BIZOMAT MAKEOS · Mechanics
Vee Engine Shaking Force, Shaking Moment and Inertia Torque (Shared Crank Throws)
For a vee engine with two cylinders on each crank throw, my vee angle and my crank phase angles, are the primary and secondary shaking forces and moments balanced, does an unbalanced primary rotate at constant size so a crank counterweight can take it, and how large is the inertia torque?
Balance of a vee engine by Norton’s bank equations: each throw drives one cylinder in each bank, the right bank sees the crank at θ + γ and the left bank at θ − γ, where 2γ is the vee angle and θ = ωt − φi, and each cylinder’s force is mB rω²[cos(θ ± γ) + (r/l) cos 2(θ ± γ)] along its bank. The bank forces combine to Fx = (FL + FR) cos γ along the vee centreline and Fy = (FL − FR) sin γ across it, the moments with the cylinder positions zi, and the inertia torque sums ½ mB r²ω²[(r/2l) sin x − sin 2x − (3r/2l) sin 3x] over both banks. The tool follows these over one revolution and reports the peak primary and secondary force and moment, whether each is balanced or rotates at constant magnitude, and the values at a crank angle you choose.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Number of crank throws (cylinders per bank) | throws | Number of crank throws; each throw carries one cylinder from each bank. |
| Vee angle between the banks 2γ | ° | Angle between the two cylinder banks; the bank angle γ is half of it. |
| Phase angle φ2 of throw 2 | ° | Crank phase angle of throw 2 relative to throw 1 (φ1 = 0). |
| Phase angle φ3 | ° | Crank phase angle of throw 3. |
| Phase angle φ4 | ° | Crank phase angle of throw 4. |
| Phase angle φ5 | ° | Crank phase angle of throw 5. |
| Phase angle φ6 | ° | Crank phase angle of throw 6. |
| Distance of throw 1 from the moment reference z1 | mm | Distance along the crankshaft from the moment reference point to throw 1. |
| Throw spacing | mm | Distance between neighbouring throws. |
| Reciprocating mass per cylinder mB (0 = factors only) | kg | Reciprocating mass of one cylinder (piston plus the rod’s share at the wrist pin), to size the 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 and torque values. |
| Crank angle ωt for the instantaneous values | ° | Crank angle of throw 1 from the vee centreline at which the instantaneous values are given. |
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 crank throws (cylinders per bank)
- What it is
- Number of crank throws, each carrying one cylinder from each bank.
- Units and format
- Enter a number in throws: throws. The form accepts 1 throws to 6 throws. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 throws 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 cylinder pairs are summed.
- Get it right
- A V8 has four throws; a V-twin has one.
What this means: Vee angle between the banks 2γ
- What it is
- Angle between the centre lines of the two cylinder banks.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 180°. 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
- Changes how the two banks’ forces add and cancel.
- Get it right
- Common vee angles are 60° and 90°; enter 0 for an inline pair.
What this means: Phase angle φ2 of throw 2
- What it is
- Crank phase angle of throw 2 relative to throw 1, which is at 0°.
- 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 and moment cancel.
- Get it right
- Read it from the crankshaft drawing; enter 0 for throws beyond the count.
What this means: Phase angle φ3
- What it is
- Crank phase angle of throw 3 relative to throw 1, which is at 0°.
- 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 and moment cancel.
- Get it right
- Read it from the crankshaft drawing; enter 0 for throws beyond the count.
What this means: Phase angle φ4
- What it is
- Crank phase angle of throw 4 relative to throw 1, which is at 0°.
- 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 and moment cancel.
- Get it right
- Read it from the crankshaft drawing; enter 0 for throws beyond the count.
What this means: Phase angle φ5
- What it is
- Crank phase angle of throw 5 relative to throw 1, which is at 0°.
- 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 and moment cancel.
- Get it right
- Read it from the crankshaft drawing; enter 0 for throws beyond the count.
