67071_10.pdf

CHAPTER

10 ROTATING DISKS AND CYLINDERS'

S Y M B O L S 1

g r

r i

r o

h h2

a0 err az P co

y

acceleration due to gravity, m / s 2 (ft/s 2) any radius, m (in) inside radius, m (in) outside radius, m (in) thickness of disk at radius r from the center of rotation, m (in) thickness of disk at radius r 2 from the center of rotation, m (in) uniform tensile stress in case of a disk of uniform strength,

MPa (psi) tangential stress, MPa (psi) radial stress, MPa (psi) axial stress or longitudinal stress, MPa (psi) density of material of the disk, kg/m 3 (lbm/in 3) angular speed of disk, rad/s Poisson's ratio

Particular Formula

DISK OF UNIFORM STRENGTH ROTATING AT w rad/s (Fig. 10-1)

The thickness of a disk of uniform strength at radius r from center of rotation

SOLID DISK ROTATING AT co rad/s

The general expression for the radial stress of a rotating disk of uniform thickness

h = h 2 exp [-2-a-a

3 + u ~r = - - ~ p 2 Co -- r 2)

(10-1)

(10-2)

10.1

10.2 CHAPTER TEN

Particular Formula

L_ O4 II

"O

Rim

FIGURE 10-1 High-speed rotating disk of uniform strength.

..:-":.i:i. CO

°° °

"; ":;".'-" ".: • d r

¢~

FIGURE 10-2 Rotating disk of uniform thickness.

The general expression for the tangential stress of a rotating disk of uniform thickness

The maximum values of stresses are at the center, where r - 0, and are equal to each other

O- 0 - - '8 p 2 r2o 3 + u

3 + u 2 O'r(max) = O'0(max) - - 8 Pca2r°

(10-3)

(10-4)

H O L L O W D I S K R O T A T I N G AT co rad/s (Fig. 10-2)

The general expression for the radial stress of a rotating disk of uniform thickness

The general expression for the tangential stress of a rotating disk of uniform thickness

The maximum radial stress occurs at r 2 = ror i

O" F =

2r ) 3 + u 2 ro r2 8 Pea2 r ~ + r o r2

O" 0 =

2 2 3 + u 2 rori

8 POe2 rZ+r° -~ r 2 l + 3 u ) r 2 3 + u

3 + u err (max) = 8 pod 2 (ro - r i ) 2

(10-5)

(10-6)

(10-7)

ROTATING DISKS AND CYLINDERS 10.3

Particular Formula

The maximum tangential stress occurs at inner boundary where r - ri

S O L I D C Y L I N D E R R O T A T I N G AT w rad/s

The tangential stress

The radial stress

The maximum stress occurs at the center

The axial strain in the z direction (ends free)

The axial stress under plane strain condition (ends free)

The axial stress under plane strain condition (ends constrained)

= r (10-8) O'0(max) 4 PC°2 r2 + 3 + v

pco 2 °° = 8(1 - u ~ [(3 - 2u)r2 - (1 + 2v)r2] (10-9)

P c02 (31-- 217)(r2 r2 ) O" r -----~ _

P C ° 2 ( 3 - - 2 u ) 2 O'r(max) -- CrO(max) = - - 8 1 - u r°

(10-10)

(lO-lOa)

22 - v pco ro ~.,,_..,

e z = 2 E t, lO 11)

p c ° 2 ( u )(r2o - 2r2) (10 -12a) ~z - - - ~ 1 - v

__ p~02/7 _ 2 r 2 ] erz -- 4(1 _ u) [1(3 2u)r2o - (10-12b)

H O L L O W C Y L I N D E R R O T A T I N G A T w

rad/s

The tangential stress at any radius r

The radial stress at any radius r

The axial stress (ends free) at any radius r

The axial stress under plane strain conditions (ends constrained) at any radius r

The maximum stress occurs at the inner surface where r - - r i

( ) I 22 ( ) ] pco 2 3 - 2u 2 ri ro 1 + 2 v CrO = - - 8 1 -- ~ r2 + r° - + - - ~ - 3 -- 2 y r2

(10-13)

(31-217) I 2 2 ]r2 P c°2 r2i -+- r o r 2 2 ri ro

(10-14)

} 2 2r 2] 1 O-15) _ _ OCO 2 t7 [r 2 + ro \

( Crz - -4- 1 - v /

( )I 2 up¢o 2 3 -- 2U r 2 + ro -- ~z = - - - f - 1 - v

2_r2 ,] 3 - 2 v J (10-16)

_ )Ero2+ ' 2 )1 or0(max) - ~ 31- 2v 1 - 2v r2 (10-17)

10.4 CHAPTER TEN

Particular Formula

The axial strain in the z direction (ends free)

