Brigham Young UniversityBYU ScholarsArchive

All Theses and Dissertations

1966-05-02

Preliminary Design of a Hydraulic VibrationMachine with Variable Amplitude and Frequency,Using Multistage Amplification and FeedbackControlMelvin Joseph MerrellBrigham Young University - Provo

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BYU ScholarsArchive CitationMerrell, Melvin Joseph, "Preliminary Design of a Hydraulic Vibration Machine with Variable Amplitude and Frequency, UsingMultistage Amplification and Feedback Control" (1966). All Theses and Dissertations. 7157.https://scholarsarchive.byu.edu/etd/7157

G x o . o o i

H !\ \ ^

PRELIMINARY DESIGN OF A HYDRAULIC VIBRATION MACHINE

WITH VARIABLE AMPLITUDE AND FREQUENCY,

USING MULTISTAGE AMPLIFICATION ,

AND FEEDBACK CONTROL V

A T h e s i s

P r e s e n t e d t o t he

Depar tmen t o f Mechani ca l E n g i n e e r i n g

Brigham Young U n i v e r s i t y

In P a r t i a l F u l f i l l m e n t

o f t he R e qu i r e me n t s f o r t he Degree

M a s t e r o f Sc i en ce

by

Melvin Jo se ph M e r r e l l

21 December 1965

T h i s t h e s i s by Melvin Jo s ep h M e r r e l l i s a c c e p t e d

in i t s p r e s e n t form by t he Depar tmen t o f Mechani ca l Eng

n e e r i n g o f Br igham Young U n i v e r s i t y a s s a t i s f y i n g t he

t h e s i s r e q u i r e m e n t s f o r t he degree o f Ma s t e r o f Sc i en ce

( / D a t e

Typed by: Norma S c a r l e t t

ACKNOWLEDGMENTS

The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o

A s s i s t a n t P r o f e s s o r Jo s e p h C. Free f o r a d v i c e and d i r e c ­

t i o n t oward t he d e s i g n n e c e s s a r y f o r t h i s t h e s i s . Ap p re ­

c i a t i o n i s a l s o e x t e n de d t o t he o t h e r f a c u l t y members who

gave t h e i r a s s i s t a n c e f rom t ime t o t i m e . Thanks go a l s o

t o C a l v i n M e r r e l l and C l a i r S h i e l d s who h e lp e d de s ig n some

o f t he components f o r t h e v i b r a t i o n machine . The a u t h o r

i s a l s o g r a t e f u l f o r t h e encou ragemen t o f h i s w i f e , M a r i l y n .

i i i

TABLE OF CONTENTS

Page*

ACKNOWLEDGMENTS............................................................................................. i i i

LIST OF FIGURES....................................................................................... v

NOMENCLATURE ............................................................................................ v i

CHAPTER

I . INTRODUCTION ........................................................................ 1

I I . S Y N O P S I S .................................................................................. 1+

The P r o b l e m ..................................................... ....Des ign C r i t e r i a .................................. 6Method o f Approach ........................................................... 7A n a l y s i s .................................................................................. 8

I I I . DESIGN OF COMPONENTS.......................................................... 10

Ram Cyl i n d e r .............................................................................. 10Power Valve . . ..........................................................................10F l u i d S p r i n g .............................................................................. 20

IV. ANALYSIS OF S Y S T E M ...............................................................25

V. CONCLUSIONS AND RECOMMENDATIONS..................................3&

LIST OF REFERENCES..................................................................................39

A P PE N D IX ES ......................................................................................................1+ 0

A. COMPUTER PROGRAMS.................................................................... ij.1

Power Valve P r o g r a m ....................... . l\2Ram Load P r o g r a m .................................................................... lj.7

B. SAMPLE CALCULATIONS AND ASSUMED VALUES . . . 5 l

i v

LIST OF FIGURES

F i g u r e Page

1. S e r v o v a l v e .................................................................................. Ip

2 . V a r i a b l e Sp r i n g Sys tem ..................................................... 6

3 . Load L o c u s .................................................................................. 8

Ip. T e n t a t i v e S y s t e m ..................................................................... 8

5* S l o t t e d Plug and Ram C y l i n d e r ................................... 11

6 . Ampl i t ude V e r su s F r equency f o r Ram . . . . . . lip

7 . Spool Va lve and Load S c h e m a t i c .................................. 16

8 . E q u i v a l e n t C i r c u i t ............................ 16

9 . Ampl i t ude and Phase Response Diagram .................... 21

10. Ampl i t ude v s . F r e que nc y .......................... . . . . . 23

11. Sys tem ............................................. . . . . . . . . . . 25

12. KG P l u s E x t e r n a l I n p u t s ................................................ 29

13* Root Locus , U n c o m p e n s a t e d ............................................ 31

lip. RC C i r c u i t .................................. 32

15. Root Locus , Compensated S y s t e m ................................... 3̂ 4-

v

NOMENCLATURE

A

D

E

E

Ei

F

Arr.pl i t u d e

C o n s t a n t ga in o f l e ad c i r c u i t

2Maximum o r i f i c e a r e a , in

O r i f i c e a r e a

E f f e c t i v e a r e a , in

Area o f power v a lv e p r e s s u r e s u r f a c e , in

Damping f o r power v a l v e , -liL ■$.££,

Capac i t o r

Damping f o r ram, ■■■■0 .~ g'C

A m p l i f i c a t i o n f o r d i s p l a c e m e n t mo n i to r

A m p l i f i c a t i o n f o r v e l o c i t y m on i to r

A m p l i f i c a t i o n f o r a c c e l e r a t i o n mo n i to r

C o e f f i c i e n t o f d i s c h a r g e

D e r i v a t i v e w i t h r e s p e c t t o t ime

2CdW C o s Q

V ol t a g e i n , v o l t s

V o l t a g e o u t , v o l t s

F o r c e , lb

vi

r

I

J

K

KG

KGH

Kv

L

M

M 1 ‘v

o

p

Pmv

va

m

pv

C o n s t a n t f o r c e , lb

Ram f o r c e , lb

C u r r e n t , m i l l i a m p s

S p r i n g c o n s t a n t , l b / i n

Forward loop t r a n s f e r f u n c t i o n

Open loop t r a n s f e r f u n c t i o n

Sp r i ng c o n s t a n t f o r power v a l v e , l b / i n

E f f e c t i v e damping l e n g t h , in

Mass o f l o ad , lb sec* in

Mass o f power s p o o l , lb sec in

As a s u b s c r i p t i s p o i n t o f l i n e a r i z a t i o n

P r e s s u r e d i f f e r e n c e , p s i

P r e s s u r e a c r o s s Moog v a l v e , p s i

Supply p r e s s u r e , p s i

P r e s s u r e a c r o s s power v a l v e , p s i

P r e s s u r e a c r o s s power v a lv e p r e s s u r e s u r f a c e , p s i

i n^Flow,

Flow

sec

in-^ s sec

Flow from Moog v a l v e ,

Flow from power v a l v e .

in3sec

in3sec

Flow to ram, in^sec

vi i

R e s i s t o r

R e s i s t o r

Time

T o ta l e n t r a p p e d volume o f f l u i d in f l u i d s p r i n g

C i r c u m f e r e n c e of v a l v e , in

D i s p l a c e m e n t , in

D i sp l a ce me n t o f power s p o o l , in

D i s p l a ce m e n t o f Moog v a l v e s p o o l , in

Bulk modulus o f f l u i d s p r i n g o i l , — pin

Damping r a t i o

Angle o f f l ow l e a v i n g o r i f i c e

O i l d e n s i t y , lb

I n v e r s e f r e q u e n c y

F requency in r .a-̂ *,an.sS6C

N a t u r a l f r e q u e n c y in ■

vi i i

CHAPTER I

INTRODUCTION

E s s e n t i a l l y t h e r e a r e t h r e e t y p e s o f v i b r a t i o n machine s

m e c h a n i c a l , e l e c t r o d y n a m i c , and e l e c t r o h y d r a u l i c ( h y d r a u l i c ) .

Each o f t h e s e machine s ha s c h a r a c t e r i s t i c s which make i t

s u i t a b l e f o r p a r t i c u l a r a r e a s o f u s e .

Mechan i ca l ♦ The mechan i ca l v i b r a t i o n mach ine s i n c l u d e

d i r e c t - d r i v e mac h ine s , r o t a t i n g u n b a l a nc e d e v i c e m ac h in e s ,

and o t h e r s . The p ro mine n t f e a t u r e s o f t h e s e machine s a r e

l a r g e d i s p l a c e m e n t s and low o p e r a t i n g f r e q u e n c i e s . Because

t he f o r c e s on machine members i n c r e a s e a s f r e q u e n c y i n c r e a s e s ,

t he members de fo rm and t h e a m p l i t u d e v a r i e s a c c o r d i n g l y .

Each t ype of v i b r a t i o n machine i s a f f e c t e d s i m i l a r l y , bu t

v a r i e s w i t h t ype o f c o n s t r u c t i o n . Mechani ca l v i b r a t i o n

machine s a r e r a r . e l y used a s c o n s t a n t d i s p l a c e m e n t machine s

above 25 c p s .

E l e c t r o d y n a m i c . T h i s t yp e o f machine i s n o t e d f o r i t s

a b i l i t y t o v i b r a t e r e l a t i v e l y smal l masses a t h i g h f r e q u e n c i e s

V i b r a t i o n i s c aus ed by a f o r c e p roduced by t h e i n t e r a c t i o n

be tween a c u r r e n t f l o w in a d r i v e r c o i l and t he m ag ne t i c f i e l d

which c u t s t he c o i l . The f o r c e t o w e i g h t - o f - m a c h i n e r a t i o i s

q u i t e sma l l compared w i t h t h e mechan i ca l o r h y d r a u l i c machine s

Because t he v i b r a t i o n depends on t h e f r e q u e n c y o f t he c u r r e n t

2

f l o w , a wide f r e q u e n c y range o f o p e r a t i o n can be o b t a i n e d ,

b u t to o b t a i n a 2 5 ,0 0 0 pounds f o r c e v e c t o r , a machine t he

h e i g h t o f a man would be r e q u i r e d .

H y d r a u l i e . The h y d r a u l i c v i b r a t i o n machine i s a

d e v i c e wh ich d e r i v e s i t s name from the method o f t r a n s m i t t i n g

power . T r a n s m i s s i o n i s by h ig h p r e s s u r e f l u i d f l ow from a

pump to an a c t u a t o r c y l i n d e r . V a l v e s d i r e c t f l ow a l t e r n a t e l y

t o e ach s i d e o f t he doub l e a c t i n g p i s t o n . I n i t i a l f l ow i s

u s u a l l y c o n t r o l l e d by an e 1e c t r o h y d r a u l i c s e r v o v a l v e . The

p r o m in e n t f e a t u r e s o f t he h y d r a u l i c machine a r e t h a t l a r g e

f o r c e s and l a r g e d i s p l a c e m e n t s can be a c h i e v e d u s i n g sma l l

componen ts . C u r r e n t l y h y d r a u l i c machine s a r e n o t c a p a b l e o f

o p e r a t i n g above 300 to 350 cps m a i n t a i n i n g any s i g n i f i c a n t

d i s p l a c e m e n t . T h i s i s r e f e r r e d t o l a t e r a s s t a t e o f t he a r t

f r e q u e n c y .

Gen e ra l i n f o r m a t i o n on t he s u b j e c t o f v i b r a t i o n

mach ine s can be found in t he f o l l o w i n g s o u r c e s : SHOCK AND

VIBRATION HANDBOOK, V o l . 2 , by H a r r i s and C r e d e ( l ) 1, CONTROL

SYSTEM COMPONENTS by Gibson and T u t e u r ( 2 ) , and FLUID POWER

CONTROL by B l a c k b u r n , R e e t h o f and S h e a r e r ( 3 ) *

The h y d r a u l i c v i b r a t i o n machine und e r p r e l i m i n a r y

d e s i g n in t h i s t h e s i s was chosen t o f i l l a s t r u c t u r a l t e s t i n g

need in t he Mechani ca l E n g i n e e r i n g Depar tmen t and t o p ro v i d e

l a b o r a t o r y d e m o n s t r a t i o n c a p a b i l i t y o f a h y d r a u l i c c o n t r o l

sy s t em . I t was a l s o p l ann ed t o d e s i g n i n t o i t a c h a r a c t e r i s ­

t i c t h a t t he h y d r a u l i c machine does n o t n o r m a l l y h a v e - - t h a t

^Numbers in p a r e n t h e s e s r e f e r to the L i s t o f R e f e r e n c e s a t t he end o f t he t h e s i s .

3

i s h i g h e r f r e q u e n c i e s o f o p e r a t i o n . To f a c i l i t a t e t h i s

m u l t i p l e s t a g e s o f a m p l i f i c a t i o n were u t i l i z e d and a v a r i a b l e

s p r i n g was d e s i g n e d so t h a t r e s o n a n c e c o u ld be m a i n t a i n e d

t h r o u g h o u t much o f t he o p e r a t i n g r ange o f f r e q u e n c y .

