Biomechanical Modeling of the Shoulder - Fundamentals and Assessment Tools for Occupational Ergonomics - page 562

0
@

1
A

R
(

cos 1 g 3 ¼ q (27 : 3)

y
sin f

z cos f) 2 z sin f) 2 þ (R

b ¼ 0 : 5(g 2 þ g 3 g 1 þ f þ a 90) (27 : 4)

Q ¼ 2R M cos b (27 : 5)

=
;

P M A ¼

PL 2 sin a R M R ¼ 0

P F Z ¼

R S sin f R M sin (g 1 a) Q cos (b g 2 g 3 f) ¼ 0 (27 : 6)

P F Y ¼

P þ R S cos f þ R M cos (g 1 a) þ Q sin (b g 2 g 3 f) ¼ 0

The known variables are: a— glenohumeral joint angle defined as the angle between the axis of the
humerus and the Y-axis of body; f— an angle at which a tangent to a given curve of the glenoid fossa at
the point S in the frontal plane crosses the Z-axis; z — distance measured along the Z-axis between the
middle deltoid attachment and a point at which a tangent to the glenoid fossa crosses the Z-axis; g 1 —
angle between the distal tendon of the deltoid fiber and the axis of the humerus; P — external load and
reduced weight of upper limb applied at point C; L 1 —distance between the distal insertion of the middle
deltoid and center of the humerus head considered as a sphere; L 2 — distance between the center of the
humerus head and point of hand where an external load and reduced weight of upper limb were applied;
R — radius of the humerus head.
The unknown variables are: R S — the glenohumeral reaction force between the articular surface of the
glenoid fossa and the head of the humerus which is perpendicular to the glenoid articular surface
described by a tangent at its midpoint; R M — the force in a single muscle fiber of the deltoideus
lateral par, applied on the humeral insertion of the deltoideus; Q — the reaction force of the deltoideus
on the curved humerus area, which is a function of R M ; y — distance along the Y-axis at which a humerus
is balanced; b— angle of “glenohumeral pulley” defined as an angle of the curved contact area of muscle
fiber with the head of the humerus; g 2 — angle at the center of the humerus formed by the perpendicular
line of reaction force and line connected with the distal insertion of the middle deltoid fiber muscle; g 3 —
angle at the center of the humerus formed by the perpendicular line to the muscle fiber at the curved
contact area and line connected with the distal insertion of muscle. This geometric model should be
applied to each fiber of the lateral part of the deltoideus.
The variable y was calculated as a distance along the Y-axis at which a humerus was in balance. The
detailed trigonometric analysis revealed the relations into analytic form and the geometric equations were:

z 0 ¼ z sin f

z 0 ) 2

E z ¼ (R

z 0 ) 2 R 2 B z ¼ (R

z 0 )R C z ¼ (R

R sin f
L 2 sin a

G z ¼

(27 : 7)

D z ¼ G z E z þ C z

q

4D z I z ¼ 4B z G z D z þ B z

B z 2G z D z I z

J z ¼

2G z 1

J p sin fþ

y ¼ z sin fcos f