Balancing of reciprocating masses and multi-cylinder engines

Primary and secondary inertia forces of reciprocating masses, partial balancing of single-cylinder engines, and the balance state of inline twin, four, six and V-twin engines.

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Why it matters

The piston and the small end of the connecting rod accelerate back and forth thousands of times a minute, and the force needed to do that is transmitted to the engine block and mounts as a shaking force. Whether an engine feels smooth depends on how the cylinders are arranged to cancel these forces and couples: this is why an inline-six is naturally smooth, an inline-four needs balance shafts for refinement, and a single-cylinder motorcycle engine vibrates at certain speeds whatever you do.

Key ideas

Inertia force of the reciprocating parts. From the slider-crank analysis, the force needed to accelerate a reciprocating mass m is m·ω²·r·(cos θ + cos 2θ / n), with n = l/r. Its reaction acts on the engine frame along the line of stroke.

  • Primary force: m·ω²·r·cos θ, varying at crank frequency. It equals the component along the line of stroke of the centrifugal force of a mass m at the crank pin.
  • Secondary force: m·ω²·r·cos 2θ / n, varying at twice crank frequency. It exists because of the obliquity of the connecting rod (finite n); with an infinitely long rod it would vanish. It equals the line-of-stroke component of the centrifugal force of a mass m on an imaginary crank of radius r/(4n) rotating at 2ω, positioned at 2θ.

Why a reciprocating mass cannot be fully balanced by a rotating mass. A counterweight on the crank produces a force that rotates with the crank. It can cancel the primary force along the line of stroke, but it then produces an equal unbalanced component perpendicular to the line of stroke. So in a single-cylinder engine only partial balancing is possible.

Partial primary balancing (single cylinder). Balance all the rotating mass at the crank pin plus a fraction c of the reciprocating mass. The residual unbalanced force is (1 − c)·m·ω²·r·cos θ along the stroke and c·m·ω²·r·sin θ perpendicular to it. With c = 1/2 the maximum of the two is smallest; c = 2/3 to 3/4 is common in practice. Secondary forces are not touched by crankshaft counterweights.

Multi-cylinder inline engines. Treat each cylinder's reciprocating mass as a rotating mass at its crank pin (for primary effects), at its crank angle and its axial position. The conditions for primary balance are the same as for rotating masses: the force polygon (Σ m·r at crank angles) and the couple polygon (Σ m·r·l) must close. For secondary balance repeat with crank angles doubled and radius r/(4n).

Common arrangements (equal masses, evenly spaced cylinders):

  • Two-cylinder inline, cranks at 180°: primary forces balanced; primary couple m·ω²·r·a unbalanced; secondary forces add (unbalanced).
  • Inline four, cranks 0°, 180°, 180°, 0° (mirror symmetric): primary forces and couples balanced; secondary couples balanced; secondary forces add up to 4·m·ω²·r·cos 2θ / n (unbalanced). Two Lanchester balance shafts rotating in opposite directions at twice crank speed cancel it.
  • Inline six, cranks at 120° in mirror-symmetric pairs: primary and secondary forces and couples all balanced.
  • 90° V-twin on a common crank pin: the resultant primary force has a constant magnitude m·ω²·r and rotates with the crank, so a counterweight opposite the crank pin balances it completely; the resultant secondary force has a maximum of √2·m·ω²·r / n, acting horizontally.

Firing order and balance. The firing order sets the torque pattern, not the inertia balance; both must be chosen together when the crank arrangement is laid out.

Formulas

F_P = m·ω²·r·cos θ (primary force)

  • m: reciprocating mass per cylinder (kg); ω: crank speed (rad/s); r: crank radius (m); θ: crank angle from inner dead centre.

F_S = m·ω²·r·cos 2θ / n = m·(2ω)²·(r / 4n)·cos 2θ (secondary force)

  • n = l/r: ratio of connecting-rod length to crank radius. Maximum at θ = 0°, 90°, 180°, 270° in magnitude m·ω²·r/n.

B·b = m_rot·r + c·m·r (partial balancing of a single cylinder)

  • B: balance mass (kg) at radius b (m); m_rot: rotating mass at the crank pin (kg); c: fraction of the reciprocating mass balanced.

F_along = (1 − c)·m·ω²·r·cos θ and F_perp = c·m·ω²·r·sin θ

  • Residual unbalanced primary forces along and perpendicular to the line of stroke (N).

