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I. S. Fischer
Assistant Professor.
Mem.ASME
New Jersey Institute of Technology,
Newark , NJ 07102
R. N. Paul
Mem.ASME
W . R. Graces Co.,
Co lumbia , MD 21044
Kinematic Displacement Analysis
of a Double-Cardan-Joint Driveline
The input-output displacement relations of two Cardan joints arranged in series on a
driveline has be en investigated in detail, including the effects of unequa l joint
angles,
the
phase angle between the two Cardan joints and also such manufacturing
tolerance errors as nonright angle moving link lengths and offset joint axe s, A com
bined Newton-Raphson method and D avidon-Fletcher-Powell optimization
algorithm using dual-number coordinate-transformation matrices
was
e mployed to
perform the analysis. An experiment w as conducted to validate the results of the
analysis. The apparatus consisted of a double-Cardan-joint driveline whose rota
tions we re measured by optical shaft encoders that were sampled by a computer
data-acquisition system. The equipment was arranged so that the phase angle be
tween
th e
joints and the offset angles betwe en the shafts at each of
the
two joints
could be readily varied. Th e relative phase angle, the difference between the
phase angle of the two joints and the angle between the planes defined by th e input
and intermediate and the intermediate and output shafts, was found to be the
significant factor. If th e offset angles at both Cardan joints are equal the double-
Cardan-joint drive line functions as a constant-velocity coupling when the magnitude
of
the
relative phase angle is zero. If the offset angles at the two Cardan joints
are
unequal, a condition prevailing in the important front-wheel-drive automobile
steering-column
application,
then fluctuation in output ve locity for a constant input
velocity is minimized although not eliminated for zero relative phase angle.
Introduction
The do uble-Cardan-joint driveline has been extensively used
as a driveline in machinery, particularly the Hotchkiss
automotive drive. The geometry of an application is specified
by the offset angles (the angle between the input and in
termediate shafts at the driving Cardan joint and the angle
between the intermediate and ou tput shaft and the driven Car
dan joint) and by the shaft offset (the minimum distance be
tween the input and output shaft axes). In most applications
the mechanism is arranged so that the offset angles at the two
Cardan joints are equal and all the shafts are in the same
plane. Under those conditions the double-Cardan-joint
driveline will operate as a constant-velocity coupling. Analyses
of this arrangement are given in Doughtie and James (1954),
in Tyson (1954), and also by Austin, D enavit, and Hartenbe rg
(1965) who added consideration of manufacturing tolerance
effects. If the shafts are in different planes and the offset
angles are equal, then the Carda n joints can be phased relative
to one another to produce a constant velocity coupling. This
case was discussed by M abie and O cvirk (1975) and by Wagner
(1979).
Recently, the double-Cardan-joint driveline has been used
as a steering shaft in front-wheel-drive automobiles. As
reported in Cowles and Chiu (1984), this application does not
allow equal offset angles. The yokes of the two Card an joints
on the intermediate shaft are located in different planes, a pro-
Contributed by the M echanisms Comm ittee for publication in the
JOURNAL OF
MECHANICAL DESIGN.
Manuscript received September 1989.
cedure called joint phasing, in order to best approach
constant-velocity output. Saari (1954) approached this prob
lem by combining the input-output displacements for two Car
dan joints, Phillip (1960) used a spherical-trigonometry
analysis, Maziotti (1964) developed a vector procedure and
Cooney (1966) has given an expression but not its derivation.
This study has modeled the double-Cardan-joint driveline
with dual-number coordinate-transformation matrices and
has applied numerical techniques to create a procedure for
evaluating the effects of joint phasing and manufacturing
tolerance errors on the input-output displacement
relationship.
In addition, an experiment was conducted to validate the
results of the numerical analysis.
Analysis
The ideal double-Ca rdan-joint driveline is a six link spatial
mechanism w ith revolute joints. Although the G riibler equa
tion predicts zero degrees of freedom, the mechanism operates
because of its special prop ortions . If those special proportions
are lost because of manufacturing tolerance errors such as
nonright angle moving link lengths and offset joint ax es, then
it is necessary to consider one joint, in this study selected to be
the output joint, as cylindrical to accommodate the sliding
that o ccurs. This sliding m otion is similar to that exhibited by
the single Cardan joint as described in Fischer and Freuden-
Journal of Mechanical Design SEPTEMBER 1991, V o l . 113 / 263
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Fig. 1 Double-cardan joint driveline
stein (1984). Therefore, this study will model the double-
Cardan- jo in t d r ivel ine as an RRRRRC mechanism.
