Large Deflection of Thick Elastic Belts

149
Rochester Institute of Technology Rochester Institute of Technology RIT Scholar Works RIT Scholar Works Theses 1976 Large Deflection of Thick Elastic Belts Large Deflection of Thick Elastic Belts J. L. Sisson Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Recommended Citation Sisson, J. L., "Large Deflection of Thick Elastic Belts" (1976). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected].

Transcript of Large Deflection of Thick Elastic Belts

Rochester Institute of Technology Rochester Institute of Technology

RIT Scholar Works RIT Scholar Works

Theses

1976

Large Deflection of Thick Elastic Belts Large Deflection of Thick Elastic Belts

J. L. Sisson

Follow this and additional works at: https://scholarworks.rit.edu/theses

Recommended Citation Recommended Citation Sisson, J. L., "Large Deflection of Thick Elastic Belts" (1976). Thesis. Rochester Institute of Technology. Accessed from

This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected].

LARGE DEFLECTION OF THICK ELASTIC BELTS

by

J. L. SISSON

Approved by:

A Thesis Submitted

in

Partial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

in

Mechanical Engineering

Prof.

Prof.

Prof.

Prof.

(Thesis Advisor)

(Department Head)

DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

November, 1976

I

M^

n

ABSTRACT

The problem of large deflections in thick elastic belts is

studied using a curved beam approximation technique and incremental

strain energy computer technique. The analytical results are com

pared with experimental results and the incremental strain energy

xcomputer technique is demonstrated to accurately predict the

deflected shape of the belt. Recommendations for further work

are also included.

TABLE OF CONTENTS

Page

Abstract i i

List Of Figures iii

List Of Symbols iv, v

Chapter

I . Introducti on 1

II. Approximation Technique

2.1 Belt Tensile Stress 4

2.2 Belt Bending Stress 7

III. Incremental Computer Solution

3.1 Introduction 11

3.2 Step One 12

3 . 3 Step Two 21

IV. Experimental Results

4.1 Experimental Results 29

4.2 Comparison Of Experimental And Analytical Results . 29

V. Conclusions 33

Appendix A - Solution For The Position Of The Feedbelt

Neutral Axi s 35

Appendix B - Stresses By Approximation Technique 38

Appendix C - Solution For Ma 42

Appendix D - Solution Foris x And Sy 43

Appendix E - Computer Program

E.l Description Of Program 45

E.2 Listing Of Main Program 47

Page

E.3 Listing Of Subroutine Function Value (x) 57

E.4 Computer Data For E = 100 psi 59

E . 5 Computer Data For E = 1 50 psi 78

E.6 Computer Data For E = 300 psi 97

Bibliography 141

in

FIGURES

Figure Page

1.1 Friction-Retard Feeder 116

2 . 1 Sys tern Geometry Model 117

2.2 System Tension Model 118

2.3 Shift Of Neutral Axis 119

2.4 Belt In Unstressed State 120

2.5 Model For 5 ^ 121

2.6 Curved Beam Model 122

3.1 Strain Energy System Model 123

3.2 Constant Radius Feedbelt Model 124

3.3 Variable Radius Feedbelt Model 125

3.4 Feedbelt With Retard Roll Model 126

4 . 1 Drawi ng Of Fi xture 1 27

4.2 Picture Of Fixture 128

4.3 Experimental Data 129

4.4 Center Distance Vs. Load (Experimental) 130

4.5 X-Axis Deflection Vs. Load (Experimental) 131

4.6 Modulus Of Elasticity For Natural Rubber 132

4.7 Center Distance Vs. Load 133

4.8 X-Axis Deflection Vs. Load 134

4.9 Table Of Results'

135

B.l System Physical Dimensions 136

E-l Flowchart 137

E-2 Belt Deflected Shape, E = 100 psi 138

E-3 Belt Deflected Shape, E = 150 psi 139

E-4 Belt Deflected Shape, E = 300 psi 140

TV

SYMBOLS

A Cross sectional area of feedbelt

a Free end of feedbelt

B Horizontal distance between pulleys one and three

b Fixed end of feedbelt

b-,,b? Width of incremental areas

C Vertical distance between pulleys one and three

D1 R1+ t/2

D2 R3+ t/2

E Modulus of elasticity

e Distance between the centroidal axis and neutral axis

F Line connecting centers of pulleys one and three

F,,F2

Sections of line F

G Shear modulus

I Moment of inertia

LT Distance around centroidal axis of the stressed belt

LTL2'L3

L4'L5'L6 Segments of LT

Lo Distance around centroidal axis of the unstressed belt

Ma Moment at point a

Ms Moment at point s

A P Incremental load applied to belt

P'

Load due to pulley three

Q ,Q Dummy loads

R Radius of centroidal axis

Rl Radius of pulley one

R3 Radius of pulley three

R'

Initial radius of centroidal axis

T Tensile force

t Thickness of feedbelt

V Shear force

y Distance between the fibers from the neutral axis

X, ,Y, Center pulley one

X3,Y_ Center pulley three

X7,Y? Point on feedbelt

X7,Y} Point on feedbelt

XJ.YJ Point on feedbelt

X5,Y5 Point on feedbelt

0a Slope at point a

T Tensile strain

srD Strain on surface D

fF Strain on surface F

5*T Tensile stress

s 1 1 , To Distance from the center of curvature to the incremental areas

^ x Deflection in x direction

<Sy Deflection in y direction

4>y3 Deflection in y direction at point aa

Angle between y axis and line tangent to a point on the

feedbelt

CHAPTER I

INTRODUCTION

The original objective of this thesis involved the mathematical

modeling of a friction-retard paper feeder mechanism which performs the

function of separating a sheet of paper from a ream of paper prior to

being fed into a copier or printing press. The functional parts of the

friction-retard feeder are the natural rubber feedbelt and stationary

micro celluar-urethane retard roll (revolves at 1 RPH} as shown in

Figure 1. The feedbelt is used to drive sheets of paper over the re

tard roll where they are separated and fed into the system one at a time.

The rib on the belt, as shown in Figure 1, is used for tracking the feed-

belt about pulleys one and two.

The initial as molded shape of the the feedbelt is circular and

it is assumed that the feedbelt is stress free in this configuration. Bend

ing, tensile and shear stresses are induced in the feedbelt when it is de

formed about pulleys one and two. These stresses cause the feedbelt to

assume an anti-clastic shape and are increased further in the section of

the feedbelt that is wrapped about the retard roll. The contact area be

tween the anti-clastic shape of the feedbelt and the retard roll forms the

nip which is used to perform the major function of separating sheets of

paper before feeding.

It becomes apparent that to understand the operation of the

paper feeder, the nip must first be analyzed. The nip formed between the

feedbelt and the retard roll is a very difficult contact problem. The

first step in understanding this problem is the determination of the

anti-clastic shape of the feedbelt. The anti-clastic shape of the feed-

belt cannot be determined, however, until the stresses in the feedbelt

are known. These stresses result from the deflected shape of the feed-

belt and therefore, the determination of the feedbelt deflected shape is

the first step in this solution. The determination of the feedbelt de

flected shape and stresses is a major task due to the large deflections

involved and is the topic of this thesis.

Two different techniques are used to determine the deflected

shape and stresses in the feedbelt. The first is an approximation tech

nique using curved beam theory. This is divided into two steps. Chapter

II, Section 2.1 uses the system geometry to determine the tensile stress

in the belt resulting from the belt being stretched over the three pulleys

while Chapter II, Section 2.2 uses the system geometry and curved beam

theory to determine the bending stress in the section of the belt which is

wrapped about the retard roll. The tensile and bending stress are then

combined to determine the stress where the feedbelt is wrapped about the

retard roll .

The second technique gives a more exact solution by using

strain energy theory and a computer program to determine the deflected

shape of the belt. The loads are applied in small increments to ensure

that the incremental deflections of the belt do not exceed the small

deflection limitations of strain energy theory. This technique provides

a much more accurate solution because it defines the deflected shape of

the belt and resulting stresses rather than assuming a deflected geometry

and using this geometry to obtain the stresses. The solution is divided

into two steps. Chapter III, Section 3.2 details the equations and computer

techniques used to determine the shape of the feedbelt as it is deflected

about pulleys one and two. Chapter III, Section 3.3 details the equations

used to determine the shape of the feedbelt as the retard roll is inserted

into the feedbelt. The equations in Section 3.2 are solved using a computer

iteration routine while the equations in Section 3.3 are not solved due to

the excessive computer time that is required.

Chapter IV contains the experimental results and compares them

to the analytical results. Also contained in this section is a comparison

of the two different analytical techniques.

Chapter V contains the final conclusions and recommendations

for further study.

CHAPTER II

APPROXIMATION TECHNIQUE

2.1 Belt Tensile Stress:

The geometry of the system is shown in Figure 2.1. The centers

of pulleys one and three are described by the straight line F. The length

of the line F can be described as:

F = /(X, -X-)2+ (Y, 0)

Defining B and C as the horizontal and vertical distance between

the centers of pulleys one and three respectively, they can be expressed as:

B =

X,-X3 (2)

C = Y,-Y3 (3)

The angleG'

expressed in terms of C and B is:

G'=TAN"'

(C/B) (4)

The perpendicular distance from the center of pulleys one and

three to the center of the feedbelt D-| and D2, are functions of the radii

of the two pulleys R, and R3 and the thickness of the belt t.

D, = R, + */ -5-

D2=R3rt/2. (6-

The line F can be divided into distances F1 and ?2 by using similar

triangles. Since y, = Yz. then:

F;/F2=D,/Di(7)

Substituting of Fp= F - F, yields:

F, = F/(yD2/Df

In a similar fashion:

(8)

Fz=F/(\ + D,/Di) (9)

From Figure 2.1 it can be seen that:

Yz= CO S~1(DJFI(10)

^ = 2.0'_ya (11)

The final length of the centroidal axis is determined using

equations 5-11. The difference between the final length, Ly, of the

centroidal and the initial unstressed length, LQ, of the neutral axis,

divided by the unstressed length of the neutral axis is approximately

the tensile strain caused by stretching the feedbelt around the three

pulleys. The total length of the centroidal axis of the belt can be

divided into six separate segments, Figure 2.2.

