Average Control for Sinusoidal- and Random-Vibration Testing

12.3, 12.12 Received 16 September 1966

Average Control for Sinusoidal- and Random-Vibration Testing

TrmRON UsrmR, J•.

Unholtz-Dickie Corporation, Hamden, Connecticut 06518

Time-division multiplexers (TDM) have found application in providing control from the average of several acceleration magnitudes for swept-sinusoidal vibration tests and, also, control to the average of several acceleration spectral densities for wide-band random tests. The vibration testing systems using the TDM are described. Pertinent system parameters are determined mathematically in order to limit signal harmonic distortion for the sinusoidal case, and to limit test-level modulation and spectral spreading for the random case. A special case for sinusoidal testing in which a narrow-band tracking filter is used on the output of the TDM is also discussed.

INTRODUCTION

YSTEMS for controlling sinusoidal and random vibration tests have found wide application and acceptance over the past decade. A typical system for controlling a sinusoidal vibration test I contains a swept- frequency oscillator driving a power amplifier through an automatic-gain-control (AGC) circuit. 2 The amplifier, in turn, drives a vibration exciter with an attached specimen. The acceleration at an attachment point of the specimen is sensed and provides the control for the AGC circuit, so that the acceleration amplitude at the point of measurement stays nearly constant, as the frequency is swept. In this fashion, 40 dB of gain variation owing to the exciter and specimen may be compressed to 1 dB of amplitude variation at the sensing point.

A typical random vibration testing system s contains a random signal generator driving a power amplifier through a spectrum equalizer. Such equalizers consist of a large number of bandpass filters, with the inputs connected in parallel. Each filter covers a portion of the frequency spectrum. The gain of each filter channel is varied manually or by means of an AGC circuit, and the output signals of the filters are summed and fed to the power amplifier. The random acceleration at an

1 J. T. Broch, "Automatic Level Regulation of Vibration Exciters," Brtiel and Kj•er Tech. Rev. 2, 14-22 (Apr. 1958).

•' W. K. Victor and M. H. Brockman, "The Application of Linear Servo Theory to the'Design of AGC Loops," ?roc. IRE 18, 234-238 (Feb. 1960).

a C. E. Maki, "Automatic Spectral Compensation of an Audio System Operating with a Random Noise input," IRE Natl. Cony. Rec. 8, 80-94 (1960).

840 Volume 41 Number 4 1967

attachment point of the specimen is sensed and fed to a spectrum analyzer that contains a set of filters identical in bandpass characteristics to those in the equalizer. By means of a detector and low-pass filter, a signal representing the average spectral density over each filter band is derived and displayed. The spectral characteristics of the acceleration at the attachment

point are controlled by manually varying the gain of the equalizer filters or by using the signals representing the average spectral densities in order to control the AGC circuits of the equalizer filters.

Single-point vibration control, as described above, has given acceptable results when the test specimen is reasonably small, because the acceleration response functions at all attachment points are nearly equal. However, large specimens need large fixtures to join the specimen to the exciter table. The dynamic asym- metries of the specimen induce widely different frequency-response functions for the accelerations at different attachment points of the specimen. For large specimens, single-point vibration control often can not produce a valid vibration test because changing the point of control changes the spectral characteristics of the vibration applied to the specimen.

Multiple-point vibration control has evolved as a means of producing a valid vibration test for large test specimens. An instantaneous average of the accelerations at several mounting points has been used as a control signal for sinusoidal and random vibration testing with limited success. Cancellation of components of the control signal owing to the procedure of taking the

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13:48:33

SINUSOIDAL AND RANDOM VIBRATION TESTING

ACCELEROMETER,• ...... AND AMPLIFIERS

v, (t) [ • fv2(t) •; ,(t) _ g o(t)

I J FILTER ¾-•(t)

I

L_AUTOMATIC GAIN CONTROL (A___G C_) __]

Fro. 1. Block diagram fm sinusoidal-vibration-testing system showing TDM for control of average acceleration magnitude.

instantaneous average causes the specimen to exhibit excessive acceleration levels at certain frequencies.

Time-division-multiplexing techniques avoid the disadvantage cited above. For sinusoidal-vibration testing, equal periods of each acceleration signal are multi- plexed to form the control signal for the AGC circuit. A block diagram of the complete system is shown in Fig. 1. The absolute average value of the output signal of the time-division multiplexer (TDM) is proportional to the average of the magnitudes of the control-acceleration signals and is insensitive to the relative phase angles of the control signals.

