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BADI^O
1955
TH331SB125
JTSSL BSA.14S
H3.Ia^H H. B.D1II0
NATURAL FREQUENCIES
Of STEEL BEAMS
by
Herman Morris Bading//
Lieutenant, United States Navy
Submitted in partial fulfillmentof the requirements for the
CERTIFICATE OF COURSE COMPLETION
IN
MECHANICAL ENGINEERING
United States Naval Postgraduate SchoolMonterey, Cal.
19 5 5
5/2 5'
This work is accepted as fulfilling
the thesis requirements for the
CERTIFICATE OF COURSE COMPLETION
IN
MECHANICAL ENGINEERING
FROM THE
UNITED STATES NAVAL POSTGRADUATE SCHOOL
PREFACE
A difference in frequency exists between a beam mounted
under ideal conditions and a beam mounted under actual con-
ditions.
It was the desire of the author to study the variation
in frequencies between the actual and ideal beams.
An electromagnetic exciting device was used for the
investigation to excite the beams.
The work was done by the author during the period January
1955 through April 1955, at the United States Naval Postgrad-
uate School, Monterey, California,
The author is indebted to Professor E, K. Gatcombe for
much helpful assistance in making the study.
ii
TABLE OF CONTENTS
Page
Certificate of Approval 1
Preface ii
Table of Contents iii
List of Illustrations iv
Table of Symbols V
Chapter I Introduction 1
Chapter II Experimental Apparatus and Procedure 2
Chapter III Computations 5
Chapter IV Results 3
Chapter V Conclusions 9
Bibliography 10
Appendix I Oscillograph Traces 11
Appendix II Beam Dimensions 16
Appendix III Frequency Equation Taking IntoAccount Shear and Rotary Inertia IS
iii
LIST OF ILLUSTRATIONS
Page
Figure 1 Schematic Diagram of Test Setup 3
Figure 2 Sample Oscillograph Traces 11
Figure 3 Sample Oscillograph Traces 12
Figure 4 Sample Oscillograph Traces 13
Figure 5 Sample Oscillograph Traces 14
iv
TABLE OF SYI'IBGLS
(Listed in the order of their use in the text)
^M - Natural Frequency in Radians/second
f - Natural Frequency in Cycles/second
1 - Length of Beam in Inches
E - Young's Modulus of Elasticity psi
I ,- Moment of Inertia (in^)
/^ - Mass per Unit Length (I^-p^)
CHAPTER 1
INTRODUCTION
A beam mounted and held under non-ideal conditions is
known to have a natural frequency different from the natural
frequency of the same beam mounted and held under ideal con-
ditions.
It was the desire of the author to study the amount of
this difference and to investigate the discrepancy between
the observed experimental and ideal natural frequencies for
a beam clamped at both ends.
A codicillary to this investigation was the desire to
find a method other than mechanical to excite the beams,
preferably to frequencies above their natural frequencies.
CHAPTER II
EXPERIMENTAL APPARATUS AND PROCEDURE
The experimental set up was as shown in Figure 1. Beams
selected for testing were three cold rolled, mild steel beams
one-eighth inches in thickness and one-half, three-quarters,
and one inch respectively in width. Variation of length was
accomplished by starting at the longest length and drilling
holes successively inward toward the center of the beam, and
moving the cast iron blocks with the securing dowels toward
each other, fitting the dowels into corresponding holes as they
were drilled.
End conditions were as follows: At each end two cast
iron blocks four inches on a side and three inches thick con-
tained the beam, A one-quarter inch dowel fit through the
beam, into each block. Four one-quarter inch bolts passing
through both blocks were taken up to clamp the beam securely.
Each of the two bottom blocks was in turn bolted by a five-
eighth inch bolt to a heavy cast iron table,
A magnet was built by the author to operate on 115^, 10
amp. supply from a 400 cps motor-generator set. This magnet
was mounted on the cast iron table under the beam which it
excited.
An A-1 strain gage was mounted at the center on top of the
beam and its output lead to the Hathaway Oscillograph,
^3
Co
§
40
Co
The frequency of each beam was obtained by exciting the
beam to 400 cps with the magnet, then de-energizing the
magnet to allow the beam to vibrate at its natural frequency
and recording this natural frequency on the Hathaway Oscillo-
graph from the strain gage output.
