6.1

CHAPTER 6 RESONANCE AND CRITICAL SPEED TESTING

BASIC CONCEPTS AND INSTRUMENTATION

Resonance and critical speed tests are carried out to obtain information about the dynamic

characteristics of a machine and its structural support and piping. Information about resonances

and critical speeds is helpful in machine diagnostics and when a machine and its associated

structures must be redesigned to overcome chronic problems.

Resonances and critical speeds are frequencies and speeds that are governed by natural

frequencies, damping, and vibratory forces. A resonance is a condition in a structure or machine

in which the frequency of a vibratory force such as mass unbalance is equal to a natural frequen-

cy of the system. If the vibratory force is caused by a rotating machine, the resonance is called a

critical speed. At or close to this speed the vibration is amplified. Testing techniques for natural

frequencies of structures differ from machines because the machines generally have speed-

dependent dynamic characteristics. Machines are tested at critical speeds to obtain the best data.

Resonances are often artificially excited with hammers and shakers to obtain natural frequencies

of foundations, structures, and piping.

This chapter is concerned with basic concepts, techniques and instrumentation used to

determine the dynamic characteristics of machines and their associated structures, foundations,

and piping. Examples are provided for each type of test.

Natural Frequencies and Mode Shapes

The natural frequency of a machine or structure is governed by its design6.1. A machine

can be thought of as a mass-elastic body and represented by masses connected to springs, as

shown in Figure 6.1. Each machine system has a number of natural frequencies that can be ex-

cited by impact, random forces, or harmonic vibrating forces of the same frequency. In general,

natural frequencies are not multiples of the first natural frequency; exceptions are rare instances

involving simple components. A natural frequency becomes important in machine diagnosis when

a forcing frequency occurs at or close to a natural frequency. A state of resonance then occurs

that produces high vibration levels. Vibration levels — governed by vibratory forces and damping

— that might otherwise be normal are then amplified to the point that excessive vibration can

structurally damage the machine — usually its bearings.

6.2

Mode shapes of a system are associated with

natural frequencies. A mode shape is defined as the

deflection shape assumed by a system vibrating at a

natural frequency. A mode shape does not provide in-

formation about absolute amplitudes of motions of a sys-

tem. (Damping and vibratory forces must be known to

obtain absolute amplitudes.) Rather, it consists of de-

flections at selected axial locations in the system that are

determined relative to a fixed point, usually the end or

center of a shaft. Two mode shapes for a turbine rotor

are shown in Figure 6.2. Note that no absolute ampli-

tudes are provided.

Figure 6.2. Two Mode Shapes for a Turbine Rotor

The vibration characteristics of a system can be expressed in terms of a linear weighted

combination of its mode shapes (Figure 6.3). The vibration of each mode considered is added

within the applicable speed or frequency range to obtain the total vibration magnitude at each

speed or frequency. The weighting factors are dependent on damping, vibratory forces, and prox-

imity of a forcing frequency to a natural frequency. When the forcing frequency is close to a natu-

ral frequency, the weighting factor is amplified predominantly by a single mode, and the vibration

response assumes the shape of that mode. Figure 6.4 shows the forced response of a turbine

rotor to mass unbalance when the rotor is operating close to its first critical speed. Note that the

shape of the rotor is similar to the mode shape of the rotor at its first natural frequency (see Figure

6.2).

Figure 6.1. Lumped-Parameter

Modeling of Rotors and Structures

6.3

Excitation

A machine or structure

can be excited by one or more

vibratory forces. The force

may have a single constant

frequency, as occurs with

mass unbalance. Multiple fre-

quencies occur in engines and

reciprocating compressors. A

variable single frequency is

typical of a synchronous motor

on startup. An example of random frequencies is pump cavitation. Vibratory forces can be

caused by various factors including design, installation, manufacture, and wear. Common forcing

frequencies for machines were listed in Table 1.4.

Figure 6.4. Forced Response of a Turbine Rotor to Mass Unbalance

Interference Diagrams

An interference diagram is used to locate resonances and critical speeds with respect to

operating speed. The vertical axis (see Figure 6.5) contains the natural frequencies and forcing

frequencies. The horizontal axis is the operating speed of the machine. The point of intersection

of one or more forcing frequencies and the natural frequency is a critical speed. Whether or not

the point is actually a critical speed depends on the amount of force and damping in the system.

Figure 6.5 is an interference diagram for a rotor subject to mass unbalance. A single forcing fre-

quency is present. An interference diagram can be generated from computer models or test data.

Computer-generated interference diagrams are often validated using test data.

Figure 6.3. Modal Analysis

6.4

Measurement and Analysis

The tools required to

determine critical speeds and re-

sonances are excitation devices,

transducers, and analyzers. Exci-

tation devices include hammers,

shakers, and machine motions.

Transducers commonly used are

accelerometers, velocity pickups,

proximity probes, optical pickups,

and magnetic pickups. Instruments

for analyzing data include meters, oscilloscopes, tracking filters, data collectors, and FFT

analyzers.

