W H I T E P A P E R
4
Each proportionality factor (stiffness, K ) is computed based on material properties (Young's
modulus and Poisson's ratio) and geometrical properties (such as cross-sectional area and
moments of inertia).
Figure 5. Forces and moments for the linear elastic model, where F
n
is the normal force, F
τ
is the tangential force, M
ben
is the bending
moment, M
tor
is the torsional moment, s
n
rel
is the normal deformation, s
τ
rel
is the tangential deformation, θ
ben
is the angular bending
deformation, θ
tor
is the angular torsional deformation, K
n
is the normal stiffness, K
τ
is the tangential stiffness, K
ben
is the bending stiffness
and K
tor
is the torsional stiffness.
The linear elastic + viscous damping model
In order to dissipate oscillations, it is possible to add viscous damping forces (normal viscous
damping force, F
n
v
and tangential viscous damping force, F
τ
v
) and moments (bending damping
moment, M
b
υ
en
and torsional viscous damping moment, M
τ
υ
or
) to the elastic ones. For this model,
the damping terms are proportional to the relative linear velocities (normal velocity, v
n
rel
and
tangential velocity v
τ
rel
) and relative angular velocities (bending angular velocity, ω
ben
and
torsional angular velocity ω
tor
).
When using this model, forces and moments are modeled as the sum of two parts: an elastic
one and a damping one, as shown in Figure 6.
Figure 6. Viscous damping forces and moments for the viscous damping model, where F
n
v
is the normal viscous damping force, F
τ
v
is
the tangential viscous damping force, M
b
υ
en
is the bending damping moment, M
τ
υ
or
is the torsional viscous damping moment, v
n
rel
is the
normal velocity, v
τ
rel
is the tangential velocity, ω
ben
is the bending angular velocity and ω
tor
is the torsional angular velocity.
