~ I

Berliner and Riiter also used a resonance method . Astin developed and u sed a method employingyoltage resonance tha t was subsequen tly adopted by Akel'li:i f. The values repor ted by H arrington were obtained by means of a bridge metbod a t 1 me, and it is uncertain what method Kistler used at 2.1 me. The measure­ments of Furth (400 me) and others [19 to 27] were made at very high frequencies, so tb e low values re­ported are undoubtedly a ttributable to anomalous dispersion and are no t comparable to static values.

Consideration of the da ta available on sta tic values sugges ts that th e major descrepancies in th e values reported are in large par t due to high con­du ctivity as, in general, it appears tha t precautions h ave not b een tak:en to obtain and maintain low conductivi ties for these solutions. To a lesser exten t it is also probabl e that frequ ency-dependen t errors ar e associa ted wi th many of th ese values. The exten t to whi ch th ese fac tors may modify the suitability of a m ethod of m easurem ent is exemplified by Kniekamp 's [9] tudy and H a rtshol'll 's analysis [30] of resonance methods.

VI. References [1) E. Berlin er and R. R uter , K olloid- Z. 47, 251 (1929). (2) R. F ur t h, Ann . Physik. 70, 63 (1923). (3) G. Kreinina, J . E xptl. Theoret. Ph ys. (USSR) 15, 208

(1945). (4) G. Akerliif, J . Am. Che m. Soc. 54, 4125 (1932).

Journal of Research of the National Bureau of Standards

(5) A. V. Astin , Phys. Rev. 3(, 300 (1929) . (6) E. Ber li ner a nd R . Ruter , I(oll oid- Z. 47, 251 (1929). (7) E. H . I-[a rrin gto n, Phys. R ev. 8, 581 (1916). (8) . S. Ki stler, J . Phys. Chern. 35, 815 (1931). (9) H . Knieka mp, Z. Physik 51, 95 (1928).

(10) L. K ockel, Ann . Ph ysik 77, 417 (1925) . (11) E. La nd t, Cent ro Zu ckerind . 44, 723 (1936) . (1 2) R. T . Lattey, Phil. Mag. n , 829 (1921). (13) R. T . Lattey, O. Gatty, and 'N. G. D avies, Phil. Mag. 12,

1019 (1 931). (14) H . Poss ner , Ann . P hysik 6, 815 (1930). (1 5) G. Scatchard , J . Am. Chern . Soc. 48, 2026 (1926). (16) C. Schreck, Physik. Z. 37, 156 (1936). (17) Ie E. Slevogt, Ann . Physik . 36,141 (1939). (18) P. 'Vaiden a nd O. Wern er, Z. physik. Chem. 129, 405

(1928) . [19) M . Von Ard enene, O. Groos, and G. Otterbein , Ph ysik .

Z. 37, 533 (1936). (20) VV. D ahms, Ann . Physik 26,177 (1936). (21) O. Dobenecker , Ann . Physik 17,699 (1933). (22) P . Drude, Z. physik. Chern . 23, 267 (1897). (23) A. Duebner, Ann . Physik 84, 429 (1927). [24) R. F ur t h, Ann . Physik 70, 63 (1 923). (25) R. K ell er, K olloid- Z. 29, 193 (1921). (26) O. Kreinina, J . Exptl. Theoret. Phys. (USSR ) 15, 208

(1 945) . . (27) P. Wenk, Ann . P hysik 17, 679 (1933). (28) A. A. Maryott, J . Resear ch N BS 38, 527 (1947) RP1794. (29) J . Wy man , Phys. Rev. 35, 623 (1930). (30) L . H a rtshorn , R adio frequency measuremen ts by bridge

and r esonance m ethod. (J ohn Wiley & Sons, Inc., New York, N. Y. , 194m.