I t sh ou ld be s a i d h e r e t h a t t h i s t h e s i s does n o t i n c l u d e

t he t o t a l d e s i g n o f t he v i b r a t i o n mac h in e . A co mple t e d e s i g n

would r e q u i r e i n v e s t i g a t i o n o f many d e t a i l s t h a t c a n n o t be

c ov e r ed in t h i s t h e s i s . These d e t a i l s w i l l be worked o u t when

t he a c t u a l machine i s b u i l t . Some components o f t he v i b r a t i o n

machine were p a r t i a l l y d e s i g n e d by two f i f t h - y e a r m ec h an i c a l

e n g i n e e r i n g s t u d e n t s , C a l v i n M e r r e l l and C l a i r S h i e l d s .

T h i s t h e s i s e s s e n t i a l l y a c c o m p l i s h e s t h r e e t h i n g s : (1)

power ma tch ing o f h y d r a u l i c components c a p a b l e o f p e r f o r m i n g

t o t he s p e c i f i c a t i o n s s e l e c t e d . T h i s was done by u t i l i z i n g a

d i g i t a l computer t o s t u d y t he l a r g e number o f v a r i a b l e s i n ­

v o l v e d ; (2 ) an e x t e n s i o n o f t he s t a t e o f t h e a r t f r e q u e n c y

r e s p o n s e a t t h e power l e v e l chosen u s i n g a f l u i d s p r i n g to

v a r y t h e n a t u r a l f r e q u e n c y o f t h e l oad and a v a r i a b l e f l u i d

volume c y l i n d e r , b o t h o f which a r e p h y s i c a l l y r e a l i z a b l e ;

(3 ) an a n a l y s i s o f t he s t a b i l i t y o f t he t o t a l dynamic sy s t em

u s i n g r o o t l o c u s t e c h n i q u e .

CHAPTER I I

SYNOPSIS

Be fo re t he d e t a i l s o f t he p r e l i m i n a r y d e s i g n a r e g i v e n ,

an u n d e r s t a n d i n g o f t he sys tem would be o f va lue . '

The Probl em

The probl em i s e s s e n t i a l l y one o f a m p l i f i c a t i o n - - t h a t

i s , t r a n s f o r m i n g an e l e c t r i c a l s i g n a l i n p u t i n t o a r e l a t i v e l y

l a r g e d i s p l a c e m e n t o u t p u t . T h i s i s done in t h r e e s t a g e s .

The f i r s t s t a g e ( F i g . l ) i s a s e r v o v a l v e c o n s i s t i n g o f

t h r e e main p a r t s : a t o r q u e mo to r , a f l a p p e r v a lv e ( f l a p p e ry n : ^ \ \ • v \ \ ̂ \ \

Nozzl e

F l a p p e r

Feedback Sp r i n g

Spool

F i l t e r

—

> £y ^ \yy. s s yy i___ vy. a tz j —

F i g . 1 . - - S e r v o v a l v e

k

5

and n o z z l e ) , and a spool v a l v e . The s e r v o v a l v e f i r s t c o n v e r t s

an e l e c t r i c a l s i g n a l t o a d i s p l a c e m e n t o f t he f l a p p e r by mag­

n e t i c f o r c e s in t he t o r q u e mo to r . The d i s p l a c e m e n t o f t he

f l a p p e r c o n t r o l s t he f l ow in t he two n o z z l e s which c o n t r o l

f l ow a l t e r n a t e l y t o t he ends o f t he s p o o l . T h i s f l ow c a u s e s

t he spoo l t o move, and t he spool movement a l l o w s h i g h p r e s s u r e

f l o w t h ro u gh one c o n t r o l p o r t wh i l e a l l o w i n g r e t u r n f l ow from

the l oad which i s t he second s t a g e .

The second s t a g e i s a n o t h e r spool v a lv e which o p e r a t e s

s i m i l a r l y t o t he one in the f i r s t s t a g e . The smal l f low

t h ro u gh t h e n o z z l e s o f t he f l a p p e r v a l v e on the f i r s t s t a g e

r e g u l a t e s t he movement o f t he smal l spool v a l v e . T h i s move­

ment in t u r n r e g u l a t e s a much l a r g e r f l ow which now can move

t he l a r g e r second s t a g e s p o o l .

The l a s t , o r t h i r d s t a g e , i s a d o u b l e - a c t i n g ram or p i s ­

t o n . Th i s has a r e l a t i v e l y l a r g e a r e a which , when a c t e d upon

by a h i gh p r e s s u r e f l o w from th e second s t a g e , o r power v a l v e ,

p r o d u c e s a l a r g e f o r c e . The l oad or o b j e c t to be v i b r a t e d i s

a t t a c h e d d i r e c t l y t o t h e p i s t o n ro d . M o n i t o r i n g t he d i s p l a c e ­

ment o f t he ram can be used t o c o n t r o l t he sy s t em.

The v a r i a b l e s p r i n g i s a l s o d i r e c t l y a t t a c h e d t o t he

ram to f a c i l i t a t e change in n a t u r a l f r e q u e n c y o f the l o a d .p

T h i s v a r i a b l e s p r i n g ( F i g . 2) i s e s s e n t i a l l y a n o t h e r d o u b l e ­

a c t i n g p i s t o n in a c y l i n d e r , bu t h a s no i n l e t and o u t l e t a s

does t h e ram. T h i s c y l i n d e r i s f i l l e d w i th a f l u i d t h a t i s

^These a r e s i m p l i f i e d d r a w i n g s . D e t a i l d r aw ings a r e a v a i l a b l e in t h e Brigham Young U n i v e r s i t y Mechani ca l E n g i ­n e e r i n g De pa r tm en t .

6

more c o m p r e s s i b l e t han most f l u i d s . By r e g u l a t i n g t h e volume

o f t h i s f l u i d , . t h e s p r i n g c o n s t a n t can be v a r i e d o ve r a wide

r a n g e .

F i g . 2 . —V a r i a b l e S p r i n g System

The p rob l em i s n o t j u s t one o f a s i n g l e f i x e d l oad a t

a p a r t i c u l a r f r e q u e n c y and v i b r a t i o n a t a f i x e d a m p l i t u d e .

The v i b r a t i o n machine must be c a p a b l e o f many d i f f e r e n t l o a d s

a t d i f f e r e n t f r e q u e n c i e s and a m p l i t u d e s .

Design C r i t e r i a

S p e c i f i c a t i o n s . Be fo re t he p r o j e c t was s t a r t e d , a

t e n t a t i v e s e t o f s p e c i f i c a t i o n s were a g r e e d upon by t ho se

7

c o n c e r n e d in t h e Mechan i ca l E n g i n e e r i n g De p a r tm en t . These

a r e a s f o l l o w s : '

I . Fo rce and p r e s s u r e

A. A f o r c e v e c t o r o f 2 5 ,0 0 0 pounds maximum

B. Maximum o p e r a t i n g p r e s s u r e o f 3000 p s i

1 1 . Ampli t ude

A. Two- inch maximum a t f r e q u e n c i e s up t o 10 cps

B. O n e - h a l f - i n c h maximum a t f r e q u e n c i e s h i g h e r

t han 10 cps

I I I . Loading

1000 pounds maximum

IV. E l e c t r i c a l i n p u t by f u n c t i o n g e n e r a t o r

V. Feedback c o n t r o l

VI . Response f r e q u e n c i e s o b t a i n a b l e beyond s t a t e o f

t he a r t

Method o f Approach

In d e s i g n i n g and ma tch ing t he h y d r a u l i c component s , a

c h a r t c a l l e d a l oad l o c u s was u s e d . T h i s c h a r t g r a p h i c a l l y

compared the l oad power w i t h t h a t o f t he d r i v e power . The

l oad l o cu s i s a g r aph o f f o r c e v e r s u s v e l o c i t y o r p r e s s u r e

v e r s u s f l o w . I f t he v i b r a t i o n machine were d e s i g n e d f o r o n l y

one f r e q u e n c y and a m p l i t u d e , t h e r e would o n l y be one cu rve

f o r t he l oad and one cu rve f o r t h e d r i v e ; b u t s i n c e t h i s was

n o t t he c a s e , t he cu rve r e p r e s e n t i n g t h e maximum power f o r

t he d r i v i n g component would need t o c o m p l e t e l y e n c l o s e a l l

t h e p r e s s u r e v e r s u s f l ow c u r v e s f o r t h e l o a d . T h i s i s shown

in F i g u r e 3 . Us ing t h i s method, c o u p l e d w i t h o t h e r p e r t i n e n t

8

F i g . 3 . —Load Locus

i n f o r m a t i o n , ma t ch ing o f a l l t he h y d r a u l i c components was

made. Because o f t he many v a r i a b l e s i n v o l v e d , much o f t he

work was done by a d i g i t a l computer (Appendix A, p . Lpl) .

A n a l y s i s

1 . O s c i 1l a t o r 5. Power Spool Valve2 . S u b t r a c t o r 6 . Ram3.k-

Ampl i f i e r Moog Se rv o va lv e

7. Feedback

F i g . ^ . - - T e n t a t i v e System

Because o f t he c o m p l e x i t y o f t he n o n l i n e a r sy s tem, t he

e q u a t i o n s were l i n e a r i z e d . T h i s made i t p o s s i b l e to work

w i t h t he L ap l a c e t r a n s f o r m e d e q u a t i o n s .

9

With l i n e a r e q u a t i o n s i t was t h e n p o s s i b l e to f i n d t he

c h a r a c t e r i s t i c e q u a t i o n o f t h e i n p u t - o u t p u t r e l a t i o n s h i p and

p l o t a r o o t l o cu s d i ag ram o f t h i s e q u a t i o n .

T h i s n o t o n l y i n d i c a t e d t he s t a b i l i t y o f t he sy s tem,

b u t a l l o w e d compar i son o f comp en sa t i o n me thods .

I t was found t h a t a number o f e l e c t r i c a l l e a d c i r c u i t s

( d i f f e r e n t i a t i n g c i r c u i t s ) were n e c e s s a r y ahead o f t he h y d r a u ­

l i c sy s t em to a cc om pl i s h s t a b i l i t y .

L i n e a r i z i n g t he e q u a t i o n s , which r e p r e s e n t a v e r y non­

l i n e a r sy s t em , was o n l y j u s t i f i e d b eca u se t he sys t em was

l e a s t s t a b l e a t maximum g a i n . I t was a t t h i s p o i n t o f maximum

ga in t h a t t h e e q u a t i o n s were l i n e a r i z e d .

CHAPTER I I I

DESIGN OF COMPONENTS

Ram C y l i n d e r . S p e c i f i c a t i o n s f o r a ram c y l i n d e r w i t h

movable he ad s to r educe f l u i d volume and which would o p e r a t e

a t 3000 p s i and p roduce 2 5 ,0 0 0 pounds f o r c e were g iven to

C a l v i n M e r r e l l . He c a l c u l a t e d s t r e n g t h r e q u i r e m e n t s f o r the

c y l i n d e r and components and p roduced d e t a i l e d d r awings o f

t h e s e component s . These d r aw ings a r e a v a i l a b l e in the Brigham

Young U n i v e r s i t y Mechani ca l E n g i n e e r i n g Depa r tm en t .

The p i s t o n rod was d e s i g n e d t o c a r r y a maximum a x i a l

and bend ing l oad o f 2 5 ,0 0 0 pounds and 2 $ ,0 0 0 i n ch -p ou nd s

r e s p e c t i v e l y .

The ram c y l i n d e r he ad s were d e s i g n e d so t h e y c o u ld be

a d j u s t e d in and o u t . A s l o t t e d p lug i s a t t a c h e d t o the heads

and moves w i t h them. The moving heads and s l o t t e d p l u g s were

d e s i g n e d t o r educe t he volume o f f l u i d on bo th s i d e s o f the

p i s t o n . Because o f t he c o m p r e s s i b i l i t y o f t he f l u i d , t he

s t i f f n e s s , and hence t he f r e q u e n c y , i s a f u n c t i o n o f the

volume. A s i m p l i f i e d d i ag ram o f t h e s l o t t e d p l u g , p i s t o n

and c y l i n d e r a r e shown in F ig u r e 5«

Power V a l v e . Almost a l l the r e q u i r e m e n t s f o r t he

power v a lv e were s p e c i f i e d . The p r e s s u r e i s 3000 p s i , and

f l ow i s d e t e r m i n e d by t he c a p a b i l i t y o f t he pump s i n c e most

o f the f l o w from the pump goes t o t he power v a l v e . I t was

10

11

n e c e s s a r y t hen t o d e t e r m i n e from pump c a t a l o g s and o t h e r

l i t e r a t u r e what r e a l i s t i c amount o f f l ow c o u ld be g e n e r a t e d ,

o r what i s t h e l a r g e s t pump f e a s i b l e . A pump which would

p roduce 65 t o 75 g a l l o n s pe r minu te and would r e q u i r e up t o

130 ho r sepower a s i n p u t was a v a i l a b l e . T h i s Hp was a

f e a s i b l e r e q u i r e m e n t f o r a d i e s e l eng ine t o be a c q u i r e d by

t he B.Y.U. The v e l o c i t y o f t he ram i s s e t by t he maximum

f l o w from the power v a l v e . The ram v e l o c i t y i s found by

t a k i n g t he f l o w to t he ram d i v i d e d by t he p i s t o n a r e a . A

maximum v e l o c i t y o f 30 i n / s e c , which c o r r e s p o n d s to 7 2 . 3

g a l / m in ( 2614. i n ^ / s e c ) , was assumed f o r t he power v a l v e .

