I
\

VI:
,l; H

“g , y I! w H'
‘; .Hi \

I
7

l

M

I

II

*I

122
293
THS

 

DETERMINATION OF YOUNG'S
MODULUS BY THE SONIC METHOD

Theain fer the Degree of B. S.
MIGHICAN STATE COLLEGE

Norman Precoda
I941

‘ ... VW.dAF.¢-¢dw3.‘ w t.’mul

III to"

 

 

.
s n . . v I
I
.
. . t
-
V I o
_
‘ - -
.
.\s.$\,)$. _ n . u 0
‘ . .
.
I o
. . .
. .
~ .
.
.
v u l
a
l u
. v
(.2
. 4 .
\
Q
I . \ 74 '
p
. l
. A
.
. .
.
l d '
n§
.
r
.
H .
-. ..
. .
‘p I r
. .
§ v
‘
. .
. .
,
L
.
. .
. fl
_
l
I §
.
. I
I I
I a
~
“ / I \
. \
a r I...
f .
_
(-
1|
.
i O
I
'
’ . 1.
| V

 

           

 

_. ‘
5A In... aflhlj H
4. d, ..V . . I.
‘ , a voJCN..l$\ .
.v : Ffiwmfi.
. ..\‘ am 1.4 d 3
I u f. A \ I. ,.t .J. .
‘ .w‘ l5,
. <- h‘ ‘1 ‘ I- > ‘I .
. vr I . A» . . .. \. 9 . v .
aI' I
n‘ l u l ( \
c .0! ‘2 . .v\ a 4‘ ox AM: o I
v
'I D
K .o

I.
. .‘Dlaqv

‘-

Detxxnhjnatjxuw of anumj'e ;IVH1]HS

By tme Sonic hethOd

A Thesis bubmitted to

The reculty of
$ “1.011 .Lkgrnil- Lfi‘L'A'L'E U OLLEUE
of

AGRIUULTURB AVD AFFLIED SCIENCE

norflen Precofia

dendidate for the Degree of

DUChOlOT of Science

dune 1941

(THESIC

11511473 01“ C 0:: Shift” Q

PLLi‘ -rJJ

I INTRODUCTIOI‘I . . . . . . . . . . . . . . . l
fiecflground — general rroble» of
vibration - Specific groblet as
covered in tnis paper

11 DISCUSSION OF USE ULw,uu . . . . . . . . . 5
Results - possibilities

.LII J’Ll)PJI;.Rr‘.TUfj .ILI‘TD 13p ‘.C)()IJi IDIJ H i118 o o o o o o o o o 8

A-Derivation of working forwula . . . . . lO
B-Correotions for rotetory and lateral
inertia . . . . . . . . . . . . . . . 18
C—Flernrel vibration RP7WW retue . . . . . . 9
D-Lonqitudihal Vibration apparatus . . . . £4
E-Dynemic teeqs by weens of Infinoed
Vibrations . . . . . . . . . . . . . . . ?9

IN] C 1'" ‘3 IJY :3 I 0:” o o o o o o o o o o o o o o o o I‘) :1)

bIj-3LI (1'RI‘1PIT—Y- o} o o o o o o o o o o o o o o o o :16

136176

DJB‘I‘QiRTV‘IFU’UF‘IW‘T Oi.“ YULJ‘I'VPS TWDULUS 7”! ‘t."‘Fi SUTIJ Ja-uuh'hUU

The method of tefiting veteriele by vibrating
that 19' 451.1th ”7317.. PT"'“h-fl§‘iy "‘11,", of tjnr‘. '3th (3,0,1; (an
e313 olesa'is than: of iflie “UftfiljfZ ourwvmdoy, (joingztjgrp
StTUC“ on TWO tEle 0? FR? tn prove their fianuippgg
by the rietflb: tone given off. (Porhers ~ore en in

a

tiwee beet rather then in these oeye or the r H 1,

end the T-men) Flower? of tine 91399 tented their
weree to Cetect YlFWP by the ohrreetorietic sonnoe
DT“CUJCV by the good find the bro erocwwene. ’rhe vib-

Y]
.1.

etiozo of tre reeo, the firum, the rovirq colurn of

.5.

FIT in a Line —— thus Whether ren“fini7ed or not, “he

1—].
LJ.

' ‘ ' "' h I" . 1‘ ~ fit a” f‘ ‘7 . 1"? ‘ o .
Oh?1%MTBPT1CWJ'C v rwurtion ri‘cm'toby the; CL¢;?U.EVV e

pFrt in mor'e lives Farce ti~e ir~wrn"riei.

Very likaiy ”to natural frornercy of vibration
of e FlHQhRH strinfi “an the first to be studied. It
wee reietiveiy easy to Pee the lonjitneinfl “eves *et
up in a flexible core. From this, the next ster Wse
iflio °t7Njg of tbr~‘ribr€tfi<fii of p :Vwi hr rfiéfi(i oofiy,
Of Tflws eevvrel ryfifivv‘e if helfl'r¢::7 ririd_3vw§;, ire
three meet com on ere: (F) the fifeo—frefi «er or
wheqwi ore 9rwi iq fiWHiit-in eflz the fi'flfiN‘Tt end Tfiwé otb“1‘
jg; Tiwee. (13) Tfiie 1353‘ lWIiJJt-iii at; twittw err“?. ((3) T110

”r b3? Tree at Y“th PPflF. The free-frvfi

yer it; one "wt;rily ed :ILy eurnwthoo et.11=e ror:]-

f‘;

is that irxiflue oese of a String, vibrations ere
mainteineo by tension er errelly errliefl' Whereas in
the r06, they ere mainte red av tbe electioity of tne
rod 5,8“lf. Due to the varieti"n in the forces wrought
into eotion, there is 9 wide vorietion in the kind

and the Successive rates of vibration. uenerelly

‘the sounder, veneer body Will beve the hifiher refie

of fundamental frequency of vibretion.

Sir Uberdxw3‘fiheotetmuwe,xflith the gi(1.gy hjs
neleidornone, studied the eotion or a vibrating tired-
lree ber. A brignt Silvereo beeo attached to the free
erml (ii‘ a. Imyd. tits i1 Lu 'j11813e(1 VIitli E: tweeil (3f -LjfiThll.
When the roo vioretem, the beeo ewowed es en inten—
selgr brfi.ulierfl; symrt eiri‘xassesesjl f oee11. UHUlS 3i; VHS
possible to record toe fireet variety ‘md beauty of
fflle (weeizrnr: pI*“dvu3efl hy"thré *rvvirig lfieetl. lie iffliNld
tf:at; ttie 'rrmi \rjlrreiief; Sigwtulliget7Prn33J;y '3n tvn: {wfi‘ffihll—
dieulor airoonions and the curves ;roouced uy the

"I

quhtoc bead one to the COJPlfithFNS o7 the two rect—

SCHFCHGTS 10? Knowledfie, Uflxa‘vibrari“n of E boby
the iJPCTtFan of the erteots of those vibrations on

t; CriI‘ i;alie<7izatge SJIP17011P(llllgfi? 7,94352762 G179 fllfl

y...

"\

oviofnt Thlfi 19 in generfil, tpe esseoce of tbe prob—

‘or tflw t rope-Ila: HTjGT (Memoritt—

leg or vimwnrtion end

}.L

ion (H ;;ethoo
(*-7) has ween

‘
Hear y e
- . A L

{it Fly Bike "'71
8. Area. BQIIQLZLL

toy or
of the general

C?
k

\

A

ejiric
~l testinm am
2“. 1’0 C 53(- 3.1
It will

I. O“ Lilli F7

F, 7.9 «‘ TTC F 1 O
1.9Lmlr .L lirov11eII
3ND OwLFlEy 2:
11" t-‘1W'MJ- l" L 191';

Joe:lne :I Alf?
-~ w- . ~:
J_I ' '11 .0, :41 LL‘I kIEC'

.L‘Illin FltTi‘fi'J'Vl-i

IMDESJ. ycirrts

”‘1 "To rm n n r
1.3.0519 1 (?\.,]>7»7]‘7\7_J:’.

