'block diagtcm of the computer program
ah&ing the' main eteps in the: computer. procedure.. The' f t r e t major etep
a f t e r receiving the input data is to -ca lcu la te constants and t o do
several i n i t i a l value as~igmnenta. 'These i n i t i a l value assignments are
used to start the spring flexure calculations and w i l l b e discussed
subsequently. The spring calculations are n a t , followed by the r o l l e r
calculations. Pinal ly , the output data l ie ted i n Table 2 is printed.
Fig. 7 -- Carmpoter Program Generalized Block Diagram
Due t o the interdependence between spring flexure and ro l l e r
kinetics, an i t e r a t i v e calculational procedure is required t o solve
for the spring.end r o l l e r dynamics.
Spring Cole's *
. 1
lNPUT Table I
j
Roller Calc's
w b
Cosrston t Calc's and ini#iol Value -b -b
A s s ~ ~ ~ ~ ~ s +
L
OUTPUT Table 2
The calculational procedure required is:
'STEP . . . . . . . . . . . . . . . .
. . . . . . . . . . . _ . . . . . . . . - . 'PROCEDURE . .
Assume a starting value for Fsr and set F st
choose initial spring deflections (L' and 6
will bend in the proper direction.
Solve for the spring flexure (L' and 6) and
stress (a ) due to the applied forces, Fs Inax
Increplent F either negative or positive, sr
2 and 3 until the spring tip intersects the
generated by point P (refer to Figure 2).
Calculate $ and atore all pertinent values.
Rotate the rotor through a small angle, de.
Repeat steps 2 through 5 until the rotor has rotated through
one full revolution (+360°).
Recall the stored values of f3 and calculate the angular
velocities and acceleration line PC. Using finite differences
the angular velocity of PC, w at position j can be found by rpc9
where 2dt is the time difference between the 8 position and j+l
the $ position. The angular acceleration of PC, A j-1
spc' at 5
is given by
8, Us*g equation'2.10 through 2.20 calculate the roller forces
and the spring tangential:force, Fst, and store all pertinent
values,
9 , Check for roller slippage, -i.e., the -roller will slip when the
rolling friction force, Ffs exceeds the product of the normal
force, F times the'wall-to-roller friction coefficient. p. ns
If Ff (F IJ - roller does not slip 11
If Ff > F p - roller slips n
10 , Rotate the rotor through a small angle, de.
11 Repeat steps 7 through 10 until the rotor has rotated through
one .full revolution ( 8 = 360') . 12. N w , with the new values for F repeat steps 2 througha12 until st
the nth calculated force values are not significantly different
than the n-1 values,
F - ~ e . 8 i l l u s t r a t e s the s tructure of the computer program used
t o perfdm the required calculational procedure.
Subrontine BENDSP performs a l l spring calculations. Spring flexure
is solved using the f i n i t e difference method'developed i n Chapter I1
and the bending stresses a r e found by applying equation 2.9. I te ra t ion
loop n d e r e d "100" iterates on BENDSP t o es tabl ish convergence fo r
the spring deflections, WRIM and DELTA.
After the spring calculat ions.are performed, BETA is calculated
and stored f o r l a t e r use i n determining the angular velocity of l i n e PC.
N e x t , TBETA i e incremcnted by DTHET and stepping loop numbered "300"
is traveled u n t i l a f u l l 360 degree ro ta t ion is obtained.
Thus far, a l l desired spring kinet ics and other geometrical data
has been calculated and stored for each J-station (angular position) fo r
one f u l l ro tor revolution. These values are now ready fo r use i n
calculating the ro l l e r kinetics.
After the calculational procedure ex i t s the "300" loop, the program
reca l l s stored information t o calculate the accelerations, forces and
ro ta t ional speeds of the ro l ler . A check is made t o determine i f the
r o l l e r was slipping. I f the ro l l e r slipped, an appropriate message is
writtensand f r i c t i o n force, FF, is set equal t o MU * FN. The calculations
a r e printed and the procedure is repeated f o r each J-station u n t i l the
f u l l 360 degree rotat ion is achieved. Here, again a l l desirable values
are stored.
After the r o l l e r dynemics have been determined, printed and stored,
i t e r a t i on loop nrrmber "299" w i l l begin the whole calculational process
from spring through r o l l e r dynamics and over again. This time the
assign initial values
Compute coordinates of point P relative to point Q
100
Tests convergence of LPRlM and ELTA
1 Compute angular position of line PC , .BETA 1 I Increment THETA I
I Compute roller dynamics 1
1 Check for roller s l i ~ v a a e 1 w
I Compute tangential swing force. FST 1
I Com~ute housino m o l f o m
I
Print colculotions
Tests convergence of force values
F i g . 8 -- Program Structure
praceaa will hegin w&th.pre~iollaly calculated spring deflections, forces
and -lea' instead of the inittalal. values' that were used; at the start.
