Rotor Balancing: Fundamentals for Systematic Processes

Preface
Preface to the 9th (German) Edition
Contents
1 Introduction
	1.1	Preliminary Note
	1.2	Importance and Quality of Balancing
	1.3	Development of Balancing Technology and Balancing Machines
		1.3.1	Unbalance Types
		1.3.2	Balancing Machines
	1.4	Standards and Guidelines
		1.4.1	Historical Course
		1.4.2	Current Situation
	1.5	List of Current Standards
2 Physical Basics
	2.1	Preliminary Note
	2.2	Physical Quantities
	2.3	Scalar and Vector
		2.3.1	Addition
		2.3.2	Multiplication
	2.4	System of Units
		2.4.1	Basic Quantities
		2.4.2	Derived Quantities
	2.5	Physical Laws
		2.5.1	Newton’s 2nd Law
		2.5.2	Mass Attraction
	2.6	Circular Motion
		2.6.1	Plane Angle
		2.6.2	Angular Frequency
		2.6.3	Circular Speed
		2.6.4	Angular Acceleration
		2.6.5	Circular Acceleration
		2.6.6	Torque
		2.6.7	Moment of Inertia
		2.6.8	Radial Acceleration
		2.6.9	Centrifugal Force
	2.7	Vibration
		2.7.1	Single Mass Oscillator with Centrifugal Excitation
			2.7.1.1 Subcritical Area
			2.7.1.2 Resonance Area
			2.7.1.3 Supercritical Area
		2.7.2	Degrees of Freedom
		2.7.3	Dynamic Stiffness
3 Terms and Explanations
	3.1	Preliminary Note
	3.2	Rotor Balancing
	3.3	Balancing Task
	3.4	Rotor
	3.5	Unbalance
	3.6	Unbalance Condition
	3.7	Unbalance Behaviour
	3.8	Unbalance Tolerances
	3.9	Correction
	3.10	Correction Plane
	3.11	Shaft Axis
	3.12	Rotor Behaviour
		3.12.1	Rotors with Rigid Behaviour
		3.12.2	Rotors with Flexible Behaviour
			3.12.2.1 Rotors with Shaft Elastic Behaviour
			3.12.2.2 Rotors with Component-Elastic Behaviour
			3.12.2.3 Rotors with Settling Behaviour
	3.13	Rotor Concept
4 Theory of Balancing
	4.1	Preliminary Note
	4.2	General
		4.2.1	Unbalance State
		4.2.2	Rotor Concept
		4.2.3	Rotor Behaviour
			4.2.3.1 General
			4.2.3.2 Effects of Rotor Behaviour
			4.2.3.3 Principle of Order
		4.2.4	Unbalance Tolerances
		4.2.5	Balancing Task
	4.3	Unbalances and Correction
	4.4	Unbalance of the Disc-Shaped Rotor
	4.5	Unbalance of a General Rotor
		4.5.1	Resultant Unbalance
		4.5.2	Moment Unbalance
		4.5.3	Couple Unbalance
		4.5.4	Modal Unbalance
		4.5.5	Equivalent Modal Unbalance
	4.6	Overview of the Balancing Tasks
		4.6.1	General
		4.6.2	The Balanced Rotor
		4.6.3	Single-Plane Balancing
		4.6.4	Two-Plane Balancing
		4.6.5	Multi-Plane Balancing
			4.6.5.1 Example 1
			4.6.5.2 Example 2
			4.6.5.3 Example 3
			4.6.5.4 Example 4
	4.7	Conclusion of the New Perspective
		4.7.1	Significance of Resonances
		4.7.2	Significance of Flexural Resonances above the Service Speed
		4.7.3	Treatment of Flexural Resonances above Service Speed
	4.8	Handling Unbalance Tolerances
		4.8.1	Concept
		4.8.2	Procedure
			4.8.2.1 Example 1
			4.8.2.2 Example 2
			4.8.2.3 Example 3
			4.8.2.4 Example 4
5 Theory of the Rotor with Rigid Behaviour
	5.1	Preliminary Remark
	5.2	Rotor Behaviour
	5.3	Unbalanced Condition
		5.3.1	Static Unbalance
			5.3.1.1 Example 1
			5.3.1.2 Example 2
			5.3.1.3 Example 3
			5.3.1.4 Example 4
		5.3.2	Resulting Unbalance
			5.3.2.1 Example 1
			5.3.2.2 Example 2
		5.3.3	Moment Unbalance
			5.3.3.1 Example
		5.3.4	Dynamic Unbalance