Σ m·r·cos θ = 0, Σ m·r·sin θ = 0 and Σ m·r·l·cos θ = 0, Σ m·r·l·sin θ = 0 (primary balance of inline engine)

  • l: axial position of each cylinder from a reference plane (m). For secondary balance replace θ by 2θ and r by r/(4n).

Worked examples

Example 1 (standard: partial balancing of a single cylinder). A single-cylinder engine has a reciprocating mass of 2 kg, a rotating mass of 1.2 kg at the crank pin and a crank radius of 50 mm. All the rotating mass and two-thirds of the reciprocating mass are to be balanced by a mass at 60 mm radius. Find the balance mass and the maximum residual primary forces at 3000 rpm.

  1. B·b = (1.2 + (2/3) × 2) × 0.05 = 2.533 × 0.05 = 0.1267 kg·m.
  2. B = 0.1267 / 0.06 = 2.11 kg, opposite the crank pin.
  3. ω = 2π × 3000 / 60 = 314.16 rad/s; ω² = 98 696 s⁻²; m·ω²·r = 2 × 98 696 × 0.05 = 9870 N.
  4. Maximum residual along the stroke (at θ = 0°): (1 − 2/3) × 9870 = 3290 N.
  5. Maximum residual perpendicular to the stroke (at θ = 90°): (2/3) × 9870 = 6580 N.

Balancing more of the reciprocating mass lowers the vertical shake but raises the horizontal one; that is the compromise.

Example 2 (GATE level: inline four and inline twin). Each cylinder has a reciprocating mass of 2 kg, the crank radius is 50 mm, n = 4 and the speed is 3000 rpm.

(a) Inline four with cranks at 0°, 180°, 180°, 0°:

  1. Primary: two cranks at 0° and two at 180°, so Σ m·r = 0; the arrangement is mirror symmetric, so Σ m·r·l = 0. Primary forces and couples are balanced.
  2. Secondary: doubled angles are 0°, 360°, 360°, 0°, all in line. The secondary forces add: maximum = 4 × m·ω²·r / n = 4 × 2 × 98 696 × 0.05 / 4 = 9870 N, at 2 × 50 = 100 Hz.

(b) Inline twin with cranks at 180°, cylinder centre lines 0.1 m apart:

  1. Primary forces cancel, but they form a couple: maximum = m·ω²·r·a = 2 × 98 696 × 0.05 × 0.1 = 987 N·m, at 50 Hz.

Common mistakes

  • Saying secondary forces come from the angular acceleration of the connecting rod. They come from its obliquity (finite l/r).
  • Thinking crankshaft counterweights can remove secondary forces. Secondary forces vary at 2ω and need masses rotating at 2ω (balance shafts).
  • Claiming a single-cylinder engine can be fully balanced by a counterweight. Only the rotating mass can; the reciprocating part can only be traded between directions.
  • Using r instead of r/(4n) for the imaginary secondary crank, or forgetting to double the crank angles.
  • Concluding that because the forces cancel the engine is balanced; check the couples too.
  • Quoting the unbalanced frequency of an inline four as crank frequency. It is twice crank frequency.

For GATE ME

Expect calculation of primary or secondary unbalanced force at a crank angle, the balance mass for partial primary balancing of a single cylinder and the residual forces, and checks of which forces and couples are unbalanced for two-, four- and six-cylinder inline engines and V-twins. Practise drawing the primary and secondary force and couple polygons for the common crank layouts.

Quick check

  1. At what multiple of crank speed does the secondary force vary?
  2. Which unbalanced effect remains in a 0-180-180-0 inline four?
  3. Radius of the imaginary secondary crank for r = 60 mm and n = 4?
  4. Secondary force for m = 4 kg, r = 0.2 m, ω = 90 rad/s, θ = 0, n = 4?
  5. Can a 90° V-twin's primary force be balanced by a counterweight?

Answers: 1. Twice. 2. The secondary force. 3. 60 / 16 = 3.75 mm. 4. 4 × 0.2 × 8100 / 4 = 1620 N. 5. Yes, completely, because it has constant magnitude and rotates with the crank.

Interview questions

All Theory of Machines and Vibrations interview questions

Try answering each one aloud before you open it.

