4.2 Unbalance Steady-State Response Computations
4.2.2 Phase Angle Explanation and Direction of Rotation
Before demonstrating additional sample cases, the phase-angle conven- tion employed in RDA is given a careful explanation at this point because of the confusion and errors that frequently occur in general where rotor vibration phase angles are involved. Confusion concerning rotor vibration phase angles stems from a number of sources. The first source of confu- sion, common to harmonic signals in general, is the sign convention, (i.e., is the phase angle defined positive when the signal leads or lags the reference signal?). The second source of confusion stems from the visual similar- ity between the complex plane illustration of harmonic signals as rotating vectors and the actual rotation of fixed points or force vectors on the rotor, for example, high spot, heavy spot (or unbalance mass).
On real machines, the most troublesome consequence of phase-angle confusion occurs when balance correction weights are placed at incorrect angular locations on a rotor. Similar mistakes often result from the fact that the rotor must spin clockwise (cw) when viewed from one end and counter- clockwise (ccw) when viewed from the other end. Consequently, it is far less confusing to have the analysis model consistent with the actual rotor’s
rotational direction and this is accomplished by starting the shaft element inputs from the proper end of the rotor. As shown in Section 2.3 of Chap- ter 2, RDA is formulated in a standard xyz right-hand coordinate system where x and y define the radial directions and positive z defines the axis and direction of positive rotor spin velocity. Thus, if one views the rotor from the end where the rotation is ccw, the positive z-axis should point toward them and the rotor model shaft elements’ input should start from the other end of the rotor. The proper end of the rotor to start RDA shaft ele- ment inputs is accordingly demonstrated in Figure 4.1 for a three-element (four-mass-station) example.
The RDA phase angle sign convention is defined by the following specifications for unbalance force and vibration displacement components:
Fx=muruω2cos(ωt+θ), mu=unbalance mass, x=X cos(ωt+φx) Fy=muruω2sin(ωt+θ), ru=unbalance radius, y=Y cos(ωt+φy)
(4.1) These specifications define a phase angle (θ,φx, andφy) as positive when its respective harmonic signal leads the reference signal. A commonly used convenient way to visualize this full complement of synchronous harmonic signals is the complex plane representation, which illustrates each harmonic signal as a rotating vector. Figure 4.2 shows this for the RDA unbalance force and vibration displacement components.
The three complex vectors (X, Y, and F) shown in Figure 4.2 are conceived rotating at the angular velocityωin the ccw direction, thus maintaining their relative angular positions to each other. However, this is not to be confused with points or vectors fixed on the rotor that also naturally rotate ccw atω. It is only that the mathematics of complex numbers has long been recognized and used as a convenient means of representing a group of related harmonic signals all having the same frequency, such as the various
x y
z
1 2 3 4
I II
III Element numbers
Mass station numbers w
FIGURE 4.1 Proper shaft element and station input ordering.
Re Im
F X Y
+ +
-
- YcosFy XcosFx F cosQ F sinQ
F ∫ muruw2 Q ∫ wt+ q Fx ∫ wt+ fx Fy ∫ wt+ fy
Fy
Fx
Q
w
FIGURE 4.2 Complex plane representation of synchronous harmonic signals.
components of voltage- and current-related signals in alternating-current electricity. Since the unbalance force is purely a synchronous rotating vec- tor, it is easy to view it as a complex entity since its x-component projects onto the real axis, while its y-component projects onto the imaginary axis.
The same could be said of the rotor orbits for the simple three-mass rotor model given in the previous subsection, because the bearing stiffness and damping inputs are radially isotropic and thus yield circular orbits. But this is not typical.
To avoid confusion when applying the complex plane approach to rotor vibration signals, it is essential to understand the relationship between the standard complex plane illustration and the position coordinates for the orbital trajectory of rotor vibration. First and foremost, the real (Re) and imaginary (Im) axes of the standard complex plane shown in Figure 4.2 are not the x and y axes in the plane of radial orbital rotor vibration trajectory.
