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© 2007
JOHN C. VISNER
ALL RIGHTS RESERVED
ANALYTICAL AND EXPERIMENTAL ANALYSIS OF THE LARGE DEFLECTION
OF A CANTILEVER BEAM SUBJECTED TO A CONSTANT, CONCENTRATED
FORCE, WITH A CONSTANT ANGLE, APPLIED AT THE FREE END
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
John C. Visner
December, 2007
ANALYTICAL AND EXPERIMENTAL ANALYSIS OF THE LARGE DEFLECTION
OF A CANTILEVER BEAM SUBJECTED TO A CONSTANT, CONCENTRATED
FORCE, WITH A CONSTANT ANGLE, APPLIED AT THE FREE END
John C. Visner
Thesis
Approved:
Accepted:
___________________________
Advisor
Dr. Paul C. K. Lam
___________________________
Dean of the College
Dr. George K. Haritos
___________________________
Co-Advisor
Dr. Daniel C. Deckler
___________________________
Dean of the Graduate School
Dr. George R. Newkome
___________________________
Co-Advisor
Dr. Jiang Zhe
___________________________
Date
___________________________
Department Chair
Dr. Celal Batur
ii
ABSTRACT
Large deflection of a cantilever beam subjected to a constant force is
modeled. The motivation for this work is derived from an excellent example of
large cantilever beam deflection, the archery limb. With the development of a
program that models the deflection of an archery limb comes the possibility to
improve upon existing designs, which in turn could have large impacts on a
rapidly growing multi-million dollar market.
This study investigates a long,
slender cantilever beam of constant cross section with homogeneous and
isotropic material properties. The beam modeled is subjected to a concentrated
force applied at the free end.
This force has constant components in two
orthogonal directions. For this model, the weight of the beam is assumed to be
negligible. It is also assumed that the beam is non-extensible and therefore the
strains are negligible. Considering these assumptions, a second order nonlinear
differential deflection curve equation is obtained by means of a static analysis.
Because an exact analytical solution does not exist, a FORTRAN Program using
Euler’s numerical method is created to solve this equation.
The first of two
boundary conditions, the curvature at the free end, is known to be zero.
However, the boundary condition at the fixed end is unknown.
A shooting
method is employed within the program to obtain the correct curvature at the
fixed end to yield the deflected beam shape. Experiments are then performed to
verify the numerical results. Comparisons with published numerical results show
excellent agreement, and excellent agreement is also obtained between the
numerical results and experimental data.
iii
ACKNOWLEDGEMENTS
To my advisor and friend,
Daniel C. Deckler, Ph. D, P.E.,
for his motivation and support throughout
my undergraduate and graduate education.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES ...........................................................................................vi
LIST OF FIGURES ........................................................................................ vii
CHAPTER
I. INTRODUCTION ................................................................................... 1
II. BACKGROUND SURVEY.......................................................................... 3
III. THEORETICAL ANALYSIS....................................................................... 9
3.1 Deflection Curve Equation Development........................................ 10
3.2 Analytical Solution ....................................................................... 15
3.3 Numerical Solution ...................................................................... 18
3.4 Program Description .................................................................... 20
IV. RESULTS ............................................................................................ 23
4.1 Theoretical Results ...................................................................... 23
4.2 Experimental Procedure and Results ............................................. 27
V. CONCLUSION ..................................................................................... 35
5.1 Future Work ............................................................................... 36
REFERENCES .............................................................................................. 37
APPENDICES .............................................................................................. 39
A. FORTRAN PROGRAM CODE.......................................................... 40
B. FORTRAN PROGRAM OUTPUT OF EXPERIMENT 1 .......................... 44
C. FORTRAN PROGRAM OUTPUT OF EXPERIMENT 2.......................... 51
v
LIST OF TABLES
Table
Page
4.1 Changes in Tip Deflection for Different Stepsizes, Δs .............................. 25
vi
LIST OF FIGURES
Figure
Page
3.1
Cantilever Beam.................................................................................. 10
3.2
Beam Free Body Diagram .................................................................... 12
3.3
Free Body Diagram of Cut Beam .......................................................... 13
3.4
Infinitesimally Small Section of Beam.................................................... 15
3.5
FORTRAN Program Flowchart .............................................................. 22
4.1
Comparison of Belendez and FORTRAN Program Theoretical Curves........ 25
4.2
FORTRAN Program Results with Varying Force ...................................... 26
4.3
FORTRAN Program Results with Constant Force of 3.92N ....................... 26
4.4
FORTRAN Program Results with Constant Force of 5.92N ....................... 27
4.5
Experimental Beam with 3.92N Applied Vertically Downward .................. 29
4.6
Experimental Beam Measurement......................................................... 30
4.7
Comparison of FORTRAN Program Curve and Experimental Curve........... 31
4.8
Experimental Beam with 3.92N Applied at Angle of 53 degrees ............... 33
4.9
FORTRAN Program Theoretical Curves vs. Experimental Curves .............. 34
vii
CHAPTER I
INTRODUCTION
While beams receive very little recognition, they play a very important role
in our everyday life.
From bridges to cranes, decks to any roofed structure,
beams are everywhere and we most likely use them every day and never realize
it. Many types of beams exist today, however this study examined only one
type, the cantilever beam.
By definition, a cantilever beam is a beam that is fixed at one end, while the
other end is suspended and unsupported, much like a diving board.
The
inspiration for this study was derived from perhaps one of the best examples of a
cantilever beam, an archery limb. Made from highly elastic material and capable
of projecting an arrow at extremely high speeds, archery limbs represent an
excellent example of a cantilever beam made of linear elastic material that is
capable of sustaining large deflections.
The traditional long bow, which is a curved stick with a string attached to
each end that is drawn and released while the bow is oriented vertically, has
served archers of many types for thousands of years.
However, increasing
popularity in archery hunting and competitive target archery has presented a
need for increased performance in archery equipment. This need prompted the
invention of the compound bow, which uses a cam pinned to the end of the limb
along with a series of cables to provide a mechanical advantage allowing the
bow to store more energy while requiring less force from the archer to draw the
bow. The result is higher potential energy with less work.
1
Today, the archery industry has blossomed into a multi-million dollar
industry with dozens of manufacturers all competing to design and manufacture
the fastest, lightest, quietest and most cost-competitive compound bow. Today’s
archery manufacturers are utilizing the best available technology to date to
remain competitive and increase their product’s performance.
Because the
majority of the performance of a compound bow lies in the limb/cam
combination, a model that describes the relationship between the limb deflection
and the action of the cam could provide insight into the inner workings of this
system as well as reveal areas within the system that could be improved upon.
While the relationship between the limb and cam is one of extreme complexity
due to several unknown variables, the development of such a model could take
compound bow performance to the next level and revolutionize the archery
industry.
This study takes the first step to design the aforementioned model by
addressing large deflections of cantilever beams of linear elastic material
subjected to a constant force applied at a constant angle to the free end. The
goal of this study is two-fold: to develop a program that will solve a second
order, non-linear differential equation governing the behavior of a deflected
beam and then perform a series of experiments that will verify the results of the
program to build confidence in the program’s accuracy.
2
CHAPTER II
BACKGROUND SURVEY
Deflection of cantilever beams has been the subject of numerous analyses
to date.
An excellent example of a cantilever beam subjected to a vertical
concentrated force at the free end can be found in Mechanics of Materials [1], as
well as many other textbooks on physics and mechanics. In this case, the small
angle assumption is valid and an equation that describes the deflection of the
free end, showing proportionality between the deflection and the externally
applied force that is applied, can be found [1]. However, in the aforementioned
textbook, the discussion only addresses beams subjected to small deflections.
When deflections are large and the small angle assumption is no longer valid, the
problem becomes increasingly difficult and an analytical solution does not exist
due to the presence of a non-linear term in the deflection equation.
For the case of large deflection, several different solutions have been found
for cantilever beams subjected to external forces. Analyses of beams undergoing
large-amplitude free vibration have been studied in the past utilizing many
conventional and mixed finite element methods. Woinowsky-Krieger [2] used a
single-term approximation to the ordinary nonlinear differential equation to
obtain a solution in terms of elliptic integrals. Srinivasan [3] applied the RitzGalerkin technique, choosing a single-term approximation to obtain the nonlinear
free vibration responses of simply supported beams and plates.
A similar
analysis, beams subject to non-linear vibrations, has also been studied. Ray and
Bert [4] presented analytical and experimental values of natural frequencies as a
3
function of the ratio of maximum amplitude to beam thickness and initial tension
of an oscillation beam. Because this investigation will focus on a force applied
slowly to the free end thus producing a static analysis, dynamic analysis of the
beam will not be considered.
Lee et al. [5] investigated large deflection of a linear elastic cantilever beam
of variable cross-section under combined loading by means of the Runge-KuttaFalsi method.
