Page 225 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

202 BIOMECHANICS OF THE HUMAN BODY

Similarly, the term for the dissipation function can be shown to be
∂ =−cr
dr (8.17)
and the generalized force, Q = 0, since there are no externally applied forces acting on the system
r
in the radial direction. By insertion of Eqs. (8.15), (8.16), and (8.17) into Eq. (8.11), the equation of
motion in the r direction is

−
−
mr mrθ 2 − mgcos θ + k r l − cr = 0 (8.18)
)
(
Lagrange’s equation must now be solved by using the second generalized coordinate θ, where
q =θ (8.19)
2
By using the same approach as seen for the first generalized coordinate, the resulting differential
relations are
∂L = mr θ
2
∂θ (8.20)
so that
d ∂ ⎞ mrr +θ mr θ
⎛
L
2
⎜
dt ∂ ⎠ ⎟ = 2 (8.21)
⎝ θ
∂L
and =− mgr sin θ (8.22)
∂θ
The dissipation function in terms of the second generalized coordinate is shown to be
∂ = 0
∂θ (8.23)

and the generalized force, Q = 0, since there are no externally applied torques acting on the system
θ
in the θ direction. By insertion of Eqs. (8.21), (8.22), and (8.23) into Eq. (8.11), the equation of
motion in the θ direction is
2

2mrr θ + mr θ + mgr sin θ = 0 (8.24)
Through simplification and rearranging of terms, Eqs. (8.18) and (8.24) can be rewritten as

(
−
cr =
mgcosθ − k r l ) + m r rθ 2 ) (8.25)
−
(

−mgsinθ = m rθ + r )
θ
2
(
and (8.26)
respectively, yielding the equations of motion for the spring-dashpot pendulum system. Note that
Eqs. (8.25) and (8.26) are also the equations of motion obtained by using Newton’s laws.
Single Elastic Body with Two Degrees of Freedom. Next, consider a single elastic body pendulum
consisting of a mass suspended from a pivot point. Assume that the mass is suspended solely by a
spring of natural length l, as seen in Fig. 8.3, and that the system experiences no viscous damping.
As before, the system is constrained to move within the vertical, or r-θ plane, and is acted upon by
a gravitational field g, which acts in the negative vertical direction; r and θ are again determined to
be the only two generalized coordinates, since the motion of the mass is constrained to movement