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C H A P T E R

3

D Y N A M I C S T R E S S E S I N M A C H I N E

E L E M E N T S 2

S Y M B O L S 2 , 3

A

a, b

b

c

L

CT

F~

F~

Fd

Fi

Fie

Fir

Fs

g

h

J

k

K

l

m

m = m / A

M b

a rea of c ross- sec t ion , m 2

coeff ic ients

w i d t h o f b a r o r b e a m , m

d i s t a n c e f r o m n e u t r a l a x is t o e x t r e m e f i br e , m

v e lo c i ty o f p r o p a g a t io n o f p l a n e w a v e a lo n g a t h in b a r , m / s

v e lo c i ty o f p r o p a g a t io n o f p l a n e l o n g i tu d in a l w a v e s i n a n

inf in ite p la te , m /s

v e lo c it y o f p r o p a g a t io n o f p l a n e t r a n sv e r se w a v e s i n a n i n fin i te

p la te , m/s

d i a m e te r o f b a r , m

m o d u lu s o f e l a s t i c i t y , G P a

f o r c e o r l o a d , k N

f o r c e a ct i n g o n p i s to n d u e t o s t e a m o r g a s p r e s su r e c o r r e c t e d f o r

in e r t i a ef fe c ts o f t h e p i s to n a n d o th e r r e c ip r o c a t in g p a r t s , k N

c e n t r i fu g a l f o r c e p e r u n i t v o lu m e , k N /m 3

t h e c o m p o n e n t o f F a c t i n g a l o n g t h e a x is o f c o n n e c t in g r o d , k N

d y n a m i c l o a d , k N

g a s l o a d , k N

in e r t i a f o rc e , k N

in e r t i a f o r c e d u e t o c o n n e c t in g r o d , k N

in e r t i a fo r c e d u e t o r e c ip r o c a t in g p a r t s o f p i s to n , k N

s ta t i c l o a d , k N

acce le ra t ion due to g ravi ty , 9 .8066 m/s 2

d e p t h o f b a r o r b e a m , m

he ight of f a l l o f we ight , m

p o l a r m o m e n t o f i n er t ia ,

m 4

(cm 4)

r a d iu s o f g y r a t i o n , m

r a d iu s o f g y r a t i o n , p o l a r , m

kine t ic ene rgy , N m

length , m

m a ss , k g

m o v i n g m a s s , k g

r a t i o o f m o v in g m a ss t o a r e a o f c r o s s - s e c ti o n o f b a r

b e n d i n g m o m e n t , N m

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3 . 2 C H A P T E R T H R E E

M t

tl

tl

n ' = l / r

p

P

r

U , V , W

u

u , .

Umax

v

v

Vo

w

w

z

oz

.y

6

5 i

6,

E

C x , C y , C z

" Yx y , 7 y z , 7 z x

0

O i

A , #

//

p

o

(7

o-o

O'x, Oy, 0

T

Tl

"ln

T x y , r y z , r ~

03

t o rque , m N

speed , rpm

speed, rps

ra t io o f l eng th o f connect ing rod to rad ius o f c rank

pressure

p o w e r , k W

rad ius o f c rank , m

r a d i us o f c u r v a t u re o f th e p a t h o f mo t i o n o f ma ss , m

t h e mo me n t a r m o f t h e lo a d , m

time, s

d i sp lacement in x -d i rec t ion

mo dulus o f res il ience , N m/m 3

displacem ent com pon ents in x , y , and z-direct ions respectively, m

resilience, N m

internal elas t ic energy, N m

work done in case o f sudden ly app l i ed load , N m

maximum in terna l e l as t i c energy , N m

poten t i a l energy , N m

veloci ty , m/s

veloci ty of art icle in the s t ressed zon e of the bar, m/s

volum e, m 3p

ini t ial veloci ty at the t ime of impact , m/s

spec i fi c weigh t o f mater i a l , kN /m 3

to ta l weigh t , kN

sect ion modulus , m 3 ( cm 3)

angle between the crank and the centre l ine of conne ct ing rod, deg

un i t shear s t ra in , rad / rad

weigh t dens i ty , k N/m 3

def lec t ion /deformat ion , m (mm)

deformat ion /def l ec t ion under impact ac t ion , m (mm)

s ta ti c deformat ion /def l ec t ion und er the ac t ion o f weigh t , m (mm )

uni t s t ra in a l so wi th subscr ip t s , gm/m

strains in x , y , and z-direct ions, ~tm/m

shear ing-s t ra ins in rec tangu lar coord ina tes , rad / rad

ang le be tween the c rank and the cen t re l ine o f the cy linder

measured f rom the head-end dead-cen t re pos i t ion , deg

stat ic angular deflect ion, deg

angle of twis t , deg

angular def l ec t ion under impact load , deg

L a m~ ' s c o n s t a n t s

Po i s son ' s ra t io

mass dens i ty , kg /m 3

normal s t ress (a l so wi th subscr ip t s ) , MPa

impact s t ress (also with subscripts) , MPa

ini t ial s t ress at the t ime of impact and veloci ty V o , M P a

normal s t ress componen t s para l l e l to x , y , and z -ax i s

shear ing s t ress , MPa

t ime o f load app l i ca t ion , s

per iod o f na tu ra l f requen cy , s

shear ing s t ress componen t s in rec tangu lar coord ina tes , MPa

angular ve loc i ty , rad / s

No te : a , a n d % wi th f ir s t su b sc r ip t s d e s ig n a t e s t r e n g th p r o p e r t i e s o f m a te r i a l u se d in th e d e s ig n wh ic h wi l l b e u se d a n d f o l lo we d th r o u g h o u

b o o k . O th e r f a c to r s i n p e r f o r m a n c e o r i n sp e c i a l a sp e c t s wh ic h a r e i n c lu d e d f r o m t im e to t im e in t h i s b o o k a n d b e in g a p p l i c a b l e o n ly i n t

im m e d ia t e c o n te x t a r e n o t g iv e n a t t h i s s t a g e .

