International Journal of Industrial Ergonomics 32 (2003) 13–23
Evaluation of meat-hook handle shapes
Yong-Ku Kong*, Andris Freivalds
The Harold & Inge Marcus Department of Industrial and Manufacturing Engineering, 310 Leonhard Building, Pennsylvania State
University, University Park, PA 16802 USA
Received 26 August 2002; received in revised form 18 November 2002; accepted 3 December 2002
Abstract
Seven meat hooks including two current designs and five new designs with flat-rectangular, frustum, double frustum,
and cylindrical shapes were evaluated. A two-phase study was conducted. In the first experiment, maximum pulling
forces were measured by load cell to evaluate the effects of handle shapes on subjective discomfort, maximum pulling
force, and muscle activity. Two pulling forces, 15 and 30 kg, were employed to a pulley mechanism to simulate pulling a
beef carcass horizontally in the second experiment. FSR (force sensitive resistor) glove was used to measure the pulling
forces on the meat hooks. The glove has 12 sensors that result in placement on the pulpy regions of each phalange. In
addition, a biomechanical hand model was developed and applied to predict tendon forces. Double frustum shaped
handles produced significantly larger maximum pulling forces and best force efficiency when normalizing forces with
EMG. In terms of external forces as measured by the FSRs, the averages of finger force contributions to the total finger
force were 27%, 32%, 32%, and 10% in order from index finger to little finger. The averages of phalange force
contributions to the total finger force were 20.9%, 33.7% and 45.4% for the distal, middle and proximal phalanges,
respectively. A Chi-square analysis indicated that the phalange force distribution for double frustum handles deviated
least from the average contributions for all hooks. Double frustum handles showed the least predicted tendon forces
and normalized tendon forces per unit external force. The optimality of double frustum shaped handles was also
supported by the lowest discomfort ratings. Therefore, based on both empirical physiological measurements and
theoretical biomechanical calculations, a double frustum handle is most efficient for pulling task, producing the least
amount of tendon forces.
Relevance to Industry
Results obtained from this handle study will help to reduce the ergonomic risk factors of finger/hand disorders for
meat hook workers who use the meat hooks with a long daily duration and high force.
r 2003 Elsevier Science B.V. All rights reserved.
Keywords: Hand tool design; Meat-hook handle shapes; Biomechanical hand model
1. Introduction
*Corresponding author. Robert A. Taft Laboratories, 4676
Columbia Parkway, MS C-24, Cincinnati, OH 45226, USA.
Tel.: +1-513-533-8312; fax: +1-513-533-8596.
E-mail address: [email protected] (Y.-K. Kong).
Poor design and excessive use of hand tools are
associated with increased incidences of workrelated disorders or cumulative trauma disorders
0169-8141/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0169-8141(03)00022-2
14
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
(CTDs) of the finger, hand and forearm (Armstrong, 1983; Aghazadeh and Mital, 1987). Improperly designed hand tools lead the workers to
high grip forces, awkward hand/wrist postures,
and more fatigue. Therefore, the proper design
and evaluation of hand tools are important issues
in industry. To design better hand tools, several
characteristics of hand tools such as handle size
(Putz-Anderson, 1988), shape (Tichauer and Gage,
1977; Scheller, 1983; Cochran and Riley, 1986),
surface type (Lewis, 1987; Fellows and Freivalds,
1991), and texture (Pheasant and O’Neill, 1975)
have been considered.
Handle size is one important hand tool
design factor to maximize grip strength, reduce
stress on the digit flexor tendons, and avoid
stresses to the first metacarpal ulnar collateral
and carpometacarpal ligaments (Chaffin and
Anderson, 1984).
Khalil (1973) compared an elliptical, a spherical
handle, and cylindrical handles and found that a
cylindrical handle of 32 mm diameter was less
integrated EMG than other handles in static
torque tasks. Pheasant and O’Neill (1975) tested
the commercially available screwdriver handles
with smooth and knurled cylinders of the same
diameter. Evaluation of handle shape on grip
fatigue in manual lifting did not indicate any
significant difference in shape when the effect of
handle diameter was eliminated. Finger grooves
designed to provide form-fitting handles are
generally undesirable (Tichauer, 1973; Mital,
1991; Johnson, 1993). Although the finger grooves
may look as though those were molded to the
hand, those are only fit to one particular hand size.
