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All Theses and Dissertations

1961-8

The Design of a Six Specimen Tensile FatigueMachineCharles Mark PercivalBrigham Young University - Provo

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BYU ScholarsArchive CitationPercival, Charles Mark, "The Design of a Six Specimen Tensile Fatigue Machine" (1961). All Theses and Dissertations. 7170.https://scholarsarchive.byu.edu/etd/7170

THE DESIGN OF A SIX SPECIMEN

TENSILE FATIGUE MACHINE

A Thesis

Presented to the

Department of Mechanical Engineering

Brigham Young University

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

by

Charles Mark Percival

August, 1961

i i

This th esis , by Charles Mark Percival, i s accepted in i t s present

form by the Department of Mechanical Engineering of Brigham Young

University as satisfy ing the th esis requirement for the degree of Master

of Science.

Typed by Carolyn P. Lloyd

ACKNOWLEDGEMENTS

The author wishes to express h is gratitude to Professor W, M.

Beebe for h is advice and guidance in th is project and the preparation

of the th esis , and to W. Ticker for his efforts in the preliminary

design. The encouragement and advice of Professor J. M. Simonsen i s also

appreciated. The author dedicates the th esis to his w ife, Janette, and

to h is parents, Mr. and Mrs. C. Guy Percival, whose encouragement and

understanding made th is thesis possible.

i i i

TABLE OF CONTENTS

iv

CHAPTER PAGE

I. INTRODUCTION ..................................................................................... I

The Problem 1

Definitions of Terms Used . . . . . . . . . . . . . . . 1

Survey of Metal Fatigue Studies . . . . . . . . . . . . 2

Design Decisions . . . . . . . . . . . . . . . . . . . 3

General Description of Fatigue Machine . . . . . . . . 4

II . TABLE ASSEMBLY . . . . . . . . . . . . . . . . ................. 7

The E n g in e ........................... 7

The Table « . . . ................ . . . . . . . . . ................. 9

The Drive Unit . . . . . . . . . . .................................. . 9

III, LEVER ARM ASSEMBLY AND TOP PLATE ............................................ 12

Spring and Spring Cups ................................... 12

Lever Arm . . . . . . . . . . . . . . . . . . . . . . . 16

Pedestal 16

Flexures .................... . . . . . . . . . . . 18

Flexure Mounts . . . . . . . . . . . .................... . . . 23

Tbp Plate . . . ........................ . . . . . . . . . . . . . 25

IV. INSTRUMENTATION............................................ 29

T im in g ....................................... 29

Load Measurement .................... . . . . . . . ..................... 31

Console . . . . . . . . . . . . . . . ................................. 33

V. OPERATIONAL ANALYSIS AND RESULTS ........................................... 36

Specimen S tress F ie ld Analysis 36

V

CHAPTER PAGE

Vibrational Analysis . . . . . . . . . ............................. . 39

Actual Operation . . . * ................... . . . 39

VI. SUMMARY............................................................................ 40

Conclusions * ............................................................ 40

Recommendations . . . . . . . . . . . ..................... . . . 40

BIBLIOGRAPHY .............................................................................. 41

APPENDIX . .......................................................................... 42

A. Spring Specifications . . . . . . . . . . . . . . . . 43

B. Calculations for Lever Arm ..................... . . . . . . . . 44

C. Forces in Specimen and Flexures . . . . . . . . . . . 45

D. Pin Shear Stress In the Pedestal Side and FlexureMount Joint . . . . . . . . . . . . . . . . . . . . 46

E* Calibration Procedure and Load Measurement . . . . . . 48

F. Specimen Stress Field Data ................................................ 50

G. Vibration Calculations .................................... . . . . . . 51

LIST OF FIGURES

vi

FIGURE PAGE

1. Fatigue Machine with Instrumentation . . . . . . . . . . 5

2. Fatigue Machine Showing the Following Assemblies*

(1) Table Assembly, (2) Lever Arm Assembly and

(3) Top Plate ....................................................... 6

3. Fatigue Machine, Table Details .................................... . . . 10

4. Fatigue Machine, Table Assembly ................................................. 11

5. Lever Am Assembly Showing the Following Parts*

(1) Spring and Spring Cups, (2) Lever Am, (3) Pedestal,

(4) Flexures, and (5) Flexure Mounts . . . . . . . . . . 13

6. Various Springs and Compressed Spring with Spring

Clamp Installed 15

7. Fatigue Machine Lever Am D etails, Sheet 1 of 2 Sheets . 19

8. Fatigue Machine Lever Arm D etails, Sheet 2 of 2 Sheets . 20

9. Fatigue Machine, Lever Arm Assembly 21

10. Am Flexure Mount, First Design ................................................ 24

11. Top Plate D etails, Sheet 1 of 2 S h e e t s ........................ 27

12. Top Plate D etails, Sheet 2 of 2 Sheets . . . . . . . . . 28

13. Shut Off B lo c k ................................... 30

14. Specimens Installed in Lever Arms ........................ . . . . . 30

15. Wiring Diagram for Timing S y s te m ........................................ 32

16. Wiring Diagram and Location of Strain Gages for

Load Measurement System . . . . . . . . . . . . . . . . . 32

17. Calibration Specimen . . . . . . . . . . ................. . . . 34

v ii

FIGURE PAGE

18. Calibration Setup . . . . . . . . . ........................................ 34

19. Console Shoving the Folloving Components:

(I) Timing Instrumentation, (2) Strain Gage Plug

Board, (3) Bridge Amplifier and Meter, and

(4) Switch and Balance ....................................... 35

CHAPTER I

INTRODUCTION

Although metal fatigue has been recognized for many years and

during that time many persons have tried to find the explanation for i t ,

there i s s t i l l not a complete theory concerning th is phenomenon. Metal

fatigue i s one of the major problems the modem designer must face. If

the design techniques are to progress as other technological advances

are made, i t i s evident that i t w ill involve a better understanding of

metal fatigue.

I. THE PROBLEM

The purpose of th is study was to develop a high force, ten sile

fatigue machine capable of testing large s ta t is t ic a l groups of specimens.

The machine represents a major addition to f a c i l i t ie s at Brigham Young

University for the study of metal fatigue.

II. DEFINITIONS OF TERMS USED

Metal fatigue. Metal fatigue i s the failure of a machine part

or te s t specimen under the action of repeated stresses. The maximum

values of these stresses may vary from a value sligh tly le s s than the

ultimate strength of the material to values well below the yield

strength of the material. The distinguishing characteristic of a

fatigue fa ilure i s that the stress i s usually repeated many times before

failure occurs.

2

Fatigue machine. A Fatigue machine i s a machine capable of apply­

ing cyclic loads or stresses to a specimen for testing purposes. These

loads may be simply tension, compresion, bending, torsion or a combina­

tion of any of these loads.

Ultimate strength. The ultimate strength of a material i s that

stress which applied to the material under sta tic conditions would

rupture the material.

Yield strength. The yield strength of a material i s that stress

above which permanent deformation of the material occurs.

