Download Kinetics: Work, Energy and Power 193 8.6 Spatial (3D) Mechanical

Kinetics: Work, Energy and Power
193
8.6 Spatial (3D) Mechanical Energy, Work and Power
Since energy, work and power are scalar quantities, it is rather simple to compute
these quantities in three dimensions. To compute translational or rotational
kinetic energies just add the following terms (assuming that the Y-axis is
vertical), ½ m vz2 and ½ I wz2, respectively. Work done in the Z direction is
similar to calculating work done along the other axes, that is, Wz = Fz sz. Thus, the
total mechanical energy (E) of a rigid body is calculated as follows:
(8.24)
where the first term is the gravitational potential energy, the next three terms are
the translational kinetic energies and the last three terms are the rotational kinetics
energies.
In a similar fashion, the work done by a force on a particle or body is the
scalar product of the force vector (F) and displacement vector (s) of the body’s
centre of gravity. That is,
(8.25)
Note, that this definition of work only applies when the particle or body moves in
a straight line. The work done by a force is path dependent so a better definition
uses the concept of a path integral, where the work may be evaluated by
knowing the instantaneous force at every location along the path. The total work
done would then be the integral of the scalar product shown in equation 8.25, but
measured in small steps (dW) from some starting point, a, to some ending point,
b. I.e.,
(8.26)
Similarly, the instantaneous power of a force acting on a particle or a body is,
(8.27)
where v is the instantaneous velocity of the particle at the point of application of
the force. This equation is often used to compute the transfer of energy from one
segment to another through a joint. The joint’s velocity would be v and the force
would be force exerted by one segment on the other through the connective
tissues in the joint (ligaments, cartilage, joint capsule, etc.).
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Spatial (3D) Mechanical Energy, Work and Power
The work (Wmoment) and power (Pmoment) of a moment of force on a rigid body are
defined as,
(8.28)
(8.29)
where q and w are the 3D angular displacement and angular velocities of the rigid
body, respectively. Note, even though angular displacement is not a vector, q,
represents the angular displacement of the body about each axis. In other words,
qx, represents the amount of rotation about the X axis.
The total mechanical work done on a rigid body is therefore either the
change in total mechanical energy or the sum of the works done by all the forces
and moments of force acting on the body. That is,
(8.30)
Laboratory (University of Sydney) equipped with a 10-camera infrared motion
capture system and multiple force platforms for biomechanical analyses