Engineering
Axial piston pump
Tilt the swash plate of a nine-piston pump and watch line A fill to the pressure the load orifice demands: the displacement follows tan of the swash angle, reversing it sends the flow round the other way, zero swash leaves the motor idling at its no-load speed. Count the flow ripple of nine pistons and compare it with the closed form, block the load and let the relief valve take the whole flow, lower the charge pressure and see the return line starve. Shaft and charge work are booked against the pressure energy in the lines and the losses in the orifice, the leaks and the relief.
What happens
A motor turns the cylinder block of an axial piston pump against a tilted swash plate: each of nine pistons strokes with cos θ, the kidney ports of the valve plate hand a chamber to line A while its volume shrinks and to line B while it grows, so the displacement per turn is n·A·2R·tan α and the delivered flow ripples at the kinematic rate of nine pistons. The two lines are compressible oil volumes joined by a load orifice, external and cross-port leaks, a charge pump topping up whichever line falls below its pressure and relief valves opening over a 10 bar band; a shunt motor drives the shaft. RK4 at 20 µs; the shaft and charge work meet the stored pressure energy and every loss to rounding.
Try this
Tilt the swash and read line A climb; reverse it and B takes over; set it to zero and the motor runs free. Compare the ripple with the closed form, then block the load and watch the relief take the flow.
The model and its limits
Rigid pistons without slipper leakage or barrel tilt, a lumped line volume with constant bulk modulus, a linear relief band, a straight motor line. Pressures and ripple are those of this idealized pump.
Equations used in the module
Equations V_g = n·A·2R·tan α, dV_k/dφ = A·R·tan α·cos θ_k dp_A/dt = (β/V_L)·(Σ Q_k − Q_load − Q_leak − Q_relief + Q_charge) J·dω/dt = T_s·(1 − ω/ω_0) − Σ p_k·dV_k/dφ − T_f − c·ω δ_odd = (π/2n)·tan(π/4n), δ_even = (π/n)·tan(π/2n)
