BIZOMAT MAKEOS · Mechanics
Engine Piston Position, Velocity and Acceleration (Exact and Fourier Approximation)
For a single-cylinder engine or compressor slider-crank at a given crank angle and speed, where is the piston and how fast is it moving and accelerating, by the exact equations and by Norton’s two-harmonic approximation, and what is the peak piston acceleration?
Piston motion of the in-line slider-crank by Norton’s engine dynamics equations: exact position x = r cos ωt + l √[1 − (r/l sin ωt)²] with its exact derivatives, and the Fourier approximations from the binomial expansion x ≈ l − r²/4l + r (cos ωt + (r/4l) cos 2ωt), ẋ ≈ −rω (sin ωt + (r/2l) sin 2ωt) and ẍ ≈ −rω² (cos ωt + (r/l) cos 2ωt), with the crank-conrod ratio r/l and the peak acceleration at top dead centre rω² (1 + r/l).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| 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. |
| Crankshaft speed | rpm | Crankshaft speed, taken as constant. |
| 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: 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: Crankshaft speed
- What it is
- Rotational speed of the crankshaft, taken as steady.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3400 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Accelerations and inertia forces rise with its square.
- Get it right
- Use the speed you want to check, often the highest running speed.
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
- ω = 2π rpm ÷ 60; x = r cos ωt + l √[1 − (r/l sin ωt)²] (eq. 13.1d); ẋ and ẍ are its exact time derivatives at constant ω (eqs. 13.1e, 13.1f).
- Approximations: x ≈ l − r²/4l + r (cos ωt + r/4l cos 2ωt); ẋ ≈ −rω (sin ωt + r/2l sin 2ωt); ẍ ≈ −rω² (cos ωt + r/l cos 2ωt) (eqs. 13.3c–e).
- Peak acceleration at top dead centre = rω² (1 + r/l); g = 9.80665 m/s².
- Example: the piston accelerates at -5,789.5 m/s² at this angle; the peak is 747.91 g.
Limitations
- The crank is assumed to turn at constant speed; the cylinder axis passes through the crank centre (no offset).
- Norton recommends a crank-conrod ratio between about 1/3 and 1/5; above 1/2 the transmission angle becomes too small.
- The approximation drops the fourth and higher harmonics; its error is a fraction of one per cent at practical ratios.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the piston speed and acceleration of an engine or compressor
- Check the error of the two-harmonic approximation
- Compare crank-conrod ratios
Who it is for
- Engine and compressor designers
- Machine designers
- Mechanical engineering students
What you get
- Piston position, velocity and acceleration (exact)
- The same by the Fourier approximation
- Crank-conrod ratio
- Peak acceleration at top dead centre in g
Interpreting the result
- The second harmonic, twice per revolution, is what makes the acceleration curve flat-topped near bottom dead centre.
- Peak acceleration rises with the square of speed.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 605–610. 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.
Useful alongside · BIZOMAT original book
The crank press and its force curve
In Power Presses by the Numbers
What it covers: Why a crank press loses force with height above the bottom of the stroke, the rating point, Press 4's curve from 400 kN at 3 mm to 184 kN at 25 mm, and a pad it could not carry.
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