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28.4 Real-Time Vibration Analyzer

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The development of an alternative, low-cost approach towards real-time monitoring and analysis of

machine vibration (Vierck, 1979; de Silva, 2000) is described in this section. The main idea behind

this approach is to construct a vibration signature based on pattern recognition of “acceptable” or

“healthy” vibration patterns. The vibration analyzer can operate in three modes: learning, monitoring,

or diagnostic. The learning mode, to be initiated first, will yield a set of vibration signatures based on

which the monitoring and diagnostic modes will operate. In the monitoring mode, with the machine

FIGURE 28.23 Experimental results using a notch filter: (a) error (mm); (b) desired trajectory (mm); (c) control

signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press.

With permission.)

SUMMARY

An approach utilizing an adaptive notch filter in the control system is elaborated upon in this

section. The notch filter is able to adaptively identify resonant frequencies present in the motion

system and suppress vibration signal transmission into the system at these frequencies. The FFT is

used to obtain the frequency distribution of the vibration signals. Experimental and simulation

results are provided to illustrate the effectiveness of the adaptive notch filter.

Vibration Suppression and Monitoring in Precision Motion Systems 28-17

under normal closed-loop control, the analyzer

only uses a naturally occurring vibration signal

to deduce the condition of the machine. No test

excitation is deliberately added to the input

signal of the machine. More than one criterion

may be used in the evaluation of the condition

of the machine, in which case a fusion approach

will generate a combined output (machine

condition) based on the multiple inputs. In the

diagnostic mode, explicit input signals are

applied to the machine and the output signal

(vibration) is logged for analysis with respect to

the associated vibration signature. In what

follows, the details of the various components

and functions of the analyzer will be described

systematically.

The block diagram of the real-time vibration analyzer system that has been developed is shown in

Figure 28.24. It consists of an accelerometer, which is mounted on the machine to be monitored.

The accelerometer measures a multifrequency vibration signal and transmits it to an intelligent DSP

module after performing appropriate signal conditioning. This module can be a standalone device

(Figure 28.25), or one integrated to a personal computer (PC) host. The vibration analysis

algorithm is downloaded to this DSP module. With this algorithm, it can be established as to

whether the condition of the machine is within a predetermined acceptable threshold. If the

condition is determined to be poor, the DSP module will trigger an alarm to the operator who

would enable a corrective action, or automatically activate a corrective action (e.g., change the

operating conditions of the machine, modify the parameters of the controller or shutdown

the machine).

The construction of the real-time vibration analyzer is inexpensive and requires only

commercially available, low-cost components. The installation can be hassle free, as the

accelerometer is able to gather vibration signals independent of the machine’s own control system.

Thus, there is no need to disrupt the operation of the machine. In the prototype reported here, a

DSP emulator board TMS320C24x model (TMSS320C24x DSP Controllers Evaluation Module

Technical References, 1997), from Texas Instruments, is used as the standalone DSP module (Figure

28.25). This C24x series emulator board is built around the F240 DSP controller, operating at

Machine

DSP module

Signal conditioning

Accelerometer

Vibration signals

Vibration

monitoring

program

Activate alarm or

corrective action

FIGURE 28.24 Schematic diagram of the real-time vibration analyzer. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S.,

Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.25 Hardware setup for the standalone

DSP module. (Source: Tan, K.K., Tang, K.Z., de Silva,

C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001,

IOS Press. With permission.)

28-18 Vibration and Shock Handbook

20MIPS with an instruction cycle time of 50 ns. It is optimized for digital motor control and

conversion applications. Other key components supported on this DSP module are analog-to-digital

converters (ADCs), dual access RAM (DARAM), on-chip flash memory, and an RS-232 compatible

serial port. This DSP module and the accelerometer unit (with signal conditioning) constitute the

only hardware requirements of the real-time vibration analyzer.

The vibration-analysis algorithm, which is described in the sequel, will be downloaded to the flash

ROM on the DSP module after satisfactory evaluation and tests on the PC. The algorithm is programmed

using a Visual Cþþ-based compatible environment, that is, a Code Composer integrated environment

(IDE). The C-based source code is then compiled into assembly code, using the built-in compiler

available in the IDE.