Now t h a t t he maximum f l o w (26i | i n -3 / sec )and maximum

p r e s s u r e d rop ( o p e r a t i n g p r e s s u r e ) r e q u i r e d from the power

v a lv e a r e known, t he q u e s t i o n t hen i s , wha t m a t h e m a t i c a l

r e l a t i o n s h i p i s obeyed be tween t h e s e two e x t r e m e s (what a r e

t he o t h e r p o i n t s in t he f l o w - p r e s s u r e c u r v e ) ? The f l ow -

p r e s s u r e r e l a t i o n s h i p s can be d e r i v e d a n a l y t i c a l l y . The

a n a l y s i s i n v o l v e s the f o l l o w i n g a s s u m p t i o n s (4 ):

I d e a l f l u i d

12

1.

2 . I d e a l f l u i d sou rc e

3* I d e a l geome t ry o f v a lv e

1;.. S t e a d y s t a t e c o n d i t i o n s

The a s s u m p t io n o f an i d e a l n o n v i s c o u s and i n c o m p r e s s i b l e

fluid is nearly correct under most conditions. At the peak f l o w r a t e s and w i t h r e l a t i v e l y v i s c o u s f l u i d s , t he e f f e c t i v e

s u p p l y p r e s s u r e a t t he v a lv e i n t a k e may f a l l o f f s l i g h t l y

beca use o f p r e s s u r e d rop in t h e su p p l y l i n e s ; bu t t h i s shou ld

n o t be more t han 5> t o 10 p e r c e n t in we 11- d e s i g n e d s y s t e m s .

The i n c o m p r e s s i b i l i t y a s su m pt io n i s a l s o j u s t i f i e d a s f a r a s

phenomena i n s i d e t he v a lv e a r e c o n c e r n e d . At normal o p e r a t i n g

p r e s s u r e s , t he f i n i t e c o m p r e s s i b i l i t y o f r e a l f l u i d s has o n l y

a n e g l i g i b l e e f f e c t upon t he f l ow t h r o u g h t he o r i f i c e s .

An " i d e a l ” c o n s t a n t - p r e s s u r e f l u i d sou rc e i s one in

which t he sou rc e s u p p l i e s f l u i d t o t he i n t a k e o f t he v a lv e a t

a c o n s t a n t p r e s s u r e i n d e p e n d e n t o f t he f l ow r a t e . A c o n s t a n t -

f l ow so u r c e s u p p l i e s f l u i d a t a c o n s t a n t f l ow r a t e i n d e p e n d e n t

o f t he change in p r e s s u r e . I t i s p o s s i b l e to b u i l d s o u r c e s

f o r wh ich t he a s s u m p t io n o f i d e a l i t y i s f a i r l y a c c u r a t e , even

f o r r a p i d changes in l o a d s ; b u t even t hough t h e sou rce may n o ttbe a b s o l u t e l y c o n s t a n t , the e f f e c t on t he sys tem i s n o t e x c e s ­

s i v e in most c a s e s .

" I d e a l geomet ry" means t h a t t h e edges o f t he o r i f i c e s

a r e p e r f e c t l y s h a r p and t he c l e a r a n c e s a r e z e r o so t h a t t he

geome t ry o f t h e o r i f i c e i s n o t a f u n c t i o n o f v a l v e - s t e m

p o s i t i o n . T h i s a s su m pt io n i s u s u a l l y a c c e p t a b l e e x c e p t f o r

d i s p l a c e m e n t s be low one o r two m i c r o - i n c h e s .

13

The a s s u m p t io n o f s t e a d y - s t a t e c o n d i t i o n i s v a l i d f o r

t he v a l v e a l o n e .

I f t he sy s t em i s such t h a t t h e s e f o u r a s s u m p t i o n s a r e

v a l i d , t he f l o w - p r e s s u r e r e l a t i o n s h i p f o l l o w s t he o r i f i c e

e q u a t i o n

« = CdAoV 2p/ P [l]where q = Flow

= C o e f f i c i e n t o f d i s c h a r g e P = P r e s s u r e d i f f e r e n c e

AQ = O r i f i c e a r e a P = O i l d e n s i t y ,

i s de pen de n t somewhat upon R e y n o l d ' s number , bu t i t has

been found t h a t in we 11- c o n s t r u c t e d v a l v e s , a v a l u e between

.6 and .65 can be u s e d ( 5 )»

The o r i f i c e e q u a t i o n i s t h a t o f a p a r a b o l a , and t he

f l o w - p r e s s u r e cu rve be tween maximums i s now d e f i n e d . The

curve t hen r e p r e s e n t s t he l i m i t i n g c o n d i t i o n f o r the ram;

( i . e . , maximum a v a i l a b l e power ) .

The p r e v i o u s c o ve r ag e o f f l o w - p r e s s u r e c u r v e s makes

i t p o s s i b l e t o examine t he l oad l o c u s . A l oad l o cu s curve

i s a l s o a f l o w - v e r s u s - p r e s s u r e cu rve s i n c e f l ow i s eq u a l t o

the v e l o c i t y m u l t i p l i e d by t he a r e a and p r e s s u r e i s the f o r c e

d i v i d e d by t he a r e a . I f the f o r c e o f a l oad of some sys tem

i s p l o t t e d v e r s u s the v e l o c i t y o f t h a t sy s t em, t h i s i s a l oad

l o c u s . The l o c i v a r y a c c o r d i n g to t he t ype of l o a d i n g .

The l oad of t h e ram, which i s a l oad on the power v a l v e ,

was a n a l y z e d a s a s p r i n g - m a s s - d a s h p o t sy s tem, and the dynamics-

can be d e s c r i b e d m a t h e m a t i c a l l y in t he f o l l o w i n g way:

F = MD2X + CDX + KX. [2]

The s p r i n g w i t h s p r i n g c o n s t a n t K i s t he v a r i a b l e s p r i n g

m en t i on ed ; t h e mass M i s t h e mass o f t h e l o a d ; and l e a kag e

and f l o w r e s i s t a n c e a c t a s a d a s h p o t w i t h a c o e f f i c i e n t o f

damping C. D i s t h e d e r i v a t i v e w i t h r e s p e c t t o t i m e . I f

t h e e q u a t i o n i s r e a r r a n g e d and a s o l u t i o n o f X = A s i n C J t

is assumed, the solution isF = MA6J ^ ( OJ n 2 - 1)

U) 2

F = Fo rce

M = Mass

A = Ampl i t ude

sin6u)t + 2 S (jJ n c o s 6 J t ) . CO

Cl) n - N a t u r a l f r e q u e n c y

Cl) = O p e r a t i n g f r e q u e n c y

^ = Damping r a t i o

0

I f t he l oad l ocu s i s p l o t t e d f o r t he e q u a t i o n , i t t u r n s o u t

t o be an e l i p s e , and i t s r e l a t i v e shape depends on t h e v a r i ­

a b l e s a m p l i t u d e , f r e q u e n c y , and n a t u r a l f r e q u e n c y . A v a lu e

f o r damping was o b t a i n e d f o r t he ram by compar ing v a l u e s o f

o t h e r h y d r a u l i c rams and s c a l i n g t o f i t t h e p a r t i c u l a r ram

( s e e Assumed V a l u e s , p . 53)* A compu te r p rogram (Appendix A,

p . i |7 ) was w r i t t e n which v a r i e d each o f t h e s e v a r i a b l e s and

e l i m i n a t e d a l l load l o c u s c u r v e s e x c e p t t ho se unde r t he maxi ­

mum o u t p u t cu rv e o f t h e power v a l v e . Each l o c u s can be com­

p a r e d t o t he maximum o u t p u t cu rve o f t h e power v a lv e s i n c e we

know the e q u a t i o n which d e s c r i b e s t h a t o u t p u t ( E q u a t i o n l ) .

The computer r e s u l t s were t h e maximum a m p l i t u d e f o r e ach o p e r ­

a t i n g f r e q u e n c y . The c a l c u l a t i o n s were made assuming t he

n a t u r a l f r e q u e n c y was eq u a l t o t he o p e r a t i n g f r e q u e n c y ( F i g . 6 ) .

Now t h a t t he f l o w - p r e s s u r e r e q u i r e m e n t s f o r the power

va lv e a r e known, s i z e , shape and o t h e r d e t a i l s f o r t he v a lv e

can be d e t e r m i n e d . A g e n e r a l f o u r - w a y v a lv e ( F i g . 7) a l o n g

w i t h i t s e q u i v a l e n t e l e c t r i c a l c i r c u i t ( F i g . 8 ) i s i l l u s t r a t e d

t o f a c i l i t a t e t h e f o l l o w i n g d e r i v a t i o n . The c i r c u i t i s t h a t

o f a l oaded Whea ts tone b r i d g e w i t h squ a re law a rms . These

a r e sq u a r e z i g z a g i n s t e a d o f saw t o o t h to show t h a t h y d r a u l i c

c o nd uc t anc e i s n o t ohmic, b u t i s p a r a b o l i c . The e l e c t r i c a l

an a logue o f an o r i f i c e i s n o t a r e s i s t o r , bu t a n o n l i n e a r

v a r i s t o r . The c o n t r o l v a lv e does n o t c o r r e s p o n d t o an ohmic

r h e o s t a t , b u t c o r r e s p o n d s t o a t r i o d e .

16

F i g . 7 ‘ -~ Spool Valve and Load Schema t i c

F i g . 8 . - - E q u i v a l e n t C i r c u i t

The a p p l i c a t i o n o f K i r c h h o f f ’ s f i r s t law to t h e sys tem r e s u l t s

in t h e f o l l o w i n g e q u a t i o n s :

17

P 1 + P2 = Ps M P 3 + P k = Ps [5]

P1 “ Pl, = Pm P3 " P2 = Pm W

The P ‘ s r e p r e s e n t p r e s s u r e d ro p s a c r o s s t h e c i r c u i t e l e m e n t s .

The s u b s c r i p t s s and m r e f e r t o t h e s u p p l y and motor ( o r

h y d r a u l i c l o ad ) r e s p e c t i v e l y . K i r c h h o f f ’ s second law r e q u i r e s

t h a t t h e f o l l o w i n g e q u a t i o n s h o l d :

q l + ql(. = q s [®] q 2 + q3 = q s

q2 - q l = qm [10] % - q 3 = qm

The q ’ s r e p r e s e n t r a t e s o f f l o w in t h e v a r i o u s b r a n c h e s .

The p r e s s u r e i s a s c a l e r q u a n t i t y in t h e f i r s t law,

and K i r c h h o f f ' s s econd law i s t h e e l e c t r i c a l a na l o g u e o f t h e

law o f c o n s e r v a t i o n o f mass in t h e h y d r a u l i c c a s e .

F i n a l l y , u s i n g t h e o r i f i c e law y i e l d s f o u r more eq u a -

t i o ns :

q i = sitP T M q2 ” S2V p2

q3 = 93"V p3 N % = SJ+V pk MThe g*s a r e h y d r a u l i c c o n d u c t a n c e s o f t he o r i f i c e s A0C ^\j2 /p .

By a s suming a c o n s t a n t p r e s s u r e sys tem and , h e n ce ,

i g n o r i n g e q u a t i o n s 8 and 9> t h e e q u a t i o n s ij. t h r o u g h lj? a r e

t r a c t a b l e . Assuming a z e r o l a p v a l v e , q^and q^ a r e z e r o i f

t h e spoo l i s t o t h e l e f t ; and == = q g , so t h e s u b s c r i p t s

on q a r e u n n e c e s s a r y . S ince

Pm = P 1 - Pi; 1[!6] - q2/ s 2

Pm ~ P 1 q2/ q2 1M P1 = Ps - P2 M

h e n c e , P2 = q P/ g P N | and pm = p s - 2c?2/ g 2 . M

The power i n t o t h e l oad i s t h e p r o d u c t o f l oad p r e s s u r e and

f l ow o r

Hra = = <3p s - 2<!3/ 9 2 - [22]

Maximum power w i l l e x i s t when

DV DcJ = 0 o r 3pm = 2ps pm “ 2 / 3 p s N

o r a t maximum power t h e l oad p r e s s u r e Pm i s 2 / 3 t h a t o f s u p p l y

p r e s s u r e .

