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Analysis of Systems in Static
Equilibrium
Procedure:
 Draw a schematic diagram of the system
 Draw the free-body diagram (FBD)
 Chose a cartesian coordinate system (x,y,z).


 For each FBD apply: F  0 & M  0
 Include Directions and Physical Units
 If # of equations equal # of unknowns; Solve!
Vector Operations
• The parallelogram law;
• The commutativity principle;
• Vector addition:
c=a+b=b+a
(commutativity)
• Vector subtraction: d=a-b
• Vector products
• The dot product: e=ab=ba
(commutativity)
• The cross product: f=axbbxa
• Vector products are distributive
z
k
i
x
j
y
Vector Products
• The dot product yields a scalar
a  b  c F  Fx i  F y j  Fz k
• Cross product in explicit form:
F1  F2  R
i
F1  F2  F1 x
F2 x
a
a
b
k
 Rx 
 
F1z   R y 
F2 z  Rz 
j
F1 y
F2 y
• Geometrical description of the cross product would
be a vector perpendicular to the plane formed by F1
and F2 and its absolute value would be:
 
F  F  F  F Sinea
1
2
1
2
12
Moment of a force
The moment of a force about a
point is the applied force times the
distance between the point and the
line of action of the force.
M  r F
[ Moment ]  [ Position]  [ Force]
or in a planar case, the
magnitude of a moment:
M  r  F  Sina
r
Scalar Representation of a Moment
M  r  F  Sina  d  F
and the direction is determine
as described below
Moment of a force is represented by a
vector perpendicular to the plane of the
force and the position vector .
Moment Vector
Three-Dimensional Case:
M  rF
i j k
M

M  r r r  M
M
F F F

x
y
x
z
y
z
M  Mx i  My j  Mz k
x
y
z





Moment Addition
Collinear moments addition
M  M1  M2  M3
Gravitational Force (Weight)
The force exerted by the Earth on an object
The gravitational force is
some times expressed as: W  mg
Where:
g = 9.81 m/s2
The Ground Reaction N
(Newton’s 3rd Law)
N W  0


N  W or as vectors N  W
Gravitational Force (Cont.)
Often the relationship:
W  mg
Is substituted based on the fact that the
dynamic equation for a free object in the
gravitational field is:



W  ma ; where a  g
Problem #1
A person standing in a bus and holding single-handedly to
an overhead rail experiences the forces depicted in the
figure. The mass of the person is 75 Kg and the
coordinates of the points of application of the forces
R1, R2, P, in meters are:
R1(-0.2;-0.15;0)
R2(+0.2;+0.15;0)
P(0.25;0.35;1.85)
And the measured forces (in Newtons) are:
R1= -5i-75j+300k
R2= 75i+25j+375k
The person is in static equilibrium which is satisfied by the
balancing action of the force P.
Determine:
The force vector P
The magnitude of the moments exerted by the forces R1,
R2 and P, on the lumbar junction O, located at:
O(-0.1;-0.15;+0.8)
The moment acting on the lumbar junction
Wt=50% Body weight (Weight of the upper body)
D=15 cm
Z
P
Wt
O
X
R1
Y
R2
6
Position
5
7
8
L1 vertebrae
4
r4
3
9
r3
rLumbar
2
r2
z
r1
1
x
y
0
The Modeling Concept (i)
A rigid body in static equilibrium is subjected to
active and reactive forces and moments.
Active forces & moments: External and Gravitational.
Reactive forces & moments: Constraint forces
Hinge
Constrains:
Roller
The Modeling Concept (ii)
Constraints and Reactions
Constraints and Reactions (iii)
Flexible Member (cable, chain, rope, muscle):
Can apply tensile forces only.
Fm
Quad
Fm
Calf
Fn
Fm
Achil
Distributed Force or Pressure
When one object is in contact with another, the force of interaction is
distributed over the contact surface. This will be described as a
“distributed force”, “bearing stress”, or “pressure”.
[ Force]
[Pr essure] 
[ Area ]
Pressure in Biomechanics (i)
Pressure in Biomechanics (ii)
“Pressure Ulcers” or “pressure sores” affects people who are
confined to bed for extended period of time, or wheelchair users, or
those with sensory deficiency such as diabetics.
Pressure in Biomechanics (iii)
Bed Sores
Skin
A bedsore is breakdown of skin
and underlying tissue to form Bone
sores and ulcers due to
persistant static pressure.
Bedsores are also known as
pressure sores, decubitus ulcers,
pressure ulcers and pressure
wounds.
Stage 4
Stage 1
Stage 2
Stage 3
Muscle
Pressure in Biomechanics (iv)
The Prosthetic Interface
Frictional Force
f  N
Class Problem #6
A person stands upright and is holding a
metal ball weighing 10Kgs. Considering the
weight of his body segments, their positions
and their dimensions given in the table,
determine the force and the moment exerted
on the Sacro-Illiac junction designated by the
point D. The following points are given:
Z
N
K
G
F
L
J
E
M
A(0.15; 0; 0)
B(0.15; 0.1; 0.4)
C(0.15; 0; 0.9)
D(0; 0; 0.9)
E(0; 0; 1.0)
F(0; -0.15; 1.38)
H
H(0.45; -0.1; 1.36)
I(0.45; -0.1; 1.05)
J(0.45; 0.34; 1.08)
K(-0.45; -0.1; 1.38)
L(-0.45; -0.05; 1.05)
M(-0.45; -0.02 ;0.81)
I
D
10kg
C
B
Y
A
X
Anthropometric Data on Body Segments
Body Segment
Relative Length
% of Height
Relative distance **
of C.M. in % of li
Relative Mass
% of Body Mass
Head
9.4
50
5.7
Neck
4.5
61
1.3
Thorax
48
Abdomen
50
Thorax+Abdomen
25.0
30.3
Upper Arm
18.0
46
2.6*
Forearm
26.0
37
1.9*
Hand
.7*
Pelvis
9.4
50
14.0
Thigh
31.5
55
12.8*
Shank
23.0
66
5.1*
Foot
16.0
60
1.3*
* Data are per single limb
** Relative distance of C.M. (measured from the lower numbered joint)
Homework # 2
The person described in the figure weighs 700 Newtons
and is 170 centimeters tall. He is subjected to the
gravitational forces and the ground reaction forces F1 and
F2 . Considering that F1 has the following components:
F1z  500 N
F2 z  20 N
F1x  25N
F1y  120N
F2 x  10 N
F2 y  60N
(1) Assign angular positions to the body segments and
obtain their centers of mass and masses, from the table
used in assignment #1 (assume the transverse distance
between the hips to be 25 centimeters) and determine the
following:
• Position of the right hip joint (the leg that is ahead);
• The net forces acting on an imaginary section through
the hip;
• Moment acting at the knee;
• Moment and force acting at the hip.
(2) Check if the person is in a static equilibrium.
(3) Determine the moments of the ground forces with
respect to the center of mass.
Z
X
Y
F2
F1
Homework Assignment #2, Cont.
• Get a textbook of anatomy, identify the major muscle
groups that activate the hip, knee and ankle.
• Determine which muscle are responsible for the
flexion- extension of the knee and draw a
geometrical model that will help you calculate the
forces in these muscles when a person puts his full
weight on one leg.
See you next time