The Vibration Analysis Handbook
TABLE OF CONTENTS
CHAPTER ONE: INTRODUCTION TO MACHINERY VIBRATION
TheoryofVibration . 1
HarmonicMotion . 1
Periodic Motion 1
The Relationship between Time and Frequency 3
Time . 3
Frequency . 4
Amplitude Measurement . 6
Sources of Frequencies . 8
Generated Frequencies . 8
Excited Frequencies . 9
Frequencies Caused by Electronic Phenomena 17
ForcingFunction . 20
Combinations of Machine Problems 21
Mixing Frequencies . 22
Electrical and Mechanical Relationship 23
Time and Frequency Domain . 24
Relationship between Velocity. Displacement. and Acceleration . 25
Units of Measurement . 27
Relationships . 27
Ways of Measuring Vibration . 32
Relation between Diameter. Speed. and RPM . 32
How To Determine Machine Speed in FPM from the Vibration Data . 33
Conclusion and Efficiencies . 34
CHAPTER TWO: TIME AND FREQUENCY ANALYSIS TECHNIQUES
Single Frequency . 36
Single Frequency with Harmonics 41
Natural Frequencies 55
Multiple Frequencies.Linear Systems 55
High Frequency Riding a Low Frequency 55
Multiple Frequencies.Nonlinear Systems . 56
Sum and Difference Frequencies .61
Conclusion .68CHAPTER THREE: HARDWARE AND SOFTWARE REQUIRED FOR ACCURATE
Personal Computer . 72
Real-time Analyzer . 73
Displacement Transducers . 79
Velocity Transducers 81
Pressure Transducers . 83
Once-Per-Revolution Markers . 84
Multiplexer . 85
Gauss Meter 87
Software . 87
Group 1. Toolbox Software . 90
Signal Analysis Program . 90
Vibration Calculation Program 90
Resonance and Deflection Calculator (RADC) . 92
Bearing Calculation Program . 92
Gears Program 95
Roll Ratio Program and Rusch Chart . 97
Group 2. Machine Doctor (MACHDOC) . 98
Polar Plot . 100
Time Plot . 102
Balancing . 102
Diagnostic Database . 103
Diagnostic Modules . 108
Roll Quality Assurance Program . 109
CHAPTER FOUR: ACCURATE EVALUATION OF MACHINERY CONDITION
Frequencies Generated . 116
Transducer Selection .118
Continuous Monitoring . 118
CommonProblems 119Imbalance 119
SoftFoot . 122
Misalignment . 122
Bearings Loose on the Shaft 125
Bearings Loose in the Housing 126
Common Forms of Looseness . 128
Noise . 128
Diagnosis of Looseness 129
Problems That Cause Pulses 130
Analysis of Electric Motors . 135
Motors Out-of-Magnetic Center . 136
Broken Rotor Bars . 139
Turn-To-Turn Shorts in Windings 143
Vibration Problems in Synchronous Motors 144
SirenEffect . 146
Solo Data on Motors 146
Steam Turbines 148
Impeller Contact .150
Special Tests 155
Startup/Coast Down Data . 155
Bump Tests .158
Noise Recording .160
Synchronous Time Averaging (STA) 161
Relative Motion Measurements (RMM) 164
CHAPTER FIVE: ACCURATE DIAGNOSIS OF ANTIFRICTION BEARINGS
Transducer Selection 167
Generated Frequencies 167
Fundamental Train Frequency 168
Ball Pass Frequency of Outer Race .169
Ball Pass Frequency of Inner Race .170
Ball Spin Frequency 170
Application of the Bearing Formulas .172
Outer Race Analysis 175
Inner Race Analysis 176
Ball Spin Frequency Analysis .176
Fundamental Train Frequency Analysis . 177VCI Bearing Calculation Program 177
Bearing Defects . 178
Raceways . 178
Outer Race 178
InnerRace . 181
Outer and Inner Race Amplitude . 185
Modulation of Ball Pass Frequency 185
Rolling Elements. Balls. and Rollers . 186
Cage . 189
Multiple Defects 189
Progressive Bearing Failure 193
Defectseverity . 196
Bearing Behavior 196
Inner Race Defect Length . 197
Defect Length Calculation . 201
Deep Fatigue Spalls vs.Shallow Flaking 203
Problem Sources 208
Acid Etching . 208
Fluting . 210
Inadequate Lubrication . 214
Looseness . 216
Bearings That Have Excessive Internal Clearance . 216
Bearings That are Turning on the Shaft . 219
Bearings Loose in the Housing . 220
Testing for Bearing Frequencies 220
CHAPTER SIX: ACCURATE EVALUATION OF GEARS
Data Acquisition 225
How To Take Data 225