Excitation Devices. Regardless of the excitation device selected, amplitude and frequency must

be controlled. If parameters are being evaluated, a steady or transient excitation is necessary; the

vibration response to the excitation must be measured and analyzed. Sweep excitation can be

induced artificially by mounting shakers on bearing caps or casings. Sweep excitation can be

produced at synchronous frequency by mass unbalance, a bowed rotor, and/or misalignment dur-

ing startup or coast down; the ma-

chine is thus producing its own ex-

citation. Figure 6.6 shows the na-

ture of excitations caused by mass

unbalance — the magnitude in-

creases as the square of the speed

— and by misalignment — the

magnitude remains constant rela-

tive to speed.

s

Figure 6.5. Interference Diagram

Figure 6.6. Comparison of Harmonic Vibratory Forces

=

=

6.5

Excitations that occur at nonsynchronous frequencies can also be used for condition anal-

ysis and identification of parameters. Nonsynchronous frequencies are produced by oil whirl at

subsynchronous frequencies; by excessive misalignment at two and three times synchronous

speed; by transient torsional excitation, which varies from 120 Hz at startup on 60-cycle machines

to zero at operating speed, depending on the slip frequency; and by rolling element bearing exci-

tations, which consist of non-integer multiples of shaft speed. The magnitudes of many of these

excitation sources are difficult to quantify because they arise from faults, but quantification is nec-

essary for modal analysis.

The fact that impact excitations are produced naturally in reciprocating machines is advan-

tageous for parameter identification but bad for normal operating conditions. Engines can be ex-

cited at frequencies up to 12 times the operating speed. Half orders can exist in four-stroke cycle

machines.

If naturally-occurring impact excitation is not available in a machine, a hammer or shaker

can be selected to excite natural frequencies. However, it is necessary to exercise caution when

these tools are used to excite operating machines to avoid bearing damage. Figure 6.7 shows a

modally tuned hammer; the sensor in the head of the hammer measures force. Phase infor-

mation, which is useful in identifying natural frequencies, can be obtained when a hammer is used

in conjunction with a dual-channel analyzer.

Hammers are also used to conduct im-

pact, or bump, tests on structures and piping.

Hammer tips (see Figure 6.7) with various lev-

els of resiliency are used to obtain the wave-

forms required to provide the necessary data.

If a hammer is not available, a four by four tim-

ber about five feet long is adequate for bump

tests. Sledge hammers with steel heads, un-

less covered by an elastomer, are not ade-

quate for impact testing. These hammers

bounce off the structure and do not put enough energy into the system. Shakers and calibrated

hammers provide the controlled excitation needed for modal testing.

Figure 6.7. Modally Tuned Hammer

6.6

Sensors. Sensors are selected to obtain vibration signals that provide information about the dy-

namic characteristics of a machine or structure. Permanently mounted proximity probes are used

to obtain information about the critical speeds of a rotor. Seismic sensors, velocity pickups, and

accelerometers are placed on the casing, foundation, piping, and other structures to determine

natural frequencies and mode shapes. Moving a sensor to different locations while applying a

fixed force with a hammer or shaker provides information about mode shapes.

Analyzers. The instrumentation used to measure and analyze a resonance or critical speed de-

pends on the goals of the test and the instrument available. Meters and oscilloscopes are ade-

quate for simple structural testing to obtain natural frequencies.

A tracking filter is best for rapid run-up/coast-down tests. Vibration is indicated in the fil-

tered frequency band, which is governed by a reference signal generated by a proximity

probe/notch or optical pickup/reflective tape. Peak vibration levels and phase changes indicate

critical speeds.

The single-channel FFT analyzer or data collector can be used for impact tests in either

the time or frequency domains. Triggering can be free or from a hammer source. Vibration peaks

indicate resonances. During impact tests a uniform (none) window or a Hanning window with a

pretrigger should be used on the analyzer. Some analyzers have special windows for impact

tests.

The dual-channel analyzer is used for impact tests when force and vibration measure-

ments are available. The force (source) is usually recorded in channel one; vibration (response)

is recorded in the second channel. The resulting transfer function of vibration/force is classed as

a mobility (velocity/force). Phase and coherence are generated simultaneously. Changes in

phase provide confirmation that a natural frequency is present at a vibration peak. Coherence is a

measure of the amount of measured force coming from the impact hammer and the relationship

between the source (excitation) and the response (vibration).

Most analyzers have special windows that force the signal to zero at the end of the FFT

sampling process (see Chapter 3) — one for the source and one for the response.

The sampling time required for the FFT process is relatively long. When FFT spectrum

analyzers are used for critical speed tests, rapid coast downs or startups will result in loss of data

— the analyzer may not be able to take samples fast enough to obtain sufficient data points for a

continuous vibration versus frequency curve. If the transient time is known, the number of data

points can be calculated.

