Friction in hydraulic cylinders

Hydraulic Cylinders

The Cylinder Friction block simulates friction in the contact between moving bodies in hydraulic cylinders and is intended to be used primarily as a building block in combination with both the double- and single-acting cylinders to develop a cylinder model with friction. The friction force is simulated as a function of relative velocity and pressure, and is assumed to be the sum of Stribeck, Coulomb, and viscous components. The Coulomb friction force consists of the preload force, caused by the seal squeeze during assembly, and the force proportional to pressure. The sum of the Coulomb and Stribeck friction forces at zero velocity is often referred to as the breakaway friction force. For more information, see the Translational Friction block reference page.

The friction force is approximated with the following equations:

$$F={F}_{C}\xb7\left(1+\left({K}_{brk}-1\right)\xb7\mathrm{exp}\left(-{c}_{v}\left|v\right|\right)\right)sign\left(v\right)+{f}_{vfr}\xb7v$$

$${F}_{C}={F}_{pr}+{f}_{cfr}\left({p}_{A}+{p}_{B}\right)$$

where

F | Friction force |

F_{C} | Coulomb friction |

F_{pr} | Preload force |

f_{cfr} | Coulomb friction coefficient |

p_{A}, p_{B} | Pressures in cylinder chambers |

K_{brk} | Breakaway friction force increase coefficient |

c_{v} | Transition coefficient |

v | Relative velocity in the contact |

f_{vfr} | Viscous friction coefficient |

To avoid discontinuity at *v *= 0, a small
region |*v*| ≤ *v*_{th} is
introduced around zero velocity, where friction force is assumed to
be linearly proportional to velocity:

$$F=K\xb7v$$

$$K=\frac{{F}_{C}\left(1+\left({K}_{brk}-1\right)\xb7\mathrm{exp}\left(-{c}_{v}{v}_{th}\right)\right)+{f}_{vfr}\xb7{v}_{th}}{{v}_{th}}$$

where

K | Proportionality coefficient |

v_{th} | Velocity threshold |

Connections R and C are mechanical translational conserving ports associated with the rod and case, respectively. Connections A and B are hydraulic conserving ports to be connected to ports A and B of the cylinder model, as shown in the following illustration. The force generated by the block always opposes relative motion between the rod and the case.

**Preload force**The preload force, caused by the seal squeeze during assembly. The default value is

`10`

N.**Coulomb friction force coefficient**Coulomb friction coefficient, which defines the proportionality between the Coulomb friction force and the pressure in cylinder chambers. The default value is

`1e-6`

N/Pa.**Breakaway friction increase coefficient**The friction force increase over the Coulomb friction. The Coulomb friction force, multiplied by this coefficient, is referred to as breakaway friction force. The default value is

`1`

.**Viscous friction coefficient**Proportionality coefficient between the viscous friction force and the relative velocity. The parameter value must be greater than or equal to zero. The default value is

`100`

N/(m/s).**Transition approximation coefficient**The parameter sets the value of coefficient

*c*_{v}, which is used for the approximation of the transition between the breakaway and the Coulomb frictions. Its value is assigned based on the following considerations: the Stribeck friction component reaches approximately 5% of its steady-state value at velocity`3`

/*c*_{v}, and 2% at velocity`4`

/*c*_{v}, which makes it possible to develop an approximate relationship*c*_{v}~=`4`

/*v*_{min}, where*v*_{min}is the relative velocity at which friction force has its minimum value. By default,*c*_{v}is set to`10`

s/m, which corresponds to a minimum friction at velocity of about`0.4`

m/s.**Linear region velocity threshold**The parameter sets the small vicinity near zero velocity, within which friction force is considered to be linearly proportional to the relative velocity. MathWorks recommends that you use values in the range between

`1e-6`

and`1e-4`

m/s. The default value is`1e-4`

m/s.

Use the **Variables** tab to set the priority
and initial target values for the block variables prior to simulation.
For more information, see Set Priority and Initial Target for Block Variables (Simscape).

The block has the following ports:

`A`

Hydraulic conserving port connected to the cylinder inlet.

`B`

Hydraulic conserving port connected to the cylinder outlet.

`R`

Mechanical translational conserving port associated with the cylinder rod.

`C`

Mechanical translational conserving port associated with the cylinder clamping structure.

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