One Degree of Freedom Spring Elements (TYPE4)

One degree of freedom (DOF) spring elements are defined as a TYPE4 property set. Three variations of the element are possible:

Spring only
Dashpot (damper) only
Spring and dashpot in parallel

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

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

There are three other options defining the type of spring stiffness with the hardening flag:

Linear Stiffness
Nonlinear Stiffness
Nonlinear Elasto-Plastic Stiffness

Likewise, the damping can be either:

Linear
Nonlinear

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.

Time Step

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

For a spring only element:(1)

Δ t = \sqrt{\frac{m}{k}}

$Δ t = \sqrt{\frac{m}{k}}$

For a dashpot only element:(2)

Δ t = \frac{m}{2 c}

$Δ t = \frac{m}{2 c}$

For a parallel spring and dashpot element:(3)

Δ t = \frac{(\sqrt{m k + c^{2}}) - c}{k}

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

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 $f (δ)$ $f (δ)$ .

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 (l - l_{0}) + c \frac{d l}{d t}

$F = k (l - l_{0}) + c \frac{d l}{d t}$

The stability condition is given by Equation 3:(5)

Δ t = \frac{(\sqrt{c^{2} + k m}) - c}{k}

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

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

$F$ , to the displacement of the spring, (

l - l_{0}

$l - l_{0}$ ). It is defined as:(6)

F = f (l - l_{0}) + c \frac{d l}{d t}

$F = f (l - l_{0}) + c \frac{d l}{d t}$

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

Δ t = \frac{(\sqrt{c^{2} + k^{'} m}) - c}{k^{'}}

$Δ t = \frac{(\sqrt{c^{2} + k^{'} m}) - c}{k^{'}}$

Where,(8)

k^{'} = \max [\frac{\partial}{\partial (l - l_{0})} f (l - l_{0})]

$k^{'} = \max [\frac{\partial}{\partial (l - l_{0})} f (l - l_{0})]$

Nonlinear Elasto-plastic Spring: Isotropic Hardening

H=1

The hardening flag must be set to 1 in this case and

f (l - l_{0})

$f (l - l_{0})$ is defined by a function. Hardening is isotropic if compression behavior is identical to tensile behavior:(9)

F = f (l - l_{0}) + C \frac{d l}{d t}

$F = f (l - l_{0}) + C \frac{d l}{d t}$

Nonlinear Elasto-plastic Spring: Decoupled Hardening

H=2

The hardening flag is set to 2 in this case and f

f (l - l_{0})

$f (l - l_{0})$ is defined by a function. The hardening is decoupled for compression and tensile behavior:(10)

F = f (l - l_{0}) + C \frac{d l}{d t}

$F = f (l - l_{0}) + C \frac{d l}{d t}$

Nonlinear Elasto-plastic Spring: Kinematic Hardening

H=4

The hardening flag is set to 4 in this case and

f_{1} (l - l_{0})

$f_{1} (l - l_{0})$ and

f_{2} (l - l_{0})

$f_{2} (l - l_{0})$ (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 (l - l_{0}) + C \frac{d l}{d t}

$F = f (l - l_{0}) + C \frac{d l}{d t}$

Nonlinear Elasto-plastic Spring: Nonlinear Unloading

H=5

The hardening flag is set to 5 in this case and

f ​ (δ)

$f (δ)$ and

f 2 (δ_{\max})

$f 2 (δ_{\max})$ (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 (l - l_{0}) + C \frac{d l}{d t}

$F = f (l - l_{0}) + C \frac{d l}{d t}$

With

δ = l - l_{0}

$δ = l - l_{0}$ .

Nonlinear Dashpot

The input properties for a nonlinear dashpot are very close to that of a spring. The required values are:

Mass, $M$ $M$ .
A function defining the change in force with respect to the spring displacement. This must be equal to unity:
$f (l - l_{0}) = 1$ $f (l - l_{0}) = 1$
A function defining the change in force with spring displacement rate,
$g (d l / d t)$ $g (d l / d t)$
The hardening flag in the input must be set to zero.

The relationship between force and spring displacement and displacement rate is:(13)

F = f (l - l_{0}) g (\frac{d l}{d t}) = g (\frac{d l}{d t})

$F = f (l - l_{0}) g (\frac{d l}{d t}) = g (\frac{d l}{d t})$

A nonlinear dashpot property is shown in Figure 11.

The stability condition for a nonlinear dashpot is given by:(14)

Δ t = \sqrt{\frac{M}{C^{'}}}

$Δ t = \sqrt{\frac{M}{C^{'}}}$

Where,(15)

C^{'} = \max [\frac{\partial}{\partial (d l / d t)} g (d l / d t)]

$C^{'} = \max [\frac{\partial}{\partial (d l / d t)} g (d l / d t)]$

Nonlinear Viscoelastic Spring

The input properties for a nonlinear viscoelastic spring are:

Mass, $M$ $M$
Equivalent viscous damping coefficient $C$ $C$
A function defining the change in force with spring displacement
$f (l - l_{0})$ $f (l - l_{0})$
A function defining the change in force with spring displacement rate
$g (d l / d t)$ $g (d l / d t)$

The hardening flag in the input must be set to equal zero. The force relationship is given by:(16)

F = f (l - l_{0}) g (\frac{d l}{d t})

$F = f (l - l_{0}) g (\frac{d l}{d t})$

Graphs of this relationship for various values of

g (d l / d t)

$g (d l / d t)$ are shown in Figure 12.

The stability condition is given by:(17)

Δ t = \frac{(\sqrt{{C^{'}}^{2} + k^{'} M}) - C^{'}}{k^{'}}

$Δ t = \frac{(\sqrt{{C^{'}}^{2} + k^{'} M}) - C^{'}}{k^{'}}$

Where,(18)

K^{'} = \max [\frac{\partial}{\partial (l - l_{0})} f (l - l_{0})]

$K^{'} = \max [\frac{\partial}{\partial (l - l_{0})} f (l - l_{0})]$

(19)

C^{'} = \max [\frac{\partial}{\partial (d l / d t)} g (d l / d t)]

$C^{'} = \max [\frac{\partial}{\partial (d l / d t)} g (d l / d t)]$