For random-vibration testing, the TDM processes equal periods of the control-acceleration signals to provide an input signal for the spectrum analyzer. A block diagram of the complete system is shown in Fig. 2. With proper selection of system parameters, the output signal of the TDM has a spectral density that is very nearly equal to the average of the spectral densities of the input signals. The cross-spectral densities of the input signals have very little effect on the expression for the spectral density of the output signal. Such is not the case for a device which provides an instantaneous average of the input signals. Either manual or automatic control of the spectral density of the output of the TDM is then possible.

In following sections, the effects of the TDM on the operation of the swept-frequency sinusoidal-vibration- testing system are analyzed. The special case of a narrow-band tracking filter inserted in the system following the TDM is also treated. The analysis provides help for selecting values of system parameters that give meaningful vibration tests.

The effects of the TDM on the operation of a random- vibration-testing system are also analyzed. This subject has been treated recently by Ratz, 4 but the following results are more extensive.

Another technique for obtaining average control for random vibration testing would be to provide fairly large time-delay differences for different vibration channels and to utilize the sum of the resulting signals as a control signal. One realization for a system of this sort would require tape-recording equipment and pro-

4 A. G. Ratz, "Automatic Random Equalization Control," J. Environmental Sci. 9, 26-28 (April 1966).

vision for transport lags. • The time-division multiplexer offers an economic advantage over this technique and is also more flexible.

Other types of magnitude-averaging devices have found application for sinusoidal-vibration testing only, and, thus, they lack the versatility of the TDM.

I. AVERAGE CONTROL-SINUSOIDAL- VIBRATION TESTING

The suppression of distortion components of the signal at the input to the power amplifier is of prime importance, since these distortion components can be magnified many times by the frequency-response characteristics of the vibration exciter and specimen. The gain-modulating action of v4(t) in Fig. 1 always produces a certain amount of waveform distortion in v2(t) because v4(t) contains harmonic or subharmonic components related to the fundamental frequency of vz(t), in addition to the desired nonzero average value. This section is devoted to the derivation of an expression for the relative distortion of the output voltage of the variable-gain amplifier, under worst possible conditions.

The output signal of the TDM, go(t), in terms of the input functions is given by

N

go(t)= •. a,(t)g,(t),

where

for •(t) = 0 [(•- 1)+Np]T_<t<En+Np]T

elsewhere

n=l, 2,---,N (2)

p=0, 1, 2, 3 ---.

The function a• (t) is simply a periodic rectangular pulse function that represents the gating action of the TDM, and T is the gating time.

For sinusoidal operation, T is chosen to be exactly one period of the fundamental frequency of the input functions (T--2•r/w). The input functions to the TDM are given very closely by

g,(t)=G, sin(wtq-O,). (3)

In Eq. 3, the small distortion components are neglected. The gating time is not chosen to be less than one period of the fundamental frequency, mainly because limita- tions in the frequency response of the amplifier pre- ceding the full-wave detector in Fig. 1 prevent fast gating. Larger gating times require longer averaging times and thus slower response in the AGC circuit.

a A. J. Curtis, H. T. Abstein, and R. J. Varga, "On the Use of Multiple (Multi-Point) Random Excitation with Application to Surveyor Spacecraft Tests," Shock Vibration Bull. 35, Part 2 (Office of the Director of Defense Research and Engineering) (Jan. 1966).

The Journal of the Acoustical Society of America 841


13:48:33

T. USHER, JR.

The output va(t) of the full-wave detector in Fig. 1 is

v3(t)=K• • la,•(t)G,•sin(oot+O,•)l, (4)

in which K• is the gain of the amplifier-detector circuit. The average value of the output of the low-pass filter is easily shown to be

21 •r

(v4(t)>a,•=(v3(t))•,•=K• --- • Gn. (5) 71' N n--1

Thus, the average value of the control voltage of the variable-gain amplifier is proportional to the average of the magnitudes of the acceleration amplitudes Gn and is not a function of their phase angles,

In addition to the average value, the rectified waveform va(t) contains a subharmonic component of the order 1IN and all integral multiples of 1IN relative to the fundamental frequency, w. The worst case for subharmonic distortion occurs when every input to the TDM, except one, is zero. Attention will be centered on the component of order I/N, since the higher-order components are smaller and are also attenuated more severely by the low-pass filter in Fig. 1.