There are a few rotor vibration academics who have joined the complex plane and the rotor x–y trajectory into a single illustration and signal man- agement method by using the real axis for the x-signal and the imaginary axis for the y-signal. This can be accomplished by having the component Y cosΦyprojected onto the imaginary axis by definingΦyrelative to the imaginary axis instead of the real axis. The author does not embrace this approach since all the rotor orbital trajectory motion coordinates then entail complex arithmetic. The author does embrace the usefulness of the com- plex plane, as typified by Figure 4.2, to illustrate the steady-state rotor vibration harmonic signals specified by Equation 4.1.
Figure 4.3 is an addendum to Figure 4.2, illustrating the x-displacement, x-velocity, and x-acceleration in the complex plane. The same can be done for the y-direction signals. As in the previous complex plane illustration,
X -wXsinFx
- w2XcosFx
Re XcosFx
X = wX
X= w2X
Fx = wt + fx
x = X cos(wt + fx)
˙x = - wX sin(wt + fx)
¨x = - w2X cos(wt + fx) w
FIGURE 4.3 Complex plane view of x-displacement, velocity, and acceleration.
all the shown vectors in Figure 4.3 rotate ccw at the angular velocityω, thus maintaining their relative angular positions to each other.
4.2.3 3-Mass Rotor Model + 2 Bearings/Pedestals and 1 Disk
The previous 3-mass model is augmented here with the addition of bearing pedestals, as formulated in Section 2.3 of Chapter 2. The inputs here differ from those in the previous example only by the addition of a pedestal at each bearing. For creating this input file from the Keyboard Input option, only the following inputs are added to the previous sample’s input as prompted by RDA. The following numerical inputs are used in this example.
Number of Pedestals: 2
Pedestal Data Pedestal No. (Prompt) Station No. Weight (lb)
1 1 5.0
2 3 5.0
Pedestal Stiffness and Damping Coefficients
Pedestal No. (Prompt) Kxx(lb/in.) Kyy(lb/in.) Cxx(lb s/in.) Cyy(lb s/in.)
1 2000 2000 0.5 0.5
2 2000 2000 0.5 0.5
This example has 16 DOFs, four more than the previous example, because each of the two pedestals has two DOF, that is, x and y. An abbreviated output summary follows. Since the model is symmetric about station 2,
rotor and pedestal responses at station 3, being the same as at station 1, are not shown here.
Response of Rotor Station No. 1
X-Direction Y-Direction
Speed (rpm) Amplitude (mils) Phase Angle (◦) Amplitude (mils) Phase Angle (◦)
100.0 0.001 −0.8 0.001 −90.8
300.0 0.007 −2.5 0.007 −92.5
500.0 0.020 −4.2 0.020 −94.2
700.0 0.044 −5.9 0.044 −95.9
900.0 0.087 −7.9 0.087 −97.9
1100.0 0.174 −10.4 0.174 −100.4
1300.0 0.398 −14.7 0.398 −104.7
1500.0 1.803 −36.5 1.803 −126.5
1700.0 1.196 −173.9 1.196 96.1
Maximum amplitudes of station 1 occurred at
1500.0 rpm for the X-direction with 1.8 mils and a phase angle of−36.5◦. 1500.0 rpm for the Y-direction with 1.8 mils and a phase angle of−126.5◦.
Response of Rotor Station No. 2
X-Direction Y-Direction
Speed (rpm) Amplitude (mils) Phase Angle (◦) Amplitude (mils) Phase Angle (◦)
100.0 0.003 −0.2 0.003 −90.2
300.0 0.031 −0.6 0.031 −90.6
500.0 0.091 −1.0 0.091 −91.0
700.0 0.201 −1.5 0.201 −91.5
900.0 0.396 −2.3 0.396 −92.3
1100.0 0.780 −3.7 0.780 −93.7
1300.0 1.762 −7.1 1.762 −97.1
1500.0 7.843 −28.0 7.843 −118.0
1700.0 5.083 −164.6 5.083 105.4
Maximum amplitudes of station 2 occurred at
1500.0 rpm for the X-direction with 7.8 mils and a phase angle of−28.0◦. 1500.0 rpm for the Y-direction with 7.8 mils and a phase angle of−118.0◦.
