BIZOMAT MAKEOS · Mechanics
Inline Multicylinder Engine Balance (Shaking Force, Inertia Torque, Shaking Moment, Firing)
For an inline engine with my crankshaft phase angles and cylinder spacing, are the primary and secondary shaking forces, the inertia torque harmonics and the shaking moments balanced, how large are the unbalanced ones, and what crank spacing gives even firing?
Balance of inline multicylinder engines by Norton’s phase-angle sums: shaking forces cancel through the second harmonic when Σ cos φi, Σ sin φi, Σ cos 2φi and Σ sin 2φi are all zero (and the fourth and sixth with 4φi and 6φi), inertia torque when the sums of φi, 2φi and 3φi are zero, and shaking moments when Σ zi cos nφi and Σ zi sin nφi are zero for n = 1, 2, 4 and 6; with the reciprocating mass, crank radius, rod length and speed you enter it gives the amplitudes of the unbalanced primary and secondary forces and moments, and the even firing interval 360° ÷ n (two-stroke) or 720° ÷ n (four-stroke).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Number of cylinders | cylinders | Number of cylinders in line. |
| Stroke cycle | choice | Two-stroke or four-stroke cycle, which sets the even firing interval. |
| Phase angle φ2 of cylinder 2 | ° | Crank phase angle of cylinder 2 relative to cylinder 1 (φ1 = 0). |
| Phase angle φ3 | ° | Crank phase angle of cylinder 3. |
| Phase angle φ4 | ° | Crank phase angle of cylinder 4. |
| Phase angle φ5 | ° | Crank phase angle of cylinder 5. |
| Phase angle φ6 | ° | Crank phase angle of cylinder 6. |
| Phase angle φ7 | ° | Crank phase angle of cylinder 7. |
| Phase angle φ8 | ° | Crank phase angle of cylinder 8. |
| Distance of cylinder 1 from the moment reference z1 | mm | Distance along the crankshaft from the moment reference point to cylinder 1. |
| Cylinder spacing | mm | Distance between neighbouring cylinder centres. |
| Reciprocating mass per cylinder mB (0 = sums only) | kg | Reciprocating mass of one cylinder (piston plus conrod wrist pin share), to size the unbalanced forces. |
| Crank radius r | mm | Crank throw, half the stroke. |
| Connecting rod length l | mm | Connecting rod length. |
| Crankshaft speed | rpm | Crankshaft speed for the force amplitudes. |
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: Number of cylinders
- What it is
- Number of cylinders arranged in one line along the crankshaft.
- Units and format
- Enter a number in cylinders: cylinders. The form accepts 1 cylinders to 8 cylinders. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 cylinders as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets how many throws are summed.
- Get it right
- Enter up to eight; phase angles beyond this count are ignored.
What this means: Stroke cycle
- What it is
- Whether each cylinder fires every revolution or every second revolution.
- Units and format
- Choose one of 2 options from the list: Four-stroke; Two-stroke.
- Example
- The form opens with “Four-stroke” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Sets the even firing interval.
- Get it right
- Two-stroke engines fire once per revolution, four-stroke once per two.
What this means: Phase angle φ2 of cylinder 2
- What it is
- Angle of crank throw 2 relative to throw 1, measured round the crankshaft.
- 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 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Phase angle φ3
- What it is
- Angle of crank throw 3 relative to throw 1, measured round the crankshaft.
- 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 180° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Phase angle φ4
- What it is
- Angle of crank throw 4 relative to throw 1, measured round the crankshaft.
- 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 270° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Phase angle φ5
- What it is
- Angle of crank throw 5 relative to throw 1, measured round the crankshaft.
- 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 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Phase angle φ6
- What it is
- Angle of crank throw 6 relative to throw 1, measured round the crankshaft.
- 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 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Phase angle φ7
- What it is
- Angle of crank throw 7 relative to throw 1, measured round the crankshaft.
- 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 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Phase angle φ8
- What it is
- Angle of crank throw 8 relative to throw 1, measured round the crankshaft.
- 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 0° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Decides which harmonics of force, torque and moment cancel.
- Get it right
- Measure in the direction of rotation; unused cylinders are ignored.
What this means: Distance of cylinder 1 from the moment referenc…
- What it is
- Distance from the point moments are taken about to cylinder 1.
- Units and format
- Enter a number in mm: millimetres. The form accepts -100000 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the moment sums but not whether they cancel.
- Get it right
- Use 0 to take moments about cylinder 1, or the distance to one end of the crankshaft.
What this means: Cylinder spacing
- What it is
- Distance along the crankshaft between neighbouring cylinder centres.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales the shaking moments.
- Get it right
- Measure the bore centre spacing of the block.
What this means: Reciprocating mass per cylinder mB (0 = sums on…
- What it is
- Reciprocating mass of one cylinder: piston plus the wrist pin share of the rod.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sizes the unbalanced forces and moments.
- Get it right
- Use the engine inertia force tool to find it, or 0 for the sums only.
What this means: Crank radius r
- 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
- 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 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.
Method and formulas
- φ1 = 0; zi = z1 + (i − 1) × spacing.
- Shaking force balanced in harmonic k (k = 1, 2, 4, 6) when Σ cos kφi = Σ sin kφi = 0 (eq. 14.3); inertia torque balanced for k = 1, 2, 3 (eq. 14.5); shaking moment when Σ zi cos kφi = Σ zi sin kφi = 0 (eq. 14.7).
- Unbalanced amplitudes from eqs. 14.2d and 14.6b: primary force mB rω² √[(Σ cos φi)² + (Σ sin φi)²], secondary force (r/l) mB rω² √[(Σ cos 2φi)² + (Σ sin 2φi)²], and the same with zi for the moments.
- Even firing interval Δφ = 360° ÷ n (two-stroke) or 720° ÷ n (four-stroke) (eq. 14.8).
- Example: the primary force sums are 0 and 0.
Limitations
- All cylinders are assumed alike in mass and displacement and equally spaced; the expressions keep the harmonics Norton includes.
- The rotating masses are assumed counterweighted on each throw.
- Vee and opposed engines need the bank angle as well and are not covered.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a crankshaft layout for primary and secondary balance
- Find the unbalanced shaking forces and moments of an inline engine
- Find the even firing interval for a cylinder count
Who it is for
- Engine and compressor designers
- Machine designers
- Mechanical engineering students
What you get
- Force sums for the 1st, 2nd, 4th and 6th harmonics
- Inertia torque sums for the 1st to 3rd harmonics
- Moment sums for the 1st, 2nd, 4th and 6th harmonics
- Unbalanced force and moment amplitudes and the even firing interval
Interpreting the result
- The primary terms matter most because of their size; the secondary terms are multiplied by r/l and are smaller.
- A layout can balance forces and torques yet leave a shaking moment that pitches the engine, as Norton’s 0, 90, 180, 270° four-cylinder does.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 645–653. Topic reference; no source tables, figures or examples are reproduced.
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