This may result in discomfort, nerve compression
and impairment of circulation for a person with an
exceptionally large or small hand (Meagher, 1987).
A better approach is to change the handle
diameter according to the length of the fingers.
Konz (1974) and Bullinger and Solf (1979)
suggested a new handle design using a hexagonal
cross section, instead of circular cross section,
shaped as two truncated cones joined at the largest
ends. The shape fits the contours of the palm and
thumb best in precision and power grips, and
produced higher torque than conventional handles
(Freivalds, 1996).
Finger/phalange force contributions with various cylindrical handles have been studied by
many researchers (An et al., 1978; Ejeskar and
Ortengren, 1981; Amis, 1987; Lee and Rim, 1990;
Radhakrishnan and Nagaravindra, 1993). They
reported that the average percentage contributions
of finger forces to the total grip force, from index
to little fingers, were 28.3%, 31.0%, 23.2%, and
17.4%, respectively. They also found that the
average contributions of distal, proximal, and
middle phalanges to the total grasp force were
46.9%, 32.9%, and 20.2%, respectively.
Many researchers have studied tendon forces
of the index finger in pinching actions (Smith et al.,
1964; Chao et al., 1976; Berme et al., 1977;
An et al., 1979; Weightman and Amis, 1982) using
2D or 3D biomechanical finger models. From
the solutions of force/moment equilibrium equations, for an external load of P; 2:124:3P;
0:73B2:5P and 1:127:1P values were reported
for the flexor digitorum profundus (FDP),
flexor digitorum sublimis (FDS) and intrinsic
tendons, respectively. Chao et al. (1976) applied
their 3D finger model to the tip, lateral, ulnar
pinching as well as grasp actions. They reported
that the ranges of unit tendon forces of FDP, FDS
and UI were 2:5325:95P; 3:0524:23P and
3:3725:2P for the index, middle and little fingers,
respectively. These biomechanical models, however, focused on only understanding a force
distribution of the fingers and were not used to
evaluate hand tools.
The meat hook is a common hand tool in
meat packing to pull heavy beef carcasses. Use of
such hooks over long-time periods and with high
forces may induce work-related disorders for the
hand and fingers such as tendonitis in the ring
finger.
The objective of this study was to evaluate the
effect of handle shapes on maximum pulling task,
muscle activity, finger force distribution, handle
discomfort and predicted finger tendon force and
to determine an optimal handle shape by use of
flexor muscle activity and pulling force.
In this study ‘efficiency’ was defined as the ratio
of pulling force to normalized flexor muscle
activity. An ‘optimal handle’ is one which results
in the maximum efficiency ratio.
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
2. Method
2.1. Meat hooks
Two meat hooks currently used in the meatpacking industry and five new designs were
evaluated. Hook A, a current model, is a 15 mm
thickness piece of flat polyethylene with a hook
inserted in the center of the piece (termed center),
and Hook B, also a current model, is a polyethylene frustum. The cross-sectional shape of
Hook B is a circular and diameters of the handle
were gradually decreased from bottom (35 mm) to
top (30 mm) with the hook inserted at 30 mm from
15
the top-end (termed off-center), whereas Hook C
has a circular cross-sectional shape and diameters
were gradually increased from bottom (30 mm) to
top (35 mm) with the hook inserted off-center. The
hooks of handle D and E are placed at the center
and off-center of the handle, respectively. Both
hooks are double frustum shapes, which are the
diameters of the handle gradually decreased from
the center (38 mm) to both ends (26 mm). Hook F
and G are cylindrical shapes with the hook offcentered and centered, respectively (Fig. 1). Note
that for off-center handles, the hook projects
between the index and middle fingers, while for
the center handles, the hook projects between the
middle and ring fingers.
2.2. Measurement system
Fig. 1. Meat hooks.
Fig. 2. FSR force glove.
Silver–silver chloride surface electrodes were
used to record the electrical activity of the flexor
digitorum superficial muscle under the maximum
pulling task. The EMG signal was acquired using
the FlexComp system (NexGen Ergonomics,
Montreal, Canada). The signal was recorded and
then was amplified, rectified, and filtered using a
low-pass filter with 60 Hz cut-off frequency. For
each test condition, the EMG data were collected
for a period of 5 s at a rate of 496 samples/s.
