stress distribution in splined shafts in torsion by the
TRANSCRIPT
Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1960
Stress distribution in splined shafts in torsion by the membrane Stress distribution in splined shafts in torsion by the membrane
analogy analogy
Charles L. Edwards
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Part of the Mechanical Engineering Commons
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Recommended Citation Recommended Citation Edwards, Charles L., "Stress distribution in splined shafts in torsion by the membrane analogy" (1960). Masters Theses. 5576. https://scholarsmine.mst.edu/masters_theses/5576
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i
T~ t 2\tc·~ c. i . \ .
STRESS DISTRIBUTION IN SPLINED SHAFTS
IN TORSION BY THE ME1'1BRANE ANALOGY
BY
CHARLES L. EDWARDS
A
THESIS
submitted to the faculty of the
SCHOOL OF MINES.AND METALLURGY OF THE UNIVERSITY OF MISSOURI
in partial fulfillment of the work requi"±ed for the
Degree of
1960
Approved by
ACKNOWLEDGEMENTS
The author wishes to thank Dr. A. J«> Miles for his assist ..
anoe and for suggesting the thesis subjecto The author also
wishes to thank Professor Go L. Scofield for his.:suggestions
on experimental procedure.
ii
PREFACE
The purpose of this investigation is to detennine torsional
shearing stress distribution in various shapes of shaftso
The project will include shafts with circular boundarieso
No attempt will be made to show the relation between stress
and applied torqueR
iii
iv
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS •••••••••••••••••••••••••••••••••••• a•••••••o••••V
PA.~T I: Derivation of the Equation for Torsional Shearing Stress and Methods. of Solution ••••••••••••••• o o ••••• 1
PART II: The Membrane Analogy And its Applications •••••••••••••••••••• 6
PART III: Procedure and Experimental Da:ta•••••••••••••••••••••••••••••8
PART IV: Conclusions ..... · .............................................. 34
BIBLIOGRAPiff o • o o o o ••a• o o • • o • o • o o a o o o a• • .o • a o ••a o • • • o o o • • • • • • o o a o • • a••• 36
VITA•••••••••••••••••••••••••••••••••••••••••••••••••••••••o••••••••o37
v
Lisr OF ILLUSTRATIONS
FIGURE Page
1 Sketch of infinitesimal body in pure torsion ••••••••.••••••••• !
2 Stress distribution in a non-circular shaft ............. u •••• l
3 Angular deflection of a plane•••••••••••··~··••••••••••••••••2
4 Sketch of non-circular prismatic shaft ••• o ••••••••••••••••••• 2·
5 Sketch of an element turning about a center without distortion ••••••••••••••••• a••·······················2
6 View of element A in the ZX plane••••••••••••••••••••••••••••3
7 Free body diagram of three dimensional element in-pure torsion••••••••••••••••••••••••••••••••••••···~··••••3
8 Geometrical representation of phi ••••••••.•••••••••••••••••• o. 4
9 Homogeneous membrane blown up over a. given cross-section ••••• 6
lOA Templates used in experiment••••••••••••••••••••oao•••••••••a8
lOB Templates used in investigation to simulate circular and splined shafts {Photographs) •••••••••••••••••••• 9
11 Component parts of.apparatus used in the investigation {Photograph} •••••••••••••••••••••••••• 110
12 Membrane apparatus with templates in place and glass plate removed {Photograph) .............. o•••o•o•••oll
13 Membrane apparatus with templates in place and glass plate in place (Photograph)ooo••o•••••••••••••••••ll
14 Membrane apparatus with templates and glass plate in place and the target in the marking position ·(Photograpijol2
15 Illustration of blowing up of films with small pressure from underneath shell (Photograph) ••••• o••••o••••••l4
16 Sketch of membrane on a circular cross-section •••••••••••••• 15
17 Membrane over circular template {Curve)•••••••o•••••••••ooo21
FIGURE
18
19
LIST OF ILLUSTRATIONS (Continued}
Lines of constant phi,circular
vi
Page
template (Theoretical), (Plate).••••••••••••••••••••••••••••22
Lines of constant elevation,oircular template (Experimental), (Plate) •••••• booQODoo••············23
20 Stress distribution in a circular shaft •••••••••••••••••••• D24
21 Circular shaft with semi-circular spline •••••••••••••••••••• 16
22 Lines of· constant elevation, one-spline template, (Experimental), (Plate) ••••••••••••••••••••••••••• 25
23. Lines o.f constant phi, one-spline template, (Theoretical)~ (Plate) ••••• o •• o ••••• o •••••••••••• a 2.6
24 Graphical -solution for fourth order equations, (Curve} ••••••• a••••••••••o•••••••••••••••••••••o•27
25 Membrame over circular one-spline 1:ernplate8 (Curve) o o •••••••••••••••••• .•••••••••••••• o •••••••• a 28
26 Stress distribution in a Gircular one-spline shaft I (Curve) a O O II O O • 0 ••• 0 a O • 0 a O ••• 0 ••• a a O II O a a • . a O 2 9