	5.4	Display of the Unbalance Condition
		5.4.1	Unbalances
		5.4.2	Position of the Axis of Inertia
		5.4.3	Overview
6 Theory of the Rotor with Flexible Behaviour
	6.1	Preliminary Note
	6.2	Settling Behaviour
	6.3	Component-Elastic Rotor Behaviour
	6.4	Shaft Elastic Rotor Behaviour
		6.4.1	Idealised Rotor with Shaft-Elastic Behaviour
		6.4.2	Influence of Bearing Stiffness
		6.4.3	Flexural Resonance Speeds at Standstill
		6.4.4	General Rotor with Shaft-Elastic Behaviour
		6.4.5	Unbalance Effects on the Rotor with Shaft-Elastic Behaviour
			6.4.5.1 Modal Unbalances
			6.4.5.2 Equivalent Modal Unbalances
		6.4.6	Balancing a Rotor with Shaft-Elastic Behaviour
			6.4.6.1 First Flexural Mode
			6.4.6.2 Second Flexural Mode
			6.4.6.3 Third Flexural Mode
		6.4.7	Choice of Correction Planes
			6.4.7.1 Variety of Rotors
			6.4.7.2 Example 1
			6.4.7.3 Example 2
			6.4.7.4 Example 3
			6.4.7.5 Example 4
			6.4.7.6 Example 5
			6.4.7.7 Example 6
			6.4.7.8 Example 7
7 Tolerances for Rotors with Rigid Behaviour
	7.1	Preliminary Note
	7.2	Basics
		7.2.1	Tolerance Planes
		7.2.2	Correction Planes
		7.2.3	Limitation of the Permissible Residual Unbalance
	7.3	Similarity Considerations
		7.3.1	Rotor Mass and Permissible Residual Unbalance
		7.3.2	Service Speed and Permissible Residual Unbalance
			7.3.2.1 Special Cases
	7.4	Determining the Permissible Residual Unbalance
		7.4.1	General
		7.4.2	Balancing Grades G
			7.4.2.1 Classification
			7.4.2.2 Special Cases
			7.4.2.3 Permissible Residual Unbalance
		7.4.3	Experimental Determination
		7.4.4	Limits from Specific Targets
			7.4.4.1 Limitation by Bearing Forces
			7.4.4.2 Limitation Through Vibrations
		7.4.5	Proven Experience
			7.4.5.1 Almost Identical Rotor Size
			7.4.5.2 Similar Rotor Size
	7.5	Allocation to Tolerance Planes
		7.5.1	Rotors with a Single Tolerance Plane
			7.5.1.1 Practical Review
			7.5.1.2 Acceptance
		7.5.2	Rotors with Two Tolerance Planes
			7.5.2.1 Restrictions on Inboard Rotors
			7.5.2.2 Restrictions on Outboard Rotors
	7.6	Assignment of the Unbalance Tolerance to the Correction Planes
		7.6.1	Single-Plane Case
		7.6.2	Two-Plane Case
	7.7	Assembled Rotors
	7.8	Unbalance Readings for the Balancing Process
		7.8.1	Example
	7.9	Checking the Residual Unbalance
		7.9.1	Acceptance Criteria
		7.9.2	Unbalance Readings in Tolerance
		7.9.3	Unbalance Readings Outside Tolerance
		7.9.4	Region of Uncertainty
		7.9.5	Particularities when Measuring Unbalances
		7.9.6	Checking on a Balancing Machine
		7.9.7	Checking Without a Balancing Machine
8 Tolerances for Rotors with Flexible Behaviour
	8.1	Preliminary Note
	8.2	General
		8.2.1	Balancing Target
		8.2.2	Balancing Procedures
	8.3	Tolerance Criteria
		8.3.1	Vibrations
			8.3.1.1 Vibrations According to ISO 21940-12
			8.3.1.2 Problems with Vibrations
			8.3.1.3 Conclusion
		8.3.2	Unbalances
			8.3.2.1 Total Permissible Unbalance
			8.3.2.2 Tolerance Planes
			8.3.2.3 Distribution of the Total Permissible Unbalance
			8.3.2.4 Modal Influence on the Permissible Unbalances
	8.4	Unbalance Tolerances for Procedures A to I
		8.4.1	Tolerances of Low-Speed Balancing Procedures