LT=

L]+

L2+

L3+

L4+

L5+

L6 (12)

Segment L-, is the distance between the points of tangency of

pulleys two and three. Since the configuration under investigation is

symmetric, L-, = L3. From Figure 2.1 it can be seen that:

L, = L, = R SIN* + F, SIN*, (13)

Again, due to symmetry, the length of wrap about pulleys one and two,

L^ and Lg respectively are equal, and can be expressed as:

L4=

L6=

D] (TT +e'

) (14)

The distance L5 is simply the center distance between pulleys one and two,

which is given by:

L5=

x]~

h (15)

Finally, the length of wrap about pulley three, L2, is:

L2=

2D2 e'2 (16)

In order to approximate the tension in the feedbelt, the tensile

stress is assumed to be uniform over its entire length and consequently

the strain due to the increased length is:

L-

L

yyQ

(17)T

From Hooke's law, the direct tensile stress in the feedbelt:

<tt= (:te (is)

Where E is the modulus of elasticity of the belt material. Finally, the

direct tensile force in the feedbelt is obtained by multiplying (TT by

the cross-sectional area, A, of the feedbelt:

T=Arr (19)

2.2 Belt Bending Stress:

Stresses due to bending are also induced in the feedbelt when it

is stretched about the three pulleys. Curved beam theory is used to approx

imate the maximum bending stress in the feedbelt where it is wrapped about

pulley three. This point in the feedbelt contains the maximum bending

stress due to the feedbelt undergoing the largest change of curvature at

this point. The reversal of the feedbelt curvature as it is wrapped over

pulley three will cause the center of the curvature to shift from one

side of the feedbelt to the other. The position of the neutral axis of a

curved beam is a function of its radius of curvature. As the center of

curvature shifts from one side of the feedbelt to the other side due to

bending, the neutral axis will also shift to the other side of the cen

troidal axis as shown in Figure 2.3.

Curved beam theory is generally limited to small deflections

since it is restricted to small strains and rotations. However, since in

this case the deflected shape is known, it appears reasonable that a good

approximation of the bending stresses can be obtained from the deflected

shape. This bending stress is determined in two steps. The first step

determines the bending stress due to the belt being stretched over pulleys

one and two. The second step determines the additional stress caused by

the feedbelt being wrapped about pulley three.

Consider an element of the belt in its unstressed as molded

circular configuration, Figure 2.4. The distance e between the neutral

axis and the centroidal axis can be determined from the following equation :

e=R-A^(20)

-

where:

R = radius of centroidal axis

A = cross-sectional area of feedbelt

f = the distance from the center of

to the incremental area

The value of the integral in equation 20 is evaluated by approximating the

cross section of the feedbelt as shown in Figure 2.5. The equation for the

integral is:

*--rV! -L,.f,L , ... h <21'

^ilf^fff=^i^'lAppendix A contains the values of e for the specific case under consideration.

2The following equations will be used to determine the strains on

-

the outer surfaces of the feedbelt:

L =7^__ (22)

-F

=(R-e +^d0

(23)

]Harold A. Rothbart, Mechanical Design and Systems. Handbook , p.15-28

2Ferdinand Singer, Strength of Materials, p. 496

where:

y= the distance of the fibers from the neutral

axis, Figure 2.6

60 =angle of rotation of one section of the feedbeltrelative to another section caused by bending,Figure 2.6

<*= Angle between the two sections of the feedbeltbefore bending, Figure 2.6

In step one of the solution the radius of curvature for the belt

is equal to oo in the area between the two pulleys. For this case

d0 = d9 and the equations reduces to the following form:

CD=

(R-e-y) (24)

In step two, the maximum bending stress occur in the portion of

the feedbelt which is wrapped about pulley three. These additional bending

stresses are calculated utilizing the final deflected shape of the feedbelt.

From Figure 2.6 it can be seen that the length of a segment of the feedbelt

on the neutral axis is (R3 +fc)dQ . Where R3 is the radius of pulley

three and y is the distance from surface F to the neutral axis (see Figure 2.6)

As the feedbelt is deformed over pulley three, the segment of the

feedbelt on the neutral axis receives no additional strain due to the bend

ing stress and therefore, is used as a base to determine the stresses at

the extreme fibers.

10

The length of a belt segment on surface F, Figure 2.7, is

R3d9

. While the length of a feedbelt segment on surface D is, (R3+ t)d9.

Where t is the thickness of the feedbelt.

The strains on the surfaces F and D are expressed by the following

two equations.

, _R~--(R3+y-)

_ -fr (26)s+

yo 3

+ y!

,+ i) -(R3'Jd), ^'^d (27)

The calculations of the belt stresses for the specific case under

investigation are contained in Appendix B. This approach is compared with

the analysis using strain energy as well as the experimental results in

Chapter IV with further discussion found in the conclusions of Chapter V.

11

CHAPTER III

INCREMENTAL COMPUTER SOLUTION

3.1 Introduction:

In this chapter, strain energy theory is used to determine the

stresses in the feedbelt and its deflected shape as the feedbelt is de

formed over the three pulley system. In the previous chapter, the deflec

ted shape of the feedbelt is assumed and the tensile and bending stresses

are determined separately based on this deflected shape. In this chapter,

the deflected shape is not assumed, but instead, it is predicted by strain

energy theory. This same theory is also used to predict the stresses in

the feedbelt. To accomplish this, a system of equations describing the de

flected shape of the belt is written. These equations are then incorporated

into a computer program and by using an iteration routine, incremental loads

are applied until the pulleys reach the proper center to center distance.

This process is used for two reasons. First, the feedbelt must comply to

the outer surface of pulleys one and two as it is deformed. The iteration

procedure is used for this purpose. Secondly, the strain energy equations

are limited to small deflections, and, therefore, it is reasonable that an

accurate prediction of the feedbelts deflected shape is obtained if small

incremental loads are applied so as not to exceed this deflection limitation.

After each loading, the equations are initialized so each new loading uses

the previous loading as its initial values.

12

In applying strain energy theory, the following assumptions will

be used:

1. The neutral axis of the belt is the same as the

centroidal axis. This allows the belt to be

modeled as a thin ring of the same radius as the

centroidal axis.

2. The decrease in belt thickness due to di/ect tension is

neglected.

3. Once a point on the belt contacts the pulley surface, it

will not slide on the surface of the pulley and therefore

it has reached its maximum stress level.

The modeling of the system is divided into two steps. Section 3.2

develops the equations necessary to deform the feedbelt about pulleys

one and two. These equations are then solved by means of the computer pro

gram. Section 3.3 develops the equations necessary to move pulley three

into place. These equations are not solved on the computer due to the ex

cessive computer time that is required.

3.2 Step One:

In this step, the feedbelt is stretched about pulleys one and two.

The initial feedbelt model is shown in Figure 3.1. Since the initial un

stressed shape of the feedbelt is a molded circular shape, it will be the

starting point. Pulleys one and two are shown contacting the inside of

the feedbelt in this initial feedbelt shape. Equations must now be written

to allow the two pulleys to be separated to the proper center to center

13

distance by applying incremental a P loads at the centers of pulleys. As

the feedbelt is deflected over the pulleys, it must also be fitted to the

surface of the pulleys. This requires a computer iteration routine and

incremental loading as this fitting cannot be done in one step with any

degree of accuracy.

The feedbelt will be fitted to only pulley two in the computer

program, due to the symmetry of the problem. To fit the belt to the sur

face of pulley two, the deflected shape of the feedbelt must be calculated.

To do this, dummy loads Q and Q are applied at the point where the de-

y x

flection is to be calculated. Due to the symmetry of the model in Figure 3.1

only the left hand semi -circle will be considered. This can further be re

duced to the upper left quadrant by making the following observations:

1. As the feedbelt is deformed, there will be no

change of slope at point a.

2. There is a tensile force at point a equal to P/2.

3. There is no shear force at point a.

4. There is a moment,. M at point a required to maintain

cl

the condition of no change of slope. This moment

is unknown and must be calculated.

Figure 3.2 contains the model of the belt that will be used to

write the deflection equations. The first step in the formulation of the

deflection equations is the development of an equation for Mg The equa

tion describing the internal bending moment, Mg at any point s in the

feedbelt is:

Ms=

M&- '(1 -C0S9) (28)

Using Castigliano's theorem, the rotation at point a in the belt

14

is1:

-1

9 = MXrML i^sd (29)

o

/Y1It can be seen from equation 28 that

yjyp= 1. Equation 29

can now be rewritten for point a as:

C*2[/VV-(')(l-C0S9)jRd9, ,9^.=

^-5= s d

(30)Jo EI

Solving for MQ yields the following equation (see Appendix C):

M^C-i, (3,)

The general equation for M at any point in the feedbelt must now

be rewritten to include dummy loads Q and Q It can be seen that thesey x

loads only effect the moment in the feedbelt from the point of application

to the fixed end (point B). Thus, the internal bending moment is given by

(see Figure 3.2.)

Ms=hAa.-^ci-cose)-Q3[R/(cosei-cose>]-qx[^siw-si^92)] (32)

Substituting Ma from equation (31) into equation (32) and com-

a

bining terms yields:

^&=^^COS9 7.|)-Oa[R,(COS9z-COSe)]-QXi[R'(SirV9-Slh/e2)J (33)

Deflections caused by tension and shear forces must also be in

cluded. Thus, the equations describing the tension and shear forces at any

1S. Timoshenko, Strength of Materials, P- 380

15

point must be derived. The tension, T, at any point in the feedbelt is

given by:

(34)

The shear, V, at any point is given by:

(35)

The x direction is given by

(36)

where:

The ^s are evaluated in zone B-C

A = Area of cross section

E = Young's modulus

G = Shear modulus

I = Moment of inertia

Evaluating the partial derivatives of equations 33, 34, and 35

yields:

(37)

(38)

(39)

Ferdinand Singer, Strength of Materials, P. 465

16

Substituting equations 33, 34, 35, 37, 38, and 39 into equation

36 and setting dS=Rde yields: ,

% %

^ = C (P/2MC0S8K-SHY8)R'd8 C (P/a)(SltV9)(C0Se)Rd9I AE j. ca

{Z(PR'i/)(C0S9- 2/TT)(R'(SI>/e;L-SlK9)) R'd9

EI(40)

Performing the integration from q-, toj yjelds the

solution for x (see Appendix C) :

PF-V. .... t. PR'r, ..^i PR^SItfe,

(41)

The solution for the deflection in the y direction, <, , is

similar to the solution in the x direction. The deflection in the y

direction is (see Appendix D):

$ fH.es _ sifKei -.Elni..9x sifY92i3 EAEL4 2 ~1 i 2SAH X

~

-1

PR/3cos9-r -,

4.PR/rn.e-_sirvaeil

(42)

EEILbnve2 'J +

2EIL4 i 1 J

Equations 31, 41, and 42 are valid for small deflections and when

R'is a constant. Since the feedbelt will be undergoing large deflections,

the technique which is used consists of deflecting the feedbelt in stages

by increasing the load P in small increments so as not to exceed the small

deflection limitation. After the first load increment, R is no longer con

stant and the deflection equations must be modified to reflect this change.