A force glove was developed by overlaying
force-sensing resistors (FSRs) on a glove to
measure finger forces. Twelve FSRs were placed
on the pulpy parts of each phalange (Fig. 2). The
active area of the FSR sensor is 5 mm diameter
and 0.3 mm thickness. The output signals from the
FSR were sent to a custom-made voltage division
circuit box which was designed to provide 0 to
75 V DC to the A/D converter (DASH 16-F
METRABYTE, 1989). A data acquisition software system was also developed.
A biomechanical hand model was developed
based on the assumption of a constant moment
arm of the tendons at the finger joints. In case of
constant tendon moment arms, Landsmeer’s
(1962) first tendon model was used to express the
relationship between tendon excursion and joint
angles. Equilibrium equations were derived using
the principles of virtual work model (Storace and
Wolf, 1979). Three tendon forces (flexor digitorum
16
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
profundus, FDP; flexor digitorum sublimes, FDS;
ulnar interosseous, UI) were derived from the
equilibrium equations (Appendix A).
2.3. Experimental design
2.3.1. Experiment 1: Maximum pulling force
A total of 6 subjects were recruited from the
Pennsylvania State University to take part in this
study. The subjects were all male, aged 23–34. All
subjects were screened by questionnaire for any
hand and wrist injuries or surgeries, which may
limit their physical activities. Subjects were provided with a brief description of the goals and
procedures. Then, they signed an informed consent
form prior to experimentation.
The subject was asked to use his left hand to pull
the load cell equipment for 3 s with his maximum
voluntary contraction. All seven hooks were tested
for each subject. A handle of the load cell was
attached to the 1.5 m height custom-made frame.
The average maximum pulling force during task
was shown on the digital screen of the load cell
equipment. EMG signals were collected from the
FlexComp system. To eliminate any effects of
background noise and minimize inter-subject and
inter-electrode differences, the EMG was normalized (EMGN) as follows: EMGN=(task EMGrrest EMG)/(maximum EMG rest EMG).
‘‘Efficiency’’ is defined as the ratio between
maximum pulling force and flexor muscle activity
measured (Ayoub and Lo Presti, 1971), was used
to determine the optimal hook handle. The
subjective ratings of handle discomfort (Borg,
1973) were also recorded for each hook handle.
The scale ranges were from 6 (very very light) to 20
(very very hard). Each subject performed four
repetitions for each hook with 3 min resting time
between each trial. All 28 trials (7 hooks 4 trials)
were tested in a completely randomized order.
Subjects and trials were considered random
factors. Dependent variables were maximum pulling force, flexor muscle activity, and subjective
ratings of discomfort. The ANOVA was performed to evaluate all possible effect of handle
type on the dependent variables.
2.3.2. Experiment 2: Finger force distribution
Four different males participated in the study.
They were all right-handed university students,
ranging in age from 25 to 30. All subjects were
screened by questionnaire for any hand and wrist
injuries or surgeries, which may limit their physical
activities. Subjects were provided with a brief
description of the goals and procedures and signed
an informed consent form.
The subjects used the force glove to evaluate
finger and phalange force distributions. Two
different hanging weights, 15 and 30 kg, were
employed. Pulling these weights with hooks, acting
horizontally through a pulley system, was comparable to average and maximum forces for pulling a
beef carcass at a meat packing company (Fig. 3).
Three trials were collected for each condition. A
total of 42 trials (2 weight 7 hooks 3 trials)
were completely randomized.
Dependent variables were finger/phalange forces
during pulling, flexor muscle activity, and subjective ratings of discomfort. The effect of handle
type, finger, phalange, and weight on the dependent variables was analyzed.
3. Results
3.1. Maximum pulling force
Maximum pulling forces on the hook handles
differed significantly at the a ¼ 0:05 level, with
Hooks D and E (double frustum handles) showing
significantly higher forces as compared with other
hook handles (F6;153 ¼ 5:89; po0:001). Fig. 4
shows the average maximum pulling force for
each hook handle.
Electromyography (EMG) was also analyzed to
objectively determine the optimal hook handle
shape for maximum pulling force. Fig. 5 presents
the results of the efficiency for each hook.
Although the results of ANOVA showed that no
statistically significant difference between hook
handles, Hooks D and E showed higher efficiencies
than the other hook handles.
Analysis of subjective discomfort ratings
showed that Hooks D and E (double frustum
handles) and B (oval handle) had ratings of 9.8,
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
17
Fig. 3. Experimental equipment.