Baker [6] obtained large deflection profiles of linear elastic
tapered cantilever beams under arbitrary distributed loads by means of a
weighted residual solution of the Bernoulli-Euler bending moment equation.
Dado and AL-Sadder [7] presented a new technique for large deflection analysis
of non-prismatic cantilever beams based on the integrated least square error of
the nonlinear governing differential equation in which the angle of rotation is
represented by a polynomial. Shatnawi and AL-Sadder [8] studied exact large
deflection of non-prismatic, nonlinear bimodulus cantilever beams subjected to a
tip moment by applying a power series approach to analytically solve highly
nonlinear simultaneous first-order differential equations.
Shvartsman [9]
examined large deflections of a cantilever beam subjected to a follower force by
reducing a nonlinear two-point boundary-value problem to an initial-value
problem by change of variables, then solving without iterations. AL-Sadder and
AL-Rawi [10] developed quasi-linearization finite differences for large deflection
analysis of non-prismatic slender cantilever beams subjected to various types of
continuous and discontinuous external variable distributed and concentrated
loads in horizontal and vertical global directions. Ibrahimbegovic [11] studied
large displacement of beams by implementing finite element analysis to threedimensional finite-strain Reissner beam theory, where beam element reference
4
axes are represented by arbitrary space-curved lines.
These papers offer
similarities to this study, however are not directly applicable because they
consider cantilevers of varying cross-section.
Cantilever beams of non-linear materials have also been studied. Lewis
and Monasa [12] numerically studied large deflections of cantilever beams made
of non-linear materials subjected to one vertical concentrated load at the free
end using a fourth order Runge-Kutta method.
K. Lee [13] examined large
deflection of cantilever beams of non-linear elastic material under the effects of
combined loading by using Butcher’s fifth order Runge-Kutta method. Baykara et
al. [14] obtained numerical results to large deflections of a cantilever beam of
nonlinear bimodulus material subjected to an end moment, showing that
bimodulus behavior has a significant effect for the case of large deflection.
Rezazadeh [15] developed a comprehensive model to study nonlinear behavior
of
multilayered
micro
beam
switches
for
the
application
of
micro-
electromechanical mechanical systems (MEMS), in which the derived nonlinear
equation was numerically solved using the nonlinear finite difference method.
Antman [16] studied large lateral buckling of nonlinearly elastic beams subjected
to flexure, torsion, extension or shear.
This configuration is described by a
position vector function and an orthonormal pair of vector functions of a real
variable which is interpreted as a scaled arc length parameter of the straight line
of centroids of a beam in its natural reference configuration. C. Cesnik et al.
[17] presented a refined theory of composite beams. The basis for the theory is
the variational-asymptotical method, a mathematical technique by which the
three-dimensional analysis of composite beam deformation can be split into a
5
linear,
two-dimensional,
cross-sectional
analysis
and
a
nonlinear,
one-
dimensional beam analysis.
Large deflection of cantilever beams that are prismatic and made of linear
elastic material have been the subject of numerous studies in which the beam is
subjected to a uniformly distributed load. Seames and Conway [18] presented a
numerical method for calculating large deflections of cantilever beams under
uniform loading. This numerical method assumed that the elastic axis of the
beam could be approximated by a number of circular arcs tangent to one
another at their points of intersection, using the Bernoulli-Euler equation to
determine the radius of each circular arc. Rhode [19] obtained an approximate
solution for the large deflection of a cantilever beam subject to a uniformly
distributed load by expanding the slope in a power series of the arc length. Lee
et al. [20] analyzed stresses and displacements experimentally in largely
deflected cantilever beams subjected to uniformly distributed loads by means of
photoelasticity. This analysis demonstrated that for the case of a beam material
having a small modulus of elasticity value with gravity acting alone as a uniform
load that large deflections would occur.
Belendez et al. [21] analyzed large
deflections of a uniform cantilever beam under the action of a combined load
consisting of a uniformly distributed load and an external vertical concentrated
load applied at the free end. This analysis obtained a numerical solution using
an algorithm based on the Runge-Kutta-Felhberg method and compared the
numerical results with experimental results. In reference [22], Belendez et al.
experimentally and numerically investigated deflections of a cantilever beam
subjected to combined loading.
Further literature review reveals that, while the
work of Belendez et al. [21], [22] offer many similarities to that of this study,
6
earlier work performed by Belendez et al. provides a more relative model to
follow and is described in detail below.
Frisch-Fay [23] solved for the large deflection of a cantilever beam under
two concentrated loads in terms of elliptic integrals. Barten [24] and Bisshopp
and Drucker [25] solved for the large deflection of a cantilever beam subjected
to one concentrated load, acting vertically downward at the free end of the
beam, also in terms of elliptic integrals. The work of [23], [24] and [25] are all
based on the fundamental Bernoulli-Euler theorem which states that the
curvature is proportional to the bending moment.
In this study, large deflection of a cantilever beam subjected to a
constant, concentrated load applied at the free end will be analyzed, and the
work of Belendez et al. [26] will be closely followed. Belendez et al. attempts to
find an exact analytical solution, however upon discovering that one does not
exist, proceeds to apply a mixed numerical and analytical approach along with
the program Mathematica to solve for the deflected beam shapes. This study
will also attempt to find an exact analytical solution, and then will utilize Euler’s
numerical method along with the employment of a shooting method in the
program FORTRAN to find the deflected beam shape.
What sets this work apart from work performed by the aforementioned
references is that this analysis will incorporate a constant, concentrated force
applied to the free end at a constant angle, thus not limiting the analysis to only
a vertical downward concentrated end force. An analytical approach to solving
the problem will be attempted. However, due to the presence of a non-linear
term, an exact analytical solution does not exist. A FORTRAN Program using
Euler’s numerical method will be created to solve for the shape of the deflected
7
beam, and a series of experiments will be performed to reproduce the FORTRAN
Program results.
8
CHAPTER III
THEORETICAL ANALYSIS
A long, slender prismatic cantilever beam of rectangular cross section made
of linear elastic material is modeled.
Figure 3.1 shows a cantilever beam of
length L with a concentrated force F applied at the free end. In this figure, δx
and δy are the horizontal and vertical displacements at the free end, respectively,
and φ0 represents the maximum slope of the beam. The constant angle at which
the force is applied is represented by α, and is measured positive downward
from the horizontal axis. The origin of the Cartesian coordinate system shall be
at the fixed end of the beam and (x,y) will represent the coordinates of point A.
The arc length of the beam, s, shall be measured between the fixed end and
point A.
For this study, it will be assumed that axial strains are negligible because any
change in length will be assumed to be a small fraction of the original length.
This will imply that the beam is inextensible.
It will also be assumed that the
cross section of the beam remains constant across the length of the beam,
meaning that the effect of Poisson’s Ratio, or the ratio of axial elongation to
lateral contraction, can be neglected [27]. Next, it is assumed that the BernoulliEuler theorem is valid, which states that the curvature of the beam is
proportional to the bending moment. Lastly, it is assumed that the deflection
due to the weight of the beam is negligible.
9
L
x
L-x
L - δx - x
δx
X
A(x,y)
s
δy
φ0
α
F
Y
Figure 3.1 – Cantilever Beam
3.1 Deflection Curve Equation Development
The analysis begins with a free body diagram, shown in Figure 3.2 that
describes the forces acting on the deflected beam. At the fixed end of the beam,
labeled as O, Mo is the reaction moment and Rx and Ry are the reaction forces
acting on the fixed end of the beam in the x and y directions, respectively. The
force F is resolved into a horizontal component, noted as Fx, and a vertical
component, noted as Fy. Summing forces in the x and y direction yields the
following equations
10
∑F
x
=0
Rx = F cosα
∑F
y
=0
R y = F sin α .
(3.1)
(3.2)
(3.3)
(3.4)
Taking a counterclockwise moment as positive and summing moments about
point O, the moment acting at the fixed end of the beam becomes
∑M
O
=0
M o = F (sin α )( L − δ x ) + F (cos α )(δ y ) .
(3.5)
(3.6)
The Bernoulli-Euler bending moment-curvature equation for a uniform cross
section rectangular beam of linear elastic material is
EI
dϕ
= M ( x, y ) .
ds
(3.7)
Where M(x,y) is the bending moment as a function of the distances x and y, φ
represents the curvature at any point along the length of the beam, E is the
modulus of elasticity and I is the moment of inertia of the beam cross section
about the neutral axis. It is necessary to find M as a function of x and y to
11
obtain the moment at any point along the length of the beam. This can be done
by cutting the beam at an arbitrary point and summing moments about the cut.