27 Circular shaft with four semi-circular splinesa•••••••••··~·20
28 Lines of constant elevation1 four-spline template, (Experimental), {Plate).0000••••••••••••••••••0•0~30
29 Membrane over circular four-spline teinpla te, {Curve) •••••••••• a •• ~ ....... o ••••.•••••••••••••••••• 31.
30 Stress distribution in a circular four-spline shaft1 (Curve)•••••••••o•••••••oooo•••••••••••o~••••••••••••32
Sl Stress distribution in· a · circular, one-splin~, and four ... spline shaft, (Curve} D ~ 0 0 0 o_o O O O O O O O ~ 0 0. 0. 0 0 0 0. 0 0 0 a O 33
32 Effect of an additional spline on a membrane ••••• o • ·• •••• a o o o 35
PART I
DERIVATION OF TiiE EC.UATION FOR
TORSIONAL SHEARING STRESS AND METHODS
OF SOLUTION
The stress distribution for a shaft with a circular cross-
section in pure torsion is a linear function of the radial distance
from the geometric center {s = T,.) . The stress distribution in a s 7T
non-circular shaft is not this simple. Since many- shafts encountered
in Mechanical Engineering are.of ~he non-circular sort, and since
these shafts tr3nsmit a relatively large torque, the ability to
predict torsional shearing stres·s distribution becomes important.
Den Hartog describes the derivation of Saint Venants compat-
ibility equation for a non-circular prismatic shaft in pure torsion(l).
In this discussion Den Hartog points out:
1. Shearing stresses on sections normal to the axis of the
shaft must act parallel to the surface of the shafto
l Ssn
Fig. 1. Sketch of e;p T
infinitesimal
Does Not Exist .: Ss =Sst
body in pure torsion.
2. Plane cross-sections do not remain plane since the shearing
stress is not distributed in a circular manner about the
center of rotation. el:> r
Fig. 2. Stress in a non-cfrcular shafto
(1) All references are in bibliography.
1
3. The stresses acting as shown in Fig~ 3 will rotate and
warp the cross-section. The projected shape of the section reIDnins
unchanged. ~ y
x x
Fig. 3. Angular deflection of a plane.
Saint Venant assumes the displacements of an element (A Fig. 5) in a
shaft to be: ~
G{=- 9,r. y V=-~lX
W= f fr,!/}
Fig. 4. Sketch of non-circular prismatic shaft.
Fig. S. S"~etch of an element turning about a center without distortion.
2
(1)
;-/here u is the displacement in the X directio7 v the displacement in the
y direction, w the displacement or warpage in the Z direction and ~,
the unit angular displacement of the element after the torque is applied.
The displacements now must be changed to strains. The total shear
ing strain ~of element A is the . sum of . angles C and D, (fig. 6).
Fig. 6. View.of element A in the ZX planeo
.Thus:
Within the elastic limit the shearing strain angle is small so that
Yx z= = e.u -1- £Yx o~ ax The same· method when applied to the ZY plane yields:
Yy~ - eY -1- .ow ~ oy·
Substitution of equation {l) produces Yx:2. == &/y t- ow Yyc = -~Xt-Jw BX ay
From Hooke's Law V 55 =.BI
Where G is the shearing modulus of elasticity and Ss is the
shearing stress. Putting/this into e~ations {4) gives .(Ss)xr= 6,9t~.1-. ~J 1t fl/- ax .
... 15sJu:c. ~ l7A .. ::-:-~.,( r.QW I _J o_j('/. .
A relation between the str~sses in equation {5} can be found by
equilibrium conditions. The fact that stre~s changes from one
side of an element to the parallel side a small distance away is
taken into account. Thus the stress in the ZY plane changes from
Fig. 7 Free body diagram of three dimensional element in pure torsion.