			8.4.1.1 Procedure A: Single-Plane Balancing
			8.4.1.2 Procedure B: Two-Plane Balancing
			8.4.1.3 Procedure C: Balancing Individual Components Before Assembly
			8.4.1.4 Procedure D: Balancing After Limiting the Starting Unbalance
			8.4.1.5 Procedure E: Sequential Balancing During Assembly
			8.4.1.6 Procedure F: Balancing in Optimal Planes
		8.4.2	Tolerances of High-Speed Balancing Procedures
			8.4.2.1 Procedure G: Multiple-Speed Balancing
			8.4.2.2 Procedure H: Balancing at Service Speed
			8.4.2.3 Procedure I: Balancing at a Fixed Speed
	8.5	Unbalance Tolerances for Procedure G
		8.5.1	Unbalance Tolerances According to ISO 21940-12
		8.5.2	Unbalance Tolerances According to DIN ISO 21940-12, Beiblatt 1
			8.5.2.1 Distribution to Several Unbalances
				8.5.2.1.1 Even Distribution
				8.5.2.1.2 Weighted Distribution
	8.6	Tolerances for the Balancing Process
	8.7	Assessment of the Unbalance State Achieved
		8.7.1	Assessment by Vibrations
			8.7.1.1 Assessment in a High-Speed Balancing Machine
			8.7.1.2 Assessment in the Test Field
			8.7.1.3 Assessment in Service Condition
		8.7.2	Assessment by Unbalances
			8.7.2.1 Assessment in a Low-Speed Balancing Machine
			8.7.2.2 Assessment in a High-Speed Balancing System
			8.7.2.3 Assessment in the Test Field
			8.7.2.4 Assessment in Service
	8.8	Susceptibility and Sensitivity of Machines to Unbalance
		8.8.1	Classification of the Susceptibility of Machines
		8.8.2	Modal Sensitivity Ranges
		8.8.3	Limit Curves
			8.8.3.1 Example 1
			8.8.3.2 Example 2
			8.8.3.3 Special Case Acceleration
			8.8.3.4 Example
		8.8.4	Experimental Determination of the Modal Sensitivity
			8.8.4.1 Example 1
			8.8.4.2 Example 2
9 Procedures for Balancing Rotors with Rigid Behaviour
	9.1	Preliminary Note
	9.2	Bodies Without Own Bearing Journals
		9.2.1	General
		9.2.2	Unbalances Due to Assembly
		9.2.3	Index Balancing
			9.2.3.1 Single Plane with Unbalances
			9.2.3.2 Single Plane with Fit-Related Errors
			9.2.3.3 Generalisation
		9.2.4	Further Use of the Index Balancing Method
		9.2.5	Auxiliary Shafts, Adapters
	9.3	Assemblies
		9.3.1	General
		9.3.2	Procedure
			9.3.2.1 Example 1
			9.3.2.2 Example 2
		9.3.3	Interchangeability of Parts
		9.3.4	Correction of the Assembly Error
		9.3.5	Dummies (Substitute Masses)
	9.4	Rotors with Parallel Keys
		9.4.1	General
		9.4.2	Shaft with Complete Key
		9.4.3	Shaft with Half Key
		9.4.4	Influence on the Unbalance Condition
		9.4.5	Bias for a Parallel Key
		9.4.6	Constructive Measures
10 Procedures for Balancing Rotors with Flexible Behaviour
	10.1	Preliminary Note
	10.2	General
	10.3	Rotor Configurations
		10.3.1	Basic Elements Of Shaft-Elastic Rotors
		10.3.2	Balancing Principles
		10.3.3	Rotor with Discs
			10.3.3.1 A Single Disc
			10.3.3.2 Two Discs
			10.3.3.3 More than Two Discs
		10.3.4	Rigid Sections
		10.3.5	Rolls
		10.3.6	Integral Rotor
		10.3.7	Combinations
		10.3.8	Repairs
	10.4	Balancing Procedures
		10.4.1	Procedure A: Single-Plane Balancing
		10.4.2	Procedure B: Two-Plane Balancing
		10.4.3	Procedure C: Balancing Individual Components Before Assembly