L - δx
Rx
Mo
δy
Ry
X
Fcosα
Y
Fsinα
Figure 3.2 – Beam Free Body Diagram
Figure 3.3 shows the cut beam with the reactions and moments acting on it.
The arc length of the beam, which is measured between the fixed end (O) and
point A, is represented by s. At point A, M(x,y) is the moment M as a function of
the distances x and y while v represents the shear force. At the fixed end of the
beam, Mo=F(sinα)(L-δx)+F(cosα)(δy) is the reaction moment, and Rx=F(cosα) and
Ry=F(sinα) are the reaction forces in the x and y directions, respectively.
Summing moments about A to obtain the moment M as a function of x and y
yields
∑M
A
12
=0
(3.8)
M ( x, y ) = F (sin α )( L − δ x ) + F (cos α )(δ y ) − F (sin α )( x) − F (cos α )( y ) = 0 (3.9)
M ( x, y ) = F (sin α )( L − δ x − x) + F (cosα )(δ y − y ) .
(3.10)
x
Rx
Mo
s
y
Ry
X
A
v
Y
M(x,y)
Figure 3.3 – Free Body Diagram of Cut Beam
Equation (3.10) provides a useful expression for the moment M as a function of x
and y which can be substituted into Equation (3.7), to yield
EI
dϕ
= F (sin α )( L − δ x − x) + F (cos α )(δ y − y ) .
ds
(3.11)
Taking the derivative of Equation (3.11) with respect to s
d ⎡ dϕ ⎤ d
EI
=
F (sin α )( L − δ x − x) + F (cos α )(δ y − y )
ds ⎢⎣ ds ⎥⎦ ds
[
]
(3.12)
d ⎡ dϕ ⎤ d
d
EI
= [F (sin α )( L − δ x − x)] +
F (cos α )(δ y − y ) .
⎢
⎥
ds
ds ⎣ ds ⎦ ds
[
13
]
(3.13)
Noting that L, δx and δy are constants, yields the following
EI
dy
d 2ϕ
dx
= −( F sin α ) − ( F cos α ) .
2
ds
ds
ds
(3.14)
The right side of Equation (3.14) is written in terms of x and y while the left side
is written in terms of φ.
Now a relationship between x, y and φ must be found.
Figure 3.4 shows an infinitesimally small section of the cantilever beam, of
which the arc length can be approximated as a straight line.
Using
trigonometry, the following relationships can be established
cos ϕ =
dx
ds
(3.15)
sin ϕ =
dy
.
ds
(3.16)
Substituting Equations (3.15) and (3.16) into Equation (3.14) yields
EI
d 2ϕ
= − F (sin α )(cos ϕ ) − F (cos α )(sin ϕ ) .
ds 2
(3.17)
Equation (3.17) is the non-linear differential equation describing the deflection
curve of a cantilever beam made of linear elastic material subjected to a
concentrated end load as shown in Figure 3.1. An attempt will be made to find
an exact analytical solution to Equation (3.17), however, should an exact
analytical solution not exist, a numerical solution will be developed.
14
ds
dy
φ
dx
Figure 3.4 – Infinitesimally Small Section of Beam
3.2 Analytical Solution
Now that the non-linear differential equation describing the deflection curve
of a cantilever beam made of linear elastic material subjected to a concentrated
end load has been found, it must be solved in order to obtain an expression for
both the x and y coordinates along the length of the deflected beam.
To obtain an analytical solution to Equation (3.17), both sides will be
multiplied by dφ/ds to obtain
EI
dϕ
dϕ d 2ϕ
dϕ
+ F (sin α )(cos ϕ )
+ F (cos α )(sin ϕ )
= 0.
2
ds
ds ds
ds
(3.18)
Rewriting each term of Equation (3.18) as a derivative with respect to the arc
length yields the following
EI
2
dϕ d 2 ϕ d ⎡ 1 ⎛ dϕ ⎞ ⎤
=
EI
⎜
⎟ ⎥
⎢
ds ds 2 ds ⎢⎣ 2 ⎝ ds ⎠ ⎥⎦
15
(3.19)
dϕ d
= [F (sin α )(sin ϕ )]
ds ds
(3.20)
dϕ d
= [− F (cos α )(cos ϕ )] .
ds ds
(3.21)
F (sin α )(cos ϕ )
F (cos α )(sin ϕ )
Substituting Equations (3.19), (3.20) and (3.21) into Equation (3.18) yields
2
⎤
d ⎡ 1 ⎛ dϕ ⎞
⎟ + F (sin α )(sin ϕ ) − F (cos α )(cos ϕ )⎥ = 0 .
⎢ EI ⎜
ds ⎢⎣ 2 ⎝ ds ⎠
⎥⎦
(3.22)
Equation (3.22) is immediately integrable taking into account that at the free
end, the following boundary condition is valid
ϕ ( L) = ϕ 0
(3.23)
where φ0 is the unknown, maximum slope at the free end of the beam.
Integrating Equation (3.22) yields
1 ⎛ dϕ ⎞
EI ⎜
⎟ + F (sin α )(sin ϕ ) − F (cos α )(cos ϕ ) + C = 0
2 ⎝ ds ⎠
2
(3.24)
and rearranging yields
1 ⎛ dϕ ⎞
C = − EI ⎜
⎟ − F (sin α )(sin ϕ ) + F (cos α )(cos ϕ ) .
2 ⎝ ds ⎠
2
16
(3.25)
Applying the following boundary conditions
⎛ dϕ ⎞
⎜
⎟=0 @ s=L
⎝ ds ⎠
(3.26)
ϕ = ϕ0 @ s = L
(3.27)
to Equation (3.25) yields the constant of integration (C)
C = − F (sin α )(sin ϕ 0 ) + F (cos α )(cos ϕ 0 ) .
(3.28)
Substituting Equation (3.28) into Equation (3.24) and rearranging yields:
2F
⎛ dϕ ⎞
[(sin α )(sin ϕ 0 − sin ϕ ) − (cos α )(cos ϕ 0 − cos ϕ )] .
⎟ =
⎜
EI
⎝ ds ⎠
2
(3.29)
Taking the square root of both sides of Equation (3.29)
dϕ
=
ds
2F
(sin α )(sin ϕ 0 − sin ϕ ) − (cos α )(cos ϕ 0 − cos ϕ )
EI
(3.30)
and separating variables yields
ds =
EI
2F
dϕ
(sin α )(sin ϕ 0 − sin ϕ ) − (cos α )(cos ϕ 0 − cos ϕ )
17
.
(3.31)
Solving for ds from Equation (3.15) and substituting the result into Equation
(3.31) yields
dx =
EI
2F
(cos ϕ )dϕ
(sin α )(sin ϕ 0 − sin ϕ ) − (cos α )(cosϕ 0 − cos ϕ )
.
(3.32)
Likewise solving for dy from Equation (3.16) and substituting the result into
Equation (3.31) yields
dy =
EI
2F
(sin ϕ )dϕ
(sin α )(sin ϕ 0 − sin ϕ ) − (cos α )(cosϕ 0 − cos ϕ )
.
(3.33)
Ideally, Equations (3.32) and (3.33) would be integrable, thus yielding
equations that would describe the horizontal and vertical deflections at any point
along the neutral axis of the cantilever beam. Unfortunately, there is not an
exact analytical solution to the integrals on the left side of equations (3.32) and
(3.33). This necessitates finding a numerical solution to Equation (3.17) to find
the deflected shape of the beam.
3.3 Numerical Solution
Using Euler’s method, the second order non-linear differential Equation
(3.17) can be reduced into two first order non-linear differential equations. The
curvature of the beam, denoted as κ, can be written as
dϕ
=κ .
ds
Taking the derivative of both sides with respect to s yields
18
(3.34)
d 2ϕ d κ
.
=
ds
ds 2
(3.35)
Substituting Equation (3.35) into Equation (3.17) yields
EI
dκ
= − F (sin α )(cos ϕ ) − F (cos α )(sin ϕ )
ds
(3.36)
and rearranging
dκ
F
F
=−
(sin α )(cos ϕ ) −
(cos α )(sin ϕ ) .
ds
EI
EI
(3.37)
Numerically integrating Equation (3.34) using Euler’s method
⎛ dφ ⎞
⎟
⎝ ds ⎠ n
ϕ n +1 = ϕ n + Δs⎜
ϕ n+1 = ϕ n + Δsκ n .
(3.38)
(3.39)
Numerically integrating Equation (3.37) using Euler’s method
⎛ dκ ⎞
⎟
⎝ ds ⎠ n
(3.40)
F
⎛ F
⎞
(sin α )(cos ϕ n ) −
(cos α )(sin ϕ n ) ⎟ .
EI
⎝ EI
⎠
(3.41)
κ n +1 = κ n + Δs⎜
κ n+1 = κ n + Δs⎜ −
19
Equations (3.39) and (3.41) represent two first order differential equations that
can be used to numerically solve Equation (3.17). This will be done by creating
a FORTRAN Program as explained in Section 3.4.