Endurance lim it. The endurance lim it of a material i s that stress

which the material can endure for an in fin ite number of cycles without

failure.

One of the f ir s t to carry out te s ts of materials under repeated

loads was a German Engineer, Wilhelm August Julius Albert. Albert (1 ),

in 1829, bu ilt a machine to te s t mine hoisting chain with repeated

loading. Interest in metal fatigue increased when the use of wrought

iron and steel became common in construction of the early railroads.

In England, a Commission was appointed in 1847 to study the su ita b ility

of iron for railway structures.

During the middle and la tter portions of the nineteenth century,

A German, August Wohler (2 ), published several reports and papers con­

cerning the fatigue problem. Wohler conducted experiments on many types

of m aterials, and as a result of h is work, put forward what became known

as Wohler’ s law. Wohler in itia ted the practice of testin g specimens

instead of using fu ll scale te s ts , and his work has served as an

III. SURVEY OF METAL FATIGUE STUDIES

the ultim ate goal of th is work was to produce an operational

machine with a minimum amount o f rework and design changes. This,

coupled with the fac ts th a t only one of these machines was to be fab ri­

cated, and th a t many of the ideas fo r the machine were un tried , sug­

gested th a t the machine should be overdesigned in those areas where

i t s proper operation would not be affected . I t was f e l t th a t i t was

simpler and more economical to have the p a rts made too large than to

risk fa ilu re s in c r i t ic a l areas. For those p a rts where size was an

important fac to r, e ffo r t was made to keep the working s tre sse s well below

a c r i t ic a l lev e l. Wherever possib le the jo in ts and connections were

designed to take the loads with compression o r shear forces in the mem­

bers. These types of jo in ts are fa r le ss susceptable to fatigue fa ilu re .

I t should be s ta ted here, however, th a t certa in fea tu res on the

machine were incorporated a f te r in i t i a l fab rica tion to correct poor

c h a ra c te r is tic s observed in the i n i t i a l phases of shake-down operation.

The drawings presented with th is paper include these changes. Discussion

IV. DESIGN DECISIONS

3

example fo r many la te r investiga tions.

Since Wohler*s time, over 5,000 papers on the subject of fatigue

have beet published. (3) In te re s t in fatigue was g rea tly increased by

the two world wars and the advent of the a irp lane . I t was estim ated in

1950 th a t as high as 80 per cent of current mechanical fa ilu re s in

service were due to metal fa tigue. (4) From th is , i t i s seen tha t

g rea ter e ffo r t must be put fo rth to achieve a b e tte r understanding of

th is problem. The l i te ra tu re survey fa iled to reveal any information

d ire c tly applicable to th is p ro jec t.

o f these changes w ill be made in those sections concerning the parts

involved.

4

The fatigue machine i s bu ilt around a s ix cylinder automotive

engine. The head was removed and compression springs attached to the

top surface of the pistons. The engine i s driven through the crank shaft,

and the reciprocating action of each of the pistons i s used to compress

a spring against one end of a lever arm. The action of the spring

applies a force to that end of the lever arm. The arm i s pivoted about

a fulcrum and the specimen attached to the arm on the opposite aide of

the fulcrum such that the moment caused by the spring i s resisted by a

ten sile force in the specimen. Thus, the force of the spring i s transmit­

ted through the lever arm to the specimen.

The machine w ill te s t s ix specimens simultaneously, one for esch

cylinder, at approximately 350 cycles of stress per minute. Hie arrange­

ment i s such that the maximum load applied to the specimens can be

varied and the machine i s capable of loading some of the specimens at

one load level while the other specimens can be tested at another load

lev e l. Instrumentation i s Included to measure the time a particular

specimen has been stressed and to monitor the force applied to the

specimen.

The fatigue machine is divided into four assemblies! (1) table

assembly, (2) lever am assembly, (3) top p late, and (4) instrumentation.

(See Figure 1 and Figure 2 .) These assemblies and their Individual

components w ill be discussed in the following chapters.

V. GENERAL DESCRIPTION OF FATIGUE MACHINE

Figure 1. F atigue machine w ith in strum entation .

Figure 2. F atigue machine showing th e fo llo w in g a ssem b lies: (1 ) ta b le assem bly, (2) le v e rarm assem bly, and (3 ) top p la te .

ON

'■ u

CHAPTER II

TABLE ASSEMBLY

The table assembly i s composed of three main components*

(1) the engine, (2) the table, and (3) the drive unit. This assembly

serves as the base for the machine and provides the power and necessary

reciprocating action for the operation of the lever arm assembly.

I. THE ENGINE

As mentioned earlier , the reciprocating action necessary for the

operation of the lever arm assembly i s achieved by driving a s ix cylinder

automotive engine through the crank. 'Hie choice of an automotive engine

and, in particular, the type having s ix cylinders in lin e , was made for

several reasons:

1. The fatigue machine has no moving parts except those necessary

to obtain reciprocating action. The engine contains i t s own lubrication

system, thus eliminating lubrication problems.

2. Automotive engines are designed to res is t fatigue failures

for an in f in ite number of cycles. Replacement parts are readily available.

The loads on the top of the piston, which are representative of the loads

in the connecting rods, crank shaft, and bearings, are not as high in

the present application as they are in the actual operation of the

engine.

The mean e ffectiv e pressure for th is type of engine i s approxi­

mately 150 psi which represents an average force of 1,500 pounds on the

8

top of the piston. Maximum firing pressure i s approximately 600 psi

which represents a maximum force of 6,000 pounds. Hie largest springs

which w ill be used give a maximum of 2,000 pounds force when deflected

3.5 inches* Thus, the maximum force exerted by the spring i s one-third

of the maximum design load for the engine. The average force exerted by

the spring i s 1,000 pounds, which i s two-thirds of the force caused by

the average operating pressure. This means that l i f e as represented by

number of operating cycles of the engine used in the fatigue machine

should be as long as the l i f e o f the engine in actual operation.

machine*

3. The s ix cylinder automotive engine has what i s referred to as

a balanced crank shaft. This means that the connecting rods from the

pistons attach to the crank shaft at 120° intervals with each other.

Thus, the moment about the caster o f the crank shaft i s zero, except

for fr ic tion and the unbalance due to variation in spring constants,

i f springs having the same spring constant are attached to one of the

two sets of three pistons whose connecting rods attach to the crank

shaft at 120° intervals. This fact greatly reduces the peak power re-

quirement for the fatigue machine and eliminates the need for a large

f ly Wheel. Thus, the only power required i s that necessary to overcome

the moment due to fr ic tion .

The engine i s a 1954 Ford, 6 cylinders in -lin e type. This engine

has a 3.625 inch bore and 3.5 inch stroke. The engine had been used

previously, but i t has been overhauled and the head, valves, and

associated components removed prior to being in sta lled in the fatigue

I I . THE TABLE

The maximum power necessary to drive the fatigue machine was

estimated to be le s s than 5 horsepower. A 5 horsepower e lectr ic motor

was chosen because i t was readily available. The e lec tr ic motor i s

rated at 440 v o lts and 6.95 amps. I t operates on a 3 phase e lectr ica l

supply and turns at 1,160 rpm. A belt drive was chosen because of ease

of in sta lla tion and sim plicity of maintainence. The power i s trans­

mitted to the engine using two WAM type b e lts . A 5 inch adjustable

sheave i s attached to the e lectr ic motor and a 20 inch pulley i s

attached to the dutch end of the engine crank shaft. The pitch diameter

of the sheave can be adjusted by changing the width of the belt grooves.