28.4.1 Learning Mode

In the learning mode, with the machine operating under normal conditions, the vibration signals

are acquired by the accelerometer and stored in the DSP module. A suitable vibration signature

(Ramirez, 1985) is then extracted from the vibration signals. There are many types of vibration

signatures that are adequate for the purpose of machine monitoring. For example, one form of

vibration signature may be based on the amplitude of the vibration; another form may use a timeseries

analysis of the vibration; yet another form may employ the spectrum of the vibration, which

can be efficiently obtained using the FFT algorithm. Regardless of the type, these vibration

signatures are dependent on the nature of the input signals driving the machine. For example, a

square wave input will produce a vibration spectrum that can be quite different from that resulting

from an input of a chirp signal (i.e., a repeating sine wave of increasing frequency) or a pure

sinusoid. Thus, a particular input signal will produce a unique spectrum based on which a unique

vibration signature can be derived. Multiple vibration signatures corresponding to the natural

vibrations of the machine (useful for the monitoring mode) or corresponding to different input

signals (useful for the diagnostic mode) can thus be captured for subsequent diagnosis and

monitoring of the machine.

28.4.2 Monitoring Mode

In the monitoring mode, the vibration signals are sampled periodically from the machine to

monitor the condition of the machine. No deliberate or additional input signal is required, so the

machine operation is not disrupted. The updated spectra are analyzed against the relevant vibration

signatures. The analysis and comparison may be done in terms of the shift in the frequency or the

amplitude of the spectrum, or a combination of the two. For example, one evaluation criterion

(EV) may be based on the mean-square (ms) value of the error between the current real-time

vibration spectrum and the vibration signature:

EV1 ¼

q¼1 ðSq 2 Spq

Þ2

M ð28:18Þ

where Sq is the discretized current real-time vibration spectrum, the superscript p represents the

vibration signature of the “healthy” machine, subscript q is the index for the data points, N is the

total number of frequency points, and M is the total number of data points. Another EV may be

formulated based on the difference in the amplitude of the current time series vibration pattern and

its corresponding vibration signature:

EV2 ¼

maxðTqÞ 2 maxðTp

q Þ

M ð28:19Þ

Vibration Suppression and Monitoring in Precision Motion Systems 28-19

where maxðTqÞ represents the highest amplitude of the current time series vibration pattern Tq; and

maxðTp

q Þ is the highest amplitude of its corresponding vibration signature.

More than one evaluation criterion may be used in the determination of machine condition. In this

case, a fusion technique is necessary. The key idea of fusion is to associate the machine with a HEALTH

attribute, which is computed from multiple evaluation criteria. These criteria are expected to influence,

to a varying degree, the HEALTH of the machine. The HEALTH attribute is thus an appropriate function,

I; of the various criteria (EVis); that is

HEALTH ¼ IðEV1; EV2; …; EVnÞ ð28:20Þ

where n refers to the number of criteria being evaluated.

A fuzzy weighted approach may be used to realize the I function as follows. The HEALTH attribute is

treated as a fuzzy variable (i.e., HEALTH [ ½0; 1􀀉). HEALTH ¼ 0 will represent absolute machine

failure, while HEALTH ¼ 1 represents a perfectly normal machine condition. This attribute may be

computed from a fuzzy operation on a combination of the evaluation criteria (EVis) obtained via an

analysis of the vibration signals against their signatures. The final decision on the condition of the

machine will be derived from the HEALTH attribute. Zadeh (1973) provides a comprehensive review on

fuzzy logic.

A Takagi and Sugeno (1985) type of fuzzy inference is used in this chapter. Consider the following p

rules governing the computation of an attribute:

IF EVi1

IS Fi

1^ · · · ^EVi

n IS Fi

n; THEN ui ¼ ai ð28:21Þ

i¼1

ai ¼ 1 ð28:22Þ

where ui [ ½0; 1􀀉 is a crisp variable output representing the extent to which the ith evaluation rule affects

the final outcome. Thus, ai represents the weight of the ith rule, Fi

j represents the fuzzy sets in which the

input linguistic variables (EVis) are evaluated, and ^ is a fuzzy operator that combines the antecedents

into premises.

