Understanding Amplitude And Phase In Rotating Machinery-Free PDF

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What is Phase and What Does it Mean, The electrical engineers are well acquainted with phase The phase of two or three voltages such as 3 phase motors is. well established and the understanding of phase and power factor is critical in electrical equipment The roll of phase. and its measurements in rotating machinery is not as well established or appreciated by many manufactures. Phase M easurements W ith Strobe Light, One of the earliest methods of phase measurements was the use of a strobe light which was triggered by a velocity. pickup Fig 1 shows the phase position before and,after the placement of a trial weight on a rotor. balancing demonstrator, The phase measurements are made from the stationary. compass The initial strobe light shows the arrow,pointing at 270 deg After placement of the trial.
weight the strobe light shows that the phase has shifted. 60 deg in the clockwise direction,Although balancing may be made from such. measurements the strobe light procedure of relative. phase measurements provides little insight to the actual. Fig 1 Phase Readings W ith 2 gm Trial W eight,rotor dynamics amplitude phase relationship. The strobe light is triggered to flash based on the signal received from the analyzer The strobe light is made to flash. on the shaft freezing it in place W hen a strobe light is used for a phase reference in balancing the determination of the. rotor high spot is not as simple or straightforward as when a noncontact probe and a keyphasor reference mark are used. Fig 2 developed by C Jackson represents the phase, angle for a typical seismic vibration analyzer in the. displacement mode In addition to the 90 deg phase, shift due to integration of the velocity signal there is. a speed dependent rolloff phase shift due to the mass. inertia effects of the velocity sensor at low frequencies. Fig 2 was developed by Jackson to help find the rotor. heavy spot for assistance in the placement of the initial. trial weight for influence coefficient balancing,Amplitude and Phase M easurements W ith.
Noncontact Inductance Probes,A noncontact probe has the distinct advantage of. observing shaft motion directly One of the earliest. types of noncontact displacement probe dating to the. 1960 s was the British Wayne Kerr capacitance, probe This probe was ideally suited for early gas Fig 2 Phase Lag From Seismic Velocity Sensor to Rotor. bearing rotor dynamic studies but was not practical for Heavy Spot Jackson 1979. industrial rotating equipment This was due to the cost of the equipment problems of contamination of the probe surface. and limited range of motion, Fig 3 Shaft M otion W ith Noncontact Displacement Probe. Fig 3 represents the direct shaft displacement as observed by a noncontact probe To view this motion the vibration. signal was displayed on an oscilloscope and the motion recorded by a Polaroid camera No phase measurements were. recorded with this early procedure By taking a series of pictures at various speeds a crude plot of amplitude vs speed. could be attained, Fig 4 Noncontact and Keyphasor Probes Fig 5 Amplitude Signal W ith Keyphasor M ark. A major advancement in the understanding of rotor dynamic amplitude phase relationships was the introduction of the. Bently noncontact inductance vibration pickup Unlike the W ayne Kerr capacitance probe the inductance probe was. much less expensive rugged and with a larger range of motion A unique innovation of Donald Bently before the. advent of the FFT analyzer was the introduction of the keyphasor. Fig 4 represents a typical setup with an inductance probe to monitor shaft motion and the addition of a second probe. referred to as the keyphasor probe A grove is placed on the shaft in line with the keyphasor probe When the shaft. notch passes under the keyphasor a signal is generated as shown in Fig 5 When the output of the keyphasor is fed into. the z axis of an oscilloscope a bright spot occurs on the vibration signature as shown in the lower waveform as seen in. The phase convention in this case is the angle phi as measured from the bright spot on the wave form of Fig 5 to the. peak amplitude This angle has a practical significance in balancing For example if one lines up the timing reference. mark with the keyphasor probe the peak amplitude A occurs at the angle phi as measured from the inductance probe. in the direction opposite to rotation This point is often referred to as the high spot The location of the unbalance is. referred to as the heavy spot, If we know the approximate critical speed of the rotor then we have an idea of where to place the first trial weight for.