The program will continue this iteration process until convergence is
obtained ,
If any of the iteration'loops are nonconvergent an appropriate
error 'message is written and the program will terminate.
(x3APTE3R rv
DISCUSSION OF ' RESULTS
Following the development of the computer program, a simulation
survey of the mechanism was conducted for a set of geometrical and
system constraints. This section of the thesis discusses the data
obtained f r m this survey.
All the data presented here are fram the computer program camputations
(see Appendix B for the program lising and a sample portion of an output
data list). Verification of these data was conducted by hand calculations
and graphical analyses at a random sampling of angular positions. However,
these techniques only serve as a check for major computational errors.
Optimum verification could only come from closely controlled laboratory
testing of an operational experimental model, but this is beyond the
scope of this.worc Also, it was deemed beyond the scope of this thesis I
to present a detailed parametric study of the mechanical system. It is
anticipated that parametrice will be a continuing effort over a lengthly
time period. Therefore, for this initial simulation survey only one set
of geometrical and system constraints were used.
Although the results presented here represent a simulation survey
conducted for a fixed set of operational constraints, various combinations
of input data were briefly investigated to verify that the computer
model fuhctiuned properly for a wide range of mechanism geometries and
operational speeds. It was believed that the results of this survey
represented a ee t of operationkl characteris t ics which are generally
typical for :. the spring constrained." hypocyclic roller mechanism.
The ge&trical and syst- constraints used i n this survey were:
Housing radius, Rss = 4.00 inches
Rotor radius, R . qo
Rotor eccentricity; ecc =
Rotor speed
2.00 inches
0.20- inches
500 rpm
Roller radius, Rr
Roller mass, Rm
Spring length, L
Spring thickness, T
Spring modulus, E
0.50 inches
,100 l b
2.50 inches
0,02 inches
30 x lo6 lbs/in2
Fr ic t ion coefficient, p r: .30
Operational characterietice, or system responses t o the above
constraints, investigated throughout a complete rotor revolution
included :
Maximum bending stress i n the spring. amax
Spring tangential force, Fst
Spring rad ia l force, F sr
Roller tangential acceleration, at
Roller tangential force, Ft
Roller f r i c t i o n force, Ff
Roller speed
The resu l t s of this investigation are presented i n Figures 10 through 16.
It -7 .be helpful a t t h i s time. t o redefine two parameters which a r e
used extena$.vely. i n the date~presentat ion, These two' parameters are the
symbol' tt@tt and the term "cycle. " . AS shown i n Figure 9, 8, used
extensively i n the graphics presentation of data, is the angular position
of l i n e PC with respect t o the'kero degree point on the housing
horizontal axls. The angle is measured counter-clockwise. Line PC is
th* radia l l i n e from r o l l e r mass center: (point P) t o .the center of
rotat ion (point C). A cycle is defined as one f u l l 360 degree rotat ion
of l i n e PC.
Fig. 9 -- Geametry Defining Angle 6
'Maximum bending stresses i n t he spring for each f i ve degree angular
position of l i n e PC a re plotted i n Figure 10. As can be seen i n the
figure, the highest bending stresses occured a t the zero degree position
of l i n e PC and the nluhnmt bending stress reached its minirrmm value a t
Spring maximum bending stress ( k s i )
- Angular position of l ine PC (degrees)
Pig. 10 -- Max- bending stress in the spring for one complete cycle of operation using: rotor speed = 500 r p , housing radius = 2.0 inches, rol ler radius = .5 inches, spring length = 2.5 inches, spr ng thickness = A20 inches, 8 spring modulus = 30 x 10 p s i , rotor eccentricity = .20 inches
180 degrees, By reviewing the mall diagram contained i n the f igure it
can b.e seen tha t these data a r e coxisistent with the rnalnnrm and minimum.
spring flexure points,
The tangential spring force Fst and t he . r ad i a l spring force Fsr
which gave rise t o these bending stresses are.shown i n Figures 11 and
12 respectively. The placement of these forces on the end of the spring
is shown i n force system diagram supplied on each figure. Generally
speaking, the maximum and minimum values of these force components matched
with the marhum and ~~ bending stresses. A deviation i n the rad ia l
spring force c u m near the 100 degree angular position is noted. A
log ica l explanation fo r this is tha t Fsr has t o momentarily supply Inore
force due t o a higher r a t e of decrease i n F st. Close examination of
Figure 10 reinforces t h i s explanation-by revealing a somewhat increased
slope i n the Fst curve near the 100 degree position. The small deviation
i n Fst appears t o have had a substant ial effect on F due t o the much sr
greater e f fec t tha t FBt has on the spring flexure (refer t o Figure 12
force system diagram). For example, i f Fst decreases some small amount,
dF, F must increase an amount greater than dF i n order t o maintain sr
the spring t i p a t point P. The reason fo r t h i s is that F ac t s through st
a longer moment arm than does F sr.
