# 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:

• $\mathrm{F}\left(\delta \right)={K}_{i}{\delta }^{i}$
• $\mathrm{M}\left(\theta \right)={K}_{i}{\theta }^{i}$
For linear dashpot, the force and moment are:

• $\mathrm{F}\left(\delta \right)={C}_{i}{\stackrel{˙}{\delta }}^{i}$
• $\mathrm{M}\left(\theta \right)={C}_{i}{\stackrel{˙}{\theta }}^{i}$
For linear stiffness and dashpot, the force and moment are:

• $\mathrm{F}\left(\delta \right)={K}_{i}{\delta }^{i}+{C}_{i}{\stackrel{˙}{\delta }}^{i}$
• $\mathrm{M}\left(\theta \right)={K}_{i}{\theta }^{i}+{C}_{i}{\stackrel{˙}{\theta }}^{i}$

### Nonlinear Behavior

The force and moment in a spring is computed as:

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

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

The variables in the force and moment equation represent:

$\mathrm{f}\left(\frac{{\delta }^{i}}{Ascal{e}_{i}}\right)$ Spring force versus displacement function input as fct_ID1i

$\mathrm{f}\left(\frac{{\theta }^{i}}{Ascal{e}_{i}}\right)$ Spring force versus rotation function input as fct_ID1i.

Scaling coefficients

$\mathrm{ln}\left(\mathrm{max}\left(1,|\frac{{\stackrel{˙}{\delta }}^{i}}{{D}_{i}}|\right)\right)$ Logarithmic function that scales the spring stiffness as the velocity increases

$\mathrm{g}\left(\frac{{\stackrel{˙}{\delta }}^{i}}{{F}_{i}}\right)$ Scale the stiffness as a function of linear input as fct_ID2i

$\mathrm{g}\left(\frac{{\stackrel{˙}{\theta }}^{i}}{{F}_{i}}\right)$ Scale the moment as a function of rotational velocity input as fct_ID2i

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

$\mathrm{h}\left(\frac{{\stackrel{˙}{\delta }}^{i}}{{F}_{i}}\right)$ or $\mathrm{h}\left(\frac{{\stackrel{˙}{\theta }}^{i}}{{F}_{i}}\right)$ Nonlinear damping function input as fct_ID4i

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 $\mathrm{g}$ and $\mathrm{h}$ both describe the damping behavior of the spring. However, the $\mathrm{g}$ function scales the spring stiffness function $\mathrm{f}$ , but the $\mathrm{h}$ function adds to the spring stiffness function $\mathrm{f}$ .

## Time Step

Time step for spring:
Time step for dashpot:
Time step for spring and dashpot:
Where,
$M$
Mass of the spring
$K$
Linear stiffness or $\mathrm{max}\left[\frac{dF}{d\delta }\right]$ for a nonlinear spring
$C$
Linear damping or $\mathrm{max}\left[\frac{\partial \mathrm{g}\left(\frac{d\delta }{dt}\right)}{\partial \left(\frac{d\delta }{dt}\right)}\right]$ for nonlinear damping

For nonlinear springs, $K$ is used for time step calculation and contact stiffness. If $K$ is not defined in the spring property, the maximum slope of fct_ID1i ( $\mathrm{max}\left[\frac{dF}{d\delta }\right]$ ) will then automatically be used to calculate the timestep. The behavior is the same for damping stiffness $C$ .