Balancing of Rotating and Reciprocating Masses

Introduction

Balancing is the process of arranging the mass distribution of a machine so that unwanted inertia forces and couples are reduced to acceptable levels. In high-speed machinery, imperfect balance produces vibration, noise, shaft deflection, bearing loads, fatigue, looseness, and premature failure. Balancing is therefore both a design requirement and a maintenance activity.

A rotating system may be balanced in one plane or in several planes. A reciprocating engine cannot generally be completely balanced because the piston motion is nonuniform and its inertia force contains several harmonics. The engineer must therefore distinguish between primary and secondary balancing and must decide which residual forces and couples are acceptable.

Learning Objectives

After studying this chapter, the student should be able to:

• explain static, dynamic, and couple balancing;
• calculate centrifugal forces and balancing masses;
• balance rotating masses in one and several planes;
• determine primary and secondary forces in reciprocating engines;
• evaluate residual couples, bearing reactions, and vibration risks;
• interpret balancing-machine measurements; and
• select practical balancing procedures while recognizing their limitations.

1. Fundamental Concepts

1.1 Inertia forces

For a particle of mass m rotating at angular velocity ω at radius r, the centrifugal force is

where F_c is in newtons when m is in kilograms, r in metres, and ω in radians per second. The force acts radially outward in a rotating reference description; the corresponding shaft or bearing loading is equal and opposite.

Angular speed is related to rotational speed by

where N is speed in revolutions per minute. Thus, centrifugal force varies with the square of speed. Doubling speed produces four times the force if the mass and radius are unchanged.

The product mr is called the mass–radius product or unbalance quantity. Its SI unit is kg·m. In balancing practice, a small mass at a large radius can produce the same effect as a large mass near the axis.

1.2 Static and dynamic balance

A rotor is statically balanced when its centre of mass lies on the axis of rotation. For discrete masses, the vector condition is

This condition eliminates the resultant centrifugal force. Static balance can be checked by allowing a rotor to rotate freely at low speed; the heavy side tends to move downward.

A rotor is dynamically balanced when both the resultant force and the resultant moment about a reference plane are zero:

Here l_i is the axial distance of mass i from the reference plane. The quantities are vectors in the transverse plane. Static balance is sufficient for a thin disc, but a long rotor may be statically balanced and still produce a rocking couple. Dynamic balancing requires at least two correction planes.

1.3 Unbalance representation

Unbalance is commonly represented by a vector U = mr, specified by magnitude and angular position. A correction mass must have equal magnitude and opposite phase:

and it must be placed 180° from the resultant heavy spot. In actual balancing, the correction angle is measured relative to a once-per-revolution reference mark, often called the keyphasor or tachometer reference.

2. Single-Plane Balancing

For a rotor with several masses in one plane, resolve the mass–radius vectors into rectangular components:

A correction mass at radius r_b is then calculated from the resultant vector. The equations assume a rigid rotor, negligible axial separation, and operation below the first critical speed or under conditions where rigid-body balancing is valid.

Worked Example 1: Single-plane correction

A disc has an unbalance of 0.018 kg·m. A correction hole is located at radius 0.15 m. Determine the correction mass.

Using m_b r_b = U,

The correction mass is 0.12 kg, placed opposite the unbalance vector. If the rotor operates at 1500 rpm, ω = 157.1 rad/s. The unbalance force before correction would be

This illustrates why a modest mass eccentricity can produce substantial bearing load.

3. Two-Plane and Multi-Plane Balancing

For a rigid rotor, two correction planes are normally sufficient. Let the correction unbalance vectors be U_A and U_B, located at axial positions l_A and l_B. The force and couple conditions are

where U_R is the resultant existing unbalance and C_R is the existing unbalance moment about the reference plane. In practice, graphical vector methods, influence-coefficient methods, or balancing software are used.

The influence-coefficient method measures vibration response due to trial weights. If the original vibration vector is V₀ and a known trial unbalance produces a change ∆V, the complex influence coefficient is

where U_t is the trial unbalance vector. For several correction planes,

and the correction vector is obtained by solving the simultaneous complex equations. The method includes machine stiffness, sensor position, phase, and cross-coupling effects, so it is more reliable than treating each plane independently.

Couple unbalance

Two equal unbalances separated axially produce no resultant force but do produce a couple. If each unbalance has magnitude U and the separation is l, the couple magnitude is

The correction must produce an equal and opposite couple. This is why a rotor may appear satisfactory on a static balancing stand but vibrate strongly at operating speed.

4. Balancing of Reciprocating Masses

A reciprocating mechanism converts rotary motion into approximately linear motion. The piston acceleration is not sinusoidal because the connecting rod has finite length. Let r be crank radius, l connecting-rod length, ω angular speed, and θ crank angle measured from the inner dead centre. For a connecting rod much longer than the crank, the piston acceleration can be approximated by

The reciprocating inertia force is

where m_r is the reciprocating mass. The sign depends on the selected positive direction; the physical force acts opposite to acceleration.