If G•-G and G•=0 for n•l, the complex Fourier coefficient of va(t) of order 1IN is ø

D•/2v=• K•a•(t)lG•sin(wt+o1)lexp -- t dt

NT

(5) Xexp -- t dt

(T= 2•r/w), (6)

where j= (--1)•. The absolute value of the sinusoid in Eq. 6 may be replaced by the average value with very little error in the result'

1 2 ft ( w ) D ---•---KiG exp -- j.•:t dt 1/:V--NT r J o

= (2/•r2)KiGe --i•'l:v sin(,r-/N). (7)

SUM SPEC.

I •s•- •'

i I •

I • L _EQ_U_A_LI_Z_ER__ ...... ANALYZER

Fro. 2. Block diagram for random-vibration-testing system showing TDM for control of avelage spectral density.

The magnitude of the subharmonic component is twice the magnitude of the complex Fourier coefficient. Thus, the relative subharmonic content of order 1IN of va(t) from Eqs. 5 and 7 is

21D/l 2 •r hl/iv=••-N sin--. (8)

The relative subharmonic content of v•(t) of order 1IN is diminished from that of va(t) by the factor N/wT•(<<I) due to the action of the single-pole low- pass filter. The time constant of the filter is T•.

The gain of the variable-gain amplifier is a function of the voltage v•(t), and can usually be approximated by Eq. 9. It is assumed that the gain is not frequency dependent'

H(v•)=v•.(t)/v•(t)= (v•(t)/K•.) e (C>>i). (9)

In Eq. 9, the constant C may be called the "compression ratio" and is usually greater than 10 for commercially available AGC circuits.

From Eqs. 5, 8, and the factor N/wTv, v•(t) is predicted by Eq. 10 for the worst case of subharmonic distortion-

,•(t)•K•- l+•- sin-- sin t+• . (10) •r w Tr •r N

Since v•(t) is assumed to be an undistorted sinusoid, a close approximation to v•.(t) is obtained by substituting Eq. 10 into Eq. 9'

v2(t) = K• sinwtK• -c

N•C 2

wTr •r s,n; (11)

• H. P. Westman, Ed., Reference Data for Radio Engineers (I. T. T. Corp., 1956) 4th ed., pp. 1006-1007.



13:48:33


From Eq. 11, it is easily seen that the subharmonic distortion of v4(t) produces distortion components of v•.(t) that are sidebands located at (I+i/N). The relative distortion of ,•.(t) for either sideband is

h•.=N2C/coTvr sin(r/N)<<l N• 1 (12)

•NC/COTv for N>>i.

Since h•. is small, the polynomial expansion in Eq. 11 is fairly accurate and the original assumption for Eq. 3 that led to the result in Eq. 8 is justified. Of course, N/coT• must be sufficiently small so that the inequality in Eq. 12 is satisfied.

Most commercially available AGC circuits for vibration testing are designed so that, when T• varies with frequency, the quantity coT• has a reasonably large minimum value over a wide frequency range. For a representative computation, let C=40, (coT•)min=104 and N=5, then h•.=0.019. Note that this value for relative distortion represents an upper limit, since it is highly unlikely that the accelerations of four out of five points on the specimen are zero.

II. SINUSOIDAL-VIBRATION TESTING-- TRACKING FILTER

In some testing situations, rattling and nonlinearities in the specimen produce a certain amount of "hash" and harmonic distortion in the acceleration waveforms at the control points. In order to eliminate the possi- bility of the level of the input w.(O (in Fig. 1) being affected by the "hash" and distortion, narrow-band tracking filters may be inserted to remove most of the unwanted components from the inputs of the TDM. This, however, requires N tracking filters. From economic considerations, it is advantageous to use one tracking filter between the output of the TDM and the input of the AGC circuit. The position is marked with "X" in Fig. 1.

The use of one tracking filter at the output of the TDM requires that some operational compromises be made. In order that Eq. 5 holds exactly, all sidebands of the functions a,•(t)G,•sin(cot-t-O,O must be present. A narrow-band filter, centered at co, at the output of the TDM would cause all of the sidebands to be severely attenuated if the gating time were T= 2•r/co. The output of the tracking filter would then be the instantaneous average (1IN times the phasor sum) of the inputs to the TDM, and the previously stated disadvantage of control from the instantaneous signal average would occur. The solution to this situation is to make the gating time T large enough so that most of the desired sidebands are included within the passband B, of the tracking filter, thus keeping the absolute average of the output of the tracking filter insensitive to phase.