The tangential component of the r o l l e r mass center acceleration
data are shown i n Figure 13. As expected, t h i s acceleration component
changed sense during the cycle and was re la t ive ly evenly dis tr ibuted
on e i t he r s ide of the zero acceleration reference l ine. As seen i n the
f igure, zero tangential acceleration values are obtained a t B equal t o
approxiPately 120 degreewand again a t 300 degrees. As expected, there
Tangentiu 1 spring force, Fst
( Lbf)
Q -Angular position of line PC (degrees)
Fig. 11 -- Tangential spring force acting at the spring's free end for: rotor speed = 500 rpm, housing radius = 4.0 inches, rotor radius = 2.0 inches, ro l ler radius = .5 inches, spring length = 2.5 inches, epring thickness = .020 inches, spring modulus = 30 x 106 ps i , rotor eccentricity = .20 inches
Radio! 8 spring 'om, Fsc (Lbf 1
, p</;-A spring
1 spring t o m ryrtem
I --
I I
0 90 180 270 360
- Angular position of line PC ( degrees)
F i g . 12 -- Radial spring force acting at the spring's free end for: rotor speed = 500 rpm, housing radius = 4.0 inches, rotor radius = 2.0 inches, rol ler radius = .50 inches, spring length = 2.5 inches, spring thickness = .020 inches, spring modulus = 30 x 106 psi , rotor eccentricity = .20 inches
g -Angular posit ion of line PC ( degrees)
F i g . 13 -- Tangential component of the ro l ler mass center acceleration for: rotor speed = 500 rpm, housing radius = 4 .0 inches, rotor radius = 2 .0 inches, ro l ler radius = .SO inches, spring length = 2.5 inches, rotor eccentricity = .20 inches
i s an appra&nate. 180 de8ree separation between the two'. zero
p o b t o ;
~ o t ' s h m in ..Figure I 3 are two' places on the cume where there
were br ie f occurrences of nmnergcal noise. This nrrmer%eal noise occured
where angle B ( re fe r t o Figure 2) eqiraled 180 and 360 degrees. Since
the noise did not upset the r ee t of the curve, it was not included
and t he source of the noise was only b r i e f ly investigated. However,
through this br ie f investigation, it appeared t h a t the source of the
noise was i n t e rna l computer ccxnputation~ involving such functions as
ATAN and ATAN2. This is the only plo t i n which the noise appeared.
Figure 14 i l l u s t r a t e s tha t the magnitude of the tangential force
due t o tangent ia l acceleration of the r o l l e r , a t the ro tor speed
surveyed, is r e l a t ive ly emall.
Simulation of the r o l l e r f r i c t i o n force f o r one complete revolution
of t he ro tor is shown i n Figure 15. Comparing t h i s curve t o the r o l l e r
tangent ia l accelerat ion curve :(Figure 13) and the r o l l e r tangential
force curve (Figure l4 ) , a very close resemblance is noted. This is
as expected because by equations 2.13 through 2.16 these a re a l l d i r ec t ly
related. It is interes t ing t o note tha t throughout the whole cycle the
required ro l l i ng f r i c t i o n force was always much less than the available
pF force. n
Due t o the re la t ive ly l ow eccentr ic i ty used i n t h i s simulation survey,
the r o l l e r speed as shown i n Figure 16 did not change dras t ica l ly
throughout the cycle.
Roller .I tongentio l fofce,h (,
(Lbf)
Fig. 14 -- Roller tangential force arising from the roller tangential acceleration for: rotor speed = 500 rpm, housing radius = 4.0 inches, rotor radius = 2.0 inches, roller radius = .so inches, spring length = 2.5 inches, rotor eccentricity = .20 inches, roller maes = .10 lb.