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Particular

I N E R T I A F O R C E

Formula

P o w e r

F / )

P = 100-----0 S I (3 -

V e l o c i t y

C e n t r i f u g a l f o r c e p e r u n i t v o l u m e

E N E R G Y M E T H O D

T h e i n t e r n a l e l a s t i c e n e r g y o r w o r k d o n e w h e n a

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

load , F ig . 3 . 1 .

T h e w o r k d o n e i n ca s e o f s u d d e n l y a p p l i e d l o a d o n a n

e l a s t i c m a c h i n e m e m b e r ( F i g . 3 - 2 )

w h e r e F i s i n n e w t o n s ( N ) , v i n m / s , a n d P i n k

F v

~

0

._I

> ,

r ~

3 3 0 0 0

U S Cus tomary Sys tem un i ts (3-

w h e r e F i s i n l bf , v i n f t / m i n , a n d P i n h p .

27rrn

v = 1---~ U S

Customary Sys tem un i ts

( 3 -

w h e r e r i n i n , v i n f t / m i n , a n d n i n r p m .

2 rrrn

v = 6-----O S I (3 -

w h e r e r i n m , v i n m / s , a n d n i n r p m .

W732

Fev =

- -

(3

rg

U p = F(5 (3

vd = Fd~ (3

LL

O

_.1

1/2 F5

Deflection, 8

FIGURE

3-1 Plot of force against deflection in case of elas-

t ic machine member subject to gradual ly applied load.

Deflection, 6

D Y N A M I C S T R E SS E S I N M A C H I N E E L E M E N T S

~ 5

FIGURE

3-2 Plot of force against deflection in case of

denly applied load on a machine member .

T h e r e l a t i o n b e t w e e n s u d d e n l y a p p l i e d l o a d a n d g r a -

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

p r o d u c e t h e s a m e m a g n i t u d e o f d ef l e ct i o n .

U p = U d ( 3 -

F a = F ( 3-

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3 . 4 C H A P T E R T H R E E

P a r t i c u l a r

F o r m u l a

T h e s t a t i c d e fo rm a t i o n o r d e f l e c t i o n

I M P A C T S T R E S S E S

Im p ac t f rom d i r ec t l oad

K i n e t i c e n e rg y

I m p a c t e n e rg y o f a b o d y f a l l in g f ro m a h e i g h t h

T h e h e i g h t o f fa ll o f a b o d y t h a t w o u l d d e v e l o p t h e

veloc i ty v .

T h e m a x i m u m s t re s se s p ro d u c e d d u e t o f al l o f we i g h t

W t h ro u g h t h e h e i g h t h f ro m re s t w i t h o u t t a k i n g i n t o

a c c o u n t t h e w e i g h t o f s h a f t a n d c o l l a r (F i g . 3 -3 )

1 / i l l , I / Z / 1 / / /

t ar

Load ,W

h i

T

Rigid collar

m h

m i I

L

F I G U R E 3-3 Striking impact of an elastic machine

mem ber by a body of weight W falling throug h a height h.

T h e m a x i m u m d e f le c t io n o r d e f o r m a t i o n o f s ha f t d u e

t o f a ll o f we i g h t W t h r o u g h t h e h e i g h t h f ro m re s t

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

T h e s t r e s s p ro d u c e d d u e t o s u d d e n l y a p p l i e d l o a d

T h e m a x i m u m d e f l e ct io n o r d e f o r m a t i o n p r o d u c e d b y

s u d d e n l y a p p l i e d l o a d

W

6st = T

(3

wh e re k = s p r i n g c o n s t a n t o f th e e l as t ic m a c h

m e m b e r , k N/ m ( l b f / i n ) .

W~32

K = ~ (3

2g

K = Wh

(3

V 2

h = - - (3

2g

W [l + T i +,2hEA] (g - l

O'i--- O'max -- ~ WL J

[ 7 2hEAl

(3-1

= a ~ t 1 + 1 +

WLJ

: O ' st 1 + + ~ s t

6rnax = ~i = ~ s t 1 + 1 + WL

[

-6st

1 + 1 +

( r Y m a x ) s u d - - 2 ( O m a x ) s t a t

(~ max)

s u d - - 2 6 s t

wh e re s u b s c r i p t

stal

= s t = s ta t i c an d

sud

= s u d d e n l y

(3-1

(3-

(3-

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D Y N A MI C S TR E SS ES I N MA C H I N E E L E ME N T S