When the gating time is increased, the time constant Tv of the AGC circuit must also be increased to prevent the AGC circuit from producing sizeable distortion

components having frequencies of (co=!=p2,r/NT), where p takes on integral values. Owing to T's being fairly large, this type of distortion appears as modulation of the amplitude of w.(t).

In this section, limits on the product B,T are determined in order to keep the average control fairly insensitive to the relative phase angles oœ the inputs. Also, limits for the ratio T•/T are determined in order to keep the relative amplitude modulation of ,•.(t) to a small value. It must be remembered that T• must not

be made unduly large, because the response time of the AGC circuit must be kept within reasonable bounds.

In order to evaluate the amount of phase sensitivity caused by the presence of a narrow-band filter on the output of the TDM, a situation giving an upper bound is examined. If N = 2 and

g• (t)= --g•.(O =G sincot (13)

as B,T approaches zero, then the filter removes all sidebands of a•(t)g•(t) and a•.(t)g•.(t) and leaves only the fundamental components, which add to zero. The control signal v4(t) is zero also, instead of having the desired average value 2K•G/r.

For purposes of analysis, the narrow-band filter is assumed to have the maximally flat frequency-response function

Zt•(u) = [ • + jV2u+ (ju7• -•, (•4)

where u is the normalized deviation of the frequency from the center frequency of the filters:

u= 2(•'--•)/B, •/B,>>I. (15)

The waveform at the input to the filter from Eqs. ! and 13 is given by

go(t)=[a•(t)-a2(t)-IG sincot. (16)

The Fourier series for a•(t) and a•.(t) are wall known 7 and are

and

i oo 2 •r

a•(t)=-+Z sin(Xp--1) t (17) 2 p=• (2p-- 1)•r • I oo 2 •r

a•(t)=--- Z sin(Xp--1) t. (18) 2 p=• (2p-- 1)r -f

Combining Eqs. 16, 17, and 18 we have

I 4 sin (2 p-- 1 )r---tlG sincot. go(t)= •'. (2p-- 1)•r T J (19)

Equation 19 represents one form of carrier-suppressed amplitude modulation. The output of the tracking filter is also a form of carrier-suppressed amplitude

7 Ref. 5, p. 1019.



13:48:33

T. USHER, JR.

modulation for this example, since only the amplitudes and phases of the series terms multiplying G sincot are altered.

For BsT<<2•r, we have u>>l, since Ico'--co] = •-/T holds for the first-order sidebands of Eq. 19. The higher-order sidebands may be neglected at the output of the filter, since they are attenuated much more severely than the first-order ones. Since u-4-2•r/BsT and H(ju)=- 1/u 2, the output of the tracking filter is

go' (t)•-- (4/•r)(BsT/2•r) • sin Ort/T) •G sincot (20)

The average output of the full-wave detector in Fig. 1. is then

K1 2T •r\ 2•r/ sin•t G sincot dt

•r\ 2•r /

The ratio of the result in Eq. 21 to the desired result, 2K•G/•r, is an indication of the sensitivity of the filtered and rectified output of the TDM to the relative phases of the input signals. This ratio for the result in Eq. 21 is

R= (8/•r •) (BsT/2•r) • B•T/2•r<<I. (22)

The small values of R for small values of B•T are expected.

For B,T>>2•r, the calculation of R is not very easily accomplished by calculating all the sidebands of the tracking filter output. It can be done, however, by determining the modulation envelope of G sincot for the output of the tracking filter. Since the Fourier frequencies of the modulation envelope in Eq. 19 corre- spond to (co'--co) in Eq. 15, the modulation envelope of go'(t) may be determined by finding the response of a filter having the frequency response function

//• (co)= [1 +V2j2co/B•+ (j2w/B•)• -• (23)

to the unit-amplitude square wave [a•(t)--a2(t)], which is the modulation envelope of the input. 8 If BsT>>2•r, the modulation envelope of the filter output go'(t) effec- tively reaches 4-1 just before the TDM switches inputs. Figure 3 shows the input go(t) and the output go'(t). Under these circumstances, only the step-function re-

a j. G. Truxal, Control System Synthesis (McGraw-Hill Book Co., New York, 1955) pp. 401-409.

T 2T I I

Fro. 3. Upper curve: Input signal to narrow-band tracking filter. Lower curve:Output signal of tracking filter. Modulation envelope shown as dashed line.

sponse, which is well known, is needed. The modulation envelope re(t) of go'(t) is

re(t)= -- 1+2V2 exp(--B•t/2V2) X cos[ (B•t/'2V2)-- 0r/4) ] (24)

and

go' (t)=m(t)G sincot.

