Phase Analysis: Making Vibration

Analysis Easier
Tony DeMatteo, 4X Diagnostics, LLC
Vibration analysis is mostly a learned skill. It is based 70 percent on experience and 30
percent on classroom training and self study. It takes years to become a confident and
competent vibration analyst. When the analysis is wrong, the recommendations for
repair also will be incorrect. No vibration analyst wants to make the wrong call. In this
business, credibility is gained in small steps and lost in large chunks.
A vibration sensor placed on a bearing housing and connected to a vibration analyzer
provides time, frequency and amplitude information in the form of a waveform and a
spectrum (Figure 1). This data is the foundation for vibration analysis. It contains the
signatures of nearly all mechanical and electrical defects present on the machine.
Figure 1. Vibration Waveform and Spectrum
The vibration analysis process involves determining the vibration severity, identifying
frequencies and patterns, associating the peaks and patterns with mechanical or
electrical components, forming conclusions and, if necessary, making recommendations
for repair.
Everybody involved in vibration analysis knows that analyzing vibration is not easy nor
automated. Have you ever wondered why? Here are a few reasons:
1) Machines Have Multiple Faults: The vibration patterns we learn in training and read
about in books just don’t look the same in the real world. We learn how mechanical and
electrical faults look in the purest form – as if there was always only that one problem on
the machine causing vibration. Machines usually have more than one vibration-
producing fault. At a minimum, all machines have some unbalance and misalignment.
When other faults develop, the waveform and spectrum quickly become complicated and
difficult to analyze. The data no longer matches the fault patterns we have learned.
2) Cause and Effect Vibration: For every action, there is a reaction. Some of the
vibration we measure is the effect of other problems. For example, the force caused by
rotor unbalance can make the machine look like it is out of alignment, loose or rubbing.
Consider all of the things that shake and rattle on your car when one tire goes out of
balance.
3) Many Fault Types Have Similar Patterns: Because machine rotors rotate at a
particular speed, and vibration is a cyclical force, many mechanical and electrical faults
exhibit similar frequency patterns that make it difficult to distinguish one fault from
another.
Learning to analyze vibration just takes time. Training courses, technical publications
and other resources such as online resources and commercial self teaching material are
available that can improve analysis skills and shorten the learning curve.
There is one diagnostic technique which quickly gets to the source of most vibration
problems. It is possibly the most powerful of all vibration diagnostic techniques. It has
been around as long as vibration analysis itself yet hasn’t gotten a lot of attention, and
it’s rare to find good information about the subject. What is this technique? It’s called
phase analysis.
What is Phase?
Phase is the position of a rotating part at any instant with respect to a fixed point. Phase
gives us the vibration direction. Tuning a car engine using a timing light and inductive
sensor is an application of phase analysis (Figure 2).
Figure 2. Engine tuning using a timing light is phase analysis.
A phase study is a collection of phase measurements made on a machine or structure
and evaluated to reveal information about relative motion between components. In
vibration analysis, phase is measured using absolute or relative techniques.
Absolute phase is measured with one sensor and one tachometer referencing a mark
on the rotating shaft (Figure 3). At each measurement point, the analyzer calculates the
time between the tachometer trigger and the next positive waveform peak vibration. This
time interval is converted to degrees and displayed as the absolute phase (Figure 4).
Phase can be measured at shaft rotational frequency or any whole number multiple of
shaft speed (synchronous frequencies). Absolute phase is required for rotor balancing.
Figure 3. Absolute Phase Measurement
 
