One Degree of Freedom Spring Elements (TYPE4)

These three configurations are shown in Figure 1 to Figure 3.

Figure 1. Spring Only

Figure 2. Dashpot Only

Figure 3. Spring and Dashpot in Parallel

No material data card is required for spring elements. However, the stiffness $k$ and equivalent viscous damping coefficient $c$ are required. The mass $m$ is required if there is any spring translation.

A spring may also have zero length. However, a one DOF spring must have 2 nodes.

The forces applied on the nodes of a one DOF spring are always colinear with direction through both nodes; refer to Figure 4.

Figure 4. Colinear Forces

Time Step

The time of a spring element depends on the values of stiffness, damping and mass.

The critical time step ensures that the stability of the explicit time integration is maintained, but it does not ensure high accuracy of spring vibration behavior. Only two time steps are required during one vibration period of a free spring to keep stability. However, if true sinusoidal reproduction is desired, the time step should be reduced by a factor of at least 5.

If the spring is used to connect the two parts, the spring vibration period increases and the default spring time step ensures stability and accuracy.

Linear Spring

Function number defining $f\text{}\left(\delta \right)$ .

N1=0

The general linear spring is defined by constant mass, stiffness and damping. These are all required in the property type definition. The relationship between force and spring displacement is given by:(4)

$$F=k\left(l-l_0\right)+c\frac{dl}{dt}$$

Figure 5. Linear Force-Displacement Curve

Nonlinear Elastic Spring

Hardening flag

H=0

The hardening flag must be set to 0 for a nonlinear elastic spring. The only difference between linear and nonlinear elastic spring elements is the stiffness definition. The mass and damping are defined as constant. However, a function must be defined that relates the force, $F$ , to the displacement of the spring, ( $l-l_0$ ). It is defined as:(6)

$$F=f\left(l-l_0\right)+c\frac{dl}{dt}$$

Figure 6. Nonlinear Elastic Force-Displacement Curve

The stability criterion is the same as for the linear spring, but rather than being constant, the stiffness is displacement dependent:(7)

$$\text{Δ}t=\frac{\left(\sqrt{c^2+k^{\prime }m}\right)-c}{k^{\prime }}$$

Where,(8)

$$k^{\prime }=\max \left[\frac{\partial }{\partial \left(l-l_0\right)}f\left(l-l_0\right)\right]$$

Nonlinear Elasto-plastic Spring: Isotropic Hardening

H=1

The hardening flag must be set to 1 in this case and $f\left(l-l_0\right)$ is defined by a function. Hardening is isotropic if compression behavior is identical to tensile behavior:(9)

$$F=f\left(l-l_0\right)+C\frac{dl}{dt}$$

Figure 7. Isotropic Hardening Force-Displacement Curve

Nonlinear Elasto-plastic Spring: Decoupled Hardening

H=2

The hardening flag is set to 2 in this case and f $f\left(l-l_0\right)$ is defined by a function. The hardening is decoupled for compression and tensile behavior:(10)

$$F=f\left(l-l_0\right)+C\frac{dl}{dt}$$

Figure 8. Decoupled Hardening Force-Displacement Curve

Nonlinear Elasto-plastic Spring: Kinematic Hardening

H=4

The hardening flag is set to 4 in this case and $f_1\left(l-l_0\right)$ and $f_2\left(l-l_0\right)$ (respectively maximum and minimum yield force) are defined by a function. The hardening is kinematic if maximum and minimum yield curves are identical:(11)

$$F=f\left(l-l_0\right)+C\frac{dl}{dt}$$

Figure 9. Kinematic Hardening Force-Displacement Curve

Nonlinear Elasto-plastic Spring: Nonlinear Unloading

H=5

The hardening flag is set to 5 in this case and $f\text{}\left(\delta \right)$ and $f2\left({\delta }_{\max }\right)$ (maximum yield force and residual deformation, respectively) are defined by a function. Uncoupled hardening in compression and tensile behavior with nonlinear unloading:(12)

$$F=f\left(l-l_0\right)+C\frac{dl}{dt}$$

With $\delta =l-l_0$ .

Figure 10. Nonlinear Unloading Force-Displacement Curve

Nonlinear Dashpot

A nonlinear dashpot property is shown in Figure 11.

Figure 11. Nonlinear Dashpot Force Curve

Nonlinear Viscoelastic Spring

Graphs of this relationship for various values of $g\left(dl/dt\right)$ are shown in Figure 12.

Figure 12. Visco-Elastic Spring Force-Displacement Curves