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344 CAM DESIGN HANDBOOK

d
motion

h

FIGURE 11.19. Sliding plate with viscous friction.

b
T = w (11.62)
r
as illustrated in the example later in this section. In this case, the units of the damping
coefficient, b r, take the form of torque/angular velocity, such as ft-lbf/(rad/s) or Nm/(rad/s).
More general damping laws can also be used. For instance, a power-law dependence upon
position may be added, giving a damping force of
n ˙
F = bdd (11.63)
as is commonly done for simulating damping due to impact (Hunt and Crossley, 1975;
Herbert and McWhannell, 1977).
The dissipated power, or the rate at which a linear viscous damper of the form in Eq.
11.60 dissipates energy, is
˙
d
P viscous = F = bd ˙ 2 . (11.64)
From this expression, it is clear that power is dissipated regardless of the direction of
motion since the damping force always acts to oppose the relative motion of the
two moving parts in the damper. No power is dissipated when the system is at rest. Fur-
thermore, the power dissipated depends only upon the velocity of the system and grows
as the square of the velocity. Damping factors are the most difficult to pin down in
analysis. The choice of the value of a damping factor of 0.03 for structural damping and
0.04 to 0.08 for well-lubricated machine (closed lubrication) components is considered
reasonable. For open lubrication systems the damping factors are higher depending on the
contaminants.

11.5.2 Coulomb or Dry Friction
In contrast to viscous friction, where the damping force depends upon the relative
velocity between the two elements, friction between dry surfaces depends only upon
the direction of the velocity and not its magnitude. The amplitude of the damping
force is determined by the normal force at the interface and the coefficient of friction
between the surfaces. For the system in Fig. 11.20, therefore, the damping force can be
represented by

F = m sgn () d ˙ (11.65)
F
n
where sgn(x) denotes the sign of x (positive or negative) since the force always acts to
oppose motion. The power dissipated by this model is
˙˙
F = m
P coulomb = d ˙ F sgn () .d d (11.66)
n
As with the viscous friction model, the power dissipated depends upon the velocity. Unlike
the viscous friction model, this dependence is linear, not quadratic. The power also scales