**The Vibration Analysis HandbookFirst EditionSecond PrintingTABLE OF CONTENTSCHAPTER ONE: INTRODUCTION TO MACHINERY VIBRATIONTheoryofVibration . 1HarmonicMotion . 1Periodic Motion 1RandomMotion 3The Relationship between Time and Frequency 3Time . 3Frequency . 4Amplitude Measurement . 6Sources of Frequencies . 8Generated Frequencies . 8Excited Frequencies . 9Frequencies Caused by Electronic Phenomena 17ForcingFunction . 20Combinations of Machine Problems 21Mixing Frequencies . 22Electrical and Mechanical Relationship 23Time and Frequency Domain . 24Relationship between Velocity. Displacement. and Acceleration . 25Units of Measurement . 27Relationships . 27Ways of Measuring Vibration . 32Relation between Diameter. Speed. and RPM . 32How To Determine Machine Speed in FPM from the Vibration Data . 33Conclusion and Efficiencies . 34CHAPTER TWO: TIME AND FREQUENCY ANALYSIS TECHNIQUESIntroduction 35Basicphysics 35Single Frequency . 36Single Frequency with Harmonics 41Clipping .52SquareWave 52Natural Frequencies 55Multiple Frequencies.Linear Systems 55High Frequency Riding a Low Frequency 55Multiple Frequencies.Nonlinear Systems . 56AmplitudeModulation 57Sum and Difference Frequencies .61Pulses .65FrequencyModulation 67Conclusion .68CHAPTER THREE: HARDWARE AND SOFTWARE REQUIRED FOR ACCURATEDIAGNOSTICSHardware 71Introduction 71Personal Computer . 72Real-time Analyzer . 73Datacollection 76Printers 78Transducers 78Displacement Transducers . 79Velocity Transducers 81Accelerometers 82Pressure Transducers . 83Microphones 83Once-Per-Revolution Markers . 84Multiplexer . 85Gauss Meter 87Software . 87Introduction 87Group 1. Toolbox Software . 90Signal Analysis Program . 90Vibration Calculation Program 90Resonance and Deflection Calculator (RADC) . 92Bearing Calculation Program . 92Gears Program 95Roll Ratio Program and Rusch Chart . 97Group 2. Machine Doctor (MACHDOC) . 98MACHDOC 98Polar Plot . 100Time Plot . 102Balancing . 102Diagnostic Database . 103Diagnostic Modules . 108Roll Quality Assurance Program . 109Group3 110Group4 110CHAPTER FOUR: ACCURATE EVALUATION OF MACHINERY CONDITIONIntroduction .111Theory .111Calibration 115Frequencies Generated . 116Datacollection .117Transducer Selection .118Continuous Monitoring . 118CommonProblems 119Imbalance 119Bentshaft 119SoftFoot . 122Misalignment . 122Looseness 125Bearings Loose on the Shaft 125Bearings Loose in the Housing 126Common Forms of Looseness . 128Noise . 128Diagnosis of Looseness 129Resonance 129Rubs 130Problems That Cause Pulses 130OilWhirl 134Analysis of Electric Motors . 135Motors Out-of-Magnetic Center . 136Broken Rotor Bars . 139Turn-To-Turn Shorts in Windings 143Vibration Problems in Synchronous Motors 144SirenEffect . 146Solo Data on Motors 146Steam Turbines 148Pumps 150Impeller Contact .150Starvation 151Cavitation 152Compressors 152Fans 154Special Tests 155Startup/Coast Down Data . 155Bump Tests .158Noise Recording .160Synchronous Time Averaging (STA) 161Relative Motion Measurements (RMM) 164CHAPTER FIVE: ACCURATE DIAGNOSIS OF ANTIFRICTION BEARINGSIntroduction 167Datacollection 167Transducer Selection 167Generated Frequencies 167Fundamental Train Frequency 168Ball Pass Frequency of Outer Race .169Ball Pass Frequency of Inner Race .170Ball Spin Frequency 170Application of the Bearing Formulas .172Outer Race Analysis 175Inner Race Analysis 176Ball Spin Frequency Analysis .176Fundamental Train Frequency Analysis . 177VCI Bearing Calculation Program 177Bearing Defects . 178Raceways . 178Outer Race 178InnerRace . 181Outer and Inner Race Amplitude . 185Modulation of Ball Pass Frequency 185Rolling Elements. Balls. and Rollers . 186Cage . 189Multiple Defects 189Progressive Bearing Failure 193Defectseverity . 196Bearing Behavior 196Inner Race Defect Length . 197Defect Length Calculation . 201Deep Fatigue Spalls vs.Shallow Flaking 203Problem Sources 208Acid Etching . 208Fluting . 210Inadequate Lubrication . 214Looseness . 216Bearings That Have Excessive Internal Clearance . 216Bearings That are Turning on the Shaft . 219Bearings Loose in the Housing . 220Testing for Bearing Frequencies 220Conclusion 222CHAPTER SIX: ACCURATE EVALUATION OF GEARSIntroduction .225Data Acquisition 225How To Take Data 225Transducer Selection . 226Gear Vibration Theory . 227Evaluation of Gear Ratios . 227Factoring 227Gearmesh Frequency . 228Fractional Gearmesh Frequency 229Hunting Tooth Frequency . 230Planetary Gears .231Digression .233End Digression .236Gear Life Expectancy 242Amplitude Modulation .243The Gears Program 244The AGMA Quality Number System 245Gear Problems and Causes 246Eccentric Gears . 246Meshing Gears That Have a Common Factor and OneGear Is Eccentric 246Gears That Do Not Have a Common Factor and Oneor Both Gears Are Eccentric . 257Gears That Are Out-of-Round or Have Several High Places 262Gears Installed on a Bent Shaft 265Loose and Worn Gears 268Misaligned Gears 269Backlash Problems or Oscillating Gears . 270Broken. Cracked. or Chipped Teeth 274Conclusion . 285CHAPTER SEVEN: ANALYZING AND SOLVING PRESS ROLL AND NIP PROBLEMSIntroduction 287Vibration Theory of Rolls In Nip 288Hardware 288Software . 289Synchronous Time Averaging . 289Hardware Setup . 292Dynamic Measurement of Rolls .294Problems Associated with Rolls In Nip 297Eccentric Rolls 297Improper Ratios of Roll Sizes . 297Resonant Frequencies . 299Installation of Improperly Ground Rolls .301Diagnosing Problems . 302Data Collection and Analysis of Overall Vibration Data . 302Relative Motion between Rolls 306Conclusions and Recommendations 328References 329AppendixA 331AppendixB .341Glossary .343Index .YAC (Alternating Current) An electric current that reverses direction in a circuit at regularintervals.