Spring aids are used to provide additional stiffness at the end of bump travel,
preventing metal to metal contact. Commonly they are represented by nonlinear
stiffness depending on displacement; however the main drawback of this approach
is that it does not show any hysteretic behavior, hence they do not produce
realistic force predictions differentiating between loading and unloading and
energy absorbed is not calculated. Although introducing damping as a function of
velocity generates some hysteresis, it does not generate realistic results for
quasi-static and dynamic events; and measured data proves that velocity does not
have a significant influence in the width of the loop. An empiric model can be
build combining nonlinear stiffness and viscous damping, as a function of
velocity, and also adding an additional term accounting for structural damping.
This structural damping, implemented similar to a friction element, is nonlinear
depending on displacement and velocity; and it is capable of representing
realistically the hysteresis present in the component behavior. The three terms
work in parallel, stiffness and structural damping capture the amplitude
dependency of the spring aid; whereas viscous damping represents the strain rate
dependency. Although measured data and previous studies suggest the strain rate
dependency is not critical, viscous damping provides a fine tuning element
generating more realistic results for the range of velocities tested. The
parameters used are obtained from measured Force-Displacement data, considering
quasi-static and different velocity conditions. The process starts using the
quasi-static curve, deriving the stiffness term from the centerline, then
structural damping is obtained based on the hysteresis loop width; and finally
viscous damping is calculated at different velocities to capture the strain rate
dependency.