P a r t i c u l a r

F o r m u l a

T h e k in e t i c e n e r g y t a k in g i n to a c c o u n t t h e w e ig h t o f

sh a f t o r b a r a n d c o l l a r

The re la t ion be tween c r , 6 , F and W

T h e m a x im u m s t re s s d u e t o f a ll o f w e ig h t W th r o u g h

th e h e ig h t h f r o m r e s t t a k in g i n to a c c o u n t t h e w e ig h t

o f sh a f t / b a r a n d c o l l a r

T h e m a x i m u m d e f l e c t io n d u e t o f al l o f w e ig h t W

th r o u g h th e h e ig h t h f r o m r e s t t a k in g i n to c o n s id e r a -

t i o n t h e w e ig h t o f sh a f t / b a r a n d c o l l a r

In te rn a l e la s t ic ene rgy of we ight W whose ve loc i ty v i s

h o r i z o n t a l

I n t e r n a l e l a s t i c e n e rg y o f w e ig h t W w h o se v e lo c i t y h a s

r a n d o m d i r ec t io n

K= WV:c 5-~J

2 g 1 + ( 3

w h e r e

Vc

= v e lo c it y o f c o l l a r a n d w e ig h t W a

the load s t r ik ing the co l la r , m/s .

w h e r e

W b

= w e ig h t o f sh a f t o r b a r

. a x m a x I

-W = c rs --~ = 6st 1 + 1 + W L (3-1

= 1 + 1 + ( 3- 1

W 2 E A h 1

O'i--O 'max--- - ~ 1 -+- -I- W L 1 + ( W b / 3 W )

(3-1

_ W 1 + 1 + (3-1

- - - A " W L J

[ j l

Ost 1 + + - ~ t J ( 3- 1

1 Wb

where a = ~ and ~ =

1 + ( ~ / 3 ) w

6max = ~ 1 + 1 + W L I + ( W b / 3 W )

(3-1

W L

2hAEa

= 1 + 1 + (3-1

A E W L J

Wv 2

U = 2----g- (3-

Wv 2

U = ~ + W 6 sin/3 (3-

wh ere /3 = angle of ve loc i ty , v , to the hor iz on

plane , deg .

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3 . 6 C H A P T E R T H R E E

Particular

Formula

v=0

Vmax

8

V

=

0

h

. . . . . . . . .

0 0 Umax

l / l / l / I I I l l l

a ) b ) c )

FIG UR E 3-4 Impact by a fall ing body

Fig. Fig.

Ene rgy 3-4a 3-4b

Up m (h - '[ - ~) W ~

Wv2

K 0

2g

U 0 0

Fig.

3-4c

Equat

W h+~)

(3-22a

(3-22b

(3-22c

T h e e qu a t i o n fo r e n e rg y b a l a n c e fo r a n i m p a c t b y a

fa l l ing body (F ig . 3 -4 )

A n o t h e r f o r m o f e q u a t i o n f o r d e f o r m a t i o n o r d e fl ec -

t i o n i n t e rm s o f v e l o c it y v a t i m p a c t

E qu i v a l e n t s t a t i c fo r c e t h a t wo u l d p ro d u c e t h e s a m e

m a x i m u m v a l u es o f d e f o r m a t i o n o r d e f l ec t io n d u e

t o i m p a c t

B E N D I N G S T R E S S I N B E A M S D U E T O

I M P A C T

I m p a c t s t r e s s d u e t o b e n d i n g

#

- - . f

.~~

~I h

FIG UR E 3-5 Impac t by a falling body on a cantilever beam

D e f l e c t i o n o f th e e n d o f c a n t i l e v e r b e a m u n d e r i m p a c t

(Fig. 3-5)

T h e m a x i m u m b e n d i n g s t r e s s f o r a c a n t i l e v e r b e a m

t a k i n g i n t o a c c o u n t t h e t o t a l we i g h t o f b e a m

(Up + K + U)a = (Up + K + U)b

~max = ~st (

1 +

= ( U ~ + K + U ) c

l + - ~ T t s

F e q = W 1 + 1 + = W 1

(3-

(3

/ -t - v2 )

(3

W l c ~ / 6 h E l l

( O ' b ) m a x - -

O b i - - 7 -

1 + 1 + W l Y

(3-

[

Wl c 1 + 1 +

I

= ~)s, l+ +E

Wlc Mbc

where (Ob)st = I I

6rnax =

(Sst 1 + +

Mb

Z b

(3-2

(3-

(3

(O'b)max -- (O'b)st[ 1- [- ~t ]

(3

mb

w h e re ff . . . .

m

Wb an d o~ =

W

1 + (3 3 f f / 140 )

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D Y N A M I C S T R E SS E S I N M A C H I N E E L E M E N T S

Particular

Formula

T h e m a x i m u m d e f le c t io n a t t h e e n d o f a c a n t il e v e r

b e a m d u e t o f a ll o f we i g h t W t h ro u g h t h e h e i g h t h

f ro m re s t t a k i n g i n t o c o n s i d e ra t i o n t h e we i g h t o f b e a m

T h e m a x i m u m b e n d i n g s t re s s fo r a s i m p ly s u p p o r t e d

b e a m d u e t o f a ll o f a l o a d / we i g h t W f ro m a h e i g h t h

a t th e m i d s p a n o f th e b e a m t a k i n g i n t o a c c o u n t t h e

t o t a l we i g h t o f t h e b e a m (F i g . 3 -6)

IW

Y h

.1. ,1,

. . . . ' . . . .