Summation of forces in the Z direction finds
d~O)'d2= le) (ss)x~-;- 2) t$.s~..c7-== 0 L ~x o_y J ·
3
(2)
(3)
{5}
0 (ss)x~ ox
The next operation in Saint Venant' s derivation of the compatibi_li ty
equation is. :to choose a ·.unique function£ such that
This f function so chosen leads to the condition of continuity • .
Substitution of equations (7} into (6) shows the continuitv. ~ o:z.l _ 2>;l.l ..
ox c;y "cJyax
.!/ Fig. 8. Geometrical representation o·f phi 0
Phi i can be plotted above an 1..'Y plane forming a curved surfacea
From equations (7) it can be seen that the slope of this surface
in the X direction is equal to the stress in ~he Y directiono
The X stress is the negative of the slope in the Y direction.
It can be shown that the slope in any direction is equal to the
stress perpendicular to that direction of slope.
Now all the equations can be written in terms of this function.
Equations (5) become
- o~ - (]{fJ,y.,. ~) c,!I - ax +~= G/-&tX-f-0:Y}
2>X . "?)y Taking a of the first of equations (9) and () of the second ~ i5x
equation and subtracting gives
4
{6)
{7}
(8)
(9)
(10)
This equation is known as the equation of compatibility because
it unites stress and strain conditionso
Equation (10) is the partial differential equa.tion for a shaft
in pure torsion. Solution of torsional stress problems thus becomes
a matter of solving equation (10)o
Ftnding the phi(f) function of equation {10) is very difficulta
Saint Venant working !°n 1855 . solved this equation for such shapes
as rectangular bars, ellipses, triangles, and semicircleso ( 2)
Many shapes such as shafts with squa·re splines cannot be readily
reduced to mathematical formulae. 'Inis fact precipitates the
use of analogies to equation (lO)o
Less than 20 years after Sta Venants work Sir William Thomson
(Lo:rd Kelvin) .devi$ed the fluid-flow analogy. {3) He pictured
the lines of . constant I {Fig. 8) as streamlines of vortical flow
within i;he boundaries of a given shape. The partial differential
equation for this flow is similar to St. Venant's compatibility
equation a The str.esses become proportional to the linear velocity
of the fluido
Lines of constant~ can also be interpreted as constant
electrical potential lineso The current flowing between these
lines is proportional to the stressa
5
P&itT II
THE MEMBRANE ANALOGY
AND ITS APPLICATIONS
A homog~neous membrane is stretched over. a hole _of given
shape with an initial tension T per unit length of periphery.
A small lateral pressure P defl.ect-$ the membrane. The tension
Twill not change appreciably if Pis i=. . .
lr-arrrrnw-T ~ p . ,·
Fig a 9. Homogeneous membrane expanded- over a
The vertical component of force Bis (4)
·-T £E. dy. ol(
The vertical component of force C is
· T~ d!J + 2-_(TOr. dy)dx The vertical ~lponents fof'l andqfare
-T oiE.. dx and atj
r li- dx + ~ (r az dx) dt respectiv~-lf. oy oy.
cross-section.
Adding the four components and equating them to the force of
the pressure - Pclx dy gives
then
0 2z + · 3-:..z - _ F ol/-:L. cJ"A::z. - T .
In 1903 15"randtl showed that equai;ions (10) and (11) were
similar and developed the Membrane Analogy for torsion. Taylor
and Griffith (5} were the first to use ~his analogy, applying
it to cylindrical bars. Since then others have used thi~ analogue
6
)(
(11)
for various odd-shaped shafts. Among these were Newbauer and Boston (6)
who in 1947 detennined the shear~ng stress distribution in twist ...
drill sections"
7
Christopherson and Southwell (7) proved.that equation (11) can
apply to any plane-potential problem, since all the partial differential
equations of this type .of problem take on this form. This fact
was previously sunnised since ¥..iles and Stephenson (8) had successfully
used equation (11) as an analogy for pressure distribution around
an oil or gas well.