It was hoped that the contribution of end conditions
might be evaluated, but time prohibited the redesigning of
the experimental setup and moving of the equipment to one of
the large hydraulic testing machines. A substitute was
devised however. Beams 3-1, 3-4 and 4 were setup in a large
testing machine and varying force as shown in Tables 2, 3 and
4 applied to a ten inch I besun spanning the support blocks,
causing them to constrain more firmly the beams. The beams
were plucked to set up the vibrations. Sample traces are
shown in Figures 3, 4 and 5. Results are considered unreli-
able because of their large deviation which is unexplained.
Whether the deviation is caused by mounting or by the method
of excitation was not determined.
CHAPTER III
COMPUTATIONS
Actual frequencies were taken directly from the Hath-
away Oscillograph, trace samples of which are shown in Figure
2. Each small division shown represents .01 second regard-
less of photographic paper speed.
For computing the theoretical natural frequency, it was
assumed that the beam vibrated in the fundamental mode. The
following formula foV a theoretical clamped-clamped beam was
taken from "Mechanical Vibrations" by Den Hartog.
Ca)' 22.4 .EI.^
" 1^^ ^7^^
Analysis of the decay in amplitude of the wave form
showed that the damping produced negligible effect on the
frequency, hence it was not considered.
COMPUTATIONS
RUN 1-1
0)n s 22.4 '^^ - ""^^^ '^ '"' ^ ""^"^
^^^°^-(11.31)?
(^OXIOO) (7.^^x10-^) (3^6) ^^^^^ ^^^^/(.2^4) (5.39 X 10'^)
sec
l^ « 1220 , 202 cpsTheor. 21^
RUN 2-3
Theor. ^9.57)^\l
(30 X 10^) (11.62 X 10"^) (336) .i77, rads/sec(.284) (3.96 X 10-^)
f = iZZi = 283 cpsTheor. 2 IT
RUN 2-4
Theor.
f
Theor.
RUN 3-1
(30 X 10^) (11.62 X 10"^) (3^6)22.4
(9.02)2 yj (.234) (3.96 X lO'^ )
^^22 . 317 cps2Tr ^
sl997 rads/sec
. 22.4 -5(30 X 10^) (1 ^.19 X 10-^) {}^6)^^263 rads/
Theor. *(11.33)^\ (.234) (11.9 x 10"^)sec
f, iS6i , 201 cps
Theor. 2ir
THEORETICAL NATURAL FREQUENCY COMPUTATIONS TAKING INTO ACCOUNTSHEAR AND ROTARY INERTIA
RUN 2-4 EMPLOYED FOR THE PURPOSE FROM KRUSZEWSKI»S WORK
Ks. = ^~- 1 (30 xl06) (116.2x10-6) _ ^^^^^ ^ ^^^3
f 3 .02 ) J (8.96 X lO-'^^) (12 X 10°)
k| z 1.6 X 10-^
^RI='(2^
Kb= U
^^^^'^ -^^"'^
-. .798 X 10-2(8.96 X 10-2)
KRI 6.37 X 10-5
f(6.59 X 10-5) (9^)^-^
= (2.8 X 10"-')^0
simply to
(30 X 10^) (116.2 X 10-6)
The equations forc^andy^ as shown in Appendix III reduce
/ J Kg if it is noted that for the frequency
with which I am concerned ( i.e. ^ 2000 rads/sec)
Kb
The frequency equation shown as a 4th order determinant
in Appendix III was solved by trial and error;
Assuming Q ^ 1997 rads/sec
Kg = 5.59 o( ^/g z .423 2K3<s 2Y.^^ 4.74
sin 2KbP = ..999
cos 2KBJgs .035
sinh 2K]f(r 57.21
cosh 2K^: 57.22
157.22
30
57.21.525
30.05
1.035
.5245
-.999.525.01^4
= -.275 +.552 - 15.75 + .553 -^901 -V15.73 -902
= -.29
FROM TIM0SHENK0»S WORK
Assume p s 1997 rads/sec
k r(1.997 X 10^)^ (g.96 X 10'^) (.234)
(30 X 10^) (116.2 X 10"^) (3^6)
.25
s .523
kl = 4.73
Substituting in the frequency equation from Appendix III
(i.e. 1 cos kl cosh kl)
1 ; (.01745) (56.65)
1 .99 (within slide rule accuracy)
Hence shear and rotary inertia show no significant effect
on theoretical frequency of the tested beams considering both
Kruszewski's and Timoshenko*s works.