Next, the average value of v4(t) is

(25)

lf0T T (26)

2 1 •.T

..•-K1G-• j I(t)lt. •r T 0 (27)

Equation 27 must be integrated in two parts since re(t)=0 at t= 2.87/B•. Thus

2 1 F ..2.87 (v4(t))•.•=-r•G--[ I

,r TLJ o

/Bs T

.87/Bs

The integration in Eq. 28 is straightforward and the result is

(v4(t))•v=(2/•r)(K•G)(1--2.11/B•T) (BsT>>2•r). (29) The ratio R is

R=l--2.11/B,T (B•T>>2,r). (30)

The results for R in Eqs. 22 and 30 are plotted in Fig. 4. Also shown are results of an experimental test with a TDM, a two-pole-derived filter having 50-Hz (314-rad/sec) bandwidth, and an average-responding ac voltmeter. It can be seen that the experimental results fall quite close to the theoretical approximations.

To keep the control voltage for the variable gain amplifier to at least 80% of the ideal value for the extreme case considered, B•T must be 10.6 or greater from Eq. 30.

To evaluate the amount of amplitude modulation on v2(t) caused by the time-varying components of v4(t), an extreme case identical to that in Sec. I is postulated for the inputs to the TDM; i.e., G•=G and G•=0 for n• 1. The analysis parallels that in Sec. I except that



13:48:33


FIG. 4:. Phase sen-

sitivity facto• R as a function of product of tracking-filter bandwidth and gating time. Theoreti- cal approximations shown as dashed lines.

20 50 I00

T is allowed to vary instead of being fixed at T= The voltages v4(t) and v2(t) are

2 G I N2T v4(t)---•Kx--- •r N ,r2 T r

and

732 • 2 C

sin-- sin t+a +... (31) N \NT

Xsincot 1-q-• sin-- sin t+a +. ß ß . (32) 7r•T• , N

From Eq. 32, the relative amplitude-modulation factor is

M= (N•rC/Tr•ørr) sin(7r/N) N• 1 (33)

•__NTC/TrTr for large N.

Rearranging Eq. 33 we have

Tv/T= (N2C/,r2M) sin(v/N). (34)

Equation 34 shows how Tv/T is a function of the number of TDM channels, compressor ratio, and relative modulation factor.

For a numerical example, it will be assumed that B,= 27r(20) rad/sec. Vibration testing frequencies may be as low as B,/2 or 2•r(10) rad/sec. For R=0.80, M=0.05, N=2, and C=40, Eqs. 30 and 34 yield T-0.084 sec and Tr= 27.3 sec. The above parameters are within the capabilities of commercial equipment, but the rather high value of Tv means that the sweep speed would have to be rather slow for the AGC circuit to work well at higher frequencies. A solution to this difficulty involves switching B8 to a larger value at a higher frequency of operation. At the same time it is possible to switch T and Tr to smaller values, to maintain the values of R and M, and also to maintain a higher frequency-sweep rate.

III. SPECTRAL DENSITY--RANDOM- VIBRATION TESTING

For random input signals, the TDM in Fig. 2 requires a gating time which is inversely proportional to

Fl. 5. Correlation function

ßmn (t) for rectangular gating functions.

,I

I

(rn-n)T

the minimum bandwidth of the peaks and notches in the spectral density functions of the input signals. If the gating time is made too small, spectral spreading occurs. The spectral density of the output signal of the TDM also becomes increasingly dependent upon the cross-spectral densities of the input signals. Since it is desired that the spectral density of the output be equal to the average of the input spectral densities, the latter effect is, of course, undesirable.

However, it is necessary to keep the gating time somewhat smaller than the averaging time of the analyzer, simply because one does not want the analyzer output to follow the spectral levels of the individual input signals. If the gating time is too large for an automatically equalized system, a "wow" in the test level may be aurally detected.

The analysis in Secs. III and IV allows the value of the gating time T to be specified, such that the spectral spreading and dependence upon cross-spectral terms is kept within definite limits. The signal at the output of the TDM is defined by Eqs. 1 and 2 in Sec. I. Changing the center term of the inequality from t to t¾-•, where /5 is a random variable with fiat distribution, makes go(t) stationary in the wide sense. The autocorrelation function of the output signal can easily be shown to be

N N

Ro (,-) = EEgo (t)go (t+ •)]= Z E R•. (•)+•. (•), (35) m----1 n=l

where

R•n,• (r) = E•gm (t)g,• (t-q- r)• (36) and

q•m,• (r) = E•a•n (t)a,• (tq- r) •. (37)

The correlation function •(r) for the rectangular pulse functions in Eq. 2 is the periodic triangular function shown in Fig. 5. This correlation function may be considered a weighting function of R• (r) in Eq. 35.

