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Lossy Gear

Gear with mesh efficiency and bearing friction (stuck/rolling possible)

Description

The Lossy Gear component models the gear ratio and the losses of a standard gear box in a realistic way, including the stuck phases that may occur at zero speed. The gear boxes that can be handled are fixed in the ground or on a moving support, have one input and one output shaft, and are essentially described by the equations

${\mathrm{\phi }}_{a}=r{\mathrm{\phi }}_{b}$

$-{\mathrm{\tau }}_{b}=r\left({\mathrm{\eta }}_{\mathrm{mf}}{\mathrm{\tau }}_{a}-{\mathrm{\tau }}_{\mathrm{bf}}\right)$

where $r$ is constant gear ratio, ${\mathrm{\eta }}_{\mathrm{mf}}$ is the efficiency due to mesh friction, and ${\mathrm{\tau }}_{\mathrm{bf}}$ is the bearing friction torque.

Loss Table

The Loss Table parameter is five-column table that defines the loss terms ${\mathrm{\eta }}_{\mathrm{mf}}$ and ${\mathrm{\tau }}_{\mathrm{bf}}$ in terms of the absolute value of the input shaft speed, ${w}_{a}$, and of the energy flow direction. Its columns are $\left[|{w}_{a}|,{\mathrm{\eta }}_{\mathrm{mf1}},{\mathrm{\eta }}_{\mathrm{mf2}},{\mathrm{\tau }}_{\mathrm{bf1}},{\mathrm{\tau }}_{\mathrm{bf2}}\right]$ with

 $\left|{w}_{a}\right|$ $\frac{\mathrm{rad}}{s}$ Angular speed of ${\mathrm{flange}}_{a}$ ${\mathrm{\eta }}_{\mathrm{mf1}}$ $1$ Mesh efficiency when input shaft is driving ${\mathrm{\eta }}_{\mathrm{mf2}}$ $1$ Mesh efficiency when output shaft is driving ${\mathrm{\tau }}_{\mathrm{bf1}}$ $Nm$ Absolute bearing friction torque when input shaft is driving ${\mathrm{\tau }}_{\mathrm{bf2}}$ $Nm$ Absolute bearing friction torque when output shaft is driving

The rows of the Loss Table are ordered according to ${\mathrm{\omega }}_{a}$, with the first row having the smallest ${\mathrm{\omega }}_{a}$. The values for ${\mathrm{\eta }}_{\mathrm{mf1}}$, ${\mathrm{\eta }}_{\mathrm{mf2}}$, ${\mathrm{\tau }}_{\mathrm{bf1}}$, and ${\mathrm{\tau }}_{\mathrm{bf2}}$ at a particular shaft speed are interpolated from the values in the table.

To add rows to the table, right-click on the value and select Edit Matrix Dimension.

With these variables, the mesh efficiency and the bearing friction are formally defined as

$\left\{\begin{array}{cc}\left\{{\mathrm{\eta }}_{\mathrm{mf}}={\mathrm{\eta }}_{\mathrm{mf1}},{\mathrm{\tau }}_{\mathrm{bf}}={\mathrm{\tau }}_{\mathrm{bf1}}\right\}& \left({\mathrm{\tau }}_{a}-{\mathrm{\tau }}_{{\mathrm{bf}}_{a}}\right){w}_{a}>0\vee \left({\mathrm{\tau }}_{a}={\mathrm{\tau }}_{{\mathrm{bf}}_{a}}\wedge {w}_{a}>0\right)\\ \left\{{\mathrm{\eta }}_{\mathrm{mf}}={\mathrm{\eta }}_{\mathrm{mf2}},{\mathrm{\tau }}_{\mathrm{bf}}={\mathrm{\tau }}_{\mathrm{bf2}}\right\}& \left({\mathrm{\tau }}_{a}-{\mathrm{\tau }}_{{\mathrm{bf}}_{a}}\right){w}_{a}<0\vee \left({\mathrm{\tau }}_{a}={\mathrm{\tau }}_{{\mathrm{bf}}_{a}}\wedge {w}_{a}<0\right)\\ {\stackrel{.}{w}}_{a}=0& \mathrm{otherwise}\end{array}$

Losses are modeled in a physically meaningful way taking into account that at zero speed the movement may be locked due to the friction in the gear teeth and/or in the bearings.

Variables

 Name Units Description Modelica ID ${\mathrm{\phi }}_{x}$ $\mathrm{rad}$ Absolute angle of ${\mathrm{flange}}_{x},x\in \left\{a,b\right\}$ flange_x.phi ${\mathrm{\phi }}_{\mathrm{support}}$ $\mathrm{rad}$ Absolute angle of support flange support.phi ${\mathrm{\tau }}_{x}$ $Nm$ Torque applied to ${\mathrm{flange}}_{x},x\in \left\{a,b\right\}$ flange_x.tau

Connections

 Name Description Modelica ID ${\mathrm{flange}}_{a}$ Flange of left shaft flange_a ${\mathrm{flange}}_{b}$ Flange of right shaft flange_b $\mathrm{support}$ support $\mathrm{heatPort}$ heatPort

Parameters

 Name Default Units Description Modelica ID $r$ $1$ $1$ Transmission ratio ($\frac{{\mathrm{\phi }}_{a}}{{\mathrm{\phi }}_{b}}$) ratio $\mathrm{Loss Table}$ $\left[0.,1.,1.,0.,0.\right]$ Matrix for mesh efficiencies and bearing friction depending on speed lossTable Use Heat Port $\mathrm{false}$ True (checked) means heat port is enabled useHeatPort Use Support Flange $\mathrm{false}$ True (checked) enables the support flange useSupport

Constants

 Name Value Units Description Modelica ID $\mathrm{Unknown}$ $3$ Value of mode is not known Unknown $\mathrm{Free}$ $2$ Element is not active Free $\mathrm{Forward}$ $1$ w_a > 0 (forward rolling) Forward $\mathrm{Stuck}$ $0$ w_a = 0 (forward rolling, locked or backward rolling) Stuck $\mathrm{Backward}$ $-1$ w_a < 0 (backward rolling) Backward $\mathrm{unitAngularAcceleration}$ $1$ $\frac{\mathrm{rad}}{{s}^{2}}$ unitAngularAcceleration $\mathrm{unitTorque}$ $1$ $Nm$ unitTorque

 Modelica Standard Library The component described in this topic is from the Modelica Standard Library. To view the original documentation, which includes author and copyright information, click here.