6.7

RESONANCE TESTING TECHNIQUES

Resonance and critical speeds are dependent on natural frequencies and forcing frequen-

cies. The purpose of resonance and critical speed tests is to obtain information about the dynam-

ic properties of a machine or structure. Natural frequencies are as much a property of a machine

as its weight or size. The condition of resonance can amplify a vibration in a machine that would

otherwise be considered normal. Critical speeds are a special case of resonance in which the

vibrating forces are caused by the rotation of the rotor.

Critical speed testing is often more complicated than resonance testing because the natu-

ral frequencies encountered are functions of stiffness and mass which may be dependent on ma-

chine speed. Resonance tests can be conducted on piping, machine support structures, or ped-

estals during shutdown, but the machine must be rotating during critical speed tests. Therefore,

different techniques and instrumentation must be used to determine resonances and critical

speeds.

Conducting a Resonance Test

The purpose of a resonance test is to determine the natural frequencies of any structure

that does not have dynamic physical properties; that is, properties that change with speed or time.

Such structures include piping, casings, and machine supports. The natural frequencies of the

entire system are compared with the frequencies of the vibratory, or forcing, frequencies on an

interference diagram (Table 6.1) to determine whether or not the system is resonant.

Table 6.1. Constructing an Interference Diagram6.3

1. Determine the important sources of vibratory forces; e.g., mass unbalance, misalignment, gearmesh, engine gas, and inertia forces. Important engine forces can be determined from

a star diagram6.2.

2. Plot the frequency of the vibratory force in CPM or Hz against speed in RPM. For example, a mass unbalance has a forcing frequency equal to the speed of the rotor, sometimes called a once-per-revolution (1x) vibration.

3. Measure or calculate the natural frequencies at a number of different speeds. The machine can be struck prior to operation to determine the natural frequency when shut down. If the natural frequency does not vary with speed, the plot is a straight line (see Figure 6.8). If the natural frequency varies with speed, as is the case with fluid-film bearing, impact tests must be conducted at various speeds. However, the natural frequency at a critical speed can be obtained with a coast-down test. If this test is not possible, the natural frequencies are cal-

culated from a validated model. A mass-elastic model6.4 is constructed from drawings or

sketches of the hardware and validated using test results either at the critical speed or zero speed. The natural frequencies are calculated at a number of speeds so that the curve of natural frequency versus rotor speed can be plotted.

6.8

The level of sophistication in reso-

nance testing depends on the available

instrumentation and the complexity of the

system. The testing technique is outlined

in Table 6.2. The first consideration is the

type of force that will be used to excite the

natural frequencies of the system. The

forcing frequency must either match the

natural frequency or be impulsive in na-

ture.

Table 6.2. Conducting a Resonance Test

1. Determine the vibrations of the structure at a number of known points during operation; see Figure 6.9. These points should allow an approximate modal description of the vibrations in the structure. Reference the vibrations to the forces by using a strobe light or key phasor if possible. If the machine speed is variable, do a waterfall, or cascade, diagram of vibration versus speed (Figure 6.10) to obtain data for the interference diagram.

2. Construct an interference diagram from the best available data. The waterfall diagram will

show resonant points. 3. Strike the structure with a 4 x 4 timber, mallet, or hammer with a soft head in the direction of

the desired mode. If the desired mode is not known, strike the structure in several directions. The direction shown diagrammatically in Figure 6.9 — vertical and horizontal — usually pro-vide useful data.

4. Striking rate is determined by the data acquisition time of the FFT setup. Strike only once per data acquisition period.

5. Striking force is determined by the response (damping) in the structure. Do not overdrive the source — strike lightly and increase if required. This means a bit of trial and observation of the response.

6. Measure and record the vibration levels at a number of reference points on the structure (see Figure 6.9). The peaks on the spectrum of vibration levels at various measurement points in-dicate the natural frequencies of the structure (Figure 6.11). Some natural frequencies are not seen at all measurement points. These are nodal points.

Figure 6.8. Interference Diagram

6.9

Sinusoidal vibrating forces can excite natural fre-

quencies. However, these forces must be swept through

the frequency range until a frequency is found that

matches a natural frequency. The shape of a vibratory

force need not be sinusoidal (Figure 6.12), but is must

have the same period as the natural frequency. A non-

sinusoidal force will generate orders of the fundamental

forcing frequency that will in turn excite natural frequen-

cies (see Figure 6.8). This phenomenon often occurs in

normal machine operation.

Figure 6.10. Waterfall or Cascade Diagram

Figure 6.9. Sites of Impact

and Reference Points

IPS

0

Y: .452 IPSrms

.

Sen

sors

6.10

Figure 6.11. Natural Frequency of a Vertical Pump from an Impact Test

It should be noted that natural frequencies of large structures

below 15 Hz are difficult to excite with an impact test. For this reason

a shaker, used in conjunction with a power amplifier and a signal gen-

erator (Figure 6.13), may be required to excite the natural frequency of

interest. The natural frequencies are determined by sweeping through

the frequency range of interest, focused energy is introduced to the

structure. An example of a mechanical shaker is a variable-speed

motor with a double-ended shaft with disks for mass unbalance. Oth-

er excitation sources, including random forces, can be added to the

shaker to excite natural frequencies instantaneously.

