Methods of static and dynamic balancing of an engine, and first, second and higher order moments:

Static balancing: The crankshaft (or rotating assembly) is balanced so that the resultant of all the centrifugal forces of the rotating masses is zero when the shaft is not rotating. The shaft is supported on knife edges/rollers and if it has a heavy point it will roll to bring the heavy side down. Static balance is achieved by adding or removing mass (counterweights on the crank webs, or drilling/grinding) so the centre of mass lies on the axis of rotation. Static balance alone does not ensure dynamic balance.

Dynamic balancing: A shaft can be statically balanced yet have a couple acting because the unbalanced masses lie in different planes along the shaft, producing a rocking couple when rotating. Dynamic balancing is done on a balancing machine where the shaft is rotated and the vibrations at the two ends are measured; correction masses are added/removed at calculated positions in the two end planes so that both the resultant force and the resultant couple are zero. For an engine, the reciprocating and rotating masses are balanced by counterweights on the crank webs and by the arrangement of the crank throws (firing order) so the forces and couples of the different cylinders cancel.

First order moments: The primary (first order) reciprocating force varies once per revolution (at engine speed). It arises from the acceleration of the reciprocating masses (piston, rings, small-end of the connecting rod). The first-order moment is the moment (couple) produced by the first-order forces of the different cylinders acting at different positions along the crankshaft; it is balanced by arranging the crank throws so the forces cancel and by using counterweights/balance shafts.

Second order moments: The secondary (second order) reciprocating force varies at twice engine speed (2x). It arises because the connecting rod is of finite length, so the piston acceleration has a second harmonic. The second-order moment is the couple produced by the second-order forces of the cylinders; it is balanced by using counter-rotating balance shafts running at twice engine speed (Lanchester-type) or by the arrangement of the cylinders.

Higher order moments: Third and higher order forces/moments arise from the higher harmonics of the piston acceleration (due to the connecting rod geometry) and from the gas-pressure and inertia effects. They are small compared with the first and second order, and are usually not balanced by design (they are accepted as residual vibration) because balancing them would be impractical; they are minimised by the choice of cylinder number and firing order and by the use of vibration dampers/isolators.

In practice, for a multi-cylinder engine, the crank throws are arranged (e.g. 6-cylinder inline with 120 deg spacing) so that the primary and secondary forces and moments largely cancel, and any residual is handled by counterweights and balance shafts; the higher-order components are small and are damped by the engine mounting and vibration dampers.