By performing a Fourier transformation on Ro(r) in Eq. 35, we obtain the spectral density function"of the output signal'

Rø(r)-•dr N2•=_•' 2• • /

X E E x• •- •(•-•)/• (38) m=l n=l •] • ' 9 W. B. Davenport and W. L. Root, Random Signals and Noise

(McGraw-Hill Book Co., New York, 1958) Chap. 6.



13:48:33

T. USHER, JR.

In Eq. 38, S,•,•(co) is the spectral density function associated with g•(t) and g,(t) and is the Fourier transform of

Using the symmetry properties m of S•,(w) we may rearrange Eq. 38 to obtain

So (c0) N 2y"Sk sin k=0 n =1 ( 2•rk• ( 2•rkh] x :v I 2,rk(m--n) S,•,• co+ +S,•n co- +2 5'. 5'. cos N T/ NT/ ,•=• •=•+• N

XRe{S.,•(coq_2,rkhq_S.,•(co 2•rkh 2•rk(m--n) [ / 2•'k\_s.,•(co_2•'k • , •, --NT/} q-sin ImlS"•wq-•) NT,}I) (39) N

where 8o=-} and 8k= 1 for k•O, and Re and Im denote "real part of" and "imaginary part of," respectively. Also, sincx- (simrx)/(•rx). The infinite summation over k in Eqs. 38 and 39 results from the Fourier components of

The interpretation of Eq. 39 lends some insight into the properties of the output signal of the TDM. Decreasing the gating time T is seen to cause spreading of the individual spectra as evidenced by the arguments of all the terms being either (coq-2•-k/NT) or (co--2•-k/NT). For sufficiently large values of T, however, spreading effects can be kept small.

Furthermore, Eq. 39 is broken up into groups o[ terms involving spectral density terms of each of the input signals (S•) and cross-spectral density terms (S•,•, m•n). The second and third group of terms contain contributions to So(w) from only the cross- spectral density functions of the input signals. If T is sufficiently large, these contributions should be negligible.

The identity found in Eq. 40 helps to confirm this supposition'

• ( •)" 2•rk(m-n) IN/2 (m=n) (40) Y'. 8• sinc cos -= •o N 0 (m • n).

The results of the identity are easily explained because the terms are proportional to the Fourier coefficients of the functions [•,•(r)+•,•(r)]/2, and to the sum of the values of the functions at r=0. As T--*m the set of terms in Eq. 39 involving Re{... } approaches zero because of the results of Eq. 40 for m•n. The set of terms involving Im{-..} approaches zero as T--•m because the arguments vanish and the coefficients form an absolutely convergent series. Thus,

1 N

lim So(cO)=-- Y'. S,,(cO), (41) T-->oo N

which is the desired result.

•0 j. H. Laning and R. H. Batin, Random Processes in Automatic Control (McGraw-Hill Book Co., New York, 1956) p. 135.


In order to examine quantitatively the amount of spectral spreading owing to the multiplexing operation, it is necessary to assume specific nonuniform spectral densities for the input signals, since spectral spreading for flat wide-band spectra is undetectable, except near the cutoff frequencies. This is easily seen, since the mean-square value of the output signal, •o •, is constant regardless of T and also has the value

o-•/= f_ So (cO)dcO. (42) In Sec. IV the effects of spectral spreading and

cross-spectral density terms are quantitatively evaluated for several specific narrow-band situations. Some experimental results are also presented.

IV. SPECTRAL SPREADING--CROSS- SPECTRAL DEPENDENCE

A. Case A

In order to separate the spectral-spreading effect from the dependence of So(cO) upon the cross-spectral terms of the input signals, input signals with S•(cO) =0(m•an) are postulated. This removes the corre- sponding terms from Eq. 39. Furthermore, it is assumed that each of the input signals have the rectangular spectrum

o (wo--B/2)<cO< (Woq-B/2)

elsewhere

n=l, 2, 3, ..., N. (43)

Equation 39 is easily evaluated for N=2 and for large values of N. For N= 2

oo

$o½)=«$½)+- z (2p-

X S co--[ q-S cO-• , (44) T T

where the spectra in the brackets are the shifted


13:48:33


counterparts of S(co). For large values of N, the summation over k may be replaced by an integral, and Eq. 39 reduces to

So(.)=

« (BT/2) (u+l)

S [ (sincx)2dx for J aX(BT/2)(u--1)

ii z (BT[2) (uq-1) S • (sincx)2dx (45)

« (BT/2) (--u+l) nu I (sincx)2dx for 0<u<l, where u=2(co--•o)/B. Note that So(cO) is symmetrical about co=COo. The expressions in Eqs. 44 and 45 have been evaluated and are plotted in Fig. 6 as functions of u for several values of BT.