W AS H INCTON, April 13, 1950

Vol. 45, No.4, October 1950 Research Paper 2138

Response of Accelerometers to Transient Accelerations By Samuel Levy and Wilhelmina D. Kroll

Cur ves an d tables a re shown for the response of acce lerometers to t l'ans ient exc it ing accele rations. Three t .vpes of accelerat ion- t im e re la t ions a re co nsid ered . \ \Then plotted , t hey have square, t ria ngu la r, a nd half-sin e-wave shapes. The natura l pe riods of t he accel­e rometers f OI' which t he co mpu ta t ions were made were approx im ately one, o ne-t hird , and one-fi fth of t he durat ion of t he accelerat ion pulse. The da mpin g coeffi cients of t he acce l­e rometers were 0, 0.4, 0.7, and 1.0 t imes t he cri t ical valu es. It is indicated t hat, to obtain a n acc uracy of bet ter t ha n 5 pe rcent of the peak accelerat ion in measurin g accelerat ion pulses hav ing t he genera l cha racteristics of t he t riangula r or s inuso idal pulses, a n accele romete r must have a na t ural period of about one-third t he durat ion of t he a cce le rat io n pulse, a nd a da mpin g constant of a bou t 0.4 to 0.7 of t he critical valu e.

I. Introduction

Accelerometers are widely used to measure os­cilla tory and transien t vibrations.

The fideli ty with which th ese instrumen ts respond in the case of oscilla to ry stimuli has been thoroughly studied [1 , pp . 61 to 70] . It is found that , when the damping is between 0. 6 and 0.7 of the critical value and th e natural pcriod of the accelerometer is less than about half of th e period of. the applied accelera­t ion, the accuracy is satisfactory.

In th e case of exciLation of th e accelerom eter by a transien t vibration, only scattered information is available regarding th e reliabili ty of Lh e response ob tained . IV eiss [2] gives th e response to a tri-

angular pulse of accelera, tion for an accelerometer whose natural period is 0.3 th e duration of th e pulse and whose damping is 0, 0.3, and 0.7 of th e cri tical value. H e also gives the response to a suddenly applied constan t acceleration for accelerometers wi th a damping ra tio of 0, 0.3, 0.7, and 1.0 times the cri tical value. IV elch [3] has d etermined , on th e VV' es tinghouse transien t analyzer , th e response to several kinds of impulses of a 50 cis single-degr ee-of­freedom shock m easuring instrumen t h aving various amounts of damping. On th e basis of these scat tered data, and information for undamp ed accelerometer derived by Franldand [4], Bio t and Bisplingboff [5], and oth ers, i t has b een common practice to assume that an accelerometer will be acceptable in a given

303

applicatio~ i~ its damping i.s O.? to 0.7 of the critical value and If Its natural penod IS less than about half of the duration of the acceleration pulse.

The curves presented in this report were computed to obtain more systematic information !'egarding ~he accuracy of damped accelerometers m measurmg transient phenomena.

II. Theory

The usual accelerometer is a single-degree-of-free­dom mechanical system. Such a system is shown in figure 1. Means are provided to indicate the relative motion x of the internal mass with respect to the frame. This relative motion is taken as a measure of the acceleration, d2y/dt2, of the frame. . The equation of motion for the mass m, figure 1, IS

d2z (d z d y ) m dt2 +c Tt - dt +k(z- y) = O. (1)

With (2)

eq 1 becomes

We wish to k no w hOlY faithfully the response x of the accelerometer reproduces the time history of the applied acceleration d2y/dt2 for pulses of accel~ration of finite duration and arbitrary shape. To gIve the analysis a wider range of useflflness, eq 4 is. written in dimensionless form by makmg the followmg sub­stitutions:

where

=(d 2Y)j(d 2y ) a dt2 dt2 ma x

r = t/T

( d2y ) ~=-kx/m -2 dt ma x

D= c/2..jmk= c/cc

R = 21T'..jm/k/T

(5)

cc= 2 mk, critical value of damping co­efficient,

T = duration of acceleration pulse to be measured,

2--;;'..jm/k= "undamped" period of accelerometer ,

(dd2~) = peak value of acceleration. t max

Substituting eq 5 into eq 4 gives

(6)

For a relatively high frequency accelerometer, R is a small number. Under these circumstances, th0 first two terms in eq 6 become negligible, and the dimensionless response ~ is equal to the dimension­less acceleration a. As R becomes larger, the first and second terms start to have an effect. The primary effect of the second term is to introduc~ a time lag between the response ~ and the acceleratlOn a. The primary effect of the first term is to tend to make the response ~ oscillate in value above and below the value of the acceleration a.

III. Results

Equation 6, giving the relation between the di·· mensionless r esponses ~ and the dimensionless ac­celeration a" was integrated numerically for three values of the natural period ratio having approxi­mately the values, R = l , 1/3, 1/5; for four values of the damping ratio, D = O, 0.4, 0.7, 1.0 ; and for the tlu'ee time-histories of acceleration pulse shown in figure 2. Numerical integra tion, instead of direct integration, was used to give results that could be plotted directly. Small variations f!'om t~e nominal values of R were used for convernence m computing. These values of R are given in table l. A spot check of the results was made using the analytical solution of eq 6.

The numerical integration was carried out using a time increment of 1/ (2 01T') times the natural period of the accelerometer. Eight decimal figures were used in the computation .

The results are plo t ted in figures 3 to 11. Figures 3 4 and 5 give the r esponse to a sinusoidal pulse of a~c~leration. Figure 3 gives the r esponse when the natural period is about equal to the duration of the acceleration pulse. Figures 4 and 5 give similar resul ts with the natmal period about one-third and one-fifth , respectively, of the dur~tion <;Jf the accele~'a­tion pulse. In each figure , the dnnenslOnless applied acceleration, a, is shown by a do tted line; the r e­sponse, ~, with the damping .ra tio D = 0 by curve 1; with D = 0.4 by curve 2; WIth D= 0.7 by cw've 3; and with D= 1.0 by curve 4.

Figures 6, 7, and 8 show the r esponse to a tri­angular pulse of acceleration, and figw'es 9, 10, and 11 show the response to a r ectangular pulse. In each figure the set of curves brings out the effect of vary­ing o~ly the damping ratio D .

IV. Discussion

It is eviden t from an inspection of the figures that for non e of the accelerometers considered does the time history of the dimensionless response ~ coincide with the time history of the dimensionless accelera-

304

I, .)

r

l :r

>

TABLE 1. -Error rOT accelerometers and acceleration pulses considered

(1) (2) (3) I (4) I (5) (6) (7) I (8)

-- - - --EITor Plottcd

R D

~m,,-a""' 1 {;.T

I€-al .. u I After shift Fig. / Cur ve I€-al"", ure

IlALF·SlNE· \V A VE PULSE

1. 014 0 0.74 1.36 1. 36 0. 17 3 1 1. 014 0.4 .15 0.62 0.24 .22 3 2 1.014 .7 -.05 .67 . 18 .25 3 3 1. 014 1.0 -. 18 .65 .24 .29 3 4 0.338 0 .17 .35 .35 .07 4 1

.338 0.4 .02 .20 . 06 .05 4 2

.338 .7 .00 .24 .06 .07 4 3

.338 1.0 -.03 .32 . 10 . 11 4 4

.203 0 . 10 .20 .20 . 01 5 1

.203 0. 4 . 07 .12 .05 .02 5 2

.203 .7 .00 . 15 . 03 . 04 5 3

.203 1.0 -.01 .20 . 06 .06 5 4

T RrA NGULAR PULSE

1.014 0 0.51 1. 29 1.29 0.15 6 1 1. 014 0. 4 .00 0.56 0.21 .21 6 2 1.014 .7 -.20 .51 .20 .22 6 3 1. 014 1.0 -.32 .. 57 .32 .25 6 4 0.338 0 . 17 .43 .43 .04 7 1

