Understanding the N+2 Method in Multi-Plane Balancing

Devices

Portable balancer & Vibration analyzer Balanset-1A

Parts

Optical Sensor (Laser Tachometer)

Complete vibration diagnostics and balancing kit from Vibromera, including four vibration sensors with cables, a signal interface module, laser tachometer with magnetic stand, digital scale, USB cable, and a carrying case. The system is designed for field rotor balancing and condition monitoring on industrial equipment.

Devices

Dynamic balancer “Balanset-1A” OEM

Definition: What is the N+2 Method?

The N+2 method is an advanced balancing procedure used for multi-plane balancing of flexible rotors. The name describes the measurement strategy: if N is the number of correction planes required, the method uses N trial weight runs (one for each plane) plus 2 additional runs—one initial baseline measurement and one final verification run—for a total of N+2 runs.

This systematic approach extends the principles of two-plane balancing to situations requiring three or more correction planes, common in high-speed flexible rotors such as turbines, compressors, and long paper machine rolls.

The Mathematical Foundation

The N+2 method is built on the influence coefficient method, extended to multiple planes:

The Influence Coefficient Matrix

For a rotor with N correction planes and M measurement locations (typically M ≥ N), the system can be described by an M×N matrix of influence coefficients. Each coefficient αᵢⱼ describes how a unit weight in correction plane j affects the vibration at measurement location i.

For example, with 4 correction planes and 4 measurement locations:

α₁₁, α₁₂, α₁₃, α₁₄ describe how each plane affects measurement location 1
α₂₁, α₂₂, α₂₃, α₂₄ describe effects on measurement location 2
And so on for locations 3 and 4

This creates a 4×4 matrix requiring determination of 16 influence coefficients.

Solving the System

Once all coefficients are known, the balancing software solves a system of M simultaneous vector equations to find the N correction weights (W₁, W₂, … Wₙ) that minimize vibration at all M measurement locations simultaneously. This requires sophisticated vector mathematics and matrix inversion algorithms.

The N+2 Procedure: Step-by-Step

The procedure follows a systematic sequence that scales with the number of correction planes:

Run 1: Initial Baseline Measurement

The rotor is operated at the balancing speed in its initial unbalanced condition. Vibration amplitude and phase are measured at all M measurement locations (typically at each bearing and sometimes at intermediate positions). These measurements establish the baseline unbalance vectors that must be corrected.

Runs 2 through N+1: Sequential Trial Weight Runs

For each correction plane (from 1 to N):

Stop the rotor and attach a trial weight of known mass at a known angular position in that specific correction plane only
Run the rotor at the same speed and measure vibration at all M locations
The change in vibration (current measurement minus initial) reveals how this specific plane influences each measurement location
Remove the trial weight before proceeding to the next plane

After completing all N trial runs, the software has determined the complete M×N influence coefficient matrix.

Calculation Phase

The balancing instrument solves the matrix equations to calculate the required correction weights (both mass and angle) for each of the N correction planes.

Run N+2: Verification Run

All N calculated correction weights are installed permanently, and a final verification run confirms that vibration has been reduced to acceptable levels at all measurement locations. If results are unsatisfactory, a trim balance or additional iteration may be performed.

Example: Four-Plane Balancing (N=4)

For a long flexible rotor requiring four correction planes:

Total Runs: 4 + 2 = 6 runs
Run 1: Initial measurement at 4 bearings
Run 2: Trial weight in Plane 1, measure all 4 bearings
Run 3: Trial weight in Plane 2, measure all 4 bearings
Run 4: Trial weight in Plane 3, measure all 4 bearings
Run 5: Trial weight in Plane 4, measure all 4 bearings
Run 6: Verification with all 4 corrections installed

This generates a 4×4 matrix (16 coefficients) that is solved to find the four optimal correction weights.

Advantages of the N+2 Method

The N+2 approach offers several important benefits for multi-plane balancing:

1. Systematic and Complete

Every correction plane is tested independently, providing a complete characterization of the rotor-bearing system’s response across all planes and measurement locations.