The value of the attribute is then evaluated as a weighted average of the uis:

HEALTH ¼

i¼1

wiui

i¼1

wi ð28:23Þ

where the weight wi implies the overall truth value

of the premise of rule, i, for the input. It is

computed as

wi ¼

j¼1

mFi

j ðEVij

Þ ð28:24Þ

where mFi

j ðEVij

Þ is the membership function for

the fuzzy set, Fi

j ; related to the input linguistic

variable, EVij

; (for the ith rule). For example, in

this application, EVj may be the maximum

error, MAX_ERR, and Fi

j may be the fuzzy set

HIGH. The membership function is represented

as mHIGHðMAX_ERRÞ: It may have the

characteristic shown in Figure 28.26. The decision

as to whether any corrective action might be

mHIGH(MAX _ ERR)

MAX _ ERR

FIGURE 28.26 Membership function for the fuzzy

input MAX_ERR. (Source: Tan, K.K., Tang, K.Z., de

Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst.,

2001, IOS Press. With permission.)

28-20 Vibration and Shock Handbook

necessary can then be based on a simple IF – THEN – ELSE formulation as follows:

IF HEALTH # g, THEN STRATEGY ¼ TRIGGER ALARM

ELSE STRATEGY ¼ CONTINUE TO MONITOR

Here, g is interpreted as a threshold value. Suitable values for g may be in the range 0.6 # g # 0.9.

Here, STRATEGY is stated to trigger an alarm to the operator who will enable a corrective action, or

automatically activate a corrective action (e.g., change the operating conditions of the machine, modify

the parameters of the controller, or shutdown the machine).

Under this framework, it is relatively easy to include additional criteria for analysis and decision

making to the system. The procedure will involve setting up the membership functions for the criterion,

formulating the additional fuzzy rules required, and adjusting the scaling parameters (the a terms in

Equation 28.21) to reflect the relative weight of the new criterion as compared with the existing ones. In

this manner, in the monitoring mode, foreboding trends can often be spotted long before the vibration

condition reaches a level that is seriously detrimental to the machine.

28.4.3 Diagnostic Mode

In the diagnostic mode, the current vibration signal corresponding to each input signal (with

standardized amplitude and frequency) is analyzed against the associated signature obtained earlier in the

learning mode, depending on the type of machine (also see Chapter 25). Similar to the monitoring mode,

there can be multiple evaluation criteria used in the diagnostic mode, so that the fusion technique

described earlier is also applicable. The input signals applied to the machine must be designed carefully so

as to yield as much information of the machine condition as possible in the operational regime of

interest. Two important considerations are the choices of amplitude and frequency.

Machines may have constraints in relation to the amount of travel that is possible. Too large an

amplitude for the input signal may be not be viable for the machine owing to the limit of travel, or

may even damage the machine. Also, the frequency range of the input should be chosen so that it

has most of its energy in the frequency bands that are important for the system. Where input

signals cannot be applied to the system in the open loop, the setpoint signal will serve as the input

for the closed-loop system since it may not be possible to directly access the system under closedloop

control. Careful considerations of the mentioned issues will ensure that significant information

can be obtained from the machine.

The input signals considered here are square wave input (Figure 28.27), chirp input (Figure 28.29), and

sine wave input (Figure 28.31), standardized in amplitude to 1 V and in frequency to 5 Hz. The

corresponding vibration signatures are shown in Figure 28.28, Figure 28.30, and Figure 28.32,

respectively.

28.4.4 Experiments

A shaker table (Figure 28.33) is used as the test platform for the experiments presented here (also see

Chapter 15). The shaker table can be used to simulate machine vibrations and evaluate the performance,

for example, of active inertial dampers. The shaker table is driven by a high-torque direct-drive motor

(which has a maximum torque of 1.11 N m, a maximum design load of 11 kg and generates a maximum

force of 175 N). The maximum linear travel of the table is ^ 2 cm.