Roller friction force, F' ( Lbf 1
Geometry defining angle Q
.I C
i4 sy! I ;$
I I
I 1
I I
I I
0 90 180 270 360
-Angular position of line PC I degrees)
Fig. 15 -- Roller friction force arising from the angular acceleration of the rol ler for: rotor speed = 500 rpm, housing radius = 4.0 inches, rotor radius = 2.0 inches, roller radius = .SO inches, rotor eccentricity = .20 inches, roller mass = .I0 l b .
Rol ler s m d
- Angular position of line PC ( degrees)
Fig. 16 -- Roller rotational speed for: rotor speed = 500 rpm, housing radius 4 .0 inches, rotor eccentricity = .20 inches
CHAPTER 9
CONCLUSIONS AM) RECClWENDATTONS
Reiterating briefly, the objective of this ,work was to simulate
the operation of a spring constrained hypocyclic roller mechanism. To
this end a oathematicala~odel was developed and coded for use on the
digital camputer and an initial simulation survey was conducted to
establish the operational characteristics of the mechanism. Based on
the meaningful results of this sumey, simulation of the mechanism
was considered a succees.
There are, however, a couple of system characteristics which merit
further investigation. One is the roller tangential acceleratio~, t
particularly the locations at which zero values are attained. Because
of the camplex interdependence between spring flexure and roller kinetics
it appears that the only sure way to verify this acceleration character-
istic is through experimentation. Therefore, it is recommended that
future investigations include an experimental model built specifically
for verifying the tangential acceleration of the roller mass center.
The other operational characteristic which warrants closer
examination is the spring flexure. Specifically, reference is made to
the deviation in the radial spring force curve (see Figure 11). Future
work should include finding a sound explanation for the irregular
portion of this curve. Here again, ultimate verification of data will
come from laboratory tests on a suitable model.
The cumputer model could.be used to-conduct parametric studies which
could clpearhead the design and application'of machinery utilizing the
spring co~trained'hypocyclic roller mechanism. Minor changes to the
model such as adding a bearing to the roller'would be desirable in
these studies,
Additionally, if high operational speeds are to be investigated in
future studies it is recommended that the spring dynamics be extended
to include the nume of the spring.
This appendh contains the'lortran IV source l i s t ing of the
computer program developed to shs!late the spring constrained.hypocyclic
roller mechanism. Pertinent coPlputer nomenclature appears before the
program listing and ca~rment cards arc placed throughout the program
to aid in the interpretation of the flow. of the computations.
'PRO- 'SYMBOL
Madmum number'.of iterations allowed for
subroutine BENDSF to stabilize on the finite
difference eolutions to the beam equation.
Code for printing the step-by-step calcula-
tions of BENDSP (omdon't print, l=print)