(a) (b) (c)

i r• BT/2w:2 . .• , 1.0 , • • , '"• ' ' / l, BT/•2w=I BT/2rr=0.5 '\

•-•0 u=2 •

B

Fro. 6. Normalized spectral density of TDM output for rectangular-input spectral densities. Stepped curves for N=2; con- tinuous dashed curves for large N,

From Fig. 6, it can be seen that spectral spreading is not a function of N, but is simply dependent upon the BT product. Increasing the value of N merely smoothes the spectral-density function So(w). As BT decreases, not only does the effective width of the output spectral density increase, but the maximum value also decreases. For small values of BT [i.e., 2•r(0.5) and smaller•, the existence of auxiliary peaks at odd multiples of u= 2•/BT becomes evident.

B. Case B

Since the effect of the number of channels on the

rather nonrealistic rectangular spectral density of Case A is primarily a smoothing effect, it will be assumed that N=2 and that each channel input signal has the realistic spectral-density function

Sn(u)=S,.,.(u)=S(u)=S/(l+u•), (46)

and also thatS•,.(u)=S,.•(u)=O, as in Case A. Equation 46 is a good approximation for the spectral density due to a simple mechanical resonance. Equation 44 has been evaluated in conjunction with Eq. 46 and the results are shown in Fig. 7 for several values of BT. The

So(u) s

.2

.I

.05

'020 2 4 6 u: 2 øø-tø9

B

Fro. 7. Normalized spectral density for TDM output Sn (u) = 39•9• (•/,) =S/(I -]-•/fi), Si: (•/,) =391 ('/,/,) --0.

conclusions for Case B are essentially the same as those for Case A.

C. Case C

The effect of spectral spreading and the effect of cross-spectral terms in Eq. 39 are evaluated by examin- ing an extreme situation. It is assumed that N= 2 and g•(t)=--g2(t). Furthermore, the spectral densities are given by Eq. 46, but the cross-spectral densities are both real and are

S12 (/,g) -- S21 (u) = -- S (I//). (47)

For the above conditions, Eq. 39 becomes 4

Sc0,) =- 12 71-2

X Su+ +Su- . BT BT

The results using Eq. 46 with Eq. 48 arc shown for various values of BT in Fig. 8.

1.0

So(U•) • 12-• - - s • 0.5 _

.05 •_• '020 2 4 6

•=2•-•o B

FIG. 8. Normalized spectral density for TDM output 811(u) = s,.,. (u) = - s,: (u) = - s,• (u) = s/(• +u:).



13:48:33

T. USHER, JR.

BAND iNARROW AMR

SPECTRAL ',BAND DETECTORS: ANALYZER '•I_L_T__ _E•__-

FIG. 9. Block diagram of experimental test.

Note that the effects of depressing the peak value and widening of the spectral peak still occur as in Cases A and B for decreasing values of BT. However, these effects are more severe in Case C. Auxiliary peaks are easily noticeable for BT=2•r(0.5). As BT--->O, S(o•o)=0, which does not occur for Cases A and B, because the unshifted spectral term appearing in Eq. 44 is not present in Eq. 48. The disappearance of the S (u) term is related directly to the initial assumption that

= = - = -

D. Case D

An experiment was conducted in order to test the ideas presented in the previous cases, particularly Case C. A block diagram of the experimental apparatus, all of which is commercially available, is shown in Fig. 9. A random-noise generator fed one filter of a set of equalizer filters to provide a narrow-band signal. The signal and its inverted replica were fed to two channels of a TDM. The output of the TDM was then fed to a multichannel spectral analyzer containing 40 filters, each 50-Hz wide. The narrow-band filter at the input was also 50-Hz wide. The mean-square value of the output of each analyzer filter was estimated and assumed to be a measure of the average spectral density over each 50-Hz band.