The learning mode is first initiated to obtain the vibration signals with the shaker table operating

under normal conditions. It is assumed in the experiments that the normal condition corresponding

to the input is a square wave signal (with a standardized amplitude of 1 V and frequency of 5 Hz).

For the purpose of implementing the diagnostic mode, the vibration signals are also obtained for

the input signals of the sinusoidal and chirp type, with standardized amplitudes of 1 V and

Vibration Suppression and Monitoring in Precision Motion Systems 28-21

frequencies of 5 Hz (see Figure 28.27 to Figure 28.32 for the inputs and their corresponding

vibration signatures).

28.4.4.1 Input Variables — the Evaluation Criteria

Different types of EV can be used as input variables for the determination of the machine condition. For

the present vibration-analysis application, the input variables chosen for the computation of the

HEALTH attribute are given below.

28.4.4.1.1 Monitoring Mode

EV1 ¼

q¼1 ðSsq;q 2 Sp

sq;qÞ2

M ð28:25Þ

EV2 ¼ ðmaxðTsq;qÞ 2 maxðTsq;qÞpÞ2

M ð28:26Þ

EV3 ¼

q¼1 ðTsq;q 2 Tp

sq;qÞ2

M ð28:27Þ

where Ssq;q represents the vibration spectrum for

a square wave input (driving the machine) at the

qth frequency point and Tsq;q represents the timedomain

signal of a square wave input at the

qth time instant. N refers to the total number of

points in the FFT spectrum and M is the number

FIGURE 28.27 Square wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K.,

Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.28 Vibration signature of the square wave

input, with a standardized amplitude of 1 V and

frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de

Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst.,

2001, IOS Press. With permission.)

28-22 Vibration and Shock Handbook

of time-series data points over an operational cycle. Hence, EV1 refers to the ms deviation between the

vibration spectrum and its signature; EV2 refers to the square of the difference between the amplitude of

the vibration signal over one operational cycle compared with its signature; EV3 refers to the ms

deviation between the vibration signal and its signature (time domain) over one operational cycle. The

superscript p represents the signature of the healthy machine.

28.4.4.1.2 Diagnostic Mode

EV4 ¼

q¼1 ðSsq;q 2 Sp

sq;qÞ2

M ð28:28Þ

EV5 ¼

q¼1 ðScp;q 2 Sp

cp;qÞ2

M ð28:29Þ

EV6 ¼

q¼1 ðSsn;q 2 Sp

sn;qÞ2

M ð28:30Þ

Here, cp denotes a chirp input signal and sn denotes a sine input signal.

For the monitoring mode, the input attributes are related only to the square input owing to the

assumption that the input signal, under normal operating conditions, is the square wave signal (with a

standardized amplitude of 1 V and frequency of 5 Hz).

Time domain signal

Voltage (V)

Time (secs)

0.8

0.6

0.4

0.2

–0.2

–0.4

–0.6

–0.8

–1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

FIGURE 28.29 Chirp wave input, with a standardized amplitude of 1 V and initial frequency of 5 Hz. (Source: Tan,

K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

Vibration Suppression and Monitoring in Precision Motion Systems 28-23

28.4.4.2 Evaluation Rules

The three rules for the computation of the HEALTH attribute are given below.

28.4.4.2.1 Monitoring Mode

IF EV1 IS LOW, THEN u ¼ a1

IF EV2 IS SHORT, THEN u ¼ a2

IF EV3 IS LOW, THEN u ¼ a3

The values of the scaling parameters, that is, a terms in Equation 28.21, reflect the relative importance

of the fuzzy rules in the determination of the HEALTH of the machine. The scaling values of a1; a2; and

a3 are set at 0.7, 0.2, and 0.1, respectively. The respective membership functions are

miðEViÞ ¼ e2nðEVi Þb

ð28:31Þ

where n and b are scaling factors for normalization of EVi. Here, they are selected to be n ¼ 10

and b ¼ 0:5:

28.4.4.2.2 Diagnostic Mode

The three evaluation rules for the computation of the HEALTH attribute in the diagnostic mode are

IF EV4 IS LOW, THEN u ¼ a4

IF EV5 IS LOW, THEN u ¼ a5

IF EV6 IS LOW, THEN u ¼ a6

FIGURE 28.30 Vibration signature of the chirp wave input, with a standardized amplitude of 1 V and starting

frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001,

IOS Press. With permission.)