S ince Pm i s z e r o a t maximum f lo w , by a p p l y i n g e q u a t i o n

21

^max “ SmaxV ^ s / ^

and w i t h most h y d r a u l i c f l u i d s t h i s i s a p p r o x i m a t e l y

^max = 70 Am a •

Amax o c c u r s when v a l v e s tem d i s p l a c e m e n t i s maximum.

From t h e above e q u a t i o n t h e maximum o r i f i c e a r e a i s

A0 = .0689 i n ^ . T h i s a r e a can be o b t a i n e d by m u l t i p l y i n g

t he p e r i p h e r a l l e n g t h o f t h e o r i f i c e by t h e d i s p l a c e m e n t (one

sh a r p edge from t h e ma t in g sh a r p e d g e ) . I t can be seen t h a t

many such c o m b i n a t i o n s c o u l d r e s u l t in an a r e a o f .0689 in .

To f i n d such a c o m b i n a t i o n , f u r t h e r c r i t e r i a f o r d e t e r m i n a t i o n

must be found . F a c t o r s i nv o lv e d a r e f o r c e - d i s p l a c e m e n t

r e l a t i o n s h i p s , w e i g h t - s i z e r e l a t i o n s h i p s , and c h o i c e o f an

e l e c t r o h y d r a u 1ic s e r v o v a l v e t h a t w i l l f i t t h e r e q u i r e m e n t s

o f t h e power v a l v e . F i n d i n g t h e l oad l o cu s c u r v e s f o r t he

power v a l v e , and a t t h e same t ime c h o o s i n g a s e r v o v a l v e w i t h

an o u t p u t t h a t ma t ches t h e power v a l v e r e q u i r e m e n t s , w i l l

g ive some o f t h e s e f a c t o r s .

The d i f f e r e n t i a l e q u a t i o n t h a t d e s c r i b e s t he f o r c e s

19

o f t he v a lv e s tem i s (6)

F = MvD2Y + bDY + (EPpv + KV)Y + p L D q r .

F = Force

Mv = Mass o f v a lv e spool

b = C o e f f i c i e n t o f damping

E = 2CdW C osg

Q = Angle o f f l o w th rough or i f i c e

P = P r e s s u r e d rop a c r o s s p power v a lv e

Kv = Power v a lv e s p r i n g c o n s t a n t

L = Damping l e n g t h

q r = Flow to ram

As was done p r e v i o u s l y , the f l o w - p r e s s u r e cu rv e o f t he s e r v o ­

v a l v e must e n c l o s e t he l oad l o cu s c u r v e s o f t he power v a l v e .

A v a l u e f o r damping was assumed f o r t he v a lv e in s i m i l a r

f a s h i o n a s was done f o r t he ram.

A computer p rogram was w r i t t e n which t ook the p o i n t s

c a l c u l a t e d by t h e p r e v i o u s computer program and r e l a t e d them

to the power v a l v e . By assuming a f i r s t s t a g e f l ow , t he

power v a lv e l oad l o c u s was compared t o t he f l o w - p r e s s u r e

cu rve o f t he assumed f i r s t s t a g e . T h i s p rogram had q u i t e

a few v a r i a b l e s and some c o n s t a n t s t h a t had t o be changed

t o f i n d t he c o m b in a t i o n d e s i r e d , such a s f r e q u e n c y , a m p l i t u d e ,

s p r i n g c o n s t a n t s , mass and f l o w .

The program d id s e v e r a l t h i n g s . I t d e t e r m i ne d t he

maximum d i s p l a c e m e n t o f a c e r t a i n v a lv e mass f rom which a

s i z e c o u l d be c a l c u l a t e d , t h e mass o f t he v a lv e be in g a

f u n c t i o n of t he va lve l e n g t h and d i a m e t e r , b o t h o f which

had to be f e a s i b l e . The program a l s o d e t e r m i n e d what f l ow-

r a t e c a p a b i l i t y was needed f o r t he f i r s t s t a g e v a l v e . T h i s ,

20

t o o , had t o be f e a s i b l e from the s t a n d p o i n t o f a v a i l a b i l i t y

o f such s e r v o v a l v e s .

The r e s u l t s o f the computer p rogram a r e a s f o l l o w s :

t h e spool a t the o r i f i c e must be one i nch in d i a m e t e r w i th

360 deg ree p o r t ; the w e i g h t o f t he spool can be no more than

one pound; t he f r e q u e n c y can be a s h i g h a s 300 cps b e f o r e

t he a m p l i t u d e o f d i s p l a c e m e n t o f t he spool d e c r e a s e s ; t he

p r e s s u r e s u r f a c e o r s p o o l - e n d d i a m e t e r must be o n e - h a l f i nch ;

and f l ow r a t e t o t he spoo l end must be a t l e a s t 15*^ i n ^ / s e c ;

s p r i n g c o n s t a n t o f t h e spool v a l v e can be 30 t o 100 pounds

p e r i n c h . Damping l e n g t h used in t he p rogram was .2 i n c h e s .

T h i s , l i k e t h e s p r i n g c o n s t a n t , i s n o t c r i t i c a l , e x c e p t t h a t

i t c a n n o t be n e g a t i v e f o r s t a b i l i t y ( 7 ) •

Du c t i n g shou ld be a t l e a s t f o u r t im e s a s l a r g e a s the

maximum a r e a o f t he o r i f i c e s in o r d e r t o a v o i d s a t u r a t i o n in

t he v a l v e ( 8 ) .

Upon i n v e s t i g a t i n g s e r v o v a l v e s , i t was found t h a t Moog

S e r v o c o n t r o l s , I n c . c o u ld mod i fy t h e i r 31-010A s e r i e s v a lv e

so t h a t ga in was e s s e n t i a l l y z e ro t o ab ou t 700 cps on t he

f r e q u e n c y v e r s u s d e c i b l e s g r a p h . T h i s v a lv e would g ive the

f l ow n e c e s s a r y and o p e r a t e a t 3000 p s i . (For ga in d i ag rams

see F i g . 9 • )

F l u i d S p r i n g . S p e c i f i c a t i o n s f o r a v a r i a b l e s p r i n g

were g iven t o C l a i r S h i e l d s who c o n s i d e r e d d i f f e r e n t d e s i g n s

f o r a v a r i a b l e s p r i n g and co n c l u d ed t h a t a f l u i d s p r i n g would

be most e a s i l y o b t a i n e d . He t hen d e s i g n e d and p roduced

d r aw ing s o f t h e f l u i d s p r i n g .

F i g . 9 . — Ampl i t ude and Phase Response Diagram

22

2The e q u a t i o n o f mot ion f o r t he l o a d , MD X + CDX + KX,

s o l v e d f o r a m p l i t u d e i s

A = ______________ Fc/K________________________ [27]

" \ / ( l - ( W / W n ) 2) 2 + ( 2 5 6 0 n/ W ) 2

Where Fc i s a c o n s t a n t f o r c e a p p l i e d to t h e mass in t h e sy s t e m .p

I f GJ n i s much g r e a t e r t han GO i t makes (2S?6Jn/GO) the o n l y

s i g n i f i c a n t t e rm u nd e r t he r a d i c a l s i g n ; t h u s

A = Fc A = Fc / ^ n ^ „ FcM2 S W „ / W 2 S W n/ u ) 2 S 6 J 3

[28]

E q u a t i o n 28 s a ys t h a t a m p l i t u d e i s p r o p o r t i o n a l to o p e r a t i n g

f r e q u e n c y when 60 i s much l e s s than OJ n , b u t because 60 i s

smal l in co mp a r i so n , a m p l i t u d e i s s m a l l .

I f 60 i s much g r e a t e r t han Go t hen t he s i g n i f i c a n t

f a c t o r i s (C0 /00n )^ and

A = Fc/MOl>2 .

T h i s s a ys t h a t a m p l i t u d e i s i n v e r s e l y p r o p o r t i o n a l to the

N

squ a re o f f r e q u e n c y . On t h e o t h e r hand , i f GOn = GO then

A = Fc/CW . [30]

Now a m p l i t u d e i s i n v e r s e l y p r o p o r t i o n a l t o t he f r e q u e n c y to

t he f i r s t power o n l y . T h i s c o n t r a s t can be s een r e a d i l y in

F i g u r e 10.

Ampl i t ude d e c r e a s e s w i t h f r e q u e n c y , b u t t he r a t e can

be changed s u b s t a n t i a l l y i f t he n a t u r a l f r e q u e n c y can be

changed t o c o r r e s p o n d t o t he o p e r a t i n g f r e q u e n c y . T h i s then

shows t he need f o r a v a r i a b l e s p r i n g .

23

Both mec han i ca l and f l u i d s p r i n g s were c o n s i d e r e d f o r

t h i s v a r i a b l e s p r i n g . The mechan i ca l type c o n s i d e r e d was a

doubl e-wedge c a n t i l e v e r . T h i s sy s t em was f e a s i b l e s i n c e s p r i n g

c o n s t a n t depends on l e n g t h . However, t h i s d e s i g n was b u l k y ,

and many p rob l ems were i n vo lv e d in a t t a c h m e n t and p o s i t i o n i n g .

The f l u i d s p r i n g a l s o p r e s e n t e d p ro b l e m s ; however , t h ey

c o u ld be overcome more e a s i l y . I f t he f l u i d u sed were incom­

p r e s s i b l e , t h e r e would be no s p r i n g i n v o l v e d . T h i s no t be ing

t he c a s e , i t can be shown(9) t h a t t h e s p r i n g c o n s t a n t r e l a ­

t i o n s h i p i s

K = [31]

where Ap i s t h e e f f e c t i v e a r e a o f t he p i s t o n .

24

Thus t he s p r i n g c o n s t a n t (K) i s i n v e r s e l y p r o p o r t i o n a l t o t he

volume (V^) o f f l u i d e n t r a p p e d on b o t h s i d e s o f t he s p r i n g

p i s t o n . T h i s s ays t h a t i f t he volume i s c o n t r o l l e d , t he s p r i n g

c o n s t a n t can be c o n t r o l l e d . Us ing a s i l i c o n e o i l which ha s

a b u l k d e n s i t y ( ) o f ab ou t 140 ,000 pounds p e r squa re i nch ,

a s i z e and c o n f i g u r a t i o n f o r t h e f l u i d - s p r i n g p i s t o n and

c y l i n d e r was c a l c u l a t e d . An o th e r computer p rogram^ was w r i t t e n

t o f a c i l i t a t e v a r y i n g t h e a r e a and a l s o c a l c u l a t e volume,

f r e q u e n c y , and s p r i n g c o n s t a n t f o r e ach a r e a c h o s e n . From

t h i s a f i v e - s q u a r e - i n c h e f f e c t i v e p i s t o n a r e a was c h o se n ,

and a maximum volume o f 9«4 g a l l o n s , and an o p e r a t i n g range

from 30 cps t o 450 cps were o b t a i n e d . S ince VK/M e q u a l s

t he n a t u r a l f r e q u e n c y , t h en by c o n t r o l l i n g t he volume, and

t h u s c o n t r o l l i n g t h e v a lu e o f K, t he n a t u r a l f r e q u e n c y i s

a l s o c o n t r o l l e d .

^ A v a i l a b l e in the Brigham Young U n i v e r s i t y Mechan i ca l E n g i n e e r i n g De pa r tm en t .

CHAPTER IV

ANALYSIS OF SYSTEM

The sys tem i s shown in b l o c k d i ag ram be low.

1. Osc i 1l a t o r 5- Power Spool Va lve2. S u b t r a c t o r 6. Ram3-k*

Lead C i r c u i t Moog Se rv o va lv e

7- Feedback

F i g . 1 1 . - - S y s t e m

The e l e c t r i c a l c i r c u i t s were n o t d e s i g n e d ; however ,

t he m ag n i tu de s o f a m p l i f i c a t i o n and t he f r e q u e n c i e s n e c e s ­

s a r y were e v a l u a t e d and c o n s i d e r e d p o s s i b l e .

F i r s t c o n s i d e r e d in t he a n a l y s i s were t he r e l a t i o n -

s h i p s be tween i n p u t and o u t p u t f o r t he Moog S e r v o v a l v e . Th i s

v a l v e was chosen because i t was t h e b e s t a v a i l a b l e c o n s i d e r i n g

f l ow r a t e and f r e q u e n c y r e s p o n s e . From a f r e q u e n c y r e s po n se

c u r v e d i t was obv io us t h a t t he t r a n s f e r f u n c t i o n was n o t a

s imple second o r d e r e q u a t i o n w i t h .5 t o .7 damping a s Moog

i n d i c a t e d f o r t he r e g u l a r v a l v e . To a pp rox im a te t he c u r v e ,

^■Bode d i ag ram f o r m o d i f i e d s e r v o v a l v e .