B -~x

/ ~? ~st /

t

~max = ~s, 1 + 1 + 8~t J

(3-2

(f ib )max-- (Ob)s t 1 + 1 +&~t 1 + (17 ( /35 )

(3-2

(3-2

1 Wb

wh ere c~ = an d

(

=

1 + ( 1 7 ( / 3 5 ) W

FIGU RE 3-6 Simply supported beam

T h e m a x i m u m d e f le c ti o n f o r a si m p ly s u p p o r t e d b e a m

d u e t o f a l l o f a we i g h t W f ro m a h e i g h t h a t t h e m i d -

s p a n o f t h e b e a m t a k i n g i n t o a c c o u n t t h e we i g h t o f

beam. (F ig . 3 -6 )

~max --(~st

1 J r - 1 - n t - - ~ s t j

(3-

T O R S I O N O F B E A M / B A R D U E T O I M P A C T

(Fig. 3-7)

T h e e q u a t i o n f o r m a x i m u m s h e a r s tr e ss i n th e b a r d u e

t o i m p a c t l o a d a t a r a d i u s r o f a f al l in g we i g h t W f ro m

a h e i g h t h n e g l e c t in g t h e we i g h t o f b a r

/-max -- 7 - s t

1 + 1 +

(3-

. . . . . R - . ,

s u ; 0 o / i r - L e v e r

.Bar W b

FIGU RE 3-7 Twist of a beam /bar

Y

J L ~ Ou

u ~ F

AE~-.~ a-~+~(~)dx ]

. . . . . .

,x- t I

FIG U R E 3-8 Displacements due to forces acting on an

me nt o f an elastic media.

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3 . 8 C H A P T E R T H R E E

Particular

Formula

The equa t ion for angula r de f lec t ion or angula r twis t

of ba r due to imp ac t load W a t radius r and fa l l ing

throug h a he ight h neglec t ing the we ight of ba r

O m a x - - O s t 1 + +

(3-

L O N G I T U D I N A L S T R E S S - W A V E I N

E L A S T I C M E D I A

( F i g . 3 - 8 )

One-dimens iona l s t re ss -wave equa t ion in e la s t ic

media (Fig. 3-8)

For ve loci ty of prop aga t ion of longi tudina l s t re ss-

wave in e last ic media

The solut ion of stress-wave Eq. (3-33a)

The va lue of c i rcula r f requency p

The f requency

L O N G I T U D I N A L I M P A C T O N A L O N G

B A R

The ve loc ity of pa r t ic le in the com press ion zone

The uniform ini t ia l compressive stress on the free end

of a bar (Fig. 3-9)

The va r ia t ion of s t re ss a t the end of ba r a t any t ime t

0 2 U - - C 0 2U

(3-3

O t 2 - - O X 2

where c = ~ / ~ = ~--Ep (3-3

--- veloci ty of pr opa gatio n of stress

waves, m/s.

Refer to Table 3-1.

x = ( A

s i n e x + B c o s P x ']

( c s inp t + D c o s p t ) (3 -

\

c c

J

where A, B , C and D a re a rb i t ra ry cons tants

which can be found f rom in i t i a l or boundary

condi t ion of the problem.

n T r c _ n T r v ~ n T r ~

(3-3

P= t -T =T

where n is an integer = 1,2, 3, . . .

f = P - - - = n V ~ - C 2 7 r1 - A (3-3

where A = wave length =

2 1 / n ,

c -- speed of

sound or stress wave veloci ty, m/s.

V = a~/~-~7 = V /~a (3-

a 0 = V0 ~ / -~ = V 0X / ~ (3-

whe re V0 = ini t ia l veloci ty of the moving w eig

mass a t the t ime of impac t , m/s .

o = ao exp - --M-- t 0 < t < -- (3-

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DYNA MIC S T RE S SE S IN MACH INE E LE ME NT S

Particular

Formula

The equa t ions of mot ion in te rms of three d i splace -

ment components a ssuming tha t the re a re no body

forces.

w I w

i l l i l l l. . . .

rrrlrtr

t ]..-i-.L Con stant x'~'o

~rL ~ cross

s e c t i o n

1 / 1 / 1 / 1 / H / / / / / / / / /

FIG UR E 3-9 Prismatic bar subject to suddenly applied

uniform compressive stress

(A + G )

0e 0 2 u

-+- G V 2 u = p Ot 2

(3-3

0e 02v (3-3

(A + G ) -~ y + G ~ 7 2v = p Ot 2

O c 0 2 w

(3-3

(A + G ) O z + G ~ 7 2 w = p O t 2

where

- - C x + Cy -Jr-C

0 2 0 2 0 2

V = ~ + ~ + ~ z2 = the Laplac ian op era tor

u E

A = a nd

(1 + u)(1 - 2u)

# = G = ~

2 ( 1 + u )

a re Lam6's cons tants

D i l a t a t io n a l a n d d is t o r t io n a l w a v e s i n

i s o t r o p i c e l a s t i c m e d i a

From the c lass ica l theory of e la s t ic i ty equa t ions for

i r ro ta t iona l or d i la ta t iona l waves

Equa t ions for d i s tor t iona l waves

Equations (3-40) to (3-41) are one-dimensional stress

wa ve e q ua t i ons o f t he fo rm

The ve loc i ty of s t re ss wave propaga t io n for the case of

no ro t a t i on

0 2 U __ /~ + 2 G V 2 u

O t 2 p

02 v /~ --I- 2 G V2 v

Ot 2 p

0 2 w _ _ /~ n - 2 G v 2 w

Ot 2 p

0 2 u = G ~ 2 u

O t 2 p

02 V = E ~72 v

O t 2 p

0 2 W - - E ~ ' 2 w

O t 2 p

020

= a 2 V 2 0

Ot 2

)~+ 2G ~, E(1 - u)