The speed reduction from th is arrangement i s approximately 4 to 1 and i t

i s possible to make adjustments in the speed of the fatigue machine.

The table serves as a base to which the other components of the

fatigue machine attach. The center to center distance for the pistons

i s 4.25 inches. This space lim itation requires that the lever arm

assemblies be alternately placed on one side and then the other. Thus,

the table was b u ilt using the centerline of the engine as a centerline

for the table. The working drawings for the table are shown in Figure 3

and Figure 4. The table was fabricated from materials heavy enough to

ensure r ig id ity . The weight of the machine i s such, and the vibrations

are so small, that i t i s not necessary to bolt the machine down.

I I I . THE DRIVE UNIT

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CHAPTER III

LEVER ARM ASSEMBLY AND TOP PLATE

The lever am assembly Is that portion of the fatigue machine

which m ultiplies the spring force produced by the reciprocating action

of the pistons and applies the resulting ten sile force to the specimen.

As shown in Figure 5, the lever arm assembly can be divided into five

parts or groups of parts; (1) springs and spring cups* (2) lever arm,

(3) pedestal, (A) flexures, and (5) flexure mounts.

I. SPRINGS AND SPRING CUPS

Springs. The force in it ia l ly applied to the end of the lever

arm i s developed by compressing a helica l spring with an engine piston.

From the mechanical arrangement of the machine, i t i s seen that the

load in the specimen i s d irectly proportional to the force applied by

the spring. Thus, one of the methods used to vary the load in the

specimen i s to change the spring used to apply the force. To give a

wide range of loads, four spring sizes were chosen. These springs,

supplied by Rockwell-Standard Corp., Logansport, Indiana, were specially

designed to give 300, 1,000, 1,500, and 2,000 pounds force when

deflected 3.5 inches. Since the pistons have a fixed displacement of

3.5 inches, each spring i s compressed th is distance, and thus, the

maximum spring force i s fixed when a particular spring i s Installed .

All of the springs have the same outside diameter but the lengths are not

the same. A l i s t of spring specifications i s Included in Appendix A,

Figure 5. Lever arm assem bly showing th e fo llo w in g p arts: (1 ) spring and spring cups,(2) le v e r arm, (3 ) p e d e s ta l, (4 ) f le x u r e s , and (5 ) f le x u r e mounts.

w

14

Spring cups. The attachment of the spring to the piston and to

the end of the lever arm la accomplished through the use of spring cups.

The arm spring cup i s attached to the arm with one large bolt to allow

easy removal for changing springs. A f ia t was milled on the end of the

arm and an arm plate rig id ly insta lled to provide a true bearing surface

for the spring cup. Each spring cup i s attached to a piston by grinding

the top o f the piston f la t to mate with the base of the cup, and securing

the two parts with four bolts.

The cups have a one-inch lip which extends over the outside

diameter of the spring. The springs are secured in the cups by a small

amount of In itia l compressive load. Due to the difference in length of

the springs, i t was necessary to provide spacer blocks which are inserted

between the armplate and the arm spring cup when the shorter springs are

used. The spring cups, arm p late , and spacer blocks are shown in

Figure 8.

In sta lla tion . The springs are in sta lled in the assembled fatigue

machine by rotating the crank shaft by hand until the piston to be worked

on i s in the bottom dead center position. The spring i s set in the

piston spring cup, the arm spring cup insta lled on the top and the spacer

block and bolt in sta lled . When the larger springs are in sta lled , they

must be in it ia l ly compressed about one inch. The spring i s compressed

in a press and secured with clamps provided. A compressed spring with

clamps i s shown in Figure 6, The clamps are wired in place as a safety

precaution* The spring can then be Inserted in the piston spring cup

and the arm spring cup in sta lled . The crank shaft i s rotated until the

spring i s compressed s lig h tly beyond the in it ia l compression for

1,000 lb 1,500 lb 2,000 lb 2,000 lb . w ith clamp

I-igure o. Various sp rin gs and compressed spring w ith sp rin g clamp in s t a l le d .

K-»Ln

500 lb .

16

Installation and the wire and clamps removed. Springs are removed by

reversing the in sta lla tion procedure.

II. LEVER ARM

The lever arm i s the portion of the assembly which transmits the

moment caused by the force of the spring acting about the fulcrum and

produces the force on the specimen. It consists of a steel bar, 3 inches

in diameter, machined to assembly with the other parts. The d eta ils for

the lever arm is shown in Figure 7, Detail 5. The maximum bending

stress on the arm and the maximum deflection of the arm were calculated

and found to be below c r it ic a l values. The calculations are given in

Appendix B.

The specimen attaches to the arm by inserting through one of two

holes in the specimen end aid securing with a nut. These two holes

represent a means of adjusting the force in the specimen. The farther

the specimen i s from the fulcrum, the le s s the force required to counter­

act the moment caused by the spring. Thus, the specimen, i f inserted in

the hole closer to the fulcrum, w ill be subjected to a higher force.

Calculations were made of the force exerted on the specimen and on the

flexures when the specimen is in either of the two specimen holes and

any of the four springs are used. The tabulated resu lts are included in

Appendix C.

III. PEDESTAL

Each pedestal i s composed of a pedestal flexure mount and two

pedestal sides. (See Figure 7, Details 7 and 8 .) At the top, the

pedestal sides are joined with a pedestal flexure mount, as shorn in

Figure 9. The pedestal flexure mount has two specimen holes drilled in

i t which* on assembly, lin e up with the specimen holes in the lever

arm. Thus, the specimen i s attached to ground at the pedestal and

stressed by the action of the lever arm. The pedestal also serves as a

grounding device for the fulcrum. This means the moment caused the

spring must be taken by the pedestal and by the jo in t between the

pedestal sides and flexure mount. These parts were f ir s t positioned and

joined by two half-inch b o lts, a fter which four 3/4 inch holes were

drilled and lin e reamed in each side and precision stee l dowels inserted.

Calculations Included in Appendix D verify that th is jo int i s su ffic ien tly

strong. The height of the pedestal was raised using pedestal space

blocks such that the largest spring can be in sta lled without the use of

spring spacer blocks. The in sta lla tion of springs i s discussed in

Section I* The pedestal attached to the table with eight 1/2 inch bolts.

When the machine was f ir s t assembled, each pedestal was secured

only to the table. No other connections were made between the lever

arm assemblies. When the machine was running, the pedestals experienced

large deflections each time the spring applied force to the end of the

lever arm. This deflection was such that the top of the pedestal moved

In a direction away from the center of the table while the base of the

pedestal remained fixed to the table. An attempt was made to reduce the

deflection by typing together a ll pedestals on each side with a contin­

uous t ie bar across the top. I t was f e l t that since high loads were not

applied to the three pedestals on the same side at the same time, that

those pedestals in the low end of the load cycle would help res is t the

deflection of the pedestal with the high load. Upon operation i t was

18

observed that the deflections were reduced considerably but were s t i l l

present to the extent that they were undesirable.

