# A novel performance model given by the physical dimensions of hydraulic axial piston motors: Model derivation

## Abstract

Since Wilson (1948) had firstly developed steady-state flow rate and moment loss model for hydraulic piston machines, several authors tried to make precise performance models. However, Huhtala (1997) discussed the strength and weakness of some existing models comparing with measurement data and concluded unfortunately that any model is not accurate enough for wide operating ranges. For a hydraulic axial piston motor of swash plate design with rotating cylinder, new performance formula is derived in this paper through theoretical study on leakages and friction losses of three facing gaps and other non-negligible losses. And efficiency surface of an example motor is estimated and extension of the formula for hydraulic pumps is discussed. A novel feature of the derived model is that all coefficients are given by the physical dimensions of a motor, hence allowing calculation and analysis of the performance of a motor in mind.

## Key Words

Leak Flow Loss Hydro-Mechanical loss Volumetric Mechanical and Overall Efficiency Performance Coefficient Model Hydraulic Axial Piston Motor## Nomenclature

*A*_{o}Opening area of a piston port

*a*_{P}Piston acceleration

*w*^{2}*R*_{ P }tan α cos θ*A*_{P}Cross-section area of a piston, π

*d*_{p}^{2}/4*A*_{VN}Opening area of the valve port notch

*A*_{Sp}Side view area of a slipper and piston outside of cylinder block

*b*_{V1},b_{B2}Inner, outer breadth of the valve plate sealing ring

*C*Performance coefficient related with leak-age

*C*_{d}Discharge coefficient of an orifice

*C*_{V}Breadth of the valve delivery/return port

*C*_{w}Drag coefficient of a cylindrical bar

*d*_{B}Diameter of a piston bore

*d*_{Br}Diameter of the support bearing

*d*_{P}Diameter of a piston

*e*_{P}Breadth of the piston port edge

*F*Force acting on several parts of a machine

*f*Friction coefficient

*f*_{P},*h*_{P}Friction coefficient and gap height between a piston and cylinder hole

*h*_{s}Gap height between slipper and swash plate

*h*_{R}Gap height between valve plate and cylinder

*K*Performance coefficient related with moment loss

*l*Length of several parts

*l*_{B}Length of a piston bore

*l*_{C}Length of the cylinder block

*l*_{F}Length of piston guide,

*l*_{ Fo }or*l*_{ Fo }+*Z*_{ P }*l*_{Fo}Length of a piston guide or bushing at ODP

*M, M*_{L}Moment, moment loss produced by several forces

*M*_{2}Output moment of a motor

*m*_{P}Mass of a piston

*n*Rotational speed of a motor in HZ,

*w/2π**p*Piston chamber pressure

*p*_{1}Inlet pressure of a motor

*p*_{2}Outlet pressure of a motor

*p*_{e}Pressure inside of a motor enclosure

- Δ
*p* Pressure difference,

*p-p*_{ e }or*p*_{1}-*p*_{2}*Q, Q*_{L}Flow, leak flow through several parts

*r*_{si},*r*_{so}Inner, outer radius of the slipper sealing ring

*r*_{V1},*r*_{V2}Inner, outer radius of the valve plate inner sealing ring

*r*_{V3},*r*_{V4}Inner, outer radius of the valve plate outer sealing ring

*R*_{c}Radius of the cylinder block

*R*_{H}Inner radius of a motor housing

*R*_{P}Pitch circle radius of pistons

*V*_{g}Motor geometric displacement

- 2A
_{P}R_{P}tan α V_{P} Piston velocity,

*wR*_{ P }tan α sin θ*z*Number of pistons in a motor

*Z*_{o}Piston number located at delivery port side

*Z*_{P}Piston displacement,

*R*_{ P }tan α(1−cosθ)- α
Tilting angle of the swash plate

- β
Bulk modulus of hydraulic oil

- ρ
Density of hydraulic oil

- Δθ
_{Vn} Angle of the valve port notch opened

- θ
Angular position of a piston or motor

- η
_{hm} Hydro-mechanical efficiency of a motor

- η
_{t} Overall efficiency of a motor

- η
_{v} Volumetric efficiency of a motor

- λ
_{P} Equivalent friction coefficient of a piston

- μ
Viscosity of hydraulic oil

- ω
Rotational speed of a motor in rad/sec,

*2πn*

## Subscript

*aP*Piston acceleration

*Br*Support bearing

*c*Fluid compressibility

*C*Cylinder block

*ch*Churning, swirl

*CH*Cylinder and motor housing

*fP,fS*Friction force on a piston, slipper

*o*Miscellaneous

*p*Pressure dependent term

*P*Piston

*pBr*Pressure on bearings

*pP*Piston pressure

*PP*Piston port

*PSP*Piston port, slipper and piston

*S*Slipper pad or swash plate

*SP*Slipper pad, slipper and piston

*V*Valve plate

*VN*Valve port control notch

*vBr*Velocity of bearings

*vP*Piston velocity

- μ
*P*, μ*S*, μ*V* Viscosity in a piston, slipper, valve plate

- ω
*P* Centrifugal force of a piston

- μ
Viscosity dependent term

- ρ
Density dependent term

## Superscript

- ∼
Fluctuating value of total flow rate or moment

- -
Average value of total flow rate or moment

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## References

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