17

In addition, as the point on the feedbelt nearest to the left of point b de

flects some amount, it will contact pulley two. This new contact point causes

the upper integration limit to be decreased by one feedbelt segment.

Load P is incremented until the feedbelt has been stretched

the proper amount.

Since in succeeding steps R no longer remains constant, equations

a

31 - 42 must now be modified. In addition, the limit of integration must

reflect the change in feedbelt contact point as the load is increased. If the

feedbelt is not circular, a line tangent to a point on the feedbelt will

not be perpendicular to the radius R. Therefore, the equations for shear

and tension components at any point in the belt must be modified. The angle

9 in equations 34 and 35 is the angle between the radius R and the x axis.

Since the feedbelt is no longer circular and R is not perpendicular to the

tangent at a point on the feedbelt, 9 is not the angle between the tangent

and the y axis. The angle between the tangent and the y axis, as shown in

Figure 3.3 will be defined as . This angle will vary as the location of

the point on the feedbelt changes. For any point X7I7 on the feedbelt

$ is:

Since R is no longer constant, the expression for Ma as a function

of P/2 must also be modified. To save computer time, the substitution,

ds = Rd e , made in equation 36 is not used. The program will begin by

defining the x and y coordinates of n number of equally spaced points

around the belt. These points are then retained throughout the calculation

18

with the x and y deflections of each point being determined for each in

crease in load P. Knowing the x and y coordinates of each point, the

moment equation can be obtained by subtracting x and y position instead of

the trigonometric manipulation used previously. Also, since the x and y

coordinates of each point are known, the distance between points, ds, can

be calculated by assuming a straight line. This approximation is acceptable

provided the spacing of the points is small.Therefore,*

from the free-body

diagram of Figure 3.3:

s l*JS\ .s.

(Slvrjds_ f

[Mol- a(x--xt.)]ds

EI*ro =

JT-Ei=

X

HHd- -^j"cx;-x,,ds=o

0

S

pj\(X',- x-.) ds

Ma =~

2 rs

(48)

(49)

i ds

From this point, the final solution for Ma is obtained using

numerical methods.

The equations for tension, shear and moment can now be written for

any pointX'-,Y7'

between points X5,Y5 and b on the belt. Dummy loads Qx

19

and Q are applied at point X5,Y5 as this is the point where the deflections

are being calculated:

T=-COS? + Q^ COS?

-

QXS\M (50)

Y=-SI*0 4-

Q^SIN^ +Q*COS0 (51)

f>/

ws = f-V-

j (*>-

x< )- q- (x;- X$)

-

Qx ( %-

Ys') (52)

In obtaining the corresponding equations for the deflections, the partial

derivatives of the axial and shear forces, and bending moment with respect

to Q and Q are determined. From equations 50, 51 and 52:x y

-T

^ -COS?

(53)

(54)

aiv)s_-cy;->v)

^M5= -U7-Xs) (55)

Substituting equations 53, 54, and 55 into equation 36 yields:

Q*.O

Qjj-fO

(P/2)(COS(?)C-SlK(?)dS

AE GA

"j EI

20

which reduces to

P- -_ Ccosp suv^ds + --

} siwflcosedss* =

ss

In a similar manner, Su is found to be:

"a "<3

-jiCcx'7-x5^ds +-^.^cx;-x6)(x;-x^d

(56)

(57)

Equation (56) and (57) will now be solved using a computer iter

ation routine. The first step in program involves the solution of the in

tegrals of equations (56) and (57) by the Newton-Cotes quadrature numerical

methods technique. The second step requires fitting the feedbelt to the

pulley surface. This is done by using an iteration routine and incrementing

the load A P until the feedbelt conforms to the pulley surface. Appendix E

contains a listing of the computer program, flow chart, detailed description

of program and the results. Figures E-2, E-3, and E-4 show the deflected

shape of the feedbelt for the various values of E and G. These results are

compared with the experimental results and results of Chapter II in Chapter IV.

Daniel McCracken and William Dorn, Numerical Methods and Fortran Programming,p. 173

21

3.3 Step Two:

Equations 56 and 57 are used in Step One to stretch the belt un

til the required center distance between pulleys one and two is achieved.

In Step Two, pulley three is moved into its proper position in incremental

stages as shown in Figure 2.1.

As the load from pulley three, P , is applied at point a, the

feedbelt will continue to deflect. Point a, however, is fixed in the y

direction due to the symmetry of the belt system. The previous boundary

condition that there will be no change of slope at point a in the belt

must also still be satisfied. To meet the boundary conditions, a load Paa

and a moment M are applied at point a.

yAn expression for P3 and M, as a function of is developed by

setting5--^= o and9a.= o at point a and solving for P and M The assump

tions in this formulation are as follows:

1. Point b is considered as a fixed end condition.

2.P/

acts at the last point of contact on pulley

three at angle3-

3. The deflection of point a. in the y direction is

zero.

4. The change of slope of point a is zero.

The first step is to write the equations for tension, shear and

moment at any point in the belt (Figure 3.4):

22

Ms=fKx7-x6) - jCose3(Y?-Y- )+ | siN93(X7-x-)-iYfa (58)

p' P'

T= P^COS? + SIN^COS?--

C0S93 SIN? (59)

P' P'

v= R-s>n? +

ysing-, swp

- cos 9-

cas$(60)

Taking the partial of M with respect to M :

a Ms

afvio-

= I (61)

Taking the partials of equations (58), (59), and (60) with respect

to P.

iL-Po,

= cos(62)

*P*.

= SlN

dlY)

>R,L

= CX7-X6)

(63)

(64)

Substituting values into equation 29:

P'

( P'^(ffUx7-X4v(a) cose^cY-- Y6^ + (.T/ sme-fyx^-^a.) ds

(65)

23

Solving for M. :a

fSPg(X--XQds rVcoseaCVyWs+

f*

P'S1N93 (X--A&) ds

Ma=X J

"

Jd 2T X 2ET (66)s

as

r EI

This equation cannot be solved because it contains two unknowns

NL and P ; therefore, a second equation for P must be written. This equa

tion is obtained by using the second boundary condition, Sya= o . Sub

stituting equations (58), (59), (60), (61), (62), (63), and (64) into the

equation forS".*. yields:

cfc&coss + la isi//ejCOs<?-V2/cose3 sin?) cosgds

he,-) ^

JS

+< (BlSik3+ \zJs\uezs\HG -\2/COS9z COS^) SlKff ds

J, &A

'(Po,(x--x-.)+(t)cos93(Y--y-,h (I) s\He3(h-*6) - mo a7-x6) ds -QM

X(67)

0

Equation (66) can be substituted into equation (67) which yields

an equation with only one unknown P . This new equation can be solved for

P by using a computer program which combines a numerical methods routine

a

and an iteration process to find the value of P . Pa can then be substituteda a

into equation (66) to obtain M . Once P and M are known, the generalequa-

a a a

tions for the deflection at any point on the belt can be written. Dummy

loads Q and Q will again be applied at the point where the deflection is tox y

24

be evaluated, Figure 3.4 Equations (58), (59), and (60) are then rewritten

to include the dunmy loads. This results in:

IY\s=Po. C X7- X-V (|) COS 93 (.Yi-Yk )+(|) S I tf9j ( X--X6)- M*

+ QxCY.,-Ys) + CI^XtXj)

(68)

T=RLCOS(?+(-|')s>Ne3COS(?-(|-')COS93S/Np-Qx!5lfV3->-^COS? (69)

V=RxSlM^+^SlK/e3S\sp-/Z'VoS93 COS + qxCOsp + Q-SlN(3(70)

Taking the partial derivatives of equations (68), (69), and (70), with re

spect to Q and Q ,yields:

x y

* = iv,-r |f=>-(71)

= sm? & = cos?<72)

*Y_ ^__/(2 (73)

&-COS* -"*

Substituting equation (68), (69), (70), (71), (72), and (73) into equation

25

(36) yields the expression for ;

C[R- C0S^(t)siN93COS?- (z) C0S93S!rt] SlKg di

o

<3<<-o

AE

+[RiSIK? +(T)siN93S>Ng-(T) COS&3COspl C05? ds

SA

P*

4

CcFbiCx,-xH^)cos3CY;-Y*)+(T)siwej(X---x^-M-.]<:YrY^ d<

"J (74)

Simplifying the above results in:

s s s

Sx-\\cOS?SIN^cJs + P^jpiCcOS?SIN?as- |^ifsiN^d<

p^osej \Cos2g<is h (o<7-x6)cVY*) ds

P_^OSei\cY,-Y6)(Y1-Y^ds +P/-,i^((yx6)CV^--

2EI 42EI

-I

^f- (Y,-Yr)ds

(75)

26

A similar equation for S3 can be developed. The general equation for de

flection in the y direction S<j can be written as follows :

srPTUoy/ds Cv(TQii)6%~\ AE GA

/W\ S ,iM,-aMa\

MaUoaJds

EI

(76)

Substituting equations (68), (69), (70), (71), (72), and (73) into equation

(76) yields:

fD^C0Sp + (T/SIN93C0Sp-(T)cQS93SlNP3COSpds

AE

Q*-* o

Q<j r O

+''(fcSliyp + (a)cOS93Sirtfft*(-g)cOS93COSg3SlNgds

GA

+

"^(X,-X<)t^)cQS93(Yr>6H^I)siN/8,(X,-Xg)-INA<0a-,-Xs)<li

El

(77)

^Ferdinand Singer, Strength of Materials, p. 465

27

Simplifying this results in:

s s s

^= fifcos^ds + !i2fcos2(Sds - P'^SQ-fsir/pcosfids

aJ,"

26A X

s

+Mcx--*Xx,-xV<la +B^2^ CcY,-Y*Nx,-x^dFt 2EI J,

o

^PSlh/93

Wx,-X6)(X7-X^d5 _ I^0x,-X^d;

(78)

The deflection equations of Step Two are not solved by the incre

mental computer technique due to the excessive computer time required to

perform the solution. The causes of the increased computer time are:

1. The solution for P and M require the solution ofa a

fourteen integrals by numerical methods and then an

iteration on P3 . This solution is required every-

a.

time the load is incremented another A p .