Some frequencies in random forces will be the same as, and

thus excite, natural frequencies. This phenomenon can be seen in

signatures taken from operating machines. Random energy in the Figure 6.12. Excitation

Shapes

6.11

system will excite natural frequencies. Figure 6.14 shows a gearbox in which a natural frequency

is excited by all of the random vibration in the system.

Figure 6.13. Shaker Test Setup

A reciprocating compressor sometimes

rings the piping around it. (Figure 6.15). Engines

behave in a similar way. Four-stroke cycle engines

provide half-order excitations in addition to orders

of operating speeds. Even a broken gear tooth will

generate natural frequencies. Machine motion

causes an impact-like force that excites natural fre-

quencies that might or might not be desirable.

The most popular method for exciting natu-

ral frequencies is to strike (excite) a machine or

structure with a timber or a hammer. The range of

frequencies excited is dependent upon the duration

of the impact. Hard-faced steel hammers tend to

bounce off a structure and so provide a short-lived Figure 6.14. Gearbox Vibration

6.12

impact. As a result only high frequencies are excited. To excite natural frequencies of 10 Hz or

less a soft tip must be used on the hammer6.3.

For resonance testing, the structure, piping, or machine should be as close as possible to

its operating state. Parts of a machine cannot arbitrarily be removed and tested. For example,

the natural frequencies of a gear not mounted on its shaft differ from those when the gear is

mounted. Similarly, the natural frequencies of a machine mounted for shop testing differ from

those of the machine mounted on its normal foundations.

Determination of Natural Frequencies from Impact Tests

The level of sophistication and detail of resonance testing varies. A simple resonance test

often provides the necessary structural natural frequency information. However, better infor-

mation and greater detail can be obtained from more sophisticated dual channel tests and instru-

mentation — Chapter 8.

Figure 6.15. Ringing of Piping by a Reciprocating Compressor

6.13

Storage analog or digital oscilloscopes can be used to observe instantaneously the vibra-

tion resulting from an impact to a structure. A trigger level must be set on the oscilloscope in the

single sweep mode. The vibration signal from the impact will be held on the screen of the oscillo-

scope (Figure 6.16). The natural period is obtained by counting the divisions in one period of vi-

bration and multiplying that number by the time-per division setting on the time base (Chapter 3).

In Figure 6.16 it can be seen that two divisions are contained in the period, or vibration

cycle. The oscilloscope was set at 5 milliseconds per division. The natural period, therefore, is

= 2 div. x 5 m sec./div. = 10 m sec. = 0.01 sec.

The natural frequency is one divided by the natural period.

f n = 1/ = 1/0.01 sec./cycle = 100 cycles/sec. (Hz) = 6,000 cycles/min.

The procedure with a digital oscilloscope is similar except that the period is read directly with a

cursor.

Figure 6.16. Vibration Signal Shown on an Analog Oscilloscope

6.14

FFT Setup and Impact Procedure

Table 6.2 outlines the steps required for impact testing while Table 6.3 relates to the FFT

setup.

Table 6.3. FFT Setup for Impact Testing

1. Set Fmax so that natural frequency of interest will be included in the spectrum.

2. Set number of lines to obtain required resolution.

3. Take data collector off autoranging and approximate range required to not over or under load the analyzer.

4. Select a uniform or Hanning window.

5. Set the trigger on the vibration — use a 1/3 sample pretrigger, if the Hanning window is used.

6. Select one to four (4) averages — depending on the noise on the data.

7. Select scale factors to reflect transducers or recorded data.

8. Set autoscaling off — select scales for displays during impacting or processing of recorded data.

9. Set display to view time and frequency during test.

10. Readjust setup and/or striking force as required after initial data are analyzed.

Set the FFT frequency span, Fmax, wide enough to capture the natural frequency of inter-

est and any other natural frequencies in the vicinity — generally FMAX = 2 x FN. Select the num-

ber of lines that will provide adequate resolution. If the test is complete and resolution is not ade-

quate to separate natural frequencies, it will have to be repeated unless it has been tape record-

ed.

Set the range of the analyzer so it will not overload or have insufficient dynamic range. Do

not use autoranging. Just because the response is low in amplitude does not mean that it is not

important.

Select a uniform or Hanning window. If the vibration does not ring down to zero at the end

of the data sampling time, leakage will occur and a uniform window is not appropriate. If a Han-

ning window is used select a pretrigger of 1/3 to 1/2 of the data sampling time (Figure 6.11). Oth-

erwise the Hanning window will destroy the data. In Figure 6.17, an impact test on a vertical

pump was conducted with a uniform window. Note the time waveform is intact.

Set the trigger on the vibration signal. Use a pretrigger with a Hanning window — other-

wise data will be lost (Figure 6.18). The reduction of amplitude by a Hanning window if the data

6.15

are not suitably positioned in the buffer are shown in Figure 6.19. The upper trace is not win-

dowed. The lower trace is windowed — this is what the FFT uses to calculate the spectrum.