a = c l = =

p (1 + u)(1 - 2u) p

(3-4

{ , ' ~

A

to-,+

(3-4

(3-4

(3-4

(3-4

(3

(3

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10/16

3 . 1 0 C H A P T E R T H R E E

Particular

Formula

T h e v e l o c it y o f s t re s s w a v e p r o p a g a t i o n f o r t h e c as e o f

z e r o v o l u m e c h a n g e

Th e r a t i o o f c l t o c 2

T h e v e l o c it y o f s t re s s w a v e p r o p a g a t i o n f o r a t r a n s -

ver se s t r ess wave , i . e . d i s t o r t i ona l wave i n an i n f i n i t e

p l a t e

T h e v e l o c i t y o f s t r e s s w a v e p r o p a g a t i o n f o r p l a n e

l o n g i t u d i n a l s t r e ss w a v e i n c a se o f a n i n f i n it e p l a t e

T O R S I O N A L I M P A C T O N A B A R

E q u a t i o n o f m o t i o n f o r t o r s io n a l i m p a c t o n a b a r

(F i g . 3 -10 )

T o r s i o n a l w a v e p r o p a g a t i o n i n a b a r s u b j e c t e d t o

t o r s i o n .

F o r v e l o c it y o f p r o p a g a t i o n o f t o rs i o n a l s t r e s s- w a v e

i n a n e l a s t i c b a r

y,

/ - I / -

\ J ~~ J . / ' Y. YM, , _

~ 4 I c = c t t ~ .

~F x , . . . ~ d x - - ~

I

_}

I

FIGURE

3-10 Torsional impact on a uni form bar showing

torque on two faces of an element

a

= c 2 - - -

2(1 - u ) p

(3-

C 1 / 2 (1 - u )

fo r Po i sso n ' s r a t i o o f u = 0 .25 (3 -

CT

= = (1 +

u ) p

(3-

/ 4 a ( a + a ) ~ / E

CL

= V )-() ~ ~- }.~--~ = p(1 -- u 2) (3-

0 2 0 - c 2 0 20

(3-

O t 2 - - O x 2

3

R e f e r t o T a b l e 3 - 1 .

A a rg eo tating o dy

- ~ ~ l l lp s itivec lu tc hb ] ~ . ; . _ . f 2 x

L I . ~ ~ ~ I L [ I A large

FIGURE 3-11 Torsional s t r ik ing impact

T h e a n g u l a r v e l o c i t y o f t h e e n d o f a b a r s u b j e c t t o t o r -

s i o n r e l a t i v e t o t h e u n s t r e s s e d r e g i o n

T h e s h e a r s t r e s s f r o m E q . ( 3 - 5 0 )

T h e i n i t i a l s h e a r s t r es s , i f t h e r o t a t i n g b o d y s t r ik e s t h e

e n d o f t h e b a r w i t h a n a n g u l a r v e l o c i t y coo

( 2rt '~

c o = 0 = \ d ~ , ] t = 2 r t

(3

t t dv / -pG

cod

r = - ~ - x / ~ ( 3 -

w o d

r 0 = - - ~ X / ~ ( 3 -5

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11/16

D Y N A M I C ST R E SS E S I N M A C H I N E E L E M E N T S 3

Part i cu lar Formula

T h e m a x i m u m s h e a r s t r es s f o r t h e c as e o f a s h a ft f i xe d

o r a t t a c h e d t o a v e r y la r g e m a s s / w e i g h t a t o n e e n d a n d

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

m e a n s o f s o m e m e c h a n i c a l d e v i ce s u c h a s a j a w

c lu tch ( F ig . 3 - 11)

~ ,, I S tr i ki n g r o t a t i n g w e i g h t

i . . . . . . . . . . . . - - - ~ x

o l l a r

I

F I G U R E

3-12 A s tr iking rota t ing weight with mass-

mo men t of iner t ia I rota t ing a t a~0 engages with one end of

shaf t and the other end of shaf t f ixed to a mass-moment of

iner t ia I f

"/-m ax - - 7-0 ~ 1 + = T

( 3 -

w h e r e = - ~ .

I

I b = m a s s m o m e n t o f i n e r t i a o f b a r = m b ~ - )

I = m a s s m o m e n t o f i n e r ti a o f s t ri k i n g r o t a t i n g w e i

Ib a n d I c o r r e s p o n d t o W b a n d W o f t h e w e i g h t o f

b a r a n d t h e r o t a t i n g m a s s o r w e i g h t r e s p e c t i v e l y .

T h e m o r e a c c u r a t e e q u a t i o n f o r t h e /'m ax w h i c h i s

b a s e d o n s t r e s s w a v e p r o p a g a t i o n

T h e i n i t i a l / m a x i m u m ( ~ - i

= 7 - m a x )

s h e a r s t r e s s f o r t h e

c a s e o f a s y s t e m s h o w n i n F i g . 3 -1 2

A s i m i l a r e q u a t i o n t o E q . ( 3 - 54 ) fo r m a x i m u m s t re s s

f o r l o n g i t u d i n a l i m p a c t

A c c u r a t e m a x i m u m s t r e s s f o r l o n g i t u d i n a l i m p a c t

s t r e s s b a s e d o n s t r e s s w a v e p r o p a g a t i o n a s s u g g e s t e d

b y P ro f . B u r r

A c c u r a t e m a x i m u m s t r e s s f o r t o r s i o n a l i m p a c t s h e a r

s t r e s s b a s e d o n s t r e s s - w a v e p r o p a g a t i o n a s s u g g e s t e d

b y P r o f . B u r r

/ ~(1 + )

7 -i = 7 - m a x - - T o V ( i _ ~_ _ .~ _ ,~

/b I

w h e r e = I ' A = ~ f a n d I b = p J1.

