Stiffness Formulation

The stiffness of the spring can be defined multiple ways with each degree of freedom defined differently.

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 $f$ .

Time Step

Time step for spring:(3)

Δ t = \sqrt{\frac{M}{K}}

Time step for dashpot:(4)

Δ t = \frac{M}{2 C}

Time step for spring and dashpot:(5)

Δ t = \frac{\sqrt{C^{2} + K \cdot M} - C}{K}

Where,

$M$: Mass of the spring
$K$: Linear stiffness or $\max [\frac{d F}{d δ}]$ for a nonlinear spring
$C$: Linear damping or $\max [\frac{\partial g (\frac{d δ}{d t})}{\partial (\frac{d δ}{d t})}]$ for nonlinear damping