Unless you have a good math and physics background it may be tough to understand, but a mathematical expression can be derived for the forces generated by a crank and rod assembly as a function of crank angle and rod angle. This equation is then expanded using the binomial theoem to an infinite series that has individual terms that are a unique function of crank angle and even multiples of crank angle - 2,4,6... These terms are sinosoidal and the coefficients indicate the frequency. The "2" coefficient indicates the frequency is twice engine speed, 4 is four times, etc.

Magnitude drops rapidly with frequency. Second order forces are on the order of about 25 percent of primary and fourth order are only about four percent. For this reason, only first and second order forces are considered significant.

The calculation is only approximate because an assumption is usually made that half the rod weight is reciprocating and half rotating, but in actual fact the amount of rod mass that is reciprocating and rotating varies throughout the cycle.

The forces for each crank and rod are vector added for the cylinder arrangement and then moments are computed to determine any residual shaking forces or rocking couples. These calculations have all been done and the basic derivation and resultants for most common engine configurations are listed in various references, one of which is Taylor's IC engine textbook, which is still available both new an used.

Engines generate a wide spectrum of vibrations, but torsional, valve gear, and combustion generated vibrations are usually low in magnitude compared to first and most second order reciprocating/rotating vibrations of the bottom end, and engine mounting schemes are usually used to isolate higher order reciprocating/rotating bottom end vibrations and the others I mentioned. Second order rocking couples from V6 configurations are usually handled quite well with good engineering of the engine mounting.

The toughest to control is the second order vertical shaking force generated by inline fours. The twin counter-rotating balance shaft scheme was invented and patented by an Englishman named Lanchester in 1914, but was seldom, if ever, used during his patent protection because with the relatively crude technology of the time, it was cheaper just to make an inline six rather than the drive and extra machining and bearings required for the balance shafts. Engine "packaging" and size weren't that important back then.

Somehow Mitsubishi managed to patent what was basically the same scheme in the seventes (I've never figured out how.) and Porsche bought a license for their 944 engine. After about 1993 when the patent expired the scheme found wider use in inline four's, especially those over 2 liters, but Mercedes never adopted it.

Duke