The first term is the primary component:

The second term is the secondary component:

The primary force occurs once per revolution. The secondary force occurs twice per revolution and results from connecting-rod angularity. Its amplitude relative to the primary amplitude is approximately r/l. The approximation is most useful when l/r is large, commonly about 3 to 5 in many engines.

4.1 Rotating and reciprocating portions of the connecting rod

The connecting rod has both rotary and reciprocating motion. A common engineering approximation assigns one-third of the rod mass to the reciprocating end and two-thirds to the crankpin end, although the exact ratio depends on mass distribution and the chosen dynamic model. The reciprocating mass may therefore be estimated as

where m_p is piston mass. This approximation must be stated because different textbooks and engine models may use different mass-allocation conventions.

4.2 Partial balancing

A rotating counterweight cannot completely balance a reciprocating mass without introducing a transverse force. If a fraction c of the reciprocating primary force is balanced by a rotating mass, the counterweight produces a force with horizontal and vertical components. The selected value of c is a compromise between reducing shaking force along the line of stroke and increasing transverse shaking force.

For a single-cylinder engine, complete primary balance is generally impossible using only crank counterweights. Multi-cylinder arrangements use crank phasing and cylinder spacing to cancel forces and couples.

5. Multi-Cylinder Engine Balance

For each cylinder, primary-force vectors rotate at crank frequency and secondary-force vectors rotate at twice crank frequency. Balance requires vector summation of both force and moment:

where x_i is the cylinder location along the crankshaft. The second pair ensures that the forces do not cancel while leaving a rocking couple.

A typical inline four-cylinder engine can be arranged so that primary forces cancel, but secondary forces may add. An inline six-cylinder engine with suitable crank phasing can achieve excellent primary and secondary force balance, although crankshaft torsional effects and firing-order constraints remain important. V-engine balance depends on bank angle, crank arrangement, and the use of balance shafts.

The balancing analysis must distinguish:

• shaking force: resultant translational force transmitted to the frame;
• shaking couple: resultant moment tending to rock the engine;
• rocking couple: a couple caused by separated cylinder forces; and
• torque fluctuation: variation in driving torque, which is not the same as inertia-force balance.

6. Bearing Loads and Vibration

For a rotating unbalance U, the harmonic force amplitude is

If the machine is modelled as a single-degree-of-freedom system with stiffness k, mass M, damping coefficient c_d, and forcing frequency ω, the steady displacement amplitude is

where

The model assumes linear stiffness and damping and harmonic excitation. Near ρ = 1, resonance can greatly amplify vibration. A balancing grade that is adequate at low speed may not be adequate near a flexible-rotor critical speed.

Bearing reactions depend on both force and couple. For a simple rotor with a transverse force F applied between bearings separated by distance L, the static reactions are found from equilibrium. If the force is at distance a from the left bearing,

Dynamic rotating loads vary with phase and must be treated as vectors or harmonic quantities. Bearing selection should consider the combined radial load, axial load, speed, lubrication, fatigue life, and allowable vibration.

7. Balancing Machines and Measurement Practice

A balancing machine measures rotor response while the rotor rotates. Soft-bearing machines measure force transmitted through compliant supports; hard-bearing machines measure forces or moments at stiff supports. Both can provide correction magnitude and angular position when properly calibrated.

A practical procedure is:

1. inspect the rotor for damage, contamination, loose parts, and correct assembly;
2. measure mass, geometry, and correction-plane radii;
3. install the rotor with correct adapters and support conditions;
4. establish a repeatable angular reference;
5. run the rotor at a safe balancing speed;
6. record amplitude and phase in each measurement channel;
7. add trial weights when using the influence-coefficient method;
8. calculate correction weights and angles;
9. apply corrections by adding, removing, or relocating material;
10. repeat the run and verify residual unbalance.

A balancing result should report speed, correction planes, correction radius, initial and final vibration, residual unbalance, phase reference, and measurement uncertainty. Phase errors can arise from a slipping tachometer mark, sensor orientation, cable problems, or incorrect trial-weight data. A large correction prediction should be checked before material is removed permanently.

The quality grade commonly used for rigid rotors is expressed as a permitted residual specific unbalance, often written as e_per in mm/s. The relationship between permissible residual unbalance U_per and balance quality grade is

where M is rotor mass. In SI-consistent form, e_per is in metres; industrial standards often express it in mm. The selected grade depends on rotor type, speed, bearing arrangement, and applicable balancing standard. A standard designation must never be treated as a substitute for a machine-specific vibration assessment.