Angular increment between each J-point
Maximunn allowable error between the nth and
n-lth iteration in calculating the deflection
of the spring
Maximum allowable error between the, nth and
n-lth iteration in calculating the line PC
Rotor speed, (rev/min)
ecc ECC Eccentricity, rotor offset (inches)
Housing radius (inches)
Roller radius (inches)
RSS
RR
TSPR Spring thickness (inches)
LSPR
DELF
Spring length (inches)
Force increment used by BENDSP to cause
bending of the spring (lbs)
Incremental x-distance used by BENDSP in the
finite difference solution of the beam
equation (inches)
' Rotor ' radius (inches)
Roller mass (lbm)
Coefficient of f r i c t i on between the roller
and housing
Hadmuin alluwable e r ro r between the nth and AERROR
n-1 calculation of BETA
output
VARIABLE ' ' SYMBOL DEFINITION
Station number
Number of solution i t e ra t ions required t o
eat is f y error cr i t e r ion (AERROR) i n calcula-
t ion of BETA
Number of solution i t e ra t ions required t o
sa t i s fy er ror c r i t e r ion (PCERR) i n the
NBD
calculation of PC i n BENDSP
Angle from housing horizontal axis t o l i n e
QO (degrees)
THETA
BETA Angle from housing horizontal axis t o line
PC (degrees)
Exact distance between point P and point C
(RSS-RR) (Inches)
RPC
Actual calculated dietance between point P
and point C (RPC - + PCERR) (inches)
PROGRAM
DELTA
AR
SPDR
FF
FST
FSR
BSMU
DEFXNXTION
X-Coordinate of point C relative to spring
coordinate system origin (inches)
Y-Coordinate of point C re la t ive t o spring
coordinate system origin (inches)
X-Coordinate of point P re la t ive t o spring
coordinate system origin-(SPR-PRIM is the
x-direction spring deflection) (inches)
Y-Coordinate of point P re la t ive t o spring
coordinate system origin ( th i s is the
y-direction spring deflection) (inches)
Normal cmponent of the ro l le r mass center
2 acceleration (in. /sec )
2 Angular acceleration of ro l le r (radlsec )
Roller speed (rpm)
Friction force required t o maintain a
non-slip rol l ing condition o i the ro l l e r
against the housing surface (lbf)
Centrifugal force on the ro l l e r due t o
the normal acceleration component ( lbf)
The force acting normal to the housing wall
due t o the radia l ro l le r force componeats,
(lbf
Tangential caponet of spring force (lbf)
Radial component of the spring force (lbf)
Max- bending stress i n the spring (psi)
'V-LE. - PROGRAM SIMmmD SYMBOL * DEPZNITION
Angle spring's x-axis is rotated with respect
to line QO (degrees)
4 Spring nrarment of inertia (in )
Young's Modulus fot spring material (psi)
Angular velocity of line PC (rad/eec)
X-cumponent of the force system at the end
of the spring (lbf)
y-carmponent of the force system at the end
of the spring (lbf)
THIS COMPUTER PROGRAM WAS DEVELOPED M SIMULATE THE OPERATI ON OF A SPRING CONSTRAlNEO HY POCYCLIC ROLLER MECHANISM
