Question 3 of 7: Shaking force of a two-cylinder opposed engine

Question 3: Shaking force of a two-cylinder opposed engine (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Two opposed pistons ($180^\circ$ apart) sharing a single crankpin in one transverse plane; crank $r$, rod $l$, reciprocating mass $m$ each, crank speed $\omega$ (constant). Rotating masses already balanced.

Find. The net shaking force of the two reciprocating masses and a balancing verdict for the first (primary) and second (secondary) harmonics.

Approach. Write each piston displacement from the crank centre, use the standard rod approximation, add the two reciprocating inertia forces, and read off the harmonic content.

1. Piston displacements. Measuring each piston position along the common axis from the crank centre $O$ (piston 1 outward $=+$, piston 2 outward $=-$), the crankpin sits at angle $\theta=\omega t$: $$x_1=r\cos\theta+\sqrt{l^2-r^2\sin^2\theta},\qquad x_2=r\cos\theta-\sqrt{l^2-r^2\sin^2\theta}.$$
2. Sum of the two paths (the key cancellation). The awkward radical is identical in both, so $$x_1+x_2=2r\cos\theta$$ exactly — every secondary and higher harmonic (which live entirely in the $\sqrt{\;}$ term) cancels between the two pistons.
3. Net shaking force. The reciprocating inertia (shaking) force is $F_s=-m\ddot x_1-m\ddot x_2=-m\dfrac{d^2}{dt^2}(x_1+x_2)$. With $\omega$ constant, $$F_s=-m\frac{d^2}{dt^2}\bigl(2r\cos\theta\bigr)=\boxed{2\,m\,r\,\omega^2\cos\theta}$$ directed along the cylinder axis.

So the shaking force is a pure first harmonic of amplitude $2mr\omega^2$; the algebra shows the secondary (and all even-plus-higher) harmonics are identically zero, not merely small. Symbolic differentiation confirms $F_s=2mr\omega^2\cos\theta$ with no residual $\cos2\theta$ term.

Balancing verdict

• Second harmonic (secondary): already identically balanced — the two opposed pistons cancel each other's secondary forces. No secondary balancer is needed.
• First harmonic (primary): not self-balanced; a reciprocating force $2mr\omega^2\cos\theta$ remains along the cylinder axis. It cannot be eliminated by rotating counterweights alone: a counterweight of $2mr$ at crank radius removes the axial term only by introducing an equal $2mr\omega^2\sin\theta$ force transverse to the axis. The usual compromise is a half-balance (counterweight $\approx mr$), which halves the primary but leaves a residual rotating shake; complete primary balance requires a pair of contra-rotating balance shafts.