Graphics Reference
In-Depth Information
|| y i y j ||
= r . By Hooke's Spring Law, the restorative spring force experienced
by object i is
y i
y j
F s ( y , i , j )= k s ( r
−||
y i
y j ||
)
.
(35.42)
||
y i
y j ||
Some springs are stiffer than others. The spring constant k s describes the
stiffness of the spring in kg/s 2 (this is another case where subscript s is part of
the name, not a proper index). Larger constants describe springs that exert higher
restorative forces, which we call stiffer, and smaller constants exert smaller forces,
which we call softer. When mass is attached to the spring, a stiffer spring will
accelerate that mass faster.
Like a pendulum under gravity, a spring will overshoot the rest position and
oscillate. Energy lost to friction within the spring or between the mass and air or a
surface will cause it to gradually decelerate and come to rest. For some combina-
tion of initial conditions and forces the spring could be critically damped so that
it exactly comes to rest without oscillating, but the oscillating behavior is more
typical. This means that a stiffer spring will not necessarily resume its rest length
sooner than a soft one if disturbed—it may merely oscillate with higher frequency,
if the frictional forces are too small.
Oscillators under numerical simulation can increase in energy due to roundoff
and integration errors, making them unstable. This is exacerbated by the fact that
ropes are often simulated as chains of very stiff springs and cloth as networks of
stiff springs. Those yield very large forces and high-frequency oscillations, which
require very accurate integration to handle stably.
Most springs in the real world lose significant energy due to their own mass
and material deformation. It is common to add an explicit damping term to springs
to model that loss. This term has the form of a frictional force because it opposes
velocity. The damping term has the same form as the original force term, but it
applies to velocity instead of position. This means it acts like a “higher order”
spring. This makes the problem of tuning springs for stability easier, but does not
eliminate it. In the case of a numerical integrator with fixed time steps, an overly
large damping constant can actually increase oscillation rather than reducing it
when the integral of the acceleration due to damping exceeds the original velocity.
35.6.4.4 Normal Force
An apple on a table experiences a downward force from gravity and negligible
downward buoyancy from the surrounding air. Yet the apple is at rest relative to
the table, so there must be a force opposing gravity to prevent the apple from
experiencing a net downward force. The source of the force is electrostatic repul-
sion between the molecules of the apple and the table, which keep their surfaces
from interpenetrating. We call this a normal force because its axis is perpendicu-
lar to the interface between the surfaces. In this case, the interface is the horizontal
tabletop, so the force vector experienced by the apple is directed vertically upward
along the table's normal vector. When we consider the case of an object on a tilted
surface (Figure 35.24), the normal force is directed away from the surface, but not
directly opposing gravity, so it can create a net horizontal acceleration.
F Normal
ˆ
F g
Figure 35.24: The normal force
prevents penetration. It is in the
direction of the adjacent sur-
face's normal and has magnitude
dependent on all other forces.
The force is trivial to describe but challenging to compute: A normal force is as
large as it needs to be to prevent interpenetration of (rigid) objects. It has direction
along the normal to the interface and magnitude that is equal to the magnitude of
the net sum of all other forces, projected onto that axis.
 
Search WWH ::




Custom Search