CHAPTER IV
RESULTS
(1) Results are tabulated in table No. 1. Errors from
the theoretical natural frequency ranged from about 9 - 30%.
(2) Shear and rotary inertia had negligible effect on
the natural frequencies of the beams tested.
(3) Deviation was almost uniformly greater for longer
beams than for shorter beams of the same cross section.
Beam f f $No. Theor. Obs. Diff. Error
1-1 202 142.5 59.5 29.51-2 263 237.5 25.5 10.322-1 226 135 41 13.152-2 256 210 46 17.962-3 233 240 43 15.22-4 317 267.5 49.5 15.632-5 357 325 32 3.963-1 201 140 61 30.43-2 219.5 200 19.5 3.333-3 243 210 33 13.533-4 263 230
TABLE NO.
33
1
14.13
3
CHAPTER V
CONCLUSIONS
(1) Uniform beams of rectangular cross section, small in
thickness and clamped at both ends, with the thickness in the
plane of loading, can be expected to vary on the average
approximately 14% lower than the theoretically predicted
value. Of this, practically none can be attributed to shear
and rotary inertia for frequencies under 270 cps.
(2) Excitation can be accomplished electro-magnetically,
(3) It is felt by the author that the end conditions
account for the remaining variation from the theoretically
predicted value of the natural frequency. The ideally built
in beam was not closely enough Sipproached in this experiment -
clamping two faces between two large blocks and doweling the
beam does not approach closely enough a beam contained on all
sides and the end. Also, there is undoubtedly an elastic
deformation of the supports in the area where the beam enters
the supports, which again moves us away from the ideal, which
assumes infinite rigidity of supports,
(4) In trying to reconcile the data with theoretical
frequencies, the following possibility was considered:
the bending moment at the end was proportional to the
slope of the beam at the end. This could be the case if
the bolts stretched.
This assumption does not seem to be borne out by the data.
BIBLIOGRAPHY
(1) Den Hartog, J. P. MECHANICAL VIBRATIONS, 3rd EditionMcGraw-Hill, 1947
(2) Timoshenko, S. VIBRATION PROBLEMS IN ENGINEERING,2nd Edition, D. van Nostrand, 1937
(3) National Advisory Committee for Aeronautics, TechnicalNote 1909 of July 1949
10
APPENDIX I
OSCILLOGRAPH TRACES AND TABLES 2, 3, AND 4
^vmMmmmmmm
RUNi-2-C.= 257^ CPS
•"ftfwwmmmmim.-.v.:
PUM2-S
/ =357 "
RUM3-4-F^^ ^30 CPS
I FIG. 2
.-.VAWvU'iVvyVviVAv.VvV.-Avvv.vv\.-
1000 Lb LO/\D ON .SUPPORT BLOCKSf = nocp^
A ,', \
;
I W V VvV,VyVAV//A^AVA\^VWv\\
>5i?(9^ ^fe lOI^O 0^/ ^UPFORT BLOCKS-L^ IIDCPS
B£m NO s-i
I-/&.3
T <r> .
1000 Lb LOPiO ON SUPPORT BLOCKS
.
'AV>'A%WW^^^W^^MWV^»^xx»<w<ww««x»»»»»*<i»«
400016 LOAD OhJ ^UPpORT BLOCKS
i^-^lOCP3
BSAM NO 3--^-
F//;. f
\ A .A
lOOOLb. LOI\D ON SUPPORT BLOCKS
-t.- I40CP5
B£m f^O. 4
r/&. v5"
BEAM No. 3-1
Load onSupports
fAct.
fTheor. Diff. Error
300 lbs1000 lbs3000 lbs5000 lbs
10000 lbs
110110110110105
201201201201201
TABLE Ko.
9191919196
2
45.345.345.345.347.8
BEANi No. ->'k
Load onSupports
fAct.
fTheor., Diff.