2. Accounts for Complex Cross-Coupling

In flexible rotors, a weight in any plane can significantly affect vibration at all bearing locations. The N+2 method captures all these interactions through its comprehensive coefficient matrix.

3. Mathematically Rigorous

The method uses well-established linear algebra techniques (matrix inversion, least-squares fitting) that provide optimal solutions when the system behaves linearly.

4. Flexible Measurement Strategy

The number of measurement locations (M) can exceed the number of correction planes (N), allowing for overdetermined systems that can provide more robust solutions in the presence of measurement noise.

5. Industry Standard for Complex Rotors

The N+2 method is the accepted standard for high-speed turbomachinery and other critical flexible rotor applications.

Challenges and Limitations

Multi-plane balancing using the N+2 method presents significant challenges:

1. Increased Complexity

The number of trial runs grows linearly with the number of planes. For a 6-plane balance, a total of 8 runs are required, significantly increasing time, cost, and machine wear.

2. Measurement Accuracy Requirements

Solving large matrix systems amplifies the effect of measurement errors. High-quality instrumentation and careful technique are essential.

3. Numerical Stability

Matrix inversion can become ill-conditioned if:

Correction planes are too close together
Measurement locations don’t adequately capture the rotor’s response
Trial weights produce insufficient vibration changes

4. Time and Cost

Each additional plane adds another trial run, extending downtime and labor costs. For critical equipment, this must be balanced against the benefits of superior balance quality.

5. Requires Advanced Software

Solving N×N systems of complex vector equations is beyond manual calculation. Specialized balancing software with multi-plane capabilities is essential.

When to Use the N+2 Method

The N+2 method is appropriate when:

Flexible Rotor Operation: The rotor operates above its first (and possibly second or third) critical speed
Long Slender Rotors: High length-to-diameter ratios that undergo significant bending
Two-Plane Insufficient: Previous attempts at two-plane balancing failed to achieve acceptable results
Multiple Critical Speeds: The rotor must pass through multiple critical speeds during operation
High-Value Equipment: Critical turbines, compressors, or generators where investment in comprehensive balancing is justified
Severe Vibration at Intermediate Locations: Vibration is excessive at locations between the end bearings, indicating mid-span unbalance

Alternative: Modal Balancing

For highly flexible rotors, modal balancing can be more effective than the conventional N+2 method. Modal balancing targets specific vibration modes rather than specific speeds, potentially achieving better results with fewer trial runs. However, it requires even more sophisticated analysis and understanding of rotor dynamics.

Best Practices for N+2 Method Success

Planning Phase

Carefully select N correction plane locations—widely spaced, accessible, and ideally at locations matching rotor mode shapes
Identify M ≥ N measurement locations that adequately capture the rotor’s vibration characteristics
Plan for thermal stabilization time between runs
Prepare trial weights and installation hardware in advance

Execution Phase

Maintain absolutely consistent operating conditions (speed, temperature, load) across all N+2 runs
Use trial weights large enough to produce clear, measurable responses (25-50% vibration change)
Take multiple measurements per run and average them to reduce noise
Carefully document trial weight masses, angles, and radii
Verify phase measurement quality—phase errors are magnified in large matrix solutions

Analysis Phase

Review the influence coefficient matrix for anomalies or unexpected patterns
Check matrix condition number—high values indicate numerical instability
Verify calculated corrections are reasonable (not excessively large or small)
Consider simulation of expected final result before installing corrections

Integration with Other Techniques

The N+2 method can be combined with other approaches:

Speed-Stepped Balancing: Perform N+2 measurements at multiple speeds to optimize balance across the operating range
Hybrid Modal-Conventional: Use modal analysis to inform correction plane selection, then apply N+2 method
Iterative Refinement: Perform N+2 balancing, then use reduced influence coefficient set for trim balancing

← Back to Main Index

What is the N+2 Method in Multi-Plane Balancing?

Published by admin on October 31, 2025 October 31, 2025