2. The solution of the deflection equations require the

solution of twenty additional integrals by numerical

methods and a more complex iteration onAP because

the feed belt must be fitted to two pulleys instead

of one.

28

3. The spacing of the points on the feedbelt will

have to be at closer intervals to accurately

fit the belt to two pulleys.

4. All the above require more core space.

As a comparison, Step One has only ten integrals and a sim

pler iteration routine. The computer time necessary to solve Step one was

approximately three minutes. It is expected that the solution for Step two

may run more than fifteen minutes and the cost for debugging and running

Step two was judged to be prohibitive. It is felt that the validity of

the technique can be demonstrated satisfactorily with the solution for

Step one.

29

CHAPTER IV

EXPERIMENTAL RESULTS

4.1 Experimental Results:

To check the analysis, an experiment was conducted to measure the

deflected shape of the feedbelt as a function of load. The fixture is de

tailed in Figure 4.1 and 4.2. Pulley one was mounted on a stationary por

tion of the fixture while pulley two was mounted to a Hunter Force Gauge,

which was mounted to a lead screw. When the lead screw was advance, the

center distance between the two pulleys increased and the resulting load

was indicated on the force gauge.

The center distance between the two pulleys was monitored using

a scale graduated in .010 in. increment while the deflection along the

X-axis was monitored with a micrometer which could be adjusted so it was

half-way between the two pulleys. The deflection of the .feedbelt along

both the x and y axis was determined by advancing the lead screw until some

predetermined incremental center distance was reached between the two pulleys.

This procedure was repeated until the required center distance was reached.

Figure 4.3 contains the data obtained from this portion of the experiment.

Figures 4.4 and 4.5 are plots of this data.

In addition to the deflection measurements, several attempts were

made to measure the stress in the feedbelt using strain guages. This was not

successful due to the low modulus of elasticity of the feedbelt. The adhesive

used to mount the strain guages was several times stiffer than the feedbelt

and would not allow the belt to deflect properly.

4.2 Comparison of the Experimental and Analytical Results:

Before comparing the analytical and experimental results, the

properties of natural rubber must be understood. Natural rubber does not

have a single value of modulus of elasticity as some materials. Instead, as

30

shown in Figure 4.6, the modulus of elasticity increases as the strain in

creases. The natural rubber compound used in the feedbelt has a modulus of

elasticity of approximately 100 psi. when it is first strained and reaches

a modulus of elasticity of approximately 150 psi. at the peak strain for the

feedbelt. The modulus of elasticity will vary as much as t 20% due to var

iations in compoinding and molding procedures. The belt used in the experi

ment was tested and the modulus of elasticity was found to be very close to

the 100 psi. and 150 psi. values.

The analytical equations are capable of treatirTg only one value of

modulus elasticity- Therefore, the computer program was used to evaluate

three values of modulus of elasticity, 100 psi., 150 psi., and 300 psi. The

300 psi. point is used to show the effect on the deflected shape of doubling

the peak modulus of elasticity. The results of these computer evaluations are

contained in Appendix E.

Figures 4.7and-

4.8 compare the experimental and analytical results

for the x and y deflections of the feedbelt. Each Figure contains four curves.

One for each of the three values of the modulus of elasticity and one for the

experimental results.

It can be seen in Figure 4.7 that during the initial loading, the

experimental results agree with the analytical results for E = 100 psi. As the

load is increased, however, the experimental data crosses the analytical curve

for E = 150 psi., and tends to parallel this analytical curve but at a slightly

higher value of E. This variation could be due to either an error in the exper

imental measurement technique, the estimation of the E, or the analysis. The

main problem encountered in the experimental measurements was the creep charac

teristic of natural rubber. When a natural rubber specimen is deflected to a

fixed dimension and held, the material will creep and cause the load reading

to decrease. To overcome this, the readings were taken as soon as possible

31

and the loads were applied slowly to limit any effect due to the strain rate

sensitivity of natural rubber. This decrease in load would cause a shift in

the curve as shown in Figure 4.7. Considering the problems encountered in

working with a material like natural rubber, the correlation between the two

curves in Figure 4.7 is excellent and sufficient to demonstrate the validity

of the incremental computer technique.

The curves in Figure 4.8 follow the same trends as those in Figure

4.7 and many of the same conclusions can be reached from these curves. The

main difference in these curves is that there is a deflection limit for the

X-axis. That limit occurs when the feedbelt is stretched straight between the

two pulleys. At this point, no further deflection will occur in the X-axis for

a corresponding deflection in the Y-axis. None of the cases studied reached

that point.

The correlation between the 100 psi. curve and the experimental curve

during the initial deflection is excellent in both cases with almost no error.

At the larger deflections there is about 7% error between the analytical and

experimental results.

Figure 4.9 contains the values for tensile force and bending stress

for the three different approaches.

The tensile force required to stretch the feedbelt to the required

center distance agree very well between the experimental and the incremental

computer technique. The approximation technique predicts a value for tensile

force which is significantly lower. This is caused by the assumption of con

stant tension in the belt which allows more of the belt to carry the strain.

The assumption that was made in the incremental computer technique

that the belt reaches its maximum stress level when it contacts the pulley was

backed up by the experimental observation. When the belt was being stretched

on the fixture, it was observed that once the belt contacted the pulley surface.

32

it did not slide on the surface when the load was increased further.

The bending stress obtained by the two different analytical methodst

is the same. No experimental data is available to verify these results. This

does, however, demonstrate that the stress can be accurately determined from

the deflected shape as assumed in the approximation technique.

33

CHAPTER V

CONCLUSIONS

The problem of large deflections in a thick elastic belt has been

studied using a curved beam approximation technique and an incremental strain

energy computer technique. This study has shown that the bending stress at

point a in the feedbelt is the same for both analytical techniques. The

assumption in the approximation technique that the bending stress can be deter

mined from the deflected shape proves to be as accurate asthe incremental com

puter technique. No experimental data is available to verify these analytical

results. Further work should be conducted to determine these stresses experi

mentally.

The load required to stretch the belt/pulley system and the final de

flected shape of the feedbelt are approximately the same for the incremental

computer technique and the experimental results. The small percentage of error

could be due to experimental error, an error in the modeling technique, or the

modulus of elasticity not being exactly 150 psi. These results confirm that

the incremental computer techniques can predict the final deflected shape and

loads of a system subjected to large deflections within the elastic limit.

The approximation technique does not accurately predict the tensile

force in the feedbelt. This is caused by the assumption that the tensile strain

is uniformly distributed around the feedbelt. As discussed in section 4.2, the

assumption is not correct due to the feedbelt not sliding on the pulley surface.

The incremental computer technique simulates the belt/pulley system

^ery well. Applying strain energy to large deflection problems by incremental

loading technique is a valid technique and could be applied to many types of

large deflection problems. The solution of the equations by .numericaltechniques

is possible but more difficult problems such as the one in Section 3.2 require

34

a faster computer to hold the cost down. Otherwise, the technique is valid

with no major problems observed.

Further work should be carried out applying the incremental computer

technique to a system with a constant modulus of elasticity to verify more

closely the accuracy of the technique. Additional modifications should also

be made to the program to allow systems with variable moments of inertia and

other varying physical properties to be studied. This is a major change to

the program, but something that could be accomplished.

To proceed with the solution of the total feeder problem, the follow

ing suggestions are made:

o Determine the additional stress in the feedbelt due to the

dynamics of the belt running over the retard roll. The ten

sile stress would be the stress mainly effected by the dy

namics of the system and could be approximated by the

capstan equation.

o Determine the pressure profile in the nip under dynamic

conditions. This could be approximated by existing

Hertzian contact stress solutions and should be verified by

experimental techniques.

o Determine the interactions of several sheets of paper

entering the nip and the shear forces which are available to

separate them. This would yield an understanding of the re

lationship between the belt and retard roll coefficient of

friction which perform the operation of separating the sheets

of paper.

APPENDIX A

Solution For The Position Of The Feedbelt Neutral Axis

35

The value of e is given by the following equation :

Ae = R-

idf(IA)

SURFACE F

SURFACE D.

CENTER OF

CURVATURE

.60

-j-

1.00

-, 091.84

.7S ,

I \ "

by:

For the cross-section under consideration, the integral is given

f,7 H*'"f.

(2A)

WHERE i

b,=

b2

/>

091u-

.60 irv

.75

ii-

f=.84

f3= '.*<>

1Harold A. Rothbart, Mechanical Design and Systems Handbook, p. 15-28

36

Thus

C ^A,>

.84 1.00

)~

=.091 U + .60 In -~ = JH /v.

(3A)

The cross sectional area is,

A= b2 h^, -*- b, h( =(.60)(./6) + C. 091) (.091) -.104 ,y

The distance from surface F to the centroidal axis is,

v Qz^z ^2"f"

^ H, Xt*

A

X=.0898 in.

Thus the initial radius of curvature of the belt is,

R= 1.00 -

.0898=.9\0Z w.

Therefore, the distance between the bending neutral axis and

the centroidal axis is,

e=.9102

- JiA =. . 00037, -t.J/4

Next consider the case after the belt has been deformed into its

final shape over pulley three. The position of the bending neutral axis

will have shifted to the other side of the centroidal axis due to the shift

of the center of curvature to the other side of the centroidal axis.

37

SURFACE D-

SURFACE F-

.091

.60

CENTER OF

CURVATURE

.91

1.75

1

T1.00

where

b, - .dO iv

b2-

. 09/ -v

?i-

.757;".

fz-.91 ,*

f = 1.00 irv

and,

?

Evaluating the integral

iA = J24

f

e= ---^7-

=.0004 /^

.124

38

APPENDIX B

Stresses By Approximation Technique

Figure B-l details the belt pulley system and shows the physical

dimensions of the system. Step one in determining the stresses is the cal

culation of the tensile stress due to stretching the belt over the pulleys.

Referring back to Section 2.1 the various parts of the system geometry will

be evaluated.

8=.78/ /v C = LOSS t*.

thus,

F=\f(.78\)z +-a03S)v

also,

6 =TAN"'

M. 035/.78D -

SL.96'

39

It follows that,

D, = . 46 5"+ .5,<95'i~.

Di-.500

+-.080=.580 in/.

F, - /.2 97/H + . 5*80/ 54 5) =.628 ,.

F2= 1.297/(1 +.54 5/. 5130)= .6 69 /-v.

