This revision replaces the fixed lever approximation and universal 35 degree takeover claim with one explicit planar construction. A rigid body, declared centre of gravity and pin-ended air cylinder remain connected as the body tips.
Pneumatic Dump-Car Geometry Explorer
Enter a declared body-and-load weight, CG radius, body cylinder-pin radius, fixed frame-pin coordinates, body angle and gauge pressure. The worksheet reports the vertical-weight moment, exact cylinder moment arm, ideal static push and ideal pressure-area bore.
Equation Used
- The body and contents are represented by one rigid planar body with declared total vertical weight acting at radius r.
- The body rotates about one fixed ideal pivot and theta is measured upward from horizontal.
- The cylinder is a two-force member with ideal pins at fixed frame point B and rotating body point A.
- Required push uses the exact perpendicular moment arm of the cylinder line at the entered angle.
- Ideal bore uses gauge pressure uniformly over the full circular piston area and omits back pressure, seal friction and design margin.
- Contents movement, changing CG, impact, inertia, friction, pin and frame loads, cylinder buckling, stability, wheel/rail reactions, structure, controls, ratings and safe operation are excluded.
Replaces fixed millimetre lever scalars and the invented 35 degree takeover threshold with exact declared pivot, cylinder-line and pressure-area geometry.
A pivoted air cylinder tips a pivoted car body
US patent 2,588,133 describes a railway dump-car body pivotally supported on a frame, with a cylinder pivotally supported by that frame and its piston rod pivotally connected to the body. An air reservoir and inlet/exhaust valve mechanism supply or vent the cylinder.
The animation retains exactly that minimum topology: fixed body pivot, fixed frame cylinder pin, rotating body pin and a pin-ended cylinder whose length changes as the rigid body tips.
The same topology does not imply one universal design
Historical patents show side-dumping railway bodies with one or more air cylinders, but actual pin locations, body shape, load distribution, valves, reservoirs, stops and return arrangements vary. The declared planar worksheet is not a reconstruction or rating of a particular car.
Use the instantaneous perpendicular moment arm
The vertical weight produces a moment equal to W times the horizontal CG coordinate r cos(theta). With body pin A and unit cylinder direction e, the scalar cross product gives the perpendicular cylinder moment arm. Parker's cylinder data gives theoretical force as gauge pressure times effective piston area, allowing an ideal full-piston bore to be reported separately.
A transparent declared-geometry example
For 50 kN declared weight at 1.10 m CG radius, a 1.30 m body pin, frame pin at [0.80,-0.40] m and 35 degree body angle, gravity moment is 45.05 kN·m and the exact cylinder moment arm is about 0.870 m. Ideal equilibrium push is about 51.81 kN. At 6.0 bar gauge, the ideal full-piston bore is about 331.6 mm before losses or design margin.
Static ideal force is not cylinder or vehicle sizing
US patent 1,810,639 notes that the greatest effort is generally needed early, before the load becomes overbalanced. The transition depends on actual body and contents geometry; there is no universal 35 degree takeover angle.
Contents can shift during dumping, changing the CG and dynamics. The worksheet also omits friction, back pressure, acceleration, impact, pin and frame loads, buckling, rail reactions, vehicle stability, structure, controls and ratings. Unsupported pressures, cycle times, capacities, clearances, maintenance limits, field installations and historical claims are removed.
Pneumatic dump-car worksheet questions
Why does the cylinder lever arm change?
Both the body pin position and cylinder line direction change as the body rotates.
Is 35 degrees a universal gravity-takeover angle?
No.
Is the reported bore a selected cylinder size?
No. It is only ideal pressure-area arithmetic.
Does this check stability or structure?
No.
Pneumatic dump-car and cylinder references
- US 2,588,133, Pneumatic operating means for railway dump cars: pivoted body, frame-pivoted cylinder, body-connected piston rod, reservoir and inlet/exhaust control.
- US 1,810,639, Dump car: compressed-air cylinders tipping the body and changing effort as the load overbalances.
- Parker Pneumatic Engineering Data: theoretical cylinder force F=P A and push/pull effective-area distinction.
- Parker Cylinder Formula Reference: force components and lever moment-arm relationships.
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