A dou ble-C ardan -joint driveline is i l lustrated in Fig. 1. It
has six joints nu mbe red 1 throu gh 6. At joint i—
1
there is a
coordinate frame fixed on the l ink connecting joints i — 1 and
/. Its position is specified by a z-axis aligned along the axis of
joint /—1 and an x-axis aligned along the l ine of shortest
distance between the axes of joints / — 1 and / . The t ransforma
tion between the coordinate frame on the l ink connecting
joints i — 1 and /' and the coo rdinate frame on the l ink connec
ting joints / and /+ 1 can be expressed as a dual-number matrix
where the dual-num ber identity matrix [7 | is defined as
M =
/
0
0
0
1
0
0~
0
; _
+ €
0 0 0
0 0 0
0 0 0
(4)
IA,] =
C(0 , ) - 5 ( 0 , ) 0
c(a
i
_
l
)s(d
i
) c ( a , _ , ) c ( 0 , ) - . y ( a , _ i )
sia^Jsid,) • s ( a ,_ i ) c (0
/
) c(&,-_,) (1)
E quatio n (1) can be expand ed into prima ry and dual com
ponents
[A,]-
c{d,) -3(6,) 0
c ( a
/
_ , ) s ( 0
;
) c ( a , _ , ) c ( 0 , ) -s(a,_
l
)
s(a
i
_
l
)s(e
i
) s(a
i
_
l
)c(B
i
) c ( a , _ , )
+ e
-d
iS
(d,)
df i
(a,-_, )c (
6,
) - a,-_ s(a,-_,
)s (0,
)
d,s (a,-_ i )c (6;) a,_, c (a ,_ , )s (6,)
-d,c(e
t
) 0
-djC^a^sidi) -a , -_ i .s(a i_ i )c(0 i ) -« ,•_ ,cC a,- . , )
-d
i
s(a
l
_
l
)s(6
i
) +a
i
_
l
c(ot
i
_
l
)c(6
i
) - ^ . ^ ( a , . , )
(2)
The six l inks comprising the double-Cardan-joint driveline
are connected serially to form a closed loop. This infers that
the six transforms associated with the mechanism are related
by the condition of loop closure:
[A
{
\[A
2
] . . . [A
6
] = [f\ (3)
E qua tion (3) states tha t transforming from the frame fixed on
one link in the mechanism throu gh the frames fixed on each of
the other l inks will result in a return to th e original co ordina te
frame. This equation governs the kinematics of the double-
Cardan-joint driveline.
The solution to the general form of equation (3) will be
found by nume rical me thod s. Duffy (1980) l isted the solutions
to var ious s ix- l ink mechanisms such as RCRRRR, RCRRRP,
RRRCRR, and RRRCRP, bu t the d isp lacement analysis o f
the RRRRRC by Ju and Duffy (1985) had not yet been
published when it was decided to model the double-Cardan-
joint driveline as an RRRRRC and obtain the displacements
by a numerical scheme.
D u a l - N u m b e r F o r m u l a t i o n o f N u m e r i c a l M e t h o d s
The numerical m ethods used to determine the d isp lacements
of the double-Cardan-joint driveline are based on the
algorithm introduced by Hall , Root, and Sandgren (1977) that
combines the Davidon-Fletcher-Powel l (DF P) op t imizat ion
rout ine with the N ewton-Ra phson (N R) method as applied to
kinematics by Uicker, Denavit , and Hartenberg (1964). These
procedures were developed for use wi th 4 x 4 homogeneous
transformation matrices, but for this study were adapted to
3 x 3 d u a l -n u m b er m at ri ces .