.338 0.4 .00 .18 .06 .07 7 2

.338 . 7 - .06 . 17 .06 .08 7 3

.338 1.0 -. 11 .22 . 12 .08 7 4

.203 0 .10 .25 .25 .01 8 1

.203 0.4 .00 . 11 .03 .04 8 2

.203 .7 -.04 . 10 . 04 .05 8 3

.203 1.0 -.07 .13 . 07 .07 8 4

SQUARE P ULS E

1.014 0 1.00 1.00 1. 00 0.04 9 1 1.014 0.4 0.25 1. 00 0. 49 . 19 9 2 1. 014 .7 .05 1. 00 .53 .22 9 3 1. 014 1.0 -.01 1. 00 .50 .28 9 4 0.334 0 1. 00 I. 00 I. 00 . 01 10 1 .334 0.4 0.25 1. 00 0.50 . 06 10 2 .334 .7 . 05 1.00 .50 . 08 10 3 .334 1.0 .00 1. 00 .50 . 10 ]0 4 . 203 0 1.00 1. 00 1. 00 . 02 11 1 . 203 0.4 0. 25 1. 00 0.50 . 03 11 2 . 203 . 7 .05 1. 00 .50 . O~ 11 3 . 203 1.0 .00 1. 00 .50 . 06 11 4

j tion a. In Inany ea CS , h.o\vcvel' , tIle coincidence can be markedly improvcd by considcring the responsc CUl'ves to be shifted a small d istance to the left. They can also be improved, in those cascs where oscillatory response is presen t, by fairing a line t hrough the oscillatory response. Both of these

l, methods of record improvemcnt are commonly em­( ployed.

The errors of the various accelerometcrs for the acceleration pulses considered are given in table 1. In COlUIilllS 1 and 2, respectively , are given th e accelerometer characteris tics: H, ratio of na tural period to pulse dura tion ; and D, mtio of damping constant to cri tical value.

In column 3, table 1, is given the difference be­tween th e maximum value of dimensionless response, ~, and the maximum value of the dimensionless ap­plied acceleraLion , a. The en ol' varies from 0 to 100 percent.

In column 4, table 1, is given the largest absolu to value of the difference ~-a where ~ and a are evalu­aLed at the same dimens ionless time. The erro l' vn,ri cs from a minimum of 10 percent to a maximum of 136 percent.

In column 5, table 1, is given the largest absolute value of the difference ~- a after shifting the ~ CLl1've to the left by the amolm t t:.T given in column 6. The crror in this case is typical of the usual way of interpreting accelerometer records. This error varies from a minimum of 3 perccnt to a maximum of 136 percent. If only accelerometers with damping are considered (D > O), the largest error is 32 pCl'cenL when the accelerometer is subjected to accelerat on pulses of tr iangular or sinusoidal time his t01·ies.

On the basis of the few cases investigated, an optimum value of damping is indicated to be between 0.4 and 0.7 of the cri tical value. It is also indicated that, to obtain an accmacy of better than 5 percen t of the peak acceleration in m casming acceleration pulses having the general characteristics of the tri­angular or sinusoidal pulses, an accelerometer must have a natUl'al period of les than abo ut one-third the duration of the acceleration pulse.

Acknowledgment is due to the BUl'eftu of Aero­nautics, Navy Department, whose research pro­jects on vibration pickups have provided the impeLu for the work presented in this paper. The au thors also extend thanks to L. W. Roberson and I. Smith for assis tance in computing the many response curves and preparing the figure and table.

c

FIG URE: 1. Single-degl'ee-of-freedom system representing accelerometel·.

Displacement o[ frame is y, displacement of internal mass is z, relative displace. ment of internal mass with ['espect to frame is X= z- y.

305

1"-

1.0 /---

/ "- (0 )

/ \ / \

x .5 / \ 0 / \ E

/ \

~ \_---0-0 0

~ 1.0 1\ ( b) N~ / \ -0-0

/ \ II / \ 0 .5 / \ 0 / \

+- / \ d / L ~---0

c 0

+- 1.0 -----, d (C) L I <II

Q) I u r u .5

<t: I I I

0 1_-0 .5 1.0 I.S

Tim e rat io , L = tiT FIG U RE 2. Pulses of acceleration for which integration was

carried out: (a) half-sine-wave pulse, (b) triangular pulse, ( c) square pulse.

2.0,..------,--------,-------,

I.S

"" 0

+-d L

1.0 Q) / If)

c / 0 n. f If) Q)

/ L

'D .5 / c d I

/ 0

0 0 +-a L

c 0

+- - .5 a L <lI

Q)

<.J V

<t:

-1.0

-1.5 ~-----'------'-----~ o .S 1.0 1.5

Time ratio, l'

FIGU RE 3. Response to a half-sine-wave pulse of acceleration, dashed curve, of an ac.celerometer whose natural period i s about equal to the duratwn of the pulse, R = 1.014.

Curve (1), damping coefficient zero, D = O; cur:,c (2) , d amping coe mcient O~4 of tbe critical, D = O.4 ; curve (a), dampmg coe(ficlCnt 0.7 or the CrItICal, D =O. I; curve (4) , damping coeffici ent equal to the critical, D = l.O.

306

-------~--- -- ~-

\ I

1

)

I ~

,

~

l 1.5

"" 0

+-0 '-

(l) if)

c 0 0-if) Q)

\...

-0 C a

.5

" <5

+-

" '-

c 0 Q

+-" '-Q)

QJ v u «

-. 5 0 .5 1.0 1.5

1ime ratio . T

FIG U RE 4,. Response to a haIJ-sine-wave pulse oj acceleration, dashed em've, oj an accelerometer whose nat1,rai period is about equal to one-third of the duration of the pulse, R = 0.338.

Curve (1), d amping coemcicnt zero, D = O; cun 'C (2), damping coemcicnt 0.4 of the clitical, D =0.4; curve (3), damping cocmcicn l 0.7 of thc criLical , J) = 0.7;

, curve (4), damping coefli cient eq ual to the criLical, D = l.O. ,

J.

"" 0 1. 5

+= 0 "-

OJ if)

c 0 D-

'" OJ "-

-0 C 0

..... 0

+= a L

c 0

+-a \...

'" Q)

u u «

-.5 0 .5 1.0 1.5

Time ratio, T

F I CUUE 5. Response to a halJ-sine-wave p1dse of acceleration, dashed CUI've, oj an accelerometer whose natural period is about equal to one-fifth oj the duration oj the pulse, ] = 0.203.

Curve (1) , damping coemeicnt zero, D =O; curve (2), damping cocflicicnt 0.4 of t ile eril ical, D = O.4; curve (3), damping cociflcicn t 0.7 of thc critical, D =0.7; curve (4), d amping coe rneien! equal to the critical, D =l.O.

oJ'

0 +-a 1.0 \...

Q) If)

c 0 0.. if)

Q) .5 I-

u C 0

0 0

0 +-0 l-

e 0

- .5 +-0 L Q)

Q)

u u «

- 1.0

- 1. 5 L........ ____ .....L _____ L-____ ~

o .5 1. 0 1. 5

T im e rati o, T

FrcuuE 6. Response to a triangular p1dse of acceleration, dashed CUTve, of an accelerometer whose natural period is about equal to the duration oj the pulse, R=1 .014.

Ourve (I), dam ping coe rn cient zero, D =O; curvc (2), damping cocmcient 0.4 of the critical, D = O.4 ; curve (~), dampi ng coeiflcient 0.7 of the critical, D =0.7; curvc (4), dampin g coc rncicnt equal to tllc critical, D =l.O.

307

l

r I

W\ 1.5

o· += d L

C\! 0/)

C 0 0.. 0/) QI ....

U c 0

.5 Cl

o' +-Cl !....

C 0 0 +-Cl L ())

C\! V u «

- .5 a .5 1. 0 I.S

Tim e rat io ,e

F I Gl::Rl, 7. Response to a triangulal' pulse of acceleration, dashed curve, of an accelerometer whose natural period is about equal to one-third of the duration of the pldse, R = 0.338.

Curve (I), damping coefficien t zero, D =O; curve (2) , damping coefficient 0.4 of the critical, D =O.4; curve (3), dampi ng coefficient 0.7 of the critical, D =0.7; curve (4), damping coefficient equal to tbe critical, D =l.O.

w-

ei 1.5 +-0 L

C\! 0/)

C 0 0.. III

'" L

-0 C Cl

Cl .5

ci

W"\

0

+-d L

C\! I/l C 0 0..

'" C\! L

-0 C Cl

.; o

+= Cl L...

c o

2.0

1.5

1.0

.S

~ O~----------L-----------~------~--~ L

'" '"

~.

~ ~ «

-.5 L-____ -L. _____ L.... ____ ....J

o .5 1.0 1.5

Time ratio, l

FIGURE 9. Response to a square pulse of acceleration, dashed curve, of an accelerorneter whose natural period is about equal K to the duration of the pulse, R = 1.014. -

~ Curve (1 ), damping coefficient zero, D=O; curve (2), damping coefficient 0.4 L of tbe critical, D =O.4 ; curve (3), damping coeffi cient 0.7 of the critical, D = 0.7;

C Q O IL-----L-----~~+_I_-+_l

..... o L

'" '" o o « -.5 L-____ .l..-____ .l..-___ --l

o .5 1. 0 1.5

Time ratio. 'r

FIa uRE 8. Response to a triangular pulse of acceleration, dash curve, of an accelerorneter whose natural period is about equal to one-fifth of the duration of the pulse, R = 0.203.

, Curve (I), damping coe fficient zero, D =O; curve (2), damping coefficient 0.4 of tbe criti cal, D =O.4; cur ve (3), dampi ng coofficient 0.7 of the critical, D =0.7; curve (4), damping coefficieut equal to the critical, D = l .0.

curve (4), damping coefficient equal to tbe critical, D =l.O.

308

~ j

2 .0

oJ'

'2 1.5 +- (I ) 0 L

Qi

'" C 0 0-V)

QJ 1.0 L

-0 c d

d

0' .5

d I....

e 0

+-0 0 L QJ

Q) u u <t

-.5 0 .5 1. 0 1. 5

Time r atio, T

FI GUR E 10_ Response to a square pulse of acceleration, dashed curve, of an accelerometer whose natural period i s about lequal to one-thi rd of the duration of the pulse, R = O.334.

C ur ve (1) , damping cocmcie n t zcro, D =O; cur ve (2), dampi ng coefftcicn t 0.4 of the critical, D = O.4 ; curve (3) , damping coeffi cien t 0.7 of t he cri t ical, D = 0.7; curve (4), damping coeffi cient cellial to t be cri ti cal, J) = 1.0.

V. References [1] J . P . D en H artog, M echanical vibrations (McG raw-Hill

Book Co., I nc. , New York, N . Y ., 1940). [2] D . E . Wei s, D esign and a pplication of accelerometers,

Proc. SE A, vol. I V, no . II, p. 89 (1947) . [3] W. P. Welch, A proposed new shock measuring instru­

m ent, Proe. SEi:iA, vol. V, no. I , p . 39 (1947).

2 0 ~ -- -- -- --(I)

J\

0 .;:: d 1. 5 -L

OJ if)

C 0 (\ (2)

0-V)

OJ L 9)

1.0

-0 C 0

1 K; -

II I d I - (4)

' (4) 0 .5

+- n (\ -0 L

e 0

+= d L 0 QJ

Q)

u u

<t

(~

\, t"'.

~) - .5 I 1\

(I ) \

o .5 1.0 1.5

Time rat io, T

F I GU R E 11. Response to a square pulse of acceleration , dashed curve, of an accelerometer whose natural period is about equal to one-f ifth of the duration of the pulse, R = O. 203.

Curve (1), cl am ping coo lJi cient ze ro , D = O; curve (2), dam ping coeffici en t 0.4 of tbe critical, D = O.4 ; curve (~ ), damping coe fficient 0.7 of t he critical, D = O.7; curve (,I), dam ping cocfli cien t eq ual to the critica l, D = J. O.

[4] J . M . Franklan d , Effects of impact on simple clastic struct ures, Taylor M odcl Basin R cp. 481 (Apri l 1942).

[5] M . A. Biot an d R. L . Bisplingh off, D ynamic loads on ai rplane st r uct urcs d uring landing, rACA ARR No. 4H lO (Oct. 1944).

W ASHI N G'rO N , May 11 , 1950.

309

l