field balancing for a relatively simple rotor system If the rotor is operating in the subcritical speed range well below. the first critical speed then the high spot and the heavy spot are in the same vicinity The first initial trial weight would. be placed 180 deg from the high spot If the rotor were operating well above the first critical speed for the case of the. simple single mass Jeffcott rotor the mass center is out of phase to the maximum displacement or high spot In this case. the initial trial weight would be placed at the angle phi as measured counter rotation from the inductance probe This. placement seems against intuition If we should attempt the unusual balancing procedure of balancing at the critical. speed then in this case the mass center is leading the mass center by 90 deg In this case one would line up the keyphasor. probe with the notch and place the first trial weight at an angle of 90 phi deg as measured opposite of rotation from the. inductance probe Such a direct understanding of the relationship between the peak amplitude and the approximate. location of the unbalance is not possible with the strobe light method. Amplitude and Phase Relationships For the Jeffcott Rotor. The first English publication to describe the dynamics of the flexible single mass rotor was presented by H H Jeffcott. in 1919 He correctly expressed the equations of motion of the single mass rotor with unbalance and damping and. presented an explanation of how the rotor could pass successfully through the critical speed He also showed the. displacement phase relationship at speeds below at and above the critical speed To properly understand rotor dynamics. amplitude phase relationships and their implications we must have a good understanding of the simple Jeffcott rotor. One of the earliest accounts on rotor dynamics was the article presented in 1869 by W A Rankine entitled On the. Centrifugal Force of Rotating Shafts Engineer Vol 27 London Rankine introduced the elementary concept of. indifferent rotor equilibrium Rankine examined the equilibrium conditions of a frictionless uniform shaft disturbed. from its initial position Because he neglected the influence of the Coriolis force he concluded that motion is stable. below the first critical speed is neutral or in indifferent equilibrium at the critical speed and is unstable above the. critical speed, During the next half century this analysis lead engineers to believe that operation above the first critical speed was. not possible It was not until 1895 that DeLaval demonstrated experimentally that a steam turbine was capable of. sustained operation above the first critical speed. Fig 6 represents the single mass Jeffcott rotor In this. model the mass is concentrated at the shaft center and the. disk is considered as massless The bearings at taken as. simple supports The damping is applied at the disc center. This assumption may also be considered as a form of. model damping The unbalance in the disk is created by. the small offset e u of the disk center from the axis of. The equations of motion of the single mass Jeffcott rotor. may be combined into one complex vector equation by. means of the complex variable transformation Z X I. Y as follows,Fig 6 Single M ass Jeffcott Rotor, The steady state synchronous unbalance response due to unbalance is given by. The amplitude and phase may be expressed in dimensionless form by using the following variables. natural frequency on rigid supports frequency ratio. critical damping d a m p in g r a tio A dim amplitude. The relative phase angle lag of the deflection vector from the rotating unbalance load is given by. Amplitude and Phase of the Jeffcott Rotor Below At and Above The Critical Speed. There are 3 distinct ranges of motion for the Jeffcott rot In each of these ranges the phase angles are distinctly. different The first speed range of interest is the speed range referred to as the subcritical speed range. In this speed range we have the condition that the operating speed is much lower than the critical speed. Case 1 Subcritical Speed Operation, Thus we have the condition that f 1 and the dimensionless amplitude and phase angle is approximately given by. This implies that at low speeds well below the first critical speed the amplitude of motion due to unbalance excitation. is increasing as speed squared This region is characterized as rigid body motion region Many current industrial. fans are designed to operate in this region as rigid body rotors often with highly undesirables effects The location. of the mass unbalance is in phase with the maximum deflection In this low speed region we say that the high spot. and the heavy spot coincide,Case 2 Critical Speed Operation. The second region is a unique region in which the rotor is operating exactly at the critical speed. In this case f 1 and the dimensionless amplitude of motion is given by. The mass center is leading the maximum deflection by. W here Acr is referred to as the critical speed amplification factor on rigid supports This is a finite value depending. upon the modal damping acting upon the rotor It should be noted that with the Jeffcott rotor all of the damping is. concentrated at the disk The bearings are providing no damping since they are node points. W ith a rotor supported on rolling element bearings the amount of aerodynamic drag damping available at the center. disk can be quite small This could lead to dangerously high levels of vibration if such a rotor were to be. continuously under these conditions,Case 3 Supercritical Speed Operation.
