Spring-Damper-Backlash


Symbol:
Identifier: Mechanics.Translation.SpringDamper
Version: 3.7
File: Mechanics.cat
Connectors: Mechanical Connector (linear) ctr1 connectable to elements of the Linear Mechanics library
Mechanical Connector (linear) ctr2 connectable to elements of the Linear Mechanics library
Parameters: Properties kind
  • "Spring"
  • "Damper"
  • "Spring-Damper"
  • "Spring-Backlash"
  • "Damper-Backlash"
  • "Spring-Damper-Backlash"
Stiffness of Contact k only visible, if kind=="Spring" or "Spring-Damper" or "Spring-Backlash" or "Spring-Damper-Backlash"
Damping of Contact b only visible, if kind=="Damper" or "Spring-Damper" or "Damper-Backlash" or "Spring-Damper-Backlash"
Stiffness of Backlash kL only visible, if kind=="Spring-Backlash" or "Spring-Damper-Backlash"
Damping of Backlash bL only visible, if kind=="Damper-Backlash" or "Spring-Damper-Backlash"
Backlash L only visible, if kind=="Spring-Backlash" or "Damper-Backlash" or "Spring-Damper-Backlash"
Rigid rigid if true, the element behaves as rigid
Selection of the rigid model kindR if true, the element behaves as rigid
Coefficient of Restitution ci if true, the element behaves as rigid
Steady State Analysis: Damping kindDamping kind of damping in Steady-State Simulation and Transient Simulation
Relative Damping psi damping losses per period divided by maximal elastic deformation energy (nonlinear approximation if enabled for transient simulation)
Lehr's Damping Factor D Lehr's damping factor (nonlinear approximation if enabled for transient simulation)
Advanced: Initial Preload of Spring kindPS
  • "No Preload or Preload from Environment"
  • "Preload Displacement Difference"
  • "Preload Force"
Initial Displacement Difference dx0 only visible if kindPS=="Preload Displacement Difference"
Initial Spring Force Fs0 only visible if kindPS=="Preload Force"
Initial Preload of Damper kindPD
  • "No Preload or Preload from Environment"
  • "Preload Velocity Difference"
  • "Preload Force"
Initial Velocity Difference dv0 only visible if kindPD=="Preload Velocity Difference"
Initial Damping Force Fd0 only visible if kindPD=="Preload Force"
Advanced Settings adv only visible, if rigid=="true" and kindR<>"without Backlash"
Minimum Separation Force eps_Fi only enabled, if rigid=="true" and kindR<>"without Backlash"
Maximum Velocity after impact for permanent contact eps_dv only enabled, if rigid=="true" and kindR<>"without Backlash"
Results: Internal Force Fi
Spring Force Fs
Damper Force Fd
Displacement Difference dx
Velocity Difference dv
Change of Potential Energy Pp not visible, if rigid==true
Power Loss Pl not visible, if rigid==true
External Power Pe only visible, if rigid==true and kindR==Without Backlash
State stop only visible, if rigid==true and kindR <> Without Backlash
  • Steifigkeit, Dämpfung und Spiel zwischen zwei Körpern (Massen) oder gegen Umgebung
  • Verwendbar als Feder, Dämpfer, Feder-Dämpfer, Feder-Spiel, Dämpfer-Spiel, oder Feder-Dämpfer-Spiel
  • Lineares oder nicht-lineares Verhaltens, alle Parameter (auch Spiele) dürfen auch zeitabhängig sein
  • Unterstützung spezieller Dämpfungsberechnungen für Analysen im Frequenzbereich, Approximation von Frequenzbereichsparametern für den Zeitbereich
  • Optional starr schaltbar, zur einfachen Reduktion von Freiheitsgraden
  • Stiffness, damping and backlash between two bodies (masses) or against the environment
  • Can be used as Spring, Damper, Spring-Damper, Spring-Backlash, Damper-Backlash, or Spring-Damper-Backlash
  • Linear or non-linear behavior, all parameters (including backlash) may also be time-dependent
  • Support of special damping calculations for frequency domain analyses, approximation of frequency domain parameters for time domain
  • Option: Switch to rigid behavior, for an easy reduction of degrees of freedom

Description:

This model implements elastic and/or damping behavior which may also contain a backlash. It is applied between two linear mechanics components. If the backlash is activated then the stiffness and damping can be independently parameterized on a user-defined interval centered on the origin of the displacement difference. The springs are always acting in parallel with the dampers.