( 3 -

(3-

~ ~( 1+) (3

' i : O'm ax - - - O'0 ( 1 -~ - -4- , ~ )

w h e r e = Wb = m___b_b n d A = m

W m m f

O i = C r m a x = O ' 0 I 1 i _4_ V / / ~ ( 1 , - l - - if )

( 1 + + A ) J

T i - - - T m a x = 7 0 1.1 + (1 + + A )

( 3-

( 3-

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12/16

3 . 1 2 C H A P T E R T H R E E

P a r t i c u l a r

F o r m u l a

I N E R T I A I N C O L L I S I O N O F E L A S T IC

B O D I E S

Wh e n a b o d y h a v in g w e ig h t W s t r i k e s a n o th e r b o d y

th a t h a s a w e ig h t W ' , im p a c t e n e r g y

W h

i s r educed

to

n W h ,

according to law of co l l i s ion of two pe r fec t ly

ine las t ic bodies , the formu la for the va lue of n

R E S I L I E N C E

The express ion for r e s i l ience in compress ion or

t e n s io n

T h e m o d u lu s o f r es i li e n ce

The a rea under the s t r e ss- s t r a in curve up to y ie ld ing

poin t r ep resents the mod ulus of r es i l ience (F ig . 1 .1)

The res i l ience in bending

T h e m o d u lu s o f r es i li e n ce in b e n d in g

Resi l ience in d i rec t shea r

The modulus of r e s i l ience in d i rec t shea r

Res i l ience in tor s ion

T h e m o d u lu s o f r es i li e n ce i n t o r s io n

T h e e q u a t io n f o r s t ra in e n e r g y d u e t o sh e a r in b e n d in g

T h e m o d u lu s o f re s il i en c e d u e t o sh e a r i n b e n d in g

t / - -

1 + a m

(1 +

b m ) 2

(3-

W '

where m = - -~- ; a and b a re taken f ro m T able

0 - 2 V 1 0 - 2 A L

U = 2 E = 2 ~ (3-

0 - 2

u - 2---E (3 -

u = 0-e (3-

U b = ( k ) 20-2AL6E (3-

( k ) 2 0-~ (3-

uh = 6---E

w h e r e ( k / c ) 2 = i for r ec tangula r c ross- sec t i

for c i r cu la r sec t ion

c = d is tance f rom ext reme f ibre

neut ra l ax is

~ V

UT = 2---G- (3 -

(3-

uT = 2G

= 2----G-- - - (3-

w h e r e k o = - v / D 2 - D 2 / 8 a n d c - 1 D o f o r h o l

shaf t .

uT = ~-~ - - (3-

j

2

U-~b = k~ -F ~-

o ~ d x (3-

k~-e

(3-

u"-b = 2G

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13/16

D Y N A M I C S T RE S SE S I N M A C H I N E E L E M E N T S 3

Part i cu lar

F o r m u l a

T h e e q u a t i o n f o r s h e a r o r d i s t o r t i o n a l s t r a i n e n e r g y

p e r u n i t v o l u m e a s s o c i a t e d w i t h d i s t o r t io n , w i t h o u t

c h a n g e i n v o l u m e

T h e e q u a t i o n f o r d i l a t a t i o n a l o r v o l u m e t r i c s t r a i n

e n e r g y p e r u n i t v o l u m e w i t h o u t d i s t o r t i o n , o n l y a

c h a n g e i n v o l u m e

F o r m a x i m u m r e s il i en c e p e r u n i t v o l u m e ( i. e. , f o r

m o d u l u s o f r e s i l ie n c e ) , r e s i li e n c e in t e n s i o n f o r v a r -

i o u s e n g i n e e r i n g m a t e r i a l s a n d c o e f fi c i e n t s a a n d b ;

v e l o c i ty o f p r o p a g a t i o n c a n d

c t .

1

[0 2 + 0-2 _nt_0-2 _ (0 10 n - 0 20 nt- 0-30- 1)

u,=g-d

( 3 - 7

1

-- 12-----~ (o 1 - 0-2)2 --I- (0-2 - 0-3)2 + (o-3 - o-1)2]

( 3 - 7

Uv = (1 - 2u)

6- - - f f~ [ (cr l + o2 + 03) 2 ] (3-

R e f e r t o T a b l e s 3 - 1 t o 3 - 4 .

