Balancing Primary and Secondary Forces in Reciprocating Engines: Derivation of Complete Balancing Conditions

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Nov 1, 2026

Reciprocating engines generate vibration because the inertia (accelerating) of the slider–crank (piston–connecting rod) does not fully cancel at the crankshaft bearings. The resulting “shaking” effects are commonly decomposed into keyword primary (1× crank speed) and keywordsecondary (2× crank speed) components, plus associated keywordprimary and keywordsecondary couples. A complete balance requires that both (i) forces and (ii) couples for each harmonic cancel.

For a single cylinder’s reciprocating mass $m$ , using the standard approximation for piston acceleration in a slider–crank mechanism, the inertia force can be expressed as a sum of a primary term proportional to $\cos\theta$ and a secondary term proportional to $\cos 2\theta$ (with $\theta$ crank angle and $\omega$ crank angular speed). A commonly used result is that the unbalanced inertia force has the form 2: $F(\theta)=F_p+F_s=m r\omega^2\left(\cos\theta+\frac{r}{l}\cos2\theta\right)$ and the corresponding unbalanced couples for a multi-cylinder layout depend on the vector sum of cylinder positions in the relevant plane (axial/offset geometry).

The key balancing idea is therefore harmonic superposition: you must choose engine cylinder phasing (crank angles) such that the phasor sums of $\cos\theta$ and $\cos2\theta$ components cancel, and likewise the moment (couple) sums cancel.

Survivor Library: The balancing of engines (primary/secondary discussion) - Explains primary vs secondary effects and balancing context. ↩
Balancing of reciprocating masses (course notes PDF) - Provides the theoretical basis for identifying primary/secondary components from reciprocating motion. ↩

Primary & Secondary Engine Balance (reciprocating forces overview)

Harmonic decomposition: where primary (1×) and secondary (2×) come from

In a slider–crank mechanism, piston displacement is not a simple harmonic function of crank angle; however, its acceleration can be decomposed into dominant harmonics. In many dynamics-of-machines treatments, the piston acceleration is approximated so that the inertia force contains:

a keywordprimary component at frequency $\omega$ ,
a keywordsecondary component at frequency $2\omega$ .

One engineering explanation expresses the two harmonic components explicitly in terms of the piston’s approximate acceleration contribution, identifying the “first order” and “second order” (secondary) terms and noting that secondary acts at twice the crank frequency .

A key implication: even if you arrange cylinder pairs to cancel the primary forces, the secondary forces may still remain (because they have different phase relationships and occur at $2\omega$ ). This motivates “separate” conditions for primary and secondary complete balancing.

EPI - Reciprocating Engine Forces & Inertia Explained - Discusses how primary and secondary acceleration forces arise and how they combine. ↩

Modeling force and couple for multiple cylinders (vector/phasor viewpoint)

Consider an $N$ -cylinder arrangement where each cylinder i has:

reciprocating mass $m_i$ ,
common crank radius $r$ ,
common connecting rod length $l$ (often assumed for deriving general conditions),
crank angle of cylinder i: $\theta_i=\theta+\phi_i$ with phasing offsets $\phi_i$ determined by crankshaft design,
a perpendicular distance $d_i$ from the chosen reference point in the plane so that inertia forces produce keywordcouples.

Let each cylinder’s unbalanced force be decomposed as: $F_{i}(\theta)=m_i r\omega^2\left(\cos\theta_i+\frac{r}{l}\cos 2\theta_i\right)$ The total unbalanced force is then the algebraic sum of forces; the total unbalanced couple is the algebraic sum of couples produced by each cylinder. Many sources formulate “complete balance” as closure of both the force polygon and the couple polygon for each harmonic (primary and secondary) .

So we can express complete balancing requirements as:

Primary force closure: algebraic sum of primary components equals zero.
Primary couple closure: algebraic sum of primary couples equals zero.
Secondary force closure: algebraic sum of secondary components equals zero.
Secondary couple closure: algebraic sum of secondary couples equals zero.

These conditions are the operational meaning of “complete balancing” of primary and secondary effects .

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩ ↩²

Deriving the complete balancing conditions (primary and secondary)

1
Step 1
Use $F_i(\theta)=m_i r\omega^2\left(\cos\theta_i+\frac{r}{l}\cos 2\theta_i\right)$ . This identifies the primary term ( $\propto\cos\theta$ ) and the secondary term ( $\propto\cos2\theta$ ) at frequencies $\omega$ and $2\omega$ respectively. 2

Footnotes

Survivor Library: The balancing of engines (primary/secondary discussion) - Explains primary vs secondary effects and balancing context. ↩

Balancing of reciprocating masses (course notes PDF) - Provides the theoretical basis for identifying primary/secondary components from reciprocating motion. ↩
2
Step 2
Total primary force $\sum F_{p,i} \propto \sum m_i\cos(\theta+\phi_i)$ . Expand using $\cos(\theta+\phi_i)=\cos\theta\cos\phi_i-\sin\theta\sin\phi_i$ . For cancellation for all $\theta$ , coefficients of $\cos\theta$ and $\sin\theta$ must vanish.
3
Step 3
From coefficient matching: $\sum m_i\cos\phi_i=0$ and $\sum m_i\sin\phi_i=0$ . Equivalently, the vector (phasor) sum $\sum m_i e^{j\phi_i}=0$ .
4
Step 4
Each primary force acts at an offset producing a couple. If the couple arm vector is represented by cylinder location angles, complete cancellation for all $\theta$ similarly requires both the cosine and sine harmonic moment coefficients to vanish. This is stated in polygon form as closure of the primary couple polygon (algebraic sum of couples about any point in the plane is zero).