Transducer Selection . 226
Gear Vibration Theory . 227
Evaluation of Gear Ratios . 227
Gearmesh Frequency . 228
Fractional Gearmesh Frequency 229
Hunting Tooth Frequency . 230
Planetary Gears .231
End Digression .236
Gear Life Expectancy 242
Amplitude Modulation .243
The Gears Program 244
The AGMA Quality Number System 245
Gear Problems and Causes 246
Eccentric Gears . 246
Meshing Gears That Have a Common Factor and One
Gear Is Eccentric 246Gears That Do Not Have a Common Factor and One
or Both Gears Are Eccentric . 257
Gears That Are Out-of-Round or Have Several High Places 262
Gears Installed on a Bent Shaft 265
Loose and Worn Gears 268
Misaligned Gears 269
Backlash Problems or Oscillating Gears . 270
Broken. Cracked. or Chipped Teeth 274
Conclusion . 285
CHAPTER SEVEN: ANALYZING AND SOLVING PRESS ROLL AND NIP PROBLEMS
Vibration Theory of Rolls In Nip 288
Software . 289
Synchronous Time Averaging . 289
Hardware Setup . 292
Dynamic Measurement of Rolls .294
Problems Associated with Rolls In Nip 297
Eccentric Rolls 297
Improper Ratios of Roll Sizes . 297
Resonant Frequencies . 299
Installation of Improperly Ground Rolls .301
Diagnosing Problems . 302
Data Collection and Analysis of Overall Vibration Data . 302
Relative Motion between Rolls 306
Conclusions and Recommendations 328
AC (Alternating Current) An electric current that reverses direction in a circuit at regular
Acceleration The rate of change of velocity with respect to time.
Accelerometer Any of various devices used to measure acceleration.
AC-Coupling A type of input that blocks the DC portion of the signal.
AC voltage measurements Peak-to-peak is the value of the total swing of the wave. Peak
is one-half of peak-to-peak. Average is 0.637 x peak. RMS is 0.707 x peak. These values
hold true for a true sine wave only. The more the signal deviates from a true sine wave,
the greater the error in these calculations.
AGMA gear quality number An established gear quality rating system for specifying
gear quality numbers for different pitch line velocities (in feet/minute).
Amplitude The maximum value of a periodically varying quantity.
Analyzer The hardware unit used to analyze data.
Analytical technique Solution of shock and vibration problems using mathematical
Angular Velocity A vector quantity describing rotational motion, the magnitude of
which is the time rate of change of angle and the direction of which is along the axis of
Auto Abbreviation used for automatic.
Average The arithmetic mean, as determined by the summation of the x’s over n.
Balls The balls in a bearing made from hardened metal alloys, sometimes used
interchangeably with rollers when calculating bearing frequencies.
Ball Pass Frequency The frequency balls or rollers pass over a single point on the inner
or outer raceway of the bearing.Ball Pass Frequency of the Inner Race (BPFI) The frequency the balls or rollers pass
over a single point on the inner raceway of a bearing.
Ball Pass Frequency of the Outer Race (BPFO) The frequency the balls or rollers pass
over a single point on the outer raceway of a bearing.
Ball Spin Frequency (BSF) The number of revolutions per second made by a ball or
roller in an antifriction bearing.
Bandwidth refers to the overall range of frequencies, also refers to the range of
frequencies between the half power points.
Band-pass filter An electronic device used to filter all signals in a predetermined
Baseline Data The first or initial vibration data taken from a machine system.
Beat The process of two frequenciesadding and subtracting as the signals go in and out
Bump Test Measures the response of a component, machine, or structure when enough
energy is introduced to excite the natural frequencies.
Calibration The process of measuring the accuracy of an instrument.