T A B L E 3 -1

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

D e n s i t y

p ),

M o d u l u s o f M o d u l u s o f c = = c t = =

e last i c i t y , E r ig id i t y , G

Mate r ia l g / cm 3 Ibm/ in3 k N / m 3 G P a M p s i G P a M p s i m / s f t / s m / s f t / s

A lum in um a l loy 2 .71 0 .098 26 .6 71 .0 10 .3 26 .2 3.8 5116 16785 3110 1046

Brass 8 .55 0 .309 83 .9 106 .2 15 .4 40 .1 5 .82 3523 11560 2165 7106

C arb on s tee l 7 .81 0 .282 76 .6 206 .8 30 .0 79 .3 11 .5 5145 16887 3200 1048

Ca s t i ron , g ra y 7 .20 0 .260 70 .6 100 .0 14 .5 41 .4 6 .0 3727 12223 2407 7865

Co pp er 8 .91 0 .320 87.4 118 .6 17.7 44 .7 6 .49 3648 12176 2240 7373

Glas s 2 .60 0 .094 25 .5 46 .2 6 .7 18 .6 2 .7 4214 13823 2675 8775

Le ad 11 .38 0 .411 111 .6 36 .5 5 .3 i3 . i i .9 1796 5879 1073 3520

Inc one l 8 .42 0 .307 83 .3 213 .7 31 .0 75 .8 11 .0 5016 16452 2987 9800

Sta in less s tee l 7 .75 0 .280 76 .0 190 .3 27.6 73 .1 10 .6 4955 15972 3071 10074

Tu ngs ten 18 .82 0 .680 184 .6 344 .7 50 .0 137.9 20 .0 4279 14039 2707 8880

#No te: p = Mass density, g/cm 3 (Ibm/in3), 7 = weight density (specific weight), kN /m 3 (lbf/in3), g = 9.8066 m/s 2 in SI units, g = 980 in/s 2 = 32.2 f

in fps units.

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14/16

3 . 1 4 C H A P T E R T H R E E

T A B L E 3 - 2

M a x i m u m r e s i li e n c e p e r u n i t v o l u m e ( 2 , 1 )

,

Type of loading Modulus of resi l ience, J ( in l

T e n s i o n o r c o mp r e s s i o n

Shear , s imple t r ansverse

Be n d i n g i n b e a ms

W i t h s i mp l y s u p p o r t e d e n d s :

C o n c e n t r a t e d c e n t e r l o a d a n d r e c t a n g u l a r c r o s s - s e c t i o n

C o n c e n t r a t e d c e n t e r l o a d a n d c i r c u la r c r o ss - s e c ti o n

C o n c e n t r a t e d c e n t e r l o a d a n d I - b e a m s e c t i o n

U n i f o r m l o a d a n d r e c t a n g u l a r s e c t i o n

U n i f o r m- s t r e n g t h b e a m, c o n c e n t r a t e d l o a d , a n d r e c t a n g u l a r s e c t i o n

F i x e d a t b o t h e n d s :

C o n c e n t r a t e d l o a d a n d r e c t a n g u l a r c r o s s - s e c t i o n

U n i f o r m l o a d a n d r e c t a n g u l a r c r o s s - s e c t i o n

C a n t i l e v e r b e a m:

E n d l o a d a n d r e c t a n g u l a r c r o s s - s e c t i o n

U n i f o r m l o a d a n d r e c t a n g u l a r c r o s s - s e c t i o n

So l i d r o u n d b a r

T o r s i o n

H o l l o w r o u n d b a r wi t h D o g r e a t e r t h a n D i

La m i n a t e d wi t h f l at l e av e s o f u n i fo r m s t r e n g t h

Spr ings

Fla t sp i ra l wi th rec tangular sec t ion

Hel ica l wi th round sec t ion and axia l load

Hel ica l wi th round sec t ion and axia l twis t

Hel i ca l wi th rec tangular sec t ion and axia l twis t

2

O

2 E

2G

2

O

1 8 E

2

O

2 4 E

3 2 E

4O-e

4 5 E

d

6 E

1 8 E

2

O

3 0 E

2

( 7 e

1 8 E

,)

3 0 E

4 G

D i

l + 2

4G

2

7 e

6 E

2

7 e

2 4 E

4G

2

O

8 E

2

( 7 e

6 E

Sources :

K. Lingaiah and B. R. Narayana Iyengar,

Mac hine Des ign Da t a Handbook ,

Vol I

(S I a nd Customary M et r ic Un i t s ) ,

Sum a Publish

Bangalore, India, 1986; K. Linga iah, Mac hine Design Da t a Ha ndbook , Vol II (S l a nd Customary Met r i c Un i t s) , Sum a Publishers, Bangalore, In

1986; V. L. M aleev and J. B. H artman,

Machine Design,

International Textbook C ompany, S cranton, Pennsylvania, 1954.

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15/16

D Y N A M I C S T R E SS E S I N M A C H I N E E L E M E N T S 3

T A B L E

3 - 3

C o e f f i c i e n t s i n E q . ( 3 - 5 8 ) ( 1 )

Ty p e o f im p ac t

L o n g i t u d i n a l i m p a c t o n b a r

C e n t e r i m p a c t o n s i n g le b e a m

C e n t e r i m p a c t o n b e a m w i t h f ix e d e n d s

E n d i m p a c t o n c a n t i l e v e r b e a m

1 1

3 2

17 5

3 5 8

13 1

3 5 2

4 3

n _

17 8

T A B L E 3 - 4

R e s i l i e n c e i n t e n s i o n

Elast ic l im it , cr

M at e r ia l M P a k p s i G Pa

Modu lus o f e las t ic i ty , E Modu lus o f res i l ience , u Impac t

s trength

Mps i J in lbf ( Izod no

C a s t i r o n :

