Shaking forces of four engines by Martino Peretto

Examples
In the following examples, the blue arrow represents the first inertia force and the red one the second inertia force.

DUCATI DESMOSEDICI GP5 TWINPULSE
APRILIA RSV4 1000
BMW S1000R YAMAHA FZS 1000 FAZER

And now the theory

Let us consider the schematic drawing of a four-cylinder engine, as shown in the following figure:

The reciprocating motion of pistons generates primary (first order) and secondary
(second order) inertia forces, expressed as


where i = 1 , 2 , 3 , 4 ;
m_C = reciprocating mass located at the wrist pin (composed of piston, rings, wrist pin masses and of a portion of connecting rod mass);
w= crankshaft angular velocity;
r = crank radius;
l= crank radius vs connecting rod length ratio;
q = w t= = crankshaft angular position (where t is the time coordinate);
a i = i th - cylinder phase.
The components of inertia forces are respectively

and

where
bi = i th - cylinder to x axis inclination. Le componenti delle forze risultano:
The first order ones generate a vector which rotates with the same velocity as the
crankshaft:

The second order ones generate a vector which rotates with a double velocity:

Both vectors trace an ellipsis or a circle in the xy plane, depending on the amplitudes and phases of the respective components. Shaking forces do not act in the same plane, but along cylinders axes.
Consequently x components generate yawing couples (i.e. : they tend to rotate the engine around y axis)

and
while y components generate pitching couples (i.e. : they tend to rotate the engine
around x axis)
e ;
where
bi = distance between i th - cylinder axis and origin O .

			Scheme of four-cylinder engine and resultant shaking forces