3.4 Program Description
To solve Equations (3.39) and (3.41), the values of φ and κ must be known.
While the angle φ at the fixed end is known to be zero, the curvature κ at the
fixed end is not known. Since the curvature at the free end is known to be zero,
a shooting method will be employed to find the appropriate initial curvature at
the fixed end of the beam. The correct initial curvature is one that will produce
zero curvature at the free end of the beam and, as a result, an accurate
deflected beam shape.
Once the initial parameters are input, the program is executed and Euler’s
numerical method is used to calculate the slope and the curvature across the
length of the beam. The program is particularly interested with the curvature at
the two end points of the beam – the fixed end and the free end. The curvature
at the fixed end of the beam begins with some unknown value of kappa that
gradually decreases across the length of the deflected beam until it reaches zero
at the free end of the beam. Since the curvature at the fixed end is unknown, it
must be determined in order to produce a curvature of zero at the free end of
the beam, therefore providing an accurate deflected beam shape. The FORTRAN
Program Flowchart can be found in Figure 3.5.
Calculating the correct initial curvature is accomplished within the program
by using the bisection method.
The unknown curvature at the fixed end is
assumed to fall between a specified initial curvature range that the user guesses;
20
the low value of this range being KLOW and the high value being KHIGH. The
program then uses KLOW and Euler’s numerical method to calculate the
curvature at the free end of the beam, KLOWEND.
Next, the program
determines the average of KLOW and KHIGH to calculate the mid-range value of
the curvature, KMID.
If the difference of KLOW and KHIGH falls within the
specified tolerance, then KMID is used to calculate the final deflected beam
shape and the program ends. However, if the difference of KLOW and KHIGH
does not meet the required accuracy, the program then uses KMID and Euler’s
numerical method to calculate the curvature at the free end of the beam,
KMIDEND. The program then compares the values of KLOWEND and KMIDEND
to determine if a curvature of zero exists between these two values. If so, KMID
becomes the new value of KHIGH and the value of KLOW remains the same. If
a curvature of zero does not exist between the range of KLOWEND and
KMIDEND, then a curvature of zero exists between KMID and KHIGH so KMID
becomes the new value of KLOW and the value of KHIGH remains the same.
At this point, the half of the range that contains a curvature of zero is kept
and the other half of the range is discarded, and the bisection method is again
employed to find a new value of KMID within the reduced range and the process
repeats itself.
The initial curvature range is continually narrowed using this
process until the difference of KLOW and KHIGH reaches the desired accuracy at
which point KMID becomes the unknown initial curvature and is used to calculate
the final deflected beam shape. The FORTRAN Program Code can be found in
Appendix A.
It is important to note a few limitations of the FORTRAN Program. First,
the program is limited to constant angles applied to the free end for angles
21
greater than zero and less than or equal to ninety degrees, measured from the
horizontal.
Also, the program can only compute deflections of beams with
constant cross sections. Lastly, if the force applied to the end becomes too large
for a given beam geometry and material to support, the result will be an unusual
and unrealistic deflected beam shape.
Enter Inputs & Initial
Conditions
Calculate KLOWEND from input
value of KLOW
Calculate KMID
YES
Is KMID within specified
tolerance?
NO
Use κ = KMID and φ,s = 0 as initial
conditions to find φ, Κ , X & Y over
the length of the beam
Calculate KMIDEND using
KMID from above
KLOWEND x KMIDEND < 0 ?
NO
KMID = KLOW
YES
KMID = KHIGH
Stop
Figure 3.5 – FORTRAN Program Flowchart
22
CHAPTER IV
RESULTS
4.1 Theoretical Results
The FORTRAN Program previously described will now be used to produce
several deflected beam curves. The first theoretical beam curve developed by
the FORTRAN Program will be compared to both the Belendez [26] theoretical
and Belendez experimental beam curves presented in said research paper in an
attempt to reproduce their results. In this example, a force of 3.92N is applied
vertically downward at the end of the beam. The beam exhibits a length of
30cm, a width of 3.04cm and a height of 0.078cm. The beam is made of lowcarbon steel consisting of exhibiting a modulus of elasticity of 2.0x1011 Pa and an
area moment of inertia of 1.2022x10-12 m4 [26].
Figure 4.1 displays the Belendez experimental and Belendez theoretical
curves, along with the theoretical curve obtained from the FORTRAN Program.
The FORTRAN Program theoretical curve compares well with the Belendez
experimental curve, exhibiting a maximum Y direction error of 3.84%.
The
FORTRAN Program theoretical curve compares very well to the Belendez
theoretical curve having a maximum Y direction error of 1.35%. To establish a
means of comparison, the Belendez experimental curve when compared to the
Belendez theoretical curve yields a maximum Y direction error of 4.19%.
Figure 4.2 shows five deflection curves calculated by the FORTRAN
Program. These curves show how the deflection of the beam changes as the
angle is held constant and the force is increased. In this example, the force is
23
applied to the end of the beam vertically downward, or 90 degrees to the
horizontal, and the force is increased from 3.92N to 7.92N in 1.0N increments.
As expected, the beam deflection increases as the force is increased.
Figure 4.3 shows six deflection curves calculated by the FORTRAN Program.
These curves show how the deflection of the beam changes as the force is held
constant and the angle is varied. In this example, a force of 3.92N is applied to
the free end and the angle is gradually decreased from 90 degrees to 15 degrees
in 15 degree increments.
Figure 4.4 also shows six deflection curves calculated by the FORTRAN
Program and shows how the deflection curve changes as the force is held
constant and the angle is varied. In this example, a force of 3.92N is applied to
the free end and the angle is gradually decreased from 90 degrees to 15 degrees
in 15 degree increments.
Figures 4.3 and 4.4 both display similar phenomena in that as the angle of
the force decreases from 90 degrees, the beam deflection increases, but only to
a point at which the deflection reaches a maximum and then begins to decrease
as the angle of the applied force reaches 15 degrees. This can be explained by
the fact that at 90 degrees, the force is acting only in the y direction and not in
the x direction.
As the applied angle is decreased from 90 degrees, the y
component of the force begins to decrease and the x component begins to
increase. This occurs until some combination of the x and y force components
and deflected geometry produce a maximum deflection.
To ensure accuracy and convergence of the FORTRAN Program, stepsizes
of the arc length s were varied from 1(10-2) to 1(10-6). From the results shown
in table 4.1, only a .0051% difference in tip deflection occurs when the arc
24
length stepsize is changed from 1(10-5) to 1(10-6).
The percent error is
calculated using the tip deflection for a 1(10-6) stepsize as the “true” value. The
result using a stepsize 1(10-3) is shown in Appendices B and C.
Table 4.1 – Changes in Tip Deflection for Different Stepsizes, Δs
Δs
Y
% Error
1(10-2)
0.1424643
4.6481%
1(10 )
0.1367674
0.4634%
1(10-4)
0.1360728
0.0469%
1(10 )
0.1361296
0.0051%
1(10-6)
0.1361366
0.0000%
-3
-5
o
Belendez vs. FORTRAN, F = 3.92N & α = 90
x (m)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.02
y (m)
0.04
0.06
0.08
0.10
0.12
0.14
Belendez Experimental Curve
Belendez Theoretical Curve
FORTRAN Program Curve
Figure 4.1 – Comparison of Belendez and FORTRAN Program Theoretical Curves
25
FORTRAN Program Results at α = 90o and Varying F
x (m)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
F
F
F
F
F
y (m)
0.05
0.10
=
=
=
=
=
3.92N
4.92N
5.92N
6.92N
7.92N
0.15
0.20
Figure 4.2 – FORTRAN Program Results with Varying Force (F=F)
FORTRAN Program Results at F = 3.92N and Varying a
x (m)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
a = 90
a = 75
0.05
a = 60
y (m)
a = 45
a = 30
a = 15
0.10
0.15
Figure 4.3– FORTRAN Program Results with Constant Force of 3.92N (a=a)
26
FORTRAN Program Results at F = 5.92N and Varying α
x (m)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
a=90
0.05
a=75
y (m)
a=60
a=45
0.10
a=30
a=15
0.15
0.20
Figure 4.4– FORTRAN Program Results with Constant Force of 5.92N (a=a)
4.2 Experimental Procedure and Results
It has been shown in Section 4.1 that the FORTRAN Program is capable of
producing an accurate beam deflection curve when compared to the Belendez
theoretical and Belendez experimental curves as shown in Figure 4.1. To further
validate the FORTRAN Program, two experiments will be performed. The first
experiment will reproduce the Belendez experimental curve and then compare
the results to the theoretical curve produced by the FORTRAN Program. The
second experiment will mirror the first experiment; however instead of applying
the force vertically downward, the force will be applied at an angle.