JléwIlz‘e

.roiiLiorI

r—v

.. 3-

Il ;U Bfilfifi LS

of

’Jlli C1. 170 F

.; 7- - F11
(_I_ (V. 1:1: I L].

0]

gr!" LLLE‘II

." _.. 1, ' .
0L IAllP T
T‘IJ‘J_S ::o
be Ft ”n

O

in

Cortirxrule

internel

L1 .L p l

i. S a i

V1 Lhi‘c'l

»"ow it

<91 l '1. TFWF’I
‘./O‘ L'“ he ("0

1 T: 1 (‘TI

be melonlet
MP ireo E7 y

c (' , 1" V ,.
(. ’]:L l 130 ' 9..

,<.|.

5‘ T‘Im
m‘ Man
O k
0L I

'n~ -. . 7 “
I I”? F‘HIFTIIC ‘24’

(I

Cot*rm

l\.

I") }

/..‘
- r1 4-
- n. L

777‘: 3.

too

I 0 j
2777‘"! 7",” VPI‘l-iéf‘
. .7 . , . ) \ I .

D ,0 I'll VT 5 TT‘

’3

': ‘lr‘bw(\1)

‘
(-11]

(.1
{I

I“
t; n

no

'I‘"
-J

goreible to reach a corresyonoing resonant freouonoy
irI tlma sgxeoixvorl. LLf flirttu2r, :3 dwrvitna is: attm7obfiwl
to Cheoiwer (oreferJIlv rt the ends Piroe the U'Tllt-

of emrlifying vibfetione produced -— then it "ill be
roseible to select the fnrflemehtel natural freemangy
of the ereoiwon, since wt reconenoe the ewtlitufie of

vibration Will be e merimnw.

 

 

AMPUTUDE VAR/ABLE FREQ -
METER L UE/vcy DRIVER

A it?

lafiflAl SFEIMNflSA/

7A” 774;”-

' l
‘0. £24 Lfi #0234 L—u
~ 1.

 

 

 

 

 

 

 

 

 

 

 

 

 

DISUU‘S IO? 0? UEWFVLUEBS

Experi.ents Co.tletefi have shOWed the feasibil-

jty of oel_oul'tin? E by Vibretions Witho t toe slow

3

I)

suress—strEin weesurenents. Values of b
quite 010991? with those flbtfiired ry static methofis

ENJ.VWBT€ shrwui to [we Felifflllé. Ewmue of tfius_weterjl.

I
F...
C')

which rsve been tested successfully by use sonic
Ilelliod.z3re txile, sletmz, tUfiiCK5 .leaul, etvuiiteeiiall. borixi,
pert rfir17xer, :roo::.:snrm.lt, aisnhzfilt twistixz, slurrtz

bars, Steel, niosle, and rm lrle alloys. (*—ll)

Internrvtstion of observed elestio ste veloc-

ities in rook of the eerth's crust is tte nesis ior

'r—10

rterrjni structure at incocecsible derths. duoh use
renuires Knowledge of elestio noduli Which ere onereot-
eristic of tyniorl roe? when subjected to the ginute
alternating stresses produced by earthquakes or explo-

on Weves. No.: 3t imrortent or the elestic consM nts

x)?
H v

to knoW is Young's yodulus of Elestioitv. Studies \ -o

U

have shown tnet the flirti,7 1V determined 3 varies

'_th tue stress at Whiou it is measured.

Q3
W1
.J
-J
.2:
'D
O
:-
«4
J
Y
r—l
c A
“
d
-<
”J.

(isnslftlly'ifllis is: ettzfilrrtod in: tirry o {MZKS EJNl oevilxies
'on tend to inoressi yielding until tney ere Closed.
This uncert tein ty, inherent in all static methots, raises
the cuestion of nOW accurately seismic values of s are
Obteined, Iona e tre Tress“t trend towers oynfiuio

*rfi

wetboos for fietewarnrion of m for TOCVS.

nor those uFtOTiFlS with a tendency to SHOW e

permereht are rltstic flow or the—yield, (com only
called creep) the high steeo of the Gynshlc method

will prove to :jve wetter VFlHQS for E. This SPOAS
to have been verified by the fact that tests by Sonic
liet3ru1s ITNflll€UTLy ikmgnd i3 t(>'he e } i§3?3? vs 119 t11xh

founfi by the usual static rethod for the se;e ester—

ILJI

Y‘
1"

Y

isl. Jensen and iichert (*—9) so.e compression
studies if concrete, round that leesuretle creee

occurred even at loses for e serwod of only 9 low

M

\

seconds. Hence in slower tests it snould we recor-

I

nixed that error esters due to creep.
hometices it taco es useful to know E of s

:neterisl uhfler load and its veristion Iron the m of

the UNLOFFPG speciuep. yOWHTF THiQ 9rd, Lenngfd
Obert \*-10) investigetefi the E of builiing wBtDTiTlS

under tressure. (”o Worh has yet been found to have

been done on steelm

.1.)

ns under eny loed but comrression)
First the E was lound my the Sonic method using the
unloemed free-Free her. Then usjns the SImB SD901-

men, it was subjected to pressure (see figure) end

its m found eLein.

 

SFEIVA¢EAI

 

 

 

[\x\\\;\JL\\\\\\J

 

 

|/’/’/’/’///’////

-7-

Thfia'FisuTt*,n1st anvcefliti: is r? sfetxfii of lute jjffitle
f I ~ - ~ _ 0 _ _V‘ -. .0 V -‘ t -_ _ O
{aliversel ifVWLUEQ wachyne xuvwi'to rrofluce (my oressjon

-« . ' ' ”'uvn 7'\ c" ‘a . r‘I ‘ \ I -. .C' ‘N n" ' . ' r- ' n ' 'u .
1u:1fol.L weurijt3 on TT7u (331: o- (IOQFM Viblkrtol Chevice
ermi Difficuf'sxve 117th lxica AME 5%; tin? cevdmer (H‘ beset,

but are oriented such that the pickur is perpendicular

s of been.

cf
0
,—J-.
:3
~4
1
H
’3’
(D
O
"b
d
H 0
(VJ
} —J
Fl;
(7*
L.) I
Q
I3
, 3
’3;
5'
O
”x
J
"‘ 1
L I
l J 0

With this setun it was possible to deterhine the E of

s reforiel under Wood, i.e., controssion. Hhe work cone

on stone, uith the excertfior of li~estone, showed a to

w
}_1
Q

increese reticly in thm refion o pressures end to

;- —- - — ~ ~ I . ' 7V7 ~.e.» , .. - - - . ~ v ["1 . . .I 1‘ V '
siumnjec}: aocwe consul HtlfjuffflgtflilCV'llgr. lilis Th e clxlto

siniler to the chense in concrete beams ercont thet the
chence Yrs much lerfier. t Wes else shown thet the no-
ioed n of stone oid not choc? With the modulms fieter—
mineu under lose. gut lor concrete the no-loefi s

E: th=v££ ohtefiiiec Inuier 1 xui —-

the Variation not exceeding 10? for normel credes of

concrete Stressed up to i of the ulti ate stress.

I’D

1&)PJLE{J£L‘LJL$ 49:11:.) f-)()U-1.:1_)\_AL_.1L"JL

,
. "- .V ’31'
. .'r ._ -, .l 41;.)-

L.-
H
O
Ho
'7
5
:1
CD
‘1
f)
"D
\3
D
—+
O
3

DireCLly fo

l!(h1 1 (J. ’l'fuflflj Y 5.x: da‘x"1ty (H. e rgéec] e, ’“hnnc
relvtdxflvb als One LJHE {Leis loI'Cnflculct‘g urnitu“ «i
el}.<tixé‘ttr il’ t’v3 (iULCI‘ JF"t(fl“ \' n, we ( eirf“ 1“c ILLPO

. L
"N
J
_)
_.<
._J
O
-.~t
._J
l—
O
9.. l
J
_..J
k
”D

Cl 4290 fVL in“? PIKE i? 9‘ -QTAWNW fr'
€l1f¥:18 LN rutfiixity 8:Ii lPtefififil ineIWJf"7CQFQ.?}§3FQCW}HL19.