Y2=COS*

(.5&0/.669) =29.89

and,

2=9o-e -rz =7js

Now calculating the various lengths of the belt segments around

the centroidal axis.

L,=L3=. 628 SIN (29. 89*)+. 669 SIN (29. 89) = .646 //?

L4=L^ = .545(1T+-J25)- 1.78 /^

L5 = /. 5*6 2/,-

|_2.= M/

LT= 6.559/v

40

Initial distance around the centroidal axis is,

L0= (\.8Z)(71)- 5.7Z,

_

6,5-579-5:72 -

H S ya STRAIN^T 5. 7Z

TT= C.MS) (ISO) - 21.76 psi

where,

Young's Modulus = E = 150 psi

T = (.10419) (21.76) = 2.27 lbs.

The bending strain and bending stress caused by bending the

feedbelt around pulleys one and two are calculated by using the equations

from Section 2.2.

*b~

(R-e-y)6p

(R-e + 3)

D.84

^ 1.00

CTD_ (.079) (150)= 11.85 ps, 0>r l.O94)(/50)= Uf psi

41

The bending strain caused by bending the feedbelt around pulley three is,

a) Length of a segment on neutral axis =.594d

b) Length of a segment on surface F = (.594 -

.094)d 9

c) Length of a segment on surface D = (.594 -

.156)da

thus,

F =(-594 -

.094)de. -

.594j|,

5R.594 de

""

6-n -

(-594 -

.156)de

-

.594de

,,,

.594 69

"

The total strain due to bending the belt over pulley three is

equal to the sum of the two previous calculations.

F =-.252

d =.342

Superimposed on this bending strain will be the tensile strain

calculated in step one (14.5%). The resulting strains are:

F =-.107

^D =.497

The stress resulting from these strains are:

*"F = 16.05 psi

5"D = 73 psi

42

APPENDIX C

Solution For Ma

Equation 33

72 , DD\

e = ( (Mg-(-rJ(i-cose)) Rde

This equation may be rewritten as follows:

% % ^'

M^Rde r ptfdQ, r PR'cosede

Z ] 2EX 1EL

-

0

After integration:^

TCc -i'

MaRe

y/i

EI

PR29

2EI

_ PR*S1N9

2EL

V,

-0

Jo

Substituting the limits and reducing:

2EI

prmt _ Pir04 EI 2 EI

Multiplying by 2EI and solving for Ma:

43

APPENDIX D

Solution For * And So

Equation 40^

" z

(l)(cos)c-siN)Rd0 C (t) sirvecosg de

1 +*

J AE GA

r (i^,)(cose-^:XR(siiv92-sirve))Rde#

+ \

\

Removing the constants from inside the integral and expanding:

% %

PR PR /-- \ cose si/ve de -4- I cose sirve de

PRSIWe^frn.saMfl . BM COSSINJ

8EI J 2ErJ

Performing the integration yields the following equation:

^ --ER-fl-Sm^^4.B-(l--;1^^^PR3SINQ2 d-SlNG^)

A 4AE ^ 4SA 2.EX

Equation for y

S = C Tpcoseco^eRde f (-g-)siNe sinq J9

^\

"

AE \ GA

r /2(5)(cose-^^cose-cose2)Rd9

44

Removing the constants from inside the integral and expanding:

s^=7^\cos2e d9 +2M Sl^e de + -~r[coszeA6

"i 2AE\ ^ ^ r

26R\ 2EI

>2^-

^@ "'2

?RscoSQ2(^COec|Q __*!(coseie

2EI \ ^^J.

PR C0S92.

TTEI

Performing the integration yields the following equation:

_

PR /TT &z SWV2^_) R_ / 7T SIN29^^~

2AE^4

"

2 4'

2GA^4 2 4'

H- ifLV _ S. _. I*&4 + PR'C0S6WSlr/9x - I ^2EIV.4 2 4

'2EI v. l

y

45

APPENDIX E

Computer Program

E.l Description of Program

Figure E.l is a flow chart which details the structure of the

computer program INTER. The program has three parts. The first part is the

main program which does the iteration and fits the belt to the pulley while

calculating the stresses in the belt. The second part is a subroutine

named function value (x), which performs the extrapolation between data

points for integration routing INTGR1. The third part is the integration

routine INTGR1 which is a canned program available on the UNIVAC 1106.

INTGR1 uses the Newton Cotes quadrature method to evaluate the integrals.

The program starts by defining the system parameters form input

supplied by the user. Once this is completed, the program performs a check

to make sure the belt is not deflected around the pulley to point A

(Figure 3.3). The belt cannot touch the pulley past this point so if the

belt is at point A, the program takes the left-hand path on the flow chart

and iterates on load P until the proper center distance is achieved. If

this is not the case, the program proceeds on the main path of the flow

chart. The program now determines the values of the angle $ and the

terms inside integrals for each point on the belt. INTR1 and Function Value

(x) are called and the integrals for the point on the belt nearest point b

are solved. Once the values of these integrals are know, the deflection

equations are iterated on load P until that point on the belt coincides

with the pulley. When this has been completed, the deflection and stresses

for all the remaining points on the belt are calculated using the value

of A P obtained from the iteration process. The program now

46

changes the limits on the integrals so the next point on the belt can be

fitted to the pulley. This process is continued until either the pulleys

reach the proper center distance or the belt reaches point A. If the belt

reaches point A first, the program uses the left-hand branch of the flow

chart and continues the loading until the proper center distance is reached.

Figures E-2, E-3, and E-4 are plots of the deflected shapes of the

feedbelt for various modulus of elasticity. For E = 150 psi, the tensile

force in the belt is 3.3 lbs. and the bending moment is .145

in- lbs. at

point a (point of maximum stress) when the belt reached the proper center dis

tance. The stress due to this bending moment is,

(5 F =MY =

(-145) (.094) = u }.

hI .000964

p

The total load on the pulleys is twice the belt tensile load or 6.6 lbs.

47

E.2 Listing of Main Program

48

DEFINE RADi:X) = i:3.l4159265X).--180.

LiEF INE DEG (X ) = (X* 1 80 . ) -3 .141 59265

COMMON .-ELK 1 .-XUAL ( 1 SO ) YUAL ( 1 OO )

D IMENS I ON X ( 1 OO ? 25 ) j Y ( 1 OO < 25 ) j AK ( 50 ) * RJ i 50 ) * fl I ( 50 )

D IMENS I ON S ( 1 OO ) Q ( 1 00 ) ? AMP ( 1 OO ) j DX 1 ( 1 OO ) < DX2 ( 1 OO ) j DX3 C 1 OO )

D IMENS I ON DY 1 ( 1 OO ) , DY2 ( 1 OO ) ? AA I fl OO 25 ) AT ( 1 OO > 25 )

H IMENS I ON DY3 ( 1OO ) j HY4 ( 1 OO ) > T 1 1 OO j U ( 1 0

10) f

D I MENS I ON TM ( 1 OO ) , FlU ( 1 OO j 25 ) ATM ( 1 OO 25 )

D IMENS I ON AREA 1 ( 1 OO ) j AREA2 ( 1 OO ) < AREAS ( 1 OO J < AREA4 ( 1 OO ) j AREA5 ( 1 OO ) <

1 AREA6 ( 1 OO j j AREA? i 1 OO ) ? AREAS i 1 OO ) > AREAS ( 1 OO )

EXTERNAL UALUE

REAL INTGRIj II -L

READ (5 j 1 ) XI j Yl jCD

1 FORMAT OFlS.tVi

read i 5? 2) ri

2 FORMAT CF 15. 6)

READ i 5 3 J XjYjDIS1

3 FORMAT (3F15.6)

READ C5 4) R3

4 FORMAT CF15. 6)

RERDiS-S) R4.E1<I1-G1

5 FORMAT C4F15.6)

SKIP=0.

N=90

DO 6 K=0-N-3

TH2=N-K

TH2=RADTH2)

ITH3=N-K+1

X ( ITH3, 1 ) =R3^ (-COS ( TH23 )

Y f I TH3 1 ) =SQRT (R32-X ( I TH3 * 1 ) ^s=2 J

flI(ITH3)=0

AKCITH3)=0

6 A-ICITH3)=0

IP1 = 1

ITH4=91

IFCITH4 .LT. 91) GOTO 15

Y0=0

Y3=Y2

Ai=-xi:ijiPi)+x2

D1=0

.'.4

rtC

AX=0

AX1=Y1-Y3

GO TO 202

15 Y0=RAD(90. )-ATAN( ( (Y ( ITH4-3* IPI ) -Yl ) / (Xl-X ( ITH4-3? IPI) ) ) )

WRITE (6 j 4O0)

400 FORMAT 1 1H 4X 2HYO > 6X < 4H I TH4 > 4X > 1 1 HY ( I TH4 - I P 1 ) X 3H IPI? 7X >HY 1

1 < 9X, 2HX1 ? 4X> 1 1HX ( ITH4> IPI ) )

WR I TE ( 6 40 1 ) YO I TH4 - Y ( I TH4 5 IPI ) j IPI j Y 1 > X 1 j X ( I TH4 * IPI)

401 FORMAT (FIO. 5? [email protected] I105 3F1O.5)

Y3=Y2-AK ( 1 )

Ai=-Xi:i? IPl)+X2+fYl-Y3).-TAN(RAD(90)-Y0)

Iil=Al^SIN(Y0)

X4=X2+ ( TAN C YO ) ) ( Y 1 -Y3 ) -Dl *=COS ( RAD t 90 . ) -YO )

Y4=Y3+D 1 =*S I N f RAD ( 90 . ) - YO )

AX= (X (1 j IPI) -X2 ) *TflH ( YO )

AX1=Y1-Y3

2fi2 IFCAX.LE.AX1) GO TO 7

49

Z1=Y1-Y(i, IPI)X5=X i: 1 j I P 1 ) +Z 1 ."TAN ( YO )

Y5=Y(1, IPI).

Z2=C0S ( PAH (9 . ) -YO ) * (X2-X5 )

X4=X5+ (X2-X5 ) -Z2*C0S ( RAD ( 90 .