THESISaFORT OOOIO C om20 C 00030 C 00040 C 00050 C 00060 C 00070 C 00080 REAL*8 BETAA I ACHECK, BTAA 00090 REAL LSPRIMOIIMUILPRIMILPR 00100 DIMENSION WRPC(300l~ARPC(300l~WR(300),AR(300)~FF(3~~ 001 10 1 .FT(300) .ATRMC(300) .ANRMC<3001 . B E T A A ( 3 9 ) 00 120 2.FN(300) ,FC(300) ,FSr(300-)9FSR(300) ,FYSI'R(3m) 00130 3.DELTA(300) .LPRIM(300),ANGL6(300) ,ACHECK(300) 00140 4,PCC(300),BDSMAX(30D),X22(300),Y22(300) 00150 S.ICOUT(300) .NBDD(300) gNTT(300) 00160 C 00170 C READ INPUT DATA FROM DATASETCTHESISeDATA) 00180 -C 00 190 READ(4r*,ENDe999)ITERrLP,KPRTrMHEfrDELERRrPCERR 00200 READ(4,*9ENDr999.)RPM,ECC,RSS,RR,THETAS.THETAE 00210 READ( a,*, ENb999-ITSRR, LSPR DELF. DELXVSFR 00220 READ(~,*,ENDP~~~)RQO~RM~MU~AERROR~ERROR 00230 C 00240 C PRINT INPUT DATA 00250 C 00260 PRINT 26 00270 WRITE-( 6,1) IT ER 00280 WRITE,(6,41 )LP 00290 WRITE(6.2)KPRT 00300 WRITE(6,3)DTHET 00310 WRITE(6,4)RPM 00320 WRITE(6.5)ECC 00330 WRITE(6.6)RSS 00340 WRITE(6.7) RR 00350 WRITE(6.8 ITSPR 00360 WRITE(6,9)LSPR 0037 0 WRfTE(6.101DELF 00380 WRITE(6.11 IDELX 00390 WRITE(6,IZ)RQO 00400 WRITE(6.13IRM 00410 WRITE(6.14)MU 00420 WRITE( 6.32.) AERROR 00430 WRITE(6,33)ERROR 00 440 WRITE( 6.35 DELERR 00450 WRITE(6,36)PCERR 00460 WRJTE(6,38)SFR 00470 WRITE( 6,39.)THETAS 00480 WRITE(6940.)f HETAE 00490 C 00500 PIe3e14159 . -
005 1 0 3 ~ 1 9 4 5 *P1/18D. 00520 W lrRPM*2e*PI/.60e 00530 THETM(.I )=THETAS*PI/ 180. 00540 DTHFTIMHET*P I / 1 80 00550 LPR=.8WLSQR 0 5 6 0 KSTOP=O 00570 Kobo 00580 LOOP=O 00590 DEL= ,75 00600 WRPC(l)=W1 0061 0 ARPC(l)=OeO 00620 lXIME=DTHETfW I 00630 ~90=90.+~1/ 180. 00640 A180~180~*PI/t80. 00650 A360t360e*PI/180e 00660 A6~2e*PI/180e 00670 RPCsRSS-RR 00680 Ei30 r E6 00690 B=l 00700 MOI=B*TSPRw3/-12 00710 EI=E*WI 00720 SFTsO , 0 00730 PRINT 15 00740 PRINT 16 00750 PRlNT 17 00760 PRINT 18 00770 PRINT 20 00780 PRINT 21 00790 PRINT 22 008 00 PRINT 23 008 1 0 PRINT 24 00820 PRINT 25 00830 299 J=O 00840 300 J=J+1 00850 JA=J 00860 I COUIU=O 00870 IS=O 60880 TH=THETAA ( J * 1 80 ,f P I 00890 NT=O 00900 IF(KOD.GT.O.AND~J.EQ.~)FST(J)=F~(J+I) 0091 0 IF(KOD.GT.O.AND.J,EQ.JB)FS(J)rFST(J-I) 00920 IF(KOD.~,0.ANDeJ.EQ.I)FSR(J)rFSR(3+1-1 00930 IF(KOD.m.O.AND.J.EQ,JB)FSR(J~=FSR(J-1) 00940 IF(KDD.GT.O)SFli=FST(Jl 00950 IF(KOD.GT.O)SFR=FSR(J) 00960 I F-( KOD. GT, 0 1 DELmDELTA ( J 0097 0 IF(KOD.GT,O)LPR=LPRIM( J) 00980 IF(KOD.GTeO)A6=ANGL6( J) 00990 ,C 01000 C COORDINATES OF POINT C (X2,Y2)
01130 C ' 01140 100 CALL BENDSP.(TSPRILSPRISFT9SFR,DELF9DELX*X2*Y21MOI 01 150 1 .BSMAX.DELrLPRgRPC,PC,EI ,A6gKPRTr ITER*NBD,DELERR
DELTA ( 3 )=DEL LPRIMf 3 b L P R FSR(J)=SFR .
PCCa( 3 )=PC BDSMAX(J)=BSMAX I coum ICOUW+ 1 ICOUT(J)=ICOUNT NBDD(J)=NBD ANGLZPATAN~(LPR-X~.YZ-DEL) DD=ABSC DEL-Y 2> ANGL6( J)oATAN2( DD,LPR-X2) A6=ANGL6(J) IF((Y2-DEL)eLE.Oe.O)ANOL2=APO+ANGL6(J)
104 Xl=LPR Y 1 =DEL IF(ICOUKTmOr*30)00 TO 102 GO TO 1 0 0
102 WRITE(6*27.)TH KSTOP=KSTOP+I IF(KSTOPmGE. 1O)STOP I S=I S+ 1
103 NT=NT+l NTT( J =NT BTAA=ACHECK( J BETAA ( J ) =BTAA IF(NTeEQ.l)BTAA=O.O ACHECK(J)=THETAA(J).+SYI-ANGL2 IF(NT.LT.2)GO TO 103 IF(IS.GT.O)GO TO 201 IF(DABS(BTAA-ACHECK(J)).LE.AERROR)GO TO 20J IF(NTmGTmlO)PRINT 34 IF.~NT*m.lO)GO TO 20.1 GO TO 1 0 0
20 t THETAA ( J+ I )=THFfAA (J-1 +M'HET IF(THETAA( J+ 1.) mGTeTHETAE*PI/.ISOm )GO TO 202 GO TO 300