Figure 4. Absolute phase is calculated between the tach signal and vibration
waveform.   
Relative phase is measured on a multi-channel vibration analyzer using two or more
(similar type) vibration sensors. The analyzer must be able to measure cross-channel
phase. One single-axis sensor serves as the fixed reference and is placed somewhere
on the machine (typically on a bearing housing). Another single-axis or triaxial sensor is
moved sequentially to all of the other test points (Figure 5). At each test point, the
analyzer compares waveforms between the fixed and roving sensors. Relative phase is
the time difference between the waveforms at a specific frequency converted to degrees
(Figure 6). Relative phase does not require a tachometer so phase can be measured at
any frequency.
Figure 5. Relative Phase Measurement
Figure 6. Relative Phase Calculated Between Two Vibration Waveforms
Both types of phase measurements are easy to make. Relative phase is the most
convenient way to measure phase on a machine because the machine does not need to
be stopped to install reflective tape on the shaft. Phase can be measured at any
frequency. Most single-channel vibration analyzers can measure absolute phase. Multi-
channel vibration analyzers like the Pruftechnik VibXpert illustrated in Figure 7 have
standard functions for measuring both absolute and relative phase.
Figure 7. Pruftechnik VibXpert 2-Channel Vibration Analyzer
When to use Phase Analysis
Everyone needs phase analysis. A phase study should be made on problem machines
when the source of the vibration is not clear or when it is necessary to confirm suspected
sources of vibration. A phase study might include points measured only on the machine
bearings or it can include points over the entire machine from the foundation up to the
bearings. The following are examples of how phase can help analyze vibration.
Soft Foot
The term soft foot is used to describe machine frame distortion. It can be caused by a
condition where the foot of a motor, pump or other component is not flat, square and
tight to its mounting, or many other things, such as machining errors, bent or twisted feet
and non-flat mounting surfaces. Soft foot increases vibration and puts undue stress on
bearings, seals and couplings. Soft foot on a motor distorts the stator housing creating a
non-uniform rotor to stator air gap resulting in vibration at two times line frequency.
A good laser shaft alignment system should be used to verify soft foot by loosening the
machine feet one at a time.
Phase can be used to identify soft foot while the machine is in operation. Measure
vertical phase between the foot and its mounting surface. If the joint is tight, the phase
angle is the same between surfaces. If the phase angle is different by more than 20
degrees, the foot is loose or the machine frame is cracked or flimsy. Figure 8 is an
example of the phase shift across a soft foot.
Figure 8. A phase shift between the foot and mount may indicate soft foot.
Cocked Bearings and Bent Shafts
Phase is used to detect cocked bearings and bent shafts. Measure phase at four axial
locations around the bearing housing. If the bearing is cocked or the shaft is bent
through the bearing, the phase will be different at each location. If the shaft is straight
and the bearing is not twisting, the phase will be the same at each location (Figure 9).
Figure 9. Phase identifies in-plane or twisting bearing motion.
Confirm Imbalance
A once-per-revolution radial vibration usually means rotor unbalance. Use phase to
prove imbalance is the problem. To confirm imbalance, measure the horizontal and
vertical phase on a shaft or bearing housing. If the difference between the phase values
is approximately 90 degrees, the problem is rotor unbalance (Figure 10). If the phase
difference is closer to zero or 180 degrees, the vibration is caused by a reaction force.
An eccentric pulley and shaft misalignment are examples of reaction forces.
Figure 10. Horizontal to Vertical Phase Shift of about 90 Degrees Confirms
Unbalance
Looseness, Bending or Twisting
Phase is used to detect loose joints on structures and bending or twisting due to
weakness or resonance. To check for looseness, measure the vertical phase at each
mechanical joint as indicated by the arrows in Figure 11. When joints are loose, there
will be a phase shift of approximately 180 degrees. The phase angle will not change
across a tight joint.
Figure 11. A phase shift between bolted joints indicates looseness.
Shaft Misalignment
Shaft misalignment is easily verified with phase. Measure each bearing in the horizontal,
vertical and axial directions. Record the values in a table or bubble diagram as shown in
Figure 12. Compare the horizontal phase from bearing to bearing on each component
and across the coupling. Repeat the comparison using vertical then axial data. Good
alignment will show no substantial phase shift between bearings or across the coupling.
The machine in Figure 12 has a 180-degree phase shift across the coupling in the radial
directions. The axial directions are in-phase across the machine. The data indicates
parallel (offset) shaft misalignment.
Figure 12. Phase Data Indicates Parallel Shaft Misalignment
Operational Deflection Shapes
Instead of comparing the phase and magnitude numbers from a table or bubble diagram,
operational deflection shape software (ODS) can be used to animate a machine
drawing. An ODS is a measurement technique used to analyze the motion of rotating
equipment and structures during normal operation. An ODS is an extension of phase
analysis where a computer-generated model of the machine is animated with phase and
magnitude data or simultaneously measured time waveforms. The animation is visually
analyzed to diagnose problems. ODS testing is able to identify a wide variety of
mechanical faults and resonance issues such as looseness, soft foot, broken welds,
misalignment, unbalance, bending or twisting from resonance, structural weakness and
foundation problems.
Figure 13 is a simple ODS of three direct-coupled shafts. Phase and magnitude were
measured from permanently mounted X and Y displacement probes on a turbine
generator. The values listed in the table were used in ODS software to animate a stick
figure drawing of the high- and low-pressure turbine shafts and the generator shaft. The
picture to the right of the table is a capture from the ODS animation showing the
vibration pattern of each shaft and the relative motion between shafts at 3,600 cycles per
minute (turning speed).
Figure 13. Shaft Operational Deflection Shape
Many machines vibrate due to deteriorated foundations, looseness, resonance of the
support structure and other problems that occur below the machine bearings. A phase
study might include hundreds of test points measured all over the machine and
foundation. Good ODS software can make it easier to analyze phase and magnitude
data from a large number of test points. Analysis of an ODS involves observation and
interpretation of the machine in motion. Figure 14 is an ODS structure drawing of a
vertical pump.
Figure 14. Vertical Pump Operational Deflection Shape Structure Drawing
Conclusion
Condition-based vibration testing is a vital component of a reliability based maintenance
program. Vibration sensors, instruments and software are able to provide key
information about machine health. The weak link in the chain is the analyst’s ability to
interpret the data, accurately diagnose the problem and trend the fault until it is time to
recommend corrective action. Phase analysis is a very powerful diagnostic tool. Every
vibration analyst should be using phase to improve vibration analysis accuracy.
About the author:
Tony DeMatteo is a vibration analyst and technical training instructor with 4X
Diagnostics LLC, a service and training company providing consulting services,
mentoring and training in diagnostic measurement, analysis, operational deflection
shape testing and modal analysis. He can be reached at 585-293-3234
or www.4xdiagnostics.com. 
 Undamped Harmonic Forced
Vibrations
Often, mechanical systems are not undergoing free vibration, but are
subject to some applied force that causes the system to vibrate. In this
section, we will consider only harmonic (that is, sine and cosine)
forces, but any changing force can produce vibration.