- YO ) jY-i=Y 1 +Z*S I N C RAD ( 90 . ) -YO )

BX=SQRT ((XI -X4 )+ ( Y 1 -Y4 )2 )

ZY2=Y(1-IP1) *

IF((Y1-ZY2) .GE. CD--2.) GO TO 48

I F C Y ( ITH4-3 ? I P 1 ) . LT . . 375 ) GO TO 4O

7 N=ITH4

DO S K=l-N-3

ITH3=K

RMP(ITH3)=(X(ITH3< IP1J-XU- IPI) )

Q 1=ABS ( X ( I TH3-3 * IPI) -X f I TH3+3 -IPI))

Q2=AES ( Y ( ITH3-3 < I P 1 ) -Y ( I TH3+3 I P 1 ) )

IF(ITH3 .EQ. 91) GOTO 207

GO TO 20S

07 Q ( I TH3 ) =RRD ( 90 )

GO TO 209

20S IFUTH3 .EQ. 1) GOTO 21 0

GO TO 211

10 QCITH3)=0

GO TO 09

211 Q ( I TH3 ) =ATAN ( Q 1 /Q2 )

209 DX 1 ( I TH3 ) =COS ( Q ( ITH3 ) ) *S I N ( Q C ITH3 ) )

DY 1 ( ITH3 ) = ( (COS ( Q ( I TH3 ) ) ) )

8 DY2 ( ITH3 ) = ( ( S I N ( Q ( I TH3 ) ) ) ** )

ITH5=ITH4-3

N=ITH4

M=ITH5

DO 98 K=MN*3

ITH3=K

DX(ITH3)=(Y(ITH3 IP1)-Y(ITH5 IPI))

DX3(ITH3)=(X(ITH3j IPI ) -X ( 1 IPI ) ) * (Y ( ITH3j IPI ) -Y ( ITHSj IPI) )

DY3(ITH3)=(X(ITH35 IP1)-X(ITH5? IPI) )

98 DY4CITH3) = (XiITH3!. IPI) -Xii, IPI ) ) * (X ( ITH3 IPI ) -X ( ITH5? IPI) )

Ml=06

P=.0

N=ITH4-3

DO 999 K=0jN?3

ITH3=NK

DS=SQRT ( ( X ( ITH3+3 ? IPI) -X ( I TH3 - I P 1 ) ) **+ ( Y C I TH3+3 -IPI) - Y ( I TH3 IPI

S(ITH4)=0.O

999 S ( I TH3 ) =S ( ITH3+3 ) +DS

IP1=IP1+1

A=S(ITH4)

E=S ( 1 )

AESERR=. 1

N=ITH4

DO 997 K=l!>N3ITH'-!=k"

XUALi:iTH3)=S(ITH33

997 YUAL ( I TH3 ) =AMP ( I TH3 )

AREA= INTGR 1 (UALUE ? A ? B , AESERR .- MAX ITS j I ND )

AAREA1=AREA

A=S(ITH4)

B=S(ITH5)

N=ITH4

M=ITH5

5(1DO 996 K=M?Nj3

ITH3=K

XUAL(ITH3)=SCITH:-:i996 YUAL ( I TH3 ) =DX 1 ( I fH3 )

AREA= I NTGR 1 ( UALUE , A , B ABSERR - MAX I TS j I ND )

AAREA2=AREA

AARER3=AREA

DO 995 K=M-Nj3ITH3=H' '

XUAL(ITH3)=Si:iTH3)

995 YUAL ( ITH3 ) =DX2 ( I TH3 )AREA= INTGR 1 ( UALUE , A > E < AESERR > MAX I TS I ND )

AAREA4=AREfl

DO 994 K=MjN3

ITH3=K

XUAL(ITH3)=S(ITH3)

994 YUAL ( I TH3 ) =DX3 C I TH3 )AREA= I NTGR 1 (UALUE ,fl,E> ABSERR < MAX I TS , I MD )

AAREA5=AREA

DO 993 K=MsN3

ITH3=K

XUAL(ITH3)=S(ITH3)

993 YUAL ( I TH3 ) =DY 1 ( I TH3 )AREfl= I NTGR 1 (UALUE > A , E - ABSERR , MAX I TS t I ND )

AAREA6=AREA

DO 992 K=M?Nj3

ITH3=K

XUALCITH3)=S(ITH3)

992 YUAL ( ITH3 ) =DY2 ( I TH3 )

AREA= INTGR 1 (UALUE iAjBj ABSERR > MAX I TS j I ND )

AARER7=AREA

DO 991 K=M<N !.3

ITH3=K

XUAL(ITH3)=S(ITH3)

99 1 YUflL ( I TH3 ) =DY3 ( I TH3 )

AREA= I NTGR 1 (UALUE .AjEj AESERR MAX I TS - I ND )

AAREA8=AREA

DO 990 K=M)N'3

ITH3=K

XUAL(ITH3)=S(ITH3)

990 YUAL ( I TH3 ) =DY4 ( I TH3 )

AREA= I NTGR 1 ( UALUE - fl B j ABSERR , MAX I TS j I ND )

AAREA9=ARER

1 2 AMA= ( P.- 2 . ) * (RARER 1 / i S ( 1 ) -S ( I TH4 ) ) )

DELX 1=- ( P.'

( 2 . *FI4*E 1 ) ) *flAREA2+ ( ( 3 . *P ) ( 4 . *G 1 *A4 ) ) *AAREA3DELX2=- ( RMfl (E 1 * 1 1 ) ) wflflREfl4+ C

R,"

( 2 . *E 1 * 1 1 ) ) --flflREflS

flJ ( ITH5 ) =DELX 1 +DELX2

DELY 1 = (Fv ( 2 . afl4E 1 ) ) RflREfl6+ ( ( 3 . P) / ( 4 . -^G 1 *:fl4 ) ) KflflREfl?DELY2=- ( flMfl-' ( E 1 * 1 1 ) ) 'FlflREfl8+ (

P--'

( 2 . *E 1 * 1 1 ) ) *flflREfl9

RK ( I TH5 ) =DELY 1 +DELY2

RITH5=ITH5

ZX1=X(ITH5j IP1-1)+FIJ(ITH5)

ZY 1 =Y ( I TH5 IP 1 - 1 ) -flK ( I TH5 )

Z=SQRT ( ( ZX 1 -X 1 ) ^^2+ ( ZY 1 - Y 1 ) )

L=flBS((Rl-Z)/Rl)

IF(Z .LT. Rl) GO TO 11

IF(L .LT. .005) GO TO 14

GO TO 705

11 IFCL .GT. .0O5) GO TO 13

GO TO 14

13 P=P-.O01

51GO TO 12

705 P=R+.02

GO TO 12

14 N=ITH4

M=ITH5

DO IO K=MjNj3

ITH3=K

RI (ITH3)=flI(ITH3)+P

T ( I TH3 ) =T ( I TH3 ) + ( P.- . ) * COS ( Q ( I TH3 ) )

U ( ITH3 ) =U ( ITH31 + ( P.- 2 . ) *S IN ( Q ( ITH3 ) )

TM ( ITH3 ) =TM ( I TH3 ) +flMfl- (P.- 2 . ) * ( X ( I TH3 j I P 1 - 1 ) -X ( 1 j I P 1 - 1 ) )

X(ITH3* IP1)=X(ITH3j IP1-1 ) +flj ( ITH3)

Y(ITH3, IPl)=Y(ITH3i. IP1-1 ) -AK ( ITH3)

IO CONTINUE

505 ITH5=ITH5-3

N1=ITH4

M1=ITH5

AM1=M1

3S DO 35 K=M1>N1j3

ITH3=K

I P I = I P 1 - 1

DX1 (ITH3)=C0S(Q(ITH3) ) *SIN (Q ( ITH3) )

DY 1 ( I TH3 ) = (COS ( Q ( ITH3 ) ) ) ***

DX2 ( I TH3 ) = ( Y ( I TH3 -IPI) - Y ( I TH5 .-IPI ) )

DY2 ( ITH3) = (SIN (Q ( ITH3) ) ) ***

DX3(ITH3) = (X(ITH3i IPI ) -X ( 1 < IPI ) )* (Y ( ITH3- IPI ) -Y ( ITH5, IPI) )

DY3 ( ITH3 ) =X ( I TH3 5 I P 1 ) -X ( I TH5 I P 1 )

DY4(ITH3)=(X(ITH3J IPI ) -X ( 1 5 IPI ) )*(X(ITH3 IPI ) -X i ITH5? IPI) )

IP1-IP1+1

35 CONTINUE

ABSERR=. 1

A=S(ITH4)

E=S(ITH5)

N=ITH4

M=ITH5

DO 989 K=MN3

ITH3=K

XUAL(ITH3)=S(ITH3)

989 YUAL ( I TH3 ) =DX 1 ( I TH3 )

AREA= INTGR 1 ( UALUE .- A B , ABSERR .- MAX ITS < I ND )

AAREA=flREfl

FlflREfl3=flREfl

DO 988 K=M:.N-3

ITH3=K

XUflL(ITH3)=S(ITH3)

YUflL ( ITH3 ) =DX2 ( I TH3 )

988 CONTINUE

fiREfl= I NTGR 1 (UflLUE fl E AESERR MAX I TS - I ND )

AAREA4=ARER

DO 987 K=MjN?3

ITH3=K

XUAL(ITH3)=S(ITH3)

987 YUflL(ITH3)=DX3(ITH3)

FlREfl= I NTGR 1 (UflLUE flE> AESERR j MAX I TS j I ND )

AAREfl5=flREfl

DO 986 KMjN3

ITH3=K

XUflL(ITH3)=S(ITH3)

986 YUflL ( I TH3 ) =DY 1 ( I TH3 )

flREfl= I NTGR 1 (UflLUE j fl j B j ABSERR j MAX I TS t I ND )

AAREA6=AREfl

52

DO 985 K=MjNj3ITH3=K

XUfiL(ITH3)=S(ITH:n9S5 YUflL ( ITH3) =DY ( ITH3 )

ARER= I NTGR 1 (UALUE R-

B ABSERR - MAX I TS > I ND )

AflREfl7=flREflDO 984 K=M?Nj3

ITH3=K

XUflLCITH3)=S(ITH3)

934 YUflL ( ITH3 ) =DY3t ITH3 )FiREfl= I NTGR 1 (UflLUE fl , B ,ABSERR , MAX I TS , I ND )

AAREA8=flREfl

DO 983 K=MN3

ITH3=K

XUflL(ITH3)=S(ITH3)

983 YUflL ( ITH3 ) =DY4 ( I TH3 )flREfl= INTGR 1 (UflLUE jflsE, ABSERR < MAX I TS , I ND )

AARER9=AREflDELX1=-(P-'

(.'=*A4E1) )*flflREfl+( (3. SRi 4. Gi-fl4) ) *FlflREfl3DELX=- (

AMA,-"