Efficiency (Max. Force / EMGN)
Maximum Pulling Force
100
45
80
Efficiency
Max. Force (kg)
43
41
39
60
40
37
20
A
35
A
B
C
D
Hook
E
F
B
C
G
D
Hook
E
F
G
Fig. 5. Efficiency of each hook.
Fig. 4. Average maximum pulling force.
3.2. Force distribution
10.8 and 10.8 (very light to fairly light) whereas,
Hook A (flat) showed a 16.7 (very hard) for the
maximum pulling task. Tukey test results are
shown in Table 1. Mean subjective ratings grouped
by vertical lines do not differ significantly at a ¼
0:05: Subjects preferred double-frustum shapes
over other shape handles.
The ANOVA of the pulling force for two
different weights (Table 2) shows a statistically
significant effect for all main effects except hooks.
Although the effect of hooks is not statistically
significant, Hook D and E generally required
lower finger forces than the others.
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
Table 1
Tukey test for subjective rating in a pulling task
Finger Force Contribution
4
40
3
30
2
20
1
10
0
0
A
INDEX
Table 2
ANOVA on finger force
Chi-S
Contribution (%)
50
B
C
D
Hook
MIDDLE
E
RING
F
G
LITTLE
Chi-square
Fig. 6. Average finger force contribution.
DF
SS
MS
F
P
Subject
Hooks
Weight
Finger
Phalange
Trial
Error
Total
3
6
1
3
2
2
1566
1583
60.512
7.063
7.792
27.788
72.973
2.020
1537.2
1715.3
20.272
1.177
7.792
3.542
36.486
1.010
0.982
20.65
1.20
7.94
3.61
37.17
1.03
o0.001*
0.304
0.005*
0.013*
o0.001*
0.358
*po0.05.
Phalange Force Contribution
50
Contribution (%)
Source
4
40
3
30
2
20
1
10
0
0
A
The mean finger forces were significantly different (F3;156 ¼ 3:61; po0:013) and, in descending
order by magnitude of force, are middle, ring,
index and little finger. Total pulling force was
assessed by summing the individual sensors and
used to compute the average contribution of
individual finger forces. The average contribution,
in order from index finger to little finger was 27%
(ranges: 22.2–30.4%), 32% (ranges: 27.9–34.0%),
32% (ranges: 30.4–34.2%) and 10% (ranges: 9.2–
11.2%), respectively.
The contribution of individual finger forces to
the total pulling force and Chi-square test of
average finger contribution for all hooks are
shown in Fig. 6.
A Chi-square analysis yielded values of 0.87,
0.58, 1.87, 1.05, 1.51, 1.51, and 3.03 for Hooks A
through G, respectively. Although these values
were significantly lower than the critical value of
Chi-square w2 (0.95, 3)=7.82, Hooks A and B
showed a more uniform finger contribution to the
total pulling force.
Chi-S
18
DISTAL
B
C
D
Hook
MIDDLE
E
F
PROXIMAL
G
Chi-square
Fig. 7. Average phalange force contribution.
The mean forces of the phalanges are also
significantly different (F2;56 ¼ 37:17; po0:001).
The forces produced by the proximal phalanges
were the greatest followed by middle and distal
phalanges. The distributions of phalange force to
the total pulling force are shown in Fig. 7. A Chisquare analysis for the average of all hooks yield
values of 3.76, 0.62, 0.67, 0.48, 0.77, 1.23, and 1.12
for Hooks A through G, respectively. Although
these values are significantly lower than the critical
value of Chi-square w2 (0.95, 2)=5.99, the
phalange force contribution of Hook D is the
most evenly distributed over each phalange,
whereas Hook A has very uneven distribution
over each phalange in the pulling task.
For all seven hooks, the proximal phalanges
exerted the greatest force, followed by the middle
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
Total Tendon Force
Tendon Force (kg)
30
25
20
15
10
5
0
C
D
Hook
Total Tendon Force
FDP
3
2
1
A
A biomechanical hand model was developed
based on the principles of virtual work and
applied to evaluate hook handles (Appendix A).
Basically, the empirical FSR forces for each hook
were used in the biomechanical hand model to
predict tendon forces of flexor digitorum profundus (FDP), flexor digitorum sublimis (FDS) and
ulnar interosseous (UI) during pulling tasks.
Figs. 8 and 9 depict that the average total tendon
forces and normalized tendon forces per unit force,
respectively.
