Page 241 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

218 BIOMECHANICS OF THE HUMAN BODY

The gravitational potential energy of the system, expressed in vector form, is given as

V = ( m r + m r + m r + m r • ) − ( g)j (8.89)
AG 0 B G 1 C G 2 D G 3
The unit vector j, according to Figs. 8.4, 8.5, and 8.6, is in the inertial coordinate system and is
always directed upward. Taking a dot product between any quantity and this unit vector results in the
vertical component of the quantity. As a result of the dot products in both Eqs. (8.87) and (8.89), the
resulting kinetic and potential energies are scalar quantities. As before, these quantities can be incor-
porated into Lagrange’s equation to determine the equations of motion for the system.

8.6 BRIEF DISCUSSION

8.6.1 Forces and Constraints

Forces play an integral role in the dynamic behavior of all human mechanics. In terms of human
movement, forces can be defined as intrinsic or extrinsic. For example, a couple about a particular
joint will involve the intrinsic muscle and frictional forces as well as any extrinsic loads sustained
by the system. If the influence of intrinsic muscle activity within the system is to be considered, the
location of the insertion points for each muscle must be determined to properly solve the equations
of motion.
Conservative forces due to gravity and elasticity are typically accounted for within the terms
defining the potential energy of the system, while inertial forces are derived from the kinetic energy.
Forces due to joint friction, tissue damping, and certain external forces are expressed as nonconser-
vative generalized forces.
In biodynamic systems, motions that occur between anatomical segments of a joint mechanism
are not completely arbitrary (free to move in any manner). They are constrained by the nature of the
joint mechanism. As a result, the structure of the joint, the relative motions the joint permits, and the
distances between successive joints must be understood in order to properly determine the kinematics
of the system.
The Lagrangian approach presented within this section has been limited to unconstrained systems
with appropriately selected generalized coordinates that match the degrees of freedom of the system.
For a system where constraints are to be considered, Lagrange multipliers are used with the extended
Hamilton’s principle (Baruh, 1999). Each constraint can be defined by a constraint equation and a
corresponding constraint force. For any dynamic system, the constraint equations describe the
geometries and/or the kinematics associated with the constraints of the system. For a biodynamic
joint system, the contact force between the segments linked by the joint would be considered a
constraint force. Constraint forces may also involve restrictions on joint motion due to orientations
and interactions of the soft tissues (e.g., ligamentous, tendonous, and muscular structures) that surround
the joint.

8.6.2 Hamilton’s Principle
In cases where equations of motion are desired for deformable bodies, methods such as the extended
Hamilton’s principle may be employed. The energy is written for the system and, in addition to the
terms used in Lagrange’s equation, strain energy would be included. Application of Hamilton’s
principle will yield a set of equations of motion in the form of partial differential equations as well
as the corresponding boundary conditions. Derivations and examples can be found in other sources
(Baruh, 1999; Benaroya, 1998). Hamilton’s principle employs the calculus of variations, and there
are many texts that will be of benefit (Lanczos, 1970).