The algorithm for the numerical solution is shown in flow
chart form in Fig. 2. The two methods of solution are
available so that the most advantageous could be used de
pending upon the requirements at each position of the input
variable. The D F P me thod is slower but will almost always
find the displacements resulting in loop closure even if the in
it ial guess of the displacements is poor, a condition that is
troublesom e for the N R me thod . If the init ial guess of the
d isp lacements has been made reasonably w el l, the N R method
is much faster. Th us the DF P m ethod is used for the initial
position of the input-link rotation when it is difficult to guess
the displacements at the output and particularly the in-
264 / Vo l . 113 , SEPTEMBER 1991
Transactions of the ASME
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( S T A R T )
V£5
(ST O P)
Fig. 2 Simplified flowchart of algorithm
termediate jo in ts . The N R m ethod is used to determine the
displacements associated with subsequent steps in the input-
link rotation because if those steps are small , then the
displacements determined for the previous input position are
generally adequate as an init ial guess. The displacements
resulting in closure are usually found within a few iterations.
Occasionally a solution cannot be found using the N R m ethod
and the algorithm will then use the DF P meth od to attempt to
locate displacement values resulting in loop closure.
Details of the Newton-Raphson Method
A first-order e xpansion of eq uation (3) is
{A(\[A
2
+ Q
r
A
2
8d
2
][A
3
+ Q
r
A
3
86
3
]
[A
4
+ Q
r
A
4
56
A
][A
5
+ Q
r
A
5
86
5
][A
6
+ Q
r
A
6
56
6
+
Q
p
A
6
8D
6
] = I (5)
where A „ are each evaluated at 0„ and D„ which are guesses of
the displacement variables, dd
n
an d dD „ are the errors be
tween the guesses and exact values of the displacements, and
Q
r
and Q
p
are the partial-derivative operators corresponding
to equation (1) such tha t
[Q,]
=
and
[Qph
0 -c(a)+eas(a) -s(a) -eac(a)
c(a)-eas(a) 0 0
s(ot)+eac(ct) 0 0
(6)
0
•ec(a) — es(a)
e c ( a ) 0 0
es(a) 0 0
(7)
E qua tion (5) can be expan ded and the higher-power term s
neglected, resulting in a form which is com putationally advan
tageous:
H
2
8d
2
H
3
5d
3
+H
4
5d
4
+H
5
88
5
H
6
88
6
H
1
8D
6
=B
T
-I (8)
where
H,= {A,A
2
. . . A
i
„
l
)Q
x
(A
l
A
2
. . . A,^)
7
and
(9)
(10)
x
A
2
A
3
A
4
A
5
A
6
=B
E quatio n (10) can be written in a mor e compac t form as
P= B
T
~J (11)
Writing out each term of equation (11) produces two 3 x 3
dual-number m atr ices:
1
1,1
1.2
3,2
1,3
3,3 J
B\,\~I
2,1
B
3,1
B
1,2
B
K
B
2
,
2
4 , 3
/ B
3,2
B
3,3 '
(12)
This matrix equation comprises nine dual equations, only
three of which are independent. Using the three terms in the
upper-right triangle of each matrix and separating them into a
set of real equations and a set of dual equations produces:
F = Ft
- ' r ea l i T - °rea l -> ,
real
1,3
= B
eal
3,1
and
W
=
R
• ' r e a l j 3 r e al
dual
1,2
= B,
3,2
2 , 1
P
=
R
r
dualj 3
£
' d u a l 3 j
^ d u a l 2 3 - ° d u a l 3 2
(13)
(14)
The real set , equation (13), however, does not include the
constraint that the diagonal terms must each be equal to unity
at closure. This constrain t can be included either as three addi
tional equations as described by Uicker, Denavit , and
Hartenberg (1964) or through modification of the current
equations set as discussed in S andor a nd E rdm an (1984).
Adopting the second method modifies equation (13) to
F = R
^ r e a l j 2 r e a l'
reaij 3
^
r e a l
2 , 3
= B„
2 ,1
"3,1
+
B
n
+
B
r
,
+ B
Ti
+ B„
2 ,2
'
real
3
^ r e a l
2 ,2
' real-i
(15)
The left-hand side of equations (14) and (15) contain the
linear combination of the respective elements of the H
matrices and the error terms as can be seen from the left-hand
side of equation (8).
E quation s (14) and (15) can be combined as parti t ions of a
single matrix of real-number equations. The resultant equa
tion will have the form:
" -Mreal "
. - M d u a l .
A
=
' r ea l
. ^dual .
(16)
where
M =
'H,
H,
H,
H,
H.