Since the deflection of a ll of the pedestals was away from the

center of the table, i t was then decided to t i e the three pedestals on

one side to the three pedestals on the other side* Thus, the forces

causing the deflections would work against each other. A top plate was

in sta lled which tied the pedestals on the two sides together and, as an

added margin, the plate was tied to the table. The in sta lla tion of the

top plate reduced the deflection of the pedestals to essen tia lly zero.

The top plate i s discussed in Section VI.

IV. FLEXURES

The fulcrum about which the lever arm pivots i s composed of two

se ts o f flexures; one on each side of the lever arm. Each set of

flexures consists of a vertical and horizontal flexure. (See Figure 7,

Detail 6 .) The flexure ends are r ig id ly fastened to the arm and the

pedestal flexure mounts and the flexures cross at their mid-points making

right angles with each other. (See Figure 9 .) These flexures are thin

str ip s o f metal designed to transmit very l i t t l e force parallel to the

direction of the small dimension. The flexures are in sta lled such that

horizontal forces act in direction of the thin dimension of the vertical

flexures and vertical forces act in the direction of the thin dimension

of the horizontal flexures. Thus, the flexures o ffer very l i t t l e

resistance to the transfer of moments about mi axis through the point

where the flexures cross. Flexures accurately specify a pivot or fu l­

crum and at the same time have no moving parts which can wear out as the

ball bearing or knife edge type fulcruas have.

Figure 7 . F atigue m achine, le v e r arm d e t a i l s , sh eet 1 o f 2 sh e e ts .

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2 6 12 H O R IZO N TAL F L E X u R E

2 7 1 2 FLEXURE CORNER BLOCK VERTICAL

2 8 1 2 FLFX UR E CORNER Bt-OCK HORIZONTAL £ ? f F T r

2 1 2 4 FLEX ORE P ID E STA l. B O .T i - 2 0 N F * 1 XX.Hf T HtAO CAP SCREW

3 0 2 4 f l e x u r e a r m b o l t ^ n u t « - 2 0 N F . I S J L K fT HEAD CAR -jCHtW WITH HEX. RUT

31 4 8 CORNER BLOCK Sc R EW H U ? F IA T H .% . MACHINE SCREW I* CONG

3 2 1 2 UPPER PE D E S TA L a o d -L --2 ?O N F » l ME* HEAO B O lT

33 4 8 PEDESTAL OOWEl p in ^ OIA « i ^ wONG

3 4 12 A R M DOWEL PIN £ DIA . LONG

3 5 3 0 PEDESTAL SP A.E R G CA..SWGH, o£A LG H j« IAEA. 1.0 h ig h

3 6 4 8 P E D E S TA L BO LT’ P - Z O H F x l REX HEAD

W IIM N 1 * WALKERS

3 7 6 AR M P L A T E

3 8 1 2 SPRING SPACER BLOCK • £A ,»NIG M X GCA. I.< HIGH

3<1 6 A R M SPRING- CUP

4 0 2 4 COMPRESSION SPRING SEE NOTE- 1

4 1 6 P ISTO N SPRING- CUP

4 2 1 2 SPRING CUP B O L T x -IGNF n f X HEAObfA.-A.JNG A C A C ju M G

4 3 12 ARM Pl AFC Sl REW ^ -Z O W F » lj FLAT READ

+ + 2 4 PISTON CUP SCREW NO. K) M U S I MCAO I*LONG

PART NO. i f t 1* * *7M A T ‘ l — 4 *DIA. STEEL NO SCALE

DF I AIL 1 4 SPRING SPACER BLOCK.NOTESI. * SIZES OP St PRlNvxS AVAILABLE ( i f SOU , KJUO , l*DO , COOO lB )

OF M R T i i a , 4 0 , * Z , MUST » F AS TAAUL A fC O1 .c'AlN it SIZE j PAW I K 4U

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BRIG HAIM YOUNG UNIVERSITY MECHANICAL ENGINE CHlMx JEPARt WENT

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AS NOTED

DRAWING NO.. | F M - 14 -

FATl&UE MACHINE LEVEK ARM

DETAILS -sac**cN C tA f j>

Figure 8 . F atigue m achine, le v e r arm d e t a i l s , sh eet 2 o f 2 sh ee ts N5o

FOP PARTS LIST SEE OR AW I Mr N O FM-I4-

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DRAWING NO- | FAA-15 OH AWN BYc.»u l.f««JrATI&UF MACHINE

LEVER ARM ASSEMBLY

v ai r CHECKED BY

Figure 9 F atigue machine, le v e r a m assembly

22

By putting a set o f flexures on each side o f the am such that

each set crosses at a point coincident with the center of the lever am,

an axis of rotation for the lever am is specified. This axis i s a

lin e through the points where the flexures cross. Care was taken that

a ll force applied to the am acted in a direction perpendicular to that

axis.

Since most of the loads applied to the am are in the vertica l

direction, the vertical flexures were made heavier than the horizontal

flexure. The horizontal flexures are mainly for positioning purposes

and thus, the loads in them which may be either ten sile or compressive,

are re la tive ly low. The loads in the vertica l flexures are high but

due to the mechanical arrangement of the machine, they are always com­

pressive.

The flexures are fabricated from 17-4-PH Stainless Steel which

i s a precipitation hardening s te e l. The flexures were hardened to

Rockwell C 45, which corresponds to an ultimate strength of 200,000 p si.

The maximum stress in the flexures occurs when the 2,000 pound springs

are used and i s approximately 88,000 psi compressive. This stress i s

below the ultimate strength of the material and the buckling stress for

the part, since stresses in the flexures are compressive, there should

be no problems with fatigue fa ilures of the flexures.

The flexures are attached to the pedestal and to the lever am

through the use of flexure mounts which are discussed in the next

section. They are attached to these mounts by securing the flexure in

a corner with a bolt inserted diagonally through the comer of the

flexure and the flexure mount. Thus, in those jo in ts where the forces

flexure mounts are used to a ttach the flexures to the lever am

and to the ground o r pedesta l. The method of attachment o f the flexures

to the mounts was discussed in Section IV, The flexure mount on the

pedestal was designed as p a rt of the pedestal while the mount fo r the

am was separate from the am and in s ta lle d on the am .

Am flexure mounts. The f ir s t design of the am flexure mounts

i s show in Figure 10. This mount was in sta lled with the lever am

extending through the large vertica l s lo t and a bolt through the top of

the mount securing the two together. The bottom lip for the vertical

flexure fa iled at i t s base on two of these mounts when the 2,000 pound

spring was used. This represented a force of 11,000 pounds on the l ip .