Response of Pedestal No. 1 Located at Station No. 1
X-Direction Y-Direction
Speed (rpm) Amplitude (mils) Phase Angle (◦) Amplitude (mils) Phase Angle (◦)
100.0 0.000 −0.2 0.000 −90.2
300.0 0.003 −0.5 0.003 −90.5
continued
Continued
500.0 0.010 −0.9 0.010 −90.9
700.0 0.023 −1.4 0.023 −91.4
900.0 0.046 −2.2 0.046 −92.2
1100.0 0.093 −3.7 0.093 −93.7
1300.0 0.219 −7.2 0.219 −97.2
1500.0 1.025 −28.3 1.025 −118.3
1700.0 0.704 −165.2 0.704 104.8
Maximum amplitudes of pedestal 1 occurred at
1500.0 rpm for the X-direction with 1.0 mils and a phase angle of−28.3◦. 1500.0 rpm for the Y-direction with 1.0 mils and a phase angle of−118.3◦.
A number of observations can immediately be made from this abbrevi- ated output summary. First, the addition of pedestals has dropped the first critical speed from about 1680 rpm (previous example) to about 1500 rpm, as all the response signals here peak at approximately 1500 rpm. Second, the orbital trajectories of rotor stations as well as the pedestal masses are all circular and corotational. This is shown by the x and y ampli- tudes for a given rotor station or pedestal mass being equal, with the x-signal leading the y-signal by 90◦. This is the result of all bearing and pedestal stiffness and damping coefficients being radial isotropic, other- wise the trajectories would be ellipses. Third, the total response of rotor station 1 is almost twice its pedestal’s total response. Relative rotor-to- bearing/pedestal motions are now continuously monitored on nearly all large power plant and process plant rotating machinery using noncon- tacting inductance-type proximity probes mounted in the bearings and targeting the rotor ( journals). Part 3 of this book, Monitoring and Diagnos- tics, describes this in detail. Since a bearing is held in its pedestal, bearing motion and pedestal motion are synonymous here within the context of an RDA model. The corresponding additional computation of rotor ( journal) orbital trajectory relative to the bearing can be derived directly with the aid of the previously introduced complex plane, wherein the standard rules for vector addition and subtraction apply. This is illustrated in Figure 4.4 and specified by Equations 4.2:
xR=XRcos(ωt+φRX), yR=YRcos(ωt+φRY) xB=XBcos(ωt+φBX), yB=YBcos(ωt+φBY) xrel=xR−xB yrel =yR−yB
≡Xrelcos(ωt+φXrel), ≡Yrelcos(ωt+φYrel)
(4.2)
All the vectors in Figure 4.4 maintain their relative angular position to each other and rotate ccw atω. By considering the view shown to be at time t=0, it is clear from standard vector arithmetic that the single- peak amplitudes and phase angles for the relative rotor-to-bearing orbital
XB XR
YR
YB
Re
Im w
Xrel
Yrel
FIGURE 4.4 Rotor and bearing displacements (R, rotor; B, bearing).
trajectory harmonic signals are given as follows:
Xrel=
(XRcosφRX−XBcosφBX)2+(XRsinφRX−XBsinφBX)2 Yrel=
(YRcosφRY−YBcosφBY)2+(YRsinφRY−YBsinφBY)2 φXrel=tan−1
XRsinφRX−XBsinφBX
XRcosφRX−XBcosφBX
φYrel=tan−1
YRsinφRY−YBsinφBY YRcosφRY−YBcosφBY
(4.3)
Equations 4.3 are general, and thus applicable to the RDA outputs for any case. Substituting outputs from the simple isotropic bearing/pedestal example problem here, one may confirm that the relative rotor-to-bearing orbits are circles since the individual rotor and pedestal orbits are cir- cles. For general anisotropic systems, the total-motion and relative-motion orbits are ellipses.