Force and Moment

Linear Stiffness and Damping Behavior

The simplest formulation is a linear elastic spring stiffness, where the internal force is proportional to the relative displacement. In this case, only the constant stiffness parameter $K_{i}$ and optional damping parameter $C_{i}$ are entered.

For linear stiffness, the force and moment are:

$F (δ) = K_{i} δ^{i}$
$M (θ) = K_{i} θ^{i}$

For linear dashpot, the force and moment are:

$F (δ) = C_{i} {\dot{δ}}^{i}$
$M (θ) = C_{i} {\dot{θ}}^{i}$

For linear stiffness and dashpot, the force and moment are:

$F (δ) = K_{i} δ^{i} + C_{i} {\dot{δ}}^{i}$
$M (θ) = K_{i} θ^{i} + C_{i} {\dot{θ}}^{i}$

Nonlinear Behavior

The force and moment in a spring is computed as:(1)

\begin{array}{l} F_{i} (δ^{i}) = f (\frac{δ^{i}}{A s c a l e_{i}}) [A_{i} + B_{i} \ln (\max (1, | \frac{{\dot{δ}}^{i}}{D_{i}} |)) + E_{i} g (\frac{{\dot{δ}}^{i}}{F_{i}})] \\ + C_{i} {\dot{δ}}^{i} + H s c a l e_{i} h (\frac{{\dot{δ}}^{i}}{F_{i}}) \end{array}

Where, $i$ is the translational degrees of freedom: 1,2,3

(2)

\begin{array}{l} M_{i} (θ^{i}) = f (\frac{θ^{i}}{A s c a l e_{i}}) [A_{i} + B_{i} \ln (\max (1, | \frac{{\dot{θ}}^{i}}{D_{i}} |)) + E_{i} g (\frac{{\dot{θ}}^{i}}{F_{i}})] \\ + ​ C_{i} {\dot{θ}}^{i} + H s c a l e_{i} h (\frac{{\dot{θ}}^{i}}{F_{i}}) \end{array}

Where, $i$ is the rotational degrees of freedom: 4,5,6

The variables in the force and moment equation represent:

$f (\frac{δ^{i}}{A s c a l e_{i}})$ Spring force versus displacement function input as fct_ID_1i

$f (\frac{θ^{i}}{A s c a l e_{i}})$ Spring force versus rotation function input as fct_ID_1i.

$A_{i}, B_{i}, D_{i}, E_{i} and F_{i}$ Scaling coefficients

\ln (\max (1, | \frac{{\dot{δ}}^{i}}{D_{i}} |))

Logarithmic function that scales the spring stiffness as the velocity increases

$g (\frac{{\dot{δ}}^{i}}{F_{i}})$ Scale the stiffness as a function of linear input as fct_ID_2i

$g (\frac{{\dot{θ}}^{i}}{F_{i}})$ Scale the moment as a function of rotational velocity input as fct_ID_2i

This input can be used to model nonlinear strain rate effects of the spring stiffness.

$C_{i}$ Linear damping coefficient used to increase the spring stiffness as a function of velocity

$h (\frac{{\dot{δ}}^{i}}{F_{i}})$ or $h (\frac{{\dot{θ}}^{i}}{F_{i}})$ Nonlinear damping function input as fct_ID_4i

Linear or nonlinear damping as a function of velocity can also be added to the spring force using either a linear damping coefficient or a user-defined function.

The functions $g$ and $h$ both describe the damping behavior of the spring. However, the $g$ function scales the spring stiffness function $f$ , but the $h$ function adds to the spring stiffness function $f$ .