The transfer functions of all the filters are approximated closely by

III(jw)12- { l+[-2(w-wx)/B-]4} -• w>_O, B= 2,r(50) (49)

where wx is the center frequency of the particular filter. The spectral density of the output of the TDM is again governed by Eq. 48, but the input spectral functions are

=S{l+[-2(w--wo)/B-]•} -•. (50)

The spectral estimate from an analyzer filter centered at o•=o•a is

2f0 I•(/•)l'•So

(51) where [//(jo•) I '2 is given by Eq. 49 where o•= wa.

So--•is• .Z':'-..• 5 x FIG. 10. Normalized

average spectral den- .I sity for TDM output.

Theoretical curves are solid. Experimental

.o5 points from setup in Fig. 9.

.02

'010 2 4 6 •q-O• o

Uo=2 B

Equation 51 has been evaluated and plotted in Fig. 10 as a function of 2 (w•--Wo)/B for several values of BT.

The experimental points are also shown on Fig. 10 but are constrained to even integer values of 2(o•--O•o)/B because the analyzer filters are separated by B rad/sec.

Although the tenet of keeping the analyzer bandwidth several times smaller than the bandwidth of the

spectral peaks to be measured has not been met, this objection has been overcome by the theoretical computation of the mean-square output of the analyzer filter. From Eq. 51, any spreading effects of the analyzer filter appear in the theoretical curves in Fig. 10. The agreement of the experimental points with theoretical results in Fig. 10 is good.

The phenomena noted for Case C are also noticeable in Fig. 10.

E. Commentary

It has been shown by a comparison of the results in Cases B and C that spectral spreading is worse when the input signals display negative correlation. Although spectral spreading can be examined separately, in a sense, by assuming that the input signals are uncorre- lated, the two effects cannot be separated for correlated signals. The cases presented in Cases C and D represent an extreme situation, which rarely, if ever, is en- countered in practice. If the condition S•(o•)=S•(o•) =-S•(o•)=--S•(o•) is satisfied, it usually holds only at one frequency and not over a band. Furthermore, Su(o•) and S.2•(o•) usually have slopes of opposite signs for the above condition. Also, for more than two input signals, it is extremely improbable that

N N

Z E

at any frequency. In the situation with two signals described above,

the spreading phenomenon completely overshadows any



13:48:33


possible cancellation phenomenon occurring in So(w) due to the presence of the proper cross-spectral terms.

From Fig. 7, for a maximum of 10% error of the actual spectral density So(w) from the true average spectral density (BT--•c) at frequencies reasonably near the peak, the BT product should be somewhat greater than 2•r(2). Since the bandwidth of the analyzer should always be less than the bandwidth of resonance peaks for good resolution, B may be interpreted as analyzer filter bandwidth to yield conservative results.

In order that the analyzer measurement of spectral density be m percent accurate with 95% confidence, it is necessary that n

2 (2•-/2BT•)•= m/100, (52)

where T• is the time-constant of the R-C averaging filter and B is the analyzer bandwidth in radians per sec. For m= 10%, Eq. 52 yields BT•= 2•r(200).

For manually equalized systems, the product BNT should be somewhat less than BT• to assure that signal level fluctuations due to multiplexing are sufficiently smoothed. A reasonably wide range of parameter values allow BT> 2•r(2) and BNT<2•'(200) simultaneously.

For automatically equalized systems with compressor ratio C, the requirements are more stringent. The response time of the compressor circuit is approximately

11 C. T. MorroaT, J. Acoust. Soc. Am. 30, 456 (1958).

T•/C, so that BNT should be somewhat less than BT•/C to assure that the controlled output will not have its level fluctuating with period NT. For C= 20, and the same analyzer parameters as before, we have BNT<2•'(10). This and BT>2•r(2) may be satisfied if N is not too large.

V. CONCLUSIONS

The analyses in the foregoing Sections have shown that reasonably accurate control of swept-sinusoidal- vibration tests to the average of several acceleration amplitudes may be had with the use of a special- purpose time-division multiplexer. Harmonic and noise suppression for N channels can also be obtained with the economical addition of one tracking filter for the swept-sinusoidal test. However, proper selection of system-parameter values is necessary and slight de- gradation of performance over the situation with no tracking filter is to be expected.

Second, the utility of the time-division rnultiplexer is further demonstrated by its capability to provide a control signal for which the spectral density is essentially the average of the spectral density of several random-input signals. Again, proper choice of system- parameter values allows accurate random-vibration tests with manual or automatic control of the average of several acceleration spectral densities to be performed.



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Average Control for Sinusoidal- and Random-Vibration Testing