Page 222 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIODYNAMICS: A LAGRANGIAN APPROACH 199

other sources identified in the references. Also note that an attempt was made to be consistent with
the symbols used from figure to figure to allow the reader to correlate the demonstrated examples.

8.4.1 Brief Introduction to Lagrange’s Equation
The application of Lagrange’s equation to a model of a dynamic system can be conceptualized in six
steps (Baruh, 1999):
1. Draw all free-body diagrams.
2. Determine the number of degrees of freedom in the system and select appropriate independent
generalized coordinates.
3. Derive the velocity for the center of mass of each body and any applicable virtual displacements.
4. Identify both conservative and nonconservative forces.
5. Calculate the kinetic and potential energy and the virtual work.
6. Substitute these quantities into Lagrange’s equation and solve for the equations of motion.

A system determined to have n degrees of freedom would correspondingly have n generalized coor-
dinates denoted as q , where k may have values from 1 to n. A generalized, nonconservative force
k
corresponding to a specific generalized coordinate is represented by Q k , where k again may range
from 1 to n. The derivative of q with respect to time is represented as q . Equation (8.2) shows the
k
k
general form of Lagrange’s equation.
d ⎛ L ∂ ⎞ L ∂
⎜ ⎟ − = Q k k = 12, ,... , n (8.2)
q
dt ∂ ⎠ q ∂ k
⎝
k
The Lagrangian, L, is defined as the difference between the kinetic energy T and the potential energy V:
L = T − V (8.3)
After determining the Lagrangian, differentiating as specified in Eq. (8.2) will result in a set of n
scalar equations of motion due to the n degrees of freedom.
Since the Lagrangian approach yields scalar equations, it is seen as an advantage over a
Newtonian approach. Only the velocity vector v, not the acceleration, of each body is required and
any coordinate system orientation desired may be chosen. This is a result of the kinetic energy
expressed in terms of a scalar quantity as demonstrated in Eq. (8.4).

T = T Translation + T Rotation
1 1 2
v
T Translation = m(v v) = mv
•
2 2 (8.4)
1
T
ω
ω
T Rotation = () (I G )()
2
For certain problems where constraint forces are considered, a Newtonian approach, or the application
of both techniques, may be necessary. The following sections will present some applications of
Lagrange’s equation as applicable to human anatomical biodynamics. Other mechanically based
examples are easily found within the cited literature (Baruh, 1999; Moon, 1998; Wells, 1967).
8.4.2 Biodynamic Models of Soft Tissue
Single Viscoelastic Body with Two Degrees of Freedom. Consider a single viscoelastic body pendulum
that consists of a mass suspended from a pivot point by a spring of natural length, l, and a dashpot,