7/24/2019 67071_22.pdf

1/25

C H A P T E R

2 2

M E C H A N I C A L V I B R A T I O N S

S Y M B O L S

B

C

c~

C t

C1, C2

d

D

e

E

f

F

Fo

Fv

g

J

k

ke

k t

K

c o e f f i c i e n t s w i t h s u b s c r i p t s

f lex ib i l i ty

a c c e l e r a t i o n , m / s 2 ( f t /s 2 )

a r e a o f c r o s s s e c t io n , m 2 ( i n 2 )

c o n s t a n t

c o n s t a n t

c o e ff ic i e n t o f v i s c o u s d a m p i n g , N s / m o r

N / u

( l b f s / i n o r l b f / u )

c o n s t a n t

c r it i c al v i s c o u s d a m p i n g , N s / m ( l b f s / i n )

c o e ff ic i en t o f to r s i o n a l v i s c o u s d a m p i n g , N m s / r a d

( l b f in s / r a d )

coe f f ic ien t s

c o n s t a n t s

d i a m e t e r o f s h a f t , m ( i n )

f l e x u r a l r i g i d i t y [ = E h ~ / 1 2 ( 1 - u2)]

d i s p l a c e m e n t o f t h e c e n t e r o f m a s s o f t h e d i s k f r o m t h e s h a f t

axis , m ( in)

m o d u l u s o f e l a st i ci t y , G P a ( M p s i )

f r e q u e n c y , H z

e x c i t i n g f o r c e , k N ( l b f )

m a x i m u m e x c i t i n g f o r c e , k N ( l b f )

t r a n s m i t t e d f o r c e , k N ( l b f )

a c c e l e r a t i o n d u e t o g r a v i t y , 9 . 8 0 6 6 m / s 2

2 2

( 3 2 . 2 f t / s o r 3 8 6 . 6 i n / s )

m o d u l u s o f r i g id i t y , G P a ( M p s i )

t h i c k n e s s o f p l a t e , m ( i n )

i n t e g e r ( = 0 , 1 , 2 , 3 . . . )

m a s s m o m e n t o f i n e r t i a o f r o t a t i n g d i s k o r r o t o r , N s2 m

( l b f s 2 i n )

p o l a r s e c o n d m o m e n t o f i n e rt ia , m 4 o r c m 4 ( in 4)

s p r i n g s t i f f n e s s o r c o n s t a n t , k N / m ( l b f / i n )

e q u i v a l e n t s p r i n g c o n s t a n t , k N / m ( l b f / i n )

t o r s i o n a l o r s p r i n g s t i ff n e ss o f s h a f t , J / r a d o r N m / r a d ( l b f i n / r a d )

k i n e t i c e n e r g y , J ( l b f / i n )

2

7/24/2019 67071_22.pdf

2/25

2 2 . 2 C H A P T E R T W E N T Y - T W O

l

m

m e

M

M~

P

q

t"

R = I - T R

R 2 = D 2 / 2

t

T

TR

U

v

w

W

x

X l , X 2

X o

.2

.2

Xs,

Y

C

= C e d a m p i n g f a c t o r

6 l o g a r i th m i c d e c r e m e n t ,

de f l ec t ion , m ( in )

6st s t a t i c de f l ec t ion , m ( in )

0 p h a s e a n g l e , d e g

A w a v e l e n g t h , m ( in )

v P o i s s o n ' s r a t i o

p m a s s d e n s i t y , k g / m 3 ( l b / in 3 )

a n o r m a l s t re s s , M P a ( p s i)

r shea r s t re s s , M Pa (ps i )

p e r i o d , s

~ . an gu la r de f l ec t ions , r ad (deg)

a n g u l a r v e l o c i t y , r a d / s

q5 an gu la r acce le ra t ion , r ad / s 2

co f o r c e d c i r c u l a r f r e q u e n c y , r a d / s

l e n g t h o f s h a f t, m ( in )

mass , kg ( lb )

e q u i v a l e n t m a s s , k g ( l b )

t o t a l m a s s , k g ( l b )

t o r q u e , N m ( l b f f t)

c i r c u l a r f r e q u e n c y , r a d / s

d a m p e d c i r c u l a r f r e q u e n c y ( = v / 1 - ~ 2 )

rad ius , m ( in )

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

r a d i u s o f t h e c o i l , m ( i n )

t i m e ( p e r i o d ) , s

t e m p e r a t u r e , K o r C ( F )

t r a n s m i s s i b i l i t y

v i b r a t i o n a l e n e r g y , J o r N m ( l b f i n )

p o t e n t i a l e n e r g y , J ( l b f i n )

v e l o c i ty , m / s ( f t / m i n )

w e i g h t p e r u n i t v o l u m e , k N / m 3 ( l b f /i n 3 )

t o t a l w e i g h t , k N ( l b f )

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

ins t an t t , m ( in )

succes s ive ampl i tudes , m ( in )

m a x i m u m d i s p l a c e m e n t , m ( i n )

l i n e a r v e l o c i ty , m / s ( f t / m i n )

l inea r acce le r a t ion , m /s 2 ( f t / s 2 )

s t a t i c de f l ec t ion of the sys t em, m ( in )

d e f l ec t io n o f th e d i s k c e n t e r f r o m i ts r o t a t i o n a l a x i s , m o r m m ( in )

w e i g h t d e n s i t y , k N / m 3 ( lb f / i n 3 )

Particular

Formula

S I M P L E H A R M O N I C M O T I O N ( F i g . 2 2 - 1 )

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

R S

(Fig. 22-1)

T h e w a v e l e n g t h

x - - Xo

s i n p t

A = 2 7 r

(2

(2

7/24/2019 67071_22.pdf

3/25

M E C H A N I C A L V I B R AT I O N S 2

P a r t i c u l a r

F o r m u l a

p / , ' ~v : pXo x i . . . . . ~

_.X:XosineF X ( /

X F m ~ a)

s (0)

F I G U R E 2 2 - 1 S im p l e h a r m o n i c m o t i o n .

T h e p e r i o d i c t i m e

T h e f r e q u e n c y

T h e m a x i m u m v e l o ci ty o f p o i n t Q

T h e m a x i m u m a c c e l er a ti o n o f p o i n t Q

2 7 1

'7- ~ m

P

1 p

7 - 27r

V m a x - - pXo

a m a x : 1 )m a - - - p 2 X o

(22

(22

(22

(22

Si ng l e - deg r ee - o f- f r eedo m s ys t em w i t hou t

damp ing and wi thout extern al force (F ig . 22-2)

Li nea r s y s t em

T h e e q u a t i o n o f m o t i o n

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

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

c o n d i t i o n x = Xo a n d ~ = 0 a t t = 0

T h e n a t u r a l c i r c u l a r f r e q u e n c 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 v i b r a t i o n

T h e n a t u r a l f r e q u e n c y i n t e r m s o f s t a t ic d e f l e c t i o n ~Sst

I