In the first experiment, the beam is fastened to the top of a bench by
means of a clamp. The beam exhibits a length of 30cm, a width of 3.04cm and
27
a height of 0.078cm.
The beam is made of low-carbon steel consisting of
modulus of elasticity of 2.0x1011 pa and 1.2022x10-12 m4. Lightweight dental
floss is used to hang the weight from the end of the beam.
A force of 3.92N is
applied vertically downward at the end of the beam as shown in Figure 4.5.
Once the beam was deflected, measurements were taken along the length of the
beam using a digital caliper to capture the x and y coordinates as shown in
Figure 4.6.
The x and y coordinates were then plotted to obtain the experimental curve
of which a force of 3.92N is applied vertically downward at the end of the beam.
Figure 4.7 shows the experimental results as compared to the results obtained
from the FORTRAN Program.
It can be seen that the experimental curve
compares well to the FORTRAN Program theoretical curve very well exhibiting a
maximum relative error of 2.18%. The FORTRAN Program output of the first
experiment can be found in Appendix B.
28
Figure 4.5– Experimental Beam with 3.92N Applied Vertically Downward
29
Figure 4.6– Experimental Beam Measurement
30
o
FORTRAN Program Curve vs. Experimental Curve, F = 3.92N & α = 90
x (m)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.02
y (m)
0.04
0.06
0.08
0.10
0.12
0.14
FORTRAN Program Curve
Experimental Curve
Figure 4.7 – Comparison of FORTRAN Program Curve and Experimental Curve
In the second experiment, a force of 3.92N is applied to the end of the
beam at a downward angle of 53o measured from horizontal as shown in Figure
4.8. The beam exhibits a length of 30cm, a width of 3.04cm and a height of
0.078cm.
The beam is made of low-carbon steel consisting of modulus of
elasticity of 2.0x1011 pa and 1.2022x10-12 m4. The beam is fastened to the top
of a bench by means of a clamp. Lightweight dental floss was used to hang the
weight from the end of the beam. In order to apply the force at the proper
angle, a steel hook was used to redirect the dental floss to prevent the weight
from hanging vertically downward, thus simulating a force applied at 53o. The
smoothly polished surface of the steel hook, coupled with the addition of
31
lubricating oil to the string, allowed for smooth sliding of the string against the
hook and thus a frictionless surface was assumed.
Once the beam was deflected, measurements were taken along the length
of the beam using a digital caliper to capture the x and y coordinates as shown in
Figure 4.6.
The x and y coordinates were then plotted to obtain the experimental curve
which exhibits a force of 3.92N applied to the end of the beam at a downward
angle of 53o measured from horizontal. Figure 4.9 shows the FORTRAN Program
curve with a force of 3.92N applied vertically downward compared to the
experimental curve with a force of 3.92N applied vertically downward, and the
FORTRAN Program curve with a force of 3.92N applied at a downward angle of
53o measured from horizontal compared to the experimental curve with a force
of 3.92N applied at a downward angle of 53o measured from horizontal. It can
be seen that both the FORTRAN Program curve and the experimental curve with
a force of 3.92N applied vertically downward compare very well exhibiting a
maximum Y direction error of 2.18%.
The FORTRAN Program curve and the
experimental curve with a force of 3.92N applied at a downward angle of 53o
measured from horizontal also compare very well displaying a maximum Y
direction error of 2.34%.
The FORTRAN Program output of the second
experiment can be found in Appendix C.
32
Figure 4.8– Experimental Beam with 3.92N Applied at Angle of 53 degrees
33
FORTRAN Program Theoretical Curves vs. Experimental Results
x (m)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.02
0.04
y (m)
0.06
0.08
0.10
0.12
0.14
0.16
Experimental Curve with F=3.92N at 90 Deg.
Experimental Curve with F=3.92N at 53 Deg.
FORTRAN Program Curve with F=3.92N at 90 Deg.
FORTRAN Program Curve with F=3.92N at 53 Deg.
Figure 4.9– FORTRAN Program Theoretical Curves vs. Experimental Curves
34
CHAPTER V
CONCLUSION
Analytical and experimental analysis of the large deflection of a cantilever
beam subjected to a constant, concentrated force, with a constant angle, applied
at the free end has been studied.
An attempt to find an exact analytic
expression for both the x and y coordinates along the length of the deflected
beam was made, however the expressions for dx and dy could not be integrated.
A numerical analysis using Euler’s Numerical Method was successfully performed
and a FORTRAN Program was written that will perform Euler’s Numerical Method
to find the x and y coordinates along the length of the deflected beam for a given
combination of beam geometry, material and force. Theoretical results compare
well with data published by Belendez [26], exhibiting a maximum Y direction
error of 1.35%.
Two experiments were performed to reproduce the FORTRAN Program
theoretical results in an attempt to give the reader confidence in the accuracy of
the FORTRAN Program.
The first experiment applied a force of 3.92N vertically
downward from the end of the beam.
The results of the first experiment
compared well with the FORTRAN Program results under the same conditions
exhibiting a maximum Y direction error of 2.18%.
The second experiment
applied a force of 3.92N at an angle of 53o measured from the horizontal. The
results of the second experiment compared well with the FORTRAN Program
results under the same conditions exhibiting a maximum Y direction error of
2.34%. The second experiment not only yielded good results when compared to
35
the FORTRAN Program results under the same conditions, but also provided
confidence as to the repeatability of the experimental setup.
5.1 Future Work
In the future, several areas of the study could be expanded upon to give
the program more versatility. First, the program could be expanded to handle
beams of non-constant cross section. This could be done by adding the variable
I(s) to the deflection curve equation in place of the constant I. Next, the program
could be adjusted to incorporate a beam made of non-linear material.
This
would remove the variable E and replace it with the variable E(s) and an
expression describing the varying material properties across the length of the
beam would need to be found. A force of non-constant magnitude could also be
added to the program. This would remove the constant F and replace it with
either F(x,y) or F(s), depending on where the force was applied to the beam.
The angle at which the force is applied could also change throughout the
deflection of the beam. This would require the variable φ(x,y) to be added to the
deflection curve equation in place of the constant φ. The mass of the beam
could also be included in the deflection curve equation, removing the assumption
that the beam is mass-less. Lastly, the program could be expanded to model an
extensible beam, where the arc length s would not be equal to the length of the
beam L. Expanding the deflection curve equation to capture some or all of the
aforementioned scenarios would give the program more versatility and greatly
expand the program’s ability to study more complex beams, as well as increase
the program’s accuracy.
36
REFERENCES
[1] J. M. Gere, ‘Mechanics of Materials’, Sixth Ed., Brooks/Cole-Thomson
Learning (2004).
[2] S. Woinowski-Krieger, ‘The effect of the axial force on the vibrations of
hinged bars’, J. Appl. Mech. 17, pp. 35-36 (1950).
[3] A. V. Srinivasan, ‘Large amplitude-free oscillations of beams and plates’,
AIAA J., 3(10), pp. 167-168 (1965).
[4] J. D. Ray and C. W. Bert, ‘Nonlinear vibrations of a beam with fixed ends’,
Trans. ASME, J. Engng. Ind., 91, pp. 997-1004 (1969).
[5] B. K. Lee, J. F. Wilson and S. J. Oh, ‘Elastica of cantilevered beams with
variable cross sections’, Int. J. Non-Linear Mech., 28, pp. 579-589 (1993).
[6] G. Baker, ‘On the large deflections of non-prismatic cantilevers with a finite
depth’, Comp. Struct., 46, pp. 365-370 (1993).
[7] M. Dado and S. AL-Sadder, ‘A new technique for large deflection analysis of
non-prismatic cantilever beams’, Mech. Res. Comm. 32, 692-703 (2005).
[8] A. Shatnawi and S. AL-Sadder, ‘Exact large deflection analysis of nonprismatic cantilever beams of nonlinear bimodulus material subjected to a tip
moment’, J. Rein. Plas. And Comp., Vol. 26, No. 12 (2007).
[9] B. Shvartsman, ‘Large deflections of a cantilever beam subjected to a
follower force’, J. Sound and Vib. 304, pp. 969-973 (2007).
[10] S. AL-Sadder and R. AL-Rawi, ‘Finite difference scheme for large deflection
analysis of non-prismatic cantilever beams subjected to different types of
continuous and discontinuous loadings’, Arch. Appl. Mech. 75, pp. 459-473
(2006).
[11] A. Ibrahimbegovic. ‘On finite element implementation of geometrically
nonlinear Reissner’s beam theory: three-dimensional curved beam
elements’, Comp. Meth. In Appl. Mech. and Eng. 122, pp. 11-26 (1995).