Cavitation When a pump is not operating at the correct point on the pump curve
resulting in restricted suction intake, and the liquid tends to vaporize while coming off
Circumferential Crack A crack around the circumference of a roll.
Clipping The truncation or flattening of the positive and/or negative portions of the
signal, normally caused by overloading electronic circuits and machinery problems.
Coastdown Data Starts the process of measuring the amplitude of vibration at all speeds
from zero to operating speed.
Contacting Displacement Transducer Also referred to as a LVDT (Linear Variable
Differential Transformer). A transducer that is used to measure relative motion or
displacement in the frequency range of 0 to 200 Hz.
Crest factor One parameter used to describe the dynamic range of a voltmeter’s
amplifiers. Refers to a wave form and is the ratio of the peak to the RMS voltage with the
dc component removed. The crest factor of a square wave is I, while that for a sine wave
is 1.414. A pulse can have a crest factor of more than 9.
Critical Damping The minimum viscous damping that allows a displaced system to
return to its initial position.CF – P – DC
Critical speed Rotor speed at which local maximum amplitude occurs. A critical speed
occurs whenever the rotor speed coincides with a system’s natural frequency.
Cycle A time interval in which a characteristic, especially a regularly repeated, event or
sequence of events occurs.
Damping A factor that causes a resonance or oscillation to decay over time.
Data Acquisition The collection and processing of data.
DC (direct current) A unidirectional current in which the changes in value are either
zero or are so small that they may be ignored.
Decibel (dB) Unit for measuring relative loudness of sound and electronic signals. The
logarithmic expression of ratios, equal to one-tenth of a Bell. The voltage ratio is:
The power ratio is:
p e r output
where zero dB is always equal to one millivolt or one milliwatt.
DC Coupling Uses a choke input that blocks the ac component and passes the dc
Degrees of freedom The minimum number of independent coordinates required to
define completely the positions of all parts of a system at any instant of time.
Diameter The width of a circular object.
Disk A wheel, usually solid and axially slim, on which mechanical work is performed,
or from which work is extracted. Examples: Turbine disk, compressor wheel.
Displacement Refers to the measurement of the distance an object moves.
Duty cycle The pulse width divided by the pulse recurrence frequency or repetition rate,
used in calculating the crest factor.Dynamic A state in which one or more quantities exhibits appreciable change within an
arbitrarily short time interval.
Dynamic Measurement An accurate measurement of a component while all the
operational forces are applied. i.e. measuring the roundness of an operating roll or shaft.
Dynamic Shape The resulting shape with all or some of the operational forces applied.
Eccentric Gear A gear that does not conform to AGMA Quality Gear Standards.
Eccentric Roll A roll that is not round.
Empirical technique The measured response of a known defect.
Engineering Units (EU) General term for the type of units used: GIs, mils, IPS, etc. Used
when the type of units does not matter; for explanation or description.
Excited Frequencies Natural or resonant frequencies excited by some source.
Expanded Mode When only a portion of a signal is displayed on the grid, it is said to
Factor An integer that can be divided evenly into the number of interest and the quotient
is an integer.
Fast Fourier Transform A numerical calculation performed on a digitized time signal that
creates a limited Fourier series of cosines and phase angles of each cosine.
Fatigue Failure Failure of a mechanical component or system as a result of repeated
Felt A fabric that supports the paper in a paper machine to assist the paper making
Fiber Optic Sensor A device that uses a fiber optic light emitter and a fiber optic light
sensor that can detect a difference between light and dark surfaces.
Fluting Arcing from the inner race to the outer race through the rolling elements in a
bearing which creates defects on the bearing raceways.
Fractional Gearmesh Frequency A fraction of gearmesh frequency caused by eccentric
gears and gears with improper ratios.Frequency The number of cycles present in a specific time period, normally one second.
Measured in Hertz, which stands for cycles per second. Frequency is usually represented
by the letter “F.” The time period is represented by the Greek letter “tau” (T). The
Frequency Domain A term used to describe the magnitudes and frequencies of
sinusoidal functions that once summed together make a time domain signal.
Frequency Modulation Periodic changing of the frequency of a sinusoidal function over
Frequency Range A specifically designated part of the frequency spectrum.
Frequency Response The response (i.e., displacement, velocity, or acceleration) of a
system described in terms of frequency.
