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.


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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGRbaaaa@39C7@ and equivalent viscous damping coefficient c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGRbaaaa@39C7@ are required. The mass m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGRbaaaa@39C7@ 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.


Figure 4. Colinear Forces

Time Step

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

For a spring only element:(1)
Δt= m k
For a dashpot only element:(2)
Δt= m 2c
For a parallel spring and dashpot element:(3)
Δt= ( mk+ 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 ) .

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 d l d t


Figure 5. Linear Force-Displacement Curve
The stability condition is given by Equation 3:(5)
Δt= ( c 2 +km )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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@ , to the displacement of the spring, ( l l 0 ). It is defined as:(6)
F = f ( l l 0 ) + c d l d t


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)
Δt= ( c 2 + k m )c k
Where,(8)
k = max [ ( 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 ) is defined by a function. Hardening is isotropic if compression behavior is identical to tensile behavior:(9)
F = f ( l l 0 ) + C d l d t


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 ( l l 0 ) is defined by a function. The hardening is decoupled for compression and tensile behavior:(10)
F = f ( l l 0 ) + C d l d t


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 ( l l 0 ) and 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 d l d t


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 ( δ ) and f 2 ( δ max ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWHMbGaaGOmamaabmaabaGaeqiTdq2aaSbaaSqaaiGac2gacaGG HbGaaiiEaaqabaaakiaawIcacaGLPaaaaaa@40BA@ (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 d l d t
With δ = l l 0 .


Figure 10. Nonlinear Unloading Force-Displacement Curve

Nonlinear Dashpot

The input properties for a nonlinear dashpot are very close to that of a spring. The required values are:
  • Mass, M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C8@ .
  • A function defining the change in force with respect to the spring displacement. This must be equal to unity:

    f ( l l 0 ) = 1

  • A function defining the change in force with spring displacement rate,

    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 ( d l d t ) = g ( d l d t )
A nonlinear dashpot property is shown in Figure 11.


Figure 11. Nonlinear Dashpot Force Curve
The stability condition for a nonlinear dashpot is given by:(14)
Δt= M C
Where,(15)
C = max [ ( d l / d t ) g ( d l / d t ) ]

Nonlinear Viscoelastic Spring

The input properties for a nonlinear viscoelastic spring are:
  • Mass, M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C8@
  • Equivalent viscous damping coefficient C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C8@
  • A function defining the change in force with spring displacement

    f ( l l 0 )

  • A function defining the change in force with spring displacement rate

    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 ( d l d t )
Graphs of this relationship for various values of g ( d l / d t ) are shown in Figure 12.


Figure 12. Visco-Elastic Spring Force-Displacement Curves
The stability condition is given by:(17)
Δ t = ( C 2 + k M ) C k
Where,(18)
K = max [ ( l l 0 ) f ( l l 0 ) ]
(19)
C = max [ ( d l / d t ) g ( d l / d t ) ]