Bode and Polar Plot

Vibration Vector

•          A vibration vector plotted in the transducer response plane

•           1x vector is 90 mic pp /220^o

•           Zero reference is at the transducer angular location

•           Phase angle increases opposite to direction of rotation

 Polar Plot

 •          Polar plot is made up of a set of vectors at different speeds.

•           Vector arrow is omitted and the points are connected with a line

•           Zero degree is aligned with transducer location

•           Phase lag increases in direction opposite to rotation

•          1x uncompensated Polar Plot shows location of rotor high spot relative to transducer

•           This is true for 1x circular orbits and approximately true for 1x elliptical orbits

Shaft Orbit Plots (II)

Not- 1X Compensation of an Orbit

•          At Left orbit is the uncompensated orbit

•          At right is the same orbit with the 1X component removed

•          The remaining vibration is primarily 1/2X from a rub

Measurement of peak-to-peak amplitude of an Orbit

X transducer measurement axis is drawn together with perpendicular lines that are tangent to maximum and minimum points on the orbit

Direction of Precession in Orbits

•          In the orbit plot shaft moves from the blank towards the dot. In the plot on left the inside loop is forward precession

•          In the right orbit the shaft has reverse precession for a short time at the outside loop at bottom

Effect of Radial Load on Orbit Shape

•          Orbits are from two different steam turbines with opposite rotation. Both machines are experiencing high radial loads

•           Red arrows indicate the approximate direction of the applied radial load.

•           Red arcs represent the probable orientation of the bearing wall

Deflection Shape of Rotor Shaft

•          When keyphasor dots of simultaneous orbits at various bearings along the length of the rotor are joined an estimate of the three dimensional deflection shape of the rotor shaft can be obtained

 *   This is a rigidly coupled rotor system 

 

 

Shaft Orbit Plots (I)

The Orbit

•          The orbit represents the path of the shaft centerline within the bearing clearance.

•          Two orthogonal probes are required to observe the complete motion of the shaft within.

•          The dynamic motion of the shaft can be observed in real time by feeding the output of the two orthogonal probes to the X and Y of a dual channel oscilloscope

•          If the Keyphasor output is fed to the Z axis, a phase reference mark can be created on the orbit itself

•          The orbit, with the Keyphasor mark, is probably the most powerful plot for machinery diagnosis

Precession

Once a gyroscope starts to spin, it will resist changes in the orientation of its spin axis. For example, a spinning top resists toppling over, thus keeping its spin axis vertical. If a torque, or twisting force, is applied to the spin axis, the axis will not turn in the direction of the torque, but will instead move in a direction perpendicular to it. This motion is called precession. The wobbling motion of a spinning top is a simple example of precession. The torque that causes the wobbling is the weight of the top acting about its tapering point. The modern gyroscope was developed in the first half of the 19th cent. by the

 

Construction of an Orbit

•          XY transducers observe the vibration of a rotor shaft

•          A notch in the shaft (at a different axial location) is detected by the Keyphasor transducer.

•          The vibration transducer signals produce two time base plots (middle) which combine into an orbit plot (right)

 Probe Orientation and the Orbit Plot

•          On the left side, when the probes are mounted at 0^o and 90^oR, the orbit plot and oscilloscope display show the same view.

•           On the right, when the probes are mounted at 45^oL and 45^oR, the orbit plots are automatically rotated

•           The oscilloscope, however, must be physically rotated 45^o CCW to display the correct orbit orientation 

Examples of 1X and Subsynchronous Orbits

 

•          Orbit at left shows subsynchronous fluid-induced instability. Note the multiple keyphasor dots because the frequency is not a fraction of the running speed

•          The orbit at right is predominantly 1X. The keyphasor dots appear in a small cluster indicating dominant 1X behavior 

 

Slow Roll Vector Compensation of 1X Filtered Orbit

•          Slow roll vector compensation can considerably change the amplitude and phase of the orbit

•           Slow roll vectors of X= 1.2 mil pp /324^o and Y= 1.4 mil pp /231^o

Slow roll Waveform Compensation of a Turbine Orbit

Note how compensation makes the orbit (right) much clearer

Full Spectrum Plots

Full Spectrum

•          Half Spectrum is the spectrum of a WAVEFORM

•          Full Spectrum is the spectrum of an ORBIT

•          Derived from waveforms of two orthogonal probes

–        These two waveforms provide phase information to determine direction of precession at each frequency