Jill 'netnonis lwcxtznc SP: e ni;w~ UL ri-ne THAI? rnwfilaix*

tru‘ (tits .infldioi. ol"tne nulnoit;exlvel l refiuneguzf (it ;X;e
‘ ‘ r~v I - v v . I] 1‘ “t
G«Pol w. (”ner ll ll Lvifc leil JLUO t o IrlU clrrn-

H.
H)
|_.o
C)
L.
C)
7'5
so
1‘1"
'3
"’1
C7)
,4
fl
H
w
I—.
:5
d‘
S
y.)
-. :1
(1‘
’D
'1
'1
U
3
[—10
3
”3
L—J
O
:3
'_Jo
d
'Jo
:5
O}
1.4
L-J
%

sypsrstus lS 'erelv so;c gOdifiCPMiOA of thesn e t'o types,
it will be sufficient to describe those two as represent-
ative of their clessos.

Anrcretus for inducins vibrations in preciuers is

finest 5rd th e\*ie ch in teres tin: possibilities.

fat very Kjfiply UHCWH is R CUitQ FOVGl TBPDQT,

0T Setti s the fnnflenentel frecnency, Which does not fill

[J0
H

:to either of the above rriwary clessificntions. This

‘is 'hesaed orl‘thCé ITéléftifWTFu‘iT? b.xt“néeii fhcecuiericyr of‘ viii-

r

1

retion and the quality of tone rroducefl. in this metnoo

the soecimen is struck as to procucc an audible tone,
then that tone is approximated with one of s set of

"orchestra bolls", or tunins forks. Thus the returel

frecuency is essily deterwinoo.

h f!

-10-

‘i‘rfin_‘:ifii‘fi‘iRS«)i ‘Jiaiili’lgl’iC'US OF ELI-JJU‘IC lixiitiifi

M MMM

 

 

 

 

F'—v&'*4

 

Vina V

 

 

 

fig. II

cross-sectionsl Free of element fin .
_ncreuent of exre sicn nn to nn' When he“ is *ent

nortie of mess stout neutrel gurfpgs thru pA

-. ., (—4
3'" (1 35h

densi f of Peteriel

redins of gyretion

reflection from tengent to WA at 3 et e distance
0r from crisin

transverse sneering force

moment of horizontal and compressive forces acting
portend culer to the cross-section enc Ihout AA
inertia of cross-section about nori7~ntnl titre
thru H8. Ashcins constrnt thrnout length of hrr.
an

17' ,. g , g 15 Ta .
lourng s J OLIfiUAS cal sllni.

L—Ja

Pd 1.4 0
~ r
C?-

L~Jo

Ticity
acceleration in transverse dirfctio‘

-11-

Rotatory end Leterel Inertia

 

 

'rhe resrootive position: of the dotted element 8-8' shows
that there is not only trensletory xotion but eleo rotet-
ion fiuring vibration of the teem. Ire ennle of rotation
is vsrieble and is eqnel to the slore of the deflection
curve. inns for e smell oross-seotion-to-lenfith ratio,
effects of rotatory notion is swell Pnfl cen be assumed

neglieirle. (See essu rtion No. 3, pefie 1?. Also see

procedure for corrections on prie 18.)

 

 

 

a I
“ Enlrreeo
V ‘ 1 V I )
\ ' ' \
I I
\ , , \
‘ I I \ h .
\ , \ CTOxS-POCthN
| _. -__ __.. _. ._.__ L__.
I
\\ " I \\
I ' p
g f ; \ View OJ Ple-
\‘ L. _‘

 

 

 

 

 

 

‘ne effects cousinq leterel inertia can
be se‘n from the sketch above. solid lines who” her in
neutral position, While dotted lines snoW sctnel cross-
section of the elevent e—e' corresponding to both posit-
ions. rkus it CFU be seer that as the cross-section—to-
lenetn retiC'becones greater

, the efteots of latertl

inertia become appreciable. (See nere l?)

I1 - liUSUrJ’TIOETg [LN 13 {—73 l I???

-, ”f Unjfcrn density and cross—section
?. rhe bFT 19 not subject to consression or tension

5. 'Ihat the smnlitnde 0L Vibrpfijdp if fl”?fiCientLy
swell that rotsry eCFocts wry be npjlpcfiefl.

4. When bsr is bent its curvstnre is sufficiently

. 0
q;;,:r]_l such thf't It 7:117}; he represented my r. 4v - _1_
. . rcr'1R
5- Length Of bent “PT 18 eoual rraCticelly to tnnt

of her when straiflnt.

n - Let r = sumnetion of forces tendirf to stretch ole ent

 

 

u
l. I; = stress : E Then 3 = delta. s3
@elta strain dx

 

5 b? Geometry 6v + a
R+r pL

 

 

'1‘») C31"! AQ]1,¢' = I. 1.n€k l1 = h)( .

 

Q
54 I
P"
(«I
,1;

Monént of F about :A - -cr --Mer“

CC}

tflien EHLrgotion_lii

II
I
'2?
’1'
'\
l
l
LT.

\
.. . r)
EWVTW \Aq} tflN“‘ftirqir. = —EIpl‘fr
(7‘ ‘v‘
And -(. ’“i l - r1]: arl 1 — $.11}er V
U I _ —fi—-—-4—
"‘3‘ (1X ( 3.:

-17)-

 

Let dis re» re;r.s ent an definitely (Fifi.
rriswrticel violent CUt fro, (Fig. l) 8 bar °deGCted
to benfiinz.
Fmrln'ticns cfl“1wotion
1, 2F . Hp Then 2v a piaYaPV
UL“
r _ 0
L'o 2T 2 id 01" (AV(X - 41") II I 659
“If?
ctr?
va1 AV - 11- lwdgv
- -0 J7
Lot AV - ov, Am - d: , e = (W, (lee - adv
6X até dxdt9
:2). GUI-’{bfifljng (1.?) GV " (LA I I: 161:) = pith-'90- 61:5
ex dW dV dth‘
01‘ d‘] - (3‘01“. = T‘IJH(Av 'l'ilOT! pf‘fl?7 - 1?:3 :3
X }7{F d,.? d1): “XE' (IXV’
pAK’i=*
'r-‘é—Q
rx‘dtn
4 Cor. lininrr (83 ,co) pfid‘v + LET‘EAV - nr“’fl -
.o ' ——‘h -
de CI oihotc
Eric. simplified, becomes 6'” + n84 - }__:1n - U
drf ! 1 di‘ct“
fr‘)\r
MAL-j

lvec rotary effectc rnc

vst term of (4) invo

5. LE
be neglected \See A.)
_,I._,") __
qun \4) becore dgz + nn“d4v =
(73¢; p (1X:
lonvitwflinrl GlHStJC “sve

Let 09- E
n
O ~Ag‘r‘r = U

5.
I.)
".‘> O
:0 fl.’ YT + 1‘1L'Cr (1
6X1

when

To

*U

'1‘!

note:

Vibra

m“.
"1119

unles

of a

17'" 9 1‘9

-14-

we 0 —. 1 ~-
né: r6 or cylinoer of ‘AQiUS - r, n
tion.

constant he 0 nnot he rcgcrdod PS 8 valocity
S‘we (Elvide iJL'hy a.ruiflntiinf of tlwé dirfwn?inns

length, viz. A. AS Rfileigh points out, the

ac ission that harmonic waves can be trsns-

mittefi at all is sufficient to prove that tne

velocity Aust vary invwrsoly PS the wevelcngth.

Conse

From

quently we LUSt have

C(tTFUSVQFSO) = E£(long.) - " * ¢

A.

y“:
*dlifl
I

9- ”’7' 9‘4" r 0 . WA
‘5) __),d".’ + Ali-~03 - O or {111ch + EI_ m, - O
C: X 'p d X dX‘ dX'

.J

Introduciufi ancalerfitihn 0f HT vity into 99” tion.