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. Peakis one-half of peak-to-peak. Average is 0.637 x peak. RMS is 0.707 x peak. These valueshold 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 specifyinggear 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 mathematicalanalysis.Angular Velocity A vector quantity describing rotational motion, the magnitude ofwhich is the time rate of change of angle and the direction of which is along the axis ofrotation.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 usedinterchangeably with rollers when calculating bearing frequencies.Ball Pass Frequency The frequency balls or rollers pass over a single point on the inneror outer raceway of the bearing.Ball Pass Frequency of the Inner Race (BPFI) The frequency the balls or rollers passover a single point on the inner raceway of a bearing.Ball Pass Frequency of the Outer Race (BPFO) The frequency the balls or rollers passover a single point on the outer raceway of a bearing.Ball Spin Frequency (BSF) The number of revolutions per second made by a ball orroller in an antifriction bearing.Bandwidth refers to the overall range of frequencies, also refers to the range offrequencies between the half power points.Band-pass filter An electronic device used to filter all signals in a predeterminedfrequency range.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 outof phase.Bump Test Measures the response of a component, machine, or structure when enoughenergy 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 curveresulting in restricted suction intake, and the liquid tends to vaporize while coming offthe impeller.Circumferential Crack A crack around the circumference of a roll.Clipping The truncation or flattening of the positive and/or negative portions of thesignal, normally caused by overloading electronic circuits and machinery problems.Coastdown Data Starts the process of measuring the amplitude of vibration at all speedsfrom zero to operating speed.Contacting Displacement Transducer Also referred to as a LVDT (Linear VariableDifferential Transformer). A transducer that is used to measure relative motion ordisplacement in the frequency range of 0 to 200 Hz.Crest factor One parameter used to describe the dynamic range of a voltmeter’samplifiers. Refers to a wave form and is the ratio of the peak to the RMS voltage with thedc component removed. The crest factor of a square wave is I, while that for a sine waveis 1.414. A pulse can have a crest factor of more than 9.Critical Damping The minimum viscous damping that allows a displaced system toreturn to its initial position.CF – P – DCRMSCritical speed Rotor speed at which local maximum amplitude occurs. A critical speedoccurs whenever the rotor speed coincides with a system’s natural frequency.Cycle A time interval in which a characteristic, especially a regularly repeated, event orsequence 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 eitherzero or are so small that they may be ignored.Decibel (dB) Unit for measuring relative loudness of sound and electronic signals. Thelogarithmic expression of ratios, equal to one-tenth of a Bell. The voltage ratio is:voltage outputvoltage inputThe power ratio is:p e r outputpower inputwhere 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 dccomponent.Degrees of freedom The minimum number of independent coordinates required todefine 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 anarbitrarily short time interval.Dynamic Measurement An accurate measurement of a component while all theoperational 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. Usedwhen 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 tobe expanded.Factor An integer that can be divided evenly into the number of interest and the quotientis an integer.Fast Fourier Transform A numerical calculation performed on a digitized time signal thatcreates 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 repeatedstress cycles.Felt A fabric that supports the paper in a paper machine to assist the paper makingprocess.Fiber Optic Sensor A device that uses a fiber optic light emitter and a fiber optic lightsensor 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 abearing which creates defects on the bearing raceways.Fractional Gearmesh Frequency A fraction of gearmesh frequency caused by eccentricgears 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 representedby the letter “F.” The time period is represented by the Greek letter “tau” (T). Theformulas are:Frequency Domain A term used to describe the magnitudes and frequencies ofsinusoidal functions that once summed together make a time domain signal.Frequency Modulation Periodic changing of the frequency of a sinusoidal function overtime.Frequency Range A specifically designated part of the frequency spectrum.Frequency Response The response (i.e., displacement, velocity, or acceleration) of asystem described in terms of frequency.Forcing function The excitation of the vibration; the problem that is causing thevibration.Foundation Machine support. May be rigid or possess mass-elastic properties.Fourier series A mathematical description of a nonharmonic periodic function using alinear combination of sine and cosine functions.Full Scale The largest value indicated on the scale or in the case of instruments havingtheir zero between the ends of the scale, the full-scale value.Fundamental Frequency The first harmonics or base frequency, such as gearmeshfrequency, ball pass frequency, etc.Fundamental Train Frequency The angular velocity of the individual ball centers. Cagefrequency.G Unit of measure used for