# Mechanics and FEM estimation of gaps generated in star-ring active contacts of ORBIT motor during operation

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## Abstract

We estimate deformations or gaps occurring at all the ideally form-closed contacts, that separate the chambers, in epitrochoid (cycloidal class) generated star-ring gears of ORBIT^{®} motors. The direct measurement of these deformations and gaps is not an easy task as the contact positions are inaccessible for installing any measuring device. Therefore, we target the accurate theoretical estimation of these deformations and gaps for any pressure or torque load. The problem poses statically indeterminate, as there are multiple contacts. We use FEM in Ansys^{®} environment to model the problem and its solution. We develop the model to its final stage with gradual improvement in assessing more accurate and realistic boundary conditions as well as loading patterns. We make an elaborate presentation of this process in this investigation report. Interestingly, in ORBIT motor momentary occurrence of gap at the transition active contact, i.e., the contact separating the two adjacent chambers with one in high pressure and the other in low pressure, is unavoidable. This occurs for a substantial period of a power cycle and sometimes in both transition contacts, that separate the overall high pressure zone and the low pressure zone. The inter-chamber leakages through these gaps are partially responsible for lowering the volumetric efficiency. The present investigation is limited to establishing a method of estimating deformation and gaps at the active contacts of such units. Our proposed method can be used for any such unit of any geometric design data and kinematic model.

## Keywords

Epitrochoid star ORBIT motor Active and transition contacts Deformations and gaps Load sharing FEM## List of symbols

- ‘−’
Super score indicate dimensionless form

- \( A_{ 0} \)
(See Fig. 8) nominal arm length of star generating point in an epitrochoidal generation, also it is the pitch circle radius of the roller/lobe in the ring \( \bar{A}_{ 0} = A_{ 0} / \, R_{ 0} \)

- \( b \)
Nominal width of the star-ring/Length of roller

- \( C_{ 0} \)
Eccentricity or centre distance \( \frac{{C_{ 0} }}{{R_{ 0} }} = \frac{1}{Z} \)

- \( E \)
Young’s modulus,\( E_{t} = E_{e} = E \), (t denotes trochoid and e denotes envelope)

- \( L_{i} \)
Length of a line joining two transition contact points at any instant

- \( F_{i} \)
Resultant instantaneous separating force due to fluid pressure acting on

*L*_{i}- \( F_{n} \)
Contact force in the normal direction

*µ*Coefficient of friction at active contacts

- \( f_{n} \)
Coefficient of friction (steel)

- \( I_{ 0} \)
Instantaneous centre of rotation of the centrodes

- \( M_{i} \)
Instantaneous torque (due to fluid pressure)

- \( M_{n} \)
Reaction torque due to force \( F_{n} \)

- \( p_{i} \)
Pressure at the inlet

- \( p_{o} \)
Pressure at the outlet

- \( \varDelta p_{i} \)
Differential pressure (\( p_{i} - p_{o} \))

- \( R_{ 0} \)
Radius of outer centrode. Base circle radius of envelope/ring profile

- \( r_{ 0} \)
Radius of inner centrode. Base circle radius of epitrochoid/star

- \( r_{i} \)
Instantaneous torque arm (uniform inlet and outlet pressure)

- \( r_{m} \)
Radius of ring lobe/Roller \( \overline{{r_{m} }} = \, r_{m} /R_{ 0} \)

- \( \delta \)
The deviation of

*I*_{0}from its ideal geometric position due to fluid pressure- \( \delta_{n} \)
Deformation at nth Contact (− means gap)

- \( f_{m} \)
Material property factor

- \( \psi \)
Angle of rotation of \( A_{ 0} \) with respect to the X-axis fixed on the inner centrode

- \( \xi \)
Rotational angle of star about its own axis/Shaft rotational angle

- \( \xi_{o} \)
Phase angle (with respect to \( \xi \)) for which chambers remain in the same phase = \( \pi /Z\left( {Z - 1} \right) \)

- \( \phi \)
Leaning angle

- \( \upsilon \)
Poisson’s ratio; \( \upsilon_{t} = \, \upsilon_{e} = \, \upsilon \), (t denotes trochoid and e denotes envelope)

- \( Z \)
Number of chambers in HST unit (= the number of lobes/rollers in ring)

- \( Q_{il} \)
Instantaneous inter-chamber leakage flow rate (m

^{3}/s)- \( \eta \)
Viscosity of hydraulic oil (Kg/m s)

## Abbreviations

- FEM
Finite element method

- HPZ
High pressure zone

- HST
Hydrostatic transmission

- HSLT
High speed low torque

- LPZ
Low pressure zone

- LSHT
Low speed high torque

- ROPIMA
Rotary piston machine

- TEM
Trial and error method

## Notes

### Acknowledgements

This research work is an outcome of the general PhD program in the authors’ Institute, IIT Kharagpur, India. There is no specific financial grant for this investigation.

### Compliance with ethical standards

### Conflict of interest

The author(s) declare that they have no potential conflict of interest with respect to the research, authorship, and/or publication of this article.

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