Let I

6% :21:
d)“:-

1
my
8 Jr] 0 ‘-‘ ‘-’ t'

denlsi
frequ
lengt
bl -

vibre

,0 . r3 .. _‘ 1
= Ah“ Thom pmdfiz + LIM4V = Q 01
g gdxh die

6",
f.

0 ~ u
+ aad4v = Q where a = ET”

d X 7 CA

«- w i._ ‘3
lat inung’s Romulus, E - p flfiflLL

ty of tha body

ency of vitratior

h of tin? Sp cim£q1

roots of etuation (Dll) and dmpcnds bn node of

‘-

tion 0? tn? Steel‘et-

D .

“"3

.45

Let Ncrfel.j«mie of vibretion lxésnufin'ttet deflection
at any point varies harmonicslly with ti e.

Let X = f(x), vtoro velue of ebecieee x, determines
Shape of the normal mode of vibration being studied.

From (0?) flag f egd4v - U

, a of 3-; “,1

A!

n

(

Lot deflection y -[; ACoert + ESinri]

II
H

d
”X '1
:3
L.) o
‘f ‘3
fl

_ J
U)

substituting value of y in (l) which second

[3‘ 9 O 0 ‘ 0
then twwxxues fi“x_ - I“X - C) Lettijngir = YV~ this
dx? 8“ e

be written d4A - b4X a 0 end is e differentiel

O
0:!
H
| .1

’4.
m

equation whose solution *-1,

X - uiobx + Cgebx + 038in bx + U4COS bx

V.

or A ”oen bx + U Sinh bx + Cgsin bx + c Cos bx

2 4

ll

0
}_.|

C

in which 0 On are arbitrary const"nts fietornined in
each case by conditions et end of the bar.

- Geflection enc

:1;

Caoe l Simply supported en

t. .0
benalne moment - c X - O, div - O
A dxr

ere ll Built-in end — deflection and Slope of

3.;

\‘

oefleCIion curve - U. K = O. ' ‘3 - 0

i

:1
54

Cree III Enfl free — Lending moment and shear = O

(1?,‘7 _ O 6317' _ O

63%: GXu
Rewritinfl (5L consten,s are not identical.
U1 [E309 bx + COSh bX] + UnEQg bx _ COS}! bx] _

co[;in bx + SiLh bx] + cfi[;«u bx — Sinh bx]

(fl

“akin: origin of coordinttee et one end, then by

“X

uHSw ill, for a ber With both enfls free, there wijl

be four confiitione:

(i) dPK = 0 (ii) d3: - 0
x~= 0 dx4 X . L ox‘
(ill) 63A - 0 (iv) 65A - U
X‘I U dLX X -_- L dth

O

5, When x - 0, (conditions 1, iii), then

(Cos bx - erb bx)

ll

(Sin bx - binh bx) - 0

Hence C? = U = 0

.LP‘

7. Then X - 01(Coe bx + Uosb bx) + U3(Sin bx + Sinh bx)

8, “he“ X a L
From ii Cl(Coeh bl — Cos bl) + 05(Sinb bl - oin bl): O

9. From iv 01(Sin bl + Sinb bl) + C,(Cesh bl - Cos bl) = U

l0. bolving 8, 9, when Cl, 05, ere not zero

0

‘3
Then (Cosh bl — er bl)‘ — (Sinb” bl — sing bl) = U

c0

bittlifyins (0032 bl - Lin? bl) - (Coshfi bl — oinnfl bl -
8008 bl Cosh bl = U

or 1 + l - BUGS bl Gosh bl = 0

ll. Tten Uos bl Gosh bl = 1 whose consecutive roots

are: *-l

 

4--..3r-r '18 J. 571,18 0
C Tl . t l . OXIDE-3" . (\x;.-,"?T§_ . .
31‘; .V (:1 0 ~- - o
o ' “en t tone 0 130110 .

bl; . 1.321 ° bgl . bi]. . bgl . by; °

 

j ._T
O
' o o
O I 4.750 I 7.?ow . 10.99e . 14.137 . 14.279 :
Q 0 . 0 O .

 

. -I nv a (V) . n (.N
Univ; (0 (CV) 1[hence e‘ = ml? r“ = r-nn - cA
—.-~c-‘ — ”“-
pn e Elg

Sjncg narrw-wny g = r one m = bl
3:7,:
my enhetitution 1“." = 7?.

;Furmlav£flrtcl

A /
7 v

let. overtone

flunfiafiental — nodes are 0.??45 from the once-
emnlitude at end.= 1,34: tines thgt at
the center. m - 4.430

let. overtone - nodes fire 0.13? from ends and in center.
4 = 10.995 *-5

-l9-

‘0
*-:‘J’ U liR‘CJ'J'l‘IONS FOR LJ‘;'.'ESF{;~.L LIT) HOT-.11.”? Iii-1119.11};
A

In the derivation of the for nla for loung‘s
.1.

InOdlLLUS, the (affrmrts CH? rotmcry iruxrtie.vwx“o FWtittod

, O . .0 _ ‘ v N. .
V%:e lIUTTTIV‘CIElxle =naettz‘the

d.

for sifiplicity end since i
cross-section is reletively ewflll in conferison with
the lenzth. Pwt Where croes—Pectien of teen in a
cownvratively lance f:€ction of length, rotary inertia
"ecomee e rreci :1y 1?? e. Einilenly, the effect of
nexlectinr effect of lateral ine-tie intnofinces 10
etnrecielle error ThOT~ the crowe—eection in small but
for bane of ler e cross-eection-to-lengtn retio, the
error introduced in neflecting this term heconoe

sufficiently larte 38 to deserve ottention.

LJ.

Following is an ex ct eolut_on, ty W. P. henon,
(See regerenCc 5—5 for material on this subject 5150)
for frccnency of vibration Where lateral and rotary

inertia eye taken into account. This solution given

a for ule which in itself in very conflicrted but the
curve of the eouetion (Kim. III) Sifltlffiee solution

1(wve 1_. 02?: I ”*diflly'lfiq ‘195V30

. _ V r1" (7 L— , ‘ _ __ ‘.A’_~ 3/1 :1].
— Us[1[l + 5.4.24 ) + m’R‘] } oosh[1[ l + i. > -
G (1 ran. C.

a - .

 

 

 

. ,, 1
t“ , , V y 4 .~
.J '7 ‘ . ‘ C T |'--. (’14 ‘, "J J” \J O Y . v- ‘——« ;J

2,,12 / + ( :31,- 3 + +0”: ‘ ) Rln{-’1[ l + --—-—-;“ ) +

 

0 :1: ' ' ~ ..'..‘.)/:~: . r‘ .1
EHV‘]*' binh .m[- l +Jn7§ ) _ ,rH,-] i} _ U
* 54“ A

r

-19..

*0
Tects for rotary

4

t‘3
r—J.
O
}__l
-+
Ho
0
'13
rwJ
(/
1’ ‘
VD
O
J
O
3
'3

inertia When the her is vibr:uing in +he fundeccntel

Let ml - Velme of w irow curve

'rhen corrected for letejel inertia,

 

who inert'e effect due to expeneien 5nd con-

traction of been gay also be corrected for, simemlt-

C‘

. - . w . . . .
aneouely With inet for rotory inertia. ulNC? the
A

former is pronort 1o 61 to koi“¢on' efietio, A, Eamon

1
Let H1 =- R(l + A)“

Then let c rreenonding m from curve a mm
C .1

now then, corrected for both rotery end 1 +err_] inertia,

Ill - p<fo1fL2>2

(Yn L- B:
(J 1.10
5.1

1—3.9].(0 t i 011 1‘(‘-1:"".'..'f\el-l ,.;
for Tectfrfiular bar vjtrgt
ecentel ween,

T‘] v.- . : '
if PI- C“; 111““. {x}: 5: ,

Sultigly

Petio of :entfi to lenith)

.ng lit rill? in the func-

 

 

4.6 \\\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O ‘- . c" ‘K‘ .

Fig.