[12] G. Lewis and F. Monasa, ‘Large deflections of cantilever beams of non-linear
materials’, Comp. Struct., 14, pp. 357-360 (1981).
[13] K. Lee, ‘Large deflections of cantilever beams of non-linear elastic material
under a combined loading’, Int. J. Non-Linear Mech., 37, pp. 439-443
(2002).
37
[14]C. Baykara, U. Guven and I. Bayer, ‘Large deflections of a cantilever beam of
nonlinear bimodulus material subjected to an end moment’, J. Rein. Plas.
And Comp., Vol. 24, No. 12 (2005).
[15] G. Rezazadeh, ‘A comprehensive model to study nonlinear behavior of
multilayered micro beam switches’ 14, Micro. Tech., pp. 135-141 (2007).
[16]S. Antman, ‘Large lateral buckling of nonlinearly elastic beams’, Arch. Rat.
Mech. and Anal., Vol. 84, no. 4, pp. 293-305 (1984).
[17]C. Cesnik, V. Sutyrin and D. Hodges, ‘Refined theory of composite beams:
the role of short-wavelength extrapolation’, Int. J. Solids Struct., Vol. 33, pp.
1387-1408 (1996).
[18]A. E. Seames and H. D. Conway, ‘A numerical procedure for calculating the
large deflections of straight and curved beams’, J. Appl. Mech., 24, pp. 289294 (1957).
[19] F. V. Rhode, ‘Large deflections of cantilever beams with uniformly distributed
load’, Q. Appl. Math., 11, pp. 337-338 (1953).
[20] H. Lee, A. J. Durelli and V. J. Parks, ‘Stress in largely deflected cantilever
beams subjected to gravity’, J. Appl. Mech., 26, pp. 323-325 (1969).
[21] T. Belendez, M. Perez-Polo, C. Neipp and A. Belendez, ‘Numerical and
Experimental Analysis of Large Deflections of Cantilever Beams Under a
Combined Load’, Phys. Scr., Vol. T118, pp. 61-65 (2005).
[22] T. Belendez, C. Neipp and A. Belendez, ‘Numerical and Experimental Analysis
of a Cantilever Beam: A Laboratory Project to Introduce Geometric
Nonlinearity in Mechanics of Materials’, Int. J. Engng. Ed., Vol. 19, p. 885
(2003).
[23] R. Frisch-Fay, ‘Large deflections of a cantilever beam under two
concentrated loads’, J. Appl. Mech., 29, pp. 200-201 (1962).
[24] H. J. Barten, ‘On the deflection of a cantilever beam’, Q. Appl. Math., 2, 168171 (1944); 3, pp. 275-276 (1945).
[25] K. E. Bisshopp and D. C. Drucker, ‘Large deflections of cantilever beams’, Q.
Appl. Math., 3, pp. 272-275 (1945).
[26] T. Belendez, C. Neipp and A. Belendez, ‘Large and small deflections of a
cantilever beam’, Eur. J. Phys., 23, pp. 371-379 (2002).
[27] R. Feynman, R. B. Leighton and M. Sands, ‘The Feynman Lectures on
Physics: Mainly Electromagnetism and matter’, Vol. 2, Addison Wesley, Ch.
38 (1989).
38
APPENDICES
39
APPENDIX A
FORTRAN PROGRAM CODE
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
**************************************************
*
*
*
THIS PROGRAM CALCULATES THE LARGE
*
*
DEFLECTION OF A CANTILEVER BEAM
*
*
*
*
INPUT VARIABLES AND INITIAL CONDITIONS:
*
*
*
*
FORCE ACTING ON FREE END OF BEAM = FORCE
*
*
ANGLE AT WHICH FORCE IS APPLIED = ALPHA
*
*
BEAM MOMENT OF INERTIA = I
*
*
BEAM MODULUS OF ELASTICITY = E
*
*
BEAM LENGTH = L
*
*
ARC LENGTH TO BE USED AS STEPSIZE = S
*
*
LOW INITIAL CURVATURE VALUE = KLOW
*
*
LOW FINAL CURVATURE VALUE = KLOWEND
*
*
HIGH INITIAL CURVATURE VALUE = KHIGH
*
*
MIDDLE INITIAL CURVATURE VALUE = KMID
*
*
MIDDLE FINAL CURVATURE VALUE = KMIDEND
*
*
TOLERANCE = EPS
*
*
SLOPE OF BEAM AT FIXED END = PHI
*
*
*
**************************************************
IMPLICIT NONE
DOUBLE PRECISION PHI, F, S, L, ALPHA, FORCE, E,
+I, X, Y, KLOW, KMID, KHIGH, KLOWEND, KMIDEND, EPS
INTEGER(4) N, NVALS
OPEN(UNIT = 10, FILE='EULER.IN', STATUS='OLD')
OPEN(UNIT = 20, FILE='EULER.OUT', STATUS='UNKNOWN')
READ(10,*) PHI, S, L, ALPHA, FORCE, E, I, KLOW,
+K HIGH, EPS
WRITE(20,100) FORCE, E, I, S, L, ALPHA, KLOW, KHIGH,
+EPS
40
100
FORMAT(1X, T30, 'EULER.FOR', //'INPUTS:', //1X,
+'FORCE = ',F8.4, '
N' /1X, 'E = ', E16.4,
+' PA' /1X, 'I = ',E16.4, ' M^4' /1X, 'S = ',
+F13.5, '
M' /1X, 'L = ',F12.4, '
M' /1X,
+'ALPHA = ',F8.4, '
DEGREES', /1X, 'KLOW = ',
+F9.4, '
1/M', /1X, 'KHIGH = ',F8.4, '
1/M'
+/1X, 'EPS = ',E14.4, ' 1/M' // 'INITIAL
+CONDITIONS:')
WRITE(20,200) PHI, KMID, X ,Y
200
FORMAT(/1X, 'PHI = ',F10.4, ' RAD' /1X, 'KMID = ',
+ F9.4, ' 1/M' /1X, 'X = ', F12.4, ' M' /1X, 'Y = ',
+F12.4, ' M' //T9,'PHI (RAD)', T22,'KAPPA (1/M)',
+T41,'X (M)',T56'Y (M)'/)
ALPHA = ALPHA*DACOS(-1.0D0)/180
C
C
C
C
C
10
C
C
C
C
C
C
**************************************************
*
THE FIRST DO LOOP CALCULATES KLOWEND USING
*
*
THE INPUT VALUE OF KLOW
*
**************************************************
NVALS = L/S
KLOWEND = KLOW
DO 10 N = 1, NVALS
PHI = PHI + S * KLOWEND
KLOWEND = KLOWEND + S * F(PHI,ALPHA,FORCE,E,I)
CONTINUE
**************************************************
*
THE SECOND DO LOOP CALCULATES KMID USING THE *
*
BISECTION METHOD AS LONG AS THE DIFFERENCE
*
*
OF LOW AND KHIGH IS GREATER THAN THE
*
*
SPECIFIED TOLERANCE
*
**************************************************
DO 20 WHILE (ABS(KLOW - KHIGH).GT.EPS)
KMID = (KHIGH + KLOW)/2
KMIDEND = KMID
PHI = 0
41
C
C
C
C
**************************************************
*
THE THIRD DO LOOP CALCULATES KMIDEND USING
*
*
THE CALCULATED VALUE OF KMID FROM ABOVE
*
**************************************************
DO 30 N = 1, NVALS
PHI = PHI + S * KMIDEND
KMIDEND = KMIDEND + S * F(PHI,ALPHA,FORCE,E,I)
30
C
C
C
C
C
CONTINUE
**************************************************
*
THE IF THEN STATEMENT DETERMINES WHETHER
*
*
KHIGH OR KLOW BECOME THE NEW KMID AND THE
*
*
ITERATION STARTS OVER
*
**************************************************
IF (KLOWEND * KMIDEND .LT. 0) THEN
KHIGH = KMID
ELSE
KLOW = KMID
KLOWEND = KMIDEND
END IF
20
CONTINUE
PHI = 0
X=0
Y=0
WRITE(20,300)PHI, KMID, X, Y
C
C
C
C
C
**************************************************
*
THE FOURTH DO LOOP USES THE FINAL KMID
*
*
VALUE FROM ABOVE TO CALCULATE PHI, KAPPA,
*
*
X & Y USING EULERS METHOD
*
**************************************************
DO 40 N = 1, NVALS
PHI = PHI + S * KMID
KMID = KMID + S * F(PHI,ALPHA,FORCE,E,I)
X = X + COS(PHI)*S
42
Y = Y +
SIN(PHI)*
WRITE(20,300) PHI, KMID, X, Y
FORMAT(1X, 4(F15.7))
300
40
CONTINUE
END
DOUBLE PRECISION FUNCTION F(PHI, ALPHA, FORCE, E, I)