In the field of Heat Transfer1 Wilson (9) in 1948 used the ·
mem1::>rane analogy to solve two-dimensional steady-state heat-conduction
problems. for this problem the height of the membrane represents
the temperature distribution in the member being analyzedo
Comparing equation (10) for torsion to equation (11) for
the membrane, it is seen that~ must equal Z and P in~st . be equal T
to z (J 'Jo This plus the fact that the membrane must be stretched
over-a hole geometrically similar to the shaft in torsion leads
to impossible technological problems. No attempt is made to meet
these conditions except for geometric similarity. Thus in the
membrane {usually a soap bubble) P is equal to some function of;?.Go}u T
The height Z is equal to a function of p. _e= K :z.G~ iE= = K, f T
This leads to a relation between shearing .stress and the slope of
the buhbleo
(S5 ) y2 = !<:i. oz= -. "?))( .
(S5))(i! == K3 aJE oy
PART III
PRCCEDURE.AND EXPERIMENTAL DATA
The experimental apparatus used was the same as -that used by
Wilson (10)~ Templates were machined to the specifications of Fig.
,----l-'-3,34.5 '1--t+-
C1~c~/qr
Fig 10 Ao
o. D ..
O a e-s p//oe F o LI r-Sp //o (!T.
Templates used in experiment.
4s0
-:zz;,_ -~
AA 3, .3 4-5 /I Q.l).,
The component parts of the apparatus are shown in Fig. (ll)o
These parts consist of an aluminum frame with a removable circular
shell, a circular target with clamps, a glass plate with a micrometer
head projected through the center, and the necessary test templates·o
The micrometer has a sharp point so that it may be screwed down to
touch th~ soap film on the template. The target has a- piece of paper
clamped on it so that as the target is brought down a point on top of
the m_icrometer head makes an impression on the paper. The micrometer
reading can then be recorded next to this mark. This procedure is
illustrated in Figso {12), (13), and (14).
The soap solution used was the same as that used by Wilson {11);
2 grams of sodium oleate.and 30 c.c. of glycerin in one liter of dis-
·tilled water. Different solutions are recommended by other invest-
igators such as Neubauer and Boston (12) who used 6 c.co of glycerin
and .s gram of sodium oleate per liter of distilled wate!• The soap
films produced by Wil~9n's solution proved to be sufficiently tough ?--:. •
and durable. A qµanti i:y. of soap solution is poured into the shell.
8
With a piece of celluloid o~ plastic a film is swept over the template.
A small wire is then used to remove bubbles and most of the excess
solution from the filmo
Figa 10 Ba Templates used in investigation to simulate circular and splined shaftso
9
Figa 11. Component parts of apparatus used in the investigation.
10
FigD 12e1 Membrane apparatus with templates in place and glass plate removed.
Figo 13. Membrane appa-ratus with templates in place., and glass plate in place.
11
Fig. 14. Xembrane apparatus with templates and glass plate in place, and the target in the marking position.
12:
The glass plate is innnediately placed over the.shell to prevent.
contamination of the film by dust and carbon dioxide. The excess
soap solution in the shell maintains a moist atmosphere to prolong
the film life.
Den Hartog (13) recommends that the film on the template be
eXpanded by a small pressure underneath the shell (Fig. 15}. With
this method the film is stretched farther in the expanding process1
thus shortening the film life. Also, when the micrometer point is
brought down on the film the chances of bursting the expanded film
are ·very high. Another disadvantage to this method is that leaks
in the shell and pressure regulation -device are very difficult to
stop, causing the height of the film to vary constantly.
For these reasons the soap films in this experiment were not
expanded. They were allowed to sag downward so that the only
pressure acting on them was that caused by the weight of the film
plus the excess moisture distributed throughout the film.
With the glass plate in place contour.lines (constant phi)
can now be plotted on the target sheet. The micrometer is left
at any· desired reading qnd the glass plate moved until contact
is made with the film. By placing a light so that its reflection
is seen in the film the point ·of contact may be clearly seen~
This is possible because the reflection of the micromet~r point
in the film will come into contact with the actual tip. When
contact is made the target is brought down on the micrometer top,
marking the paper •. Numerous points are so plotted then lines
13
drawn to connect these points,. These lines thus become contour lines.
Fiq. 15. Illustration of blowing up of films with small pressure from underneath shell.
14
15
The micrometer tip is peri~dically moistened with soap solution
to prevent film breakage when contact is made.
In order to compare experimental results with theory, two
problems for which the solutions are known were investigated.
The first was a solid circular shaft. A membrane stretched over
a circular hole of radius R is blown up { or allowed to sag) Fig. (16).