.
1’“
sq.
.
/

llI

.v, 17,], l.{’-

- n

I‘

united

Sonic Apparatus used by f. P. Hornihrook in
in ireening and thawing studies of concrete for the

btetee bureau of

 

 

 

Decade e

 

4’
<

r‘ I: -~
otflnttrxs

*_4

'runeo
audio

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Condenser
\A) e v ..

f o J)

. r ‘10 twp
.LT‘HLL.» OT. ,.I' 0 ‘D

USCillograph
o «b
ungnet type loudsreeker
Specimen
e .—
L=f5 m4)», ”A"
Amplifier ‘K\‘
Cryetel pickup

 

 

 

(A)

 

N

[>40

ce fieceoe condenser With
in stere of
of free .ool -
a verirhle
.005

-- I ° r- . p. ,‘4
Lance oi IilCTOLfiiivg-

.001 microferede for e

1.11 ricrofr

:5. 1'5’ W‘Q

Eir condenser he

cevecitence VFriFhle

total rnnqe

+

P. O
L” .AlPCU cornering

vine e totel caneci—

The oscillator is .

s Iictt

)_Jo
1...).

to a x meanet-tvre

chennel-iron

r".

'mounted on

r.

(:1. . .

SlUmlDUM rod

which is screwed into

three-prongeo that

isicloir

("'

L).

snecimons at the node

cemented

to voice coil of

be ohteinefi. rickup is held to siie of snccinen ‘y
ruhi‘er hnnrl. Picknn outnnt is i'ed tnru eainlii'ier
to e cethmfleixgrcnmilloirrnh.

fireoiunicy (nitnnt to” oscillzitor “r<9 crlitwv ted

bv synchroniVinq
the 60 nyrClQ Tf‘Os'fPI' SlJITlI/‘W,

izod 60, 130, lRO cycle

condenser of oscillator is
e rixeo Ini tor of“ ‘ves is
greah screen correStonning
erect Ulti?l0 of the EVE”

rgiven by the I11.1‘i'7m9T Of

T

. <
this way Concenser n

'3
(

!

n

the
e per seconc.

then edjnsted

O O
FlPt?“Ped

0T1

L'Y‘r;
,_ k1

cuencv Wh
n frequency

conplete woven which

calibrated over u r

(..

of £00 - 2000 cycles per seconfl. This celihretion
was CManTa zt verious tines enc was never found

to very

ore then one-relf

of one per cent.

”705D Cilfifllit Of‘CNWIil]O¢ETJfli‘Sith

swoop being sync ron—

(W7
. t-

r

Apparatus and procedure used by u. mrire in the

determination of Young's modulus of building material

0')

he

The essence of this method: longitudinel

vih‘etion excited in a speci on hy tsrning one end

P.

s recorded photo;renhicelly With aid of cathode—rev
oscillosreph en, its frequency determined by comrfr—
ison vdiiiéasninerimnosed_ij we scele.

., ., 0 0
1, Then H = 4 l‘n‘p

) but for

r ,

g rrccisely, m - 41 n
frequency determination, test spe imens are generally

selected such that the hrscxeted term approscnes unity.

hence \l) is sufficiently accurate for a fire, approx-

Where 1 - length of sneciwen
n - frecuency of longitudinal vibretion
o - density
5 _ yeisson's retio
n - redjus of gyretion of cross-section stout

line of centers of nor ~l SQCtWODP.

IQTQbquxwh0 XYQ MQQTRTU U6 QxT Tut)»
\A flfloxbxtn‘vxw‘b QKQIQ QMQQQUMQ SO>~ (Qm\> szbek>0>xQV

NVSQU .0 \A Q Qth («SKSSVQQT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

@
Jf‘
/ <2 m t. l: w J the .unm E?»
w -3 .38
m a x SK ..
k Q .395 5
. . J‘MII \ A A
out w {E w
Aw r w
F. i
Illllfllll
>93 [\Io \.
l l; m.
g3§ufi0 % BMW QMDQ gauge T 2WX§UWQ6
« a

 

 

 

 

 

 

 

 

J:

 

 

 

 

 

A. - Gene

1.

“.3

f2!
1) .

l.

8.

T‘E'il. 1‘~~.I‘I‘E‘=I"..'j‘f‘ff:F‘I!13

2+
.L'J.
C)
3

bpeciren, rod or cylinder of uniform cross—sec.

is hold at its ’idroint end tanned Ylth e li:ht

Electromagnetic detector t'ensror s Ch”hiCTl
Vihretion into electrinl volteqe oscille ion,
which when amplified, controls vertical deflezcion
of fluorescent spot of cetnode-rey oscillogreih.

biflfllQ-chvorse tine-base 8 reads out vibration

pettern horizontally across screen at nroner tine,

[-Jo

ne—msrker superimnoses on the rnoto—

While a t

. f)

wrennic record, 8 bFQRK or snort duration every

by con ering vihretion pettern Tith the time

wrvrfefi, ‘rrerinencsf is (HTtFTSMlDGfi.

iled irrzngements'

‘7-

I"

mXCiter and Detector

“Toad (LI) MM??? T -or‘ur 3 qppn -- “2.1/1 on 7‘1": (1i (1‘. “[‘ltr
H‘ ‘ .. , ‘1 C. 4. L.‘ , C , k’-.IJ .. J t- ‘4 ..«' \Ji.tx‘_. - L/\A,l _| \J ,._~ I. I U.~
fiTn as to prevent nrrreCJenle translator;

movement vhen stru0n xv the lifiht shrine h5w.@r,

.ueteCLor'— I” zillijht POlt iron er.eture, {bout
l/S?" thick enl ce-ented to end of seecl on is

(EDIHET{”fPCE htr c 8 Jill- a lT-H;9}3 ?“r(vn II-Ffiiffl‘ed

ner enent ;quH3b H, wou d rfiififi calls or the

tyre used in ordinary telerhone TOCPJVPTF.

l. l

’ ‘1
O

L

I

.(
,J
,J
E._l
.J-
H\
F-1-

‘er
a. Lost suit hle is e resistance—cereCitv unit of
,5006 response in the urper eudihle renqe.

n. ln these QEpOWjJPNtS, e direct—coutieo e nlifier
of good Characteristics, vein; eveilecle *es uSew,
3. Cetflnmfie-rey Oscillhigreph and idfma-Hase

a. Oscillogrerh onereted norgall; et anode volte e

or ecu—1°00, according to intensity require onts.

D
‘0
IT

Potentionetere, P, e0“- 7ijh-tension sunrly
nrovlde control or initial position of the fluor-
escent sent.

h.‘flhere being no neeh of 8 linear tiwe-hvco, a very

r

r]; Le? Of'C)%CilJJl§T€jfll in EWJflLGS td_fi size? a);
Volta e in the usurl way. Conlenser is Cher ed
iron oscillograph nign tension supply;

0. when a ne‘e;uf is prepared ior r recorc conee Her

U

is “hora-oircwited my 2 S”it0n, S ett ch w to
fiasu-e1‘ twalrnlsc>, 1), E?fi(' L: e i1l11oiwesm;erit 53L(Vb J.s

derlectee to leit Side of screen.

1. £110 in; ten;

‘\
,4
)._4
I

r...
.1
P4

.L— ' '. ,, 1 -. , r f“
“(:10 SlJTl;-p E1ryn, l4], Era—LP(J;7GS

at Cgrrect instant to Sfireeo end the v.&TfltiOU

e.

lst £1 vriltzoge oi‘ 8;,grw:rll

l

-ete_y 500,

entree”, (iisclurr e lfiflf F

coxnecued in

CorLCienSer 1 T‘O'v’~‘ I'WI'L, 7'1 '1‘

, J' ,V . ‘2‘ _:__l.:; _t| “.O_LVI.J

I
“1
D
O

I u
‘
.J
\
.
..
\J

F-PI'l "-f S If CIT): E?