Hooks D and E (double frustum handles)
showed the least tendon forces for the total pulling
forces as well as the unit external forces. Hook A
had the highest tendon and normalized tendon
forces. The ranges of tendon forces per unit
external force (P) for all three tendons were
0:4720:84P; 0:6921:10P and 1:4522:54P for
FDP, FDS and UI, respectively and overall
tendon forces of the pulling tasks for all types of
hooks were 3.15–3.7 times for the unit external
force.
B
4
0
3.3. Biomechanical hand model
A
Normalized Tendon Forces
5
Tendon Force (kg)
phalanges and the distal phalanges. That is, an
average 45% (ranges: 43.8–46.6%) of total finger
force was exerted by the proximal phalanges, 34%
(ranges: 31.8–37.4%) by the middle phalanges and
21% (ranges: 16.3–22.5%) by the distal phalanges,
respectively.
19
E
F
FDS
Fig. 8. Average total tendon forces.
G
UI
B
C
D
Hook
Tendon Force / Unit Force
E
FDP
F
FDS
G
UI
Fig. 9. Normalized tendon forces.
4. Discussion
The double-frustum shaped handles (Hook D
and E) showed significantly larger maximum
pulling forces than the other handles. The currently used flat rectangular handle (Hook A) had
the lowest maximum pulling force. The ratio of
maximum pulling force to normalized EMG
activity was used to evaluate efficiency of hook
handles and resulted in Hook E being best,
followed by Hook D. In the second study, Hook
D and E (double frustum handles) required less
pulling force to maintain the given loads, whereas
Hook F and G (cylindrical handles) required more
force.
The subjective discomfort ratings of the double
frustum handles were also better than other
handles. Hook D provided the best subjective
discomfort ratings among these handles followed
by Hook E and B (9.8, 10.8 and 10.8, respectively).
The double frustum shapes allowed each finger to
grip at its optimum circumference. The longest
middle finger had large circumference, whereas the
shortest little finger had small circumference. It
may make users feel more comfortable during a
pulling task, if the handle size fit to users’ hand
size. Most subjects complained of a pain to the
proximal phalanges when they used Hook A
during the pulling tasks. The relatively sharp edges
of this flat handle (Hook A) dig into the palm near
the proximal phalanges of the fingers. According
to the Chi-square analyses, subjects generally
preferred uniform phalange force distribution
(i.e., 0.48, 0.77 and 0.62, for Hook D, E and B,
20
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
respectively) whereas, they did not have preference
for more uniform finger force distribution (i.e.,
0.87 for Hook A). Therefore, an optimal handle
shape should have stronger fingers providing a
larger contribution to total force than weaker
fingers, with a uniform force distribution on all
phalanges.
Finger and phalange force distributions for
pulling tasks were studied using a force measurement system. The index finger contributes between
27.8% and 30.4% to the total pulling force, the
middle finger 27.9–34.0%, the ring finger 30.4–
34.2% and the little finger 7.2–11.2%. The
contributions of index and middle fingers in the
pulling tasks are in line with average values of
27.7% and 30.9% found in the literature whereas, the contributions of ring and little fingers do
not correspond with average values of 23.5%
and 17.9% found in the previous gripping
studies (Amis, 1987; An et al., 1978; Chen, 1991;
Ejeskar and Ortengren, 1981; Lee and Rim, 1990;
Radhakrishnan and Nagaravindra, 1993). The
sum of contributions of middle and ring fingers
is 63.2% (ranges: 60.3–68.2%) in the pulling tasks,
whereas the sum of contributions of middle and
ring fingers is 54.4% in the cylindrical grip studies.
That is, the role of ring finger in pulling tasks was
significantly more important than in cylindrical
grip actions. This may explain the high frequency
of trigger finger injuries in the ring finger observed
in meat packers. According to the injury records of
one of the meat packing company, the ring finger
is the most commonly affected digit, followed by
the middle, index and little fingers in the jobs
requiring use of meat hooks.
The average phalange force contributions to the
total pulling force are 20.9%, 33.7% and 45.4%
for the distal, middle and proximal phalanges,
respectively. The high concentrations of the both
middle and proximal phalanges (77.6–83.7%) may
explain the frequency of trigger finger injuries in
pulling tasks other than static holding or grasping
tasks. According to the Karwowski and Salvendy
(1998), the localized compression in an area of the
A1 pulleys (between proximal phalanges and
metacarpals) is one of main factors in trigger
finger injuries. Flat rectangle handle (Hook A)
showed high percentage of middle and proximal
phalange forces whereas, double frustum handles
showed a lower percentage of force contributions
of these two phalanges in this study.