Figure 6.17. Impact Test on a Vertical Pump with a Uniform Window

6.16

Figure 6.18. Data from an Impact Test Processed with a Uniform (Upper) and Hanning (Lower) Window Without Pretrigger

6.17

Figure 6.19. Impact Test Results with a Hanning Window. Upper Trace not Windowed

The interval of impacting should be greater than the data sampling time so that one impact

is recorded per FFT sample. Otherwise sideband noise will occur (Figure 6.20). The choice of

using single or multiple-averaged impacts depends on the noise within the data. The choice of

averaging has to be made on a case by case basis. Usually four (4) averages provide a good

compromise. The level of impacting should be just enough to get response. Do not try to kill it!

Figure 6.21 shows data from a dual channel impact test on a fan support structure. The

trigger was on the instrumented hammer. An exponential response window (Chapter 3) was used

on the velocity signal and a force window was used on the hammer signal. The upper plot is the

transfer function mobility (IPS/lb) and the lower plot is phase. Approximate 180° phase changes

in the vicinity of transfer function peaks are used to identify natural frequencies. The real portion

(transfer function multiplied by the cosine of the phase) is used to develop mode shapes. The im-

aginary part of the transfer function (transfer function multiplied by the sine of the phase) is used

to evaluate damping.

6.18

Figure 6.20. Sideband Noise from Multiple Impacts in FFT Sample

Figure 6.21. Impact Test on a Fan Mounting Structure

Y: 33.302

6.19

CRITICAL SPEED TESTING TECHNIQUES

The natural frequencies of a machine or structure are governed by its design6.1. Critical

speed testing can be more complicated than resonance testing because the natural frequencies

encountered can be functions of machine speed [4] — e.g., when fluid-film bearings or large over-

hung masses such as fan impellers are part of a system stiffness or effective mass can change.

Natural frequencies can be measured during startup or coast down of a machine to obtain

its critical speeds. However, the values of the natural frequencies obtained during startup may be

variable because of the acceleration and loads imposed on the rotor. For this reason coast-down

tests provide the best data. In addition, some machines — turbines, for instance — do not

achieve thermal stability during startup (Table 6.4).

Table 6.4. Conducting a Critical Speed Test

1. Select one or more appropriate sensors to measure the vibration. Proximity probes meas-ure relative shaft displacement and are preferred if they are permanently installed. Other-wise, mount seismic sensors — either velocity sensors or integrated accelerometers — as close to the bearing as possible in the horizontal and vertical directions. If casing resonance is suspected — use casing mounted sensors.

2. For a permanent or temporarily mounted trigger use a proximity probe or a magnetic pickup adjacent to an indentation or mark on the rotor. A combination of an optical pickup and re-flective tape or paint (hot surfaces) may also be used.

3. Wire the vibration pickups and trigger to a tracking filter, tape recorder, or FFT analyzer.

4. If the vibration pickups and trigger are permanently mounted, a coast-down test of the ma-chine can be performed directly.

5. Run the machine at 10% to 15% overspeed if possible, then cut the power and allow the machine to coast down from normal operation as data are being recorded. Otherwise measure and record data prior to and during the time the power is cut.

6. If either sensors, trigger, or both must be mounted, record the startup. Run the machine un-til it is thermally stable before cutting power for the coast-down data.

7. Process the data and identify the critical speeds. They will be peaks in the spectrum. De-pending on the plot they will be peaks in the spectrum from an FFT analyzer, peaks from the tracking filter in a Bodé plot, or loops from the tracking filter in a polar plot.

8. It is a good idea to record coast down and startup data prior to and after an outage.

Use the Interference Diagram for Test Planning

Figure 6.22a shows that, for rotor supported on fluid-film bearings, the natural frequency

can change as changes in speed influence the stiffness properties of the machine6.4. In this case

the natural frequencies vary with speed because the stiffness of fluid-film bearings is a function of

6.20

speed. The natural frequencies of rotors mounted on rolling element bearings are constant and

not speed dependent because the stiffness of the bearings is relatively constant — it varies slight-

ly with load.

Figure 6.22b shows the variation in fluid-film bearing stiffness with rotor speed (vertical

lines) superimposed on the variation in natural frequency of the rotor with bearing stiffness (ho-

rizontal lines). Rotors with large overhung impellers have gyroscopic effects that stiffen the rotor

as speed increases, thereby raising the natural frequency6.5.

Figure 6.22. Change in Natural Frequency with Speed

Mass unbalance is typically used to excite natural frequencies during tests to identify criti-

cal speeds. If other excitation sources are present at two, three, or more times running speed,

natural frequencies may be excited at lower speeds. That is, the first natural frequency would be

excited at one half the normal first critical speed — called a half critical — if an excitation at two

times running speed is present (Figure 6.22).

Because natural frequencies can vary with speed, ringing a nonoperating rotor mounted

on fluid-film bearings during a bump test does not provide correct natural frequencies for the sys-

tem. Thus, in systems in which the natural frequencies vary with speed, the bump test is not use-

ful except when modeling a rotor to determine the natural frequencies of the machine at rest.

