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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

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AP 2260D I SPLAY

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Thesis g977iB125 Bading

Natural frtquencles ofsteel beams.

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Natural frequencies dfsteel beams*