Balancing of reciprocating parts — for primary balance in a multi-cylinder in-line engine, which conditions must simultaneously hold true for forces and couples in the plane of primary forces?

Introduction / Context: Primary balancing in reciprocating engines deals with once-per-revolution inertia forces produced by pistons and connecting rods. For a smooth and vibration-free operation, designers arrange cranks, masses, and sometimes balance weights so that these primary effects do not result in net shaking at the engine bearings or frame.

Given Data / Assumptions:

• Multi-cylinder in-line engine with reciprocating masses.
• Primary inertia forces vary with cos θ at engine speed ω.
• Primary couples arise because forces act at different axial positions along the crankshaft.

Concept / Approach: Represent each reciprocating mass by an equivalent rotating vector (equivalent crank). For true primary balance, the vector sum of primary forces must be zero, and the vector sum of their moments (couples) about any reference point must also be zero. Only when both conditions are satisfied do we avoid net shaking force and net rocking couple.

Step-by-Step Solution:

Verification / Alternative check: Classical arrangements (e.g., perfect primary balance in certain multi-cylinder layouts) demonstrate negligible once-per-rev vibration when both sums vanish.

Why Other Options Are Wrong:
(a) Only forces balanced leaves rocking couples, still causing vibration.
(b) Only couples balanced leaves translational shaking.
none of these — contradicts engine balancing fundamentals.

Common Pitfalls: Balancing forces but forgetting the axial spacing of cranks (couples); treating rotating counterweights as a cure-all without checking moments.

Final Answer: both (a) and (b).