Worked Example 2: Primary and secondary forces

An engine has reciprocating mass 2.0 kg, crank radius 0.05 m, connecting-rod length 0.20 m, and speed 3000 rpm. Determine the amplitudes of primary and secondary inertia forces.

First,

Primary amplitude:

Secondary amplitude:

Thus the secondary amplitude is approximately 25% of the primary amplitude, consistent with r/l = 0.25. These are inertia-force amplitudes; gas forces, friction, crankshaft flexibility, and structural transmission may change the actual measured frame loads.

Applied Case: Illustrative Balancing of a Motor Rotor

Background

An illustrative production plant operates a 30 kg fan rotor at 1800 rpm. Operators report increasing vibration after replacement of a hub. The rotor has two correction planes separated by 0.40 m.

Problem

A single-plane static check shows little tendency to rotate, yet the assembled fan produces strong horizontal vibration. This suggests couple unbalance, installation error, or a flexible-rotor effect rather than a simple single-plane heavy spot.

Analysis and approach

The rotor is inspected for hub seating, blade deposits, bent shaft, and loose fasteners. A tachometer and two accelerometers are installed near the bearings. Trial weights are placed successively in the two correction planes. The measured amplitude and phase changes are used to form a two-plane influence-coefficient model. The solution indicates correction vectors of equal approximate magnitude but opposite angular phase in the two planes, characteristic of a couple.

The correction weights are installed at the specified radius, followed by a verification run. The residual vibration falls substantially and the bearing reaction estimate is reduced. The rotor is then checked at its operating temperature because thermal growth and aerodynamic loading can alter alignment.

Learning points

Static balance alone does not guarantee dynamic balance. Correction angles must be referenced to the same rotational mark throughout the test. The final acceptance decision should use both residual unbalance and measured vibration under normal operating conditions. If vibration rises sharply with speed, a critical-speed or structural-resonance investigation is required rather than repeated weight correction.

Limitations, Safety, and Professional Practice

Balancing calculations are model-dependent. Rigid-rotor equations are unsuitable when the rotor bends appreciably between bearings. Piston-force equations neglect combustion pressure, gas elasticity, friction, crankshaft torsion, and structural flexibility. Mass-allocation approximations for connecting rods introduce uncertainty. Instrument readings may also be affected by sensor mounting and signal processing.

Safety is essential because a trial weight can detach at high speed. Guards, overspeed protection, remote operation, certified fixtures, and documented lockout procedures are required. Correction weights must be secured against centrifugal loading. Removing material can weaken a component, and welding or drilling may change fatigue strength, corrosion resistance, or certification status. Traceability of corrections is important in aerospace, automotive, railway, and medical-equipment applications.

Modern systems use digital tachometers, order tracking, finite-element rotor models, laser alignment, and automated balancing. Artificial-intelligence methods can classify vibration signatures or predict maintenance needs, but they do not replace correct sensor installation, physical understanding, or engineering validation. Models should be trained on representative operating data and protected against unsafe automated decisions.

Best-Practice Checklist

• Define the rotor as rigid or flexible before selecting a method.
• Use vector quantities for phase-sensitive calculations.
• Check units: unbalance is commonly kg·m, force is N, and speed must be converted to rad/s.
• Balance in at least two planes when axial separation is significant.
• Include both primary and secondary reciprocating forces.
• Evaluate forces and couples at the bearings, not only at the rotor centre.
• Verify corrections with an independent run.
• Record speed, phase reference, correction radius, residual unbalance, and uncertainty.
• Investigate resonance, looseness, misalignment, and aerodynamic effects before adding excessive correction mass.

Chapter Summary

Balancing controls the inertia forces and couples generated by rotating and reciprocating masses. Static balance requires the vector sum of mass–radius products to be zero, whereas dynamic balance additionally requires the corresponding moment sum to be zero. Single-plane balancing is adequate for thin rotors; long rotors generally require two-plane correction.

Reciprocating-engine inertia force contains primary and secondary components. The primary component varies once per revolution, while the secondary component varies twice per revolution and depends on the crank-to-rod length ratio. Complete balance of a single-cylinder engine is generally impossible with simple counterweights, so engine design uses crank phasing, counterweights, balance shafts, and optimized cylinder arrangements.

Unbalance forces increase with the square of speed and may cause large bearing loads or resonance. Balancing machines determine correction magnitude and phase through force, vibration, and influence-coefficient measurements. Reliable practice combines correct modelling, safe measurement, appropriate standards, verification runs, and investigation of non-balance faults.

Key Terms

Static balance: Condition in which the centre of mass lies on the axis of rotation.

Dynamic balance: Condition in which resultant unbalance force and unbalance couple are both zero.

Unbalance: Unequal mass distribution represented by a mass–radius vector.

