15A.5 Integration

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This Appendix discusses the integration process, including basic theory and implementation in the time

and frequency domains.

FIGURE 15A.8 Range detection test on a time-domain

signal.

FIGURE 15A.7 Range detection performed in engineering units.

FIGURE 15A.9 Test scalar measurement.

15-84 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The Sound and Vibration Toolkit contains the

following integration VIs:

* SVT Integration VI located on the Integration

palette for time-domain integration

* SVT Integration (frequency) VI located on

the Frequency Analysis .. Extended

Measurements palette for frequencydomain

integration

15A.5.1 Introduction to Integration

The conversion between acceleration, velocity, and

displacement is based on one of the fundamental

laws in Newtonian physics, represented by the

following equations:

x_ ¼

d

dt ðxÞ

x€ ¼

d

dt ð_xÞ ¼

d2

dt2 ðxÞ

Velocity is the first derivative of displacement with respect to time. Acceleration is the first derivative of

velocity and the second derivative of displacement with respect to time. Therefore, given acceleration,

perform a single integration with respect to time to compute the velocity or perform a double integration

with respect to time to compute the displacement.

When representing the acceleration of a point by a simple sinusoid, the velocity and the displacement

of the point are well known and represented by the following equations:

a ¼ A sinðvtÞ ð15A:1Þ

v ¼ 2

A

v

cosðvtÞ ¼

A

v

sin vt 2

p

2

􀀏 􀀐

ð15A:2Þ

d ¼ 2

A

v2 sinðvtÞ ¼

A

v2 sinðvt 2 pÞ

Note: The initial condition is arbitrarily set to zero in Equation 15A.1 and Equation 15A.2.

The amplitude of the velocity is inversely proportional to the frequency of vibration. The amplitude of

the displacement is inversely proportional to the square of the frequency of vibration. Furthermore, the

phase of the velocity lags the acceleration by 908. The phase of the displacement lags the acceleration

by 1808. Figure 15A.15 illustrates the relationship between acceleration, velocity, and displacement.

The integration of a sinusoid is known in closed form. Integration of an arbitrary waveform typically

requires a numerical approach. You can use several numerical integration schemes to evaluate an integral

in the time domain.

FIGURE 15A.10 Limit testing on THD measurements.

FIGURE 15A.11 Continuous mask test on power spectrum.

Virtual Instrumentation for Data Acquisition, Analysis, and Presentation 15-85

© 2005 by Taylor & Francis Group, LLC

In the frequency domain, you can define any

arbitrary band-limited waveform as a sum of

sinusoids. Because the amplitude and phase

relationships are known for sinusoids, you can

carry out the integration in the frequency domain.

15A.5.2 Implementing Integration

If you need to perform measurements on velocity

or displacement data when you have only acquired

acceleration or velocity data, respectively, integrate

the measured signal to yield the desired data. You

can perform integration either in the time domain

as a form of signal conditioning or in the

frequency domain as a stage of analysis. When

performed in the frequency domain, integration is

one of the extended measurements for frequency

analysis.

15A.5.2.1 Challenges when Integrating Vibration Data

Converting acceleration data to velocity or displacement data presents a pair of unique challenges. First,

measured signals typically contain some unwanted DC components. The second challenge is the fact that

many transducers, especially vibration transducers, have lower-frequency limits. A transducer cannot

accurately measure frequency components below the lower-frequency limit of the transducer.

15A.5.2.1.1 DC Component

Even though a DC component in the measured signal might be valid, the presence of a DC component

indicates that the DUT has a net acceleration along the axis of the transducer. For a typical vibration

measurement, the DUT is mounted or suspended in the test setup. The net acceleration of the DUT is

zero. Therefore, any DC component in the measured acceleration is an artifact and should be ignored.

15A.5.2.1.2 Transducers

Most acceleration and velocity transducers are not designed to accurately measure frequency components

close to DC (see Chapter 15). Closeness to DC is relative and depends on the specific transducer. A typical

accelerometer can accurately measure components down to about 10 Hz. A typical velocity probe can

accurately measure components down to 2 to 3 Hz. Inaccurately measured low-frequency vibrations can

dominate the response when the signal is integrated because integration attenuates low-frequency

components less than high-frequency components.

FIGURE 15A.13 Discontinuous mask test on swept-sine frequency response.

FIGURE 15A.12 Continuous mask test on a power

spectrum.

15-86 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

15A.5.2.1.3 Implementing Integration Using the Sound and Vibration Toolkit

Both the SVT Integration VI and the SVT

Integration (frequency) VI address the challenges

of converting acceleration data to velocity or

displacement data.

15A.5.3 Time-Domain Integration

This section presents examples of and discussion

about time-domain integration.