It i s f e l t that the fa ilu res were due to the high stress concentration

factor resulting from the sharp comer factor at that point. The calcu­

lated stress at that point without the consideration of the stress

concentration factor was 44,000 pounds per square inch. This high

stress indicates an error in the original design.

The am flexure mount block was redesigned as shown in Figure 7,

Details 9 and 10. Much more material was included in the flexure lip

and the whole piece was made more rigid . The sharp comers at the base

o f the l ip s giving r ise to the high stress concentration were eliminated

by including a large radius and then inserting a comer block to give

a bearing comer for the flexure. Calculations showed the stress on the

lip i s reduced considerably by the change. Additional corrective

23

between the flexures and the flexure mounts are high, compressive type

jo in ts resu lt.

V. FLEXURE MOUNTS

24

Figure 10. Arm fle x u r e mount, f i r s t design

25

measures were taken by shot-peening the radius and adjoining material*

Shot-peening (6) i s an operation whereby the surface of a material is

bombarded with small steel shot. This procedure covers the surface

with a layer of compressive stress a few thousands of an inch thick and

acts to protect the surface from the formation of ten sile cracks. If the

load at which the ten sile cracks form can be raised, the fatigue l i f e of

a material can be shown to increase.

The new arm flexure mounts were in sta lled by inserting the lever

arm through a close f it t in g hole in the arm flexure mount. The two

parts are secured together with two precision stee l dowels inserted in

holes drilled and reamed in place. The close f i t between the flexure

mount and the lever create an effective joint to transfer the moment.

The dowels are used primarily for positioning.

Pedestal flexure mounts. The pedestal flexure mounts now being

used are those originally designed. (See Figure 7, Detail 7 .) These

flexure mounts also include the sharp corner on the lip s similar to the

old arm flexure mounts. However, there i s more material in the pedestal

flexure mounts which prevent fa ilures in the lip s . When the arm

flexure mounts were redesigned, i t was decided to rework the pedestal

flexure mounts. The corners of the lip s were rounded to relieve the

stress concentrations and then the comers and adjacent material were

shot-peened.

VI. TOP PLATE

As discussed in Section III , a top plate was added to prevent the

deflection of the pedestals. This top plate i s a large stee l plate 5/8

inches thick having the same external dimensions as the overall

dimensions of the table. The center lin es of the plate coincide with

the center lin es of the table. This top plate attaches to the top of

each of the pedestals and the center of the top plate i s fastened to

the center of the table using two vertical t i e down plates. Figures 11

and 12 show the top plate d eta ils .

The mechanical arrangement of lever arm assembly i s such that

when a specimen fa i ls , the lever arm to which i t was attached i s free

to ride up and down on the spring instead of compressing i t . Unless

th is violent o sc illa tio n is suppressed i t w ill damage the flexures and

possibly other parts of the machine. To prevent th is from happening,

one and one*half inch holes were drilled and tapped in the top plate

over each of the pistons. Bolts are threaded in these holes to a point

such that the top of the lever arm during operation of the machine just

clears the bottom of the bolt. When the specimen f a i ls , the arm comes

to rest against the bottom of the bolt and the force of the spring i s

transmitted d irectly to the top plate.

Figure 11. Fatigue machine, top p la t e d e t a i l s , sheet 1 o f 2 sh ee ts

■;C3

, i C “K

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52 2 T IE DOWN P LA TE

53 « STOP BOLT AND NOT i f - » 7 { HSA MEAD

54 2 T IL BAA

5 5 2 A TO P P LA TE BOLT4 - -2 0 N E 0 i 50C.KET HEAD

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DETAIL 21

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BRIGHAM TOILRANGE DRAWING NO . I F M - 1 7 DRAWN *Y

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T O P P L A T E

D E T A I L S

N>00Figure 12. Fatigue machine, top p la t e d e t a i l s , sheet 2 o f 2 sh ee ts

CHAPTER IV

INSTRUMENTATION

There were certain features which are included with the machine

to measure certain parameters o f operation. Included in these items

are the timing system, the load measurement system, and the control

console.

I. TIMING

The measurement of the number of cycles of stress for a particu­

lar specimen i s accomplished by measuring the number o f cycles of stress

per minute and the total time of stress cycling. The number of cycles

per minute i s found by measuring the rotational speed of the engine with

a tachometer. The time of stress cycling i s measured using an e lectr ic

clock and a shut o ff device that stops the clock when the specimen fa i ls .

The shut o ff block i s shotm in Figure 13. The shut o ff device i s a

block through which the specimen i s inserted on in sta lla tion in the

fatigue machine. The specimen holes in the pedestal and t i e bar are

oversized. Thus, the specimen can be aligned in the machine by correctly

positioning the shut o ff block over the specimen hole.

To the underside of the block are attached ligh t springs Which

hold the block up o ff the t i e bar when no force is applied to the top

of the block. There i s also a micro-switch Which i s normally open in

th is up position . When the specimen i s tightened on in sta lla tion , the

load compresses the springs which seats the block firmly on the t ie bar.

30

Figure 13. Shut o f f b lock .

55

Figure 14. Specimens in s t a l l e d in le v e r arms.

31

This c loses the micro-switch and when the control switches are turned on,

starts the clock. When the specimen fa i ls , the load holding the block

down i s released and the block springs to the up position. This opens

the micro-switch and shuts o ff the clock. Control switches provided

make i t possible to turn any or a ll of the clocks o ff and on as required.

A wiring diagram of the timing system i s shown in Figure 15.

II . LOAD MEASUREMENT

The dynamic load on the specimen i s measured with e lec tr ica l,

resistance type, strain gages on the upper and lover surfaces of the

lever arm at the points of maximum bending stress. The bending stress

in the lever arm i s d irectly proportional to the load on the specimen

and th is stress can be measured with the strain gages. The strain gages

were in sta lled in a four active element Wheatstone bridge to measure

bending stress. The in sta lla tion used on the lever arms i s shown in

Figure 16,

A Bridge Amplifier and Meter, Model BAM-1, for measuring strain

at the gage locations and a Switch and Balance, Model BSG-6, to

fa c il ita te the measurement of more than one channel of information were

purchased from E llis Associates, Pelham, New York,

Using the calibrated strain gages on the lever arm i t i s possible

to make dynamic measurements of specimen loads by connecting the output

of the Bridge Amplifier to an oscilloscope or oscillograph. The strain

gage system on the lever arm i s calibrated using a calibration specimen

prepared for that purpose. The calibration specimen has a larger cross

section area than the specimens which would normally be used In the

fatigue machine, (See Figure 17.) Four resistance type strain gages

32

KEY

A CONSOLE MASTER SWITCH

B . ' CuOr K CONTROL SWITCH

C . M IC R O S W ITCH

D. E L E C T R I C C L O C K

Figure 15. Wiring diagram for tim ing system.

NOTE S1. STRAIN GAGES A R E ,T Y P E A -7

LOT NUMBER. 2 - 51G A G E F A C T O R I .T 8 * 2 ^ G A G E RE SIS TAN C E 120 OHMS

2 . COVER G A G E S WITH WAX AND PLASTIC TAPE FOR P R O TE C TIO N

G A G E S ON LEVER ARM

BRIDGE AMPLIFIER

B L O C K SCHEMATIC O F C I R C UI T

Figure 16. Wiring diagram ana lo c a t io n o f s tr a in gages for load measurement system.