25

26

Bode p l o t s ( l O ) o f a number o f f i r s t and s econd o r d e r t r a n s f e r

f u n c t i o n s were drawn on t r a c i n g p a p e r and a d d e d . When the

r i g h t c o m b i na t i o n o f t r a n s f e r f u n c t i o n s were fo und , t h e i r

a d d i t i o n compared f a v o r a b l y w i t h t h e cu rve f o r t he m o d i f i e d

s e r v o v a l v e . The c o m b i n a t i o n t h a t f i t b e s t was one w i t h two

f i r s t o r d e r t r a n s f e r f u n c t i o n s w i t h c o r n e r f r e q u e n c i e s o f

%l\.0 c p s , p l u s a second o r d e r f u n c t i o n w i t h a n a t u r a l f r e q u e n c y

o f Sb0 c p s and damping eq u a l t o .2 ( F i g . 9 , p . 2 1 ) .

The t r a n s f e r f u n c t i o n e q u a t i o n s in L ap l a c e t r a n s f o r m

n o t a t i o n a r e

G j ( s ) = z / i = i / ( ( s T + D 2 ( s 2 + 1357S + 3 3 9 3 ) ) , [32]

where l / ' f = 5ij.O c p s o r 3393 r a d i a n s / s e c .

Z = D i s p l a ce m e n t o f t h e ram

1 = C u r r e n t

T h i s a pp ro x i m a t e t r a n s f e r f u n c t i o n d e v i a t e s from th e cu rv e

more a s f r e q u e n c y i s i n c r e a s e d above lj.00 c p s j even s o , t he

d e v i a t i o n up t o 600 cps i s no t more t h a n o n e - h a l f d e c i b e l .

The f l o w r e l a t i o n s h i p o r o r i f i c e e q u a t i o n i s :

Q = ZCdW V2Pnv/P = ZCdwV2(Ps-Pva/ P • [33]

Qm = Flow from Moog v a l v e

Pmv= P r e s s u r e d rop ove r Moog v a lv e o r i f i c e

Cd = C o e f f i c i e n t o f d i s c h a r g e

W = P e r i p h e r a l l e n g t h o f v a lv e a t o r i f i c e

P = O i 1 d e n s i t y

P ^ s P r e s s u r e d rop a c r o s s power v a l v e end or p r e s s u r e s u r f a c e

o a s a s u b s c r i p t i s p o i n t o f l i n e a r i z a t i o n

L i n e a r i z i n g t h e s e e q u a t i o n s t o make them t r a c t a b l e r e s u l t s in

^m " ^mo Kj (Z ” 2Q) + K g ^ v a - ^vao NF u r t h e r r e d u c t i o n r e s u l t s in

M

where

K3 = - (Kl Zo + V '

K, = b Q / a Z1 m

K = b Q / b P •2 nr va

,_P + Q )2 vao mo [36]

[37]

[38]

and K^ were t ak en f rom the f l o w - p r e s s u r e - c u r r e n t

c u r v e s s u p p l i e d by t h e Moog s e r v o v a l v e company. They can a l s o

be e v a l u a t e d d i r e c t l y f rom t h e p a r t i a l d e r i v a t i v e s o f t h e o r i ­

f i c e e q u a t i o n .

The f l o w f rom t h e Moog v a l v e can be r e l a t e d t o t h e

v e l o c i t y o f t h e power v a l v e s i n c e t h e f l ow from t h e Moog v a l v e

(Qm) i s eq u a l t o t h e a r e a (Av ) o f t h e s u r f a c e i t i s a c t i n g

upon in t h e power v a lv e , m u l t i p l i e d by t h e v e l o c i t y o f t h a t

s u r f a c e , o r

The f o l l o w i n g i s an a n a l y s i s o f t h e power v a l v e e q u a t i o n s

The f o r c e b a l a n c e e q u a t i o n f o r t he power v a l v e i s

Q = A DYm v [39]

P A = M D2Y + bDY + (EP + K )Y + D LDq va v v pv v ' r

Ay = Area o f power v a l v e

Y = D i sp l ace me n t o f power v a l v e spool

Mv = Mass o f spool

b = C o e f f i c i e n t o f damping

PpV = P r e s s u r e d rop a c r o s s power v a lv e

= Power v a l v e s p r i n g c o n s t a n t

28

Kv

L = Damping l e n g t h

q f = Flow to ram

E = 2CdW CosQ

Q = Angle o f f l o w from o r i f i c e

The t e rm p L D q r was n e g l e c t e d s i n c e i t can be made v e r y sma l l

by making t he e f f e c t i v e damping l e n g t h s m a l l ( l l ) .

A f t e r t r a n s f o r m i n g i n t o S n o t a t i o n and combin ing eq u a ­

t i o n s 3 2 , 35, 39, and I4.0 ,

L e t ( l / A v ) ( - K 2Mv S2 + (A2 - K2b)S - ) = G2 ( S ) . [ij.3]

K j G j t S ) ! = (Y/Av ) ( - K2Mv S2 + (A2 - K2b)S - K ^ ) - K y [l|.l]

R e a r r a n g i n g r e s u l t s In

Y = Ki G1( S ) I / G 2 (S) + K3/ G 2 (S)

Now i t i s known t h a t f o r t h e ram, q„ = A DX,r rand Q = K Y + K P + K

pv $ r 1P + K = q .

5 r 6 r

Qpv = Flow f rom power v a l v e

q r = Flow to t he ram

Ar = Area o f t h e ram

k5 = j>qr / b P r

M

The f o r c e b a l a n c e e q u a t i o n f o r t h e l oad i s

F = P A = MD2X + CDX + KX.r r r

29

= P r e s s u r e d rop a c r o s s t h e ram

X = D i sp l a c e m e n t o f t h e ram

Fr = Ram f o r c e

M = Mass o f l oad

C = C o e f f i c i e n t o f damping

K = Ram s p r i n g c o n s t a n t

Combining e q u a t i o n s ij.5, ij.6, and I4.7 g i v e s

Y = ( X/K^A^) ( - K^MS2 + (A2 - K^C)S - K^K) - K6/K^. [5l]

Combining e q u a t i o n s i|l|. and 5 l r e s u l t s in

X - (K1G1( S ) / G 2 (S)G3 ( S ) ) I + K3/G 2 (S)G3 (S) -K7/ G 3 (S)[52j

G3 (S) = ( l /K^A r ) ( - K^MS2 + (A2 - K^C)S - K^K) [53]

k 7 = - W NE q u a t i o n 52 i s now an e q u a t i o n in t e rms o f d i s p l a c e m e n t o f t he

ram and i n p u t c u r r e n t t o t h e Moog s e r v o v a l v e . The e q u a t i o n i s

no t a d i r e c t r e l a t i o n s h i p be tween X and I a s would be most

d e s i r a b l e . With a l i t t l e i n v e s t i g a t i o n , however , e q u a t i o n 52

can be a n a l y z e d a s i f i t were a d i r e c t r e l a t i o n s h i p .

F i g u r e 12 shows t h e b l o c k d i ag ram o f t h e open loop

t r a n s f e r f u n c t i o n s . The two e x t r a t e rm s ( i n s i d e t he dashed

l i n e s ) can be a n a l y z e d a s e x t e r n a l s i g n a l s i n t o t he sys tem

X

F i g . 1 2 . — KG P l u s E x t e r n a l I n p u t s

and need n o t be c o n s i d e r e d in t h e s t a b i l i t y a n a l y s i s . I f t he

sys tem i s s t a b l e , i t w i l l r emain so w i t h t h e s e e x t e r n a l s i g n a l s

a d d e d ( 12 ) .

30

Now h av ing t h e l i n e a r r e l a t i o n s h i p

x / i = k1g1(s ) /g2(s )g (S), 55

t h i s r e l a t i o n s h i p can be i n v e s t i g a t e d f o r s t a b i l i t y by means

o f r o o t l o c u s . I f G j ( S ) , G ^ S ) and G^(S) a r e s u b s t i t u t e d

i n t o e q u a t i o n 55> t he r e s u l t s a r e

x = KG = Kl % ArAvA ^M yK2 _____________ x1 (S2+(A2-K^C)S+K)(S2+(A2-K2b)S+T1)

-K^H M - K ^

1__________ . [56]( s T + D 2(s2+1357S+3393)

S u b s t i t u t i n g v a l u e s f o r t h e c o n s t a n t s and f a c t o r i n g g i v e s

KQ = Kl K),Ar Av / K^K2MvMT 2_______________________ x( S+25J++J2513) ( S+25JJ.-j 2513) ( S) ( S+Jj.00000)

___________ 1_____________ .(S+3393)2(S+2.51)(S+135^)

I t can be seen t h a t t h e r e a r e v e r y low f r e q u e n c y r o o t s in t he

sy s t em t h a t need t o be compensa ted by z e r o s t o p r o v i d e h igh

f r e q u e n c y r e s p o n s e . T h i s can be p a r t i a l l y a c c o m p l i sh e d w i t h

t h e t y pe o f f e e d b a c k u s e d . I f a c c e l e r a t i o n , v e l o c i t y andp

p o s i t i o n f e e d b a c k a r e u s e d , t h en (C^S + C£S + C^ ) can be

i n t r o d u c e d i n t o t h e e q u a t i o n s . A c c e l e r a t i o n c o u ld be p r o ­

v i d e d by an a c c e l e r o m e t e r ; v e l o c i t y , by i n t e g r a t i n g the a c c e l ­

e r a t i o n , and p o s i t i o n , p o s s i b l y by an i n d u c t i o n p o t e n t i o m e t e r .

By s e l e c t i n g t h e r i g h t r a t i o be tween t h e c o n s t a n t s ( a m p l i f i ­

c a t i o n ) , two o f t h e low f r e q u e n c y p o l e s can be c a n c e l e d ; t h a t

i s , i f

= 2.51 [58] c 1/ c 3 = .01 [59]

31

th en e s s e n t i a l l y

c 3s 2 + c 2s + c x = c 3 ( s ( s + 2 . 5 1 ) ) ;

so t h a t

kah = K ^ A r A ^ j f S X S + S . S l l / K j K a l V i r 2( S+251++J 2 5 13) ( S+25I+- J 2513) ( s ) ( S+ij.00000)

__________ 1_____________ . feol(S+3393)2(S+2.£l)(S+135^) j

To e l i m i n a t e t h e o t h e r low f r e q u e n c y p o l e s , t h e r e mus t be more

c o m pe nsa t i n g c i r c u i t s in t h e f e e d b a c k loop o r in f r o n t o f t he

Moog v a l v e , one be ing t h e e q u i v a l e n t o f t h e o t h e r . The r o o t

l o c u s p l o t w i t h o u t more comp ensa t i o n i s shown in F i g u r e 13«

□ Po l e

A 2 P o l e s , 2 Zeros

@ 2 P o l e s

.2000

.1000

Pole a t IpclO^

4000 2000

,1000

.2000

F i g . 13«—Root Locus , Uncompensated

I t i s p o s s i b l e t o i n t r o d u c e z e r o s i n t o t he sys tem to

e l i m i n a t e unwanted p o l e s in two ways . The f i r s t i s u s u a l l y

32

by pa s se d b eca u se t he ne twork r e q u i r e s an e x p e n s i v e a m p l i f i e r .

The s econd i s a s imple RC c i r c u i t ( s e e F i g . lij.) which p l a c e s

a z e r o in t h e r e g i o n o f t he c r i t i c a l f r e q u e n c y . T h i s RC

c i r c u i t i s c h a r a c t e r i z e d by a p o l e and a z e ro w i t h t h e po l e

l o c a t e d a t a h i g h e r f r e q u e n c y ( a t l e a s t two o c t a v e s ) and ,

t h e r e f o r e , d o e s , n o t oppose t he e f f e c t s o f t h e z e r o . The

l e a d c i r c u i t w i l l need t o be i s o l a t e d by an a m p l i f i e r in

o r d e r t h a t t h e l o ad s o f e ach c i r c u i t may be i s o l a t e d .