Parameters:

General Parameters

The behavior of the component is defined by the enumeration kind. Depending on the selection the element models:

  • a pure spring,
  • a pure damper,
  • a spring-damper.
  • All of these selections can be combined with a backlash (in series), creating three further models.

The selected variant defines the required parameters for the element. The parameter dialog only shows the parameters required for the current setting. E.g., for a pure spring only the Stiffness of Contactk is required for the parametrization and thus only this parameter is accessible to the user. The maximum amount of parameters is present for a spring-damper-backlash model.

The Stiffness of Contact k and Damping of Contact b define the respective stiffness and damping of the element in cases where the backlash is exceeded or where no backlash is present. The BacklashL is an interval, which is centered on the point of zero displacement difference between the two connectors of the element i.e. the neutral position. Inside the backlash interval the element possesses a Stiffness of BacklashkL and a Damping of BacklashbL, which are different from the previously specified k and b and thus can be entered separately (see the figure below).

Ranges of validity for the parameters in the spring-damper-backlash

Thus a backlash, which occurs during a change from a tension to a compression load, can be modeled. When there is no reaction force inside the backlash interval i.e. a real backlash has to be modeled, then kL and bL must be set to zero.

If the damping of an element is not known, the following approximation settings can be applied:

where

0.005 ... 0.01 for the damping in metallic materials
0.1 ... 0.25 for the damping in highly elastic materials
0.05 ... 0.15 for structural and contact damping

(Source (German): Großmann, Knut: "Ein Beitrag zur Modellfindung und -berechnung für die dynamische Analyse von Werkzeugmaschinen-Strukturen". Technical University of Dresden, Faculty of Mechanical Science and Engineering, Dissertation A, Dresden 1976)

In this approach, is measured in the SI unit .

The element can behave as rigid when the stiffness and damping are deactivated. If rigid is true, the element models a rigid constraint or an end stop if backlash is defined.
The enumeration kindR contains the selection of rigid models:

  • Without Backlash: The element has no backlash and behaves as a Constraint with dx==0
  • Rigid Plastic: The element behaves as a rigid end stop with consideration of a plastic impact. The coefficient of restitution is ci = 0.
  • Rigid Elastic: The element behaves as a rigid end stop with consideration of an elastic impact. The coefficient of restitution is ci = 1.
  • Coefficient of Restitution: The element behaves as a rigid end stop with consideration of a plastic-elastic impact. The coefficient of restitution ci can be defined by the user: 0 <= ci <= 1.

Initial Values:

The group Initial Values contains the enumerations kindPS and kindPD as well as (depending on their settings) the parameters dx0, dv0 and Fs0, Fd0 for the definition of an initial preload of the spring-damper element.

  • The enumeration kindPS holds a selection of initial states of the spring:
    • "No Preload or Preload from Environment": This is the default selection. The spring is initially unloaded or will be pre-loaded by the environmental conditions (i.e. initial values at the connections at ctr1 or ctr2).
    • "Preload Displacement Difference": The spring is pre-loaded. This option activates the parameter dx0, the initial displacement difference.
    • "Preload Force": The spring is pre-loaded. This option activates the parameter Fs0, the pre-load force.
  • dx0 ... initial displacement difference of the spring
    • The value of this parameter must be consistent with the initial values of the connections at ctr1 (-> x10) and ctr2 (-> x20):
      dxSD0 == x10 - x20
      To fulfill this constraint, it is recommend to unfix x10 or x20 (-> the corresponding initial values of the connections).
    • Also the initial speeds must be consistent, because
      • dv0 = v10 - v20 (initial velocity difference)
      • der(dx0) = dv and dv = 0
  • Fs0 ... the initial spring force or pre-load
    • Here, dx0 will be computed internally by dx0 = Fs0/k taking the stiffness k into account.
    • Note: Also here the initial values must be consistent (see above).
  • The enumeration kindPD holds a selection of initial states of the damper:
    • "No Preload or Preload from Environment": This is the default selection. The damper is initially unloaded or will be pre-loaded by the environmental conditions (i.e. initial values at the connections at ctr1 or ctr2).
    • "Preload Velocity Difference": The damper is pre-loaded. This option activates the parameter dv0, the initial velocity difference.
    • "Preload Force": The damper is pre-loaded. This option activates the parameter Fd0, the pre-load force.
  • dv0 ... initial velocity difference of the damper
    • The value of this parameter must be consistent with the initial values of the connections at ctr1 (-> v10) and ctr2 (-> v20):
      dv0 == v10 - v20
      To fulfill this constraint, it is recommend to unfix v10 or v20 (-> the corresponding initial values of the connections).
  • Fd0 ... the initial damping force or pre-load
    • Here, dv0 will be computed internally by dv0 = Fd0/b taking the damping b into account.
    • Note: Also here the initial values of the velocities must be consistent (see above).