33

were attached to the specimen at i t s midpoint and equally spaced around

i t s circumference. The calibration set up i s shown in Figure 18. The

procedure for calibration and load measurement are included in Appendix G.

III. CONSOLE

The clocks and switches for the timing mechanism and a plug

board for the leads from the strain gages were insta lled on a console.

This console shown in Figure 19, provides a central location for the

fatigue machine instrumentation.

34

Figure 17. C alibration specimen.

Figure 18. C alibration setup

Figure 19. Console showing the fo llo w in g components: (1) tim ing instrum entation , (2) s tr a ingage plug board, (3) bridge a m p lif ie r and meter, and (A) switch and balance.

Ul

CHAPTER V

OPERATIONAL ANALYSIS AND RESULTS

The purpose of the work discussed In th is chapter is to provide

information pertaining to the operational characteristics of the fatigue

machine. The work involved i s divided into three areas: (1) specimen

stress f ie ld analysis, (2) vibrational analysis, and (3) actual opera­

tion.

The fatigue machine i s designed to apply an axial ten sile force

to a specimen. Due to rotation of the lever arm, deflections and mis­

alignments, bending stresses are induced in the specimen and superimposed

on the axial ten sile stresses. In order to obtain meaningful resu lts in

a ten sile te s t , i t i s necessary that th is braiding stress be maintained

at a low lev e l. Tests were conducted to determine the magnitude of the

bending stress induced by the machine.

To test for bending, a calibration specimen with four active

strain gages was insta lled in the machine. A description of the c a li­

bration specimen is included in Appendix G. Measurements were taken of

the stress at various locations on the specimen as the load was applied.

The load was applied dynamically by operating the machine and sta tica lly

by turning the machine over by hand. The read out from the various strain

gages was observed on an oscilloscope. Since the tota l resultant axial

load due to bending i s zero, the average of the stresses in various

I . SPECIMEN STRESS FIELD ANALYSIS

37

locations for each load was assumed to be due to the axial load. The

percentage difference of the values at different positions from the

average value was taken as a measure of the effect of bending at that

point.

Measurements of the stress were taken at intervals as the specimen

was rotated in the specimen hole and the same load applied each time.

From these tr ia ls i t was possible to pick the positions of maximum and

minimum stresses. These stresses usually occurred 180° apart and on a

lin e coincident with the centerline of the lever arm. The maximum stress

usually occurred at the point on the specimen farthest from the fulcrum,

which Is where the maximum displacement would occur. Representative

values of the percentage bending for both the dynamic and sta tic situa-

tlons are included in Appendix H. From these i t can be seen that there

i s l i t t l e difference between the two cases. The average percentage

bending observed was 10 per cent.

From observations taken under various conditions, the following

were observed to affect the amount of bending:

1. In itia l angle between lever arm and horizontal. The larger

the in it ia l angle the greater the bending stresses. Bending moment i s

applied as the specimen was stressed and the nuts tried to seat on non­

parallel surfaces.

2. Preload. High preload on the specimen increased the in it ia l

angle of the lever arm and thus increased the bending.

3. Maximum load. The percentage bending decreased as the maximum

load on the specimen Increased. The higher loads allowed the specimen to

deflect and resis t more of the load as ten sile stresses

38

4. Specimen hole. The percentage bending Is greater for the

inside specimen hole. This hole is closer to the fulcrum, thus, the

elongation of the specimen causes a greater rotation of the lever arm.

5. Specimen alignment. The alignment of the specimen in the

specimen holes was found to affect the bending stress.

The amount of bending was measured on a specimen 0.50 inches in

diameter. This i s a larger cross section than i s proposed for test

specimens. Test specimens having a square cross section, 0.30 inches on

a side, have been used in the machine and are proposed for future use.

The same bending moment acting on both the calibration specimen and the

test specimen causes a maximum axial bending stress In the calibration

specimen which i s three times the bending stress in the square test

specimen. Thus, i f the bending load on the two different specimens i s

the same for the same axial stress, the e ffect of bending in the test

specimen should be one-third that observed in the calibration specimen.

It i s f e l t that the e ffect of bending can be further reduced by

determining the best in it ia l lever arm angle and adjusting the

dimensions on the lever arm assembly to obtain th is angle. The lever

arm angle can be controlled by the height of the spring and spring cup

portion of the assembly. This can easily be adjusted with shims between

the arm plate and arm spring cup. The bending can possibly be reduced by

changing the method of attachment of the specimen to the pedestal and

lever arm. The resulting method of attachment should be such that the

specimen i s allowed to align properly without applying bending. A pos­

sib le solution may be something on the order of a ball and socket or a

very s t i f f spring such as a B e llev ille spring.

39

The total axial load on the specimen during actual operation was

compared to the calculated values for a ll springs and specimen hole

combinations. The resu lts were seen to compare favorably. The variation

between the actual and calculated loads i s most lik e ly due to the fact

that the springs used vary somewhat from their rated force when com­

pressed.

The trace of the tota l axial load in the specimen during actual

operation was observed. The load i s applied approximately sinusoidally

with no discontinuties or sudden load impacts.

II. VIBRATIONAL ANALYSIS

A vibrational analysis was made to determine i f the natural fre­

quencies o f the components of the machine were within the range of the

operating frequencies. It was found that a ll c r itic a l components are

su ffic ien tly rigid to have natural frequencies far greater than the

operating frequency of 5 to 7 cycles per second.

The lever arm assembly is that portion of the machine most lik ely

to vibrate at low frequencies. The lowest natural frequency of th is

system was calculated, (see Appendix I) and was found to be 2S0 cps.

These calculations and visual observation indicate that the

vibration of the machine or i t s components w ill have no e ffect on i t s

mechanical operation.

Following it* final assembly, the fatigue machine was operated

continuously for 300 hours. During operation, visual observation indi­

cated that a ll components of the machine functioned properly. The machine

runs with n eg lig ib le vibration.

111. ACTUAL OPERATION

CHAPTER VI

SUMMARY

The goal of th is work, as described in Chapter I, was achieved.

Conclusions and recommendations concerning th is study are lis ted below.

I. CONCLUSIONS

The conclusions which may be drawn from th is work are?

1. The mechanical arrangement used for th is machine is suited

for the operation.

2. The fatigue machine, in i t s final assembled form, w ill perform

sa tisfactorily ; no future failure of parts i s expected.

II. RECOMMENDATIONS

The following recommendations are made concerning the fatigue

machine described in th is paper and future fatigue machines of th is type.

1. The study of bending in the specimen should be continued and

a method of specimen attachment to reduce the effect of bending should

be devised.

2. In future machines the three pedestals on one side should be

machined from one piece of material, possibly a casting. This would

reduce machining and increase r ig id ity .

3. All flexure mounts should have the same type of flexure

attachment as in the la tes t design for the arm flexure mounts.