= R e s i s t o r EQ = V o l t a g e o u t

C = C a p a c i t o r

F i g . 1^ . - -RC C i r c u i t

Through t r i a l and e r r o r i t was found t h a t by c a n c e l i n g t h e

sys tem po l e a t 135k w i t h a n e tw or k z e r o and by c a n c e l i n g t h e

po l e a t 3393 by two z e r o s , t h e sys tem becomes s t a b l e a t a l l

v a l u e s o f ga in w i t h t he damping i n c r e a s i n g a s t h e g a in i n ­

c r e a s e s . S inc e t he p o l e l o c a t i o n s depend on many v a r i a b l e s ,

e x a c t c a n c e l l a t i o n would be d i f f i c u l t . Because o f t h i s t h e r e

w i l l be some v a r i a t i o n s in t h e sys tem t r a n s f e r f u n c t i o n which

was d e s i r e d . T h i s v a r i a t i o n w i l l add t h e e x t r a p o l e s and

z e r o s n o t c o m p l e t e l y c a n c e l e d t o t he o v e r a l l sys t em f u n c t i o n ;

b u t because o f t h e c l o s e n e s s o f t he z e r o t o t he p o l e , t he

33

r e s u l t i n g sma l l r e s i d u e a t t h e p o l e , and t h e smal l v a r i a t i o n in

r e s i d u e s a t t he o t h e r s , make such a sys t em s t i l l a d e q u a t e ( 13 )•

The sy s t em p a r a m e t e r s a r e i n h e r e n t l y u n s t a b l e . These

p a r a m e t e r s a r e : c o m p r e s s i b i l i t y , which i s a f u n c t i o n o f temp­

e r a t u r e ; t ime c o n s t a n t s and v i s c o s i t y , a l s o t e m p e r a t u r e s e n s i ­

t i v e ; c o n d u i t ( i . e . , f l u i d l i n e ) r e s o n a n c e and o t h e r s . In

u s i n g a co m p e n sa t i o n method such a s t h i s , i t i s most l i k e l y

t h a t t h e r e w i l l be m o d i f i c a t i o n s n e c e s s a r y a s p rob l ems a r i s e .

These p rob l ems canno t be p r e d e t e r m i n e d and may n o t be e a s i l y

d i a g n o s e d and s o l v e d . Some o f t h e s e w i l l be d i s c u s s e d in

t h e r ecommenda t i ons .

The method o f c r i t i c a l f r e q u e n c y c a n c e l l a t i o n w i l l n o t

c r e a t e t h e needed f o r c e t o d r i v e t he l o a d s ( v a l v e s , ram, e t c . )

a t a h i g h e r f r e q u e n c y . I t can o n l y make h i g h e r f r e q u e n c i e s

p o s s i b l e i f t he machine i s c a p a b l e o f o p e r a t i n g a t t h o s e f r e ­

q u e n c i e s . The h y d r a u l i c sys tem which i n c l u d ed t h e Moog v a lv e

was d e s i g n e d f o r t he h i g h f r e q u e n c i e s , which a r e n o t p o s s i b l e

w i t h o u t t h e e l e c t r i c a l l e a d n e t w o r k . For low f r e q u e n c i e s

and low a m p l i t u d e s t he h y d r a u l i c sys tem w i l l no t be a t peak

powe r .

The f i n a l c l o s e d l oop e q u a t i o n i s

( s + i / r ) ( s + i / T 3 r (s+2 5 iph J2 5 i3 )

i(S+251+-J2513) (s+^oMo'b)

1/ T2 = 3393 i / T _ , = I 3 $ k 1 / T = 5ooo 5 3

3>k

At a v a l u e o f damping o f . 7 , whi ch c o r r e s p o n d s t o an

a n g l e o f lj.5 d e g r e e s on t he l o c u s ( F i g . 15)» t h e r e e x i s t s a

v a l u e o f g a i n o f ab ou t 10? a s suming A^ e qua l t o 1. By i n ­

c r e a s i n g o r d e c r e a s i n g A^, t h e g a in would d e c r e a s e o r i n ­

c r e a s e a c c o r d i n g l y . T h i s means t h a t by making A^ l a r g e (10-^

t o 107 ) , t he damping w i l l a lways be l a r g e enough ( . 3 t o . 7 )

t o e f f e c t r e a s o n a b l y good r e s p o n s e . A s i n e wave i n p u t was

e n v i s i o n e d a s t he u s u a l i n p u t ; however , o t h e r wave forms

c o u l d be u s e d .

□ Pol e

V 2 P o l e s , Ze ros A 2 P o l e s , 2 Ze ros

O P o l e , Zero

P o l e s , 50 ,0 0 0

P o l e , 20 , 000 P o l e , IpclO^

k.000

1000

2000

3000

F i g . 1 5 . - - R o o t Locus Compensated System ( P l o t t e d n e a r o r i g i n on ly )

The t e r m (S2 + (S(A2 - K ^ C ) / ( - K^M)) + K/M) in Equa­

t i o n 56 i s w o r t h y o f f u r t h e r e x a m i n a t i o n . The n a t u r a l f r e ­

quency 60 wh ich i s eq ua l t o V K / M in t he sys tem can be nchanged b e cau se o f t he v a r i a b l e s p r i n g . The mass on t he

35

ram, M, can a l s o be changed which w i l l change t h e t e rm

((A^ - K^C) / - K^M). S ince 60^ i s d e c r e a s e d when i n p u t f r e ­

quency i s d e c r e a s e d , t h e r e s p o n s e i s r e l a t i v e l y unchanged .

As t h e mass i s d e c r e a s e d , t h e t e rm ( (A^ - K^C) / - K^M) i s

i n c r e a s e d and r e s u l t s in t h e complex po l e b e in g f u r t h e r away

f rom t h e im a g in a r y a x i s . Th i s makes i t p o s s i b l e t o o p e r a t e

a t h i g h e r f r e q u e n c i e s .

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

I t h a s been s t a t e d p r e v i o u s l y t h a t becau se o f t he com­

p l e x i t y and s i z e o f such an u n d e r t a k i n g , t he scope o f t h i s

t h e s i s does not* i n c l u d e t he t o t a l d e s i g n o f a h y d r a u l i c v i b r a ­

t i o n mac h ine . Because f l u i d power r e s e a r c h i s q u i t e young ,

e s s e n t i a l l y b e g i n n i n g w i t h t he demand o f World War I I , much

i s s t i l l unknown. The p rob l em o f c o n t r o l which i s s o l v e d in

t h i s t h e s i s may be i n a d e q u a t e a f t e r t he components a r e a c t u ­

a l l y f a b r i c a t e d . T h i s s i t u a t i o n would no t be s u r p r i s i n g s i n c e

many o f the c o n s t a n t s u se d in t he c a l c u l a t i o n s had t o be assumed.

When t h e s e c o n s t a n t s , such a s damping in t he v a l v e and in t h e

ram, a r e found e m p e r i c a l l y f rom the components t h e m s e l v e s ,

t h e n , de pend ing on t he d e v i a t i o n s from the assumed v a l u e s ,

some p rob l ems may need r e s o l v i n g .

Many r e s e a r c h e r s , p a r t i c u l a r l y t h o s e who have had

e x p e r i e n c e in t h i s f i e l d , i n d i c a t e t h a t in many f l u i d c o n t r o l

d e s i g n s some o f t he c o n t r o l work must be done a f t e r a p r e ­

l i m i n a r y d e s i g n o f t h e sys tem ha s been made and a machine

b u i l t . An example o f t he p rob l ems a r i s i n g and n o t a c c o u n t e d

f o r in t h i s s t u d y i s t h a t o f r e s o n a n t l i n e s between v a l v e and

l o a d . I f t h i s p rob l em i s e n c o u n t e r e d when t h e v i b r a t i o n machine

i s made, i t i s recommended t h a t compos i t e l i n e s o f d i f f e r e n t

36

37

l e n g t h s a n d / o r d i a m e t e r be t r i e d , t he smal l d i a m e t e r b e in g

p l a c e d n e x t t o t he v a l v e ( l l } J . I f p r a c t i c a l , a c o n i c a l l i n e

i s recommended(15 ) •

I t i s recommended t h a t an o i l c o o l i n g sys tem be i n c o r ­

p o r a t e d i n t o t h e low p r e s s u r e s i d e o f t he o i l sy s t em. I t i s

e s s e n t i a l t o keep t he o i l t e m p e r a t u r e down when f u l l f l ow i s

n o t u t i l i z e d and t o r educe t e m p e r a t u r e s e n s i t i v i t y e f f e c t s

a s much a s p o s s i b l e . The f l u i d s p r i n g was a l s o d e s i g n e d t o

be c o o l e d .

S ince t h e n a t u r a l f r e q u e n c y o f the l oad i s a f u n c t i o n

o f many v a r i a b l e s , t he o p e r a t i n g f r e q u e n c y w i l l n o t be p r e ­

c i s e l y matched w i t h o u t some m a n i p u l a t i o n . I t i s p o s s i b l e

t h a t u n t i l t h e sy s t em r e a c h e s a s t e a d y t e m p e r a t u r e s t a t e , a

f r e q u e n c y match w i l l be q u i t e d i f f i c u l t . When a s t e a d y s t a t e

c o n d i t i o n i s a c h i e v e d , a f r e q u e n c y match shou ld be o b t a i n a b l e

by v a r y i n g t h e volume o f f l u i d in t he f l u i d s p r i n g . I f a

s p e c i f i c f r e q u e n c y i s n o t t oo c r i t i c a l , t h e o p e r a t i n g f r e ­

quency which i s c o n t r o l l e d by t h e o s c i l l a t o r can be changed

t o match t he l oad n a t u r a l f r e q u e n c y a f t e r t h e y a r e c l o s e to

each o t h e r . A s e p a r a t e s i g n a l f rom the o u t p u t d i s p l a y e d on

an o s c i l l o s c o p e might we l l be used t o check o u t p u t a m p l i t u d e

a g a i n s t t he d e s i r e d a m p l i t u d e .

I t i s l i k e l y t h a t t h e r e w i l l be f r e q u e n c y d r i f t i n g due

t o t h e n o t e d i n f l u e n c e s . I f t h i s d r i f t i s n o t e x c e s s i v e ,

r e a d j u s t m e n t f rom t ime t o t ime w i l l be s a t i s f a c t o r y . I f t h e

d r i f t i s e x c e s s i v e , t hen f u r t h e r c o n t r o l o f t h e env i ro nm en t

w i l l be n e c e s s a r y . I f t he r e q u i r e m e n t s a r e n o t t oo s e v e r e .

38

a c o n d i t i o n o f n e a r f r e q u e n c y match may be d e s i r e d . T h i s n e a r

match would a l l o w the a m p l i t u d e to be a d j u s t e d e i t h e r up or

down a s t he d r i f t r e q u i r e s .

In t he v e r y h igh f r e q u e n c y range i t w i l l n o t be p o s s i b l e

t o c o r r e c t d r i f t by ch ang in g t he f l u i d s p r i n g a s i t i s now

d e s i g n e d s i n c e t he a d j u s t a b l e heads on t he c y l i n d e r must be

moved. T h i s w i l l r e q u i r e f r e q u e n c y ma tch ing by t he i n p u t o r

ch ang in g t h e c y l i n d e r d e s i g n .

Noi se p rob l ems a r e p o s s i b l e , bu t n o t p r o b a b l e . Flow

i n s t a b i l i t i e s somet imes cause h i g h f r e q u e n c y n o i s e , b u t r a r e l y

a f f e c t s a s y s t e m ' s p e r f o r m a n c e ( 16). E l e c t r i c a l n o i s e can be

a 60 cps n u i s a n c e some t imes , b u t can be p r e v e n t e d by us. ing a

Ij.00 cp s e l e c t r i c a l sys tem o r i t can be e l i m i n a t e d a f t e r i t

h a s become a p ro b l em. A m p l i f i e r n o i s e i s u s u a l l y much t oo

h i g h a f r e q u e n c y t o be a p rob l em in t h i s sy s t em.

I t i s recommended t h a t t he v a lv e be made by a v a l v e -

m a n u f a c t u r i n g company which h a s had e x p e r i e n c e in t h i s f i e l d

and can f a b r i c a t e t o t o l e r a n c e s s p e c i f i e d on t h e d r a w in g s .

I f t h i s i s n o t p o s s i b l e , t h e e n g i n e e r r e s p o n s i b l e f o r t h i s

p r o j e c t sho u l d f o l l o w the c o n s t r u c t i o n v e r y c l o s e l y and be

f a m i l i a r w i t h t he r e q u i r e m e n t s and p rob l ems co nn ec t e d w i t h

such a f a b r i c a t i o n . S e c t i o n s 9*5 and 9*54- ° f FLUID POWER

CONTROL a r e u s e f u l r e f e r e n c e s .

The t o t a l d e s i g n , e s p e c i a l l y t he h i g h p r e s s u r e i t e m s ,

sho u l d be checked by a r e g i s t e r e d p r o f e s s i o n a l e n g i n e e r

b e f o r e u s e .

LIST OF REFERENCES

1. C y r i l M. H a r r i s and C h a r l e s E. C r e d e , SHOCK ANDVIBRATION HANDBOOK, Vo l . 2 . New York: McGraw-H i l l ( 1 9 6 1 ) , Chap. 25 .

2 . John E. Gibson and F ra nz B. T u t e u r , CONTROL SYSTEMCOMPONENTS. New York: McGraw-Hi l l ( 1 9 5 8 ) , C h a p t e r s10 and 11.