Mass–radius product: Product mr representing unbalance magnitude.

Correction mass: Mass added, removed, or relocated to reduce unbalance.

Couple unbalance: Two-plane unbalance producing a moment with little resultant force.

Rigid rotor: Rotor whose bending deformation is negligible in the operating range.

Flexible rotor: Rotor whose bending deformation significantly affects balance.

Primary force: Reciprocating inertia force varying once per crank revolution.

Secondary force: Reciprocating inertia force varying twice per crank revolution.

Shaking force: Resultant force transmitted from an engine or rotor to its supports.

Shaking couple: Resultant moment transmitted to the frame.

Crank radius: Half the crank throw, denoted by r.

Connecting-rod ratio: Ratio r/l of crank radius to rod length.

Influence coefficient: Complex response per unit trial unbalance.

Correction plane: Axial plane in which balancing mass is applied.

Phase angle: Angular position of a vibration or unbalance vector relative to a reference.

Residual unbalance: Unbalance remaining after correction.

Balance quality grade: Specified permissible level of residual specific unbalance.

Order tracking: Analysis of vibration components as multiples of rotational speed.

Review Questions

Short Answer Questions

1. What is the difference between static and dynamic balance?
2. Why does centrifugal force increase with the square of speed?
3. Define primary and secondary reciprocating inertia forces.
4. Why are two correction planes commonly needed for a long rotor?
5. What information should be included in a balancing-machine report?

Long Answer Questions

1. Derive the conditions for static and dynamic balance of a rotor with several discrete masses.
2. Explain the influence-coefficient method for two-plane rotor balancing.
3. Derive the approximate primary and secondary inertia-force equation for a reciprocating engine.
4. Discuss the causes and consequences of shaking forces and shaking couples in multi-cylinder engines.
5. Explain how resonance, damping, and bearing stiffness affect vibration caused by unbalance.

Analytical / Application Questions

1. A rotor has a measured unbalance of 0.012 kg·m and operates at 2400 rpm. Calculate the unbalance-force amplitude and specify the correction mass required at a radius of 0.20 m.
2. An engine has reciprocating mass 1.8 kg, crank radius 0.045 m, rod length 0.18 m, and speed 3600 rpm. Calculate primary and secondary force amplitudes.
3. A rotor is statically balanced but produces opposite-phase vibration readings at two bearings. Explain the likely fault and propose a measurement and correction procedure.

Multiple Choice Questions with Answers

Question 1

A rotor is statically balanced when:

A. Its centre of mass lies on the axis of rotation

B. Its bending stiffness is infinite

C. Its bearing reactions are equal at all speeds

D. Its secondary inertia force is zero

Correct Answer: A. Its centre of mass lies on the axis of rotation

Question 2

For a rotating unbalance U, the centrifugal-force amplitude is:

A. Uω

B. U/ω²

C. Uω²

D. U²ω

Correct Answer: C. Uω²

Question 3

A rotor with zero resultant unbalance force but a nonzero unbalance moment has:

A. Static balance only

B. Couple unbalance

C. No dynamic loading

D. Zero mass

Correct Answer: B. Couple unbalance

Question 4

The secondary reciprocating inertia force varies at:

A. One-half crank frequency

B. Crank frequency

C. Three times crank frequency

D. Twice crank frequency

Correct Answer: D. Twice crank frequency

Question 5

If speed is doubled while unbalance remains constant, centrifugal force becomes:

A. Twice as large

B. Four times as large

C. Half as large

D. Unchanged

Correct Answer: B. Four times as large

Question 6

For a reciprocating engine, the approximate ratio of secondary-to-primary force amplitudes is:

A. l/r

B. r/l

C. r + l

D. r − l

Correct Answer: B. r/l

Question 7

A long rotor is dynamically balanced primarily by:

A. Changing only its paint thickness

B. Using correction in two or more axial planes

C. Reducing bearing lubrication

D. Measuring only its static tilt

Correct Answer: B. Using correction in two or more axial planes

Question 8

At resonance in a lightly damped rotor system, vibration amplitude generally:

A. Becomes independent of excitation

B. Is strongly amplified

C. Becomes exactly zero

D. Depends only on rotor mass and not stiffness

Correct Answer: B. Is strongly amplified

Question 9

A balancing-machine phase angle must be referenced to:

A. An arbitrary bearing dimension

B. A repeatable once-per-revolution reference

C. The ambient temperature only

D. The rotor colour

Correct Answer: B. A repeatable once-per-revolution reference

Question 10

A rotor of mass 10 kg has permissible residual unbalance 0.0002 kg·m. Its specific residual unbalance is:

A. 0.00002 m

B. 0.0002 m

C. 0.002 m

D. 0.02 m

Correct Answer: A. 0.00002 m