Many modern engines are fitted with balance shafts, to smoothe out the vibrations created by piston, con-rod and crankshaft movement. These shafts represent a remarkable vindication of the power of mathematical techniques to solve engineering problems, because the mathematics involved in analysing piston motion, so as to design the correct balancing mechanisms, is only an approximation of the actual motion, yet the practical results are quite amazing.

Even if the engine has been designed to be very smooth, the normal production process would still require a balancing operation to be performed on the rotating and reciprocating parts, to take care of small variations in dimensions and material density. This normally requires the removal of small amounts of metal at the correct places. It is a similar process to adding weights to a car’s road wheel that should, in theory, be balanced already.

Vibrations are usually cancelled by introducing another vibration with a mirror-image waveform, in the same way that the effect of a force can be cancelled by introducing a force of the same size, but in the opposite direction. For example, if you hold a heavy object in one outstretched arm while you’re being rotated on a platform, you’ll feel a pull on that arm. The magnitude of the pull will increase as the square of the speed, so that if the rotation is speeded up you’ll eventually be pulled off the platform. This is exactly what happens when a crankshaft is subjected to an unbalanced force, except that the crank is restrained from leaving its position by the main bearings. The force will create a vibration because it rotates with the crank, which means that it changes direction all the time. It will also stress the material.

Now imagine that you’re still on the rotating platform but have an object of the same mass in each hand, arms stretched out and pointing in opposite directions. The forces will oppose each other, and you’ll find it easy to stay on the platform, because the forces are balanced, but as the speed goes up your arms will feel an increasing pull. In fact, the platform will soon reach a speed where your arms will be pulled out of their sockets! Similarly, if a crankshaft experiences forces that balance each other, there is no vibration, but the internal stress levels are raised. Using high-strength steel can accommodate this.On in-line four- and six-cylinder engines, the crankshaft is usually balanced on its own, so that the reciprocating masses, consisting of the piston assembly masses and about one third of the con-rod masses, have to balance each other. This occurs fully on an in-line six-cylinder engine, but in-line and flat fours have some residual imbalance that I will investigate.

The exact in-and-out movement of the pistons is not easy to visualise, because of the complications introduced by the sideways movement of the con-rod. For this reason, engine balancing is usually described in terms of primary and secondary forces and couples, whose effect can be added together to get an approximate picture of whether they are in balance or not.

A primary force, or couple, is one that would arise if the con-rods were infinitely long, whereas a secondary force, or couple, is the additional effect due to the actual con-rod length. A couple can be thought of as a torque (force times perpendicular distance) that usually produces no motion.

**PISTON MOTION **

The secret to understanding the inertia force on a piston is to examine the acceleration, because this force is equal to the acceleration times the reciprocating mass (Newton’s first law of motion). If we consider a piston starting at top dead centre (zero crankshaft angle with the vertical centre line), then one would expect the piston to accelerate to maximum speed at a point halfway down the bore (90 crankshaft degrees), and then decelerate to come to a stop at bottom dead centre, before changing direction to start moving upwards again. This does not happen, because the con-rod deviation, whose magnitude depends on the ratio between the crank radius and con-rod length, has an influence on the angle where maximum piston speed occurs. In the example used for the graphs, the crank radius/con-rod length ratio is four, and the con-rod deviation speeds up the piston movement to such an extent that it reaches maximum speed at about 75 crankshaft degrees. On the up stroke, the reverse happens, and the con-rod deviation slows the piston down so that maximum speed occurs at 360-75 = 285 crankshaft degrees, instead of at mid-stroke (270 degrees).

This shows that the movement towards the crank is not the same as the movement away from the crank, purely because we’ve had to take the con-rod’s deviation into account. On an in-line four-cylinder engine, pistons one and four are moving away from the crank while the other two pistons are moving towards the crank, and vice versa. On a flat four, pistons one and two are moving away from the crank while the other two are moving towards the crank, and vice versa. These different movements play a role in determining each engine’s state of balance.

An easy way of understanding the above is to look at a graph of piston acceleration. Figure I shows the acceleration of a single piston for one revolution. The labelled curves show the primary, secondary and total acceleration (sum of the other two accelerations). The primary acceleration is the same as the acceleration in a vertical direction of a point on the big-end journal vertically below the piston centreline. This is shown as the black curve in figure I. The secondary acceleration, due to the sideways con-rod deviation, is shown as the purple curve. It has two peaks because the con-rod deviates twice per revolution. The yellow curve is the sum of the two accelerations, ie it shows the total approximate acceleration of one reciprocating mass.

The foregoing applies to one piston in an engine, and we will use a comparison between an in-line four-cylinder and a flat (or boxer) four-cylinder to illustrate the way the forces and moments are combined to decide whether the engine is in balance or not.