BIBLIOGRAPHY

41

1. H. F. Moore and J. B. Kommers. The Fatigue of Metals. New York:McGraw-Hill Book Company, Inc., 1927. p.~T0.

2. Reference 1, pp. 12-14.

3. J. Y. Mann. "The Historical Development of Research on the Fatigueof Materials and Structures," The Journal of the Australian In stitu te o f Metals, Vol. I l l , No. 3, November, 1958. p. 238.

4. M. Hetengi (ed .) , Handbook of Experimental Stress Analysis. NewYork* John Wiley and Sons Inc., 1950. p. 593.

5. J. 0. Alraen. "Shot Blasting to Increase Fatigue Resistance,"Society of Automotive Engineers Journal (Transactions).Vol. 51, No. 7, July, 19431 ^7 7 W . 6

6. Timoshenko and Cl ease* H. MacCullough. Elements of Strength ofMaterials. New York* D. Van Nostrand Company, Inc., 1949. p. 44.

APPENDIX

APPENDIX A

43

SPRING SPECIFICATIONS

Approx* 500 lbs. at 3-1/2** deflection - .375” centerless ground AISI-5160 3-7/16" o.d. +/- 1/32"8-1/2 c o ils7- 11/16" free length + /- l/8«Rate - 136 to 150-lbs. per inch

Approx. 1000 lbs, at 3-1/2" deflection - .437” centerless ground AISI-5l6o 3-7/16" o.d . + /- 1/32"8- 1/2 c o ils7 - 23/32" free length + /- 1/8"Rate - 272 to 300 lbs. per inch

Approx. 1500 lbs at 3-1/2" deflection - ,500" centerless ground AISI-5160 3-7/16" o.d. +/- 1/32"9.8 c o ils8 - 27/32" free length +/- 1/8”Rate - 408 to 452 lbs. per inch

Approx. 2000 lbs, at 3-1/2" deflection -.531" center!essground AISI-5160 steel 3-7/16" o.d . + /- 1/32"9.8 c o ils9- 1/8" free length +/- 1/8"Rate - 544 to 600 lbs, per inch

All Springs have the following Specifications;Magnaglo bar before coiling Qads closed and ground square Harden and draw Rc 48 - 52 PressShot peen and heat 450° max.Must have fu ll coverage100 per cent magnaglo inspectionMust be free from a ll tool marks, nicks and surface defects Oil fin ish

APPENDIX B

CALCULATIONS FOR LEVER ARM

The formulae for the axial bending stress in a beam and for the

deflection of a cantilever beam vith the load applied at the free end are

S - JUL

a - J L L -3 E I

where

S i s the bending stress at a particular section M is the moment at that sectiony is the distance from the neutral axis at which the stress occurs I i s the moment of Inertia of the section about the neutral axis

A i s the deflection of the free end P is the load applied at the free end L is the length of the beamE i s the modulus of e la s t ic ity of the material in the beam.

Using the above formulae mid assuming the arm to be a cantilever beam,

b u ilt in at the specimen-fulcrum end, the maximum axial bending stress

was found to be 15,000 p si. The arm i s made of tool steel with an

ultimate strength of 100,000 psi. The endurance lim it of a material i s

approximately one half the ultimate strength. Thus, the stress in the

arm i s well below the endurance lim it o f the material. The banding

deflection was found to be 0.044 inches

APPENDIX C

FORCES IN SPECIMEN AND FLEXURES

The forces on the specimen and the flexures were calculated for

the various specimen hole and spring combinations. The actual forces

on the specimen were measured after the machine was completed.

Spring Si ze

(Pounds)

2,000

2,000

1.500

1.500

1,000

1,000

500

500

Specimenhole

Inside

Outside

Inside

Outside

Inside

Outside

Inside

Outside

Calculatedflexureforce(Pounds)

22,000

14.000

16,500

10,400

11.000

7,000

5.500

3.500

Calculatedspecimenforce(Pounds)

19,900

12,000

15.000

8,900

10.000

6,000

5.000

3.000

MeasuredSpecimenforce(Pounds)

19,000

11,500

15,400

9,100

9,400

6,000

4,700

2,900

APPENDIX D

46

PIN SHEAR STRESS IN THE PEDESTAL SIDE AND FLEXURE MOUNT JOINT

Using the method outlined by Timoshenko and MacCullough (6) the

shear stress in the pins between the pedestal sides and the pedestal

flexure mount was determined. When the 2,000 pound spring i s used, the

tota l moment on the joint i s 44,000 inch-pounds. This represents the

highest load condition. The load calculations were made using the

following assumptions; (1) The jo in t on one side of the pedestal r e s is ts

half of the moment. (2) The centroid of the pin pattern is at the

center of the half inch bolt. (See Figure 9 .) and (3) The bolt takes

none of the load. The formulae used are l is te d below.

where

F* i s the force on one pin in the direction of the force causing the moment ( ie , direct shear force)

P i s the force causing the moment on the joint n is the number of pins in the jointFM is the force on a pin in a direction perpendicular to a lin e

from the centroid of the pin pattern to the pin M is the moment about the centroid of the joint % is the distance from the centroid o f the bolt pattern to any pin

The vector addition of F* and F" resu lts in the to ta l force on a particu­

lar pin. (Fn - Fn* + Fn”)

f« » - u q qo4 250 lb for a ll four pins

47

RA “ Rg " Rc “ 1»45 inches RQ * 1.30 inches

2 • R 2 + R 2 + R 2 + R 2 - 7.99 inches2 n A B C D

F” - F" - F" - 4,000 lb P • 3,580 lb A B C 0

Fa - 3,800 lb Fb - 4,180 lb Fc - 3,800 lb FQ « 3,700 lb

The cross sectional area of the 3/4 inch diameter pins used in the joint

i s 0.44 inches2. Therefore the maximum shearing stress on the pins used

in the pedestal side and flexure mount joint i s 9,500 p si. Under dynamic

loads the shear stress i s le s s than or equal to the normal stress. The

pins used have a minimum ultimate stress of 50,000 psi which means the

endurance lim it for the pins i s more than twice the applied stress.

APPENDIX E

48

CALIBRATION PROCEDURE AND LOAD MEASUREMENT

1. Calibrate the calibration specimen* by comparing the read out

from the four strain gages connected in series to a known load applied to

the specimen by a tension testin g machine,

2. Insert the calibrated specimen in the fatigue machine and

complete the required wiring using the four specimen gages in series.

Wire both the lever arm bridge and the calibration specimen bridge into

the Switch and Balance. Connect the Switch and Balance to the Bridge

Amplifier.* 2

3. Balance the calibration specimen bridge and calibrate the read

out using the proper constant.

4. Balance the lever arm bridge.

5. Using a hydraulic jack, apply a force to the bottom of the

lever arm midway between the flexures and the spring cup end. Observe

load on the calibration specimen using the meter on the Bridge Amplifier

to prevent overloading of the system.

6. When the desired load i s reached, switch to the lever arm

bridge and adjust the amplifier gain to achieve the sane reading.