(E 1 1 1 ) )AAREA4+ (Ps ( . *E 1 * 1 1 ) ) *AAREAS

FlJ C ITH5) =DELX1+DELX

DELY 1= ( P.- ( .aA4E 1 ) ) AAREA6+ ( ( 3 . *P ) t 4 . *G 1 A4 ) )AARER7

DELY=- ( FlMfl--

( E 1 *I 1 ) ) *AAREA8+ (P.-'

( . E 1 * 1 1 ) ) FlflREfl9

PK ( ITH5) =DELY1+DELY

fll CITH5)=flI(ITH5)+P

T ( I TH5 ) =T ( I TH5 ) + (P-- . ) *COS ( Q ( I TH5 ) )

U ( I TH5 ) =U ( I TH5 ) + ( P.- . ) #*. I H ( Q ( I TH5 ) )

TM ( ITH5 ) =TM ( I TH5 ) +AMFI- (R.-2 . ) * ( X ( I TH5-

I P 1 - 1 ) -X ( 1 * I P 1 - 1 ) )

X(ITH5> IP1)=X(ITH5< IP1-1 J+A-J ( ITH5)

Y ( ITH5? IPI ) =Y ( ITH5 IP1-1 ) -AK ( ITH5)

WRITE (6 j 203)

203 FORMAT ( 1 H X - SHflREfl 1 , 5X j 5HflREfl2 j 5X > 5HARER3 5X , 5HAREA4 . 5X , SHAREA5j

1 5X . 5HARER6 , 5X > SHAREA7 1 5X 5HAREA8 > SX j SHAREAS < SX < 5HDELX 1 j 5X , 5HDELX2 j

15X, 8HAJ ( ITH5) , X< 5HDELY1 )

WRITE (6 j 04) flflREfll > flflREflj flflREflS? flflREfi4* flflREflS? FlflREfl6* flflREfl7j

IflflREflS.fiflREASj DELX1 DELXj AJ ( ITH5) , DELY1

04 FORMAT (13F10.6)

WRITE (6> 205)

205 FORMAT (IH , 2Xj 5HDELYj5Xj8HAK ( ITH5) * 2X, 5HRTTH5- 3X.- SHHI (ITH5) ?6X< IH

1P6X7HT(ITH5) 5 3X.-7HUC ITH5) < S'X, 8HTM ( ITH5) <3X? 1 1HX ( ITH5-IP1 ) -4X.- 1 IH

1Y(ITH5-IP13)

WR I TE ( 6 j06 ) DELY , AK ( I TH5 ) , AM 1 , A I ( I TH5 ) . P < T ( I TH5 ) , U C I TH5 ) < TM (ITH

15) j X ( ITH5? IPI ) Y ( ITHSj IPI )

06 F0RMAT(8F10.6jF15.8)

ITH5=ITH5-3

M1=ITH5

AM1=M1

IFdTHS .LE. O) GO TO 39

GO TO 38

39 ZY3=Y1-Y(1;.IP1)

ZY4= (CD/ . -ZY3 ) / ( CD --

. )

IF(ABS(ZY4) .LE. . 005) GO TO SOI

IF(ZY4 .GT. O.) GO TO 50

P=P.98

SKIP=1.

ITH5=ITH5+3

GO TO 503

5&1 IF (SKIP -EQ. O.) GO TO 504

ITH5=ITH4-3

SK IPs-S

CO TO SOS

53

5@

31=

316

503

40

570

ITH4=ITH4-3DO 315 K=l-91-:-:

ITH3=K

AAI(ITH3-IP1)=AI r ITH:-:)

AT(ITH3.dPl)=T(ITH3)AU ( I TH3 -

IP 1 )=ur I TH3i

ATM ( I TH3 f I P 1 ) =TM f I TH:": l

ITH6=ITH4+3DO 316 N=ITH6-91 -:-:

ITH3=N

XCITH3- IP1)=X(ITH3i- I P 1 1 '

Y ( ITH3 j I P 1 ) =Y ( I TH3 j I P 1 - 1 )

GO TO 15

DELX 1=- (P.''

( . R4E 1 ) ) flflREfl+ ( ( 3 . *P ) ( 4 . G 1A4 ) ) *AAREASHELX=- (

AMA.--

( E 1 * 1 1 ) ) AARER4+ (P--

( . E 1 * 1 1 ) ) *:flflREfl5

flJ ( I TH5 ) =DELX 1 +DELX

DELY 1 = (P--

( . R4SE 1 ) ) FlflREfl6+ ( (3 .P ) ( 4 . *G 1 *fl4 ) ) flfiREfl7

DELY=- (FiMfl.-'

( E 1 * 1 1 ) ) flflREfl8+ (Fv ( . *E 1 * 1 1 ) ) flflRER9

flK ( ITH5 ) =DELY 1 +DELY

X(ITH5!.IP1)=X(ITH5j IP1-1 )+flJ ( ITH5)

Y ( ITH5-

I P 1 ) =Y ( I TH5 - I P 1 - 1 ) -flK ( ITH5 )

ITH5=ITH5-3

GO TO 39

X4=XddPl)

! *= ! 1

WRITE (6 ,570)

FORMAT ( 1 H -X > 1OHLAST PO I NT )

ITH5=ITH4-3

N=ITH4

DO 41 K=1N3

ITH3=K

AMP ( ITH3 ) =X ( I TH3 I P 1 ) -X (1 d P 1

CE=SQRT ( (X4-X ( ITH3 ? I P 1 ) ) $$+ ('

Efl=SQRT ( (X4-X ( 1 - I P 1 ) )$+ (Y4--

FlC=SQRT ( (X ( 1 j IPI ) -X ( ITH3- IPI )

*;s;

i

r'4-Y(ITH3? IPI

dldPl))SS)

l^+(Y(i. IP1)-Y(ITH3? IPI) )**)

Q 1 =flBS C X ( ITH3-3 IPI) -X ( I TH3+3 -IPI))

749

750

751

41

44

Q=flBS (YdTH3-3 ? IPI) - Y ( ITH3+3

IF(ITH3 .EQ. D GO TO 749

GO TO 750

Q(ITH3)=0

GO TO 75 i

, IPI))

Q(ITH3)=flTflN(Ql.-

DX1 (ITH3)=C0SCQC

DY1(ITH3)=((C0S(

DY(ITH3)=( (SIN(

N=ITH4

M=ITH5

DO 42 K=M>N3

ITH3=K

Q2 )

ITH3) )*SIN(Q(ITH3:

Q(ITH3) ) ))

Q(ITH3) ) )**)

) = C

i = C

Y ( I TH3d P 1 ) - Y ( I TH5dP 1 ) )

X(ITH3- IPli-Xd? IPI) )*(Y(ITH3 IPI ) -Y ( ITH5- IP!) )

XdTHS, IPl)-X(ITH5dPl))

XdTHS- IP1)-X(1- IPl))i(XdTH35 IPI ) -X ( ITH5- IPI) j

DX(ITH3:DX3(ITH3"

DY3(ITH3]

DY4(ITH3:

P=.0

N=ITH4-3

DO 98 K=0jN-3

ITH3=N-K

DS=SQRT((X(ITH3+3, IPI ) -X ( ITH3- IPI) )***+( Y CI TH3+3 - IPI ) -Y ( ITH3- IPI)

&&2 ;i

S(ITH4)=O.0

:-; C I TH3 ) =S ( ITH3+3 ) +DS

54

CONTINUE

CONTINUE98 CONTINUE

IP1=IP1+1

fl=SCITH4)

B-S ( 1 )

N=ITH4

M=l

DO 981 K=i-N-3

I TH3=K

XUALCITH3)=SCITH3)98 1 YUflL ( I TH3 ) =flMP ( ITH3 )

RREfl= I NTGR 1 (UflLUE - fl - E > AESERR - MAX I TS I ND )

AREfll (ITH5)=flREflFlMfl= ( P.- . ) C AREA 1 ( ITH5 ) C S ( 1 ) -S C I TH4 ) ) )

A=SCITH4)

B=S(ITH5)

N=ITH4

M=ITH5

DO 980 K=M-N-3

ITH3=K

XUALCITH3)=S(ITH3)

9S0 YUAL ( I TH3 ) =DX 1 ( I TH3 )

AREA= INTGR 1 (UALUE - A - E ABSERR - MAX I TS dND )

AREA(ITH5)=AREfl

AREA3(ITH5)=AREA

DO 979 K=MjNj3

ITH3=K

XUAL(ITH3)=S(ITH3)

979 YUAL ( ITH3 ) =DX ( ITH3 )

AREA= INTGR 1 (UALUE .- A - B - ABSERR , MAX I TSd ND )

AREA4CITH5)=AREA

DO 978 K=MjN-3

ITH3=K

XUAL(ITH3)=S(ITH3)

978 YUAL ( I TH3 ) =DX3 ( I TH3 )

AREA= I NTGR 1 (UflLUE -AjEj AESERR - MAX I TS , I ND )

ARER5(ITH5)=AREA

DO 977 K=M-N-3

ITH3=K

XUAL(ITH3)=S(ITH3)

977 YUAL ( ITH3 ) =DY 1 ( ITH3 )

AREA= INTGR 1 (UALUE - A - B ? ABSERR - MAX I TS - I ND )

AREA6(ITH5)=AREA

DO 976 K=M-N-3

ITH3=K

XUALCITH3)=S(ITH3)

976 YUAL ( I TH3 ) =DY ( I TH3 )

AREA= I NTGR 1 (UALUE j A - E - AESERR - MAX I TS , I ND )

AREA7(ITH5)=AREA

DO 975 K=M-N3

ITH3=K

XUALCITH3)=S(ITH3)

975 YUAL ( I TH3 ) =DY3 ( I TH3 )

AREA= INTGR 1 (UALUE - A - E , AESERR - MAX ITS 5 I ND )

ARER8CITH5)=ARER

DO 974 K=MjN-3

ITH3=K

XUAL(ITH3)=S(ITH3)

974 YUAL ( I TH3 ) =DY4 ( I TH3 )

AREA= INTGR 1 (UALUE .- ft, 2, AESERR , MAX I TS - I ND )

ARER9(ITH5)=APEACONTINUE

' CONTINUE

CONTINUE

CONTINUE

CONTINUE

CONTINUE

DELX 1=- (P.---

( . *R4^E 1 ) ) AREA C I TH5 ) + C C 3 . *P ) ( 4 . G1 fi4 ) ) FlREfl3 C I TH5 )DELX=- (

flMfl,-

( E 1 * 1 1 ) ) wflRER4 ( ITH5 ) + CP/ ( . *E 1 * 1 1 ) ) *AREAS ( I TH5 )