$Error
300 lbslOOO lbs5000 lbs
10000 lbs
133140135133
268268268268
TABLE
135128133135
No. 3
50.347.849.750.3
BEAM No. 4
Load onSupports
fAct.
fTheor. Diff. Erro:
300 lbs1000 lbs2000 lbs3000 lbs4000 lbs6000 lbs
125140210220220230
178178178178178178
TABLE Nc
5338
-32-42-42-52
'. 4
29.821.4
-18.-23.6-23.6-29.2
15
APPENDIX II
BEAM DIMENSIONS
CROSS SECTIONBEAM NO. WIDTH THICKNESS AREA
1 .4754" .124" 5.39 X 10-2in2
2 .717S" .1243" 3.96 X lO'^in^
3 .9620" .1233" 11.9 X lO'^in^
MOMENT INERTIA
7.55 X 10-5in^
11.62 X lO'^in^
15.19 X 10"^in^
SPECIFIC WEIGHT .234 lb per cu. in.
RUN NO."T-r
1-2
BEAM NO. 1
LENGTH11.31"9.92"
2-12-22-32-42-5
BEAM NO. 2
10.70"10.06"9.57"9.02"3.52"
3-13-23-33-4
BEAIvi NO. 3
11.33"10.33"10.31"9.30"
BEAM NO. 4
Width - 1.0225"Thickness s .0312"Length . 6.04"Cross Section Area -
Area Moment Inertia =
3.192.59
10"2in210-6in^
16
Cd ' ^ (22.4) (30 X 10^) (2.59 X IQ-^) 0^6) ^^^-^^ rads/secTheor. (6.04)^ \ (.234) (3.19 x 10-2)
f . iiM , 173 cpsTheor. 2TV
17
APPENDIX III
FREQUENCY EQUATION TAKING INTO ACCOUNT SHEAR AND ROTARY INERTIA
EFFECT OF TRANSVERSE SHEAR AND ROTARY
The effect of shear and rotary inertia were determined
by modifying the results of E. T. Kruszewski in his article
"Effect of Transverse Shear and Rotary Inertia on the Natural
Frequency of a Uniform Beam" published in the NACA Technical
Note 1909 of July 1949.
In his article Kruszewski includes the effects of shear
and rotary inertia to give an equilibrium equation as follows:
EI ^-i-EI (!5Hi Jr 2«i) £z -(mw2,Im2w4) y ^ qdx^ AgG A^E dx2 A-pAgG
He then finds the solution to be
y - c, cosh Kg«(x + C2 sinh K^<x -^ C3 cos Kg^x -^ C/^ sin KgAx
His definitions are as follows:
Aj - effective shear area
Am - effective total cross section
P^ - mass of beam per unit length
tO - natural frequency
E - modulus of elasticity (psi)
I - moment of inertia (in^)
G - shear modulus (psi)
L - half length of a free-free beam
IS
VEI " L JAgG
L V At
V, 2 -
2
2B
Kruszewski then applies boundary conditions to his
solution to obtain the frequency equation for a cantilever,
symmetrically vibrating free-free beam, and an antisyraetri-
cally vibrating free-free beam respectively.
It was the author* s desire to have a frequency equation
for a clamped-clamped beam; this was obtained as follows,
(1) y s C;]^ cosh Kgo^x -\- C2 sinh KgoCx ^- C^ cos KgfixL L xj
-V ^i, sin Kgj^x
(2) y = Ci Kb<=^ sinh Kb^x -^ C2KB O^ cosh<<x - 03X2^ sin Kb^x
-V C^^Kb^cos Kg^x
(3) y = CiKg^ cosh K^x ^ OzH^^ *^^ ^rf^^ - ^3^^bV17 L l7~" L L2
cos Kb^x - C /^
KB^fi^ sin Kb^x
(4) T = Ci KbV sinh KbOCc ^C2 KbV cosh Kfi^ ^ C3 KbV]J L l3 L L3
sin Kb^x - C4 KB3>g3 cos Kb^xf L iJ" • L
19
Boundary Conditions:
At X - y =
X . 2L y
At X s O"^ 1 djr a 1 ryl 1 Ka^K2v_2B 1 d3;
Kb^ dx
KRI^) d£l X »OX
I
X 2L
Applying these boundary conditions:
(A) = Ci -V C3
(B) r Ci cosh 2 Kb(^-\- C2 sinh 2KbcK •\- C3 cos 2 Kb^-VC^ sin 2Kb^
(C) = C2rrKBc(3 ^(K.2^K.T2)KB.a[ K32KB^ \ Kb^(^LL^ LJ Vl-Kg^KRi^KB'^^; Lj
-^ cjriKo^ ^ Kri2) K£^ . Kb£^'
(D) « (Ci sinh 2Kg°^) \Kbc< i /Ks^Kb^4
\^L 'V1^k7Kr?Kb
\/kb^ »(Ks^-VKri2) Kb^I
2v 2+ (C2 cosh 2KB«OrM:j^j^+^Ks^4-KRi2) KqM^^K^
Ll v^^ L M-Ka2KRi2KBB^ {C3 sin 2Kb Kri2) Kb
L
'\ (C/^ cos
/3 ) P^B^^ /K32Kg2 X/kb^^ JKs^^.' ^L -^(l-Ks^KRi^KBJi^J
^Pkb^, /Ks^Kb^ \/KBft3 _(K32^KRi2) k^\l ^1-Ks2kri^Kb^J\1J" L 1
The characteristic values are obtained by setting the
determinant of the coefficients of the "C^s" equal to zero.
