BIZOMAT MAKEOS · Mechanics
Engine Gas Force and Gas Torque at a Crank Angle
For a cylinder pressure at a given crank angle, what force does the gas put on the piston, what side force does the piston put on the cylinder wall, and what torque reaches the crankshaft, exactly and by Norton’s approximation?
Gas force and gas torque of a single-cylinder slider-crank by Norton’s equations: gas force Fg = Pg π B² ÷ 4 on a bore B, piston side force Fg tan φ with the conrod angle φ from sin φ = (r/l) sin ωt, exact gas torque Tg = Fg tan φ × x with the exact piston position x, and the approximation Tg ≈ Fg r sin ωt (1 + (r/l) cos ωt), with the cylinder pressure you enter for the crank angle.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Bore B | mm | Cylinder bore, the piston diameter. |
| Gas pressure at this crank angle (gauge) | MPa | Cylinder pressure above the crankcase pressure at this crank angle, from the pressure curve. |
| Crank radius r (half the stroke) | mm | Crank throw, half the piston stroke. |
| Connecting rod length l (centre to centre) | mm | Connecting rod length between the crankpin and wrist pin centres. |
| Crank angle ωt from top dead centre | ° | Crank angle measured from top dead centre in the direction of rotation. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Bore B
- What it is
- Diameter of the cylinder bore, equal to the piston diameter.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 76.2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The gas force rises with its square.
- Get it right
- Take it from the engine data, e.g. 76.2 mm for a 3 in bore.
What this means: Gas pressure at this crank angle (gauge)
- What it is
- Cylinder pressure above the crankcase at this crank angle.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0 MPa to 1000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The gas force and torque rise in proportion.
- Get it right
- Read it from a measured or calculated pressure curve at the same crank angle.
What this means: Crank radius r (half the stroke)
- What it is
- Radius of the crank throw, which is half the piston stroke.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 45 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Piston velocity rises in proportion and the r/l ratio changes.
- Get it right
- Measure from the main bearing centre to the crankpin centre, or halve the stroke.
What this means: Connecting rod length l (centre to centre)
- What it is
- Length of the connecting rod between crankpin and wrist pin centres.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 157.5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer rod lowers r/l and the second harmonic.
- Get it right
- Measure centre to centre of the two bores in the rod.
What this means: Crank angle ωt from top dead centre
- What it is
- Angle the crank has turned from top dead centre.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -720° to 720°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets where in the cycle the values are taken.
- Get it right
- Measure in the direction of rotation; 0° is the piston at the top of its stroke.
Method and formulas
- Ap = π B² ÷ 4; Fg = Pg Ap (eq. 13.4).
- sin φ = (r/l) sin ωt; side force on the cylinder wall Fg14 = Fg tan φ (eq. 13.5a).
- Exact gas torque Tg = Fg tan φ × x with x = r cos ωt + l cos φ (eqs. 13.1d, 13.6d).
- Approximation Tg ≈ Fg r sin ωt (1 + (r/l) cos ωt) (eq. 13.8b).
- Example: 13.681 kN of gas force gives 384.78 N·m on the crankshaft.
Limitations
- Enter the gas pressure for the crank angle analysed; the pressure curve comes from the engine’s thermodynamics or measurement.
- Friction and inertia are not included; add the inertia torque for the total.
- The cylinder axis passes through the crank centre.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Turn a cylinder pressure into crankshaft torque
- Find the piston side thrust on the cylinder wall
- Check the approximate gas torque formula
Who it is for
- Engine and compressor designers
- Machine designers
- Mechanical engineering students
What you get
- Piston area and gas force
- Conrod angle and piston side force
- Gas torque on the crankshaft (exact and approximate)
Interpreting the result
- Gas torque is zero at top and bottom dead centre, where the moment arm is zero.
- The gas force is internal to the engine and does not shake the frame; the gas torque does turn the crank.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 605–613. Topic reference; no source tables, figures or examples are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
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Open the Engine Gas Force and Gas Torque at a Crank Angle calculator