02070 C THE RADIAL COMPONENTS OF THE FORCES 02080 C 02090 200FN(J)=FC(J)+FSR(J) 02100 C 02110 C IF THE ROUE!? SLIPS, SET FF=MU*FN 02120 C 02 130 02140 C 02 150 02160 02170 C 02180 C 02190 C 02200 02210 02220 02230 C 02240 02250 02260 02270 02280 02290 02300 C
PRINT CALCULATIONS
C R SLIPPING (FF>MU*FN)
02330 IF(FF.(J)mGTmMU*FN(-J))PRlNT 19 02340 C 02350 GO TO 203 02360 C 02370 400 LOOP=LOOP+I 02380 IFcLOOPmLT.LP)PRINT 37 02390 IF(LOOPmGE.LPIG0 TO 999 02400 KODtxKUD+ 1 0241 0 GO TO 299 02420 C 02430 C PRINT-OUT FORMAT 02440 C 02450 1 FORMAT(/NIOX,5HITER=.I3) 02460 2 FORMAT(10X15HKPRT=r13) 02470 3 FOR#AT(10Xr6HDTHET=,F4.2) 02480 4 FORMAT( 1 OX1 4HRPMs9 F8 2 02490 5 FORMAT(?OX,4HECC=,F5.4) 02500 6 FORMAT.(IOX,4HRSS=,F5.3) 02510 7 FORMATI 1 OXI 3HRRt9F5.3) 02520 8 FORMAT(10Xr5HTSPt~5F54) 02530 9 FORMAT( 1OXI5HLSPR~,F5;3) 02540 10 FORMAT(IOX*5HDELF=rF5.4) 02550 1 1 FORMAT( 10X15HDEXX=,F5.4) 02560 12 FOQMAT! I OX 4HRQO=, F5.3
-02570 13 FORMfl(10X13HRM+,F5.4) 02580 $14 FORMAT( 10Xr3HMU-9F5.A) 02590 1 5 FORMAT.( N39H THIS IS THE CALCULATED OUTPUT DATA) 02600 1 6 FORMAT( 35H FOR EACH M'HET ROTATION OF THE) 02610 . I 7 FORMAT( 29H PRIMARY ROTATING ELEMENT) 02620 18 FORMAT(29 FOR 360 DEGREES ROTATI ON 1 02630 19 FORMAT(5X * * ROLLER SLIPPING * * *) 02640 20 FORMAT( U 3 7 H J ICOUNT NBD NT 02650 21 FORMAT(41H THETA BETA ANGL6 02660 22 FORMAT.(4lH RPC X 2 Y2 02670 124HLPRf M DELTA 1 02680 23 FORMAT(4IW PC ANRMC ATRMC 02690 1 12HAR 1 02700 24 FORMAT(B1H SPDR FF FT 027 1 0 1 24HFC EN 1 02720 25 FORUAT(41H F S FSR BSMAX 1 02730 26 FORHATI 1J/ 1 OH INPUT DATA * *) 027 4.0 27 FORMAT1Af30H .NDN-CONVERGENCE AT THETAS, F7.2,/) 02750 28 FORMAT( 1.7H FYSP 1 02.760 30 FORMAT(/3X94181 02770 31 FORNAT[3X16E12.41 02780 32 FORMAT( IOX17HAERR09=9F9081 02790 33 FORMAT( I OX96HERR0R=,F9.8 1 02800 35 FORWAT(10X,7HDELERR=,F9e8) 028 10 36 FORMAT( IOX96HPCERR=,F9e8) 02820 34 FORMAT( 2OH INCREASE AERROR) 02830 37 FURMAT(fl19H NEW LOOP * * *) 02840 38 FORMAT( 10X14HSFR=rF6.3) 02850 39 FORMATI lOX, 7HTHETAS=, F80 3) 02860 40 FORHAT( tOX,7KT'HETAE=9F8.3) 02870 41 FORMAT(IOX13HLP~,13) 02880 999 STOP 02890 END 02900 C 029 t 0 SUBROL?TINE BENDSP(TrL,FST,FSR,DF,DELXr X2,Y2,MOI 02920 1,BSMAX,DELTA,LPRIM,RPC,PC,EI,ANGL6,KPRTrITER,ICOUNT 02930 29 DELERRIPCERRIKOD, TCSTOP,THETA 1 02940 C 02950 02960 02970 02980 02990 03 000 03010 03020 03030 03040 03050 03060 03070
REAL L,LPRIM,MOI , DIMENSION Y ( 3 0 0 ) ,BM(300),DEL(l00)
BSMAX=O .O PC=O l 0 Y( 1 )=Om0 Y(2)=0.0 I COUNT=O FSRReFSR PRLsLPRI M bDELTA KK=O NP=O NQ=O
N=O Y(3)=0.0 C=T/2. IF(KPRTeEQe0)GO- TO 14 PRINT 50 PRINT 51 PRINT 52 PRINT 53 PRINT 54 PRINT 55
14 FY=F.ST*COS(ANGL~)+FSR*SIN(ANGL~) FX=FSR*COS( ang16 1-FST*S? FJ e liNGLh BM(3)=FY*(LPPZM-DELX)+FX*DELTA IF(KPRT*EQoIlWRITE(6,57)N K= 1
1 1 1=1 IF(KPRT.EQ.I)WRITE(6,57)K DEtS=Oo 0
10 Y~I+2)=T'lELX**2*RM(I+2)/EI*(t .+((Y(1+2)-Y(1+1 1 ) 1/!3ELX)+x2)**(3./2. )+2e*Y(I+I )-Y(f ) BSTRES=BM( I+2)*C/M9i BSMAXmAMAX I ! RSMAX, 3STYS DELS=TSELS+SQRT(DEL,(**.- iY(1+2)-Y(1+1))**2) IF(KPRTeECa I lWRITE(6,55)IVy( I+2) ,BM(I+2)eBSTPES9DELS