DOUBLE PRECISION PHI, FORCE, E, I, ALPHA
F = (-FORCE/(E*I))*(SIN(ALPHA)*COS(PHI)+
+COS(ALPHA)*SIN(PHI))
END
43
APPENDIX B
FORTRAN PROGRAM OUTPUT OF EXPERIMENT 1
EULER.FOR
INPUTS:
FORCE =
E =
I =
S =
L =
ALPHA =
KLOW =
KHIGH =
EPS =
3.9200
.2000E+12
.1202E-11
.00100
.3000
90.0000
.0000
10.0000
.1000E-09
N
PA
M^4
M
M
DEGREES
1/M
1/M
1/M
INITIAL CONDITIONS:
PHI =
KMID =
X =
Y =
.0000
.0000
.0000
.0000
RAD
1/M
M
M
PHI (RAD)
KAPPA (1/M)
X (M)
Y (M)
.0000000
.0043767
.0087371
.0130812
.0174090
.0217205
.0260157
.0302946
.0345572
.0388035
.0430335
.0472473
.0514447
.0556259
.0597908
.0639394
.0680717
.0721878
4.3767068
4.3604035
4.3441007
4.3277986
4.3114976
4.2951980
4.2789000
4.2626040
4.2463103
4.2300191
4.2137307
4.1974455
4.1811636
4.1648853
4.1486110
4.1323408
4.1160751
4.0998141
.0000000
.0010000
.0020000
.0029999
.0039997
.0049995
.0059991
.0069987
.0079981
.0089973
.0099964
.0109953
.0119940
.0129924
.0139906
.0149886
.0159863
.0169837
.0000000
.0000044
.0000131
.0000262
.0000436
.0000653
.0000913
.0001216
.0001562
.0001950
.0002380
.0002852
.0003366
.0003922
.0004520
.0005159
.0005839
.0006560
44
.0762876
.0803712
.0844385
.0884895
.0925244
.0965429
.1005453
.1045314
.1085014
.1124551
.1163926
.1203139
.1242191
.1281080
.1319808
.1358374
.1396779
.1435023
.1473104
.1511025
.1548785
.1586383
.1623821
.1661097
.1698213
.1735168
.1771962
.1808596
.1845070
.1881383
.1917537
.1953530
.1989363
.2025036
.2060550
.2095904
.2131099
.2166134
.2201010
.2235727
.2270285
.2304684
.2338925
.2373006
.2406930
.2440695
.2474302
.2507750
.2541041
.2574174
.2607150
4.0835581
4.0673073
4.0510619
4.0348222
4.0185885
4.0023609
3.9861398
3.9699253
3.9537177
3.9375172
3.9213241
3.9051385
3.8889606
3.8727908
3.8566291
3.8404758
3.8243311
3.8081952
3.7920684
3.7759507
3.7598423
3.7437436
3.7276546
3.7115755
3.6955066
3.6794480
3.6633998
3.6473622
3.6313355
3.6153197
3.5993151
3.5833217
3.5673398
3.5513695
3.5354109
3.5194642
3.5035295
3.4876071
3.4716969
3.4557992
3.4399141
3.4240417
3.4081822
3.3923356
3.3765021
3.3606818
3.3448749
3.3290814
3.3133015
3.2975352
3.2817827
.0179808
.0189775
.0199740
.0209701
.0219658
.0229611
.0239561
.0249506
.0259447
.0269384
.0279317
.0289244
.0299167
.0309085
.0318998
.0328906
.0338809
.0348706
.0358598
.0368484
.0378364
.0388238
.0398107
.0407969
.0417825
.0427675
.0437519
.0447356
.0457186
.0467009
.0476826
.0486636
.0496439
.0506234
.0516023
.0525804
.0535578
.0545344
.0555103
.0564854
.0574597
.0584333
.0594061
.0603780
.0613492
.0623196
.0632891
.0642578
.0652257
.0661928
.0671590
45
.0007322
.0008125
.0008969
.0009852
.0010776
.0011740
.0012744
.0013787
.0014870
.0015993
.0017154
.0018354
.0019593
.0020871
.0022187
.0023541
.0024933
.0026363
.0027831
.0029336
.0030879
.0032459
.0034075
.0035729
.0037419
.0039145
.0040908
.0042707
.0044541
.0046412
.0048317
.0050259
.0052235
.0054246
.0056292
.0058373
.0060488
.0062637
.0064820
.0067037
.0069288
.0071573
.0073890
.0076241
.0078625
.0081041
.0083490
.0085972
.0088486
.0091032
.0093609
.2639967
.2672628
.2705131
.2737477
.2769666
.2801699
.2833574
.2865293
.2896856
.2928262
.2959513
.2990607
.3021546
.3052329
.3082956
.3113428
.3143745
.3173907
.3203914
.3233767
.3263464
.3293007
.3322396
.3351631
.3380712
.3409639
.438413
.3467033
.3495499
.3523813
.3551973
.3579981
.3607835
.3635538
.3663088
.3690485
.3717731
.3744825
.3771767
.3798557
.3825196
.3851684
.3878021
.3904207
.3930242
.3956126
.3981860
.4007444
.4032877
.4058161
.4083295
3.2660440
3.2503194
3.2346088
3.2189124
3.2032303
3.1875625
3.1719092
3.1562704
3.1406463
3.1250368
3.1094421
3.0938623
3.0782974
3.0627476
3.0472128
3.0316931
3.0161887
3.0006995
2.9852257
2.9697673
2.9543243
2.9388969
2.9234850
2.9080887
2.8927081
2.8773431
2.8619940
2.8466606
2.8313431
2.8160414
2.8007556
2.7854858
2.7702319
2.7549941
2.7397722
2.7245665
2.7093768
2.6942032
2.6790457
2.6639044
2.6487792
2.6336703
2.6185774
2.6035008
2.5884404
2.5733962
2.5583682
2.5433565
2.5283610
2.5133817
2.4984186
.0681243
.0690888
.0700525
.0710152
.0719771
.0729381
.0738983
.0748575
.0758158
.0767732
.0777298
.0786854
.0796401
.0805939
.0815467
.0824986
.0834496
.0843997
.0853488
.0862970
.0872442
.0881904
.0891358
.0900801
.0910235
.0919659
.0929074
.0938479
.0947874
.0957260
.0966636
.0976002
.0985358
.0994704
.1004041
.1013368
.1022684
.1031991
.1041288
.1050576
.1059853
.1069120
.1078378
.1087625
.1096863
.1106090
.1115308
.1124516
.1133713
.1142901
.1152079
46
.0096219
.0098860
.0101532
.0104235
.0106970
.0109735
.0112531
.0115357
.0118213
.0121100
.0124017
.0126963
.0129939
.0132944
.0135978
.0139041
.0142134
.0145255
.0148404
.0151582
.0154787
.0158021
.0161283
.0164572
.0167889
.0171233
.0174604
.0178002
.0181427
.0184878
.0188356
.0191860
.0195390
.0198946
.0202527
.0206135
.0209767
.0213425
.0217108
.0220816
.0224549
.0228306
.0232087
.0235893
.0239723
.0243577
.0247454
.0251355
.0255280
.0259227
.0263198
.4108279
.4133114
.4157799
.4182335
.4206723
.4230961
.4255051
.4278992
.4302785
.4326430
.4349927
.4373276
.4396477
.4419531
.4442437
.4465196
.4487808
.4510273
.4532592
.4554764
.4576789
.4598668
.4620402
.4641989
.4663430
.4684726
.4705876
.4726881
.4747741
.4768456
.4789026
.4809451
.4829732
.4849868
.4869860
.4889708
.4909413
.4928973
.4948390
.4967663
.4986793
.5005780
.5024623
.5043324
.5061882
.5080298
.5098571
.5116701
.5134690
.5152537
.5170241
2.4834717
2.4685411
2.4536266
2.4387284
2.4238463
2.4089805
2.3941308
2.3792973
2.3644799
2.3496786
2.3348934
2.3201243
2.3053713
2.2906343
2.2759133
2.2612083
2.2465193
2.2318461
2.2171889
2.2025476
2.1879221
2.1733123
2.1587184
2.1441401
2.1295776
2.1150307
2.1004993
2.0859836
2.0714834
2.0569986
2.0425292
2.0280753
2.0136366
1.9992133
1.9848051
1.9704121
1.9560343
1.9416715
1.9273237
1.9129909
1.8986729
1.8843698
1.8700815
1.8558078
1.8415488
1.8273044
1.8130745
1.7988591
1.7846580
1.7704713
1.7562988
.1161247
.1170405
.1179553
.1188691
.1197819
.1206937
.1216046
.1225144
.1234233
.1243311
.1252380
.1261439
.1270488
.1279527
.1288556
.1297576
.1306586
.1315586
.1324576
.1333557
.1342527
.1351488
.1360440
.1369382
.1378314
.1387236
.1396149