' r . 1 , 2
H,
H
A
H-
r,2,3
H,
r ,2 ,3
H*
H.
H<
H,
H,
Hi,
, l , 2 ' r . 1 ,2
H-,
V, 2,3 ^ r . 2 , 3
H, H, H
A
H<
H
K
Hn
d,l,2
3
d,l,2
4
r f , l , 2
5
r f , l , 2
d,l,2 >d,l,2
H
ld,l,3
H
3d,l,l
H
*d,l,3
H
5d,l,3
H
6 r f, l , 3 ^ ' d . U
H, H,
rf,2,3
3
d , 2 , 3
H, H,
d , 2 , 3
H„
H
r
d,2,3
(17)
J o u r n a l o f M ec h a n i c a l Des ig n SEPT EM BER 1 9 9 1 , Vo l . 1 1 3 / 2 65
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•Orpal
2
j + -"reaij j
eal-) i ^ ^r ea li i ' -" real
° r e a l
3 1
+ ^ r e a ] ̂ + -^real
2,2
3,3
3,2
+
B„
B
e a U
B
A
L ^ d u a l
3 2
(18)
and
b e
50
4
86
5
86
6
5D
6
(19)
Rearrangement of equation (16) to solve for the error terms
produces the required equation for the N ewfon-R aphson itera
tion technique:
A = M~> V (20)
The guesses of the displacement variables d„ a nd D
n
are re
vised by the error terms, A, and this new set of joint variable
positions is used in the next i teration to compute a new set of
error terms. Convergence is obtained when each of the error
terms obtained from equation (20) is less than some specified
resolution limit.
After obtaining a solution for the given position of the in
dependent variable Q
{
, a new position is selected and the en tire
process repeated. The solution set from the previous position
is used as the starting values for the new iteration. The step
size between successive positions must be small enough to
allow the N ewto n-Ra phson te chnique to find closure. This
step size depends upon the specified mechanism and varies
with its position. It is usually safe to use 20 degrees as a max
imum step size.
Equations for the Davidon-Fletcher-Powell Optimization
Method.
If perfect guesses of the displacement variables are
substituted into equation (10), then matrix B will be the identi
ty matrix as required by equation (3), the condition of loop
closure. For imperfect guesses, matrix B will differ from the
identity matrix, implying lack of closure. The difference be
tween matrix B and the identity matrix can be embodied in an
objective function such as
^ = E E l t ^ e a ^ - M ^ ' + ^ d u a , ] ^
2
)
(21)
/= i y= i
which vanishes at closure but is greater than ze ro when closure
does not occur. An optimization method can solve equation
(3) by minimizing the objective function given by equation
(21).
The DF P algorithm req uires a set of gradients for each
variable in addition to the objective function. The gradients
provide the slope of the objective function w ith respect to each
variable and are found by differentiation of equation (21) to
be
j = l k = \
ax
i
3 3
+
y'=l k=\
d[ B
d
duaUjfc
dXi
(22)
E quation s (21) and (22) are used by the DF P meth od to find
a solution. The DF P algorithm is described in detail by Fox
(16) and basically consists of selecting the direction of steepest
descent of the objective function from the variable vector,
proceeding in that direction by an amount determined by the
closeness of the objective function to the target value (in this
case, zero) and then repeating thre e steps until the convergence
criteria is satisfied.
Ex per im enta l Inv es t ig a t io n
A series of experiments were conducted to validate the
results of the numerical analysis. An experiment apparatus in-
ANGLE BRACKET
OUTPLH SHAFT H00ULE
BASEPLATE
P HASE COW L I NG
1HPW SHAFT MODULE
D O l M E - C A f t O A N - J O J N T S H A F T
Fig. 3 Experimental apparatus
266 / Vo l . 113, SEP TE M BER 1991 Transactions of the ASME
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8/17/2019 263_1.pdf
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350-
325
300
275
250
225
200-J
175
150
125 -j
100
75
50 H
25
OH
-25
- 5 0 H
l h e l o - 2
l h c t o - 3
t h e l o - 4
l h e t o - 5
l h e t o
:
6
d-6, inches
T — i — i — i — \ — i — i — i — i — i — i i i i i i i i r
0 20 40 60 80 100 120 140 160 180 200 220 240 260 2B0 300 320 340 360
T HO A -I POSITION, degrees
Fig.