The biomechanical hand model predicted average unit tendon forces of FDP, FDS and UI were
0:7P; 0:94P and 1:74P in this study. According to
the previous studies, the ranges of unit tendon
forces of FDP, FDS and UI were 2:5325:95P;
3:0524:23P
and
3:3725:2P;
respectively.
Although these values are somewhat lower as
compared with previous research, they still may be
considered as a form of grip efficiency. The tendon
forces predicted from the biomechanical hand
model showed a good correlation with the results
of other measures. According to the study, double
frustum handles showed the least tendon forces,
whereas Hook A showed the highest tendon
forces. Therefore, double frustum handles would
be recommended to make better subjective discomfort, performance (pulling force) and less
finger tendon force in a pulling task.
Acknowledgements
The authors thank Taylor Packing Co., Inc. for
funding this project and providing meat hooks.
Appendix A. Formulation of hand model
A.1. Displacements of the flexor tendons (FDP,
FDS and UI)
FDP : XP ¼ PR1 y1 PR2 y2 PR3 y3 ;
ðA:1Þ
FDS : XS ¼ SR2 y2 SR3 y3 ;
ðA:2Þ
UI : XUI ¼ UR3 y3 ;
ðA:3Þ
where i is the 1, 2, 3, and 4: index, middle, ring and
little fingers, XP the FDP displacements, XS the
FDS displacements, XUI the UI displacements, y1
the angle between middle and distal phalanx, y2
the angle between proximal and middle phalanx,
y3 the angle between metacarpal and proximal
phalanx, PRI FDP moment arm at the joint i
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
(1=DIP; 2=PIP; 3=MP), SRI the FDS moment
arm at the joint i (2=PIP; 3=MP), URi the UI
moment arm at the joint i (3=MP) (Fig. 10).
If external moments are applied to the each
joint, resulting in flexor muscle forces and the
principle of virtual work yield:
FP qXP þ FS qXS þ FUI qXUI
þ ðT1 þ T2 þ T3 Þ qy3
þ ðT1 þ T2 Þ qy2 þ ðT1 Þ qy1 ¼ 0;
21
PR1 FP ¼ T1 ;
ðA:11Þ
PR2 FP þ SR2 FS ¼ T1 þ T2
ðA:12Þ
and
PR3 FP þ SR3 FS þ UR3 FUI ¼ T1 þ T2 þ T3 :
ðA:13Þ
ðA:4Þ
where qX is the increment of tendon excursion and
qy the increment of joint angle.
The virtual displacements are related by:
Therefore,
FP ¼ T1 =PR1 ;
FS ¼ ðT1 þ T2 PR2 FP Þ=SR2 ;
ð14Þ
ðA:15Þ
qXP ¼ ½ðqXP =qy1 Þ qy1
FUI ¼ ðT1 þ T2 þ T3 PR3 FP SR3 Fs Þ=UR3 :
þ ðqXP =qy2 Þ qy2
þ ðqXP =qy3 Þ qy3 ;
ðA:16Þ
ðA:5Þ
qXS ¼ ½ðqXS =qy2 Þ qy2 þ ðqXS =qy3 Þ qy3 ;
ðA:6Þ
qXUI ¼ ½ðqXUI =qy3 Þ qy3 :
ðA:7Þ
Substituting n; qXS ; qXRI and qXUI in calculate in
Eqs. (A.5)–(A.7) into Eq. (A.4) become:
FP ðqXP =qy1 Þ ¼ T1 :
ðA:8Þ
FP ðqXP =qy2 Þ þ FS ðqXS =qy2 Þ ¼ T1 þ T2
ðA:9Þ
and
FP ðqXP =qy3 Þ þ FS ðqXS =qy3 Þ
þ FUI ðqXUI =qy3 Þ ¼ T1 þ T2 þ T3 :
ðA:10Þ
For the linear displacement functions 1, 2 and 3,
the equilibrium equations yield:
A.2. Tendon moment equilibrium equations of index
finger (DIP, PIP, & MP joint)
Three moment equilibrium equations can be
obtained from each joint and calculated by the
following equations:
T1 ¼ F1 L11 =2½cosð90 þ y1 þ y2 þ y3 Þ
cosð90 y1 y2 y3 Þ
sinð90 þ y1 þ y2 þ y3 Þ
sinð90 y1 y2 y3 Þ
¼ F1 L11 =2cosð180Þ
¼ F1 L11 =2;
Fig. 10. Free-body diagrams for finger and phalanges.