3 . John F. B l ac k bu r n , Ge rha rd R e e t h o f and J . Lowen S h e a r e r , FLUID POWER CONTROL. Cambr idge , Mass: M . I . T . P r e s s( I 9 6 0 ) .

Ij.. B l a c k b u r n , p . 179

5 . I b i d . , p . l 8 l .

6 . I b i d . , p . 381.

7. I b i d . , p . 373 .

8 . I b i d . , p . 253 .

9 . I b i d . , p . 166 .

10. C. J . S a v a n t , J r . , BASIC FEEDBACK CONTROL SYSTEM DESIGN. New York: McGraw-Hil l ( 1 9 5 8 ) , p . 129.

11. B l a c k b u r n , p . 376.

12. F r a n c i s M. Raven, AUTOMATIC CONTROL ENGINEERING. NewYork: McGraw-Hi l l ( 1961) .

13. J . G. T r u x a l , CONTROL SYSTEM SYNTHESIS. New York:McGraw-Hi l l ( 1 9 5 5 ) , p . 301]..

lit.. F. D. E z e k i e l , " E f f e c t s o f a H y d r a u l i c Con du i t w i t hD i s t r i b u t e d P a r a m e t e r s on C o n t r o l Va lve S t a b i l i t y . " Sc . D. T h e s i s , Depar tment o f Mechan i ca l E n g i n e e r i n g , M . I . T . , Cambr idge , Mass . (1951]-).

T r a n s . ASME, V o l . 80 (May, 1958) , pp . 9 0 l|-9 0 8 .

15 . B l a c k b u r n , p . 390 .

16. I b i d . , p . 3 6 3 .

17. I b i d . , p . 652 .

39

APPENDIXES

APPENDIX A

COMPUTER PROGRAMS

POWER VALVE PROGRAM

1 WRIT E(6 * 2)2 FORMAT( 4 0 HO MEL MERRELL LOAD LOCUS FOR POWER VALVE)

WRITEt 6 , 5 )5 FORMAT( 3 6 H - I N P U T INFORMATION SEE PROGRAM ABOVE)

DIMENSION B ( 2 1 ) , F O G ( 2 1 ) »VOG <2 1 ) , F F ( 2 1 ) » V < 2 1 ) . P F O R C E ( 2 1 ) , 1PV<2 1 ) , F P O g < 2 1 ) , V E L P ( 2 1 )

SUPPLY=3 0 0 0 . 0 ARAM=8 •5

RATCNG = . 0 2 WTS T OP= . 2 4 FLOWCH= 3 . 8 5 FLOWSP=2 5 . 0 RAT STR= . 3 F LOW = 1 5 . 4

FWANT=1 5 0 .P I = 3 . 1 4 1 5 9 2 7 RAT I 0 = . 5

C P V V I S = . 0 4 READ( 5 , 2 0 3 6 )

2 0 3 6 FORMAT( 1 1 9 H0 1

W R I T E ( 6 »2 0 3 6 )END I AM= . 5 PORT = 3 6 0 . 0 D I A M = . 9 K= 5 . 0

2 0 1 5 p v w a t e = d i a m * d i a m * r a t i oDL= . 2 KCHNG= 5 . 0 KSTOP=3 0 . 0 BMPV=PVWATE/3 8 6 . 0 DIAMST = . 9 D I A S T P = 1 . 1 DIAMCH= . 1 0 ENDSTP= . 7 ENDCHN= . 1 0 PORT CH = 2 0 . 0 PORTSP=3 6 0 . 0 ENDSTR = . 5

KSTART = 5 .FREQ=1 5 0 . 0

42

k3

C V I S = 1 5 . 0 BULK=1 0 0 0 0 0 •

D I V I D = 2 0 . 0 WATE=1000 •F S T O P = 5 0 0 . 0 AS TRT= 0 . 0 ACHNG1=. 0 2 AP T = 3 • 0 FCHNG = 2 0 . 0 FNCHNG = 5 0 • 0 FNSTOP=500 . 0 D I V I D 1 = 5 0 . 0 DI V-I DA = DI Vi D 1 ACHNG=ACHNG1 A=ASTRT BM=WATE/ 386 . 0 F=FREQWN = F * 6 . 2 8 3 2 8 5 3 FN = FWNP=SQRT(K/BMPV)ZETA=. 1 SQI GL=•1FNP=WNP/ 6 . 2 8 3 1 8 5 3 R = F * 6 . 2 8 3 1 8 5 3 I F ( F N ) 4 0 , 4 0 » 3VOL=( 4 . 0*ApT*APT*BULK) / ( BM*WN*WN)DO 45 I = 2 » 2 0B1 = A TAN( ( - 2 . 0*ZETA*R* W N ) / ( ( WN*WN) - ( R*R) ) )BZ=ARCOS( 0 . 0 )BINC = - ( ( SORT( ( B 1 - B Z ) * ( B 1 - B Z ) ) ) / D I V I D)B ( I ) =B ( I - D + B I N C

45 CONTINUE TRANS=0•0

550 DO 55 1 = 1 . 2 0FOG( I ) = ( ( S I N ( B ( I ) ) ) * ( ( ( WN*WN) / ( R * R ) ) - 1 . 0 ) ) + ( ( 2. 0*ZETA*WN

1 ) / R ) * ( COS( B ( 1) ) )I F ( FOG( I ) . GE . . 0 ) GO TO 552 FOG( I ) = - F O g ( I )

552 VOG( I ) = COS( B( I ) )F F ( I ) = F O G ( I ) *A*R*R*BMV ( I ) =V0G( I ) * A*RP V ( I ) = S U P P L Y - F F ( I ) /ARAMF P 0 G ( I ) = S I N ( B ( I ) ) * ( ( ( ( WNP*WNP) / ( R*R ) ) 1 . 0 ) + 3 . 4 4 2*PV( I ) / (

1 B M P V * R * R ) ) + C O S ( B ( I J ) * ( ( ( 2 . *SQIGL*WNP) / R ) + DL* ( ( S Q R T ( P V ) ) 2 / ( BMPV*R) ) )

55 CONTINUEDO 65 1 = 1 . 2 0FAVAL=2 7 . 7 7 7 7 7 7 * 1 9 0 0 . 0 - (V( I ) * V ( I ) ) )I F ( V ( I ) . GE. 3 0 . . O R . F F ( I ) . GE. FAVALJGO TO 80

I F ( T R A NS . G T . O . . AND. ( I ) . GE. 2 0 ) GO TO 91 65 CONTINUE

A=A+ACHNG GO TO 550

80 A = A - A C H N G / D I V I D A TRANS=1 . 0 GO TO 550

91 PRES=( B U L K * A P T * A ) /VOLV E L M A X = A M A X 1 ( V ( 1 ) , V ( 2 ) » V ( 3 ) , V ( 4 ) » V ( 5 ) , V ( 6 ) , V ( 7 ) , V ( 8 ) , V ( 9

1 ) * V ( 1 0 ) , V ( 1 1 ) , V ( 1 2 ) » V ( 1 3 ) » V ( 1 4 ) » V ( 1 5 ) » V ! 1 6 ) » V ( 1 7 ) » V ( 1 8 )2 » V ( 1 9 ) » V ( 2 0 ) )

A P V = ( VELMAX * 8 . 4 * 3 6 0 . 0 ) / ( 3 8 3 3 . 9 * 3 . 1 4 1 5 9 3 * D I A M* P OR T )DO 260 I = 1 , 2PFORCE( I ) = F P O G ( I ) * APV* R* R* BMPV V E L P ( I ) = V O G ( I ) * APV * R

260 CONTINUEXDOTMX = F L O W / ( ( P I * E N D I AM*END I AM) / 4 . )x d o t s q = x d o t m x * x d o t m xA R E A = P I * E N D I A M * E N D I A M / 4 .FACTOR=3000 . 0 * AREA/ XDOTSQ DO 280 I = 1 , 2PVFAVA = FACTOR * ( XDOTSQ- ( V E L P ( I ) * V E L P ( I ) ) )I F ( SWI TCH. EQ. . 0 ) GO TO 275I F ( V E L P ( I ) . L E . X D O T M X . O R . P F O R C E ! I ) . L E . P V F A V A ) GOTO 286 I F ( V E L P ( I ) . GE . XD OT MX . OR . P F OR C E ( I ) . G E . P V FA V A ) GO TO 2 24 0

280 CONTINUE GO TO 125I F ( F . L T . F W A N T ) GO TO 190FORMAT( 5 3HO DIAMETER AT O RI F I CE - WE I GH T - P O R T - E N D DIAMET

1ER FLOW, 5 F 1 5 . 5 )W R I T E ! 6 , 2 0 2 1 ) DI AM, PVWATE, PORT, NNDI AM, FLOW W R I T E ! 6 , 2 9 5 )

295 FORMAT( 102HOFREQUENCY VALVE PRESS VALVEAMPLI ANGLE 1 VALVE FORCE VALVE VELOCITY SQIGL VALVE NAT FRE 2Q )

W R I T E ! 6 , 3 0 0 0 ) F , P V ( I ) , APV , B ( I ) , P F O R C E ( I ) , V E L P ( I ) , S Q I G L , F N P 3 0 0 0 F O R M A T ( F 7 . 2 * F 1 2 . 2 , F 1 5 . 7 , F 8 . 3 , F 1 2 . 2 , F 1 4 . 2 , F 1 5 . 5 » F 1 3 . 3 )

W R I T E ( 6 » 1 1 0 )110 FORMAT( 100HOFREQUENCY AMPLITUDE ANGLE F/AWWM V / A

1W VELOCITY FORCE VOLUME NAT-FREQ ZETA PRES) W R I T E ! 6 , 1 2 0 ) F , A , B ( 1 ) , F O G ( 2 0 ) , V O G ( l ) , VELMAX, F F ( 1 ) , V O L , F N

I , ZETA,PRES120 FORMAT( F 7 . 0 , F 1 3 . 7 , F 1 0 . 3 , F 9 . 3 , F 9 . 3 , F 8 . 2 , F 9 . 2 , F 9 . 1 , F 1 0 .

I I , F l 0 . 5 , F10•1 )SWIT CH = 0 • 0GO TO 190

125 A=ASTARTI F ( F - FCHNG) 1 4 5 , 1 4 5 , 1 4 5 F = F + ( FCHNG/5 • )

I F ( F-FSTOP1 1 0 , 1 0 , 1 5 5 145 F=F+FCHNG

U = F / ( F - F C H N G )I F ( F . GT . FCHNG) GO TO 149 ACHNG=ACHNg / 1 . 1

149 ACHNG=ACHNg / < U * U )I F ( F-FSTOP > 1 0 , 1 0 , 1 5 5

155 A=ASTRTO I V I D A = D I V r D l ACHNG=ACHNG1 F = F REQI F( FN- FNC HNG )1 7 0 , 1 8 0 , 1 8 0

170 FN=FN+FNCHNG/5.0I F ( FN-FNST OP 11 , 1 0 , 2 0 0

180 FN=FN+FNCHNGI F ( FN-FNSTOP11 , 1 0 , 2 0 0

190 K=K+KCHNGI F ( K . L T . K S T O P ) GO TO 2015 K=KSTARTDIAM= DIAM + DIAMCH I F ( DI AM• LE . DIASTP 1 GO TO 2 15 DIAM=DIAMST

2215 END IAM = ENDI AM ENDCHNI F ( ENDIAM. LE . ENDSTP) GO TO 2015K=KSTARTDIAM=DIAMSTENDIAM=ENDSTRRATIO=RATlO—RATCNG

IFIFLOW.LT.FLOWSP) GO TO 2015 K=KSTART D IAM = DIAMST ENDIAM=ENDSTR RATI0=RAT STR FLOW = F LOW+FLOWCH

* IF(FLOW.LT.FLOWSP) GO TO 2015GO TO 200 SWITCH=1 . 0 F=F-FCHNG A=ASTART GO TO 10

200 STOP END

SENTRY LOCI

L I ST OF MNEMONIC CHARACTERS FOR POWER VALVE PROGRAM

ARAMBMPVBULKC P W I SDIAMDIAMCHDIAMSTDIASTPD I V I DD I V I D IDLDI V I DAENDCHNENDIAME'NDSTRENDSTPFNPFPOGFWANTKKCHNGKSTARTKSTOPPORTCHPORTlSPPVp v w a t eSQIGLSUPPLYXDOTMX

AREA OF RAMMASS OF POWER VALVE SPOOL BULK MODULUS OF OILCOEFFICIENT OF DAMPING ON POWER SPOOL DIAMETER OF POWER SPOOL DIAMETER INCREMENT I N I T I A L DIAMETER OF POWER SPOOL MAXIMUM DIAMETER OF POWER SPOOL ANGLE DIVIDERI N I T I A L DECREASING AMPLITUDE DIVIDER DAMPING LENGTH

