A system is statically balanced when the resultant of all centrifugal forces due to rotating masses is zero — i.e., the centre of mass lies on the axis of rotation.
This is checked by ensuring the vector sum of all \(mr\) terms equals zero:
\[\sum mr = 0\]
A single correction mass placed in the same plane can achieve static balance.
Static balance alone is insufficient for systems with masses distributed across different axial planes. Dynamic balance requires two conditions to be satisfied simultaneously:
\[\sum mr = 0 \qquad \text{and} \qquad \sum mrl = 0\]
This generally requires two correction masses placed in two chosen planes.
For coplanar rotating masses, a graphical or analytical approach uses the force polygon. Closing the \(mr\) vector polygon determines the required balancing mass and its angular position.
For masses distributed along a shaft, the procedure is:
The piston acceleration in a crank-slider mechanism gives rise to two force components:
Primary force (first-order):
\[F_{\text{primary}} = mr\omega^2 \cos\theta\]
Secondary force (second-order):
\[F_{\text{secondary}} = \frac{mr\omega^2 \cos 2\theta}{n}\]
where \(n = \ell / r\) is the ratio of connecting rod length \(\ell\) to crank radius \(r\).
In multi-cylinder in-line engines, the angular phasing of the cranks is arranged to cancel primary and secondary forces and couples. The chapter examines which engine configurations achieve complete or only partial balance.
Complete balance is not always possible or practical — particularly for single-cylinder engines. In such cases:
| Concept | Equation |
|---|---|
| Centrifugal force | \(F = mr\omega^2\) |
| Static balance condition | \(\sum mr = 0\) |
| Dynamic balance condition | \(\sum mr = 0\) and \(\sum mrl = 0\) |
| Primary reciprocating force | \(F = mr\omega^2 \cos\theta\) |
| Secondary reciprocating force | \(F = mr\omega^2 \cos 2\theta \, / \, n\) |
Balancing is fundamentally a vector problem. Both graphical (polygon) and analytical methods are used throughout. Dynamic balancing always requires satisfying both force and moment equilibrium, and the chapter builds progressively from simple single-plane cases up to the complexity of multi-cylinder engine analysis.