		10.4.4	Procedure D: Balancing After Limiting The Initial Unbalance
		10.4.5	Procedure E: Sequential Balancing During Assembly
			10.4.5.1 Problem with Transfer Unbalances
			10.4.5.2 Solution
			10.4.5.3 Problem Assembly
		10.4.6	Procedure F: Balancing in Optimal Planes
		10.4.7	Procedure G: Balancing at Multiple Speeds
			10.4.7.1 2  +  N procedure and N + 2 Procedure
			10.4.7.2 Correction Ratio
			10.4.7.3 Recommendation
			10.4.7.4 Computer Support
			10.4.7.5 Beiblatt 1 to DIN ISO 21940-12
		10.4.8	Procedure H: Balancing at Service Speed
		10.4.9	Procedure I: Balancing at a Fixed Speed
		10.4.10	Settling Behaviour Procedure
11 Description of the Balancing Task
	11.1	Preliminary Note
	11.2	Balancing Rotors with Rigid Behaviour
		11.2.1	Rotor with Journals
			11.2.1.1 Tabular Description of a Rotor Type
			11.2.1.2 More Tables
			11.2.1.3 Maximum Data
			11.2.1.4 Additional Information on the Rotors
		11.2.2	Rotors Without Journals
	11.3	Balancing Rotors with Flexible Behaviour
		11.3.1	Low-Speed Balancing
		11.3.2	High-Speed Balancing
			11.3.2.1 General
			11.3.2.2 Tabular Overview
12 Balancing Machines
	12.1	Preliminary Note
	12.2	Quotation and Technical Documentation
		12.2.1	Horizontal Balancing Machines
			12.2.1.1 Limits for Rotor Mass and Unbalance
			12.2.1.2 Efficiency of the Measuring Run
			12.2.1.3 Unbalance Reduction Ratio RUR
			12.2.1.4 Rotor Dimensions
			12.2.1.5 Bearing Journal
			12.2.1.6 Setting Range of the Correction Planes
			12.2.1.7 Drive
			12.2.1.8 Brake
			12.2.1.9 Additional Information
		12.2.2	Vertical Balancing Machines
			12.2.2.1 Limits for Rotor Mass and Unbalance
			12.2.2.2 Rotor Dimensions
			12.2.2.3 Influence of the Moment Unbalance
		12.2.3	Non-Rotating Balancing Machines
		12.2.4	High-Speed Balancing Machines
			12.2.4.1 Drive
			12.2.4.2 Bearing Pedestals
			12.2.4.3 Measuring Device
	12.3	Technical Details and their Assessment
		12.3.1	Drive
			12.3.1.1 Squirrel-Cage Motor
			12.3.1.2 Slip Ring Motor
			12.3.1.3 DC Motor
			12.3.1.4 Drive Power
			12.3.1.5 Cardan Shaft Drive
			12.3.1.6 Belt Drive
			12.3.1.7 Induction Field Drive
			12.3.1.8 Self-Drive
			12.3.1.9 Compressed Air Drive
		12.3.2	Display Systems
		12.3.3	Functional Principle of the Balancing Machine
		12.3.4	Brake
		12.3.5	Speed
		12.3.6	Calibration and Setting of the Measuring Device
			12.3.6.1 Soft-Bearing Balancing Machine
			12.3.6.2 Hard-Bearing Balancing Machine
		12.3.7	Foundation
		12.3.8	Minimum Achievable Residual Unbalance Umar
		12.3.9	Bearing Support
			12.3.9.1 Twin-Roller Bearing
			12.3.9.2 V-Block Bearing
			12.3.9.3 Sleeve-Bearing
			12.3.9.4 Spindle Bearing
			12.3.9.5 Service Bearings
			12.3.9.6 Special Bearing Systems
		12.3.10	Mass Moment of Inertia, Number of Cycles
		12.3.11	Measuring Principle
		12.3.12	Test Rotors, Test Masses
			12.3.12.1 Test Rotors
			12.3.12.2 Test Masses
		12.3.13	Overload
		12.3.14	Environmental Influences
		12.3.15	Unbalance Reduction Ratio RUR
		12.3.16	Economic Efficiency
	12.4	Boundary Conditions
13 Tests on Balancing Machines
	13.1	Preliminary Note
	13.2	Statistics with Unbalances
		13.2.1	Circular Scatter Field
		13.2.2	Ring-Shaped Scatter Field
