10.2 Frequency Spectrum

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Excitations (inputs) to a dynamic system progress

with time, thereby producing responses (outputs)

which themselves vary with time. These are signals

that can be recorded or measured. A measured

signal is a time history. Note that in this case the

independent variable is time and the signal is

represented in the time domain. A limited

amount of information can be extracted by

examining a time history. As an example, consider

the time-history record shown in Figure 10.2.

It can be characterized by parameters such as the

following:

ap ¼ peak amplitude

Tp ¼ period in the neighborhood of the peak ¼ 2 £ interval between successive zero crossings near

the peak

Te ¼ duration of the record

Ts ¼ duration of strong response (i.e., the time interval beyond which no peaks occur that are larger

than ap=2Þ

Nz ¼ number of zero crossings within Ts ðNz ¼ 14 in Figure 10.2)

It is obviously cumbersome to keep track of so many parameters and, furthermore, not all of them are

equally significant in a given application. Note, however, that all the parameters listed above are directly

Aerodynamic

Excitations

Engine

Excitations Control Surface

Excitations

Aerodynamic

Excitations

Body

Response

FIGURE 10.1 In-flight excitations and responses of an aircraft.

Te

Time

Acceleration

ap

−ap/2

ap/2

0

Tp / 2

Ts(Nz = 14)

FIGURE 10.2 A time-history record.

10-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

or indirectly related to either the amplitude or the frequency of zero crossings within a given time interval.

This signifies the importance of a frequency variable in representing a time signal. This is probably the

fundamental motivation for using frequency-domain representations. In this context, however, more

rigorous definitions are needed for the parameters: amplitude and frequency. A third parameter, known

as phase angle, is also needed for unique representation of a signal in the frequency domain.

10.2.1 Frequency

Let us further examine the basis of frequency-domain analysis. Consider the periodic signal of period T

that is formed by combining two harmonic (or sinusoidal) components of periods T and T=2 and

amplitudes a1 and a2 as shown in Figure 10.3. The cyclic frequency (cycles/sec, or hertz, or Hz) of the two

components are f1 ¼ 1=T and f2 ¼ 2=T: Note that in order to obtain the angular frequency (radians/sec),

the cyclic frequency has to multiplied by 2p:

10.2.2 Amplitude Spectrum

An alternative graphical representation of the periodic signal shown in Figure 10.3 is given in Figure 10.4.

In this representation, the amplitude of each harmonic component of the signal is plotted against the

corresponding frequency. This is known as the amplitude spectrum of the signal, and it forms the basis of

the frequency-domain representation. Note that this representation is often more compact, and can be

far more useful than the time-domain representation. Note further that in the frequency-domain

representation, the independent variable is frequency.

10.2.3 Phase Angle

In its present form, Figure 10.4 does not contain all

the information of Figure 10.3. For instance, if the

high-frequency component in Figure 10.3 is

shifted through half its period ðT=4Þ; the resulting

signal is shown in Figure 10.5. This signal is quite

different from that in Figure 10.3 but since the

amplitudes and the frequencies of the two

harmonic components are identical for both

signals, they possess the same amplitude spectrum.

So, what is lacking in Figure 10.3 in order to make

FIGURE 10.3 Time-domain representation of a periodic signal.

Frequency f

a1

a2

0

f1 = 1/T

Amplitude

f2 = 2/T

FIGURE 10.4 The amplitude spectrum of a periodic

signal.

Vibration Signal Analysis 10-3

© 2005 by Taylor & Francis Group, LLC

it a unique representation of a signal, is the information concerning the exact location of the harmonic

components with respect to the time reference or origin ðt ¼ 0Þ: This is known as the phase information.

For example, the distance of the first positive peak of each harmonic component from the time origin can

be expressed as an angle (in radians) by multiplying it by 2p=T: This is termed the phase angle of the

particular component. In both signals (shown in Figure 10.3 and Figure 10.5) the phase angle of the first

harmonic component is the same and equals p=2 according to the present convention. The phase angle of

the second harmonic component is p=2 in Figure 10.3 and zero in Figure 10.5.

10.2.4 Phasor Representation of Harmonic Signals

A convenient geometric representation of a harmonic signal of the form

yðtÞ ¼ a cosðvt þ fÞ ð10:1Þ

is possible by means of a phasor. This representation is illustrated in Figure 10.6. Specifically, consider a

rotating arm of radius a; rotating in the counter-clockwise (ccw) direction at an angular speed of v

rad/sec. Suppose that the arm starts (i.e., at t ¼ 0) at an angular position f with respect to the y-axis

(vertical axis) in the ccw sense. Then, it is clear from Figure 10.6(a) that the projection of the rotating arm

on the y-axis gives the time signal yðtÞ: This is the phasor representation, where we have

Signal amplitude ¼ length of the phasor

Signal frequency ¼ angular speed of the phasor

Signal phase angle ¼ initial position of the phasor with respect to the y-axis

It should be clear that a phase angle makes practical sense only when two or more signals are

compared. This is so because for a given harmonic signal we can pick any point as the time reference

ðt ¼ 0Þ: However, when two harmonic signals are compared, as in Figure 10.6(b), we may consider one

of those signals that starts (at t ¼ 0) at its position peak as the reference signal. This will correspond to

a phasor whose initial configuration coincides with the positive y-axis. As is clear from Figure 10.6(b),

for this reference signal we have, f ¼ 0: Then the phase angle f of any other harmonic signal would

correspond to the angular position of its phasor with respect to the reference phasor. Note that, in

this example, the time shift between the two signals is f=v; which is also a direct representation of the

phase. It should be clear, then, that the phase difference between two signals is also a representation of

the time lead or time lag (delay) of one signal with respect to the other. Specifically, the phase that is

ahead of the reference phasor is considered to “lead” the reference signal. In other words, the

signal a cosðvt þ fÞ has a phase lead of f or a time lead of f=v with respect to the signal of a cos vt:

FIGURE 10.5 A periodic signal with an identical amplitude spectrum as for Figure 10.2.