**VERTICAL IN-LINE FOUR – FORCE BALANCE**

Figure II show the primary and secondary accelerations of the first two pistons in a vertical four-cylinder engine, plotted against crankshaft degrees, with zero degrees being equivalent to number one piston at top dead centre. The primary accelerations of pistons number one and two are mirror images of each other, so that they cancel, but the secondary accelerations are identical, so that the secondary curve is actually one curve superimposed onto another. The above means that the net primary force is zero, but the total secondary force is double one of these forces.

Pistons number three and four move in a mirror image of the first two, hence we can conclude that the complete engine will be in primary force balance, but a secondary force equal to four times the unbalanced secondary force of one reciprocating mass will remain unbalanced. Figure III shows this unbalanced acceleration.

**VERTICAL IN-LINE FOUR – MOMENT BALANCE **

Figure IV shows a diagram of the directions of the primary and secondary forces, when piston number one is just starting to move towards the crankshaft. Note that the secondary forces act in the same direction as the primary forces for pistons one and four, which are moving downward, and act in the opposite direction to the primary forces for pistons two and three, which are moving upwards. We can again see that the primary forces cancel and the secondary forces add up.

They all act at the geometric centres of the big-end journals. If we label the primary forces F1 to F4 and the secondary forces f1 to f4, for the four cylinders, and the distance between big-end journals as d, then the moments, taken about the vertical crank centreline, are:

Clockwise: F4 x (1,5).d + F2 x (0,5).d + f3 x (0,5).d

Anticlockwise: F1 x (1,5).d + F3 x (0,5).d + f2 x (0,5).d

Since all the F-forces have the same size, and all the f-forces have the same size, these moments are equal in size but opposite in direction, so they balance out. We conclude that an in-line four only has an unbalanced secondary force. This is a force that tries to bounce the engine up and down twice per revolution, but a balance shaft will cancel it out, as we shall see.

**FLAT FOUR -FORCE BALANCE **

The crankshaft of a flat engine looks just like the crank of an in-line engine, but the pistons move differently. Figure V shows the primary and secondary forces for the first two pistons of an engine where the pistons oppose each other, such as a Volkswagen Beetle or a Subaru. Zero degrees is again equivalent to number one piston being at top dead centre, but in a flat engine number two piston is also at top dead centre, but is at the other side of the cylinder block.

Primary and secondary accelerations now cancel out each other, because they’re mirror images of each other. The other two cylinders are always half a revolution away: in other words, they will be at bottom dead centre when the first two are at top dead centre. The accelerations from these two pistons will also cancel each other out, so that a flat four is in perfect primary and secondary force balance.

**FLAT FOUR -MOMENT BALANCE**

Figure VI shows the direction of the primary and secondary forces, with pistons number one and two just starting to move towards the crankshaft, and pistons number three and four starting to move away from the crankshaft. Once again, the secondary forces act in the same direction as the primary forces for pistons that are moving toward the crank, and act in the opposite direction to the primary forces for pistons that are moving away from the crank. The forces are labelled as before, and if we now take moments about the crank centre we get:

Clockwise: F4 x (1,5).d + F2 x (0,5).d + f3 x (0,5)d + f2 x (0,5)d

Anticlockwise: F1 x (1,5).d + F3 x (0,5)d + f1 x (1,5).d + f4 x (1,5)d

All the F-forces have the same size, and all the f-forces have the same size, and for perfect balance the clockwise moments must be equal to the anticlockwise moments. When we equate them, we find that the primary moments cancel out, but the secondary moments add up to 3f.d- fd = 2f.d, showing that a moment of 2f.d remains, where f is the size of the maximum secondary force and d is the distance between crankpin centrelines. This shows that a flat four has an unbalanced secondary moment, which tries to rotate the engine in a horizontal plane, first clockwise and then anticlockwise, during every engine revolution. This is hardly noticeable, and explains why flat engines are very smooth, and certainly don’t need balance shafts.

**BALANCE SHAFTS**

We can now see where balance shafts may be useful. Unbalanced secondary forces, opposite in direction to the ones on an in-line four, can be generated on an additional shaft that looks like a camshaft, but with only four fairly large lobes, rotating at twice engine speed. Unfortunately, this will produce the required forces in every direction as it rotates, while the pistons only produce the secondary forces in the up-and-down direction. The answer is to use two contra-rotating balance shafts, whose combined effect supplies the required force in the vertical direction but cancel out in every other direction. The way this is achieved is shown in figure VII.

These shafts were first patented by Fred Lanchester before WW1, and used on big four-cylinder diesel engines, before being “discovered” by Mitsubishi in the ’60s for its four-cylinder petrol engines. They are now common on many in-line fours, and are employed in combination with hi-tech engine mountings to make these engines very smooth.

The many other engine types will be dealt with in a future article.