I did some research into this topic by mathematical analysis, also wrote a program to simulate the vibration. The result is exactly the same as found in any textbooks. However, to explain to you in simpler language, let us see in this way ...
The left pictures shows an inline-4 engine. It fires once every 720° / 4 = 180° crank angle, hence 2 of the pistons are in exactly the same position and move in the same direction, while the remaining 2 pistons are also a pair. To avoid the end-to-end vibration as experienced in 3-cylinder engines, car makers always arrange the pistons as shown in the picture, that is, symmetrical. In other words, piston 1 and 4 are a pair, while piston 2 and 3 form another pair. Therefore movement of piston 1 will be balanced by the symmetric piston 4. The same goes for piston 2 and 3.
That’s just the end-to-end vibration with respect to the engine center. What about the resultant upward / downward vibration ? It seems that the movement of piston 1 is counter balanced by piston 2, while piston 3 counters piston 4. However, this is just skin-deep. More professional speaking, that just proves the balance of 1st order force. The second order force (which can be derived from equation) is normally much smaller than the 1st order force and it is rotating at twice the frequency of the 1st order force. Nevertheless, the configuration of inline-4 actually multiplies the magnitude of 2nd order force thus making it hard to be ignored, especially is for larger engines.
A simpler explanation is given in the below pictures, which compare a perfectly balanced boxer-4 engine with an inline-4 engine.
Solution - Twin-balancer shafts
The longer the stroke, the heavier the pistons and con-rods, the more second order vibration generates. Unfortunately, car makers favour straight-four engine for its advantages of low cost and compact dimensions. Since the 80s, car engineers regard 4-pot engines larger than 2 litres in capacity had better to be equipped with twin-balancer shafts to dampen the vibration. Although the strengthening of engine block, the use of hydraulic engine mount and lightweight pistons helped breaking such rule, the trend of pursuing refinement once again led to many engines larger than 2 litres to use balancer shafts.
Balancer shaft was
by British automotive engineering master Dr. Frederick Lanchester in
early 20th century. Mitsubishi obtained the patent and put it into mass
production in the 1976 Colt Celeste 2000, then Fiat group used it in
Lamda engine series, including the 1.6-litre Delta HF turbo and Fiat
/ Lancia Thema's 2-litre turbo. Meanwhile, Saab 9000 and Porsche 944
introduced it into their powerful inline fours. All these car makers
license from Mitsubishi.
To deal with second order vibration, a pair of balancer shafts is needed, driven by the engine and rotate in opposite directions to each other, at twice the speed of the crankshaft. They locate in either sides of the engine. One of them is positioned just above the crank shaft level, the other is far above. Counter weights on the balancer shafts will completely cancel the second order force, thus result in a silky-smooth rotation.
The use of 2 balancer shafts instead of a large single one is because the vibration generated by the engine is mostly in vertical direction. 2 shafts rotating in opposite direction can cancel each other’s transverse force and result in a net vertical force which is used to balance the vibration.
Without twin-balancer shaft, Porsche would have been impossible to make the 3-litre inline-four which powered the 944 S2 and 968. That’s the biggest four-cylinder engine in modern cars.
The inline-5 engine
once every 720° / 5 = 144° crank angle. As a result, the
design is as shown in below. Firing order is 1-3-5-4-2.
My mathematical analysis proved that both its resultant first order force and second order force are balanced. Therefore it doesn’t need the twin-balancer shafts as a big 4-cylinder engine. However, it generates end-to-end vibration like 3-cylinder engines, because piston 1 is not in the same position as piston 5, and piston 2 is not in the same position as piston 4. Therefore both ends of the engine will vibrate up and down with respect to the engine center.
Solution - single balancer shaft
Obviously, the solution is the same as 3-pot engines, that is, employ a balancer shaft on which there are counter weights moving in the opposite direction to the pistons. The balancer shaft is driven by the engine at the same speed as the crankshaft.
Is that enough to make 5-cylinder engine as smooth as 6-cylinder? no. For packaging reasons, the balancer shaft cannot be placed in the most optimized position, that is, right above or below the crankshaft. Therefore it has to be offset to either side of the engine, resulting in incomplete cancellation of vibration.