In the supercritical speed the rotor speed is well above the critical speed This region is referred to as supercritical. speed operation in which the dimensionless speed f 1 The dimensionless amplitude of motion is given by. In this region of operation the amplitude of motion is no longer increasing as mass center inversion has occurred. The rotor is spinning or rotating about the center of mass Since the center of mass is offset from the axis of rotation. the inductance probes observe an orbit with radius e u. The phase angle of motion is now out of phase to the displacement vector and is given by. Fig 7 shows the motion of the Jeffcott rotor in the speed ranges. In the first figure the speed is well below the critical speed Point M represents the location of the mass center Point. C is the center of rotation of the shaft The point H represents the high spot and is in line with the mass center line. C M extended At this low speed the balance correction weight would be placed opposite the high spot The center. figure represents the rotor operating at the critical speed In this case the rotor mass center is leading the maximum. amplitude or high spot H by 90 deg If one were to mark the high spot as I had once observed in industry on an. experimental rotor with caulk one would place the balance correction at 90deg from the mark opposite the direction. of rotation The third figure on the right represents the shaft motion well above the critical speed The mass center. has inverted 180 deg and is now opposite the high spot. Fig 7 Jeffcott Rotor M otion Below At and Above The Critical Speed. Development Of A Jeffcott Rotor Using DyRoBes Rotor Dynamics Software. A single mass rotor model was developed using DyRoBes to simulate an idealized Jeffcott rotor A 200 lb. disc was mounted on a steel shaft of 40 inches length with a diameter of 4 inches To simulate rigid. bearings bearing stiffness values of 1 0e7 Lb in were assumed Since the shaft weight is included in this. model the modal weight is the sum of the disk weight W d plus one half the shaft weight W s The total. Jeffcott modal weight then is approximately 271 lb. Fig 8 represents the Jeffcott rotor on stiff bearings mounted on a 40 inch shaft. Fig 8 Jeffcott Rotor on Stiff Bearings Fig 9 1 st Critical Speed of Jeffcott Rotor at. Fig 9 represents the animated 1 st critical speed of the Jeffcott rotor It should be noted that with the Jeffcott. rotor there is no bearing motion Anytime this situation occurs in a real rotor one has a dangerous. condition in which the bearings will provide no damping In order to provide damping for the Jeffcott rotor. a third bearing was applied acting at the disc center At this bearing location damping in both x y. directions of 20 lb sec in was assumed, The critical damping for the Jeffcott rotor is given by. Cc 2 M cr 2 271lb 386 4x628 rad sec 881 lb sec in, For an assumed amplification factor of 20 1 40 0 025. The amount of damping acting at the disk center is given by C 0 025 Cc 22 lb sec in. Fig 10 represents the damped mode shape of the Jeffcott. rotor with damping of 20 lb sec in added at the disk. center The addition of the damping causes a slight. reduction of the damped natural frequency to 5998 RPM. The modal damping is expressed in terms of the log. VIBRATION INSTITUTE 33RD ANNUAL MEETING HARRISBURG PA JUN E 23 2 7 2009 UNDERSTANDING AMPLITUDE AND PHASE IN ROTATING MACHINERY Edgar J Gunter Ph D Professor Emeritus Department of Mechanical and Aerospace Engineering Former Director Rotor Dynamics Laboratory University of Virginia RODYN Vibration Analysis Inc DrGunter aol com Abstract The measurement of phase is essential in

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