·IZ7
.. ~ r I . Lt'~\,.
T A-A T ~
Fig. 16. Sketch of membrane on a .circular. cross-section.
Outting a concentric circle out of the membrane and summing forces
in the Z direction gives:
T dz: 2. rrr = Prrr;z. Where 'T' is the~~rce per unit length of periphery, 'P' is the
pressure, and 'r' is the radius from the centera
Solving for z gives: ,!!: = - .J:i:r dr := :.... Pr-:,.'+ c :::t:r 4T
Z=-O wh~n r.:::. R. .. -", Z =- P {R;;;!.-;-;;e.) . 4-T
Or for a given membrane Pis constant so: T
~ -= c1
( R.-::i..- r-:2.) Where c1 is a constant.
If the membrane is adjusted so that equations (10) and {11) are
equal then Z becomes equal to phi(_p) and E.. equals .:<G~thus:
i= ~Ge, l~~r1 4-
T
e, is a function of geometry and torque so that for a given_
ahape and torque:
f - C (R~-;r-:i.) - ~
Where c2 is a constant.
(12)
(13)
16
Equation (13) is the theoretical shape for the phi~)function and
equation (12) is the theoretical shape for a membrane over a
circular templ~te Fig. (17) shown both the theore~ical curve (equation 13)
and the ~xperimental curve (equation 12). For comparative purposes
the experimental values were-multiplied by a constant. This gives
the same effect as ·equating the constants iii equations (12} and (13).
The lines of constant phi(f}for the theoretical and experiment~l
circular shaft are plotted in Figs. (18) and (19) respectively.
Fig. (20) is the stress distribution in the circular shaft. This
curve was found by graphical differentiation of Fig. (17). Fig. 20
can also be found by differentiation of equation {13).
o~ = Cz_(o-zr)=Ss
This shows the linear relation between the radius and the stress.
The second problem with a known solution is given by Den Hartog (14).
The shaft is circular with one semi-circular spline. Fig. (21).
Fig. 21. Circular shaft. with semi-circular spline.
The phi (j) function is given a·s: ·
(14)
Where C is a constant, a is the radius o~ the shaft, and b ·is the
radius of the spline.
17
This function must meet the conditions. of continuity ( equation_ 18.}
md the boundary condi tionso
Equation (a) shows the continuity of equation (14):
• * •
To meet the boundary conditions phi(~) must be _equa~ to zero
at t~e boundary:
X=-b j y=o i= c (!/-;i.qb + :!:;/-H2·):::c{-:Z.q/;+jqb).::o
x = -x. o. J ,ro 'I?= c (-1t1fo--::zq/aq)+z bQ::i.q _ H'-) -?q-:a.+o
~- C (tq~D""--4q~f 4a~-l')· = C (b~b) =o -?a-:z_
y A . .
-~ x ·polnt A Fj,,ZI _
x-= b 0d-fi <J = b ~/3
p:::c{b-·-c~=z,e,J+ b~,BJ-.zqh~t~91~~-h::)
1=c~ ( Coq~-J-~,8)-2qb<W,8+ ~t?'aC(J'l./3 -bi b~~~~t-~ff/ .
J = c [b:i.(r) -2ab Co<:1-{3 + 2-b0a eoo-/3 _ g]
. . . b::z£i]
. ~:::: C L - Z ab Cdl.f!,_ f- Zq b Co<J,8] o
18
19
By in_spection it can be seen that the maximtun . shearing stress
occurs at Y equal zero and X equal b. This can also be shown by
/':.. {x'J-J/1):zB"°a -:2h~CIX2
)
the phia) function. o~ = re (J.;<-.za i- rx2-;y-z J 2..
-ox.
o~ =-- fssllrJqt.,--=- c &-b-7..a rab~tab)-= GB; (;?. o, -b) c15i C:,7'
Taking the slope at any other point on the boundary will give
stresses of smaller valu~ than equ~tion {15).
Fig. 22 shows the experimental plot of constant elevation
for the shaft of Figo 21. Fig. 23 shows the theoretical plot
whi.ch represents lines of constant phi ~ ). Fig. 24 shows the
graphical method used to determine lines of constant phi~ for
the theoretical solution in Fig. 23. Points were chosen at Y
equals zero between X equal band X equals aQ At these points
the values of phi(~}were found from equation (14). These values
of phi{i)were then used at Y equals b to solve for the values of
Xo Solving for X involved fourth order equations such as follows:
:J ::= Oj X=~P . :2t:SQ ,:}... r- t?_ b'' f = .;.- c ( ·il?--?.c(2bf 2 b-b ) =- C '-3b - 3a u u =.b, x-= 7\ :r ==- c C3.b-:i..-3qb] 7 } -~ ~ 6:i..]
c [) 61 - :i ab] = c: (x"-...:,._q-x + if"' +- ~-,_,, ~ ~b"-) ~c J?-1qxf-::,,ab\ . . x -t'j I i-
2 O. 1--3 - X4 -t ( 2-b -3 o.) bx 2- 3 b5 (a-b) 4-= /,b72~' j b=,4S75''
-'X<I +3,31-SX 3 -/..Z/Xa== .. 3J/
Y~ e,=>ll ( Y=- -x1-+3.31-5X3-1,B1x
2
The line on which the maximum stress occurs CY-equal zero,
Xis between band a) is plotted for the experimental and theoretical
solution in Fig. 25.
20
Fig. 26 shows the stress distribution along the line of maximum
stress for both the theoretical nnd experimental solution. Since
the stress is -proportional to the slope of the phi (or Z) curve the
stress becomes:
Ss'~ ~ =-- c r:z X-;;?..a - zh~) (16) ~x . -x~
Fig. 26 was found by graphical differentiation of Fig. 25. Equation
{16} if plotted would agree with Fig. 26.
This plot is therefore a qualitative result. The actual
numerical value of the stress depends on c. This constant is
a ~u~ction of a1 ·and therefore a function of torque.
The last problem worked was a shaft as shown in Fi'J• 27.
:8 Fig. 27. Circular shaft with four semi-circular splineso
The theoretical solution to this problem is not knowno
-:-i_·, .. The experimental plot of constant elevations is eho't'm in
Fig. 280 The point at which the maximum stres·s occurs was seen to
be at Y equals zero and X equals b, or at the corresponding point
on auy of the other splines. Fig. 29.is the plot of the membrane
along the line Y equals zero and X lying between b and a. This
curve (Fig. 29) was graphically differentiated to show the stress
distribution in a qualitative m~nner {Fig. 30}o
For comparative purposes the stress distribution in the circular
shaft, the one-spline shaft, and the four spline shaft are- shown
together in Fig. 31.
. ··i·· ·.·· .
.. ,.1.
·····i· -. . . . .:!::
.. . t ..... - ••• -+. -+ ~ ••
... , I• • •- •
.. ~r-•• • 1 • .. .. : ! : .
::::t· :::.!:::. I . . +-·--sl----+~--+<~-+-~+---,1---+~-i-.~
.::l:: .... ::·:;:··· ::::1: . : : : .1: :::.:t·::.
.1 ..
'.! I
··---l-I·
j. : i
. .. ! , I
I
·I · !' I ------
i ... ~.:.+:::
. t. \ ..
. ...
·~-+-~+----if---+~-+-~-+----+----1f----+·~--+<~-+
.. ·:1<
...... !. ;., ' I
:1
~ . .. .
. . ... .
. .· .. -.. . ··:;·::: ..
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'
1 ' .. · :: . .. ---- -; _ _____ ___ ...
' .. I
. , . . . . : t ~ . . .
·-r-; '. ;; t' ; ·_ '. : .i....t,- -t- · -·· ···
::·: :.: r.: 1:
: ! ~: 1:: i: . . _ .. _. l ' - . -
, . !• •
• I •..
: ::1~:: :
PA.t.~T W
CCHCLUSIONS
34
The results obtained from the investigation were consistent in
several ways. The plot of _the membrane was always_ just below the plot
of the phi(f}function1 and repeated testing yielded consistent curves
for the investigations in which the solutions were known. It was con
cluded from th~s that the re~ults for the four-spline shaft were slight
ly under the theor~tical (unknown) solution.
Comparison of the figures for the contour lines shows that the
theoretical plots are slightly smoother-than the experimental
plots. This may be due to the fact that some of the points were
plotted using the color ·bands in the film as guides (Fig. 15).