Footnotes

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩
5
Step 5
Secondary force $F_{s,i}\propto m_i\cos(2\theta+2\phi_i)$ . Expand $\cos(2\theta+2\phi_i)=\cos 2\theta\cos 2\phi_i-\sin 2\theta\sin 2\phi_i$ and require cancellation for all $\theta$ → $\sum m_i\cos 2\phi_i=0$ and $\sum m_i\sin 2\phi_i=0$ .
6
Step 6
As with the primary case, complete secondary balancing additionally requires the algebraic sum of secondary couples to be zero, equivalently closure of the secondary couple polygon.

Footnotes

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩
7
Step 7
Because each harmonic must vanish for all crank angles, the force polygons (primary and secondary) must close, and the couple polygons (primary and secondary) must close. This is the standard textbook statement of complete balancing.

Footnotes

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩

Final complete balancing conditions (force and couple)

Let $\theta$ be the crank angle of a reference cylinder, and let cylinder i have phasing $\phi_i$ so that its crank angle is $\theta_i=\theta+\phi_i$ .

Primary (1×) complete force balancing: $\sum_i m_i\cos\phi_i=0,\qquad \sum_i m_i\sin\phi_i=0$ Equivalently, in complex form: $\sum_i m_i e^{j\phi_i}=0$

Secondary (2×) complete force balancing: $\sum_i m_i\cos 2\phi_i=0,\qquad \sum_i m_i\sin 2\phi_i=0$ Equivalently: $\sum_i m_i e^{j2\phi_i}=0$

Primary complete couple balancing: the same “closure” must hold for the moment contributions of the primary forces (forces acting with arms that produce couples), which is stated as: the algebraic sum of primary couples about any point in the plane must be zero (primary couple polygon closes).

Secondary complete couple balancing: likewise, the algebraic sum of secondary couples must be zero (secondary couple polygon closes).

These four conditions are exactly what complete balancing means in the harmonic-force-and-couple sense. The “polygon closure” formulation is a geometric re-expression of the coefficient-cancellation approach.

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩ ↩² ↩³

type="tip" title="Pro Tip: Balance harmonics independently" content="Because primary varies with $\cos\theta$ and secondary varies with $\cos2\theta$ , you must satisfy cancellation for each harmonic separately; otherwise one harmonic will reintroduce vibration even when the other is balanced."

type="warning" title="Warning: Primary balance alone is insufficient" content="Many cylinder phasings can cancel primary forces, yet secondary forces remain because the secondary terms depend on $\cos2(\theta+\phi_i)$ , shifting the phasor relations; textbook discussions emphasize complete secondary balancing requires separate force and couple conditions. "

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩

Harmonic contributions and what must be canceled

Qualitative mapping of which terms vanish under each complete-balancing requirement.

How complete balancing conditions are typically applied

Decompose inertia into harmonics

Step A

Represent each cylinder inertia force as primary ( $\propto\cos\theta$ ) + secondary ( $\propto\cos2\theta$ ). 2"

Survivor Library: The balancing of engines (primary/secondary discussion) - Explains primary vs secondary effects and balancing context. ↩
Balancing of reciprocating masses (course notes PDF) - Provides the theoretical basis for identifying primary/secondary components from reciprocating motion. ↩

Enforce primary force cancellation

Step B

Require $\sum m_i\cos\phi_i=0$ and $\sum m_i\sin\phi_i=0$ (vector sum closure)."

Enforce primary couple cancellation

Step C

Require primary couple polygon closure (algebraic sum of couples is zero). "

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩

Enforce secondary force cancellation

Step D

Require $\sum m_i\cos2\phi_i=0$ and $\sum m_i\sin2\phi_i=0$ ."

Enforce secondary couple cancellation

Step E

Require secondary couple polygon closure (algebraic sum of secondary couples is zero). "

ACS College of Engineering (MMT-M4.pdf) - Complete primary/secondary balancing conditions via polygon closure - States conditions for complete secondary balance and closure of force/couple polygons. ↩

Common checks and interpretation

Knowledge Check

Question 1 of 4

Q1Single choice

For a reciprocating engine, which harmonic corresponds to the secondary unbalanced force?

It varies with $\cos(2\theta)$ (twice per crank revolution)

It varies with $\cos\theta$ (once per crank revolution)

It is independent of crank angle

It varies with $\sin\theta$ only, with no cosine term

Knowledge Check

Question 1 of 1

Q1Single choice

Complete balancing (as commonly stated in polygon form) requires which set of closures?

Only the primary force polygon must close

Only the secondary force polygon must close

Primary force polygon and primary couple polygon close, and secondary force polygon and secondary couple polygon close

Only the flywheel mass must be adjusted; engine geometry is irrelevant

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Balancing Primary and Secondary Forces in Reciprocating Engines: Derivation of Complete Balancing Conditions

Footnotes

Primary & Secondary Engine Balance (reciprocating forces overview)

Harmonic decomposition: where primary (1×) and secondary (2×) come from

Footnotes

Modeling force and couple for multiple cylinders (vector/phasor viewpoint)

Footnotes

Deriving the complete balancing conditions (primary and secondary)

Footnotes

Footnotes

Footnotes

Footnotes

Final complete balancing conditions (force and couple)

Footnotes

Footnotes

Harmonic contributions and what must be canceled

How complete balancing conditions are typically applied

Decompose inertia into harmonics

Footnotes

Enforce primary force cancellation

Enforce primary couple cancellation

Footnotes

Enforce secondary force cancellation

Enforce secondary couple cancellation

Footnotes

Common checks and interpretation

Knowledge Check

Knowledge Check

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