Page 221 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

198 BIOMECHANICS OF THE HUMAN BODY

8.3 THE BIODYNAMIC SIGNIFICANCE OF THE EQUATIONS
OF MOTION

The equations of motion are dynamic expressions relating kinematics with forces and moments. In
a biodynamic musculoskeletal system, the forces and moments will consist of joint reactions; internal
forces, such as muscle, tendon, or ligament forces; and/or externally applied loads. Consequently,
the equations of motion can provide a critical understanding of the forces experienced by a joint and
effectively model normal joint function and joint injury mechanics. They can yield estimates for
forces that cannot be determined by direct measurement. For example, muscle forces, which are
typically derived from other quantities such as external loads, center of mass locations, and empirical
data including anatomical positioning and/or electromyography, can be estimated.
In terms of experimental design, the equations of motion can provide an initial, theoretical under-
standing of an actual biodynamic system and can aid in the selection of the dynamic properties of
the actual system to be measured. More specifically, the theoretical model is an initial basis that an
experimental model can build upon to determine and define a final predictive model. This may
involve comparative and iterative processes used between the theoretical and actual models, with
careful consideration given to every assumption and defined constraint.
Biodynamic models of the human musculoskeletal system have direct implications on device/tool
design and use and the modeling of normal and/or abnormal (or undesired) movements or movement
patterns (the techniques with which a device or tool is used). Applications of the models can provide
a better understanding for soft and hard tissue injuries, such as repetitive strain injuries (RSI), and
can be used to identify and predict the extent of a musculoskeletal injury (Peterson, 1999).

8.4 THE LAGRANGIAN (AN ENERGY METHOD) APPROACH

The equations of motion for a dynamic system can be determined by any of the following four
methods:
1. Newton-Euler method
2. Application of Lagrange’s equation
3. D’Alembert’s method of virtual work
4. Principle of virtual power using Jourdain’s principle or Kane’s equation
Within this chapter, only the first two methods are discussed. For a detailed discussion of other methods,
consult the references.
The Newton-Euler (Newtonian) approach involves the derivation of the equations of motion for
a dynamic system using the Newton-Euler equations, which depend upon vector quantities and
accelerations. This dependence, along with complex geometries, may promote derivations for the
equations of motion that are timely and mathematically complex. Furthermore, the presence of
several degrees of freedom within the dynamic system will only add to the complexity of the derivations
and final solutions.
The energy method approach uses Lagrange’s equation (and/or Hamilton’s principle, if appro-
priate) and differs from the Newtonian approach by the dependence upon scalar quantities and veloc-
ities. This approach is particularly useful if the dynamic system has several degrees of freedom and
the forces experienced by the system are derived from potential functions. In summary, the energy
method approach often simplifies the derivation of the equations of motion for complex multibody
systems involving several degrees of freedom as seen in human biodynamics.
Within this section, several applications of the Lagrangian approach are presented
and discussed. In particular, Lagrange’s equation is used to derive the equations of motion for
several dynamic systems that are mechanically analogous to the musculoskeletal system. A brief
introduction of Lagrange’s equation is provided, however, the derivation and details are left to