			13.2.2.1 Example 1
			13.2.2.2 Example 2
		13.2.3	Characteristics of One- and Two-Dimensional Normal Distributions
		13.2.4	Further Special Features
		13.2.5	Spot Checks or a Hundred Percent Check
		13.2.6	Key Figures
		13.2.7	Rejects
	13.3	Test Rotors and Test Masses
		13.3.1	General
		13.3.2	Test Rotors Overview
			13.3.2.1 Individual Test Rotors
		13.3.3	Test Masses
		13.3.4	Test Rotors in Detail
			13.3.4.1 Type A Test Rotors
			13.3.4.2 Test Rotors Type B
			13.3.4.3 Test Rotors Type C
		13.3.5	Test Conditions
	13.4	Test of the Minimum Achievable Residual Unbalance Umar
		13.4.1	Start Condition
		13.4.2	Correction
		13.4.3	Test Runs with Test Masses
		13.4.4	Evaluation of the Umar Test
		13.4.5	Abbreviated Umar Test
	13.5	Unbalance Reduction Ratio Test RUR
		13.5.1	Start Condition
		13.5.2	Test Runs with Test Masses
		13.5.3	Evaluation of the RUR Test
		13.5.4	Abbreviated RUR Test
	13.6	Test of the Moment Unbalance Influence Ratio IMU
		13.6.1	Starting Conditions
		13.6.2	Test Runs with Test Masses
		13.6.3	Evaluation of the IMU Test
	13.7	Compensation Test for the Index Process
		13.7.1	Start Condition
		13.7.2	Test Runs with Test Masses
		13.7.3	Evaluation of the Compensation Test
14 Unbalance Correction
	14.1	Preliminary Note
	14.2	Types of Correction
		14.2.1	Material Removal
		14.2.2	Relocating Material
		14.2.3	Adding Material
	14.3	Correction Time
		14.3.1	Organisation of the Correction
		14.3.2	Automation of the Correction
	14.4	Correction Errors
		14.4.1	Correction Masses
		14.4.2	Correction Planes
		14.4.3	Correction Radii
			14.4.3.1 Radial Correction
			14.4.3.2 Correction on the Circumference
			14.4.3.3 Correction by Spreading Two Correction Masses
		14.4.4	Correction Angle
		14.4.5	Permissible Correction Errors
	14.5	Unbalance Reduction Ratio
		14.5.1	General
		14.5.2	Small Correction Step
		14.5.3	Large Correction Step
		14.5.4	Series
15 Preparation and Execution of Rotor Balancing
	15.1	Preliminary Note
	15.2	Causes for Unbalances
	15.3	Design Guidelines and Drawing Specifications
	15.4	Layout of the Correction
	15.5	Work Planning
		15.5.1	Rotor Condition During Balancing
		15.5.2	Permissible Unbalance Readings for the Balancing Process
			15.5.2.1 Commonly Practiced Approach
			15.5.2.2 Current Approach of the Standards
	15.6	Automation
	15.7	Loading and Unloading
	15.8	Preparations on the Rotor
	15.9	Production Cycle Rotor Balancing
16 Errors in the Balancing Process
	16.1	Preliminary Note
	16.2	Causes for Errors
		16.2.1	General
		16.2.2	Missing Parts
		16.2.3	Additional Parts or Effects
		16.2.4	Changed Rotor State
		16.2.5	Rotor Behaviour is Not Reproducible
		16.2.6	Unbalance Measurement
	16.3	Handling of Errors
		16.3.1	Error Types
			16.3.1.1 Systematic Errors
			16.3.1.2 Random Errors
			16.3.1.3 Scalar Error
		16.3.2	Determination of Errors
			16.3.2.1 Estimation of Errors
			16.3.2.2 Measuring Errors
			16.3.2.3 Errors During Measurement
		16.3.3	Treatment of Errors
			16.3.3.1 Calculation of Systematic Errors
			16.3.3.2 Calculation of Random Errors
			16.3.3.3 Calculation of Scalar Errors
		16.3.4	Determination of the Combined Error
