End Stop


Symbol:
Identifier: Mechanics.Translation.EndStop
Version: 4.1
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: End Stop Distance 1 l1 end stop 1; end stop in contact for dx>=l1
End Stop Distance 2 l2 end stop 2; end stop in contact for dx<=-l2
Model mode
  • "Rigid End Stop"
  • "Elastic End Stop"
Kind of Impact kind plastic, elastic, or with restitution coefficient;
for rigid end stops only (mode=="Rigid End Stop")
Coefficient of Restitution (0<=ci<=1) ci only visible if mode=="Rigid End Stop" and kind=="by Coefficient"
Stiffness Stop 1 k1 only visible if mode=="Elastic End Stop"
Stiffness Stop 2 k2 only visible if mode=="Elastic End Stop"
Damping Stop 1 b1 only visible if mode=="Elastic End Stop"
Damping Stop 2 b2 only visible if mode=="Elastic End Stop"
Friction: Consideration of Friction friction
Rigid Friction with kindF only visible if friction==true
Force of Static Friction Fst only visible if friction==true
Force of Sliding Friction Fsl only visible if friction==true and kindF=="Continuous Transition"
Limit Velocity Difference dvL only visible if friction==true and kindF=="Continuous Transition"
Advanced: Advanced Settings adv only visible if mode=="Rigid End Stop" or friction=="true"
Minimum Separation Force eps_Fi only enabled if adv==true and mode=="Rigid End Stop"
Maximum Velocity after impact for permanent contact eps_dv only enabled if adv==true and mode=="Rigid End Stop"
Relative Minimum Difference of the Friction Forces dFfrel only enabled if adv==true and kindF=="Stick-Slip"
Minimum Separation Velocity Difference eps_dvsl only enabled if adv==true and kindF=="Stick-Slip"
Minimum Separation Momentum eps_dI only enabled if adv==true and mode=="Rigid End Stop"or kindF=="Stick-Slip"
Transition Shape of Force of Sliding Friction kindTrans
  • "Gevrey Approach (based on tanh-Function)"
  • "tanh Approach"

only visible if friction==true
only visible if kindF=="Continuous Transition"
only enabled if adv=="true"
Shape Coefficient (Transition Shape) eps_sigma only visible if kindTrans=="Gevrey Approach (based on tanh-Function)"
Shape Coefficient (Transition Shape) eps_tau only visible if kindTrans=="tanh Approach"
Results: Internal Force Fi
End Stop Force Fstop
Spring Force in End Stop Fs only visible if mode=="Elastic End Stop"
Damping Force in End Stop Fd only visible if mode=="Elastic End Stop"
Friction Force Ff only visible if friction==true
Displacement Difference dx
Velocity Difference dv
Stop State stop
State of Friction sf only visible if friction==true
Power Loss Pl only visible if friction==true
  • Elastischer Anschlag inkl. nichtlinearer Kontaktsteifigkeit und Dämpfung sowie veränderlichem Spiel
  • Alternativ: Starres Modell nach den Stoßgesetzen (Plastischer od. Elastischer Stoß, Vorgabe des Stoßfaktors)
  • Berechnung der Verlustleistung bzw. Energie beim Stoß
  • Optional: Reibung im Spiel
  • Elastic end stop model incl. nonlinear contact stiffness and damping as well as variable backlash
  • Alternative: Rigid model according to the impact laws (plastic or elastic impact, specification of the coefficient of restitution)
  • Calculation of power dissipation or energy at impact
  • Option: friction in backlash

Description:

This model implements a linear mechanical end stop. The end stop can be modeled either as a Rigid End Stop or as an Elastic End Stop.

If the model Rigid End Stop is selected, the contact with the stops is modeled as an ideal impact where the principle of momentum conservation applies. If one end stop is in contact and "pressed together" by the internal force, then the element behaves like a rigid connection. In case there is no contact, the end stop acts as an open connection without transmission of force.