1 ,LPRIM, PC,FY ,FX,DELTA Y(I+3)=Y(I+2)*1002 IF(L-DELS)1.1,3
2 BM(I*3)~FY*(LPRIM-FLOAT(I+I)*DELX)+FX*(~A-Y(1+2)) I = I H GO TQ l r !
1 ANGt7=ATAN(DELX/(Y(X+2)-Y(I+1))> XDIFF=r(nELS-L)*SIN(ANGL7) LPRIM=FLOAT(I)*DELX-XQIFF Y9IFF=CnELS-L)*COS(ANGL7)
, I)ELTA=Y ( I+?)-YnIFF DEL( K+ I )=;jZLTA I
0MC3 )=FY*tLPRIM-nELX)+FX*QELT/, TF(KoGTm1TER.AND.KKaLT. 1 >?RIIdT 5G IF(K.GT. ITER) KK=KK+1 K=K+ I IF(YaLTo.3)SO TO 1 1 IF(ABS(?EL(rO-nEL(Y-I 1 ) .LEanELERRa131 .KoGTaITFR)GCl T I 12 GO TO 1 1
12 PC=SQRT((LPRIM-X~)*~+!Y~-~)ELTA)**~) I COUNT= I COUNT+ 1 IF<ICOUNT*GE.3Od)GO TO 18 IF(NaEOoT))G!: 7 3 5 IF(ABS(EPC-PCi mLEoPCER17)C;0 TO 999
5 N=N+t IF(PC-RPC)3,4,4
4 IF(NQoEQoO)FSR=FSR+4.*(PC-~Pc)*----
03650 03660 03670 C 03680 C PRINT-OUT FORMAT 03690 C 03700 50 FORMAT ( // /32H THIS IS THE CALCULATED DATA 0371 0 51 FORMAT(37H FOR EACH STEP OUT ALONG THE BEAM) 03720 52 FORMAT4 23H FOR EACH ITERATION) 03730 53 FORMAT( N I 1 H N . 1 03740 54 FORMAT(/llH K 1 03750 55 FORMAT.( 47H I Y 03.760 I 60BELS LPRIM 03770 1 12HDELTA ) 03780 56 FORMATt3X114,9E12.4) 03790 57 FORMAT(/3X914) 03800 58 FORMAT(/29H BENDSP I S NON-CONVERGENT) 038 10 59 FORMAT(/ 28H INCREASE DELERR OR ITER) 03820 60 FORMAT(//5X18ET3.4,//) 03830 C
BM BSTRES PC FY FX
03840 18 PRINT 58 03850 WRITE(6,60)THETA,ANGL6,FST,FSRqFSRR,DELTA,LPRIM,DELS 03860 WRITE(6,60)DIPRLIPC93SMAX9FX9FY9X2,Y2 03870 KSTOP=KSTOP+ I 03880 IF( KSTOP .GE. 4)STOP 03890 C 03900 999 RETURN 039 10 END READY
APPENDIX B
CWUTER PROGRAM SAMPLE' OUTPUT DATA LISTING
Because of the-&reat number'of output data sheets generated by a
simulation survey, only a few sample sheets are preseqted here to show
the output data format.
INPUT DATA * *
THIS IS THE CALCULATED OUTPUT PATA FOR EACH WHET QmATPON OF THE P Q I M A R Y RDTATINO KEUMT F3R 360 DEGREES ROTATIbiU
J THETA 9PC PC SPD9 FST
ICOUNT NDD BETA X2 ANRMC FF FSQ
NT ANGL6 Y? ATRMC FT BSr-dAX
LPP I %I AP FC
-*-
OCO
++
+
UIW
U!
mw
00
C
Qd
W
ao
'c
--CU
dd
d
-d--
CO
C
++
+
Lc
wu
O
IOM
O
\c\lO
C)
~O
'Q
--nl
... O
GO
-w-
CCO
++
+
W LL! U
! o
em
-w-
COG
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FOOTNOTES
I A. Be Soni, Mechanism 'Synthesis and Analysis (New York, New York: McGraw-Ili l l , 1974), p, 124-128.
2~oeeph 8. Faupel, Engineering Design (New York, New York: John Wiley and Sons, 1964) , p. 109-111.
3~idney I?. Borg and Joseph J. Gennaro; 'Modern Structural Analysis '(New Pork, New York: Van Nostrand Reinhold Company, 1969),
Borg, Sidney F. , and Gennaro, Joseph J. ' Modern Structural AnaI)rsis. Ncw Pork, New Pork: Van Nostrand Reinhold Compeny. 1969.
Faupel, Joseph He Engineering :Design. New Pork, N w York : John Wi3ey and Sons, 1964.
Soni, A. H. .Mechanism Smthesie. axid 'Analy8.i~. New York, New York: McGraw-Hill, 1974.