32.2 Specification of Vibration Limits

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Design and control procedures of vibration have the primary objective of ensuring that under normal

operating conditions, the system of interest does not encounter vibration levels that exceed the specified

values. In this context, then, the ways of specifying vibration limits become important. This section will

present some common ways of vibration specification.

32.2.1 Peak Level Specification

Vibration limits for a mechanical system may be specified in either the time domain or the frequency

domain. In the time domain, the simplest specification is the peak level of vibration (typically,

acceleration in units of g — the acceleration due to gravity). Here, the techniques of isolation, design, or

control should ensure that the peak vibration response of the system do not exceed the specified level. In

this case, the entire time interval of operation of the system is monitored, and the peak values are checked

against the specifications. Note that, in this case, it is the instantaneous peak value at a particular time

instant that is of interest, and what is used in representing vibration is an instantaneous amplitude

measure rather than an average amplitude or an energy measure.

32.2.2 Root-Mean-Square Value Specification

The root-mean-square (RMS) value of a vibration signal yðtÞ is given by the square root of the average of

the squared signal as

yRMS ¼

1

T

ðT

0

y2dt

􀀒 􀀓1=2

ð32:1Þ

Note that, by squaring the signal, its sign is eliminated and, essentially, the energy level of the signal is

used. The period T; over which the squared signal is averaged, will depend on the problem and the nature

of the signal. For a periodic signal, one period is adequate for averaging. For transient signals, several time

constants (typically four times the larger time constant) of the vibrating system would be sufficient. For

random signals, a value that is as large as feasible should be used.

Vibration Design and Control 32-3

© 2005 by Taylor & Francis Group, LLC

In the method of RMS value specification, the RMS value of the acceleration response (typically,

acceleration in gs) is computed using Equation 32.1 and is compared with the specified value. In this

method, instantaneous bursts of vibration do not have a significant effect as they are filtered out because

of the integration. It is the average energy or power of the response signal that is considered. The duration

of exposure enters into the picture indirectly and in an undesirable manner. For instance, a highly

transient vibration signal can initially have a damaging effect. However, the larger the T that is used in

Equation 32.1, the smaller the computed RMS value. Hence, in this case, the use of a large value for T

would lead to diluting or masking the damage potential. In practice, the longer the exposure to a

vibration signal, the greater the harm caused. Hence, when using specifications such as peak and RMS

values, they have to be adjusted according to the period of exposure. Specifically, a larger specification

should be used for longer periods of exposure.

32.2.3 Frequency-Domain Specification

It is not realistic to specify the limitation to

vibration exposure of a complex dynamic system

by just a single threshold value. Usually, the effect

of vibration on a system depends on at least the

following three parameters:

1. Level of vibration (peak, RMS, power, etc.).

2. Frequency content (range) of excitation.

3. Duration of exposure to vibration.

This is particularly true because the excitations

that generate the vibration environment may not

necessarily be a single-frequency (sinusoidal)

signal and may be broadband and random.

Furthermore, the response of the system to the

vibration excitations will depend on its frequency

transfer function, which determines its resonances and damping characteristics. Under these

circumstances, it is desirable to provide specifications in a nomograph where the horizontal axis gives

frequency (Hz) and the vertical axis could represent a motion variable such as displacement (m), velocity

(m/s), or acceleration (m/s2 or g). It is not important which of these motion variables represents the

vertical axis of the nomograph. This is true because in the frequency domain

Velocity ¼ jv £ displacement

Acceleration ¼ jv £ velocity

and one form of motion may be easily converted into one of the remaining two motion representations.

In each of the forms, assuming that the two axes of the nomograph are graduated in a logarithmic scale,

the constant displacement, constant velocity, and constant acceleration lines are straight lines.

Consider a simple specification of machinery vibration limits as given by the following values:

Displacement limit ðpeakÞ ¼ 0:001 m

Velocity limit ¼ 0:01 m=sec

Acceleration limit ¼ 1:0g

This specification may be represented in a velocity vs. frequency nomograph (log – log) as in Figure 32.2.

Usually, such simple specifications in the frequency domain are not adequate. As noted above, the

system behavior will vary depending on the excitation frequency range. For example, motion sickness in

humans may be predominant in low frequencies in the range of 0.1 to 0.6 Hz and passenger discomfort in

0.001m 1g

1.0 10.0 100.0 1000.0

Frequency (Hz)

Velocity

(m/s)

0.001

0.01

FIGURE 32.2 Operating vibration specification

(nomograph) for a machine.

32-4 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

ground transit vehicles may be most serious in the frequency range of 4 to 8 Hz for vertical motion and

1 to 2 Hz for lateral motion. In addition, for any dynamic system, particularly at low damping levels,

the neighborhoods of resonant frequencies should be avoided and, hence, should be specified by low

vibration limits in the resonant regions. Furthermore, the duration of vibration exposure should be

explicitly accounted for in specifications. For example, Figure 32.3 presents a ride comfort specification

for a ground transit vehicle, where lower vibration levels are specified for longer trips.

Finally, it should be noted that the specifications we are concerned with in the present context of

design and control are upper bounds of vibration. The system should perform below (within) these

specifications under normal operating conditions. Test specifications are lower bounds. The test should

be conducted at or above these vibration levels so that the system would meet the test specifications.

Some considerations of vibration engineering are summarized in Box 32.1.

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