–        For phase accuracy they must be sampled at same time

•          Calculated by performing a FFT on each waveform

•          These FFT’s are subjected to another transform

–        Data converted to two new spectra – one for each direction of precession – Forward or Reverse

–        Two spectra are combined into a single plot

Forward to the right, Reverse to the left

Calculation of Full Spectrum Plot

 

First

Waveform and its half spectrum

Second

Waveform and its half spectrum

Combined orbit and its full spectrum

 

Circular Orbits and Their Full Spectra

<-- Forward Precession        

         Spectrum on forward side of plot

 <--    Reverse Precession

         Spectrum on reverse side of plot

         Direction of rotation – CCW

<--   Forward Precession

        Spectrum on forward side of plot

        Direction of rotation – CW

<--   Reverse Precession

        Spectrum on reverse side of plot

        Direction of rotation - CW

Full Spectrum of Elliptical Orbit

Orbit is generated by two counter rotating vectors

Forward spectrum length is twice the length of forward rotating vector

Reverse spectrum length is twice the length of reverse rotating vector

Major axis of ellipse = a +b

Minor axis of ellipse = a - b

Original orbit cannot be reconstructed from full spectrum because there is no phase information.

3 possible orbits are shown

Circular & Elliptical 1x Orbits

•         Direction of precession is indicated by dominant line of “Forward” and “Reverse” components. 

•          Flatness of ellipse is determined by the relative size of forward and reverse components

•          When orbit is circular there is only one spectrum line

•          When orbit is a line the spectrum components are equal.

•          Therefore, the smaller the difference between components, the more elliptical the orbit. 

Orbit and Spectrum of a ½x Rub

•         Orbit and spectrum of  a steam turbine with a ½ x rub

•          Full spectrum clarifies the complex orbit which is a sum of ½ x, 1x and their harmonics.

•          From the ratio of forward ad reverse components

•          1x is the largest, forward and mildly elliptical

•          ½ x and 2x orbits are nearly line orbits

•          Small component of 3/2 x is third harmonic of ½ x fundamental

 

Half and Full Spectrum Display of a ½ x Rub

 

Differentiating ½ x Rub and Fluid Instability from Full Spectrum Plots

•         Half and full spectrum display of a ½ x rub (red data) and fluid induced instability (blue data)

•         Note similarity in appearance of the two half spectrum plots

•         The full spectrum plots clearly show the difference in the subsynchronous vibration

–        The ½ x rub orbit is extremely elliptical – small difference between forward and reverse components

–        The fluid induced instability orbit is forward and nearly circular – large difference between forward and reverse 1x and ½ x components.

•         The unfiltered orbits are at the bottom

Full Spectrum Cascade Plot of Machine Start Up

•         Horizontal axis represents precession frequency

•          Rotor speed is to the left and amplitude scale is on the right

•          Order lines drawn diagonally from the origin show vibration frequencies that are proportional to running speed

•         Display of spectra plots taken at different speeds during start up

•          Base of each spectrum is the rotor speed at which the sample was taken

•          Diagonal lines are “Order” lines. Usually 1x, 2x and ½ x are plotted

•          Resonances and critical speed can be seen on 1x diagonal line

•          Sudden appearance of ½ x indicates rub which can produce harmonics.

•          Phase relationships cannot be seen on cascade plot.

•          Many harmonics at low speed usually due to scratches on shaft

 

·         Horizontal ellipse shows rub second balance resonance (critical)

·         Vertical ellipse shows ½ x rub frequency is almost equal to first critical. Slight shift to right is due to stiffening of rotor system from rub contact.

Full Spectrum Waterfall Plot

•         Displays spectra with respect to time

•         Used for correlating response to operating parameters

•          Time on left and Running Speed on right. Amplitude scale is at extreme right

•          Plot of compressor shows subsynchronous instability whenever suction pressure is high (red). 1x component is not shown on plot. 

•          Full spectrum shows subsynchronous vibration is predominantly forward.

Waterfall of Motor with Electrical Noise Problem

•         High vibration at mains frequency (60 Hz) during start up (red). 1x is low.

•          Vibration reduces when normal speed and current are reached (green)

•          When motor is shut down (blue) 60 Hz component disappears suddenly.

•          1x component reduces gradually with speed.

Summary

 •         Conventional spectrum is constructed from the output waveform of a single transducer

•         Full Spectrum is constructed from the output of a pair of transducers at right angles.