What this means: Phase angle φ6
- What it is
- Crank phase angle of throw 6 relative to throw 1, which is at 0°.
- 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 and moment cancel.
- Get it right
- Read it from the crankshaft drawing; enter 0 for throws beyond the count.
What this means: Distance of throw 1 from the moment reference z1
- What it is
- Distance along the crankshaft from the moment reference point to throw 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
- Moves the reference for the shaking moments only.
- Get it right
- Enter 0 to take moments about throw 1.
What this means: Throw spacing
- What it is
- Distance along the crankshaft between neighbouring crank throws.
- 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 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shaking moments rise in proportion.
- Get it right
- Measure centre to centre of the crank pins.
What this means: Reciprocating mass per cylinder mB (0 = factors…
- What it is
- Reciprocating mass of one cylinder: the piston plus the rod’s share at the wrist pin.
- 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
- Forces, moments and torque rise in proportion.
- Get it right
- Weigh the piston assembly; enter 0 to see the factors only.
What this means: Crank radius r
- What it is
- Crank radius from the main bearing centre to the crank pin centre.
- 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
- Forces rise with r; the secondary also with r/l.
- Get it right
- Half the piston stroke.
What this means: Connecting rod length l
- What it is
- Connecting rod length between the crank pin 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 the secondary terms.
- Get it right
- Measure pin centre to pin centre.
What this means: Crankshaft speed
- What it is
- Rotational speed of the crankshaft for the force and torque values.
- 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
- Forces and torque rise with the square of speed.
- Get it right
- Check at the highest running speed.
What this means: Crank angle ωt for the instantaneous values
- What it is
- Angle of throw 1 from the vee centre line at which to report the values.
- 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
- Changes only the instantaneous values.
- Get it right
- Step it to follow the forces through a revolution.
Method and formulas
- γ = vee angle ÷ 2; θ = ωt − φi (eq. 14.10a); zi = z1 + (i − 1) × spacing.
- Right bank: FR = mB rω²[cos(θ + γ) + (r/l) cos 2(θ + γ)]; left bank: FL with θ − γ (eq. 14.10d).
- Fx = (FL + FR) cos γ, Fy = (FL − FR) sin γ (eq. 14.10j); moments Mx = (ML − MR) sin γ, My = (−ML − MR) cos γ with the zi-weighted bank sums (eqs. 14.11).
- Inertia torque: ½ mB r²ω² Σ [(r/2l) sin x − sin 2x − (3r/2l) sin 3x] with x = θ + γ and θ − γ for the two banks (eqs. 14.12).
- The forces and moments are followed at 1,440 crank angles over one revolution; a harmonic whose magnitude does not change rotates at constant size.
- Example: the peak primary moment factor is 316.23 mm with balanced shaking forces.
Limitations
- Both cylinders on a throw are taken in the same plane along the crankshaft; any offset between the banks is ignored.
- The force expression keeps the first two harmonics, as Norton’s approximation does; rotating masses are assumed counterweighted on each throw.
- All cylinders are assumed alike in mass and size and the throws equally spaced.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a vee engine layout for primary and secondary balance
- See whether an unbalanced primary can be taken by a crankshaft counterweight
- Compare vee angles and crank phasing for shaking forces and moments
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Peak primary and secondary shaking force and whether each is balanced or rotates at constant magnitude
- Peak primary and secondary shaking moment
- Peak inertia torque
- Force, moment and inertia torque at the chosen crank angle
Interpreting the result
- A harmonic of constant magnitude turning with the crank can be cancelled by a counterweight on the crankshaft; one that changes size cannot.
- The secondary terms are multiplied by r/l and turn at twice crank speed, so crankshaft counterweights cannot balance them.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 664–669. 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
Unbalance and balancing in place
In Vibration Monitoring by the Numbers
What it covers: How unbalance makes a 1× vibration, the force on the bearings, balance quality grades, single-plane balancing with a trial weight and vectors, splitting a correction, and a trial weight left on.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.