7. Remove the load and push the calibration knob on the Bridge

*The calibration specimen te s t section i s 0.50 inches in diameter by 6 inches long. Four BLH SR-4 strain gages are attached to the middle of the specimen and equally spaced around i t s circumference. The strain gages attached are type AD-7 with a gage factor of 1.97 » 2 per cent and a gage resistance of 119.5 » 0.3 ohms.

2For complete operation instructions refer to Instruction manuals for these two instruments.

49

Amplifier and note the calibration constant for the lever arm bridge,

8, The calibration i s repeated for each specimen hole at various

load ranges.

9. After the lever arm bridge has been calibrated, the load on

a specimen i s measured by using the proper calibration constant and

measuring the output of the lever arm bridge.

APPENDIX F

50

SPECIMEN STRESS FIELD DATA

The measurements were taken in the positions shown.

To change the load measurement to a stress, divide the load by

the area of the calibration specimen, 0.196 in^.

specimenspring mad of lever arm

Position Load in Deviation Percent1,000 lb. from average Difference

RUN 1. Dynamic, 2,000 lb spring, Outisde specimen holeA 12.2 +.6 +5B 11.5 * .l -1C 10.8 8 -7D 11.9 +.3 +3

RUN 2. Static 2,000 lb spring, Outside specimen holeA 12.1 +.7 +6B 11.6 +• 2 +2C 10.5 - .9 -8D 11.4 0 0

RUN 3. Dynamic, 500 lb spring. Outside specimen holeA 3.3 +.3 +10B 3.2 +.2 +7C 2.5 *♦5 -17D 3.0 0 0

RUN 4. Static , 500 lb spring. Outside specimen holeA 3.3 **4 + 14£ 3.0 ♦ a +3C 2.4 -17D 2.8 • a -3

APPENDIX G

51

VIBRATION CALCULATIONS

The natural frequencies of the lever ana assembly were calculated

using an approximation for the system involved. The calculation was made

by assuming the lever to be a rigid mass supported on three springs. This

system was given two degrees of freedom, the vertical displacement x of

the center of gravity of the lever arm and the rotation 0 of the lever

arm about i t s center of gravity. The items represented by springs are

the specimen, the flexures and the spring attached to the end of the

lever arm.

The approximated system i s shown below. The spring constants K

for the specimen and the flexures were calculated using the e la s t ic ity

of these pieces; the spring constant for the spring was already known.

The mass M, the location of the center of gravity, and the mass moment

of inertia J about the center of gravity for the lever arm, arm flexure

mount, and spring cup combined were calculated using the weights and

shapes of the pieces.

52

Kj (specimen) « 4.5 x 10 ̂ Ib/tn.K2 (flexure) + 1,5 x 10° lb /in ,Kj (spring) • 144 lb /in .Lj - 10.7 inches L« * 7.4 inches L3 - 12.5 inches M - 0.284 lb -sec2/in . J - 17.3 lb -sec2-in .

The radius of gyration r of the lever arm combination about i t s

center of gravity i s J/M, thus r2 was found to be 61 in2.

The equations of motion for translation and rotation are

F « M aand

M* » JoCwhere

Fx i s a force in the X direction acting on a body M is the mass of the bodya i s the acceleration of the body in the X direction M* is a moment about the center of gravity of a body J i s the mass moment of inertia of the body about i t s center

of gravitycC is the angular acceleration of the body about i t s center of

gravity

Assuming small o sc illa tio n s and undamped motion, the equations

of motion of the lever arm system are as follows*

1. M x - Ki (x - Lj6) - K2( x - L28) - K3( x + L30)

and

2. J e - L j(x - Lj©) - K2 L2( x - L2e ) - K3 L3( x + L3&)

where

and © » oC » iL_® dt

Equations 1 and 2 can be reduced rearranging and substitution

3, x + AX - B© - 0

4. 6 + B — 2r

x » ce - owhere

5,M

53

6.and

7.

R . K3L3 - K2L2 " H h lM

C - K1L12 ♦ k2l 22 * H H 2

r2 M

The principal modes of vibration to occur the motions in the X and

© directions must be harmonic at the same frequency* therefore the

solutions to the two d ifferen tia l equations are of the form

8. x - X cosuJt

and

9. 0 « 6 cosoot.

Substituting equations 8 and 9 into equations 3 and 4, the fo l­

lowing equations are obtained:

(A - CO2) X + B e - 0

~ X + (C - Ui2 ) 6 “ o rzSolving for the amplitude ratio x in equations 10 and 11 one

eequation in one unknownuj i s obtained.

12. _JL " - B - . C -ou2® A "-CO2 - J L

r l2

Solving for UJ in the above equation

13. < 2 - \ <A+C) i / i < A - C ) 2 7 -5-2

Substituting in the numeric values of A, B, and C, the values for

i and (^2 were found to be 1,600 and 3,500 radians per second respec­

tiv e ly . The frequency in cycles per second i s equal to . Thus the

natural frequencies for the lever are system were found to be 250 cps

and 590 cps using a two degree of freedom approximation.

10.and

11.

THE DESIGN OF A SIX SPECIMEN

TENSILE FATIGUE MACHINE

An Abstract of

a Thesis

Presetted to the

Department of Mechanical Ehgineering

Brigham Young University

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

by

Charles Mark Percival

August, 1961

ABSTRACT

The purpose of th is study was to develop a high force, ten sile

fatigue machine capable o f testing large s ta t is t ic a l groups of specimens.

The uoxk included the design of the machine and supervision of i t s

fabrication. Load measurement devices mere designed and calibrated,

and a brief operational analyste o f the machine was made.

The system made uaa of the reciprocating action o f an automotive

engine when driven through the crank shaft. Attaching a spring holder

to the top of each piston made i t possible to use the linear displace­

ment of the piston to compress s spring, when the spring i s compressed,

i t applied a force to one end of a lever arm.

The system i s arranged to te s t one specimen for esch cylinder of

a s ix cylinder automotive engine. This type of engine was chosen be­

cause i t has a balanced crank shaft. That i s , each of the connecting

rods from the pistons are attached to the crank at 120° intervals.

Assuming that there ie no fr iction and that a ll springs have the same

spring constant, the moment on the crank due to the spring forces would

be sero. Thus, the peak power that would be required to drive the

machine i s greatly reduced.

Steel flexures are used aa a fulcrum for the lever arm. The

moment Induced by the spring forces i s resisted by attaching a specimen

to the arm at a point on the opposite side o f the fulcrum. This

arrangement allows the load level in the specimen to be varied by changing

2

the spring used to apply the force to the lever arm, and by changing the

distance of the specimen from the fulcrum.

The load on the specimen i s d irectly proportional to the strain

caused by the bending load or stress in the lever arm. Thus, measure­

ment of the load in the specimen is accomplished by applying strain gages

to the lever arm. An arrangement i s made whereby the time of stress

cycling was measured using e lectr ic clocks.

The work also included an operational analysis of the fatigue

machine. This consisted of Instrumentation calibration, an analysis of

the stresses induced in the specimen by the machine and a study of the

vibrational characteristics of the machine.

APPROVED: