(i) 1

(1
A STUDY OF BOURDON TUBE DEFLEOTIOH
by
Geza Kardos
Thesis submitted to the Faculty of Graduate Studies
and Research in partial fulfillment of the
requirements for the Degree of
Yaster of Engineering
MoGill University
August, 1957
- 1 -
Commercial Bourdon Tubes.
- iii -
ACKNOWLEDGEMENT
The author wishes to express his thanks to the
following without whose help this study could not
have been carried out.
Professer A.R. Edla, who
guided the author in his researches; The Management
and Staff of Aviation Electric Limited, whose facilities and support were essentiel for the preparation
and testing of the sample Bourdon tubes; Mr. C.J.
Jenning of Eclipse-Pioneer Division of the Bendix
Aviation Corporation, who supplied the material
from which test units were m&de and who started
the author on this study; and Miss J. Carruthere
who did the typing of the report.
- iv -
ABSTRACT
Bourdon tubes are one of the most widely used
pressure sensing deviees.
In spite of their use
for over one hundred years, the design of Bourdon
tubes is still a trial and error procedure.
In the study, the leading theories of flat oval
Bourdon tube deflection are reviewed and compared
with test data.
A Bourdon tube design procedure
based on test results, is recommended minimizing
the number of test samples required to develop a
new tube, and the directions of future research
are indicated.
- v -
INDEX
TITLE • • • • • • • • • • • • • • • • • • •
ii
ACKNOWLEDGEMENT • • • • • • • • • • • • • . iii
ABSTRACT. • • • • • • • • • • • • • • • • •
iv
INDEX • • • • • • • • • • • • • • • • • • •
v
INDEX OF FIGURES. • • • • • • • • • • • • •
vi
..• •
.......•
INDEX OF EQUATIONS. •
• • • • • • • viii
NOMENCLATURE.
• • • • • .
•
1
INTRODUCTION. • • • • • • • • • • • • • • •
4
ACTION OF THE BOURDON TUBE. • • • • • • • •
6
. . . .. . .
TEST PROCEDURE. • • • • • • • • • • • • •
24
• 35
TEST RESULTS
•
43
• • • • • • • • • • •
50
MANUFACTURE OF BOURDON TUBES. •
•
• •
CONCLUSIONS • • • •
• • • • • • • • • • •
•
APPENDIX A.
Sample Calculation of Bourdon Tube Data.
65
APPENDIX B.
Geometry of Bourdon Tube Deflection. • • 71
BIBLIOGRAPHY. • • • • • • • • • • • • • • •
76
- vi -
INDEX OF FIGURES
TITLE
-PAGE
Commercial Bourdon Tubes.
Frontispiece
Fig. (1)
Drawing of Bourdon Tube Notation.
3
Fig. (2)
Sensitivity Curves, Wolf 1 s Theory.
15
Fig. ( 3)
Sensitivity Curves, Wuest 1 s Theory.
20
Fig. (4)
Tubes Prepared for Rolling.
27
Fig. ( 5)
Rolling Fixture with Pressure Pad
Retracted.
28
Fig. (6)
Tube; During Rolling.
29
Fig. {7)
Rolling Fixture Starting to Roll
Tube.
30
Pig. (8)
Tube Being Formed in Rolling Fixture.
31
Fig. (9)
Shims Being Removed from Formed Tube.
32
Fig. {10)
Finishing Operations.
33
Fig. (11)
Sections of As Received Tubing Used
for Tests.
34
Fig. (12)
Deflection Test Apparatus.
40
Fig. (13)
Deflection Measuring Fixture.
41
Fig. (14)
Detail of Deflection Measuring Head.
42
Fig. {15)
Sensitivity Ratio Resulta for 0.1
Axis Ratio.
51
Fig. (16 ) Sensitivity Ratio Resulta for 0.2
Axis Ratio.
Fig. (17)
52
Sensitivity Ratio Resu1ts for 0.3
Axis Ratio.
53
- vii -
INDEX OF FIGURES
TITLE
Fig. (18)
PAGE
Unit Deflection vs Axis Ratio 1200
55
Series Tubes.
Fig. (19)
Unit Deflection vs Axis Retio 3200
56
Series Tubes.
Fig. {20)
Unit Deflection vs Axis Ratio 4300
57
Series Tubes.
Fig. (21)
Logrithmic Curves of Unit Deflection
58
vs Axis Ratio.
Fig. {22)
Unit Deflection vs
for Tube With
60
Common Major Axis.
Fig. {23)
Tube Deflection Geometry.
Fig. (24)
Tube Angle vs Ratio of Tip Travel to
Radius Change.
72
75
- viii -
INDEX OF EQUATIONS
EQUATION NUMBER
PAGE
1 • • • • • • • • • • 1
2
• • •
• • • • • 8
•
3 • • • • • • • • • • 8
4 • • • • • • • • • • 9
5
•
• • • • • • • • • 9
6 • • • • • • • • • • 10
1 • • • • • • • • • • 10
8 • • • • • • • • • • 11
9 • • • • • • • • • • 12
10 • • • • • • • • • • 12
11
• •
12 • •
• • • 13,22
•
• • • • •
• 13
13 • • • • • • • • • • 14
14 • • • • • • • • • • 14
15 • • • • • • • • • • 16
16 • • • • • • • • • • 17
17 • • • • • • • • • • 17
18 • •
• • • • • • • 17
19 • • • •
• •
•
• 18
20 • • • • • • • • • • 18
- ix -
INDEX OF EQUATIONS
EQUATION NUMBER
PAGE
21 • • • • • • • • • 18,22
22 • • • • • • • • • . 19
23 • • • • • • • • • 19
24 •
• • • • • • • • 19
25 .
• • • • • • • • 21
26 •
• • • • • • • • 59
27 •• • • • • • • • 61
- 1 -
NOMENCLATURE
a= Half-width of tube cross section (Fig. 1.)
A= Constant in Wuest Formula (eq • .:zs)
.? =
Half-Height of the tube cross section (Fig. 1.)
C =Center of tube curvature (Fig. 1.)
E= Young's Modulus
/, = Pressure
sensitivity ratio (eq. 6 )
;;_ = Stiffness ratio ( eq. z_ )
,C
= Fun ct ions in Wolf 1 Theory ( eq./3 )
A= Wall thickness (Fig. 1.)
~
= Geometrical Moment of Inertia of cross section
J = A function in Von Karman' s Theory ( eq. 3
)
L = Distance of tip travel (Fig. 1.)
~
~
= Linear distance from tube mounting to tip (Fig. 1.)
= Bending moment
0 = Origin of axis (Fig. 1.)
~
= Differentiai
~=Radius
s
pressure applied to tube.
of curvature of tube center line (Fig. 1.)
=Arc length of middle surface of tube section,Wo1f 1 s Theory(Eq.g)
T =Tension in tube wall, Wuest Theory (eq./6)
~
= Displacement of middle surface, Wuest Theory(eq./6)
V= Displacement of middle surface, Wuest Theory( eq. /t )
hi= Potential energy due to distortion, Wolf's theory (eq.8)
X=
Axis through cross section (Fig. 1.)
X= Direction co-ordinate (Fig. 1.)
~=Axis
through cross section (Fig. 1.)
- 2 -
Y=
Direction co-ordinate (Fig. 1).
~=Geometrie
A
?
Factor, Wuest Theory (eq. 24).
= Increase in operand
= Displacement of middle surface Wolf's Theory (eq. 8).
&()=Denote a function of operand.
~
~
= Geometrie
= Poissons
parameter
/f';{
a7
ratio.
~ = Angular length of tube (Fig. 1).
k
= Constant.
/?
= Constant
- 3 -
Fig. 1. Drawing of Bourdon Tube Notation.
-4INTRODUCTION
The Bourdon Tube is one of the most widely used elements
in the detection of fluid pressures.
It was invented in 1849
by Schinz and marketed in 1850 by E. Bourdon.
Since then it
has been widely used and developed to detect pressures from
psi to 140,000 psi.
!
Although attempts were made as early as
1890 by Lord Rayleigh to analyze and predict the performance
of Bourdon Tubes, as yet no satisfactory theory has been determined that can predict the action of Bourdon Tubes within
reasonable design limita.
Errors between theory and test
resulta differ by as muchas
5o%.
(Reference 3).
Part of
this deviation can be attributed to the fact that very little
accurate data has been published on the performance of tubes
over a wide range of geometrie parameters.
The first attempt
to do so was made by H.L. Mason (Reference 3) under the auspices
of the ASME Research Committee on Mechanical Pressure Elements.
H.L. Mason tabulates data from various Bourdon Tube manufacturera.
This data is of questionable value because it is primarily based
on production data and not limited to any particular shape of
tube element.
Most theoretical attacks on the problem are of limited usefulness as they have found the general solution too complex to
handle mathematically and have bad to limit their deriviations
to those giving mathematically soluble equations, limiting cases
where particular geometrie ratios approach zero or infinity.
Unfortunately few practical tubes manufactured fall into these
categories.
- 5It is the intenti9n of this research to made an emperical
study of the deflection characteristics of Bourdon Tubes of
the most common cross-sectional shape, the flat oval.
The
existing theories are to be examined and compared with test
data from a number of tubes manufactured to give data over
a wide geometrie range.
From this the deviation of the theories
from the emperical data can be determined, emperical curves and
formulae describing the deflection characteristics can be derived,
and the reasons for the deviations between theory and test resulta
can be evaluated.
- 6 ACTION OF THE BOURDON TUBE
General:
The Bourdon Tube consista essentially of a curved tube
of oval or elliptical cross-section which is rigidly secured
at one end while the other end is allowed to move freely. Both
ends are sealed except for a capillary tube which permits the
application of a pressure on the inside of the tube.
When a
differentiai pressure is introduced into the tube, the radius
of curvature of the tube increases and the free end of the tube
is displaced in an approximate straight line by an amount directly
proportional to the pressure differentia!.
The action of the tube may be qualitatively described as
follows, when a
positive pressure is applied to the inside of
the tube, the walls parallel to the principal axis bulge and the
cross-section attempts to become circular.
The inside wall moves
towards the center of curvature C. resulting in a compressive
stress in the wall along the length of the tube.
The outside
wall moves awày from the center of curvature resulting in a
tensile stress in the wall along the length of the tube. Therefore,
with a tube of constant section and curvature, a uniform bending
moment about the principal axis of the tube is produced along the
length of the tube which increases the radius of curvature ~
it can be stated that internal pressure decreases the radius of
curvature of the tube cross-section and therefore in accordance
with Gauss theorem (Reference
15}, the product of the curvatures
Or
- 7 must remain constant, the radius of curvature of the tube
increases.
Although this action can be easily explained, the evaluation of the moment due to the bulging becomes mathematically
difficult.
Added to this is the fact that the curved tubes do
not react to bending moments the same as curved rods of equivalent geometrie inertia.
Therefore, two mathematically different
quantities must be considered as well as their affect on one another.
Bending of Curved Tubes.
As an introduction to the deflection of Bourdon Tubes, the
bending of curved tubes is to be considered.
The bending of
curved tubes is also important when considering the deflection
of Bourdon Tubes under internai moments •
.The deflection of a curved tube under an applied moment
differa from the deflection of a curved beam of equivalent geometric moment of inertia of the cross-section, because the crosssection of the tube distorts during bending.
The deflection of a curved bearn of radius )f, geometrie
moment of inertia I and angle of curvature ~ due to a moment 1'1
can be expressed as change in angle ~ J!
: ~
E.I
(Reference 10).
(/)
- 8 This does not hold true for curved tubes.
In the case
of curved tubes, the geometrie moment of inertia must be
modified by a factor ;;__ called the stiffness ratio. (Reference
10).
(:l)
The evaluation of the stiffness ·ratio
~
is of great
practical importance in piping design and excellent data
has been collected and published by the M.W. Kellogg Co.
(Reference 11) for round tubing.
H. Von Karman 1 s values for
/~
for round tubing derived
by using the principle of minimum potential energy in the
Rayleigh-Ritz faahion is given in equation (3).
1
tl
=1
Where j
f
{'3)
-r-~--.
/ZÂ+I-:J
is a function of the geometry factor
?- as
given in Table 1.
TABLE I (Reference 11) •
Â
0
• 05
.1
.2
J'
1
• 762
.568
.307
·75
.176
.0748
.0352
1.0
.0202
- 9 -
s.
Timoshènko
(Reference 13) using the potentiel energy
of deformation approximates a formula for the stiffness ratio
of rectangular cross-section tubes.
+ [ ;r~(;·f1 [~·,n·[..t/(t/+ ~
a-•rt)(t.')j
(SI)
(; _,tL
zJ
Jr :t. (1-; I .. ) [/ -+_]_a.
I,
/.! j :t (A.~) t
f~ (il/
(~,
tfi. (f)f1•J'] J
A,
= Thickness
~a.=
~
~
of horizontal plates.
Thickness of vertical plates.
= Geometrie
= Geometrie
moment about horizontal neutral axis.
moment about vertical neutral axis.
This formula is rather unwieldy for general analysis but
quite useful for a particular problem.
Timos#henko has reduced
his formula for the special case of the tube of square crosssection.
(5)
- 10 -
Bourdon Tube
De~lection
Theory.
F.D. Jennings (Rererence 2} summarizes the leading theories
o~
Bourdon Tube behavior. The following analysis of the theories
is based on Jenning's paper,except that the papers of the original
author were consulted ror a more thorough understanding.
Jennings
discusses both flat oval and elliptical section Bourdon Tubes. In
this discussion we shall restrict ourselves to the flat oval section.
In order to have a common basis for comparison, Jennings reduces
all theories in terms of dimensionless ratios
l
/.2-
and
which
describe the action or the tube.
(6)
This is the sensitivity ratio which gives the deflection of
the tube under pressure.
(7)
This is a stiffness ratio and is the same as that in equation (2) since
43
~
If the values of
is equal to
~
and
4
~
(
appendix B.)
,{ can be determined, the deflection
of a tube under pressure and external moment can be predicted regardless of the material and absolute dimensions.
- 11 -
Jennings plots values of /, and
,{ against ~ and obtains
curves for each particular value of axis rat1o4" ,
Alfred Wolf (Reference 7) derives a theory for Bourdon Tube
deflection using energy methode based on two elements of strain,
the bending of the walls in a transverse section through the
tubing, and a longitudinal extension parallel to the axis of
the tubing.
Consider a section of Bourdon Tubing with 0 as the origin
shown in Fig. 1.
Under internal pressure, the elastic displace-
ment of the middle surface (dotted line in diagram) of transverse
section takes place along the
;;t
and · the
~
placements are functions of the middle surface
axis.
S
These dis-
which is
assumed to remain unchanged in length.
Using these displacements, Wolf derives the expression for
potential energy t0 per unit length of tubing due to bending of
the walls and to the increase in volume.
(i)
1 = Displacement
of the middle line parallel to
~axis.
- 12 -
The potential energy due to the extension of elements
parallel to the axis of the tubing is considered and evaluated
as a function of .5 only.
The affect of simultaneous change
of radius of curvature of the axis of the tubing, the length
of which remains constant and the elastic displacement of the
walls is considered.
The potential energy ~ per unit length of tubing due
to extension is derived by Wolf.
(f)
Wolf now carries out an operation which he justifies
only because it. works.
energy
two
W per
He assumes that the total potential
unit length of tubing will be the sum of the
~nergies.
(lo)
Since the system is in equilibrium the potential energy
must be a minimum, therefore, equation(/0} can be expressed as
- 13 a differentia! equation for minimum ~ •
(Il)
This is solvable by quadratures for the special case of
tube with rectangular cross-section.
Wolf, to make the equation
solvable for flat oval section tubing, further simplifies the
equation by letting
~
approach infinity.
Which is solvable by quadratures.
To determine the sensitivity of the Bourdon Tube equation
(/L) is solved wi th A~ as a pa,rameter.
sti tuted back in equation ( /O)
gives a minimtun potential energy
The solution is sub-
and the value of
W
4
~ which
is the required value of LI~
Wolf uses the methods of the calculus of variations for
solving the equations.
He then applies his method to the case
of the .flat oval tube wi th mean major axis
,:la.
and mean minor
axis 2./-and derives expressions which result in the sensitivity
- 14 ratio
1,
stiffness ratio ~ of flat oval Bourdon Tubes.
/, = ·:l 66 7 (/-,a~ /"
/,.(/-,a~ •//~.3 ~ ..Â- ,_
- t'{
:=
/,.. ( /
-~ ~. //~.J,;:; ~-2.
(11)
/JI- (/ -~ ~ <1'tf'/~ /i ~3.
Values of
0 /{
and~
are functions of the axis ratio
and are listed in Table II.
TABLE II (Reference 7)
%_
. o.o
0.1
0.2
0.3
0.4
;:-
r;
1.000
1.273
1.469
1.574
1.574
1.000
1.040
1.059
1.057
1.024
~
1.000
1.207
1.373
1.498
1.572
Alfred Wolf's values of ~ and /;__are plotted against
Â
in figure ( fl,) for various axis ratios.
Wolf in deriving his theory makes 'several assumptions
which limit its accuracy, the most important of which is that
he simplifies equation (Il) to equation ( /J..) by let ting
K
- 15 -
•
~
s:..
0
Q)
.c:
E-1
en
~
r-i
0
~
..
en
<D
1>
s:..
~
t..)
~
.p
.....
>
.....
.p
.....
O'J
c:
Q)
-;
Cl)
•
C\J
•
t()
.....
~
0
- 16 approach infinity.
This immediately limita the formula to
tubes having a large radius of curvature, i.e., tubes having
values for
Â
equal to or greater than 1.
This procedure eliminates the second term in the differential equation, which is insignificant for lsrge values of
A(
but is quite significant for small values of
~-
Dr. Walter Wuest (Reference 7) derives a theory for
Bourdon tube deflection by evaluating the stress distribution
in the walls of the tube.
Consider a section of Bourdon tubing with
origin and axis
~
and ?
as the
as shown in Figure (1).
Wuest assumes that the middle surface can be given with
?
as a function ·of
;;c
•
If the axis ratio ~ is small, that is, approaches zero,
displacement
ace
~
U
of the middle surface tangent to the middle surf-
displacement
V normal to the middle surface can be
assumed to be along the
::t
and 1/ axis, which considerably
simplifies the necessary calculations.
Wuest sets up the equat-
ions for the tangentiel and circumferential strains of the wall
elements in terms of the displacement of the middle surface
and
V
, the curvature K
can be written as
, the change in angle A~ (which
~. (See Appendix B).
~
- 17 and the distance of the element from the middle surface.
Using Hooks Law and Poissons ratio, the tangential stress
and the circumstantial stress in terms of the previously
determined strains can be derived.
Integrating over the
thickness of the wall, resulting in the tangential tension
~ the circumferential tension
~
and the moment
~~ ·
(11)
,!( 3 tf"
1:z (1-"'" z)
tl(' 2 v
t:t
':it ~
- 18 -
In the limi ting case where
%
approaches ze.ro,
~ - can be evaluated.
also approaches zero,
j{-;t-
Evaluating the sheering force in terms of "'Tc
the internal pressure
1; .
~
.1
/%:
and
two simultaneous equations are derived.
The shearing force is ell.minated and an equation relating
?: / 1'1~
and r:' is obtained.
(~o)
Substituting for ~ and
Tc
the final differentiai equation
relating the bulging to the geometry of the tube is derived.
(1{1)
Wuest derives a general solution for the equation which
yields values for
constants
c,
and
% in terms of a geometrie function, ,A boundry
c~ and geometrie and hyperbolic functions of the
middle surface function
e(%).
- 19 -
/l
~
6 (1- ..,a)"' a3,;D
,4 Y.ft- 3 E
The function · é9(~) can be given by trigometric functions,
~,
exponential functions or polynominals in
Wuest further explores the case wheree{~Jis expressed as
a sixth degree polynomial..
This gives solutions to the equations
for tubes of various shaped cross-section depending upon the constants in the polynomial and the boundary conditions.
The solution
that most closely approximates the flat oval shape is one in which
only the sides in the direction of the
elastically and are parallel ·to the
~
Â
axis are cons1dered to act
axis, the sides in the dir-
ection of the? axis are considered to be infinately rigid with
respect to bulging and infinately elastic with respect to bending.
For this shape Wuest finds values of
1:
and .lz_ in terms of tube
geometry.
(,z. Y)
- 20 ,~
0
Il
Il
~~
...O'l
•
l""\
- 21 -
-/
~
C'b.s~ :2~
5/;y~
-
;4 -
Ce>S
P'
$/#,(5
Wuest derived values of ,f, and are shown plotted against
?.. in Fis. 3 as curve for axis ratio
%
~0 ·
Wuest analysis has two principle short comings.
the assumption that
The first,
~ approaches zero limita the use of his analysis
to tubes of very small axis ratio, that is lesa than 0.1.
This excl-
udes it from being applied on tubes required to operate at medium or
high pressures because of their heavy wall which cannot be readily
made with small axis ratio.
The seoond limitation is by use of the
idealized section which excluèes the effect of the tube wall parallel
to the
~
axis; these sides do resist bending and have a restraining
effect on the bulging of the sides parallel to the ;c axis although
i ts effects drop off as ~ becomes smaller.
Thus, both limitations of Wuest's theories are applied to
flat oval tubes exclude its use with tubes of large axis ratio, but
the analysis is of qualitative value, to give an indication as to the
fonn of the function of
/,
wi th respect to /--
Wuest, in addition to deriving his theory, gives in Reference {7), curves for functions of
with respect to
values of ~ taken from an unpublished work by himself.
for finite
These
curves are reproduced in Figures (3)
Ezamination of the fundamental differentia! equations describing the action of the Bourdon Tube as derived by Wolf and Wuest
- 22 -
shows a marked similarity.
(:;..1)
The first terms on the left-hand side of each equation
are fourth differentiel of the displacement of the direction of
the~
axis. The second terms on the left-hand side of each equat-
ion are identical functions of the displacement in the direction
of the
~axis.
The terms on the right-hand side of each equation
are identical functions of the pressure except that \iolf' s theory
has an additional differential factor added.
Wuest's equation
differa from Wolf's primarily in the third term which ts simplified
by Wuest because of his assumption that the axis ratio is zero.
Wuest idealized section approaches Wolf's suggested solution
using a rectangular cross-section.
Wolf in his solution for
the flat oval section drops the second terms on the left-hand side
- 23 by assuming that
/<'
approaches infini ty.
It can be seen that the different methods of analyzing
the action of the tube, have derived the same general equations,
but only in the simplification for purposes of solution do the
answere differ.
Therefore, Wolf and Wuest's solutions can be
treated as answers to the same equation, emphasizing different
phases of the solution.
- 24 MANUFACTURE OF BOURDON TUBES
The Bourdon Tubes used for this study were especially
manufactured in accordance with the procedure developed by the
Eclipse-Pioneer Division of Bendix Aviation Corporation and
presently used by Aviation Electric Limited of Montreal.
The
tubes, in various stages of manufacture, are shown in Figures
(4), (6) and (10).
Lengths of tubing are received from the mill in round,
elliptical or flat oval cross-sections with dimensions held to
very close limita.
The tubing is eut to lengths (Figure
for forming into individuel tubes.
4
(1))
One end of the tube is flat-
tened and bent to permit it to be held by the rolling fixture
(Figure
4
(11)).
The tubing is filled with the required number
of shims to control the reduction of the minor axis during the
rolling operation.
(Figure
5).
The tube is formed in the rolling fixture
The flattened end of the tube is fitted into a slot
in the forming roll (Figure 6 (iii)) and Figure (7), around which
the tube is formed.
A pressure pad exerts a constant pressure against
the outside of the tube collapsing it onto the shims.
The forming
roll is rotated and the tube is formed around the roll (Figure 8).
The tube is removed from the forming roll (Figure 6 (iv).
The
shims are then withdrawn from the tube without distorting the tube
shape (Figure 9 and Figure 10 (v)).
The rolled tube is eut to the desired included angle and
the mounting end is formed to receive the capillary tube through
- 25 which the pressure is applied to the finished tube (Figure 10 (vi)).
The required free end fitting, the capillary tube and the
mounting bracket are soldered on, to complete the Bourdon Tube.
In cases where the tube materiel responda to heattreatment,
the tube ia heat treated either before or after aoldering depending
on the temperature of the beat treatment and the nature of the solder
uaed.
The above method of manufacture permita a wide control of
the geometrie parameters.
The shimming used, and rolling pressure,
permits a range of axis ratios limited only by the original configuration of the tube or the minimum shimming practical.
The roll dia-
mater and rolling pressure, governs the radius of curvature of the
finished tube.
The cut.ting operation governs the tube angle.
The
wall thickness and material .of the tubes is dependent upon what the
tube mill is willing and able to supply.
The principal requirement of producing good tubes for production purposes is to ascertain the correct values for the above
controlling variables, and to devise methods of holding .them to
close limita.
For this study, 33 commonly used tube sections were supplied
by Mr.· C.J. Jenningi of Eclipse-Pioneer nivision of the Bendix Aviation Corporation, in three different materials; Trumpet Brasa, Ber711ium Copper and Ni-Span
c.
The range of original tube sections are
- 26 shown in Figure (11).
Almost all of the tubes for this study
were rolled using a single size forming roll.
The rolling press-
ures were controlled to ensure that tubes of uniform quality were
produced, without wrinkling or stretching.
were selected to give desired axis ratio.
The shims in the tubes
In specifie cases a
complete range of shims from the minimum to the maximum were used.
The Beryllium Copper and Ni-Span C tubes were heat treated
in accordance with recommended heat breat practices for the particular materiel, Reference (17)and (18).
For the tubes used in this study, the fitting for the free
end was replaced by a round copper bar to facilitate test measurements.
- 27 -
Fig.
4·
TUBES PREPARED FOR ROLLING
(i) Stock Cut to Length.
(ii)End Flattened and Bent.
- 28 -
Fig.
5.
Rolling Fixture With Pressure
Pad Retra.cted.
- 29 -
Fig. 6.
Tube - During Rolling
(iii) Tube Fitted in For.ming Roll.
(iv)
Formed Tube Removed
~rom
Roll.
- 30 -
Fig. 7• Rolling Fixture Starting to Roll Tube.
-
)1 -
Fig. 8. Tube Being For.med in Rolling Fixture.
- 32 -
Fig. 9. Shims Being Removed From Formed Tube.
- 33 -
Fig. 10. Finishing Operations.
(v) Tube with Shims Removed.
(vi) Tube Cut to Required Angle.
(vii) Tube soldered with Mounting Plate
Capillary Tube and Free End Fitting.
- 34 -
Fig. 11.
Sections of As Received Tubing
Used for Tests.
- 35 TEST PROCEDURE
PURPOSE OF TESTS:
As previously stated, the present knowledge of the
deflection characteristics of Bourdon tubes is sketchy.
To
date satisfactory theoretical evaluations have not been made.
Until a satisfactory theory is determined and to provide data
against which it can be checked, empirical data of the characteristics is desired.
The Special Research Committee on Mechan-
ical Pressure Elements of the American Society of Mechanical
Engineers have taken the first steps to gather this empirical
data, (Reference 3).
Information was solicited by the Committee
from 50 manufacturera on tubes of their manufacture.
This data,
published, shows a great deal of variation, primarily because the
data supplied was of a production nature.
It is the purpose of this study to produce and determine
the deflection characteristics of a wide range of tubes of similar
cross-section, the flat oval.
Against the test resulta the lead-
ing deflection theories will be checked and an attempt made to
empirically evaluate the àeflection characteristics.
TESTING:
Bourdon tubes were manufactured in accordance with the
method previously outlined.
of A and axis ratios
~ •
Tubes were made with a broad range
The geometrie parameters were measured
and the deflection under pressure was determined.
This data was
then tabulated to give a picture of the relationship of geometrie
parameter to deflection characteristics.
- 36 The wall thickness ,( was determined on a sample of
the tubing as it was received from the tube mill.
The tube was
split and the wall was measured with a ball head micrometer.
The average of several readings over the length was recorded.
The thickness was measured ta the nearest .0005 inch.
Since
the wall thickness varied from .035 inch to .oo8,the accuracy
will be from
1.5%
ta 6% depending on the sample.
The outside major axis, the outside minor axis and
the outside diameter of the finished tube were measured with
knife-edge calipers reading ta .001 inch accuracy.
With out-
aide major axès varying from .920 inch ta .250 inch, the accuracy will be from
.11%
to
.4%.
With outside minor axes varying
from .166 inch ta .031 inch, the accuracy will be from .6% to
3.2%.
With the outside diameter approximately
1.75
inches, the
accuracy will be .06%.
The angle ~6 wa~ determined by measuring the cord on
the inside radius with knife-edge calipers.
Because of the
solder fillets at the free end and at the mounting, the reading
could only be considered to be correct within .020 i nch which
gave an accuracy of 2
lected an accuracy of
0
-
in the included angle. This in turn ref-
4%
in evaluation of ~o
•
After pressure t es ting, the free end of each tube was
eut open ta determine how the soldering ran and an estimate was
made as to the reduction in the effective angle of the tube due
ta solder running inside the tube and increasing the thickness.
The apparatus for checking the Bourdon tube deflections
- 37 is shown in Figure (12).
The tube is mounted in the deflection
measuring fixture Figure (13), which permits measurement of the
tip deflection in two perpendicular directions.
The tube is then
pressurized using an ".Am.thor" dead weight test stand.
Details of the measuring head are shown in Figure (14).
The measuring head consista of two micrometers calibrated to
.0001 inch mounted at right angles and in the same plane.
The
measuring head and the Bourdon tube are insulated from one another.
By placing an ohm meter between the measuring head and the tube,
the point at which contact is made between the tube and micrometer
is determined by the meter indicating electrical continuity.
All
readings were made at the point where electrical continuity was
broken.
tube.
This procedure ensures that there was zero load on the
Readings could be duplicated to .0001 inches.
Deflection Testing consisted of mounting the Bourdon tube
in the fixture and cycling it a minimum of
test pressure range plus
25%,
5
times through the
the test pressure range was taken
as that pressure which gave a maximum total deflection of at
least
.05
inch.
The co-ordinate of the free end of the tube
were then measured at zero pressure and at least four pressures
within the pressure range.
The difference in the co-ordinates
was calculated to determine the displacement under pressure in
the two directions, using Pythagoras' theorum and by dividing
by the pressure, the deflection rate was calculated.
Initially the direction of the tip travel was taken at
random direction with respect to the micrometers, but it was soon
- 38 found advantageous to orientate the tube by rotating the mounting
table so that the total movement of the tip in one measured direction was lesa than .002 inch.
This procedure reduced the error due
to the free end of the tube passing across the face of the micrometer, because there was then virtually no travel across the face
of the micrometer measuring the principle deflection, and although
the percentage error is increased in the small co-ordinate, the
order of magnitude of the co-ordinate was so small that it had
very little effect on the value of the total deflection.
The values of Young's modulus for this study were taken
from published values given in (References 9, 17 and 18).
CALCULATION:
The measured geometry and deflection data for each Bourdon
tube was reduced to the required parameters for evaluating the
characteristics of the tube.
for one tube
A sample of complete calculations
is given in Appendix A and the methods are self-
explanatory.
The calculated values for
.:::t. /
/;
and ~ were plotted
to compare them with the values published by Wolf and Wuest to
determine if the test resulta confirm the data derived by them.
If the resulta differ from published values, an attempt is to
be made to evaluate in what way they differ and what reliab111ty
can be expected in the published values.
The test resulta are to be examined to determine pattern
thet can be used for the design of Bourdon t'ubes or for pointing
the direction in which f .uture investigation should be carried out.
- 39 All calculations were made with a slide rule because
the accuracy of tests justified only three significant figures.
- 40 -
Fig. 12. Deflection Test Apparatus.
- 41 -
Fig. 13. Def1ection Measuring Fixture.
- 42 -
Fig.
14.
Detail of Deflection Measuring Head.
- 43 TEST RESULTS.
On the following Table III, (pages
44
to
49
inclusive),
are tabulated the results of the tests carried out.
The tubes are identified with a four digit number.
The
first two digits signify the serial of the tubing before forming
and the last two digits denote the identity of the individuel
tube.
Therefore, tubes bearing the same first two digits were
made from the same piece of tubing.
There are discontinuites in the series of the tube numbers
because tube numbers were assigned in the early stages of manufacturing, and some he.d to be dropped when the respective tube was
damaged in subsequent processing.
The values of
""-*
>r
~
were calculated from ~
using the method derived in Appendix B.
In the tabulation the following abbreviations are used
for material designations.
N.s.c. - Ni-Span
c.
Be. Cu.- Beryllium Copper.
T.B.
- Trumpet Brass.
The following are the values of Young's modulus used
in all calculations.
Ni-Span C E = 27 x 106 •
Beryllium Copper
Trumpet Brass
E
= 19
x 106 •
E ~ 15 x 106 •
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t
TUBE NO.
MATERIAL
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a
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9 ,··6"
T.~:
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31
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- 50 -
CONCLUSIONS
In Fig. (15), (16), and (17) are plotted the values
from test of the sensitivity ratio ~ with respect to
Â
for
axis ratios of approximately 0.1, 0.2 and 0.3 respectively. On
the same figures are shown the corresponding values as published
by Wuest and Wolf.
Wolf's values are in all cases lower than the test data
especially in the region of small ?\ , this was expected from
the analysis of his deriviation.
in the region of
~
It should also be noted that
greater than 1, his values are atill lower
than those from test.
Examination of the data shows a fair co-relation between
the test data and the values published by Wuest.
The co-relation
ia not sufficient to firmly eatablish that Wuest's values are
correct, but much closer agreement has been found in these tests
than had previously been found by the Special
on Mechanical Pressure Elements.
Reaea~ch
Committee
The maximum divergence of test
data from Wuest's values is 50%,most values diverge less than
this.
Wuest's resulta for finite axis ratio are given in his
paper without deriviation, therefore, it is not possible to determine if his values were arrived at empirically, in the manner
of this study, or theoretically.
In addition, the values had to
be procured from an extremely small figure which makes it difficult to ascertain their exact magnitude.
Until a more satisfactory theory is derived, it is rec-
. ~·
- 51 -
-1~~~~~--~~==~~~4=l=~====~2
--~------4
1
i
•
::l
•
m n
1)..-4
1
1
'
1
'
uu_LlJ_J--ro~l\\~+-----ttCL±=tj==~·~====~1
1
\\
\\"'
\'..
. :0·
. ...
~·
· ··- ~···.
-520
i
1
1
'
i
1
'
•
i
V)
i
+J
r-1
!j
_%~
p:: '..-i
1
Il
1
'
1
1
~
1
1
!
1
1
1
1
1
1
1
1
1
1
1
l!
1
i
'
i
~a:
~ 1
tl::
CISG'l
-~ ~
@-\
~
-
..-lC\1
1> •
\\
\
~0
-1-l
~~
ë~
.~
""
1\~
~\.
~
~~
1
1
i
04
"'
1
1
1
!
lkc
1
~'-
!
!
~
i
.'\::.
1
1
~
40 ~ f\
1
;
'
~
r~
1'-
f'
..r"',
~
P<tf' .
't'
~
........
'
i'._ '
i
......
'
1
"""
i
'
1
1
1
1~
1
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~
i
1
1
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-- ......
""-....r-..
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-1-l
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j
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1
1
i
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...::..· ....1·....
' .
··~
!:.. ·.
0
SJ
'
r--;
,.,
2
1
'
l7.l
+-'
~
l7.l
20
1
0
r
1
> •
+l
.......
~
cc.-
-~
.. •
'
'
L
\
~.
~\
~
•
~
;:_
\
~-
~ '\
1
1
~',
~
1
1
t
1
~
mo
\
1
......
..-10
\
1
~
...... C"'\
\
1 \
\
~
~-
~
-0
Q)
......
-.,..
1»<
~
i~
-_:;o.
+l
asm
\
1
•
1
1
_,
...
'
''
'' 'l.
'
"'"""'
l'.
~
...
.....
,
" "-..
......
/
............
.........
~
.........
."
........
"' "'~
"""
1~
.l/1
li
f(
1
.Jf::
t ~
l~" ·fr
!'<:
1
1-
~
j
IL
~
1
<: ~
4--
-
-
:r .
_ t,.. ,
- 54 ommended that the values published by Wuest and reproduced here!n
as Fig. (3) be used for des!gning new Bourdon tubes, but with the
understanding that the design thus arrived at will be only a close
estimate of the final requirement and that experimentation will be
required to arrive at the exact design to produce the desired deflection.
The scatter of the test resulta seems to be greater than
can be readily attributed to experimental error and consideration
should be given to a suggestion by Mr. K. Gotein, in the discussions of Ref. ( 3) , the.t the parameters used
/)
and
~ may be
inadequate to completely describe the action of the tube.
If
this were so, the scatter may be due to this error in parameter.
One of the most easily controlled parameters in the
manufacture of Bourdon tubes is the axis ratio.
A knowledge
of how deflection varies with axis ratio is essential, it would
be of special value in interpolating data from a tube of known
characteristics to a ture with required characteristics.
To evaluate the relationship between unit deflection
~: and axis ratio the graphs shown in Fig. (18), (19) and
(20) were plotted.
The resultant curves were not straight lines
nor did they have the same curvature.
The data was replotted on
log log paper as Fig. (21), here the data falls very close to
straight lines.
Because the curves fall on lines having different
slopes and origine, it is apparent that the relationship of unit
deflection to axis ratio is different for each aize tubing.
The
l
t
-
1
-T
1
1
~
-r
t
r-l
L
1
l \:(
1
...
-dl.,
1
.
.-
'
l
t
i
t~-~·
1
lj
-!
1-
1
·- 1
1
1
M
~1
.
j
j
.
t
'
1
1
-r
1
'
1
- i
.
t- . -j
w
'
~-
v.-('j
--.!'>,
' '
1
1
- -1-1
1
1
~
J
1
l
1
1
1
r
-
..
.........
J
1
l
1
---~-
r·1
1
_;_j_ .
J
·- t
'
1
1
1
~~
l
.
1
''
·-
-+
1
1
1
1
1
1
- 56 r
1
1
-
•!
1
r
t
.
...
~
1
- i
'
1
-
+-~- 1-- - --1
l
~~~
·.::.
l
-
.--0'
i
.
1
1
.....
l
1
1
i
.
·t
1
1
t
t
:
: . -i_ _!_
1
i
__ j _...
t
1
- j
~
i
1
1
-~-r ---.-,
'
1
--t'
1
1
1
1
! ; '
- --1·--.-i - -
r:. ;_
1
l
t
1
·--~--
1
~ "l' --..L1
..j.
--j "
r
•
: . ---: - '
1
1
-. '- 1
-·
1
~: e ·'
- i
~
_1
-
1
t
,./'
1
l
L
f
!
-i - i
--t
l
1
_t
~ r·
1
1
-t
1
1
__L __ _
1
1
1
-
1-
- .J -
'
+
1
-
.J.
1
J
J
• __ j__~--
J
- 57 -
1-
1
1
J
1
l
1
l-
I
;-
: -
! 1
-!
t
:
•
j
r
i
1
1
l
1
->---r-m
;
1
1
:
1
1
1
!
.s
l"
1'
1
'
;
!
:
1
1
1
1
1
1
i
1
fOL-------~--~~~--~~~~~------~----~--~~_.~~~
··0/
·05
.(
1
'
- 59 curves do indicate that the relationship for any one tube can
be expressed as a power function such as
(:z 6)
Aj
and
~,
are constants dependent on tube geometry.
Thus for any one basic design of Bourdon tube the unit
sensitivity for all exis ratios can be determined from as few
as two tests to determine the constants
;t
and n,
•
Using the above relationship and applying it to the
approximate Bourdon tube design arrived at from Wuest•s data,
it becomes possible to determine the geometry of a Bourdon tube
giving the desired deflection within 10% by manufacturing and
testing three or four tubes.
In calculating the values of the sensitivity ratio it
was noted that the geomet~ic relationship ~
~v
the deflection characteristics.
3
often outweighs
To determine if sorne more direct
relationship did not exist between unit deflection and
the
~
two were plotted against one another on a log log scale.
No
overall continuity was found but it was noted that one set of
data seemed to fall on a straight line.
Examination of this data
showed that all points having the seme initial major axis and the
same axis ratio, regardless of materiel or wall thickness fell on
approximately the same line.
This datais shown in Figure (22).
- 60 ~
Fig. 22. Unit Deflection vs
for Tubes with Common Major Axis.
IOW C
/ (/C C
+---------- ~"·---: -+:'-'\-~·.~B\ - +-+-----_--·_-_·~~ ~-----+:--_-___,t-1---~+--=~=:=~~~
1------+
' --+'-~-+--+-+!\Ir++-+------ f---·- -r;
1
1
! i !'\\'
1
1---- --+-----'
:
1
1
l
1
l
--~~--
r ! li ~----T- .---t
1
l
j
Î
l
:
1
1
'
1
1
!
i
!!
/ (1() ~----- r----~--!--+--+--+--t-+-1---
-. J ·---..,--+--:---
+--+-+-+
. ..;..r+:t,,
· .·.
1-----1--'--4---f--t-~-+--H----+'+---t--+--t--t-+-1r-t'-i
'~ .,-.: ::- :::,. '
.1.
~--+--t--+--+--+-+-+-l-+-----'~\+----lf--+-+-+-t--i-1H
-it
';'
'. ·.:":. \.i •.,.
·.. : .:· ~
,,.
\
'·' '
. ... . ~·...
~
\
..__+--4--+-+--- -- ·-- -- t---- - -+--- 1---+--+-+-+-..........
;
1
1
/ ()
__L_~ l
1
.--1..-...J-...J---L-.J...._.__
_ _ _ ___ ' - - - - - ._ _ _
/.
1',-
...J-_-.J.~---'--L-L-1
/IJ
• .
~- ·
· · ~),;l~
- 61 -
Since the major axis determines the value of a
of
~
and the value
vas approximately the same for all tubes, it was thought
that this may indicate the function of uriit sensitivity with
respect to wall thickness.
uted for
~
When the wall thickness was substit-
as a parame ter, the average straight line was main-
tained but the scatter was increased.
~
ship.
Therefore, it was felt that
was a more appropriate parameter for expressing the relationThe straight line indicates that the relationship can be
expressed as a power function.
( :z.7)
/
=-(
/fL and
~
:;;\.)
n
!l
n~ are constants dependent on tube geometry.
Data for two values of axis ratio . were plotted.
Although
the data lay in close proximity both the origin and slope of the
lines seemed to be different indic a ting tha t both IÇ_ and 17.:{
~ •.
were dependent on
Figure (22) representa data for Bourdon tubes made from
tubing of initial outside major axis of .270 inches and radius
of approximately
.84
inches.
The data plotted is for series 1000,
1100, 1200, 1500, 1600, 1700, 2000, 2100, 2800, 3000, 3100 and
3300 tubes with axis ratios .35 to .15.
For this study data for only one set of outside major
ares was availab1e.
Further study shou1d be made to verify the
va11dity of the above re1ati onship.
- 62 -
The relationship becomes of real value in practical
Bourdon tube production because the rolling dies for the tubing
are expansive and control the outside dimensions of the tube.
With the above relationship the tube designer can test a few
experimental tubes of different wall thickness but same outside
dimension and quickly establish the unit sensitivity of all
tubes with that particular outside dimension and radius regardless
of wall thicknees or materiel.
Because of the limited tests carried out, the relationship of unit deflection to radius cannot be determined, further
research should be carried out to determine the relationship.
In the testing, it was assumed that
Saint~Venact's
principle would apply and that the effect àue to end fittings
would be negligible.
On this basis, the free end fittings were
chosen as round bars of any diameter that was convenient for
soldering.
Examination of deflection resulta showed certain
discrepancies which could only be explained by the fact that
the free end fitting was markedly different.
In future testing
it is recommended that a uniform end fitting of a type that can
accommodate a wide range of tube sections be used.
Tubes produced for this study were made with an included
angle ~ for which tooling existed at Aviation Electric Limited.
On the basis of the analysis presented in Appendix B Fig. (24) it
- 63 -
la recommended that in future studies an included angle approaching
360° be used.
Since the most difficult parameter to evaluate accur-
ately is the included angle, this procedure would reduce the affect
of errer in angle on the value of
~~ •
The present method of deflection testing dld not reveal
any marked hysteresis affect.
It can therefore be assumed that
any hysteresis was outside the accuracy of the test method.
SUMMARY:
The test resulta indicate that the data published by
Wuest for finite axis ratios will give a reasonable first estimate
of Bourdon tube deflection for design purposes.
It was determined that for any one basic Bourdon tube
design, _the unit sensitivity can be expressed as a power function
of the axis ratio.
Test resulta lndicate that for tubes of the same basic
configuration the unit sensitivity can be expressed as a power
function of /\ when wall thickness is varied.
The author reels that the data contained herein contributes to the advancement of the knowledge of characteristics of
Bourdon tube deflection, in assessing existing theories, presenting methods of Bourdon tube design and pointing the directions
which future research should take.
In the pursuance of these studies, the author reels that
he has become thoroughly conversant with the methods of
manufacturing and testing Bourdon tubes.
designin~,
In addition he has
- 64 gained valuable experience in scientific evaluation and
experimental method.
- 65 -
APPENDIX A
SAMPLE CALCULATIONS OF BOURDON TUBE DATA
- 66 ~i
\
/
'·
/".,.,,. t.
C~r;;/L. ê
#A.;ot'{P
Ovr ::./PE /~1\o'-<''k
r t
·l' ,.., T .
c- ,._., :Ç
h'x;·:s
. /0/ ,
A/,,;s
- r: ~· :;
'1
-- ·- --
)
.cl_..::::::?
• :ZY':Z - •C:<S"
z
. ''; ..<·~~i .
:
·)
.
./CJI.'
' •
//;1"'/~r...x/./'/;#;TE
n/,-s,cr
AK/.5
./?
SA'.,,.,;$
Ax,;s
=
~
•JJr,/t'f.'
,0
A"Et;u/,11'~&- ?;, ;P'~oPt/ f"t-.
~~/,;O
.:;:
~. .Ir-
h
,
:; ;.' ;,
Tv6·_..
U/ ,;r,y
,Z;'E5h~é..P
- 67 -
r-'?"~""" r/cr,.;v.T
#(c,·.r
.,~T/1~ (~: ~.N'tÇ~
5r/7E:; ··
~>;E
To
#r7E~ ~~''-/Nt;
~t!"pV('"ê
_:::-chL //Yfi.
/~Cr~
/PT
/T"'tc.::#'EI"'E
//;,..Rvé"N h'.r
//_çpo~
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#ou~s
ç.r
~?1 =
. tJôf
,.,
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c
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[(o13~)-:z.'f"(.ooo9t" Jq
2
a =
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:=
-;l'i'6 - . t.J:Z ç
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::::
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36 0
't ·· ·
~.).: ..... .{•;;. .' ~ .. ·=· -~. ; ·~;. · ·.·.:·:•· . :
- :<x :l.
°-
:l
?!!f.
-? Rc·
S//V - /
k'";.;" r
- 69 -
ç2i "'"
36 CJ O - ;:<
5/N-/
/,
;l_
3o7
X,7S"3
•
: :. r. 725
-
= /'fA
ar.
Â
t
::;
-,
#,
,/,6Yj>
·//0
,2?/
-=::
--
/Y/
E:
, //tl :l
·032.
a
_.c-p_;e
= ·797 ,K rt/2.-s"
5'.P#N ("
#E,y.r
/;f?E,a; TE/?
:<. 7x /t:Jl
:::: /, 7 t5 ~ d -S' x
~725'
:z 7
1( /
~~
)( ·77'7
............ '-.i.,..·
- 70 -
~ = ~~
,.;f
=- / :< 6
::'
/'
x ·:< f/ x
(·(),2:) 3
•//0
, Y,tf
/#tf'f,.,T.N
/.-y
f (f)5
7.,-,B~
E
T;.;v~
r'.,LCt/LA<7EP
/s
//?~VLA'f"F.l)
...zz?
i·
-~,-.
- 71 -
APPENDIX B
GEOMETRY OF BOURDON TUBE DEFLECTION
- 72 1
·7,(','(
\
/-+ t:'~ z:,.,.r ,R r, s.r;~rr r·
rA't."
~;trT/o
PE,L.Err"-"
/'. ré'.P
-
·--
:
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~0
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.v:.:
T#E
.P~,-.& ~r ,.,,,.,.
.<'·
/IV
/E.'i:'~:r
t:7'
c';éP,..n!'"T~y C' .J/t!,..t~TE4?
---~0
-~tJ
-/
(lf
---
= '-f,-~
e
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.!:..5
(7.,
::
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fl.!f:.
.;#/r.N.~.:-It'.trP
.T ;;.t-·
=
sP.
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•
- 73 -
·::.
.t'".:l'
.-r,
-
A':,~
-
({J
/-;, #(>
ro'
=
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<"/l'V~
- <-~~)
tf
-
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tf'
0
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;!:?
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/
;1~:.~
dK
t"cs , -
A';., ~c
-~.-~
·•
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BIBLIOGRAPHY
1.
"BIBLIOGRAPHY ON BOURDON TUBES AND BOURDON TUBE GAUGES".
by L.M. Van der Pyle.
ASME Paper 53-lRD-1
2.
"THEORIES ON BOURDON TUBES".
by F.B. Jennings.
ASME Paper 54-A-168.
3.
"SENSITIVITY AND LIFE DATA ON BOURDON TUBES''·
by H. C. Mas on.
ASME Paper 54-A-169.
4•
"THE INFLUENCE OF THE SHAPE OF THE CROSS-SECTION ON
THE BEHAVIOR OF BOURDON TUBES".
by Walter Wuest.
Translation ASME Paper 54-A-165.
5.
"MATERIAL SELECTION FACTOR SIGNIFICANT IN BOURDON TUBES".
by J.B. Giacobbe and A.M. Bonds.
Journal of Metals. Nov. 1952. pp. 1147-1148.
6.
"DEFORMATIONS
&~D
STRESSES IN BOURDON TUBES".
by R.A. Clark and E. Reissner.
Journal of Applied Physics.
Vol. 21.
pp. 1340 - 1341.
7.
"AN ELEMENTARY THEORY OF THE BOURDON GAUGE".
by Alfred Wolf.
Journal of Applied Mechanics.
pp. A-207 - A - 201.
September 1946.
- 77 -
8.
"A DIMENSIONAL ANALYSIS APPROACH TO BOURDON TUBE DESIGN".
by K. Goitein.
Instrument Practice September 1952.
pp. 748 - 755.
9.
11
SELECTING AND WORKING BOURDON TUBE MATERIAL".
by J.B. Giacobbe and A.M. Bonds.
Instrument Manufacturing. July-August 1952.
10.
"STRENGTH OF MATERIAL PART II".
by
s.
Timoshenko.
Second Edition.
11.
pp.l07-112.
"DESIGN OF PIPING SYSTEMS".
by M.W. Kellogg Company.
Second Edition.
12.
Chapter 3.
"STRESSES AND REACTIONS IN EXPANSION PIPE BENDS".
by A. M. Wahl.
ASME
13.
Transactions. Vol. 49-1927.
ttBENDING STRESSES IN CURVED TUBES".
by S. Tlmoshenko.
ASME
14.
Transactions. Vol. 45-1923.
"ADVANCES IN APPLIED MECHANICS".
by R.A. Clark end E. Reissner.
Vol. II.
15.
PP• 93-122.
''THE MATHEMATICAL THEORY OF ELASTICITY".
by A.E.H. Love
Fourth Edition.
page 500.
- 78 16.
''STRESSES AND DEFORMATIONS OF TOROIDAL SHELLS OF
ELLIPTICAL CROSS-SECTION".
by R.A. Clark, Gilroy and
E.
Reissner.
Journal of Applied Mechanics.
March 1952.
Page 37.
17.
"CONSTANT MODULUS ALLOY FOR ELASTIC ELEMENTS".
by Marshall Ward.
Project Engineering.
18.
11
July 1956.
HEAT TREATING CURVES FOR BERYLLIUM COPPER STRIP".
Penn Precision Products Inc., Reading, Penn.
Bulletin 8, October 15, 1956.
19.
"STRESS CHARTS FOR PRESSURIZED ELLIPTICAL AND
OBLONG TUBES tt.
Product Engineering Design, Work Sheets,
llth Series, Page
20.
4,
McGraw-Hill Publication.
"ANALYZING STRESSES IN SHELLS AND RINGS".
by D.H. Conway.
Machine Design, December 1954.

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