Lever (AB-PB)

R2026b

Position-and-angle-based lever mechanism

Since R2026b

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Lever (AB-PB) block

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Libraries:
Simscape / Foundation Library / Mechanisms

Description

The Lever (AB-PB) block represents an ideal mechanical lever. Ports T1, T2, and T3 are position-based translational ports. Any of these ports can connect to a pivot support, external force source, or load.

Class 1 Lever	Class 2 Lever	Class 3 Lever

Forces at the translational ports act along the translational network direction. The fulcrum does not have to be stationary.

You can also model the lever rotational dynamics by enabling the rotational port R.

To define the coupling method of the rotational and translational motions, you can choose between a linearized small‑angle approximation or a full‑angle nonlinear geometry. Therefore, the block can model the lever in one of three ways:

• Model 1 – No rotational port

• Model 2 – Rotational port and linearized small‑angle approximation

• Model 3 – Rotational port and full‑angle nonlinear geometry

You can use any of these models to represent a Class 1, Class 2, or Class 3 lever. The table lists the equations for each of these models.

Model 1 No Rotational Port	Model 2 Rotational Port and Linearized Approximation	Model 3 Rotational Port and Full‑Angle Geometry
$$K_{\text{13}}\cdot f_{T3}=-K_{\text{12}}\cdot f_{T2}$$ $$f_{T1}+f_{T2}+f_{T3}=0$$ $$K_{\text{12}}=r_{12}$$ $$K_{\text{13}}=r_{12}+r_{23}$$ $$lengt{h}_{\text{P12}}=\frac{r_{13}}{r_{12}+r_{23}}lengt{h}_{\text{P13}}$$ $$lengt{h}_{P12}=x_2-x_1$$ $$lengt{h}_{P23}=x_3-x_2$$ $$lengt{h}_{P13}=x_3-x_1$$	$$K_{\text{13}}\cdot f_{T3}=-K_{\text{12}}\cdot f_{T2}-\epsilon \cdot \tau$$ $$f_{T1}+f_{T2}+f_{T3}=0$$ $$K_{\text{12}}=r_{12}$$ $$K_{\text{13}}=r_{12}+r_{23}$$ $$lengt{h}_{\text{P12}}=\frac{r_{13}}{r_{12}+r_{23}}lengt{h}_{\text{P13}}$$ $$lengt{h}_{\text{P13}}=\left(r_{12}+r_{23}\right)\cdot \epsilon \cdot \theta$$ $$lengt{h}_{P12}=x_2-x_1$$ $$lengt{h}_{P23}=x_3-x_2$$ $$lengt{h}_{P13}=x_3-x_1$$	$$K_{\text{13}}\cdot f_{T3}=-K_{\text{12}}\cdot f_{T2}-\epsilon \cdot \tau$$ $$f_{T1}+f_{T2}+f_{T3}=0$$ $$K_{\text{12}}=r_{12}\cdot \cos \left(\theta \right)$$ $$K_{\text{13}}=\left(r_{12}+r_{23}\right)\cdot \cos \left(\theta \right)$$ $$lengt{h}_{\text{P12}}=\frac{r_{13}}{r_{12}+r_{23}}lengt{h}_{\text{P13}}$$ $$lengt{h}_{\text{P13}}=\left(r_{12}+r_{23}\right)\cdot \sin \left(\epsilon \cdot \theta \right)$$ $$lengt{h}_{P12}=x_2-x_1$$ $$lengt{h}_{P23}=x_3-x_2$$ $$lengt{h}_{P13}=x_3-x_1$$

The block equations use these symbols:

• r₁₂ is the arm length between ports T1 and T2. This value corresponds to the Arm length T1-T2 parameter.

• r₂₃ is the arm length between ports T2 and T3. This value corresponds to the Arm length T2-T3 parameter.

• length_P12 is the orthogonal projection of the distance between ports T1 and T2 onto the translational network rail.

• length_P23 is the orthogonal projection of the distance between ports T2 and T3 onto the translational network rail.

• length_P13 is the orthogonal projection of the distance between ports T1 and T3 onto the translational network rail.

• x_T1, x_T2, x_T3 are the absolute positions on the translational network rail of ports T1, T2, and T3, respectively.

• f_T1, f_T2, f_T3 are the forces along the translational network rail at ports T1, T2, and T3, respectively.

• τ is the torque of the lever acting on the rotational network at port R.

• θ is the lever angle at port R.

• ε is the positive rotation sign:

  • If the Positive angle parameter value is Corresponds to positive projected length, ε = 1.

  • If the Positive angle parameter value is Corresponds to negative projected length, ε = –1.

length_P13 is the orthogonal projection of the distance between ports T1 and T3 onto the translational network rail. You can control it by specifying a high-priority target for the Projected length from T1 to T3 variable in the Initial Targets section in the block dialog box.

When the lever angle θ is 0 deg, the lever length is perpendicular to the translational network, and the Projected length from T1 to T3 is 0 m. Use the Positive angle parameter to change whether positive lever angles correspond to positive or negative projected lever lengths.

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.

Examples

New

Lever Analysis

Use the Simscape™ Lever (AB-PB) block to model lever systems in the position-based translational and angle-based rotational domains. Two models demonstrate the torque balance and kinematic constraints of a lever. The first model computes mechanical advantage from force measurements. The second model predicts which end of a lever contacts a wall when two forces act on it.

• Since R2026b
• Open Live Script

Ports

Conserving

T1 — First translational port on lever

position-based translational

Position-based translational conserving port associated with the first point on the lever. This point can represent the fulcrum, load, or force.

T2 — Second translational port on lever

position-based translational

Position-based translational conserving port associated with the second point on the lever. This point can represent the fulcrum, load, or force.

T3 — Third translational port on lever

position-based translational

Position-based translational conserving port associated with the third point on the lever. This point can represent the fulcrum, load, or force.

R — Lever rotational port

angle-based rotational

Angle-based rotational conserving port that represents the entire lever body.

Dependencies

To enable this port, select the Enable rotational port check box.

Parameters

Main

Arm length T1-T2 — Arm length between ports T1 and T2

0.1 m (default)

Arm length between ports T1 and T2.

Arm length T2-T3 — Arm length between ports T2 and T3

0.1 m (default)

Arm length between ports T2 and T3.

Rotational Port

Enable rotational port — Whether to model rotational dynamics

off (default) | on

To model the lever rotational dynamics, select this check box.

Positive angle — Whether positive lever angle corresponds to positive or negative projected lever length

Corresponds to positive projected length (default) | Corresponds to negative projected length

Select whether positive lever angle corresponds to positive or negative projected lever length, as shown in the Rotational Network schematic. This parameter refers to behavior for small angles, near 0 deg, as opposed to when the lever rotates by large angles that could flip the length sign.

Dependencies

To enable this parameter, select the Enable rotational port check box.

Linearized small-angle approximation — Coupling method of rotational and translational motions

on (default) | off

Whether to use linearized equations that work well for small angles, near 0 deg, or nonlinear equations that are more accurate for large angles. By default, the lever uses linearized small‑angle approximation as the coupling method of rotational and translational motions. To use full‑angle nonlinear geometry, clear this check box.

Dependencies

To enable this parameter, select the Enable rotational port check box.

Tips

• To model rotational effects, such as pivot point damping or friction, or to model a torque source, enable the rotational port R. This port represents the rotation of the entire lever body. Connect all rotational effects related to the pivot support, external force source, or load to this rotational port, even though the block models these points using different translational ports.

• If the lever operates at large angles, switch to full‑angle nonlinear geometry by clearing the Linearized small-angle approximation check box. In the linearized model with a rotational port, errors in the force and torque results scale with θ ^2, and errors in the projected lengths and angle results scale with θ ^3. For accurate results at greater angles, use the nonlinear model.

• Models 1 and 2 produce an error when the Projected length from T1 to T3 variable exceeds the sum of Arm length T1-T2 and Arm length T2-T3 because that is physically impossible for a lever. If you receive this error, it means that the lever operates at a very large angle. In this case, you can switch to Model 3, which uses full‑angle nonlinear geometry.

• When the block uses Model 3, it is more numerically sensitive than the other two models.

  • To avoid a solver error at angles of +/- 90 deg, connect a torque load to port R.

  • To maintain rotation past +/- 90 deg, connect an Inertia (AB) block to port R.

Extended Capabilities

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

Version History

Introduced in R2026b