15A.5.3.1 Single-Shot Acquisition

and Integration

The following example shows how one can use

integration to convert acceleration data into

displacement data in a single-shot acquisition

and integration. In this example, the acquired

waveform is sampled at 51.2 kHz and is double

integrated. Figure 15A.16 shows the block diagram

for the VI.

Because the integration is implemented with

filters, there is a transient response associated with

integration while the filters settle. You should take

care to avoid the transient region when making

further measurements. Figure 15A.17 shows the

results of a single-shot acquisition and integration

of a 38 Hz sine wave. You can see the transient

response in the first 200 msec of the integrated

signal.

15A.5.3.2 Continuous Acquisition and Integration

The more common case for time-domain integration occurs with continuous acquisition. Figure 15A.18

shows the block diagram for a VI designed for continuous acquisition and integration.

In this example, the high-pass cut-off frequency used for the integration is 10 Hz. Additionally, the

integration is explicitly reset in the first iteration of the VI and performed continuously thereafter. In this

example, this additional wiring is optional because the SVT Integration VI automatically resets the first

time it is called and runs continuously thereafter.

If you use the block diagram in Figure 15A.18 in a larger application that requires starting and

stopping the data acquisition process more than once, NI suggests setting the reset filter control to

“TRUE” for the first iteration of the while loop. Setting the reset filter control to TRUE causes the filter to

reset every time the data acquisition process starts. Set the reset filter control to “FALSE” for subsequent

iterations of the while loop.

FIGURE 15A.14 Discontinuous mask test on a sweptsine

frequency response.

FIGURE 15A.15 Integration of a 0.5 Hz sine wave.

FIGURE 15A.16 Block diagram for single-shot acquisition and integration.

Virtual Instrumentation for Data Acquisition, Analysis, and Presentation 15-87

© 2005 by Taylor & Francis Group, LLC

Figure 15A.19 shows the results of the continuous

acquisition and integration of the same 38 Hz

sinusoid used in the single-shot acquisition and

integration example.

As in single-shot acquisition and integration,

continuous acquisition and integration has an

initial transient response. Take care to avoid

making additional measurements until the

response of the filters settles. Once the filters

settle, you can use the integrated signals for

additional analysis.

Figure 15A.20 shows the frequency response for

time-domain single integration. Figure 15A.21

shows the frequency response for time-domain

double integration.

In Figure 15A.20, one can see the characteristic

20 dB per decade roll-off of the magnitude

response of the single integration. In Figure

15A.21, one can see the characteristic 40 dB per

decade roll-off of the magnitude response of the

double integration.

Upper and lower frequency limits exist for

which you can obtain a specified degree of

accuracy in the magnitude response. For example,

sampling at a rate of 51.2 kHz, the magnitude

response of the integrator is accurate to within

1 dB from 1.17 to 9.2 kHz for single integration

and from 1.14 to 6.6 kHz for double integration.

The accuracy ranges change with the sampling

frequency and the high-pass cut-off frequency. The

attenuation of the single integration filter at

9.2 kHz is 2 95 dB. The attenuation of the double

integration filter at 6.6 kHz is 2 185 dB. Accuracy

at high frequencies usually is not an issue.

15A.5.4 Frequency-Domain

Integration

You can use the following strategies to obtain the

spectrum of an integrated signal:

* Perform the integration in the time domain

before computing the spectrum.

* Compute the spectrum before performing the integration in the frequency domain.

The following example demonstrates the implementation of the strategies used to obtain the spectrum

of an integrated signal. Figure 15A.22 shows the block diagram for the example VI.

The high-pass cutoff frequency parameter of the SVT Integration VI is wired with a constant of

10 Hz. The SVT Integration (frequency) VI does not have a high-pass cutoff frequency parameter.

Instead, the SVT Integration (frequency) VI sets the DC component of the integrated signal to zero if

the spectrum scale is linear or to negative infinity (2 Inf) if the spectrum scale is in decibels.

Figure 15A.23 shows the results of integrating in the time and frequency domains.

FIGURE 15A.17 Transient response in single-shot

acquisition and integration.

FIGURE 15A.18 Continuous acquisition and

integration.

FIGURE 15A.19 Settled response of continuous

acquisition and integration.

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© 2005 by Taylor & Francis Group, LLC

The power spectrum is computed after the timedomain

integration filters settle. The frequencydomain

integration scales the spectrum at each

frequency line. No settling time is necessary for the

frequency-domain integration because integration

filters are not involved in the frequency-domain

integration.

Perform frequency-domain integration in the

following situations to maximize performance:

* When the integrated signal is not needed in

the time domain

* When spectral measurements are made

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