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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.
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
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
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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
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
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.