20
z
10 1
cosh 2Kg sinh 2Kb ^os 2Kb
(1)
(3) (4) (5)
sin 2Kb
(2)
(6)
(1) )M-V
(K,
u•VKri^) Kg^l (K^K.2Kb2 Kbc(
sZKjii^Kb^I
(2)e^Ks2
+ Kki2 ) Kgl - Kb£\ /'k82Kb2 X. ^B^,L L J \pc7%?%7 L
_^
K ^K 2(3) Kbo( K^\(K32-^K^,2) KBJlsinh2KB<
(4)
(5)
(6)
\ L "^\l3 L /1-Kg^I
cosh 2Kbo^
Kri Kb^
P^Bg K^^Kg^|
kb^3 . {K,2^Kri2) j^V^^^ ^KgA
^L "^l-Kg^KRi^KB^^lT" lJ} '
fKBp. Ks^KB^|
kbP^ . (Ks^-^%1^) KbMIcos 2Kb jS
!__L I-Ks^Kri^Kb^^ l7 L
/J'
Trial and error was employed by the author to solve for the
in this fourth order determinant.
It might be noted that Kruszewski in his article sets
his slope per unit length equal to the shearing strain per
unit length as a boundary condition at x and x - 2L.
Timoshenko prefers to use slope equal to zero at both ends.
Evaluation considering Timoshenko* s derivation and end con-
ditions was therefore considered advisable.
21
On pages 337-338 of Timoshenko's "Vibration Problems in
Engineering" , he derives the following differential equation
for the lateral vibration of prismatical bars:
it* "jde Vy%^&px^at % f(^ Dt^-^
if a solution y » XT is assumed where
X X Ci sin kx -V- C2 cos kx -V C3 sinh kx -^ C/^ cosh kx
T s (A cos pt 4 B sin pt)
And if we keep in mind that Kruszewski's
63 s p in Timoshenko
g, the above differential equation transforms to
Ei5^*Ex(e^;* if)!?-(--'--^*)ywhich is the same as Kruszewski has.
The symbols in Timoshenko* s equation are defined as
follows:
E s Young* s Modulus of elasticity
I s moment of inertia of cross section of beam
g s gravitational constant
G r shear modulus
5 « weight of material of bar per unit volume
k r numerical factor depending on shape of
cross section
>2At*
A s area of cross section of beam
1 z length of beam
22
dX^ s Cj^ {-k sin kx -j- k sinh kx) -V- Cg (-k sin kx -k sinh kx)
-V- C^ (k cos kx -V k cosh kx) -\- C • (k cos kx -k cosh kx)
Boxindary Conditions:
(1) lx)x.o =
(3) (X)x-1 =
(2)
¥i...o - °
(4) fdX=
l^/xsl
from (1) Cj^ s ^
(2) C3 =
from (1), (2), & (3); C2 (cos kl - cosh kl) -f C^ {sin kl
- sinh kl) 3
from (1), (2), & (4); C2 9-k sin kl -k sinh kl) -V C^ (k cos kl
-k cosh kl) s
A solution for C2 and C^^ other than zero can be had only if
(cos kl - cosh kl) (sin kl - sinh kl)
(-k sin kl -k sinh kl) (k cos kl -k cosh kl)
Evaluation of this determinant leads to the frequency
equation
1 s cos kl cosh kl
=
23
i
"It ti :)
AP 2260D I SPLAY
q ^ 6 1
Thesis g977iB125 Bading
Natural frtquencles ofsteel beams.
. AN 25AP 2260
Dl SPLAYP 5 6 1
Thesis 29771B125 Bading
Natural frequencies dfsteel beams*