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Linear conversion of pressure to actuation in an isothermal liquid system

**Library:**Simscape / Fluids / Isothermal Liquid / Actuators

The Single-Acting Actuator (IL) block represents an actuator
that converts the liquid pressure at port **A** into a mechanical force
at port **R** via an extending-retracting piston. The piston motion is
limited by a hard stop model. When the piston position is calculated internally, it is
reported at port **p**, and when the position is set by a connection to
a Simscape™
Multibody™ joint, it is received as a physical signal at port **p**.

The **Initial piston displacement**, **Fluid dynamic
compressibility**, and reference environmental pressure can be modified.
Fluid and mechanical inertia are not modeled.

The piston displacement is measured as the position at port **R** relative
to port **C**. The **Mechanical orientation**
identifies the direction of piston displacement. The piston displacement is neutral,
or `0`

, when the chamber volume is equal to the **Dead
volume**. When displacement is received as an input, ensure that the
derivative of the position is equal to the piston velocity. This is automatically
the case when the input is received from a Translational Multibody Interface block
connection to a Simscape Multibody joint.

Three models are available to model the extension limit of the actuator piston. This block uses a similar formulation as the Translational Hard Stop block and models uniform damping and stiffness coefficients at both ends of the piston stroke. For more information on the hard stop model options, see the Translational Hard Stop block.

The hard stop force is modeled when the piston is at its upper or lower bound. The
boundary region is within the **Transition region** of the
**Piston stroke** or **Piston initial
displacement**. Outside of this region, $${F}_{HardStop}=0.$$

You can optionally model cushioning toward the extremes of the piston stroke.
Setting **Cylinder end cushioning** to
`On`

slows the piston motion as it approaches its
maximum extension, which is defined in **Piston stroke**. For more
information on the functionality of a cylinder cushion, see the Cylinder Cushion (IL) block.

You can optionally model friction against piston motion. When **Cylinder
friction** is set to `On`

, the resulting
friction is a combination of Stribeck, Coulomb, and viscous effects. The pressure
difference is measured between the chamber pressure and the environment pressure.
For more information on the friction model and its limitations, see the Cylinder Friction block.

You can optionally model leakage between the liquid chamber and the piston
reservoir. When **Leakage** is set to
`On`

, Poiseuille flow is modeled between the piston and
cylinder. This block uses the Simscape Foundation Library Laminar Leakage (IL) block. The
flow rate is calculated as:

$$\dot{m}=\frac{\frac{\pi}{128}\left({d}_{0}^{4}-{d}_{i}^{4}-\frac{{\left({d}_{0}^{2}-{d}_{i}^{2}\right)}^{2}}{\mathrm{log}({d}_{0}/{d}_{i})}\right)}{\upsilon L}\left({p}_{A}-{p}_{env}\right),$$

where:

*ν*is the fluid kinematic viscosity.*L*is the piston length,**p**–*P*_{0}.*p*_{A}is the pressure at port**A**.*p*_{env}is the environmental pressure, which is selected in the**Environment pressure specification**parameter.

The cylinder diameter, *d*_{0},
is $${d}_{0}={d}_{i}+2c,$$ where *c* is the **Piston-cylinder
clearance**, and the piston diameter,
*d*_{i}, is $${d}_{i}=\sqrt{\frac{4{A}_{P}}{\pi}},$$ where *A*_{P} is the
**Piston cross-sectional area**.

At the extremes of the cushion orifice area and check valve pressure range, you
can maintain numerical robustness in your simulation by adjusting the block
**Smoothing factor**. A smoothing function is applied to all
calculated areas and valve pressures, but primarily influences the simulation at the
extremes of these ranges.

The normalized orifice area is calculated as:

$$\widehat{A}=\frac{\left(A-{A}_{leak}\right)}{\left({A}_{\mathrm{max}}-{A}_{leak}\right)}.$$

where:

*A*is the cushion_{leak}**Leakage area between plunger and cushion sleeve**.*A*is the_{max}**Cushion plunger cross-sectional area**.

The **Smoothing factor**, *f*, is
applied to the normalized area:

$${\widehat{A}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\widehat{A}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\widehat{A}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}}.$$

The smoothed orifice area is:

$${A}_{smoothed}={\widehat{A}}_{smoothed}\left({A}_{\mathrm{max}}-{A}_{leak}\right)+{A}_{leak}.$$

Similarly, the normalized valve pressure is:

$$\widehat{p}=\frac{\left(p-{p}_{cracking}\right)}{\left({p}_{\mathrm{max}}-{p}_{cracking}\right)}.$$

where:

*p*is the cushion_{cracking}**Check valve cracking pressure differential**.*p*is the cushion_{max}**Check valve maximum pressure differential**.

Smoothing applied to the normalized pressure is:

$${\widehat{p}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\widehat{p}}_{}^{2}+{\left(\frac{f}{4}\right)}^{2}}-\frac{1}{2}\sqrt{{\left(\widehat{p}-1\right)}^{2}+{\left(\frac{f}{4}\right)}^{2}},$$

and the smoothed pressure is:

$${p}_{smoothed}={\widehat{p}}_{smoothed}\left({p}_{\mathrm{max}}-{p}_{cracking}\right)+{p}_{cracking}.$$

The Single-Acting Actuator (IL) block comprises four Simscape Foundation and two Fluids Library blocks:

Translational Hard Stop

Laminar Leakage (IL)

Converter

Sensor

Cylinder Cushion (IL)

Cylinder Friction (IL)

**Actuator Structural Diagram**

Rotating Single-Acting Actuator (IL) | Single-Acting Rotary Actuator (IL) | Pilot Valve Actuator (IL) | Double-Acting Actuator (IL) | Double-Acting Actuator (G-IL)