Rotational Hard Stop

Double-sided rotational hard stop

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Libraries:
Simscape / Foundation Library / Mechanical / Rotational Elements

Description

The Rotational Hard Stop block represents a double-sided mechanical rotational hard stop that restricts the motion of a body between upper and lower bounds.

The slider connects to port R and the case connects to port C. The block transmits torque from port R to port C when the gap is closed in the positive direction.

The block computes the slider angular position, φ, from the slider angular velocity, ω = ω_R - ω_C.

θ is the slider arc angle and ψ is the slider angular position on the negative side, θ = |φ - ψ|. The block assumes the slider arc angle is 0 and does not model θ and ψ, but the schematic uses these symbols to explain the basic design of the hard stop.

g_p is the initial gap on the positive side, measured from x:

• For the positive gap to be open, g_p must be positive.

• A negative value of g_p means that the slider is penetrating into the upper bound.

• The gap is closed when g_p - φ ≤ 0.

g_n is the initial gap on the negative side, measured from y:

• For the negative gap to be open, g_n must be negative.

• A positive value of g_n means that the slider is penetrating into the lower bound.

• The gap is closed when ψ - g_n ≤ 0.

The block provides several modeling options:

• Three options based on stiffness and damping. These models use similar underlying equations and differ in how stiffness and damping are applied at bounds.

• A modeling option based on the coefficient of restitution. This model is different from the other three because it uses a mode chart to represent the hard stop behavior.

Models Based on Stiffness and Damping

In hard stop models based on stiffness and damping, the impact interaction between the slider and the stops is assumed to be elastic. This assumption means that the block represents the stop as a spring that comes into contact with the slider as the gap is cleared and opposes the slider penetration into the stop with a torque linearly proportional to this penetration. The block models this elasticity with the Contact stiffness at upper bound and Contact stiffness at lower bound parameters. To account for energy dissipation and nonelastic effects, the block uses the Contact damping at upper bound and Contact damping at lower bound parameters.

The basic hard stop model, Full stiffness and damping applied at bounds, damped rebound, uses these equations:

$$T=\{\begin{array}{ll}{K}_p\cdot \left(\phi -g_p\right)+D_p\cdot \omega \hfill & \text{for }\phi \ge g_p\hfill \\ 0\hfill & \text{for }{g}_n<\phi <g_p\hfill \\ K_n\cdot \left(\phi -g_n\right)+D_n\cdot \omega \hfill & \text{for }\phi \le g_n\hfill \end{array}$$

$$\omega =\frac{d\phi }{dt}$$

where:

• T is the interaction torque between the slider and the case.

• g_p is the initial gap between the slider and upper bound.

• g_n is the initial gap between the slider and lower bound.

• φ is the slider angular position.

• K_p is the contact stiffness at the upper bound.

• K_n is the contact stiffness at the lower bound.

• D_p is the damping coefficient at the upper bound.

• D_n is the damping coefficient at the lower bound.

• ω is the slider angular velocity.

• t is time.

In the Full stiffness and damping applied at bounds, undamped rebound hard stop model, the equations contain additional terms, ge(ω,0) and le(ω,0). These terms ensure that the block does not apply damping on the rebound.

$$T=\{\begin{array}{ll}{K}_p\cdot \left(\phi -g_p\right)+D_p\cdot \omega \cdot ge\left(\omega ,0\right)\hfill & \text{for }\phi \ge g_p\hfill \\ 0\hfill & \text{for }{g}_n<\phi <g_p\hfill \\ K_n\cdot \left(\phi -g_n\right)+D_n\cdot \omega \cdot le\left(\omega ,0\right)\hfill & \text{for }\phi \le g_n\hfill \end{array}$$

The relational functions ge (greater than or equal) and le (less than or equal) do not generate zero crossings when the angular velocity changes sign. For more information, see Enabling and Disabling Zero-Crossing Conditions in Simscape Language. However, the solver treats the ge and le functions as nonlinear. Therefore, if simscape.findNonlinearBlocks indicates that the rest of your network is linear or switched linear, use the Full stiffness and damping applied at bounds, damped rebound option to improve performance.

The default hard stop model, Stiffness and damping applied smoothly through transition region, damped rebound, adds two transitional regions to the equations, one at each bound. When the slider travels through a transition region, the block smoothly ramps up the torque from zero to the full value. At the end of the transition region, the block applies the full stiffness and damping. On the rebound, the block smoothly decreases both stiffness and damping torques to zero. These equations also use the ge and le relational functions, which do not produce zero crossings.

Model Based on Coefficient of Restitution

Unlike the models based on stiffness and damping, this model does not allow penetration of slider into the hard stops. The hard stop behavior is represented by a mode chart with three regular modes and three instantaneous modes:

• FREE — There is no torque transmission between the slider and case.

• CONTACT_UPPER — The positive gap is closed.

• CONTACT_LOWER — The negative gap is closed.

• RELEASE_UPPER — The instantaneous mode needed to transition from CONTACT_UPPER to FREE.

• RELEASE_LOWER — The instantaneous mode needed to transition from CONTACT_LOWER to FREE.

• IMPACT — The instantaneous mode used when the slider bounces.

When the slider hits the case slowly, with speed less than the static contact speed threshold, the slider and case stay in contact. Otherwise, the slider bounces. When the slider bounces, it loses the speed due to the coefficient of restitution. In either of the contact modes, slider speed ω = 0. To transition from a contact mode to free mode, a torque greater than the static contact release torque threshold must be applied to the slider, and the transition must go through an instantaneous release mode to set the initial speed.

This modeling option improves simulation performance because static contact mode does not require the block to keep computing hard stop torque when the block is in contact mode.

Variables

To set the priority and initial target values for the block variables prior to simulation, use the Initial Targets section in the block dialog box or Property Inspector. For more information, see Set Priority and Initial Target for Block Variables.

Nominal values provide a way to specify the expected magnitude of a variable in a model. Using system scaling based on nominal values increases the simulation robustness. Nominal values can come from different sources, one of which is the Nominal Values section in the block dialog box or Property Inspector. For more information, see Modify Nominal Values for a Block Variable.

Ports

Conserving

R — Slider

mechanical rotational

Mechanical rotational conserving port associated with the slider that travels between the stops installed on the case.

C — Case

mechanical rotational

Mechanical rotational conserving port associated with the case.

Parameters

Upper bound — Gap between slider and upper bound

0.1 rad (default) | scalar

Gap between the slider and the upper bound. Specify the direction with respect to the local coordinate system, with the slider located in the origin. A positive value specifies the gap between the slider and the upper bound. A negative value sets the slider as penetrating into the upper bound.

Lower bound — Gap between slider and lower bound

-0.1 rad (default) | scalar

Gap between the slider and the lower bound. Specify the direction with respect to the local coordinate system, with the slider located in the origin. A negative value specifies the gap between the slider and the lower bound. A positive value sets the slider as penetrating into the lower bound.

Hard stop model — Select the hard stop model

Stiffness and damping applied smoothly through transition region, damped rebound (default) | Full stiffness and damping applied at bounds, undamped rebound | Full stiffness and damping applied at bounds, damped rebound | Based on coefficient of restitution

Select the hard stop model:

• Stiffness and damping applied smoothly through transition region, damped rebound — Specify a transition region in which the torque ramps up from zero. At the end of the transition region, the block applies the full stiffness and damping. This model applies damping on the rebound, but damping is limited to the value of the stiffness torque. Therefore, damping can reduce or eliminate the torque provided by the stiffness, but never exceed it. All equations are smooth and produce no zero crossings.

• Full stiffness and damping applied at bounds, undamped rebound — This model has full stiffness and damping applied with impact at upper and lower bounds, with no damping on the rebound. The equations produce no zero crossings when velocity changes sign, but there is a position-based zero crossing at the bounds. Having no damping on rebound helps to push the slider past this position quickly. This model has nonlinear equations.

• Full stiffness and damping applied at bounds, damped rebound — This model has full stiffness and damping applied with impact at upper and lower bounds, with damping applied on the rebound as well. The equations are switched linear, but produce position-based zero crossings. Use this hard stop model if simscape.findNonlinearBlocks indicates that this block prevents the whole network from being switched linear.

• Based on coefficient of restitution — This model uses a mode chart with regular and instantaneous modes to represent the hard stop behavior. All equations are smooth and produce no zero crossings. This modeling option improves simulation performance. For more information, see Model Based on Coefficient of Restitution.

Contact stiffness at upper bound — Elasticity of collision at upper bound

1e6 N*m/rad (default) | positive scalar

Elastic property of the colliding bodies when the slider hits the upper bound. The greater the value of this parameter, the less the bodies penetrate into each other and the more rigid the impact becomes. Setting this parameter to lesser values makes contact softer, but generally improves convergence and computational efficiency.

Dependencies

To enable this parameter, set Hard stop model to

• Stiffness and damping applied smoothly through transition region, damped rebound

• Full stiffness and damping applied at bounds, undamped rebound

• Stiffness and damping applied smoothly through transition region, damped rebound

Contact stiffness at lower bound — Elasticity of collision at lower bound

1e6 N*m/rad (default) | positive scalar

Elastic property of the colliding bodies when the slider hits the lower bound. The greater the value of this parameter, the less the bodies penetrate into each other and the more rigid the impact becomes. Setting this parameter to lesser values makes contact softer, but generally improves convergence and computational efficiency.

Dependencies

To enable this parameter, set Hard stop model to

• Stiffness and damping applied smoothly through transition region, damped rebound

• Full stiffness and damping applied at bounds, undamped rebound

• Stiffness and damping applied smoothly through transition region, damped rebound

Contact damping at upper bound — Dissipating property at upper bound

0.01 N*m/(rad/s) (default) | positive scalar

Dissipating property of the colliding bodies when the slider hits the upper bound. The greater the value of this parameter, the more energy dissipates during an interaction.

Dependencies

To enable this parameter, set Hard stop model to

• Stiffness and damping applied smoothly through transition region, damped rebound

• Full stiffness and damping applied at bounds, undamped rebound

• Stiffness and damping applied smoothly through transition region, damped rebound

Contact damping at lower bound — Dissipating property at lower bound

0.01 N*m/(rad/s) (default) | positive scalar

Dissipating property of the colliding bodies when the slider hits the lower bound. The greater the value of this parameter, the more energy dissipates during an interaction.

Dependencies

To enable this parameter, set Hard stop model to

• Stiffness and damping applied smoothly through transition region, damped rebound

• Full stiffness and damping applied at bounds, undamped rebound

• Stiffness and damping applied smoothly through transition region, damped rebound

Transition region — Region where torque ramps up

0.001 rad (default) | positive scalar

Region where the torque ramps up from zero to the full value. At the end of the transition region, the block applies full stiffness and damping.

Dependencies

To enable this parameter, set Hard stop model to Stiffness and damping applied smoothly through transition region, damped rebound.

Coefficient of restitution — Ratio of final-to-initial relative speed between slider and stop after collision

0.7 (default) | unitless value between 0 and 1

Ratio of the final to the initial relative speed between the slider and the stop after the slider bounces.

Dependencies

To enable this parameter, set Hard stop model to Based on coefficient of restitution.

Static contact speed threshold — Threshold relative speed between slider and stop before collision

0.1 rad/s (default) | nonnegative scalar

Threshold relative speed between the slider and the stop before collision. When the slider hits the case with speed less than the value of this parameter, they stay in contact. Otherwise, the slider bounces. To avoid modeling static contact between the slider and the case, set this parameter to 0.

Dependencies

To enable this parameter, set Hard stop model to Based on coefficient of restitution.

Static contact release torque threshold — Threshold torque needed to transition from contact mode to free mode

0.001 N*m (default) | positive scalar

Minimum torque needed to release the slider from a static contact mode.

Dependencies

To enable this parameter, set Hard stop model to Based on coefficient of restitution.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2007a

R2023a: Modeling option based on coefficient of restitution

The new option for the Hard stop model parameter, Based on coefficient of restitution, models the hard stop by using a mode chart that includes free mode, where there is no torque transmission between the slider and the case, and static contact mode, where there is zero speed difference between the slider and the case.

In previous releases, only the modeling options based on stiffness and damping were available.

This change has no compatibility impact.