  1. 1.What is meant by the balancing of reciprocating masses?Concept

    The reciprocating parts (piston, gudgeon pin, small end of the rod) need an inertia force m·ω²·r(cos θ + cos 2θ/n) along the line of stroke, which reacts on the frame as a shaking force. Balancing them means cancelling that force and its couples, either by arranging several cylinders so their forces cancel or, in a single cylinder, by a crank counterweight. A rotating counterweight can only partly balance it: it removes part of the primary force along the stroke but introduces an equal component perpendicular to it, so single-cylinder engines are only partially balanced.

  2. 2.Explain why balancing is important in multi-cylinder engines.Concept

    In a multi-cylinder engine the inertia forces of the cylinders act at different crank angles and at different positions along the crankshaft, so they can be made to cancel each other. Choosing the crank angles and cylinder spacing so that the primary and secondary force and couple polygons close removes shaking forces and rocking couples without extra parts; an inline six achieves full primary and secondary balance this way. A poor arrangement leaves forces or couples that shake the mounts and body at engine or twice-engine frequency.

  3. 3.How does primary and secondary balancing differ in reciprocating engines?Concept

    The primary force m·ω²·r·cos θ varies at crank frequency and is what a reciprocating mass would produce with an infinitely long connecting rod; it behaves like the line-of-stroke component of a mass at the crank pin. The secondary force m·ω²·r·cos 2θ/n varies at twice crank frequency and is caused by connecting-rod obliquity; it behaves like a mass on an imaginary crank of radius r/4n turning at 2ω. Primary effects are handled by cylinder arrangement and crank counterweights; secondary effects need suitable crank layouts or balance shafts at twice crank speed.

  4. 4.Why are inline four-cylinder engines often equipped with balance shafts?Application

    In a 0-180-180-0 inline four the primary forces and couples cancel, but the secondary forces of all four cylinders are in phase and add up to 4·m·ω²·r·cos 2θ/n, a vertical shake at twice engine speed that grows with speed squared. Lanchester balance shafts are two shafts carrying eccentric masses, rotating in opposite directions at twice crankshaft speed; their horizontal components cancel and their vertical components oppose the secondary force. They are used mainly on larger-displacement fours, where this force becomes objectionable.

  5. 5.How can the balancing of a single-cylinder engine be achieved?Application

    A single-cylinder engine can only be partially balanced. A counterweight opposite the crank pin balances all the rotating mass plus a fraction c of the reciprocating mass, typically one-half to three-quarters. That leaves (1 − c)·m·ω²·r·cos θ along the line of stroke and introduces c·m·ω²·r·sin θ across it, so the designer only chooses the direction in which the engine shakes. A balance shaft at crank speed can cancel the remaining primary force; a flywheel does not balance anything, it only smooths torque.

  6. 6.A single-cylinder engine has a reciprocating mass of 2 kg and a crank radius of 0.1 m. What balance mass at 0.2 m radius would balance the whole primary force along the line of stroke, and why is this not usually done?Numerical

    Balancing the whole reciprocating mass needs B·b = m·r, so B = 2 × 0.1 / 0.2 = 1 kg opposite the crank pin. But this counterweight produces an equal unbalanced force m·ω²·r·sin θ perpendicular to the line of stroke, so the shaking is merely turned through 90°. In practice only a fraction, about one-half to two-thirds, of the reciprocating mass is balanced (plus all of the rotating mass) to share the residual between the two directions.

  7. 7.An engine runs at 3000 rpm. At what frequencies do its primary and secondary unbalanced forces act, and which one matters for an inline four-cylinder engine?Numerical

    The primary force varies with cos θ, at crank frequency: 3000/60 = 50 Hz. The secondary force varies with cos 2θ, at twice crank frequency: 100 Hz. In a 0-180-180-0 inline four the primary forces and couples cancel, so the remaining unbalance is the secondary force at 100 Hz, which is why the vibration of a four-cylinder engine is felt at twice engine speed.

  8. 8.What design considerations are taken into account to minimize vibrations in multi-cylinder engines?Application

    To minimize vibrations in multi-cylinder engines, designers consider factors such as the arrangement of cylinders, the use of counterweights on the crankshaft, the implementation of balance shafts, and the optimization of engine mounts. Additionally, the firing order and the use of lightweight materials for reciprocating parts can help reduce vibrations and improve engine balance.

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