6.21

Rotors are often lifted out of a machine and rung to obtain natural frequencies. In such

cases only the natural frequencies of the rotor are obtained, not those of the system. The effects

on natural frequencies and critical speeds of fluid-film bearings, pedestals, and supporting founda-

tions would be accounted for as they are in Figure 6.22.

The natural torsional excitation of

motors and engines is often used in reso-

nance and critical speed testing. The inter-

ference diagram would show no variation of

natural frequency with speed (see Figure

6.23) for an example of an interference dia-

gram with no variation in natural frequency).

In synchronous motors an excitation at

twice slip frequency is obtained during

startup that excites the first and second tor-

sional natural frequencies of the unit. The

result is a sweep of a single frequency. Be-

cause the excitation varies from 120 Hz (60

Hz power) to zero, the second critical

speed is encountered before the first (Fig-

ure 6.23). Engines are usually tested by sweeping through the speeds from low to high idle, so

that the many engine frequencies will excite the natu-

ral frequencies of the system. These frequencies are

then identified using an interference diagram (Figure

6.24).

An interference diagram for an en-

gine/generator is shown in Figure 6.24. The first tor-

sional natural frequency can be excited at engine

speeds of 1,050 RPM, 1,400 RPM, and 1,800 RPM at

6, 4.5, and 3.5 orders of excitation respectively. The

torsional test data (Figure 6.25) show that the 4.5x order torsional vibration excites the first natural

frequency as indicated on the interference diagram.

Figure 6.23. Synchronous Motor Start-Up

Figure 6.24. Calculated Torsional Critical Speeds of an En-

gine/Generator Unit

6.22

When a system experiences a

natural impact or random excitation, a

number of natural frequencies will be gen-

erated in the vibration signal. These

frequencies can often be identified as nat-

ural frequencies by changing machine

speed. If the frequencies do not change

with small changes in machine speed and

the rotor is mounted in rolling element

bearings, chances are great that they are,

natural frequencies.

Mode shapes can be measured

when several sensors are mounted on a

machine. The sensors selected identify

modal properties of the machine (Chapter 8). For example, the noncontact proximity probe is

used to assess vibration modes of the rotor; seismic sensors on the casing, pedestals, or founda-

tion are used to determine structural modes.

Using Tracking Filters

A tracking filter (Figure 6.26) can

be used to generate a Bodé plot (Figure

6.27) or a polar plot. The Bodé plot is a

record of amplitude and phase versus

speed — usually filtered. A trigger off the

shaft is used to generate the phase record.

Figure 6.27 is a Bodé plot of a coast-down test of a 4,000 HP boiler feed pump. Mass un-

balance is the excitation source. The synchronous speed tracking filter that was used tracks the

once-per-revolution amplitude and phase of the vibration as the speed of the unit changes. A

trigger must be mounted on the machine so that a reference signal can be obtained. Multiples of

operating speed can be tracked with the proper reference signal.

In Figure 6.27, critical speeds occur at 2,000 RPM, 2,600 RPM, and 3,500 RPM. The first

critical speed involves the motor. The second and third critical speeds are associated with the

pump drive unit.

Figure 6.26. Tracking Filter

Figure 6.25. Torsion Test

6.23

The vibration peaks and phase

changes — theoretically 180° — indicated

natural frequencies at the speed of the

machine. A critical speed would be indi-

cated by the interference of the natural

frequency and the forcing frequency on

the interference diagram. It can be seen

that the natural frequency at operating

speed is not necessarily the natural fre-

quency measured during the start-up or

coast-down test. An interference diagram

is helpful in visualizing the natural fre-

quencies of a machine at speed other than

critical speeds.

Using the FFT Spectrum Analyzer

The peak hold feature of an FFT

spectrum analyzer can be used to provide

data on critical speeds. However, the fre-

quency range selected must be high enough to track the coast down. The peak hold feature

holds and displays the peak values of all data after each spectrum is computed (figure 6.28a).

The acquisition time of the block of data analyzed is dependent on the frequency span selected.

The lower the frequency span, the longer the acquisition time.

The formula for acquisition time is

f

N

N is the number of lines, usually 400. The number of samples is a measure of analyzer resolu-

tion. The acquisition time can be decreased by reducing the lines of resolution. In addition, use of

overlap processing also reduces data acquisition time. With overlap processing data from the

previous sample are used in calculating the spectrum. In typical machine coast-down tests the

process of computing the FFT is faster than the data acquisition.