C lass 20 (ord inary ) 42 .8 a 62 68 .9 10 0 .22 1 .9

Cla ss 25 68.9 a 10.0 89.2 13 0.43 3.8 7 .9

Nicke l , G rad e I I 117.2 a 17.0 24 .5 18 0 .90 8 .0

Ma l leab le 137.9 20 .0 172 .6 25 0 .90 8 .0 2 .7

A l u m i n u m a l l o y , S A F 3 3 4 8 .3 7 .0 6 6 .7 9 .7 0 .2 8 2 .5

Brass , SA E 40 or SA E 41 68 .9 10 .0 82 .4 12 0 .45 4 .0

Bronz e , SA E 43 193 .0 28 .0 110 .8 16 2 .77 24 .5

M o n e l m e t a l :

H ot - r o l le d 206 .9 30 .0 176 .5 25 .5 1 .96 17.6 120

C o l d - r o l l e d , n o r m a l i ze d 4 8 2 .6 70 .0 1 76 .5 2 5 .5 1 0 .79 9 6 1 0 0

Steel:

SA E 1010 206 .9 30 .0 30 .3 1 .69 15

SA E 1030 248 .2 36 .5 206 .9 30 2 .45 22 20

S A E 1 0 5 0, a n n e a l e d 3 3 0 .9 4 8 .5 2 0 4 .8 2 9 .7 4 .2 7 3 8

S A E 1 0 95 , a n n e a l e d 4 1 3 .7 6 0 .0 2 0 4 .8 2 9 .7 6 .77 6 0

SA E 1095 , t em per ed 517.1 75 .0 204 .8 29 .7 16 .08 94

S A E 2 3 2 0 , a n n e a l e d 3 1 0 .3 4 5 .0 2 0 4 .8 2 9 .7 3 .8 2 3 4 5 2

SA E 2320, t em per ed 689 .5 100 .0 204 .8 29 .7 18 .83 167 40

S A E 3 2 5 0 , a n n e a l e d 5 5 1 .6 8 0 .0 2 1 3 .7 3 1 2 1 .5 8 1 93

S A E 3 2 5 0 , t e m p e r e d 1 3 79 .0 2 0 0 .0 2 1 3 .7 3 1 .0 72 .5 7 6 4 5 3 0

S A E 6 1 5 0 , a n n e a l e d 4 2 7 .6 6 2 .0 2 1 3 .7 3 1 6 .9 6 6 2

SA E 6150, t em per ed 1102.3 160 .0 213 .7 31 52 .47 466

R ub be r 2 .1 0 .3 1034 x 10 -9 150 x 10 -6 33 .89 300

a Cast iron has no well-defined elastic limit, but the values may be safely used anyway for all practical purposes.

Source: Reproduced courtesy of V. L. Maleev and J. B. Hartm an, Machine Design, International Tex tbook C o., Scranton, Pennsylvania, 1

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16/16

3 .1 6 C H A P T E R T H R E E

R E F E R E N C E S

1. Maleev, V. L. , and J. B. Hartman, Ma ch ine Des ign , International Textbook Co., Scranton, Pennsylvan

1954.

2. Lingaiah, K., and B. R. Nar aya na Iyengar, Ma ch ine Des ign Da ta H a ndbook , Vol I ( S I a nd C us tom a r y M e

Units) , Suma Publishers, Bangalore, India, 1986.

3. Lingaiah, K., M a c h i n e D e s ig n D a t a H a n d b o o k , Vol II ( S I a nd C us tom a r y Metr i c Un i t s ) , Sum a Publish

Bangalore, India, 1986.

4. Lingaiah, K.,

M a c h i n e D e s ig n D a t a H a n d b o o k ,

McGraw-Hi l l Book Company, New York , 1994 .

5 . Burr , A rthur H . , and John B. Cheatham ,

Mecha n i ca l Ana ly s is a nd Des ign ,

2nd edition, Prentice Hall, Eng

wood Cliffs, NJ, 1995.

6. Spotts, Merhyle F. , Im pac t Stress in Elast ic Bodies Calculated by the Energy Meth od ,

Eng ineer ing Da ta

Pr oduc t D es ign ,

edi ted by Douglas C. Greenwood, McG raw-H i l l Book Com pany, New York , 1961.

7 . Burr , A. H. , L ongi tud ina l and Tors ional Imp act in Uni form Bar wi th a Rig id Body a t One End ,

J. Ap

Mech . ,

Vol. 17, No . 2 (June 1950), pp. 209-217 ;

T r a n s . A S M E ,

Vol. 72 (1950).

8. Timoshenko, S. , and J. N. Goodier,

Theory o f Elas t ic i ty ,

3rd ed i t ion , McGraw-Hi l l Book Company, N

York, pp. 485-513, 1970.

9. Kolsky, H.,

S tress W a ves in Elas t ic So l ids ,

Dover Publicat ions, New York, 1963.

10. D urellli, A. J., and W . F. Riley,

In troduct ion to Photomecha nics ,

Prentice Hall Inc, Englewood Cliffs, NJ, 19

11, A rnold, Im pac t Stresses in a Simply Supp orted Beam ,

P r o c . I . M . E . ,

Vol. 137, p. 217.

12. Dohrenw end and Mehaffy , D ynam ic Loading ,

Ma ch ine Des ign ,

Vol. 15 (1943), p. 99.

B I B L I O G R A P H Y

Lingaiah , K. , and B. R. Narayana Iyengar , M a c h i n e D e s i gn D a t a H a n d b o o k , Engineering C ollege C o-opera

Society, Bangalore, India, 1962.

Norman, C. A., E. S. Ault , and E. F. Zarobsky, Funda m en ta l s o f Ma c h ine Des ign , The M acmil lan Com pany, N

York, 1951.