Forcing function The excitation of the vibration; the problem that is causing the
Foundation Machine support. May be rigid or possess mass-elastic properties.
Fourier series A mathematical description of a nonharmonic periodic function using a
linear combination of sine and cosine functions.
Full Scale The largest value indicated on the scale or in the case of instruments having
their zero between the ends of the scale, the full-scale value.
Fundamental Frequency The first harmonics or base frequency, such as gearmesh
frequency, ball pass frequency, etc.
Fundamental Train Frequency The angular velocity of the individual ball centers. Cage
G Unit of measure used for acceleration measurements.
Gear Life Expectancy The gear’s life expectancy is reduced by the inverse of the
common factor of the gears in mesh.
Gearmesh frequency The angular speed of the rotating gear times the number of teeth
on the gear.
Generated Frequencies All frequencies that can be calculated and vary as a fuction of
machine speed.Harmonic Exact multiples of a fundamental frequency.
Harmonic analysis Anaiysis of a periodic complex wave form using sine and cosine
Harmonic Marker Lines displayed on the data used to identify the harmonic multiples
Harmonic Motion The vibration motion of the projection upon a straight line of a point
moving uniformly along a circumference in the same plane.
He& The unit of frequency, one cycle per second.
Housing A casing used to enclose a piece of machinery such as a motor or bearing.
Hunting Tooth Frequency (HTF) The reciprocal of the time period one gear makes with
a tooth on the other gear a second time.
Imbalance A condition where there is more mass on one side of a rotating device than
the other causing an outward radial force in the direction of the additional mass when
the object is rotated.
IPS (inches per second) Units used when measuring vibration in velocity.
Journal Specific portions of a shaft surface from which rotor applied loads are
transmitted to bearing support.
Lambda the greek letter I symbol for wavelength.
Mesh The fitting of gears together, similar to fitting the fingers of two hands together.
Mil Unit of measure, equal to a one thousandth of a second.
Misalignment When two machines are coupled together and their axes of rotation are
not aligned.Misaligned Gears Gears that are not meshing evenly across the pitch line.
Msec An abbreviation for milliseconds or thousandths of a second.
Modal Analysis A vibration response analysis that uses a unique combination of
previously determined mode shapes for its mathematical description.
Model A mathematical or experimental simulation of a component system.
Modulation The act of mixing two or more frequencies. Amplitude modulation is a time
varying amplitude. Frequency modulation is a time varying frequency.
Motion Description of the displacement, velocity, or acceleration of a system as a
function of time.
Mv Abbreviation for millivolts, thousandths of a Volt.
Natural Frequency The reciprocal of the natural period of a system.
Nip The point of contact when two rolls are rotating in contact with each other.
Noise Any disturbance, especially a random and persistent disturbance, that reduces the
clarity or quality of a signal.
Noncontacting Displacement Transducer A transducer that measures motions without
contacting the target, the common name is proximity probe.
Oil Whirl The shaft ridge on an oil wedge that whirls while it rotates; occurs in some
lightly loaded, plain journal bearings.
Oscillating Gears A condition where gears move from a point of contact between teeth
in a direction of advance then bounce back in the opposite direction.
Out-of-Round Rolls A roll whose center of rotation is not the geometric center of the
p-p (peak to peak) Signal amplitude measurement, the maximum value of one cycle.
Periodic Motion Oscillatory, periodic or repeating motion.
Pitch-line Velocity The tangential velocity of a rotating gear or bearing at the pitch-line.Planetary Gears A system of gears where a central, sun gear rotates meshing with and
turning a planetary gear whose axis of rotation moves around the sun gear. The
planetary gear meshes with a ring gear which surrounds the planetary gears and the sun
Polar Plotting format which is circular. Measurements are in amplitude and degrees.
Proximity Probe Ferromagnetic dynamic displacement transducer. Nonconducting
displacement transducer used for measuring relative motion.
Pseudo RMS Technique used by most analog meters to measure RMS. These meters
multiply average by 1.11to present RMS measurements. This works for pure sine waves
Pulse A transient amplification or intensificationof a characteristicof a system,especially
of a wave characteristic, followed by return to equilibrium or steady state.
Radial Load A load that is directed toward the center of the axis of rotation.
Random Motion Motion that is not repetitive in magnitude or frequency of occurrence.