Figure 6.27. Bodé Plot of a Coast-Down Test

6.24

If a 400-line FFT analyzer is set on a frequency span of 100 Hz (6,000 CPM), four sec-

onds (400/100) are needed to acquire a sample. Because many samples are necessary to plot a

smooth curve during a start-up or coast-down test without data drop out, the frequency span of

the analyzer must be evaluated carefully prior to data collection. Resolution is lost with too wide a

frequency span. Too narrow a span may prevent the observation of the critical speed because

the data acquisition time is excessive. Table 6.5 illustrates the procedure

Table 6.5. Critical Speed Testing with an FFT

Determine approximate coast-down time

Set the FFT on peak hold

Select number of lines, FMAX, and overlap, and calculate data acquisition time

Divide coast-down time by data acquisition time to get number of data points in the peak hold curve. This will determine the resolution of the data

Trigger FFT prior to coast down so a full sample is gathered prior to coast down

Consider, for example, Figure 6.28a, a 12 second startup of a two-pole motor that operates at

3,600 RPM. If the analyzer is set on 6,000 CPM (100 Hz) and four seconds are required for each

sample, only three data points will be ob-

tained — hardly sufficient for a curve. In-

creasing the frequency span to 400 Hz

(24,000 CPM) and decreasing the number

of lines to 100 means that a sample is

taken every 0.25 second (100/400 =

0.25); 12 second span/0.25 sec-

ond/sample is equal to 48 samples, or

one sample every 75 RPM. However, the

resolution is reduced to increments of

24,000 CPM/100 lines, or 240 RPM,

which yields 3,600/240 RPM, or 15 data

points over the frequency range between

0 and 3,600 RPM. Eighty percent overlap

processing — i.e., 20% data retained —

at 100 lines with a 100 Hz (6,000 CPM)

span provides a data acquisition time of

0.2 second (100 lines per 200 Hz x 0.2), Figure 6.28a. Peak Hold Coast Down Test

N = 100

FMAX = 100 Hz

OVL = 80% DAT = .2 sec

36 Data Points

6.25

Figure 6.28a. The resulting 100 points at

about 60 RPM intervals with a resolution of

100 CPM may provide an adequate plot. It

must be recognized that the small amount

of new data contained in each point pro-

cessed compromises the analytic results.

Under these conditions a tracking filter is

necessary.

Figure 6.28b shows coast-down

data of a steam turbine analyzed by an FFT.

In this example the coast

down and startup required

so much time that data ac-

quisition is not an issue.

The dip in this spectrum at

2,400 RPM means that the

rotor is bowed.

Using Polar Plots

Figure 6.29 is a

polar plot produced by a

synchronous tracking filter

from a start-up test of a

turbine generator. The plot

used the amplitude (peak

to peak) and phase (angle

between a reference mark

and peak vibration) of vi-

bration at various speeds.

The tracking filter plots the

real (amplitude times co-

sine of phase angle) and the imaginary (amplitude times sine of the phase angle) at the various

speeds. Figure 6.29 shows an example polar diagram. The generator is operating in the coun-

ter clockwise direction — therefore phase lags are measured in the clockwise direction. The small

Figure 6.28b. Coast-Down Tests

of a Steam Turbine

Figure 6.29. Polar Plot of a Start-Up Test of a Generator Bearing

6.26

loop6.6 in Figure 6.29 identifies the first critical speed of the generator as 1,000 RPM. There is a

phase change of 90°. The large loop is the second critical speed, 2,250 RPM. It shows the op-

erating speed to be well above the second critical.

DAMPING

Damping information can be used to identify problems that might occur when a machine pass-

es through or dwells on a critical speed. If significant damping is present, it is not necessarily

harmful to operate a machine close to a critical speed, but inadequate damping could result in a

disaster.

Evaluating machines subject to mass unbalance during operation as well as those operat-

ing close to a critical speed requires data on damping. Damping for a single mode can be esti-

mated from the time domain, the frequency domain, or by phase. Modal testing techniques are

used when a system has more than one mode (Chapter 8).

Damping Ratio and Logarithmic Decrement

The time domain of the vibration of a steam turbine

casing subjected to an impact is shown in Figure 6.30. The

damped natural period, which is obtained directly from the

plot, is equal to 0.0125 second per cycle. The frequency,

which is the reciprocal of the period, is equal to 80.0 Hz.

The damping ratio can also be calculated form Figure

6.30. The damping ratio (C/CC) is a measure of modal vis-

cous damping. It is defined as the ratio of damping in the sys-

tem to critical damping. Critical damping is the lowest mag-

nitude of damping that will not allow vibration. The damping

ratio varies from zero (no damping) to one (no vibration).

Typical damping ratios are 0.001 for steel; 0.05 for rubber;

0.025 for machines with rolling element bearings; and from

0.03 to 1.0 for machines with fluid film bearings. The API

(American Petroleum Industry) recommendation is that ma-

chines passing through a critical speed should have a damping ratio greater than 0.0625.

Figure 6.30. Time-Domain Da-ta from an Impact Test on

a Turbine Casing

6.27

The level of vibration of a damped system typically decreases with time at a rate that can

be used as a measure of damping. This measure is called the logarithmic decrement. It is de-

fined in equation form as

= logarithmic decrement

n = number of cycles of vibration measured

xo = amplitude of vibration, initial cycle

xn = amplitude of vibration, cycle n

n = natural logarithm to base e

The logarithmic decrement is related to the damping ratio by the following equation.

