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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

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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)

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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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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-

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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

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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

°

S

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N

O

I

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LU

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i

6

4

2

0

-2

-4

-6

-10

. / ' ~

v

\ /'

,-v

S

-

/ / \ \ / /

/ e \ \ 7 •

.

/ / \ \ i i

/ '' \ \ / /

/

•''• V •/ /

:

\^/^

i

// V /

—•

v

^

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.

Doughtie , V. L . , and James, W. H. , Elements of Mechanism, John Wiley &

S ons , I nc , N e w Y or k , N Y , 1954 .

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Interscience, S omerset, N J, 1980.

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I. S.,

and F reudenste in , F . , " Inte rna l F orce and Moment Transmis

sion in a Cardan Joint with Manufac tur ing Tole rances ," AS ME JOURNAL OF

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R.

L ., Optimization Meth ods

for

Engineering Design, Addison-Wesley

Inc., Reading, MA , 1971.

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ing Matr ix L oop E qua t ions

for

the G enera l Three-Dimensiona l M echanism ,"

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of

Engineering

for

Industry, Aug ust 1977, pp. 547-550.

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John Wiley & Sons, Inc . , New Y ork, N Y , 1975.

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SAE SP#262, 1964.

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Join ts ," Technical Repor t 60-WA-37, ASM E , November 1960.

Saar i , O. , "How

to

Obta in Useful S peed Var ia tions With Universa l Joi nts , "

Machine Design, O ctober 1954, pp. 175-178.

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and Synthesis, Prentice Hall Inc., E nglewood Cliffs, N J, 1984, p. 621.

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H.

N ., Kinematics, John Wiley & So ns, Inc . , New York, N Y , 1954.

Uicker , J. J., Denavi t , J., and H ar tenberg, R. S., "A n I te ra tive Method for

the Displacement Analysis

of

S pa ti a l M e c ha n i sm s , " AS M E Journal of Applied

Mechanics, June 1964, pp. 309-314.

Wagner , E. R., "Useful Design Aid s ,"

Universal Joint

and

Driveshaft

Manual, C. E . Cooney, ed. , Soc ie ty of Automotive Engineers , Warrenda le , P A,

1979, pp. 411-415.

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