4 Computed joint motions offset angles both 35 deg., offset
0.867
in., phase angle SO deg.
- 9
-10
T
20
40
1 — i — i — i — i — i — i — i — i — i — i — i — i — i — r
80 100 120 140 160 180 200 220 240 260 280 300 320 340 360
THETA-1 POSITION, degrees
Fig.
5 Computed I/O relationship offset angles both 35 deg., offset
0.867 in., phase angle 50 deg.
corporating a double-Cardan-joint driveline was designed on a
modular basis so that the offset angle at each Cardan joint,
the angle between the input and output shafts, the offset
distance between the input and output shafts and the phase
angle between the two Cardan joints could be varied. It is i l
lustrated in Fig. 3. Althoug h provided for in the analysis, the
experiment apparatus did not permit variation of l ink angular
dimensions and joint-axis offsets within the two Cardan
joints. Both the input and output shafts were equipped with
optical shaft encoders to measure rotation. These were inter
faced to a micro com puter for data acq uisit ion.
Resu l t s a nd Co nclus io ns
The numerical solution is capable of producing data on the
motion of each joint in the double-Cardan-joint driveline. A
plot of the joint motions for a typical double-Cardan-joint
driveline are shown in F ig. 4. i?-type joints num bered 2
throug h 5 display single-cycle variation per revolution of the
, input shaft . M otion of prismatic joint D
6
is zero for this exam
ple. The m otion of the outpu t shaft is displayed as angle 6
6
and varies with two cycles per input-shaft revolution. This
nonuniform motion can best be visualized by the input-output
(IO) relations hip, the lead and lag of the output-shaft rota
tion with respect to the input-shaft rotat ion. F igure 5 i l
lustrates the IO re lationship for the joint motions in Fig. 3.
If tolerance errors result in offset joint axes, then sliding
will occur at the prism atic joint. F igure 6 shows the sliding
which was computed by the numerical method for a double-
Journal of Mechanical Design SEPT EM BER 1 9 9 1 , Vo l . 1 1 3 / 267
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6/9
5 . 1 5
5 . 2
H
5 . 2 5
5 . 3
5 . 3 5
5.4 i
5 . 4 5
5 . 5
5 . 5 5
5 . 6
5 . 6 5
5 . 7
5 . 7 5
0.1 inch errors in all links
zero errors in all links
1 1 I 1 1 I I I I I 1 I 1 1 I I 1 1
0 20 40 60 80 100 120 HO 160 180 200 220 240 260 280 300 320 340 360
IHE IA -1 POSITION, degrees
Fig. 6 Computed D
6
motion for a mechanism with dimensional errors
in the links offset angles both 35 deg ., offset 0.867 in., phase angle 50
i~~[ i i i i i i i i i i i i i r
7 9 II 13 15 17 19 21 2 3 25 27 29 31 33 35 37
JOINT ANCLE5. degrees
Fig. 7 Effect of relative phase angle on I/O displacement relationship
Cardan-joint driveline similar to that used to produce F ig. 5,
but with joint-axis offsets of 0.1 inches. Fo r com parison, the
motion of the same driveline but with zero joint-axis-offset
tolerance errors is also shown in F ig. 6.
Relative Phase Angle. If the input and output shafts do
not intersect, but are offset by a certain distance, then the
plane formed by the output and intermediate shafts and the
plane formed by the input and intermediate shafts are dis
placed from one another by the "angle of twist." The rela
tionship between the two planes each containing one of the
Cardan-joint yokes which are mounted on the intermediate
shaft (angle 4> in Fig. 1) is called the "phase angle." The
"relative phase angle," RPA, is defined as
RPA = (phase angle) - (angle of twist) (23)
The importance of the relative phase angle is observed in
F ig. 7. For a driveline with equal joint angles, the amplitude of
fluctuation in output displacement increases significantly with
relative phase angle to a maximum amplitude occurring at a
relative phase angle of 90 degrees. In the figure it is observed
that for joint angles of 45 degrees, about the largest experi
enced in practical applications, the fluctuation can range from
zero degrees for R PA = 0 degrees to 20 degrees when RP A = 90
degrees.