AJ ( ITH5 ) =DELX1 +DELX

DELY 1 = (P.-'-

( . A4*E 1 ) ) flREfl6 ( I TH5 ) + ( ( 3 . *P ) / ( 4 . ^G 1 A4 ) ) ARER7 ( I TH5 )

DELY=- (flMfl.--

( E 1 * 1 1 ) ) *ARER8 C I TH5 ) + C P-- ( . E1 * 1 1 ) ) AREA9 C I TH5 1

flK C ITH5) =DELY1+DELY

ITH5=ITH5-3

M=ITH5

CONTINUE

CONTINUE

IP1=IP1-1

IFCITH5 .LE. O) GO TO 43

GO TO 44

43 IF1=IPI+1

CONTINUE

ZY=YCldPl-l)-flKCl)ZY5= ( CD-' . - (Y 1 -ZY ) ) iCD.-2 . )

IFCRBS(ZYS) .LE. .005) GO TO 45

IF(ZY5 .LT. 0.) GO TO 56

CONTINUE

CONTINUE

P=P+.0

GO TO 563

56 P=P*.999

563 DO 46 -.1=1 -N? 3

ITH5=J

DELX 1=- ( py ( . *fl4*E 1 ) ) AREA ( I TH5 ) + C C 3 . *P ) / ( 4 . G 1 A4 3 ) *AREA3 ( I TH5 )DELX=- (AMA^ ( E 1 * 1 1 )) ARER4 ( I TH5 ) + (

P.'

( . E 1 * 1 1 ) ) *AREAS ( I TH5 )

A-J ( ITH5 ) =DELX 1 +DELX

DELY 1 = (py C . -=*R4E 1 ) 3 ^ARER6 ( I TH5 ) + ( C 3 . *P ) ( 4 . *G 1A4 ) ) *RRER7 ( I TH5 )

DELY=- ( AMA.--

( E 1 * 1 1 ) ) wflREflS ( I TH5 ) + (P C . E 1 * ID) *AREA9 C I TH5 )

CONTINUE

CONTINUE

46 AK C I TH5 ) =DELY 1 +DELY

IP1=IP1-1

GO TO 43

45 N=ITH4

DO 47 K=l-N-3

ITH5=K

T ( ITH5 ) =T ( I TH5 ) + CP-- . ) *COS C Q ( I TH5 ) )

U ( ITH5) =U ( ITH5) + (P-- . ) *SIN (Q ( ITH5) )

TM ( I TH5 ) =TM C I TH5 ) +AMA- ( P.- . ) * C X ( I TH5 d P 1 - 1) -X ( 1 - I P 1 - 1 ) )

X ( ITH5- IPI ) =X ( ITH5? IP1-1 ) +AJ ( ITH53

Y ( ITH5- IP! ) =Y ( ITH5- IP1-1 ) -flK ( ITH5)

47 flI(ITH5)=flI(ITH5)+P

DO 415 K=l?91,3

ITH3=K

flfl I ( I TH3- I P 1 ) =fl I ( I TH3 )

RT(ITH3- IP1)=T(ITH3)

flU(ITH3dPi)=U(ITH3)

4 1 =, ATM ( ITH3dP 1 ) =TM ( I TH3 )

DO 416 N=ITH4-91.-3

ITH3=N

56XCITH3? IP1)=X(ITH3- IP1-1)

416 Y(ITH3j IP1)=Y(ITH3-IP1-H48 DO 303 N=l-4:-:-:-!

ITH5=N

WRITE (6 j 301)

SO 1 FORMAT ( 1 H jX , 7HT ( I TH5 ) , 7HU ( I TH5 ) - 8HTM C I TH5 ) - 3X - 1 1 HX C ITH5- 1 P 1 ) - 4X -

111HY(ITH5-IP1)3WR ITE ( 6 - 30 3 T ( I TH5 ) - U C I TH5 3 - TM ( I TH5 ) - X C I TH5d P 1 3 - Y ( I THSd PD

30 FORMAT ( 3F 1 O . 6 -F 1 5 . 3 1

303 CONTINUE

WRITE (6- 3173'

3 1 7 FORMAT ( 1 H 1 , 57X , 1 8HF0RCES I N FEEDBELT 3

WRITE (653183

313 FORMAT (1HO)

WRITE (6 j 319)

319 FORMAT (1HO)

WRITE(6*30)

30 FORMAT (1H0-1X- 1HE-9X- 1HGj9X- lHfl-9X- 1HI)

WRITE (61. 31 ) El - Gl .- fl4d 1

3 1 FORMAT ( 6X - 4F30 .15)

WRITE(6-3)

3 FORMAT (1 HO)

IG1=IP1

DO 340 I=ldGld

IP=I

WRITEC6-33)

33 FORMAT ( 1 HO -. 64X , 3H IPI)

WRITE (6?34) IPS

34 FORMAT ( 1 H , 64X -13)

WRITE(6,35)

35 FORMAT ( 1 HOd 1 X - 8HP0S I T I ON - 1X - 4HBELT - 1 4X .- 4HBELT .- 1 4X 5 4HEELT

1, ISXj lHXd7Xj 1HY- 15Xj6HSYSTEM)

WRITE(6-36)

326 FORMAT [ 1 H -

1 4X > 2H0N - 1 3X , 7HTENS I LE - 1 2X - 5HSHEAR - 1 2X , 7HEEND I NG -

1 10X- 10HCOORDINATE- SX- 10HCOORDINATE.- 1 IX, 4HL0AD)

WRITE(6-37)

37 FORMAT ( 1 Hd 3X - 4HEELT - 1 OX - 1 1 HFORCE C LBS . ) - 7X - 1 1HFORCE C LBS . ) ,

15X- 17HM0MENT (IN. -LBS. ) -6X-5HCIN. ) - ISXjSHCIN. ) - 13Xj6HCLBS. ) )

DO 341 J=0-9O-3

ITH3=91-J

WR I TE (6 - 328 ) I THS , AT ( I THS .- I P2 ) 5 FlU ( I THS , IP ) - ATM C I THS - IP ) -

1 X C I THS , IP ) - Y ( I THSdP ) .- AA I ( I TH3dP2 )

38 FORMAT ( 1 OX-

14 7X - 6F 1 8 . 9 )

341 CONTINUE

340 CONTINUE

504 END

57

E-3 Listing of Subroutine Function Value (x)

58

FUNCTION UALUE (X)COMMON.--

ELK 1--XUAL f 1 flfll - YUAL ( 100)

N=91

IF CX.GT.XUALd'i i GA TA '^

DO IO ..1=1, M-:":

I=J

I F ( X . EQ . XUAL ( I ) ) Gfi TH 93

RI = I

( X . LT . XUAL ( I ) ) GO TO 1 0

GO TO 99

IO IF ( X . GT . XUAL ( I +3 ) ) GO TO SO

30 IF(I.EQ.N) GOTO 99Y= ( ( (X-XUAL ( I ) ) -

(XUAL C 1+3) -XUAL CI ) ) ) * (YUAL C 1+3) -YUAL (1)3) +YUAL C I )

GO TO 97

98 Y=YUAL ( I )

97 UALUE=Y

GO TO 40

99 WRITE (6- 100)

190 FORMAT ( IX 'INPUT DATA OUT OF RANGE')

CALL EXIT*

40 RETURN

END

59

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P-^sa

SECTION A-A

PAPER

SUPPLY

FEEDBELT

RETARD ROLL

FIGURE 1.1

Friction-Retard Feeder

117

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136

Figure B.l

Systems Physical Dimensions

YES

CALCULATE a AND

INTEGRATION

INPUTS

CALL SUBROUTINE

FUNCTION VALUE

(X)

OO INTEfrlVATION

INCREASE OR

DECREASE &P

CALCULATE

DEFLECTION (STRESS

FOR ALL POINTSON THE QELT

NO

CHECK TO SEE IP

PROPER PULLEY

CENTER DISTANCE

HAS SEEN ACHIEVED

YES

INPUT

DIMENSION

STATEMENTS

READ

STATEMENTS

137

GENERATE INITAL

SYSTEM

DIMENSIONS

DETERMINE IF

BELT HAS REACHEDSO8

POINT

NO

CALCULATE a AND

INTEGRATION

INPUTS

DQ INTEGRATION

CALL SUBROUTINE

FUNCTION VALUE

CALCULATE

DEFLECTION * STRESS

OF BELT POINT

NEAREST TO PULLEY

CHECKTO SEE IF

POINT IS ABOVE, ON

OR BELOW PULLEY

SURFACE

DECREASE

C.P

BELOW PULLEY

INCREASE &P

ABOVE PULLEY

CALCULATE

DEFLECTION (STRESS

OF REMAINING POINTS

ON THE BELT

PRINT

CHECK. TO SEE If

PROPER PULLEY

CENTER DISTANCE

HAS 3EEN ACHIEVED

J_INDEX PROtRAM TO

DEFLECT NEAT POINT

ON THE GELT TO THE

PULLEY SUftfALE

tMD

Figure E.l Flow Chart

DEFLECTED SHAPE OF THE FEEDSELT

E = /0O

S= "3

Incremental

Loading No.

Figure E.2 Belt Deflected Shape, E = 100 psi

E ISO

S 510

139

1.0

-1.0 -&

Incremental I

Loading No.

Figure E.3 Belt Deflected Shape, E = 150 psi

E-300

* I0O

-.7

^Incremental f

Loading No.

Figure E.4 Belt Deflected Shape, E = 300 psi

141

BIBLIOGRAPHY

1. McCracken, Daniel, and William Dorn. Numerical Methods And

Fortran Programming. New York: John Wiley and Sons, Inc.,

2. Rothbart, Harold. Mechanical Design And Systems Handbook.

New York: McGraw-Hill Book Company, 1964.

3. Salvadori, Mario, and Melvin Baron. Numerical "Methods In

Engineering. Englewood Cliffs, N.J.: Prentice-Hall, Inc.

1961.

4. Singer, Ferdinand. Strength Of Materials. New York:

Harper and Row, 1962.

5. Timoshenko, S. Strength Of Materials, Part I. New York:

D. Van Nostrand Company, 1958.

6. Timoshenko, S. Strength Of Materials, Part II. New York:

D. Van Nostrand Company, 1956.