.1405053
.1413947
.1422831
.1431706
.1440572
.1449428
.1458275
.1467112
.1475941
.1484760
.1493569
.1502370
.1511161
.1519943
.1528716
.1537480
.1546235
.1554981
.1563718
.1572446
.1581166
.1589876
.1598578
.1607271
47
.0267192
.0271208
.0275247
.0279309
.0283393
.0287498
.0291626
.0295776
.0299947
.0304140
.0308354
.0312589
.0316845
.0321122
.0325420
.0329738
.0334077
.0338436
.0342815
.0347214
.0351632
.0356071
.0360528
.0365006
.0369502
.0374017
.0378551
.0383104
.0387675
.0392265
.0396873
.0401499
.0406143
.0410805
.0415485
.0420182
.0424897
.0429629
.0434377
.0439143
.0443926
.0448725
.0453541
.0458373
.0463222
.0468086
.0472967
.0477863
.0482775
.0487703
.0492646
.5187804
.5205226
.5222506
.5239644
.5256642
.5273498
.5290214
.5306789
.5323223
.5339517
.5355670
.5371683
.5387556
.5403290
.5418883
.5434337
.5449651
.5464826
.5479861
.5494757
.5509515
.5524133
.5538612
.5552953
.5567156
.5581220
.5595145
.5608933
.5622582
.5636094
.5649467
.5662703
.5675802
.5688763
.5701586
.5714272
.5726822
.5739234
.5751509
.5763647
.5775649
.5787514
.5799243
.5810835
.5822291
.5833611
.5844795
.5855843
.5866755
.5877531
.5888171
1.7421404
1.7279962
1.7138660
1.6997498
1.6856474
1.6715589
1.6574841
1.6434229
1.6293754
1.6153413
1.6013207
1.5873134
1.5733193
1.5593384
1.5453707
1.5314159
1.5174741
1.5035451
1.4896288
1.4757253
1.4618343
1.4479558
1.4340897
1.4202359
1.4063943
1.3925649
1.3787475
1.3649421
1.3511485
1.3373666
1.3235964
1.3098378
1.2960907
1.2823549
1.2686304
1.2549170
1.2412148
1.2275235
1.2138431
1.2001735
1.1865145
1.1728661
1.1592282
1.1456007
1.1319834
1.1183762
1.1047792
1.0911920
1.0776147
1.0640472
1.0504892
.1615955
.1624630
.1633297
.1641956
.1650606
.1659247
.1667880
.1676505
.1685121
.1693729
.1702329
.1710921
.1719504
.1728079
.1736647
.1745206
.1753758
.1762301
.1770837
.1779365
.1787885
.1796398
.1804903
.1813400
.1821890
.1830373
.1838848
.1847316
.1855776
.1864230
.1872676
.1881115
.1889547
.1897972
.1906390
.1914801
.1923206
.1931604
.1939995
.1948379
.1956757
.1965129
.1973494
.1981852
.1990205
.1998551
.2006891
.2015225
.2023553
.2031875
.2040191
48
.0497604
.0502577
.0507566
.0512569
.0517587
.0522619
.0527666
.0532727
.0537803
.0542892
.0547995
.0553112
.0558243
.0563387
.0568545
.0573716
.0578899
.0584096
.0589306
.0594528
.0599763
.0605011
.0610271
.0615542
.0620826
.0626122
.0631430
.0636750
.0642081
.0647423
.0652777
.0658142
.0663518
.0668904
.0674302
.0679710
.0685129
.0690559
.0695998
.0701448
.0706908
.0712378
.0717857
.0723347
.0728845
.0734354
.0739871
.0745398
.0750934
.0756479
.0762033
.5898676
.5909046
.5919280
.5929378
.5939342
.5949170
.5958864
.5968422
.5977846
.5987134
.5996289
.6005308
.6014193
.6022944
.6031560
.6040042
.6048390
.6056604
.6064683
.6072629
.6080441
.6088119
.6095663
.6103074
.6110351
.6117494
.6124504
.6131381
.6138124
.6144734
.6151211
.6157554
.6163765
.6169842
.6175787
.6181599
.6187277
.6192823
.6198236
.6203517
.6208665
.6213680
.6218563
.6223313
.6227930
.6232416
.6236768
.6240989
.6245077
.6249033
.6252857
1.0369408
1.0234018
1.0098721
.9963516
.9828402
.9693377
.9558442
.9423593
.9288831
.9154155
.9019562
.8885053
.8750625
.8616279
.8482011
.8347823
.8213712
.8079676
.7945716
.7811830
.7678017
.7544275
.7410603
.7277001
.7143467
.7009999
.6876598
.6743260
.6609986
.6476774
.6343623
.6210532
.6077499
.5944524
.5811605
.5678740
.5545929
.5413171
.5280464
.5147807
.5015198
.4882638
.4750123
.4617654
.4485229
.4352846
.4220505
.4088204
.3955942
.3823718
.3691530
.2048501
.2056805
.2065104
.2073397
.2081684
.2089966
.2098243
.2106514
.2114780
.2123040
.2131296
.2139546
.2147791
.2156032
.2164267
.2172498
.2180724
.2188945
.2197162
.2205374
.2213582
.2221785
.2229984
.2238179
.2246369
.2254556
.2262738
.2270916
.2279091
.2287262
.2295429
.2303592
.2311752
.2319908
.2328061
.2336211
.2344357
.2352500
.2360639
.2368776
.2376910
.2385041
.2393169
.2401294
.2409417
.2417536
.2425654
.2433769
.2441881
.2449991
.2458099
49
.0767595
.0773167
.0778746
.0784334
.0789930
.0795535
.0801147
.0806768
.0812396
.0818031
.0823675
.0829326
.0834984
.0840649
.0846322
.0852001
.0857687
.0863380
.0869080
.0874786
.0880499
.0886218
.0891943
.0897674
.0903411
.0909154
.0914903
.0920657
.0926417
.0932183
.0937953
.0943729
.0949510
.0955295
.0961086
.0966881
.0972681
.0978486
.0984295
.0990108
.0995926
.1001747
.1007572
.1013402
.1019235
.1025072
.1030912
.1036755
.1042602
.1048453
.1054306
.6256548
.6260108
.6263535
.6266830
.6269993
.6273024
.6275923
.6278691
.6281326
.6283829
.6286200
.6288440
.6290548
.6292523
.6294367
.6296080
.6297660
.6299109
.6300426
.6301611
.6302665
.6303586
.6304377
.6305035
.6305562
.6305957
.6306220
.6306352
.3559377
.3427259
.3295173
.3163119
.3031094
.2899099
.2767132
.2635191
.2503275
.2371384
.2239515
.2107667
.1975840
.1844032
.1712241
.1580467
.1448708
.1316963
.1185231
.1053510
.0921799
.0790097
.0658403
.0526714
.0395031
.0263352
.0131675
.0000000
.2466205
.2474309
.2482411
.2490510
.2498608
.2506704
.2514799
.2522892
.2530983
.2539073
.2547161
.2555248
.2563334
.2571419
.2579502
.2587585
.2595667
.2603747
.2611827
.2619907
.2627985
.2636064
.2644141
.2652219
.2660296
.2668372
.2676449
.2684525
50
.1060162
.1066021
.1071883
.1077748
.1083615
.1089485
.1095357
.1101231
.1107107
.1112986
.1118866
.1124748
.1130632
.1136517
.1142404
.1148292
.1154182
.1160073
.1165964
.1171857
.1177751
.1183645
.1189540
.1195436
.1201332
.1207228
.1213124
.1219021
APPENDIX C
FORTRAN PROGRAM OUTPUT OF EXPERIMENT 2
EULER.FOR
INPUTS:
FORCE =
E =
I =
S =
L =
ALPHA =
KLOW =
KHIGH =
EPS =
3.9200
.2000E+12
.1202E-11
.00100
.3000
53.0000
.0000
10.0000
.1000E-09
N
PA
M^4
M
M
DEGREES
1/M
1/M
1/M
INITIAL CONDITIONS:
PHI =
KMID =
X =
Y =
.0000
.0000
.0000
.0000
RAD
1/M
M
M
PHI (RAD)
KAPPA (1/M)
X (M)
Y (M)
.0000000
.0047156
.0094182
.0141077
.0187840
.0234471
.0280970
.0327335
.0373568
.0419666
.0465631
.0511460
.0557155
.0602714
.0648137
.0693425
.0738575
.0783588
4.7156453
4.7025787
4.6894663
4.6763087
4.6631061
4.6498592
4.6365681
4.6232335
4.6098556
4.5964349
4.5829718
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