DECREASING AMPLITUDE DIVIDER SPOOL END DIAMETER ENCREMENTSPOOL end d i a m e t e r i n i t i a l s p o o l end d i a m e t e rMAXIMUM SPOOL END DIAMETER POWER VALVE NATURAL FREQUENCY F/AWWM FOR POWER VALVE I N I T I A L FREQUENCY SPRING CONSTANT FOR POWER VALVE K INCREMENTI N I T I A L SPRING CONSTANT MAXIMUM K PORT INCREMENT MAXIMUM PORTPRESSURE DROP ACROSS POWER VALVE ORIFICE

WEIGHT OF POWER SPOOL DAMPING OF POWER VALVE SUPPLY PRESSUREMAXIMUM VELOCITY OF POWER SPOOLFOR OTHERS SEE L I ST OF MNEMONIC CHARACTERSPROGRAM

FOR RAM

RAM LOAD PROGRAM

WRIT E (6 * 2 )2 FORMAT( 29HOMEL MERRELL LOAD LOCUS THREE)

R E A D ( 5 , 4 ) F R E Q , C V I S » 3 U L K » D I V I D » V 0 L C H » W A T E » F S T 0 P , V L S T P » A S T 1RT, ACHNG, VOLST,APT, FCHNG

4 FORMAT( F 3 •0 , F 2 . 0 , F 6 . 0 , F 3 . , F 3 . 0 » FA . * F 3 . 0 » F 3 . , F 5 . 4 *1 F 5 • 4 * F 5 . 2 , F 2 . 1 , F 3 . 0 )

W R I T E (6 » 506 )506 FORMAT( 18H+INPUT INFORMATION)

WRIT E ( 6 > 6)6 FORMAT( H3HOFREQUENCY VI SCOUS- K BULK MOD DELTA A D

1ELTA VOL WEIGHT FREQ-STOP VOL STOP I N I T I A L A POT AR 2 -F )

W R I T E ( 6 , 8 ) F R E Q , C V I S , B U L K , A C H N G , V O L C H , W A T E , F S T O P , V L S T P , A S TRT »A P T , FCHNG

8 F O R M A T ( 2 F 8 . 0 , F 1 3 . 0 » F 1 0 . 4 , F 1 3 . 0 , 2 F 9 . 0 , F 1 2 . 0 , F 8 . 3 , F 1 1 . 2 , F 7 1 . )

FNCHNG=50•0 F N S T 0 P = 5 0 0 . 0 A=ASTRT BM = WAT E/ 3 8 6 • 0 F=FREQ FN = 0 . 0

14 R = F * 6 . 2 8 3 1 8 5 W N = F N * 6 . 2 8 3 1 8 5 Z E T A = C V I S / ( 2 . *BM*WN)WN = F N * 6 . 2 8 3 1 8 5 / ( S Q R T ( 1 . 0 - ( ZETA*ZETA ) ) )B1 = ATAN( ( ~ 2 • 0 * Z E T A * R * WN ) / ( ( WN*WN) - ( R * R ) ) )RN = W N / 6 . 28-3185 B2=ARCOS( 0 . 0 )

30 B=B1B I N C = - ( ( SORT( ( B 1 - B 2 ) * ( B 1 - B 2 ) ) ) / D I V I D )I F ( F N ) 3 5 • 3 5 * 3 2VOL = ( 4 . 0 * A p T * . A P T * B U L K ) / ( BM*WN*WN)

35 F O G = ( ( S I N ( B ) ) * ( ( ( WN*WN)/ ( R * R ) ) - 1 . 0 ) ) + ( ( 2 . 0 * Z E T A * W N ) / R ) * ( 1 C 0 S ( B ) )

I F ( FOG) 3 5 6 , 4 0 , 4 0 FOG=-FOG VOG=COS( B )V=VOG*A*RPRES=( B U L K * A P T * A ) /VOL F F=FOG* A* R*R* BM

k-7

k.Q

FAVAL = 2 7 . 7 7 7 77 7 * ( 9 0 0 . 0 - ( V * V ) )I F ( FAVAL) 2 0 0 , 6 5 * 6 5

65 I F ( FF-FAVAl ) 7 , 9 5 , 2 0 0200 A=A-ACHNG/20.

WRIT E ( 6 , 9 0 9 )909 FORMAT(2H+5)

6= B1201 F 0 G = ( ( S I N ( B ) ) * ( ( C W N * W N ) / ( R # R ) ) - 1 . 0 ) ) + ( ( 2 . 0#ZETA*WN) / R ) * (

1COS( B ))I F ( FOG) 20 12 » 2 2 , 2 0 2

2012 FOG=—FOG202 VOG=COS( B )

V=VOG*A*RFF=FOG*A*R*R*BMPR ES=( B U LK* APT* A) /VOL FAVAL=27. 7 7 7 7 7 7 * ( 9 0 0 . 0 —( V*V ) )I F ( F F . L E . O . O . A N D . V . L E . 2 9 . ) GO TO 208WRI TE ! 6 , 9 0 1 )

901 FORMAT( 2H+1 )I F ( FAVAL) 2 0 8 , 2 7 , 2 0 7 WRITE( 6 , 9 0 3 )FORMAT( 2 H +2 )Z=SQRT( S * B )I F ( F F . L E . F A V A L . A N D . Z . G E . B 2 ) G 0 TO 95 WRI TE ! 6 , 9 0 5 )

905 FORMAT( 2 H +3 )208 B=B+8INC

I F ( ( SORT( B * B ) ) 3 2 ) 2 0 1 , 2 0 1 , 2 0 0 B = B + BINCI F ! ( SORT( B * B ) ) 3 2 ) 3 5 , 3 5 , 8 A=A+ACHNG

907 FORMAT! 2 H + 4 )W RI T E ! 6 , 9 0 7 )GO TO 30

95 WRITE( 6 , 1 0 0 )100 FORMAT( 100HOFREQUENCY AMPLITUDE ANGLE F/AWWM V/A

1W VELOCITY FORCE VOLUME NAT-FREQ ZETA PRES)105 W R I T E ( 6 , 1 1 0 ) F , A , B , FOG »VOG,V»FF»VOL,RN,ZETA,PRES 110 FORMAT( F 7 . 0 , F 1 . 5 , F 1 0 . 3 , F 1 1 . 3 , F 7 . 3 , F 1 . 2 , F 1 0 . 0 , F 9 . 1 , F 10.

1 1 , F 1 0 . 5 , F 1 0 •1 )A= ASTARTI F ( F - F C H N G ) 1 1 7 , 1 2 0 , 1 2 0

117 F = F + (FCHNG/5 . )I F ( F - F S T O P ) 1 A , 1 A , 130

120 F=F+FCHNGI F ( F - F S T O P ) 1 A , 1 4 , 1 3 0

130 A=ASTRT F=FREQI F ( FN- FNCHNG) 1 3 5 6 , 1 3 6 , 1 3 6

1356 FN=FN+FNCHNG/5.01357 I F ( F N - F N S T O P ) 1 4 , 1 4 , 1 5 0

136 FN=FN+FNCHNGI F ( FN-FNSTOP)1 4 , 1 4 , 1 5 0

150 STOP END

SENTRY LOCUS

L I ST OF MNEMONIC CHARACTERS FOR RAM PROGRAM

A AMPLITUDEa s t r t i n i t i a l a m p l i t u d e3 ANGLEB1 STARTING POINT IN 1ST QUADRANTB2 ENDING POINT IN FIRST QUADRANTBINC ANGLE INCREMENTBM MASS OF LOADCVIS COEFFICIENT OF DAMPING FOR RAMF OPERATING FREQUENCY CPSFAVAL AVAILABLE FORCE CALCULATED FROM CURVEFF FORCEFN NATURAL FREQUENCY CPSFNSTOP MAXIMUM ON NATURAL FREQUENCY FOG F/AWWM FOR RAM PRES PRESSURERN NATURAL FREQUENCY RADIANS PER SECONDVOG V/AWVOL VOLUME OF FLUIDZ ABSOLUTE BZETA DAMPING RATI O FOR RAM

APPENDIX B

SAMPLE CALCULATIONS

AND

ASSUMED VALUES

SAMPLE CALCULATIONS

Amax f o r Power Valve

q = 70 A V P max maxv s

A = q / ( 7 0 «V P ) max ^max7 v ' v s

Amax = 2 6 ^ / 7 0 - 5 ^ . 7 7 = .0689

L i n e a r i z a t i o n C o n s t a n t s f o r Power Valve

K, = b q / ^ Y = C ¥ V 2 (P - P ) / D 4- r d s r '

= .6 * 3 . Ii|.l5 * V 2 ( 3000 ) / 7 .Ii.8 - 10- ^

= 16 ,869

K5 = b q r / b P r = ( - Y 0Cdw V 2 / p ) / ( 2 V P s - P r )

= ( - . 0 2 * . 6 - 3 . Uj.15 V 2 / 7 . i+ 8 * 1 0 " ^ ) /2 V 3000

= - .0 5 6

Volume o f F l u i d in F l u i d S p r i n g f o r a S p e c i f i e d

F re q u en cy and Mass

K = IjA I j 3 / V t = ^

v t =

= 4 . 2 5 - 1 0 0 , 0 0 0 / ( 6 2 8 . 3 ) 2 -2 .S9

= 9 . 8 in^

52

ASSUMED VALUES

The c o e f f i c i e n t o f damping f o r t he ram was e s t i m a t e d

to be l p lb s e c / i n on the b a s i s of a s i m i l a r d e s i g n ( l 7 ) * The

a r e a was d i f f e r e n t ; h e nc e , a s c a l i n g p r o c e s s was n e c e s s a r y .

The c o e f f i c i e n t o f damping f o r t he power v a lv e was

assumed t o be .03 which would p l a c e t h e damping n e a r .1 f o r

a n a t u r a l f r e q u e n c y of 100 c p s .

The a c t u a l b u l k modulus of t he commonly used s i l i c o n

o i l s i s a p p r o x i m a t e l y l ij .0,000 to 160 ,000; however , because

o f b u b b l e s which a r e c r e a t e d in t he o i l , a v a l u e of 100,000

was assumed.

53

PRELIMINARY DESIGN OF A HYDRAULIC VIBRATION MACHIN

WITH VARIABLE AMPLITUDE AND FREQUENCY,

USING MULTISTAGE AMPLIFICATION

AND FEEDBACK CONTROL

An A b s t r a c t o f A T h e s i s

P r e s e n t e d to t he

Depar tmen t o f Mechani ca l E n g i n e e r i n g

Brigham Young U n i v e r s i t y

In P a r t i a l F u l f i l l m e n t

o f t he Req u i r eme n t s f o r t he Degree

Ma s t e r o f Sc i en ce

by

Melv in Jo s ep h M e r r e l l

21 December 1965

A3STRACT

T h i s t h e s i s c o v e r s t h e p r e l i m i n a r y d e s i g n o f a h y d r a u l i c

v i b r a t i o n machine and was u n d e r t a k e n t o f i l l t h e need f o r such

a machine in t h e dynamics a r e a o f t h e Mechan ica l E n g i n e e r i n g

Depar tmen t o f t h e Brigham Young U n i v e r s i t y .

Power ma tch ing of s e v e r a l s t a g e s o f a m p l i f i c a t i o n , b o t h

e l e c t r o h y d r a u 1 ic and h y d r a u l i c , was a c c o m p l i sh e d as was t h e

d e s i g n i n g o f t h e m echan i ca l components i nv o lv ed and the de ­

t a i l e d d r aw ing s of t h e s e . An a n a l y s i s n e c e s s a r y t o examine

s t a b i l i t y was a l s o a c c o m p l i s h e d . The t h e s i s does n o t cov e r

t h e f a b r i c a t i o n o f t h e v i b r a t i o n machine , and many d e t a i l s

w i l l a r i s e when t h i s i s u n d e r t a k e n .

The main s p e c i f i c a t i o n s a r e : 25 ,000 pounds maximum

f o r c e v e c t o r , maximum o p e r a t i n g p r e s s u r e o f 3000 p s i , two inch

maximum d i s p l a c e m e n t , and 1000 pounds maximum l o ad .

The sys tem i n c l u d e s an e l e c t r i c a l i n p u t by a f u n c t i o n

g e n e r a t o r o r an o s c i l l a t o r , a c om pe nsa t i n g l e a d c i r c u i t , a

Moog s e r v o v a l v e , a power v a l v e , and a ram a c t u a t o r . In con ­

n e c t i o n w i t h t he ram i s a f l u i d s p r i n g which e n a b l e s o p e r a t i o n

a t r e s o n a n c e o v e r much o f t h e f r e q u e n c y r a n g e . The o u t p u t ,

a c c e l e r a t i o n , v e l o c i t y and d i s p l a c e m e n t a r e m o n i t o r e d and

a r e u se d a s a f e ed bac k c o n t r o l .

2

A l i n e a r a p p r o x i m a t i o n was made t o s t u d y s t a b i l i t y .

APPROVED: