Hydraulic motor displacement, measured in cm³/rev (metric) or in³/rev (US), is the volume of fluid that turns the output shaft one full revolution, and it sets the torque constant of the machine: larger displacement means more torque per bar of pressure drop, with speed falling as a result [S3]. A 250 cm³/rev motor running at 250 bar with 90% mechanical efficiency delivers roughly 895 N·m of shaft torque, while the same unit at 450 rpm and 92% volumetric efficiency demands about 122 L/min of pump flow [S1].
Across the four common motor architectures, displacement and pressure capability bracket the use case: gerotor/orbit units typically sit in the 50 to 630 cm³/rev range at 140 to 210 bar, vane units in 25 to 250 cm³/rev at 140 to 175 bar, axial-piston units in 28 to 250 cm³/rev at 280 to 450 bar, and radial-piston units in 200 to 4000 cm³/rev at 350 to 450 bar [S2]. Sizing a hydraulic motor for any real machine therefore starts from required torque and speed, not from a catalog part number.
Three Core Formulas That Drive Every Sizing Calculation
The SI torque equation is T (N·m) = (D × ΔP × 0.1) / (2π) × ηm, with D in cm³/rev and ΔP in bar; doubling either displacement or pressure drop doubles shaft torque, which is why displacement selection is effectively torque selection [S3]. A 100 cm³/rev motor at 200 bar and 60 L/min of flow produces the same hydraulic power as a 50 cm³/rev motor at the same conditions, but the larger-displacement unit splits that power toward torque while the smaller-displacement unit splits it toward speed [S3].
Rotational speed follows n (rpm) = (Q × 1000 × ηv) / D, with Q in L/min and D in cm³/rev: halving flow halves rpm, and doubling displacement halves rpm at constant flow [S3]. Hydraulic power is P (kW) = (T × n) / 9549 in metric, or P (HP) = (T × n) / 63025 in US units, and a practical rule of thumb used in calculators is P (kW) = Q (L/min) × P (bar) / 600 [S2]. Imperial torque in lb-in is T = (D_in3 × ΔP_psi) / (2π) × ηm, and flow is n = (Q_GPM × 231 × ηv) / D_in3 [S3].
Displacement-to-Torque and Speed Chart by Motor Family
Typical overall efficiencies cluster between 75% and 90% for gerotor/orbit and vane units, 90% to 97% for axial-piston units, and 92% to 98% for radial-piston units, with each family anchored to a speed band that matches its displacement range [S2]. Gerotor/orbit motors suit 50 to 800 rpm in conveyors and feeders, vane motors cover 200 to 2500 rpm in fans and light drives, axial-piston units handle 500 to 6000 rpm in mobile and high-speed circuits, and radial-piston units deliver 1 to 300 rpm in winches and wheel drives [S2].
Standard commercial displacements follow the family: gerotor/orbit at 50, 80, 100, 160, 250, 400, 630 cm³/rev; vane at 25, 45, 70, 100, 160, 250 cm³/rev; axial piston at 28, 45, 63, 80, 100, 125, 180, 250 cm³/rev; radial piston at 200, 400, 600, 1000, 2000, 4000 cm³/rev [S2]. Power density is the headline reason to pick hydraulics at all: a hydraulic motor of a given frame size produces roughly 5 to 10 times the torque of an electric motor of the same size, and it can stall indefinitely under pressure without thermal damage [S3]. The trade is overall efficiency of 80% to 92% versus 90% to 97% for electric drives, plus noise, leakage, and the cost of the supporting construction machinery and equipment hydraulic power unit.
Sizing Procedure: From Load Requirement to Pump Flow
Step one is calculating displacement from required torque, in metric: D (cc/rev) = T (N·m) × 20π / (ΔP (bar) × ηm), or equivalently D (cc/rev) = T × 62.83 / (ΔP × ηm) [S2][S1]. A worked case: 900 N·m continuous torque at 250 bar with 90% mechanical efficiency requires 900 × 62.83 / (250 × 0.90) = 251.3 cm³/rev, which points at a standard 250 cm³/rev unit; back-checking the 250 cm³/rev motor gives 250 × 250 × 0.90 / 62.83 = 895 N·m, slightly below the target and recoverable by a modest displacement bump, by a higher permitted continuous pressure, by a gearbox reduction, by trimming load torque, or by selecting a motor with better measured mechanical efficiency at the duty point [S1].
Step two is converting displacement into flow at the required speed: Q (L/min) = N (rpm) × D (cc/rev) / (1000 × ηv); in US units, Q (GPM) = N × D / (231 × ηv) [S2]. For 450 rpm at 250 cm³/rev and 92% volumetric efficiency, the required flow is 450 × 250 / (1000 × 0.92) = 122.3 L/min, and the pump must deliver that flow at working pressure, not just at no load [S1]. Verifying with an interactive calculator, a 10 in³/rev motor at 2000 psi and 5 gpm with 85% efficiency produces 225.5 lb-ft of torque at 115.5 rpm and 4.96 hp of output, a quick sanity check before any spec is signed off [S4].
Mechanical and Volumetric Efficiency: Why the Two Numbers Must Be Separated
Mechanical efficiency, sometimes called torque efficiency, captures bearing drag, seal friction, and viscous shear between fluid and internals: it reduces the pressure-generated torque that actually reaches the shaft, and it degrades sharply as case pressure and oil temperature climb [S3]. Volumetric efficiency captures internal leakage past the wear plates, pistons, or gear clearances: it reduces the flow that converts into shaft rotation, and it is the dominant loss at low speed and high pressure [S3]. Overall efficiency is the product η_overall = ηm × ηv, with typical overall figures of 75% to 90% for gerotor and vane units and 90% to 97% for axial-piston units, falling at low speed where volumetric loss dominates [S2][S3].
Ignoring one of the two efficiencies is the most common sizing error, and it surfaces as either a motor that cannot start the load or a motor that runs hot at normal duty [S3]. Three practical safeguards appear consistently in published guidance: use measured breakaway torque rather than running torque where the mechanism has high static friction, apply a documented service factor keyed to the mechanism rather than a blanket percentage, and build a small duty table covering breakaway, continuous, and maximum-speed points with time share, oil temperature, and whether the load is driving or overrunning the motor [S1].
Decision Matrix: Which Motor Family Fits Which Duty
For low-speed, high-torque winch and track-drive service at 1 to 300 rpm, radial-piston motors at 200 to 4000 cm³/rev and 350 to 450 bar with 92% to 98% overall efficiency are the right call, accepting the higher cost and footprint [S2]. For mid-speed conveyors and feeders at 50 to 800 rpm, gerotor/orbit units are described as compact, economical, and offering moderate efficiency [S2]. For fans, light drives, and process rollers at 200 to 2500 rpm, vane motors at 25 to 250 cm³/rev and 140 to 175 bar deliver smooth output with 80% to 92% efficiency [S2].
For mobile-equipment main propulsion and high-speed circuits at 500 to 6000 rpm, axial-piston units at 28 to 250 cm³/rev and 280 to 450 bar set the efficiency benchmark at 90% to 97%, which is why they dominate excavator swing and travel drives, and the same architecture pairs naturally with a variable speed drive style electronic control loop on the pump displacement [S2].
Failure Modes, Limits, and What a Displacement Chart Cannot Tell You
Peak torque is not the only data point that matters, and a motor chosen only from maximum torque is frequently larger than necessary and runs inefficiently at normal load, while a motor chosen only from normal speed may fail to start the machine [S1]. A published case study describes an $18,000 high-torque hydraulic motor that arrived unusable on the workbench because the engineering team specified it from peak numbers without checking breakaway-to-running ratio, case pressure, flushing flow, and the conveyor gearbox back-tooth load [S6]. The lesson is to size at three duty points, then verify pressure, flow, case pressure, mounting, and overspeed limits before sign-off [S1].
Other limits a displacement chart cannot capture: case drain pressure must be kept below the seal rating on most piston motors, return-line pressure adds to the case pressure on motors with internal drain, and hot-oil efficiency at 80 °C is typically 3 to 5 percentage points below the cold-rated figure used in catalog curves [S1][S3]. The supporting torque sensor and torque wrench tester chain on the driven shaft should be sized to the same breakaway and continuous numbers used in the motor calculation, since a mis-sized transducer masks the real load and can drive the next motor replacement on the same error.
Two trackable signals will tell you whether the sizing worked in the field: first, the motor case temperature stabilises below 90 °C at continuous duty, with the reservoir within 10 °C of the case, which indicates volumetric loss is in the expected band [S3]; second, the pump flow at working pressure matches the calculated Q (L/min) within 5% after warm-up, which indicates the line losses, valve pressure drop, and volumetric efficiency inputs were not over-optimistic [S1][S4]. If either signal misses its band, revisit ηm, ηv, and the duty table before increasing pressure or displacement, since simply raising the relief setting moves the failure mode to hoses, valves, pump power, cooler duty, and structural loads rather than fixing the original mismatch [S1].
See also our earlier report, Rate-of-Rise Heat Detector: 15°F/min Threshold and Vent Behavior.