28-24 Vibration and Shock Handbook

The scaling values of a4; a5; and a6 are set at 0.4, 0.2, and 0.4, respectively. Similar membership

functions as for the monitoring mode are used here. The machine condition attribute HEALTH is then

computed as in Equation 28.23.

28.4.4.3 Tests

28.4.4.3.1 Monitoring Mode

In the monitoring mode, the normal input signal

(i.e., the square wave with standardized amplitude

of 1 V and frequency of 5 Hz) is applied to the

shaker-table system. At t ¼ 5 sec, a sinusoidal

signal (with amplitude 0.4 V and frequency

f ¼ 5 Hz) is also applied to the system to simulate

a fault arising in the machine. The time-domain

signal of the machine (corresponding to the square

input) is shown in Figure 28.34. The spectra of the

machine before and after t ¼ 5 sec are shown in

Figure 28.35. The vibration-analysis algorithm is

able to detect the fault in the machine. Before the

introduction of the fault, the HEALTH attribute of

the shaker table is found to be 0.98. After the

introduction of the fault, the HEALTH attribute

falls to 0.63, which is below the threshold value, set

at 0.7. As a result, the alarm is triggered.

FIGURE 28.31 Sine wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K.,

Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.32 Vibration signature of the sine wave

input, with a standardized amplitude of 1 V and

frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de

Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst.,

2001, IOS Press. With permission.)

Vibration Suppression and Monitoring in Precision Motion Systems 28-25

28.4.4.3.2 Diagnostic Mode

In the diagnostic mode, three input signals (i.e., sine, square, and chirp wave with standardized

amplitude and frequency) are selected to be applied to the shaker table system in turn. To simulate a fault

arising at t ¼ 5 sec, the input gain is increased by a factor of 1.4 times at t ¼ 5 sec. The time-domain

FIGURE 28.33 Test platform: the shaker table. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin,

S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.34 Time-domain vibration signal corresponding to the square input, with a standardized amplitude of

1 V and frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated). (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H.,

and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

28-26 Vibration and Shock Handbook

vibration signal of the machine (corresponding to the chirp signal, with a standardized amplitude of 1 V

and starting frequency of 5 Hz) is shown in Figure 28.36. The spectra (corresponding to the chirp signal)

of the machine before and after t ¼ 5 sec are shown in Figure 28.37. The time-domain vibration signal of

the machine (corresponding to the sinusoidal wave input, with a standardized amplitude of 1 V and

FIGURE 28.35 (a) Vibration signature corresponding to the square input, with a standardized amplitude of 1 V and

frequency of 5 Hz; (b) spectrum of the machine corresponding to the square input after fault occurs. (Source: Tan,

K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 28.36 Time-domain vibration signal corresponding to the chirp input, with a standardized amplitude of

1 V and starting frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated).

Vibration Suppression and Monitoring in Precision Motion Systems 28-27

frequency of 5 Hz) is shown in Figure 28.38. The spectra (corresponding to the sinusoidal input) of the

machine before and after t ¼ 5 sec are shown in Figure 28.39.

The vibration-analysis algorithm is able to detect the fault in the machine. Before the introduction of

the fault, the HEALTH attribute of the shaker table is found to be about 0.97. After the introduction of

the fault, the HEALTH attribute falls to 0.58, which is below the threshold value, set at 0.7. The alarm is

triggered as a result.

FIGURE 28.37 (a) Vibration signature corresponding to the chirp input, with a standardized amplitude of 1 V and

starting frequency of 5 Hz; (b) spectrum of the machine corresponding to the chirp input after a fault occurs.

FIGURE 28.38 Time-domain vibration signal corresponding to the sinusoidal input, with a standardized

amplitude of 1 V and frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated).

28-28 Vibration and Shock Handbook

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