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1. Forces
- Internal force and external force
(1) Internal force: Forces acting between body parts
(2) External force: Forces acting between the body and environment. It can
be distant forces (gravity) or contact forces.
- Contact force
Contact force is composed of normal and tangential forces; normal forces
are normal to the surface of objects and usually unidirectional (e.g. only
pushing not pulling by sticking on an object surface) and tangential
(shear) forces are tangential to the surface
- Generalized contact force
A generalized contact force  has 6 components (3 orthogonal forces and 3
orthogonal moments)
- Resultant force
When there are two concurrent forces (forces meeting at the same point),
addition of the forces can be obtained by a parallelogram. However, it is
inconvenient to use a parallelogram when there are many forces. It is
rather simpler to add the rectangular components to get R
(resultant force)
2. Moment of force at a point
- The moment of force F at a point O can be defined, using vector
algebra, as
M0 = F
X r (when the position vector is defined from the force to
the axis) (2.1-1)
M0 = r X F
(when the position vector is defined from the axis to the force)
(2.1-2)
The equation 1.1-2 can be restated with the rectangular components
of F and r as
M0 = F X r
= (Xi + Yj + Zk) X (FXi
+ FYj + FZk)
=
(YFz - ZFY)i
+ (ZFx - XFz)j
+ (XFy - TFx)k
=
Mxi + Myj
+ Mzk
(2.2)
- The moment of force a point can be expressed as a determinant form as
 (2.3)
3. Moment of force at an axis
Figure 3-1.
- The force F which generates the moment at the axis 0-0 can be
resolved into two forces, Fp
and Fn. Fp
is a force which is parallel to the axis of moment and does not produce
any moment effect at the axis. Fp
is a force that is perpendicular/normal to the axis and the moment at the
axis is generated by this force. The magnitude of the moment at the axis M00
is F·(d·sinq).
Since F·sinq
is equal to Fn, the following
equation is also valid.
M00=Fn·d
(3.3)
- the moment of force about an axis can also be written as a form of
determinant as
 (3.1)
where cosX, cosY, and cosZ are directional cosines. Px, Py, and Pz are
coordinates of force application point.
4. Couples
- A force couple/couple consists of two equal, opposite, and
parallel forces. Those two forces generate a perfect torque which has no
force effect (resultant force = 0) at the midpoint between those two
forces.
- As far as the orientation of a couple is remained the same, the
couple can be moved freely in space. Therefore, a couple is a free
vector.
- Any given force/forces can be replaced with a force and a couple
5. Varignon's Theorem (Theorem of moments)
- "If a set of forces acting on a rigid body is reduced to one
resultant force, the moment of the resultant force about any point O
is equal to the moment of the original force system about O" (French
scientist Pierre Varignon, 1654-1722).
- That it, the sum of the moments of each force is equal to the
moment of resultant force
r X R = r X F1 + r X F2 + r
X F3 + ... + r X Fn = r X (F1 +
F2 + F3 + ... + Fn) (5.1)
- The theorem is valid for both coplanar and parallel force (e.g.
gravity) systems.
- The theorem is valid for concurrent forces (concurrent forces:
orthogonal components of a force)
- The theorem can be used to determine the point of resultant force
application
- The resultant force can be calculated if, and only if, the
direction of resultant force R is perpendicular to Mc. However,
in 3 dimensions, this may or may not the case
6. Wrenches
- Any arbitrary sets of forces can be reduced to a resultant force and a
corresponding couple.
- F
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