Page 224 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

BIODYNAMICS: A LAGRANGIAN APPROACH 201

where m is mass and v is the velocity vector. Substituting Eq. (8.5) into Eq. (8.6) yields
m 2 2
T = [ r + ( ) ] (8.7)
θ
r
2
The potential energy of the system is determined to be
k
−
V =− mgr cosθ + ( rl) 2 (8.8)
2
Rayleigh’s dissipation function is applied in order to properly handle the viscous component of the
given dynamic system. In this case, it is assumed that the viscous nature of the dashpot will be
linearly dependent upon the velocity of the mass. More specifically, the viscous damping force is
considered to be proportional to the velocity of the mass and is given by the following relation,

F =− r c (8.9)
d
Equation (8.9) is expressed in terms of the generalized coordinate r, where c is defined as the coef-
ficient of proportionality for viscous damping. Rayleigh’s dissipation function is defined as
1
=− cr 2 (8.10)
2
Further information concerning the definition and use of Rayleigh’s dissipation function can be
found in the references.
Lagrange’s equation can be modified to accommodate the consideration of the viscous damping
forces within the dynamic system. Subsequently, Eq. (8.2) can be rewritten as

d ⎛ L ∂ ⎞ L ∂ ∂
⎜
q
dt ∂ ⎠ ⎟ − q ∂ k + q ∂ k = Q k (8.11)
⎝
k
where the Lagrangian, L = T − V, is determined by using Eqs. (8.7) and (8.8),
m k
2
2
rθ
−
L = [ r + ( ) ] + mgr cosθ − ( rl) 2 (8.12)
2 2
By applying Lagrange’s equation (8.11), and using the first generalized coordinate r, where
q = r (8.13)
1
each term of Eq. (8.11) can easily be identified by differentiation of Eq. (8.12). The resulting rela-
tions are
∂L = mr
∂r (8.14)

so that
d ∂ ⎞ mr
⎛
L
⎜
⎝
dt ∂ ⎠ r ⎟ = (8.15)
∂L
−
(
and = mrθ 2 + mgcos θ − k r l) (8.16)
∂r