	16.4	Permissible Indications for the Residual Unbalance
	16.5	Acceptance Criteria
		16.5.1	General
		16.5.2	Unbalance Readings in Tolerance
		16.5.3	Unbalance Readings Outside the Tolerance
		16.5.4	Region of Uncertainty
	16.6	Special Methods for Measuring Errors
		16.6.1	General
		16.6.2	Measuring Systematic Errors
		16.6.3	Measuring Random Errors
		16.6.4	Measuring Scalar Errors
	16.7	Examples of Errors and Their Handling
		16.7.1	General
		16.7.2	Examples
			16.7.2.1 Movable Parts
			16.7.2.2 Liquids or Solids in Cavities
			16.7.2.3 Thermal Influences and Effects due to Gravity
			16.7.2.4 Windage Effects
			16.7.2.5 Magnetized Rotor
			16.7.2.6 Tilted Service Ball Bearings
			16.7.2.7 Incomplete Assembly
			16.7.2.8 Coupling Face on Rotor
			16.7.2.9 Fitting Errors
			16.7.2.10 Relative Rotation of Mounted Parts
			16.7.2.11 Adapter Unbalance
			16.7.2.12 Unbalance of the Drive Shaft
			16.7.2.13 Adapter Run-out
			16.7.2.14 Eccentricity of Balancing Bearings
			16.7.2.15 Systematic and Random Errors of the Measurement Chain
		16.7.3	Specialties When Measuring Unbalances
			16.7.3.1 Errors When Measuring on a Balancing Machine
			16.7.3.2 Errors when Measuring Without a Balancing Machine
17 Protective Measures for Balancing
	17.1	Preliminary Note
	17.2	General
	17.3	Dangers Due to the Rotor
	17.4	Protection Classes
		17.4.1	Protection Against Contact
		17.4.2	Protection Against Particles or Parts
			17.4.2.1 Area-specific Energy
			17.4.2.2 Absolute energy
			17.4.2.3 Impulse
	17.5	Choice of Enclusures
	17.6	Examples of Protection Classes
		17.6.1	Class 0
		17.6.2	Class A
		17.6.3	Class B
		17.6.4	Class C
		17.6.5	Class D
	17.7	Protection Class C for Universal Balancing Machines
		17.7.1	Design of the Protection
		17.7.2	Marking of the Protection
	17.8	Hazards and Safety Requirements
18 In-Situ Balancing
	18.1	Preliminary Note
	18.2	Vibration Limits
	18.3	Task
	18.4	Theory of In-situ Balancing
		18.4.1	Causes for Unbalances
		18.4.2	Difficulties
		18.4.3	Methodology
			18.4.3.1 Correction in a Single Plane
			18.4.3.2 Correction in Two Planes
			18.4.3.3 Correction in More Than Two Planes
	18.5	Practice of In-situ Balancing
		18.5.1	Measuring Techniques
		18.5.2	Measuring Planes
		18.5.3	Boundary Conditions
	18.6	ISO 21940-13
19 Symbols, Vocabulary and Definitions
	19.1	Preliminary Note
	19.2	Symbols
	19.3	Vocabulary and Definitions
		19.3.1	Mechanics—Mechanik—Mécaniques
		19.3.2	Rotors—Rotoren—Rotors
		19.3.3	Unbalance—Unwucht—Balourd
		19.3.4	Balancing—Auswuchten—Équilibrage
		19.3.5	Balancing Machines—Auswuchtmaschinen—Machines à équilibrer
		19.3.6	Flexible Rotors—Nachgiebige Rotoren—Rotors flexibles
		19.3.7	Rotating Rigid Free-Bodies—Rotierende starre freie Körper—Corps-libres rigides en rotation
		19.3.8	Balancing Machine Tooling—Zubehör zu Auswuchtmaschinen—Outillage de machine à équilibrer
20 Documents for Calculations
	20.1	Preliminary Note
	20.2	Decimal Multiples and Decimal Parts
	20.3	Conversion Factors for SI Units and Inch/Pound Units
	20.4	Nomogramms, Diagramms
		20.4.1	Nomogramms
		20.4.2	Diagrams
Image Sources
References
Index