'a )9 1...! F'e o

thrr~ly«5 culfiwwit, {‘mC *Vitfz it late Eirot Wwel~ciiqy
eecreeee e Ton¢nti#lly as cond‘LP'r Cu.r ‘9.
Pete of rec 3"? ror 3r:n -'"m 01 ~ r u er~
ceC”, if qua e s ll. tiPC’ lire, "crt ‘3

[any :3 7? V, gi‘oLJe L isu Tin t nr?e(: z,o ixe v42oii

”a
5“

J
’2‘
rr‘

V >». _-‘ __ n , I ~« . ~ ‘-
fJJ LlUlDu 1.10 ~ (L . n, L_|_ Flap-LY , ' Pf} '5 ‘Wvllfk l

‘ r ----- " -H-\ -v . r- W M' Q r
Eif'l’jfif‘: 5‘13'1FC» Hun LII 5‘; The lJFl‘_‘(,-':1‘kglam‘ ",
CWT-”C I ll: C01".
v a; _‘ '0 v . , ‘ _ ‘O ‘

ml' f "5 1 ll ’31) (l 1‘ Ll- , ' {(7. fl 1 llV'CCE‘G C(,)I"E""i.31,l‘f
r; 1(1’xw Cz‘q ' , '0‘ r ".H vr ' r ,,,-‘ (.14
r4 ' WA I: HI; Lv , I l1 1 I r . .13: ’(7
I I U
—‘ J- P\ _ ' . fl

0 511*U1 mar L tn“ .9 t <u» “a , r urge ml

- .1:.1° .9 1 .00! ' .,. .
anal lier ls cullJCJeht to

evreri
......

output of oscilletor on ties eozo

' .4. _. ,] ‘,, - '7
‘1 21‘ ': C I l
'T IJ‘(_.1 ‘L'I‘"..L_. ‘Jrj‘TICL

(5:010:55

control Of

O 1
.. _ c,‘ -
1‘", l’fltFQCOT.a. 1‘5

('1') ,

EretJMNtcy (flf'tiwe-ymwflier “In;
neriod of several month

-‘ r”: bwr 01 7‘ timr (‘5 - ‘f-i‘ w-nr‘T-‘nrc
I 01;; ,3 |_"\]_~I‘O.T _:_. ._3__.On.l_nf‘, J)“. 0 Jill

o? eined frcn

13 ‘ ‘V
110$? 3

5,000 per

to h_'7 LL ,'

Six records

61 I‘ F}

l; h (‘3

1
(ring 6:
4.I/;- JAY

taken

elate-Tolder vertic

L

_ensity Determir

0')

Q. .

tio

'fozfii.

” checked over a

‘_' "f
| ‘ C

9
,

ffirst c lC?¢ turin:

record

.1

found

Ct

:"PeCiUiFOK"”Fs
‘

constant Within 0.95%.

f-l.9 lens Fitted to e
ores of frecuencieo un to

made without recourse

or) canch “Plifitfl hyr tinnv*flrsiii€

ally.

“S

)—

Dry porous materiels — hulk density by reduler

'J

“(r F) T)

'c T) I

"y

norm] E“. ‘V
1.. ,. 3L- . ‘

l—Jo

ll

0

8

7.
u

with yet

I‘

t

iel

‘r‘o
_.K/

V

,. not
. - Tueee heve/heen worked

Water in the

":3

art pleyed by

translission of vibration is yet unknown.

CZWVFRWQAQML/QWQCES

 

 

-29...

1 'x r~ -, - w!" . -, - , *
lJINJL_Ib {DECK}; Ffiflfimyihj CW? TFWTTCVTW VIPHtfiwllr’S >k—7

This netloc of testinx is vzeev on the nrincinle
that precticmllv ell systems in the field 0? enfiineer-

inq cen he set in :otion hy elternetin: forces of e

i
(‘1

\ily rerroducihle hv

bin

(D

form. Such forces ere re

“:1

("W

1 L

two revolvinr leeees eccentricellv smnprrted end
I .

novir: in orroeitc directions. is shown tv the dieqrew

telow, (ijiudiich the e {VHL solid_(3:rcles ere the

}_Jo

eccentricclly surpirtod revolvin: ”fRSflf) et inte rel
velues of pi for the these Fuele, centrifufiel forces
ect in such e “e6 99 to cencel one another. At other

values of pi, resultent ection is snow as to produce

9 CUIWIe (3? Tile {jirvn fr)? L.

 

 

FVflflSE INVGLE'

-30..

lmoUClrU VIBRflTIONS

H‘- ‘w «a A ~.‘ . f“ v .‘L‘. I v - I' ' . -. 3- 1
imicn iniit enticigflna ei Thi? T£flMltVJDPs(i‘Se&W

e electric deter driving, inst_u‘ents to newsure

“S

D
C
c+
Q
d
.

frcwniencx; of din: s any? to Furire Tmrver irnn1t tn’;»‘

Drivinfi actor *

 

 

 

 

 

 

\,
Speci .mn I.

 

 

 

AKUeflecto—
rxét€?r

V\\ \\ \ \ \\ \ \\\ \\ \\\\\\\\‘

 

 

 

I/lj/l/l|
.1
I’ll/J

 

 

 

971
do or
(c) G E

 

 

 

 

 

 

 

 

lorsion Test

 

 

 

 

 

(d) Gimmice ill T011 rive ivnsitf ll of‘IreVQLAWLng rewaens

J
(D
"5
(3
(D
T!
is

C. ‘

"‘
Q
<
H.
.4
3.)
__$
"‘3
...L

; in is v: icrle, os ill-
etors have heen built ior frequenCi,s u; to 100 cvcles

U

Per 3533”“‘3- Umi'l'll'ufiéil forces have been iiw'ctmced

of iirom l.- b“,fflh3 poruuis. Eihice inie 1U13t jiSEittETfiled

rig ixilbr tr) tine inidgf Tile .etri;nun ivoiuiiite .toIUEe

i) 5

produced i£%}frcfortionrl_ixi twc F‘Flifnj’”7 TFCROT
of tie svste . For or: rle: in irid eq “n “V“Pfl‘u

“*3
J
O
('9'
’3
‘3
C

w
‘3
Zn
' )
r—J
i +
.1
1.,
7*
'L—Jo
O
,3
VJ.
'1
I).
“D
(1'
Si)
d
(D
"3
A)
.L‘"
{D
:1"
7)..
CD
fl
4"
l

icum eltwrnetinfi force produced will hecome as high
as 40 X 50,000 or 2,000,000 pounds.
Procedure

With the oscillator rigidly fired to vitreting

system, the corresponcinr power innut

--.)

- v f‘ Ny~ j J“ f ‘1"r‘ '. '1 ['1 ‘J :I .C) V -."I r " a '
11encgi<m n rm; cetimriihet_ ant.zi Lreoiuuicv—fw Mir c U“f€

rlcddxu . Eor':xmit re ult&:ljie frecwmut'r should SI)

..

O

bCyond up least one resonance Foint end the irecuenCY-

I
‘J

' u ’1 f '-i 1' I" 1 1 ‘ i "' fi' ". 1‘ ‘. ‘1 ~ r”: "
rower CUlVO ciecnuc Ly BlOWlJ cec.eesin~ FCQB‘ of

l. AI1 oeutilliytoi' is -leGE? er? ermiivev t iii e :?EVYXH3T0

fetieue testinc 'FCJlflP of eouel cerrcitj.

-‘ x ,- . -.‘-' -'”“!
,n laboratory Pbfl in fiolt-

I51 ifl‘d<érrjiu1ei:t of“ t.= 7<eiéefuirvi, ' eritiiei’ {7P(l rwttwir

’

Ln
‘5
‘D
:3
(D
5
c :»
-J
,0
U)
m
:1

Adwmiflna:es cacti Anni

«i. Dyn€.ixz lands (N11 Fe c finfh:cfl With :flx tic loadsu
fiatiinc and 89in? continund Si sultnnecunly to
(ieSTJfilOfijINl, iflin::_zciwi clxianl;r F?fiffi0ncfiiinj,iirt_
urnl confiiticns.

Dinndvvntnjcc

r 19W crnericncc rcccrfls are nvrilnnle.
Until this is rcmecied, continusl checks Snonld
1m) :zmie ifiifil {Bjilnifilfii ”csrtivvt r1*csf

3. ln field tests, it may be dili
~n*ij losses thrr ra(iaticn. Exafinloz 1n

Vibrating a floating hull, energy was lost

thru tlfé<flm1rninc Cfl‘iflm>ixater cdirmmvvt to the

fflfhl‘l0£*tlfTQF cl‘ tfiis: cifiicfi. 91%: nt1i+m= n11 -

_.--4 ‘

l .L

31 ~ ' ~\ ~..— Ta‘ '7- . r . ‘ ' -- ‘ —.. ,.
QTOLS. VJJTHLlOiS bros can by _;3_1ncr7 lanc b¢

~~ "r-~.- WV- . L ‘ i ‘- —, . : m -4.
VQJJ Qcflgӣ04r3 .Prcc pro-octor igcflcn cl rcccnkpc

‘3 ‘ ‘~' 'v '. ‘ 'l 'V " N WI I. ‘1“ ' V "- . —‘ ' ’
Trcnncncy 1n eaxly antics Ol OOuRt?lCujvn YOUld
m .- — w-u'. (x _.'C— 3;, r r- - ‘v - 1‘» . .. l ‘ . -. -\ 7- . 1
icronca WVJJLijfa. l5; .«mxnn,'if {$118 calla, 3t -15 7w3r57_cle
o -. ~ ~ r ‘ w ‘ r . “ ‘ ‘ -: ‘ 4. _‘

to lOFTJQdCQ path cl Vibrat arc cl eniivcs Pnn

. r" - . r —--— 1 3,: < v - . q - 1 . -‘ - - -. . ~n A» ‘- > I- —~p ,\
xnzeng Icnrcuan; e ‘4F£._fOUJM1 :11 su7l’QUlWJ7li,S '3rfi1'5+3

Cauld we made Fefcnc cc fleticn of ccnctructicn or

“,4- ,2

0 fl 0 - .g'l .L 7-. 1-, 3. : _\ _“ 1" (A ‘ . .
lntbclluthfl cl hc ~Ccplncry. l?ls ypa bécn cone

(—

‘1 -I- '1 1 r' 1. 1 'l - ‘ J‘- - a - '1 - <
an 2P7, (I‘CIS one ills. One 01 {im> wiry iicsb
19"nl.|r.t iY-11.=fl t‘“?W1 i‘cricr ctr? iril rrvtirin 01' ii 9 ci‘fivfl‘—

hccn n “ *nln -9tt9r to corrnct. Si ilcrly Nari?—

.. k L} -

5
.t
.

GUtFJN, V”?%;1cclc. a r 'tfilW7300F]_ V‘CULU?fiClfi¥E of (xvil-

inifi walls, townrs, nnfi founioticnc ccnlc be found.

DOWCLKSIOU

From stuoies of e number of artiCles on the subject,
"Determination of Young‘s nooulus by the sonic Method"

the following general statements seem to he indiceted.

J

U

1. In the derivetion of tne reletionsnin tetween E ena
frequency or vibretion, the materiel is assumed perfectly
elastic. by the Sonic method, emplituées of Vibration
UP to 10“6 incnes per inch are produced, Whisk seems to
be sufi

loientlr sushi as to remain Witnin the elastic

li its for the nsteriel and thus Verify the accuracy of

T1?

. whet Specimens cen be tested Without destroying,

de aging, or otherWise alterinfi the saunle. hence tre

(a
6
C)”
T“)
’1
<
“
:3

be used for nrozressive tests at ell sees.
This is conGHCiVe to greater eccurecy OJ results since
the rnynber'cyf vars! bles ill'tbfi tweet sergile cer1lme held.

to the onenée in the intetuel structure of the been it—

o. in the freezing ens thawing tests of concrete, tfle

largest cnange in E occurs curing the Iinst few cyCles

L

l

CFC BVQD 3 0f Eood concrete Will slow a oecneeso. Poor

Speci e s can usually be fletegtec mjtnjn 20 cycles “MN
the 5 of the mood and the bed ereciwens st tnet tine is

quite different.
Ufliziwe SPQMIE'CO in? :vart nifxwise ijl'the lulilfiijt:

heterisl field wtwere 52in? ocours thnu eyeles of free?—

L—J.
f5
1
fl:
:‘3
pi
6+
:3.
D
H.
H
VT
3
d
U)
0
CD
CD
:3
O
L.

similsr cycles.
4. Temperature chon cs have little effect on results.

sstureteo or near saturated

U
,3
0!
IL.)
[.4
O
,_J
2’)
cf
s
’D
O
4
k ,5
CD
’3’)
‘ .J
:5

Specimen have little effect on tte results. Hence e
seturetee condition is Well suited for obteinins center-
stive neasurewonts. fine sonic wofiulus is very sensitive

to s (ll omen es of moisture in e reletively dry sample.

14‘ 5

Yet unexrlsined - That the oifrererce in or the
saturated enfi the dry Specimen is severel ti es ttet cue
to the chenge in density, the wofiulus meins frester for

tile tset; STV§ClTLQ ,s.

5. That oven drying will cruse e conniooreole cuanze

(

in natural frequency. Obert (t-lO) found tnnt His best
results were ozteinev when the sample was cured
in water, 28 days in air and then dried for 24 hours in

en even; this lest step being etsoluteLy necessary. nor
testins of stone, two week crying st ordinary room eur-
eratures ves suffiCJenp to give consistent results.

1w nrocess troceers, sneciuen toxins to dis—

‘ _.‘ "' 3;) l

intefirate ens causes the resonfince DQFK to stresd out,

it; brute {is . 01“? crni nmirew ci‘tific. it ‘to E?Cli?0tl fiuidelweniu l
frocuency.

.UTOKEW}<QT c Jagnm‘ FFCClwleE or witi11< y ot&MUf irreg-
ulvrities, tend to increase the tronyble error or tne
7. rho sigwxlicicy, accuracy aim;5fix¥wa of the sonic

l-

LCSL {routine teets can we run in es little as l-

(D
__J
.4
r...)
(vi
C+
!.’.
' '1
it
_J¢
*)
l.—

unes ter test eemnle) mev trove of thu

' ' 'm 'a *0 1‘ 4 . <~ '“r ("1
Stud 1‘28 there (.1 LET e grout" 0L EFHIM'LQS’ 18 CI‘U';UTZ:,';37_"23(_3_.

t. where See.e to he little if elv inter Paiei eveilable

(3‘
U)
0
r“~
“J

«rioue 917ee CH'EVifiTOfiEQES

concerning tne ei‘ec
in concrete 8 COL ens. Uere e:oulo we ttken no the gex-
i gnu Rive enzytecufixe will_110t tr? eucn_aws p0 plwui;ce e
yrixrz (if (a terrnr te {nectuionrz ot’ SECWle zu1e ' oi'tei‘ 0-:uce
tL'Q Twe~iilftiji.' l :Wicileiicgr “7011l(1 FF? ciii.fi_c. ll. i;o lzitnel'yjfietL.
9. rue nigh eneec oi YlFrFt10DS in such ee to ell ln-

ate tne ezlezte of creep 1n tue uPDPllCl.

I‘D

,. . - -. ' x. 1 1- - ' r~ v- '\ u‘- I‘ ‘- , ~ I' ' 1‘ ‘F 1 ~.'|
lu. the eaten {CCQPWCJ ttainaule 'ev not ochvC nu CUM-
meneutately valuetle exce t In? lahorato?y Sturjefi of
Stress-etreln relationsnzfifi. Accurate values lot 3 Will

-« —. 1~ ~ . . -, . .' -,_- N 1 .._..'
r prbv()ljselHL liingflVROlfL9Z3 _ooo lrfll(wqp10n or

ll. LS an Ric to co rariecn between rceulte of dilietent

~: ‘v... ,-- ‘4— K ’ .' ' ‘.' ,-- - 4-~ ~. ' »
lnvestlgfters, lb Shoulc P9 very enventegeoue to esteLlISn

routlue i0? te“tiN4 peterifllt by 503i? m“tm069-

G
"3
Ms
.‘ ‘5
,_
'v—
:7
G)
p...‘
d
f“
,.. u

”D

Tiuoenen

1.0,

r] f?

-(:'t)-

13’ IEILIOGIZAP" T

('1‘
L)

lfibrvrtion.l?retdjrns iI1FUn3iTVWerirn", 197??

D.

VT!)

. .

y ‘I ' '1 I‘I
.Jorut, :1. I3
A
. i
1'1 Fl (3, f

‘27 4.... -c N '1 ,. (jg ,_, nrl {M713
"n :OQLII“. er. UO., lC):, n99? [I -."r/(_,

., (—1 ‘ ‘ -<
~t nTOK of uflilfl

I'LrQCIJllf‘n 00., lggr,

P.
"fiction of e n”
cludini the 5'

n ind in Flexure ln-
fa 0+ of Roterv :nfi titercl

lnertle"
Journal ACOUmthPl uOClOtY of hrorice, vol.

6,

T\
L) ~
J.

v 'I
(no 21/ -'—\
.33.: L- ,'._,

Mornitrook, F. B.

8 T‘Cl
/\

-‘O‘S.L

F71 r.
lhoueon,

"“nrllcw’lfin “L SODiJ
30$

1 l

m

my
ii

\
1— .- —, 5‘ 2‘ -
.ne”ir- otuv.
l

0 Hctbcfi to hreev ing
11 . _ of nvlpnp-f-P"
.33. Elul__etyin.lTo. ltYL :ppg:e g3, lhéc. 1€V‘9

m

.
O -o— C

"The Effect of Poiery enfl Letorel Inertia on
flflxrrurel VilT"°tiru' of Prfifirr tic inure"
Tournfil A.on”tic l Society of Anerica, Oct.

959

"Leeeurine Caerfiee in

Concrete by the Uynemi
A

Proc.

f}rjtfe, (%

1 '2 )‘l

igj1rflj nr1l 1):):OY‘CWL1-‘L 1-98 of
c Lethofl"

an, 1940, pefe 1115

.._J.A

"; C‘ l'" T: '-
.‘Lo‘w'. .1..J-'O, \u’O-l-0

In
L.

"The Dete .inetio on of Voucc'" Hoculus for

_ ilo:n
Pli llOCO”N

vol

rernherd

,

"IJ/ \V‘n {.
rztione"

Pr

QC.

1:- 5? :38

1(1‘ 9 , {I O

H.

icnl fegcfiine no uouin"l o SClCDCn
20, l¢55, rail filA

O
mic testing by ueene of lnducoo vio—

A.S.T.M., PTTC 11, Vol. 57, 1957,
‘33 :7". .4:

"Comnerieon of Stetically and Dynewicelly
netmflm;ined Etnung'e Looifibis of Pfln_eticitgfl'
Proc. getionel Aceoetj of Sciences, Vol. F?,
do. 9, thruery l 38, nege Fl

*-9 Jensen,

"SHOP

TESS].

Proc.

10 Ubert, L.

"SODjO uptbod 0f Uoterninini the nooulue
Elanticjty of n ildin;

l-‘I'QST-f

Prcc.

ll. lkr“eIT3, E.
'Cfefif
1.99.719
Proc.

nege

Lord Relei

Theory of
RPCMlJlQD

hr
1
.‘l .

men,
"Conn

Meter

Proc.
1?

r7 I.

.go ,

l‘llll, E. H

" oocrn Aids

(”(7
lg '37 ,

r)”—

-o_;' I

Q. who
t -Tl me

A C‘ F! r
J‘s- . L,) .

l .121. ,

0 Y
uTe'
' ‘- P1 ’1
lLoLJQl .1“. ,

Vol.

0
urirnjlfountflss

Sonic
L.,

of
A.S.T.
460

Vol.

fh

C“ v “I
. r. "

(J O O , 3- O (a C)

fl .

0
erieon

of

mined Eleetic
netionfl nceoe
, e 21

nege 151

Steti.

Teete of

.'" r:
J‘_a'.

239 ,

twmfiilue
Vibreti

7‘ "

78
. a ,

ally
nstr
- x

to Vibration
Journel of Arpliefi LQChRDiCQ,

‘fi
4
._| .

Concrete

1059, neqe

of

O
_ ul)

1111C. C' 1‘

terie

19'9, rwu"

‘17

NW, yr».

0? Rocks"

Sciences, Vol. l9,
{Stilflfif"
Vol. 4, NO. 4,

 

 

o I u . .3..- o I .III I I- I‘ll .I1"I.I‘Irl I I- I'- II .I O I. o I nII lo I I I I
D- o I I n I I IuI II n I
I u U
I I a 4 I \ O, I o
n! I I ...l I I I. . l— I I . . . I . .n t
$ I 4 I II 5 I I 'l' I II I « ‘ ' I I
I I d I U I ‘ a I I not ’I $I- ‘ O f I r. O
. . - .. L. . .. - tun . 2:. L .. “In“ .... -. 9..- .. .... I
I. II I ., .r. (. ”on... «a. z... .L...o.- ’Igcz ..ka«?..l...xa...n....mw.or no; .N . _
I . . .M. ...u‘.0 Dannaraamfl. Flinn“ Igurv!‘ $0.: 4. 3... :1 . 9. a. .II. a ..
. . . -1 .........;. .. .I V 4, - a: -. .
. I a .
. . k.

   
 

. .21.». ....\ 2.... f; :35.-. .. +9.. . ......_-J...v, .c _ . . . .. , .. .. . . _, . . .r. . - . .
. o . .Nt&av1. . o I. throw .r. , 3. .3“ KW. ryws ‘1. ?.lo«<..’..).. .. . .
. L. . . . .

 
 

 

7
. A I o I
I _ I. . . c I. A I: l . y . . .5 .. . . I
_ J. . I. I.\0\(....1 .. . .. . t. a... :.(.. .L. . .. . - . .
I . . I I I- o. . a I. . I I .7 I \ 1 . 0. r‘, ¢ . . I I
v I .L h * 2‘54 a“: \a . « .vIVI. A. I III. ‘. VII'K 1.)”! r..PO-I|t .I. r I . L . u\ I
‘ . \ I ‘ . 1 n .u... ... II... . ...‘ QIW .‘ .I. . o I. .Iyln.._ “AI. 0 . ,I . .5 . I. I.» 3‘
o I . . .,I . .
I\ . . . A I
I o 4 O
a I I. I
_ l l _ It
I. In I
I ~ I \i . . s
I I . I r I 4
I O I . I
I I! I I
~ V
I n . I! . I I \ .o _ 9
p .Q I I o I I i
I
n. _ I a I . I u at ~ I I I . I
Q Q. h I II
I I v I I l
I n ‘ II- . I. I . r I I I I g
. I _
I L I \ I I I I ( I. I
I l ' I o
I U I\ I I
I l 'h I 'D ID D
I . .I I " h > P k'. P k ' I l A. I c — I r _ .
I i r
' I . t r O A I
‘I'IPIV 7. tr L ‘
l . . . x . . -
I ~
.
\ I. I . I
_
I _

 

.
. a.
.
_
m
.
_
o
'|
u
C D
I .
‘ IL.
.
. -
_ n
I
I
t -_
S .
.l. a
II‘ I
I
. _ Z.
n.
_ at
M . ..I
u ....
u .
I I. I
\ III. II‘ I 0"
no! J0.“ _ I. I «I 4.. II I I
A . .. - .
P O I .
. . Q . . .. I.
. r I . I. I _. I;
I. p .\. .
II
._ II . I I
I . I ..
I I I
. I. .1 I. I I I I I I I . v I\
I w v a
p ~ h I 0
. .. . L . . I . . .I .
I I
II I. I' . I! \~ I I I (
a a n _ i
u 0\ I J I . I t
\ .I‘ t h
. OI I I
I J I ’ . «
I: 5 . ~ I . . I
u I \ I p o |.. i I .
. I I I
.\ Iv

WW"

0

‘P

 

 

MICHIGAN STATEI UNIVERSITY LIBRARIES

LU.

 

3|

‘1