  • If a pre-loaded spring is specified (setup of dx0 or Fs0) an initial constraint will be taken into account. Therefore, the initial values of the connected elements at ctr1 and ctr2 must be consistent with this initial constraint (dx0==x10 - x20).
    Only one of the initial displacements and initial velocities of the environmental structures at ctr1 or ctr2 must be fixed (see also Constraint model).
  • If a pre-loaded spring is specified (setup of dv0 or Fd0) an initial constraint will be taken into account. Therefore, the initial values of the connected elements at ctr1and ctr2 must be consistent with this initial constraint (dv0==v10 - v20).
    Only one of the initial velocities of the environmental structures at ctr1 or ctr2 must be fixed (see also Constraint model).

If rigid=="true" and kindR<>"without Backlash" the element models a rigid end stop. In this case some further advanced parameters are offered to the user: eps_Fi and eps_dv, which are tolerance limits, and adv, which enables these tolerance limits. A detailed description can be found in the help of the End Stop element.

Steady State Analysis:

The parameter settings and extra results on tab Steady-State Simulation are documented in the section Special Properties of Model Elements, which is part of the description of the topic Periodic Steady-State Simulation.
If the element acts as damper kind=="Damper", then the switch onlyRelDamp appears. For a detailed description of the meaning of this switch, see chapter Damper, section "Steady State Analysis".

Results:

  • The Internal Force Fi, the Spring Force Fs, and the Damper ForceFd are computed using the following formulas (see also the figure above):


  • The Displacement Difference is the difference between the current displacements of the elements which are linked to the connectors ctr1 and ctr2:
  • The Velocity Difference results from the speeds at the connectors:
  • The Change of Potential Energy (power delivered to the spring) is computed with:
  • The Power Loss (power dissipated in the damper) is:
  • If rigid==true and kindR is set to "Without Backlash", the external power Pe is computed (Constraint behavior):

    If this power is positive, the element acts as a power source, otherwise it consumes power.
    If the element is switched to rigid and a backlash is considered, the element behaves as a rigid end stop. In this case the state variable stop can be analyzed. For a description of stop, see the help of the Stop element.

Remarks:

If, after a complete parameterization, the element is changed to a kind with less parameters e.g., from a Spring-Damper-Backlash to a Spring, the following simulation runs are, of course, performed with a reduced set of parameters. Nevertheless the disabled parameters continue to carry their last values. So, after changing back to the previous kind all hidden parameters are restored to their original values.

By entering a negative backlash a pre-tensioned system can be modeled. In the backlash interval, which becomes the pretension interval here, a twice as high stiffness and damping are obtained. Outside the interval stiffness and damping are as specified, see figure below.

Pre-tensioned spring with k = 1 Nm/mm, b = 1 Ns/mm, L = -1 mm.

If the parameters for stiffness and damping are formulated using conditional expressions, materials which behave differently under expansion and compression can be modeled. For instance, the elastic behavior of ropes, which can only be tensioned, could be specified with the expression "if (dx<0) then 1e6 else 0".
Backlashes, which are non-symmetric with respect to the neutral position of the spring-damper, can be modeled by combining the Spring-Damper-Backlash with a Constraint of type "Displacement Difference", which has the center offset as its parameter.