These bands changed with membrane age and were also shifted by
the micrometer tip, since· the tip seemed to attract the film at
a distance of approximately .001 of an incho
A. slightly moist membrane was not a deterent factor in this in
vestigation. Wilson (15) found that with 'age' the membrane dried
out and shrank upwards towards th~ desired theoretical curve. How~
ever in this problem the theoretical curve was ·slightly below ev~n a
moist membrane. Therefo~e immediately after the excess solution was
removed measurements were taken. Although this eliminated the
variable of membrane age, it.gave best results.
The four-spline shaft shows a slightly higher maximum stress than
the one-spline and also a sharper drop in stress. At first this seems
unreasonable. ~ig. 32 illustrates the effect an additional spline
could have on .a membrane. . The s .lope is fncreased at the boundary and
decreased elsewhere.
· · ... --. · ~ •o/d 1/Aembrcme ~, 'Shqpe · ·
i-,.-..,1,_z v~ s s s. . 1/ew Me>7lb,z:,"e · · ·
Fig. 320 Effect of an Shqp~
additional spline on a : Membranea
Fig. 31 is a comparison betwa'en the various shaf::ts assuming
the angular deflections ~ to be equal o The stress ratio:"L between ·
the one-spline and dirctilar shafts is 2o54o . The ratio between the
four-spline and circular shafts is 2.7.60 These are not legi:timate
stress concentration factors because the relation between torque
and angular deflection is not the . 1same among the shaftso
This suggests that a technique for finding the relation between
stress and torque in this ·problem should be developedo An accurate
method for measuring the volume under the membrane would :have to be
used since a very small volume is involve:d• The purpose for fin~ing
the volume is that it can be shown {16} .that the torque is proportional
to the volume under the membrane.· Neubauer and Boston (17) used
graphical intergration for.best resultsn
35
BIBLI03ru\PHY
.1. Den J-{artog, J.P • ., M.:_ Strength tl :Materials, McGraw-Hill 1952, pp. 3:..10.
2. Ibid. Page 19.
3. Den Hartog, J.P., 11.1• Strength .2f Materials1 McGraw-Hill 1952, page 21.
4. W'ilson, L.H., Thesis., The Application of the Membrane Analogy to the Solution of Heat Conductidm Problems, Missouri School of Min-;;; 1948, -page7_o __
5. Taylor, Ga I. and Griffith, A.A~Adv. Como for Aeronautics, T.ech. Repto -(British_) 1917-1918, ~ 3M.
6. Neubauer, T.P. a.nd Boston, DaW •. , Transactions .A..S.M.E. Vol. 69, No. 81 November, 1947.
7. · christopherson, B.A., and Southwell, R.V., Proceedings of the Royal Society, S€ries A, Vol. 168, 1938~ pp. 317~350.
s. ~riles, J~.J. and Stephenson,. E.A., A.i.--uer. Inst. of lFi.in. and ~· Engrs. ~!. Pub., No~ 919# May.,_ 19~ --
9o W1lson3 L.H., Thesis, ~ Application of ~- r-1ew.hrane Analogy . to the Solution of Heat Conduction Problems. r!issouri School of llin~ 1948$1 --
10. Ibid. Page 16.
11. Op. Cit .. vlil"son L.H., Page 11.
12. Neubauer1 T.P.,. and Boston1 D.W., Transa·ctions A.S.H.E. Vo., 69 1 no. 8.,. November., 1947, Pc1:ge ~97D
13. Den Hartog, J.P., Ad. Strength of Materials, McGra,.11-Hill · 1952, pcige 13.
14. Ibid~ Page 337.
. . ' 15. Wilson, L.H., Thesis, ~Application £!.~Membrane Analogx:
to the.Solution of Heat Conduction Problems, Missouri School of Mines';' 1948, page2~ .
16. Den Hartog# J.P., M• Strength .2£. Materials:, McGraw-Hill 1952', page 9.
17. Meubauer, T.P., and Boston, D.W., Transactions A.S.M.E. Vol. 69, Noo 8, November 1947. Page 9020
1f7 VITA
The author w2.s born January 21, 1934, at Hiawatha, Kansas.
His high school edudation was completed in 1952 at Lafayette
High School in St. Joseph, Hissouri, at which time he entered the
St. Joseph Junior College at st. Joseph, Missouri and attended there
fn:un 1952 to 1954. In September, 1954 he entered Missouri School
of Hines a.nd Hetallurgy at Rolla, Missouri. In June1 1957 he
received his Bachelor of Science in Mechanical Engineering. He
be came an instructor ·in Mechanical Engineering September, 1957
· and entered graduate school.·