ðA:17Þ
22
Y.-K. Kong, A. Freivalds / International Journal of Industrial Ergonomics 32 (2003) 13–23
T2 ¼ F1 ½L12 sinðy1 þ y2 þ y3 Þ cosðy2 þ y3 Þ
þ F2 ½L12 =2 sinð2y2 þ 2y3 Þ
þ L11 =2 sinðy1 þ y2 þ y3 Þ cosðy1 þ y2 þ y3 Þ
þ L13 sinðy2 þ 2y3 Þ
F1 ½L12 cosðy1 þ y2 þ y3 Þ sinðy2 þ y3 Þ
þ L11 =2 cosðy1 þ y2 þ y3 Þ sinðy1 þ y2 þ y3 Þ
þ F3 ½L13 =2 sinð2y3 Þ;
þ F2 L12 =2 ½sinðy2 þ y3 Þ cosðy2 þ y3 Þ
þ cosðy2 þ y3 Þ sinðy2 þ y3 Þ
¼ F1 f½L12 sinðy1 þ y2 þ y3 Þ cosðy2 þ y3 Þ
cosðy1 þ y2 þ y3 Þ sinðy1 þ y2 þ y3 Þ
þ L11 =2 sin2ðy1 þ y2 þ y3 Þg
þ F2 L12 =2 ½sin2ðy2 þ y3 Þ
¼ F1 f½L12 sin y2 þ L11 =2 sin 2ðy1 þ y2 þ y3 Þg
þ F2 L12 =2 ½sin2ðy2 þ y3 Þ;
ðA:18Þ
T3 ¼ F1 ½L12 sinðy1 þ y2 þ y3 Þ cosðy2 þ y3 Þ
þ L11 =2 sinðy1 þ y2 þ y3 Þ cosðy1 þ y2 þ y3 Þ
þ L13 sinðy1 þ y2 þ y3 Þ cosðy3 Þ
þ F1 ½L12 cosðy1 þ y2 þ y3 Þ sinðy2 þ y3 Þ
L11 =2 cosðy1 þ y2 þ y3 Þ sinðy1 þ y2 þ y3 Þ
þ L13 cosðy1 þ y2 þ y3 Þ sinðy3 Þ
þ F2 ½L12 =2 sinðy2 þ y3 Þ cosðy2 þ y3 Þ
þ L13 sinðy2 þ y3 Þ cosðy3 Þ
þ F2 ½L12 =2 cosðy2 þ y3 Þ sinðy2 þ y3 Þ
þ L13 cosðy2 þ y3 Þ sinðy3 Þ
þ F3 ½L13 =2 sinðy3 Þ cosðy3 Þ
þ cosðy3 Þ sinðy3 Þ
¼ F1 L12 ½sinðy1 þ y2 þ y3 Þ cosðy2 þ y3 Þ
þ cosðy1 þ y2 þ y3 Þ sinðy2 þ y3 Þ
þ F1 L11 =2 ½cosðy1 þ y2 þ y3 Þ sinðy1 þ y2 þ y3 Þ
cosðy1 þ y2 þ y3 Þ sinðy1 þ y2 þ y3 Þ
þ F1 L13 ½sinðy1 þ y2 þ y3 Þ cosðy3 Þ
þ cosðy1 þ y2 þ y3 Þ sinðy3 Þ
þ F2 L12 =2 ½sinðy2 þ y3 Þ cosðy2 þ y3 Þ
þ cosðy2 þ y3 Þ sinðy2 þ y3 Þ
þ F2 L13 ½sinðy2 þ y3 Þ cosðy3 Þ
þ cosðy2 þ y3 Þ sinðy3 Þ
þ F3 ½L13 =2 sinð2y3 Þ
¼ F1 ½L12 sinðy1 þ y2 þ y3 Þ
þ L13 sinðy2 þ y2 þ y3 Þ
ðA:19Þ
where T1 ; T2 ; T3 is the moments acting on the DIP,
PIP, and MP, FI the external force acting
vertically upward on the phalange (1=distal;
2=middle; 3=proximal), L11 ; L12 ; L13 : the length
of the proximal, middle, and distal phalanges of
index finger.
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