If you have ever driven a car with a 4 cylinders engine and 3 cylinders one, maybe you felt that in the second case more vibrations are transmitted from the engine to the car; furthermore, the engine torque is less smooth. On older three-cylinders engines this difference can be also eared.

Of course, a part of this issue is due to one cylinder less respect to a four-cylinder engine, with a more variable engine torque as result.

In this post I would like to focus on another aspect, maybe less intuitive than fire spacing, but as much important: engine balancing, strictly related to engine vibrations.

## First and second-order forces and moments

Let’s start with the more simple forces to visualize. Looking at the picture and imagine the dynamic mechanism, we can guess that a centrifugal force acts on the connecting rod big end.

Without talking about mathematical details, it is shown that alternative forces generated during the combustion phase can be expressed with the following mathematical expression:

\[

F_r=m_rR\omega^2[cos\theta+(R/L)cos2\theta]

\]

This formula means that the generic alternative force is a function of crank angle $\theta$, so its module is variable. In particular, it can be thought as a sum of two forces (of which we take only the projection on cylinder axis): the first one variable as the crank angle, the second one as the double of the crank angle. They are respectively first order and second-order forces, that create first order and second-order moments.

Why are they so important? Because if not balanced, vibrations are transmitted to the chassis.

## First-order forces and moments on three cylinders engine

Let’s analyze in-depth the forces in using the image above. Cylinders are all in line and cranks are equally spaced each other by 120°. If we consider centrifugal forces, it is easy visualized that they have the same intensity and form a closed triangle: this means that are equilibrated themselves. Also first-order reciprocating forces are in the same condition, so another closed polygon of force is created and they are self equilibrated. In the following picture we can see this condition, where cylinder n°1 is taken as reference.

The same result is obtained if crank angles are calculated each one respect to the corresponding cylinder axis. In order to repeat the analysis on second-order forces, we just need to double the angle values, obtaining the following scheme.

As we can see forces on cranks n° 2 and n°3 are inverted, so the polygon remains closed and forces are self equilibrated.

## Vibrations reduction: moments balancing

**I order moments**

The analysis of moments needs more attention. If we evaluate the equilibrium respect to point 2, we obtain the following scheme, more readable if we have a look also to the initial engine scheme:

Force $F_{1,p}$ create a moment $F_1C$ around point 2; force $F_{3,p} $ create a moment $F_3C$. The balancing of this moment needs an additional moment $\sqrt{3}FC$. How is it possible to create this? One solution is the use of two additional masses on the crankshaft, spaced by 180°, with a double effect: the total balance of centrifugal forces, the moment of which points in the same direction as the first-order one, and balance the last one. This solution has a problem: consider cylinder n°1 for simplicity. The intensity of reciprocating forces (that generates the moment) is a function of crank angle, instead the intensity of the balancing moment is always the same. It means that a part of the balancing moment became an imbalance! If we accept to balance only a part of the first-order moment the solution is to add the masses in order to balance totally centrifugal forces and only half of first-order moments. This is a compromise solution between balancing and simplicity.

**Alternative balancing**

Another (more efficient) solution is the following: If we consider the compromise solution previously analyzed, it is possible to add a countershaft rotating at the same angular speed of the crankshaft but in opposite direction, and adding two additional masses on it is possible to balance the remaining half part of the unbalancing moment without any additional negative effect. Of course, this is a more complicated solution, in terms of engineering and production.

**II order moments**

The analysis is the same as the previous, so we can calculate moments respect to point 1. Also this system needs a balancing moment $\sqrt{3}Fc$.

Despite the previous case, now is impossible to try to balance second-order moments using the crankshaft in anyway, because they change direction as twice the crank angle. A solution is to add two countershafts with balancing masses, rotating at twice the angular speed of the crankshaft, counter-rotating respect each other.

There are many disadvantages in the use of this solution. It is an important complication in the engine design, friction is increased (organic efficiency reduced) and the balancing of the secondary moment is less important than the primary moments one, because the intensity is lower.

The four cylinders engine, for example, needs to be balanced only respect to first-order reciprocating forces, using two countershafts with the same angular speed of the crankshaft.

The six-cylinder engine is inherently self-balanced, both statically and dynamically; in this way no vibrations are transferred to the chassis.

The following video shows how countershafts are composed and where are mounted in a four cylinders engine.