Range The set of values lying between the upper and lower limits.
Real-time Pertaining to the actual time during which a physical process transpires.
Relative Motion The motion of one body with respect to another disregarding any
motion relative to a third point or reference.
Relative Motion Measurement The measurement of the motion of one body with respect
to another usually performed with a contacting or noncontacting displacement
Resonance The enhancement of the response of an electric or mechanical system to a
periodic driving force when the driving frequency is equal to the natural undamped
frequency of the system.
Resonant Frequency The frequency of the resonance.
Root Mean Square (RMS) A method of measuring the true energy under the curve. The
half power point of a sinusoid or .707times the peak value of a pure sinusoid.Rotor Bars The ferromagnetic bars in the rotor of an electric motor.
RPM An abbreviation used for Revolutions Per Minute.
Shallow Flaking A defect in a bearing raceway where the bearing raceway is missing
shallow flakes of metal.
Shock A nonperiodic excitationof a mechanical system characterizedby sudden loading.
Shock absorber A device that dissipates energy in order to modify the response of a
mechanical system to applied shock.
Shock isolater A resilient support that isolates a system from a shock loading.
Shock pulse A substantial disturbance characterized by rise and decay of acceleration
in a short period.
Shock spectrum The maximum response (acceleration, velocity, or displacement) of a
seriesof damped or undamped singledegree-of-freedom systems resulting from a specific
shock excitation. An independent mass-spring-damper system is associated with each
Signal-to-noise Ratio The ratio of signal levels to noise level.
Sinusoidal A description for a phenomenon that follows a sine function or a cosine
Siren Effect A generated frequency equal to the number of bars/slots in a rotor times
the rotating speed.
+slope Refers to the portion above zero in the y-axis direction.
-slope Refers to the portion below zero in the y-axis direction.
Spectra Plural of spectrum.
Spectral lines An isolated peak of intensity in a spectrum.
Spectrum The distribution of amplitude as a function of frequency.
Speed The distance covered by a point divided by the time required to cover that
distance defined for an instant in time.
Square Wave A rectangular wave or near rectangular wave that rises to a positive level
flat peak for a period of time then falls to a negative level flat peak for a period of time
and repeats itself. Contains odd harmonics sometimes called odd fractions.StartupData Vibration data taken on a machine as the machine goes from zero rotational
velocity up to operating speed.
Starvation This occurs in pumps when there is not enough liquid present to fill each
vane on the impeller every revolution.
Stiffness The description of the elastic properties of a system given in terms of pounds
force per inch of deflection.
Synchronous Time Averaging Method of time averaging to average out nonsynchronous vibration.
Thrust Load A load on a machine that is in the axial direction.
Time Delay The time interval between the starting point of a signal and the detection
of the trigger.
Time Domain The signal level with respect to time.
Transducer An electronic device that converts a mechanical vibration or motion into an
Transducer Sensitivity Used to transform the voltage output of a transducer to the
appropriate engineering units (V/E).
Trigger A pulse or signal used to initiate data collection. A triggered sweep or delay
Truncation Signals that have the peaks cut off at some level.
Time The inverse of frequency. The continuous passage measured in seconds, minutes,
Vane pass frequency The frequency at which the blades of a pump pass a particular
Velocity The speed or how fast an object is moving.
Velocity Transducer Unit used to measure the vibration velocity of an object.
Vibration The physical motion of a rotating machine.
Volt The difference of electric potential between two points of a conductor carrying a
constant current of one ampere, when the power dissipated between these points is equal
to one watt.Voltage The dot product line integral of the electric field strength along its path.
Measured in volts.
Wavelength In a periodic wave, the distance between two points of corresponding phase
in consecutive cycles.
X-axis The horizontal direction on the grid.
X-scale The minimum and maximum scaled values used on the x-axis of the grid.
Y-axis The vertical direction on the grid.
Y-scale The minimum and maximum scale values used on the y-axis of the grid.
Z-p (zero to peak) Signal amplitude measurement from the zero reference to the
maximum or minimum value of the signal.
Zoom Also called frequency translation. When data is collected on a range other than
between zero and the bandwidth, it is called a zoom. A zoom is different from an expand.
The zoom mode increases the resolution, where the expanded mode does not.
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