acceleration measurements.Gear Life Expectancy The gear’s life expectancy is reduced by the inverse of thecommon factor of the gears in mesh.Gearmesh frequency The angular speed of the rotating gear times the number of teethon the gear.Generated Frequencies All frequencies that can be calculated and vary as a fuction ofmachine speed.Harmonic Exact multiples of a fundamental frequency.Harmonic analysis Anaiysis of a periodic complex wave form using sine and cosinefunctions.Harmonic Marker Lines displayed on the data used to identify the harmonic multiples(harmonics).Harmonic Motion The vibration motion of the projection upon a straight line of a pointmoving 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 witha tooth on the other gear a second time.Imbalance A condition where there is more mass on one side of a rotating device thanthe other causing an outward radial force in the direction of the additional mass whenthe 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 aretransmitted to bearing support.LLambda 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 arenot 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 ofpreviously 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 timevarying amplitude. Frequency modulation is a time varying frequency.Motion Description of the displacement, velocity, or acceleration of a system as afunction 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 theclarity or quality of a signal.Noncontacting Displacement Transducer A transducer that measures motions withoutcontacting the target, the common name is proximity probe.Oil Whirl The shaft ridge on an oil wedge that whirls while it rotates; occurs in somelightly loaded, plain journal bearings.Oscillating Gears A condition where gears move from a point of contact between teethin 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 theroll.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 andturning a planetary gear whose axis of rotation moves around the sun gear. Theplanetary gear meshes with a ring gear which surrounds the planetary gears and the sungear.Polar Plotting format which is circular. Measurements are in amplitude and degrees.Proximity Probe Ferromagnetic dynamic displacement transducer. Nonconductingdisplacement transducer used for measuring relative motion.Pseudo RMS Technique used by most analog meters to measure RMS. These metersmultiply average by 1.11to present RMS measurements. This works for pure sine wavesonly.Pulse A transient amplification or intensificationof a characteristicof a system,especiallyof 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 anymotion relative to a third point or reference.Relative Motion Measurement The measurement of the motion of one body with respectto another usually performed with a contacting or noncontacting displacementtransducer.Resonance The enhancement of the response of an electric or mechanical system to aperiodic driving force when the driving frequency is equal to the natural undampedfrequency of the system.Resonant Frequency The frequency of the resonance.Root Mean Square (RMS) A method of measuring the true energy under the curve. Thehalf 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 missingshallow 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 amechanical 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 accelerationin a short period.Shock spectrum The maximum response (acceleration, velocity, or displacement) of aseriesof damped or undamped singledegree-of-freedom systems resulting from a specificshock excitation. An independent mass-spring-damper system is associated with eachfrequency.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 cosinefunction.Siren Effect A generated frequency equal to the number of bars/slots in a rotor timesthe 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 thatdistance defined for an instant in time.Square Wave A rectangular wave or near rectangular wave that rises to a positive levelflat peak for a period of time then falls to a negative level flat peak for a period of timeand repeats itself. Contains odd harmonics sometimes called odd fractions.StartupData Vibration data taken on a machine as the machine goes from zero rotationalvelocity up to operating speed.Starvation This occurs in pumps when there is not enough liquid present to fill eachvane on the impeller every revolution.Stiffness The description of the elastic properties of a system given in terms of poundsforce 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 detectionof the trigger.Time Domain The signal level with respect to time.Transducer An electronic device that converts a mechanical vibration or motion into anelectronic signal.Transducer Sensitivity Used to transform the voltage output of a transducer to theappropriate engineering units (V/E).Trigger A pulse or signal used to initiate data collection. A triggered sweep or delayramp.Truncation Signals that have the peaks cut off at some level.Time The inverse of frequency. The continuous passage measured in seconds, minutes,hours.Vane pass frequency The frequency at which the blades of a pump pass a particularpoint.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 aconstant current of one ampere, when the power dissipated between these points is equalto 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 phasein 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 themaximum or minimum value of the signal.Zoom Also called frequency translation. When data is collected on a range other thanbetween 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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