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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, bomstadwayne@aol.com

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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

. . . CCO

0.

.

OCO

. --*c\C

----rc

\C

----rclo

a

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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.