With the consideration of momentum conservation, the transition between these two states appears at a single time instant. A detailed description of the states and the transitions between them is given in the Results section. The rigid end stop causes force peaks (impulses). The computation considers these correctly, but they do not appear in the results due to their infinite height. As a consequence of the state change and due to the instantaneous change in kinetic energy, the connections and masses connected with the end stop display steps in their velocities.

In the Elastic End Stop model, the contact in the end stops is implemented through spring-damper elements. They can be parameterized for each of the stops separately. After reaching the end point and as long as the inner force is directed towards the end stop, the contact behaves like a spring-damper. The connection is opened when the internal force changes its sign and the end stop does not transmit any drag forces. The detailed description of the states and the transitions among them is given in the Results section.

If correctly parameterized, the Elastic End Stop model displays a realistic behavior. However, as the correct parametrization of the spring-damper behavior can be challenging, the idealized Rigid End Stop model with its straightforward parametrization via a restitution coefficient is easier to handle.

The end stop model also provides a friction sub-model. If the element exhibits backlash, rigid friction can be taken into account. This friction acts in parallel direction to the end stop.

Parameters:

  • The fundamental parameters for the geometric description of the end stop are the distances of the stops to the neutral position of the element. The neutral position is defined as the point where the Displacement Difference dx = ctr1.x-ctr2.x equals zero. These distances are given with the parameters Stop 1 (l1) and Stop 2 (l2). For positive values of l1 and l2, the stop is free in the neutral position. Negative values for l1 and l2 indicate that at least one of the stops is in contact in neutral position. In general, the slider of the end stop is free if the condition is fulfilled.

    Figure 1: Definition of l1 and l2 for the End Stop model
  • The model that is used in the simulation is specified through an option for the enumeration mode; either "Rigid End Stop" or "Elastic End Stop".
  • If an "Elastic End Stop" is used (default), the Stiffnesses (k1 and k2) as well as the Damping (b1 and b2) of the two contacts must be given. Since the damping is hard to obtain experimentally, selecting the appropriate damping is usually not straightforward. In this case, it is useful to refer to typical approximations which are described in the help section of the Spring-Damper Backlash element type.
  • If a "Rigid End Stop" is used, the Kind of Impact influences the behavior of the end stop. During an impact, the momentum is conserved, i.e. the masses and speeds before and after the impact must fulfill:
  • Depending on the Kind of Impact which is described by the Coefficient of Restitution ci, the speeds after the impact are:
  • For an elastic impact (Kind of Impact is set to "elastic"), ci equals 1. For a plastic impact (Kind of Impact is set to "plastic"), ci is 0. These are set for the chosen Kind of Impact automatically. If the Kind of Impact is set to "by Coefficient", the user can adjust the Coefficient of Restitution as they see fit.

Friction:

The Stop model can also take rigid friction in parallel direction to the end stop into account. The friction acts when the stop exhibits backlash. The friction models can be activated with the Boolean friction (Consideration of Friction).
There are two rigid friction models which can be selected with the enumeration kindF, "Rigid Friction" with:

  • Continuous Transition and
  • Stick-Slip:

Both friction models are also available to the Rigid Friction element. For a detailed description of these models and their parameters, see the help chapter of the Rigid Friction model.

Advanced Parameters:

Advanced setting for rigid end stop models

In rigid end stop models, there are further parameters which are needed for the tolerance limits of the zero point. In previous releases, they used to use internal default values.

  • eps_Fi: This parameter sets the tolerance for the internal force Fi in the case that the stop element changes its state from stop to loose (separation). Figure 3 shows two diagrams for this tolerance:

    Figure 3: Meaning of the eps_Fi tolerance for the change of state from stop to loose. The state stop sets the two constraints dv==0 and da==0 (conditions).
    For stop==2: If the element is supposed to move away from the stop, a force is needed and it has to be greater than eps_Fi, the condition is then Fi>=eps_Fi.
    For stop==1: If the element is supposed to move away from the stop, a force is needed and it has to be smaller then -eps_Fi, the condition is then Fi<=-eps_Fi.
    If eps_Fi was 0, then da would be 0 when the state is changed from stop to loose, but this conflicts with the stop conditions. The end stop model needs this limit to avoid frequent discontinuities and cycles in the event iteration.
  • eps_dv: This parameter defines the tolerance around zero so that a mass remains in the stop state. Figure 4 shows two diagrams for the jumping ball example (discover the belonging sample model: BallRigid.isx).

    Figure 4: Displacement and velocity differences of a bouncing mass over time
    In the case that eps_dv=0, the mass would bounce indefinitely. The smaller eps_dv is, the smaller the calculation step size.
  • eps_dI:
    Similarly to the Rigid Friction model: End stops and friction points can be combined without any restrictions. Reaching static friction or hitting an end stop causes changes in momentum which is transferred to adjacent end stops and friction points causing them to lose contact if the minimum impulse eps_dI is exceeded. Smaller changes may be related to numerical errors, which is why the minimum impulse eps_dI must be specified to distinguish this variance from an actual impulse. Usually, the default value is sufficient for most cases.
  • If Rigid Friction with stick-slip is taken into account, advanced parameters are available. These parameters are also described in the help of the Rigid Friction model.

Advanced settings for friction models with continuous transition

The enumeration kindTrans provides two continuous transition models which can be selected if adv is set to true.

  • "Gevrey Approach (based on tanh-Function)": This approach and the meaning of the corresponding shape parameter eps_sigma are described in chapter on the Continuous transition between two signals by Gevrey approach.
    The advantage of this approach is that the output becomes a real 0 before the transition and real 1 after the transition (see figure in the linked chapter).
    One drawback is however that there are small discontinuities at the transition's start and end. If these discontinuities cause any problems (in rare occasions), please try using the next transition approach ...
  • "tanh Approach": This approach and the meaning of the corresponding shape parameter eps_tau are described in the chapter on the Continuous transition between two signals by a hyperbolic tangent approach.
    The advantage of this approach is that there are no discontinuities (see figure in the linked chapter).
    The downside is that the output lies only in proximity of 0 or 1.

Results:

  • The internal force Fi, the displacement difference dx and the velocity difference dv between the connectors of the element as well as the state stop of the end stop are provided as result quantities.
    The internal force Fi is always the sum of the end stop force and the friction force: Fi = Fstop + Ff).
    If the end stop has no contact on either side and friction is not considered, Fi is zero.
    In the case of contact and no friction, Fi results from the spring-damper force for the "Elastic End Stop" model. For a "Rigid End Stop" model, the internal force is equal to the force occurring with a rigid connection at the two connectors ctr1 and ctr2 and which is determined by the system surrounding the element. Depending on the contact side, the internal force is always greater than zero (end stop on side 1) or smaller than zero (end stop on side 2).
  • The end stop force Fstop is the part of the internal force which only refers to the end stop. In case of an "Elastic End Stop" model, the model provides the spring force (Fs) as well as the damping force (Fs) separately. The rule is always Fstop = Fs + Fd
  • The friction force Ff is the part of the internal force which only refers to the friction model. It is only visible if the element takes friction into account.
  • The displacement difference dx is computed from the displacements at the connectors as follows: dx = ctr1.x - ctr2.x.
  • The velocity difference dv is obtained with: dv = ctr1.v - ctr2.v.
  • The result quantity state (stop) describes the current state of the end stop and thus explains the valid situation and relationships for the model.
  • The dissipated power is only computed for the friction model (Pl = Ff*dv). So it is only visible if the element take friction into account.

For the "Elastic End Stop" model, there are the following states:

State and value for stop: Behavior: State Transitions:
free (stop = 0) The element resides between the end stops. The internal force Fi is zero and there is no connection between ctr1 and ctr2. If one of the stops is reached, the state changes to "At End Stop 1" or "At End Stop 2" accordingly.
at end stop 1 (stop = 1) The element acts as a spring damper with the parameters given for the end stop at side 1. The internal force is greater than zero. The element moves away from the stop as soon as the internal force becomes zero (or less). If the element is still located behind the end stop at side 1, then the state changes to "Detaching from End Stop 1", otherwise to "Free".
at end stop 2 (stop = 2) The element acts as a spring damper with the parameters given for stop 2. The internal force is smaller than zero. The element moves away from the stop as soon as the internal force becomes zero (or greater). If the element is still located behind the end stop at side 2, then the state changes to "Detaching from End Stop 2", otherwise to "Free".
detaching from end stop 1 (stop = 11) The element behaves in the same way as in the state "Free". Geometrically speaking, however, it is still located behind end stop 1, but is already detached, because the internal force has become smaller than zero. If stop 1 has been passed, the state changes to "Free". If the sign of the speed difference changes, however (i.e. the element is driven into the stop again), the state changes to "At End Stop 1".
detaching from end stop 2 (stop = 21) The element behaves in the same way as in the state "Free". Geometrically speaking, however, it is still located behind end stop 2, but is already detached, because the internal force has become greater than zero. If stop 2 has been passed, the state changes to "Free". If the sign of the speed difference changes, however (i.e. the element is driven into the stop again), the state changes to "At End Stop 2".
both sides at stop (stop = 3) The end stop is parameterized in such a way that l1 + l2 <0 . In this case, both stops are in contact. The two spring dampers act with the respective pre-load. As soon as l1 + l2 becomes greater than zero again, the state changes to one of the above based on the circumstances.

For the Rigid End Stop Model, there are the following states:

State and value for stop: Behavior: State Transitions:
free (stop = 0) The element resides between the end stops. The internal force Fi is zero, and there is no connection between ctr1 and ctr2. If the stops are reached, an impact occurs based on the restitution coefficient. If the absolute velocity difference after the impact is smaller or equal eps_dv, the state changes to "end stop 1" or "end stop 2" respectively, and the velocity difference becomes zero. If the absolute velocity difference is greater than eps_dv, the state remains "free".
at end stop 1 (stop = 1) The element acts as a rigid connection as long as the internal force is greater than -eps_Fi. The element moves away from the stop as soon as the internal force becomes -eps_Fi (or less). The state changes to "Free".
at end stop 2 (stop = 2) The element acts as a rigid connection as long as the internal force is smaller than eps_Fi. The element moves away from the stop as soon as the internal force becomes eps_Fi (or greater). The state changes to "Free".
both sides at stop (stop = 3) The end stop is parameterized in such a way that l1 + l2 <0. In this case, both stops are in contact. The element behaves like a rigid connection which can transmit positive as well as negative internal forces. As soon as l1 + l2 becomes greater than zero again, the state changes to one of the above based on the circumstances.

For the Friction Model (if considered), the state variable sf is defined in the same way as in the Rigid Friction model (see Table 1 in chapter Rigid Friction of the Linear Mechanics library).

  • The elastic end stop moves if the internal force changes its sign (see above). In case of damping, the internal force can become zero before the actual end stop point has been passed again in reserse direction. This situation can be interpreted as follows: When getting into contact, the end stop is elastically deformed. This deformation is reversed less quickly than the two ends of the element move away from each other and thus the stop is detached, before the end stop point is reached. If you want to model a "sticky" end stop which can also transmit certain drag forces (e.g. something is dragged out of a liquid), it is recommended you use the Spring-Damper-Backlash element type. This element is detached only after the end stop point has been passed, i.e. depending on the position instead of force.
  • If an end stop element is parameterized in such a way that it is "end stop 1" or "end stop 2" at the beginning of a simulation, the elastic end stop produces internal forces due to its elastic deformation. These forces help detach the contact. In contrast to that, the rigid stop does not develop internal forces as a consequence of its deformation. In order for the contact to be detached, external forces must be present.