Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 478

B.7.17 Damping
The damping force can be modeled after Stokes's law, in which resistance is assumed to be linearly
proportional to velocity ( Eq. B.134 ). This is usually valid for oscillations of sufficiently small ampli-
tude. The damping force opposes the motion. The constant, k d , is called the damping coefficient . Add-
ing the damping force to the spring force produces Equation B.135 . Dividing through by m and
collecting terms results in Equation B.136 , where b ¼ k d / m and a

2

¼ k / m.

F d ¼k d dx
dt

(B.134)

m d

dt ¼kx k d dx

2

x

(B.135)

dt

d

dt þ b dx

2

x

2

dt þ a

x ¼ 0 (B.136)

If there is a spring force but no damping, the general solution can be written as x ¼ C cos( atþ y 0 ). If
there is damping but no spring force, the general solution turns out to be x ¼Ce bt

þ D , with C and D
constant. If both the spring force and the damping force are present, the solution takes the form shown
in Equation B.137 .

x ¼ Ce bt cos

ðdt þ y 0 Þ

s

a

b

2
4

where d ¼

2

(B.137)

for b < 2 a

B.7.18 Angular momentum
Angular momentum is the rotational equivalent of linear momentum and can be computed by
Equation B.138 , where r is the vector from the center of rotation and p is momentum (mass

velocity).
The temporal rate of change of angular momentum is equal to torque ( Eq. B.139 ) . Angular momentum,
like linear momentum, is conserved in a closed system ( Eq. B.140 ) .
L ¼ r p (B.138)

dL
dt

t ¼ (B.139)

SL i ¼ constant (B.140)

B.7.19 Inertia tensors
An inertia tensor ,or angular mass , describes the resistance of an object to a change in its angular momen-
tum [ 5 ][ 13 ]. It is represented as a matrix when the angular mass is related to the principal axes of the
object ( Eq. B.141 ) . The terms of the matrix describe the distribution of the mass of the object relative to a
local coordinate system ( Eq. B.142 ) . For objects that are symmetrical with respect to the local axes, the
off-diagonal elements are zero ( Eq. B.143 ). For a rectangular solid with mass M and dimensions a , b ,and
c along its local axes, the inertia tensor at the center of mass is given by Equation B.144 . For a sphere with