The damping ratio can be estimated from the following relationship.

= logarithmic decrement

= damping ratio

C = actual damping, lb. sec./in.

CC = 2mn = critical damping

m = modal mass, lb. sec./in.

n = 2 fn = natural frequency, radians/sec.

fn = natural frequency, Hz

The logarithmic decrement for the dominant pedestal mode at 83 Hz in Figure 6.30 is (1/7)

n (5.5/0.8), or 0.275. The damping ratio is 0.275/2, or 0.043.

n

o

x

xn

n

1

2

12

C Cc

C Cc

/

/

2 0 5C

Ccfor

C

Cc.

2C

C

C

C

C c

6.28

Figure 6.31. Shaft Vibrations of a Turbine during Coast Down (Courtesy C. Jackson)

Half-Power Method

Damping can be evaluated for a single mode of vibration with data from the frequency

domain (Figure 6.31). Half-power points of vibration response close to a critical speed or reso-

nance are used. A half-power point is equal to peak amplitude multiplied by 0.707. The following

equation is used.

or

Q = quality factor

AF = amplification factor

fn = natural frequency or critical speed; CPM, Hz, or RPM

f1 = lower half power point obtained from spectrum — Figure 6.32

f2 = second half-power point; i.e., higher frequency at which a vibration response of

0.707 times peak vibration occurs consistent with fn.

Q AFfn

f f C

Cc

2 1

1

2

C

Cc AF

1

2

6.29

The half-power method can be used to calculate a damping ratio of 0.16 from the turbine data

shown in Figure 6.31. The critical speed is 5,800 RPM, the first half-power point is 5,200 RPM,

the second half-power point is 7,100 RPM, and the amplification factor is 3.05.

Coast-down data can be used to calculate the damping of a rotor (Figure 6.32). The rotor

is part of a grinding machine that is mounted on a structural platform. The peak vibration at 0.245

inch per second can be used to identify the critical speed, which is 1,238 RPM.

The half-power points are used to obtain the amplification factor. At 0.707 times peak vi-

bration, or 0.17 inch per second, the speed N1 and N2 at the half-power points are 1,350 RPM

and 1,140 RPM (see Figure 6.32). The damping ratio for a calculated amplification factor of 5.9 is

0.084. This relatively high value is surprising because the rotor is supported in rolling element

bearings.

The actual damping value can be calculated from single-degree-of-freedom theory. The

rotor weighs 525 pounds. The critical damping is calculated as follows.

m = W/g, mass of rotor, 525 lbs./g, or gravitational constant, 386.1 in./sec.2

= 525/386.1 = 1.36 lb. sec.2/in.

n = natural frequency

= 20.6 Hz x 6.28 = 129.57 radians/sec.

CC = 2 x 1.36 lb. sec.2/in. x 129.57 radians/sec.

= 352 lb. sec./in.

C = 0.084 (352) = 29.6 lb. sec./in.

The structural natural frequency (28 Hz) of the platform could not be excited by a bump test but

was seen during a coast-down test.

Cc m n 2

6.30

Figure 6.32. Coast-Down of a Grinding Rotor

Phase-Angle Method

The phase information associated with a frequency-response plot or a start-up or coast-

down test can be used to evaluate damping. The Bodé plot shown in Figure 6.31 contains a

change in phase angle with respect to speed as the rotor passes through a critical speed. The

formula for the amplification factor is

AF = amplification factor

NC = critical speed, RPM

d/dN = rate of change of phase angle with respect to speed around critical speed

The rate of change d/dN is approximated by the finite difference /dN. From Figure 6.31, N is

equal to 5,800 RPM and the finite difference is (458-380)/(6250-5250), which is 0.078. The ampli-

fication factor is

AF = (5,800/360)(0.078) = 3.95

Nd360

dNFA

c

N1

6.31

REFERENCES

6.1. Eshleman, R.L., “Resonance and Critical Speed Testing. Part I: Basic Concepts and In-

strumentation,” Vibrations, 6(3), pp 3-7 (1990). 6.2. Den Hartog, J.B., Mechanical Vibrations, McGraw-Hill Book Co. (1968) [Dover Books]. 6.3. Eshleman, R.L., “Resonance and Critical Speed Testing. Part II: Resonance Testing Tech-

niques,” Vibrations, 6(4), pp 3-7 (1990) 6.4. Eshleman, R.L., “Modeling Turbomachinery for Design and Analysis,” Rotor Dynamics and

Balancing Course, Syria, VA, May 15-19, 1989, Vibration Institute, Willowbrook, IL (May 1989).

6.5. Eshleman, R.L. and Eubanks, R.A., “On the Critical Speeds of a Continuous Shaft-Disk

System,” J. Engrg. Indus., Trans. ASME, 89 (4) (November 1967). 6.6. Halloran, J.D., “Vibration Analysis of Rotating Machinery Using Polar Diagrams,” Vibrations,

1(4), pp 2-10 (1986).