–        Displays frequency and direction of precession

–        Forward precession frequencies are shown on right side

–        Reverse Precession frequencies are shown on left side

•         Full spectrum is the spectrum of an orbit

–        Ratio of forward and reverse orbits gives information about ellipticity and direction of precession

–        However, there is no information about orientation of orbit

•         Cascade and Waterfall plots can be be constructed either from half or full spectra

Half Spectrum Plots

Spectrum Plot-1

•          Machines can vibrate at many different frequencies simultaneously  1x, 2x, 3x, vane passing etc.

•          Timebase  and orbit have frequency information but only a couple of harmonics can be identified – impossible to identify nonsynchronous frequencies

•         Using an analog tunable analyzer the amplitude and phase at each individual frequency can be identified but only one at a time. 

–        All frequencies cannot be seen simultaneously.

–        Trend changes in individual frequencies cannot be followed

–        Each frequency sweep may take one minute during which short duration transient events may be missed

   •          A Spectrum Plot by a FFT Analyzer shows all frequencies instantaneously.

Spectrum Plot-2  

•          Spectrum plot is the basic display of a Spectrum Analyzer. It the most important plot for diagnosis

•          Spectrum plot displays the entire frequency content  of complex vibration signals in a convenient form.

–        It has frequency on X-axis and amplitude on Y-axis

–        It is constructed from sampled timebase waveform of a single transducer – displacement, velocity or acceleration

•          Fast Fourier Transform (FFT) calculates the spectrum from the sample record which contains a specific number of waveform samples

•          Spectrum plots can be used to identify harmonics of running frequency, rolling element bearing defect frequencies, gear  mesh frequencies, sidebands 

Periodic motion with more than one frequency

Above waveform broken up into a sum of harmonically related sine waves

 

Illustration of how the previous signal can be described in terms of a frequency spectrum.

Left        - Description in time domain

Right      - Description in frequency domain 

 

Spectrum Frequency as a Function of Pulse Shape

Construction of Half Spectrum Plot - 1

•          Raw timebase signal (red) is periodic but complex.

•           Fourier transform is equivalent to applying of a series of digital filters

•           Filtered frequency components are shown as sine waves (blue)

•           Phase for each signal can be measured with respect to trigger signal

•           We can see components’ amplitude, frequency and phase 

  
Construction of Half Spectrum Plot - 2

•          If we rotate the plot so that the time axis disappears we see a two dimensional spectrum plot of amplitude v/s frequency

•           Component signals now appear as  series of vertical lines.

•           Each line represents a single frequency

•           Unfortunately, the phase of the components is now hidden.

•           It is not possible to see phase relationships  in spectrum plot.

 

These plots show why it is impossible to guess the frequency content from the waveform.

Vertical lines in top plot show one revolution

It is clear that 2x and higher frequencies are present

But 3x and 6x could not be predicted from the waveform.

A Fourier spectrum shows all the frequencies present

Linear and Logarithmic Scaling

•          Amplitude scaling can be Linear or Logarithmic

•          Logarithmic scaling is useful for comparing signals with very large and very small amplitudes.

–        Will display all signals and the noise floor also

•          However, when applied to rotating machinery work

–        Log scale makes it difficult to quickly discriminate between significant and insignificant  components.

•          Linear scaling shows only the most significant components.

–        Weak, insignificant and low-level noise components are eliminated or greatly reduced in scale

•          Most of our work is done with linear scaling

 

Illustration of Linear and Log scales

•           Log scale greatly amplifies low level signals

•           It is impossible to read 1% signals in linear scale

•           It is very easy to read 0.1% signals on the log scale

Limitations of Spectrum Plots

•          FFT assumes vibration signal is constant and repeats forever.

•          Assumption OK for constant speed machines .

–        inaccurate if m/c speed or vibration changes suddenly.

•          FFT calculates spectrum from sample record

–        Which has specific number of digital waveform samples

–        FFT algorithm extends sample length by repeatedly wrapping the signal on itself

–        Unless number of cycles of signal exactly matches length of sample there will be discontinuity at the junction

–        This introduces noise or leakage into the spectrum

•          This problem is reduced by “windowing”

–        Forces signal smoothly to zero at end points

–        Hanning window best compromise for machinery work

Effect of Windowing

•          Figure shows a timebase plot with a mixture of 1/2x and 1x frequencies. 

 Two examples of half spectrum plots are shown below

•          Without window function the “lines” are not sharp and widen at the bottom

•           This “leakage” is due to discontinuity at sample record ending

•          When “Hanning” window is applied to the sample record 1/2x spectral line is narrower and higher

•           Noise floor at base is almost gone.