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© 2005 by Taylor & Francis Group, LLC

Another important observation may be made with regard to the phasor representation of a harmonic

signal. A phasor may be expressed as the complex quantity

yðtÞ ¼ a e jðvtþfÞ ¼ a cosðvt þ fÞ þ ja sinðvt þ fÞ ð10:2Þ

whose real part is a cosðvt þ fÞ; which is in fact the signal of interest. It is clear from Figure 10.6 that, if

we take the y-axis to be real and the x-axis to be imaginary, the complex representation 10.2 is indeed a

complete representation of a phasor. By using the complex representation 10.2 for a harmonic signal,

significant benefits of mathematical convenience could be derived in vibration analysis. It suffices to

remember that practical vibrations are “real” signals, and regardless of the type of mathematical analysis

that is used, only the real part of a complex signal of the form 10.2 will make physical sense.

10.2.5 RMS Amplitude Spectrum

If a harmonic signal yðtÞ is averaged over one period T; the negative portion cancels out the positive

portion, giving zero. Consider a harmonic signal of angular frequency v (or cyclic frequency f ), phase

angle f; and amplitude a; as given by

yðtÞ ¼ a cosðvt þ fÞ ¼ a cosð2pft þ fÞ ð10:3Þ

Its average (mean) value is

ymean ¼

1

T

ðT

0

yðtÞdt ¼ 0 ð10:4Þ

f/w

Period 2p/w

t

y

a

0 x

f

w

(a)

y

a

y = a cos (wt+f)

t

y

a

0 x

f

(b)

y

a cos wt

a cos (wt+f)

FIGURE 10.6 Phasor representation of a harmonic signal. (a) A phasor and the corresponding signal;

(b) representation of a phase angle (phase lead) f:

Vibration Signal Analysis 10-5

© 2005 by Taylor & Francis Group, LLC

which can be verified by direct integration, while

noting that

T ¼ 1=f ¼ 2p=v ð10:5Þ

For this reason, mean value is not a measure of the

“strength” of a signal in general. Now let us define

the root mean square (RMS) value of a signal. This

is the square root of the mean value of the square

of the signal. By direct integration, it can be shown

that for a sinusoidal (or harmonic) signal; the

RMS value is given by

yRMS ¼

1

T

ðT

0

y2ðtÞdt

􀀒 􀀓1=2

¼

affiffi

2 p ð10:6Þ

It follows that the RMS amplitude spectrum is obtained by dividing the amplitude spectrum by

ffiffi

2 p : For

example, for the periodic signal formed by combining two harmonic components as in Figure 10.3, the

RMS amplitude spectrum is shown in Figure 10.7. This again is a frequency-domain representation of a

signal, and the independent variable is frequency.

10.2.6 One-Sided and Two-Sided Spectra

Mean squared amplitude spectrum of a signal (sometimes called power spectrum because the square of a

variable such as voltage and velocity is a measure of quantities such as power and energy, even though it is

not strictly the spectrum of power in the conventional sense) is obtained by plotting the mean squared

amplitude of the signal against frequency. Note that these are one-sided spectra because only the positive

frequency band is considered. This is a realistic representation because one cannot talk about

negative frequencies for a real system. But, from a mathematical point of view, we may consider negative

frequencies as well. In a spectral representation it is at times convenient to consider the entire frequency

band (consisting of both negative and positive values of frequency). It then becomes a two-sided spectrum.

In this case the spectral component at each frequency value should be equally divided between the

positive and the negative frequency values (hence, the spectrum is symmetric), such that the overall mean

squared amplitude (or power or energy) remains the same.

We have seen that for a harmonic signal component of amplitude a and frequency f (e.g.,

a cosð2pf þ fÞ) the RMS amplitude is a2=2 at frequency f ; whereas the two-side spectrum has a

magnitude of a2=4 at both the frequency values 2f and þf :

Note that, even though it is possible to interpret the meaning of a negative time (which represents the

past, previous to the starting point), it is not possible to give a realistic meaning to a negative frequency.

This concept is introduced primarily for analytical convenience.

10.2.7 Complex Spectrum

We have shown that for unique representation of a signal in the frequency domain, both amplitude and

phase information should be provided for each frequency component. Alternatively, the spectrum can be

expressed as a complex function of frequency, having a real part and an imaginary part. For instance, for a

harmonic component given by a cosð2pfi þ fÞ the two-sided complex spectrum can be expressed as

Y ðfiÞ ¼

ai

2 ðcos fi þ j sin fiÞ ¼

ai

2

e jfi

and,

Y ð2fiÞ ¼

ai

2 ðcos fi 2 j sin fiÞ ¼

ai

2

e2jfi ð10:7Þ

in which j is the imaginary unity as given by j ¼

ffiffiffiffi

21 p : Note that the spectral component at the negative

frequency is the complex conjugate of that at the positive frequency. This concept of complex spectrum is

Frequency f

f1 = 1/T f2 = 2/T

a1/√1

RMS Amplitude

a1/√2

FIGURE 10.7 The RMS amplitude spectrum of a

periodic signal.

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© 2005 by Taylor & Francis Group, LLC

the basis of (complex) Fourier series expansion (FSE), which we shall consider in detail in a later section.

It should be clear that the complex conjugate of a spectrum is obtained by changing either j to 2 j or

v to 2v (or f to 2f ).

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