When we consider the free-body diagram of the system, we now have

an additional force to add, namely the external harmonic excitation.

A mass-spring
system with an external force, F, applying a harmonic excitation.

The equation of motion of the system above will be:

m¨x+kx=Fmx¨+kx
=F

Where F is a force of the form:

F=F0sinω0tF=F0sin⁡ω0t

This equation of motion for the system can be re-written in standard

form:
¨x+kmx=F0msinω0tx¨+kmx=F0msin⁡
ω0t

The solution to this system consists of the superposition of two

solutions: a particular solution, xp (related to the forcing function), and
a complementary solution, xc (which is the solution to the system
without forcing).
As we saw previously, the complementary solution is the solution to
the undamped free system:

We can obtain the particular solution by assuming a solution of the

form:

xp=Dsin(ω0t)xp=Dsin⁡(ω0t
)

Where ω0 is the frequency of the harmonic forcing function. We

differentiate this form of the solution, and then sub into the above
equation of motion:
¨xp=−ω20Dsin(ω0t)xp¨=−ω02Dsin⁡(ω0t)

−mω20Dsin(ω0t)+kDsin(ω0t)=F0sin(ω0t)
−mω02Dsin⁡(ω0t)+kDsin⁡(ω0t)=F0sin⁡(ω0t)

Solving for D, we find D and the the particular solution, xp:

D=F0k1−(ω0ωn)2D=F0k1−(ω0ωn)2

xp=F0k1−(ω0ωn)2sin(ω0t)xp=F0k1−
(ω0ωn)2sin⁡(ω0t)

Thus, the general solution for a forced, undamped system is:

xG(t)=F0k1−(ω0ωn)2sin(ω0t)
 +Csin(ωnt+ϕ)xG(t)=F0k1−(ω0ωn)2sin⁡(ω0t)
+Csin⁡(ωnt+ϕ)

The
complementary solution of the equation of motion. This represents the natural response of the
system, and oscillates at the angular natural frequency. This is the transient response.

The particular
solution of the equation of motion. This represents the forced response of the system, and oscillates
at the angular forced frequency. This is the steady-state response.
 The general
solution of the equation of motion. This represents the combined response of the system, and the
sum of the complementary (or natural) and particular (or forced) responses.

The above figures show the two responses at different frequencies.

Recall that the value of ωn comes from the physical characteristics of
the system (m, k) and ω0 comes from the force being applied to the
system. These responses are summed, to achieve the blue response
(general solution) in the third figure.

Steady-state Response

In reality, this superimposed response does not last long. Every real
system has some damping, and the natural response of the system
will be damped out. As long as the external harmonic force is applied,
however, the response to it will remain. When evaluating the response
of the system to a harmonic forcing function, we will typically consider
the steady-state response, when the natural response has been
damped out and the response to the forcing function remains.

Amplitude of Forced Vibration

The amplitude of the steady-state forced vibration depends on the
ratio of the forced frequency to the natural frequency.
As ω0 approaches ωn (ratio approaches 1), the magnitude, D,
becomes very large. We can define a magnification factor:
 MF=F0k1−(ω0ωn)2F0k=11−(ω0ωn)2MF=F0k1−
(ω0ωn)2F0k=11−(ω0ωn)2

The magnification factor, MF, is defined

as the ratio of the amplitude of the steady-state vibration to the displacement that would be
achieved by static deflection.

From the figure above, we can discuss various cases:

 ω0 = ωn: resonance occurs. This results in very large amplitude

vibrations, and is associated with high stress and failure to the
system.
 ω0 ~ 0, MF ~ 1: The forcing function is nearly static, leaving
essentially the static deflection and limited natural vibration.
 ω0 < ωn: Magnification is positive and greater than 1, meaning the
vibrations are in phase (when the force acts to the left, the system
displaces to the left) and the amplitude of vibration is larger than the
static deflection.
 ω0 > ωn: Magnification is negative and the absolute value is
typically smaller than 1, meaning the vibration is out of phase with
the motion of the forcing function (when the force acts to the left, the
 system displaces to the right) and the amplitude of vibration is
smaller than the static deflection.
 ω0 >> ωn: The force is changing direction too fast for the block's
motion to respond.

Rotating Unbalance
One common cause of harmonic forced vibration in mechanical
systems is rotating unbalance. This occurs when the axis of rotation
does not pass through the centre of mass. In this situation, instead of
the centre of mass remaining stationary, it experiences some
acceleration. This causes a force on the axle that changes direction as
the center of mass rotates. We can represent this as a small mass, m,
rotating about the axis of rotation at some distance, called an
eccentricity, e. The forced angular frequency, ω0, in this case is the
angular frequency of the rotating system.