Offset of Shaft Axes. As seen in F ig. 8, for a given RPA,
268 / Vo l . 113, SEP TE M BER 1991
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a l p h a - 6 = 0 . 0 0 0 i n
a l p h a - 6 = 0 .8 6 7 in
a l ph a -6 = 2 .1 69 i n
1 1 1 1 I 1 1 1 1 I F 1 1 1 1 1 1 I I
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
RELATIVE PHASE ANGLE, degrees
Fig. 8 Effect of offset on I/O relationship
10
9
8
7
6
5 -
4 -
3
2 -
1 -
0 -
A —
A
sma l ler
* — • sm a lle r
# - • - £ s m a l l e r
9-— 9 sma l ler
/
« ,
.-'*''
S s*
,'S' .-•-
/ ̂ ...- .
j o i n t
j o i n t
j o i n t
j o i n t
/
/
'
ong le = 5 degrees #
a n g l e = 15 degrees /
ong le = 25 degrees /
a n g l e = 35 degrees , •
' / .•
-t/
y -••' ^
*' •••-• JS
' . . - • * ^ ^ ^
••-'*' ' ^ J f ^ ^
.•
~ 1 1 I I I I I 1 I I I I I I
6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
DIFFERENCE BETWEEN JOINT AN GLES, degrees
Fig. 9 Effect of unequal joint angles on I/O relationship
variation of the offset of the input and output shafts has
negligible effect on the IO displacement relationship.
Unequal Joint Angles. When both joint angles in the
driveline are equal, then arranging the mechanism so that the
relative phase angle is zero will cause fluctuation in the input-
output relationship to vanish. Thus for RPA = 0, when the
phase angle equals the angle of twist, the double-Card an-joint
driveline will function as a constant-velocity coupling. This
agrees with the results reported by Wagner (1979).
However, it has been observed in this study that unequal
joint angles results in a situation where the joints cannot be
phased to arrange for constant-velocity motion. It is apparent
that the amplitude of input-output fluctuation is minimized
for zero relative phase angle. This is seen in F ig. 9 which pro
vides a measure of the input-output fluctuation amplitude for
various combinations of joint angles at zero RPA . L arger
amplitudes would be obtained for non-zero RPA's. Several
characteristics can be observed in F ig. 8:
(1 ) Increasing the difference between the two joint angles
results in larger amplitudes.
(2) L arger joint angles, with the difference between them
held constant, produce greater amplitudes.
These facts point to the conclusion that the need to
minimize joint angle difference becomes ever more importa nt
with larger joint angles.
Journal of Mechanical Design
SEP TE M BER 1991 , Vo l . 113 / 269
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Experimental Results. E xperiments were conducted for a
variety of arrangements. The results for several phase angles
in a typical arrangement of the input and output shafts are
shown in F ig. 10. The irregularities in the curves are observed
to have a magnitude of 0.36 degrees, equal to the resolution of
the encoders. To evaluate the error, the experiment for which
the joint angles were 10 and 30 degrees, the phase angle 60
degrees and the input-output shafts offset 0.867 in. was
repeated four times to arrive at a standard deviation in the
measure output of 0.0323 degrees. In that the measured
amplitude of the fluctuation was about 4 degrees, the error
had a m agnitude of less than
1
percent of the peak signal. The
small magnitudes of these errors indicated that the quality of
the measurements was good.
A comparison of experimental and computed results for a
typical arrangement of input and output shafts with several
phase angles is shown in F igs. 11 and 12. It is observed that
there is close correlation between the experimental and
analytical results.
6
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e x p e r i m e n t a l , p h a se a n g l e = 30 deg
- c o m pu ted , pho s e a n g l e
= 30 deg
e x p e r i m e n t a l , p ha s e a n g l e
= 60 deg
c o m pu ted , pha s e a n g l e = 60 deg
1
1
i
1 1 1 1 1 1 1 1 1 1 1
• k
N
\ \
\ \
\ \
\ \
\ \
\ ^
V- .
1 1 1 1 1
0 20
40 60 80
100 120 140 160 180 200 220 240 260 280 300 320 340 360
T HE T A -1 POSIT ION, deg ree s
Fig.
12 Comparison of experimental and computed data phase angles
- 20.1 an d 0 deg. Offset an gles both 30 d eg., offset 0.867 in.
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