The displacement u at any radius r of a thin hol low rota t ing disk

~ ' 2 ~ -

2 up~ (r/2 + r2o) (10-18) 2E

[p J r (3 + u)(1 - u)

u = E 8

2 2 )] 2 1 + u ror i 1 + u r 2 (10-19) × r2o+ri 4 1 - - u r 2 3 + u

SOLID THIN UNIFORM DISK ROTATING AT w rad/s UNDER EXTERNAL PRESSURE Po (Fig. 10-3)

The radial stress at any radius r

The tangential stress at any radius r

The max imum radial stress at r = 0

The max imu m radial stress at r = ro

The max imum tangential stress at r = 0

The displacement u at any radius r

3 + u)(r2o _ r2 ) O'r = -Po + p~2 8 (10-20)

Cro -- -Po + P co2 8 r2° l + 3 u ) 3 + u r2 (10-21)

3 + U ) 2 (7r(max) = --Po + P c02 8 ro (10-22)

ar = --Po (10-23)

or0(max) -- O'r(max) (10-24)

r { P ¢02 2 } u = ~ ( 1 - u) - P o + - - ~ [(3 + U)ro - (1 + u)r 2]

(10-25)

9o

Po

FIGURE 10-3 Rotating disk of uniform thickness under external pressure.

ROTATING DISKS AND CYLINDERS 1 0 . 5

Particular Formula

H O L L O W CYLINDER OF UNIFORM THICKNESS ROTATING AT co rad/s. SUBJECT TO INTERNAL (Pi) AND EXTERNAL (Po) PRESSURES (Fig. 10-4)

The general expression for the radial stress of a hollow cylinder of uniform thickness rotating at co rad/s under internal (Pi ) and external (Po) pressure at any radius r

The general expression for the tangential or hoop stress of a hollow cylinder of uniform thickness rotating at co rad/s under internal (Pi ) and external (Po) pressure at any radius r.

The tangential or hoop stress in a hollow cylinder rotating at co rad/s under Po and Pi at r = ri (Fig. 10-4)

B pco2 (31- 2/J ) °'r -- A - ~ + - ' 8 -- b'

f 22 J 2 ri ro × r2i + ro r 2 r2 (10-26)

A + _ + ,, 1 - u or 0

2 ri ro 1 + 2u x r2i + r o + - - ~ - 3--2u r2 (10-27)

2 2 2 = -- r i r o ( P i - - P o ) where A Pi r2 poro ; B -

r2o - r2i r2o - r 2

O'(0 max)r = r i " - -

2 2 pi(r2i + ro) - 2poro

- q

pco2 ) [2r2 + ( 2 _ 4u) 1 r2

(lo-2aa)

= pi(r2i + r2o) - 2por2o

r2o - r2i

pco2( _ u ) I r 2 o + ( 3 2u) 1 + - 4 31-2u 1 - 2 u r2

(10-28b)

o•• Po

h

F I G U R E 10-4

h

Internal Pressure, Pi

F I G U R E 10-5

External Pressure Po

F I G U R E 10-6

10.6 CHAPTER TEN

Particular Formula

The tangential or hoop stress in a hollow cylinder rotating at co rad/s under Po and Pi at r = ro (Fig. 10-4)

0"(0 max)r=ro ~ -

2pi r2 - po(r2o + r2i ) r 2 __ r 2

(31)E 1-2 1 pco2 -- 2U r2 + r2 +-4-- -z, 3-2z,

(10-29)

The tangential stress in a cylinder rotating at co rad/s at any radius r when subjected to internal pressure (Pi) only (Fig. 10-5)

The tangential stress in a cylinder rotating at co rad/s at any radius r when subject to external pressure (Po) only (Fig. 10-6)

(crO)p °=° = p¢~Z((r~2+ r 2) pw2(31- 2 u ) - d ) +-7- - ~

2 ri ro 1 + 2u x r 2 + r o + - ~ - 3 - 2 u r2 (10-30)

_por2o(r2 + r2i ) pco2 ( 3 1 _ 2 u ) (aO)p;=O = r2(r 2 - r 2) + - - 4 - - u

I 22 < )1 2 ri ro 1 + 2u x r 2 + r o + - 7 - 3 - 2 u r2 (10-31)

ROTATING THICK DISK AND CYLINDER WITH UNIFORM THICKNESS SUBJECT TO THERMAL STRESSES

The hoop or tangential stress in thick disk or cylinder at any radius r rotating at co rad/s subject to pressure Po and Pi

The radial stress in thick disk or cylinder at any radius r rotating at co rad/s subject to pressure Po and Pi

a o = A + ~ ( 3 + u ) r2o - r 2 - ~ 3 + u

- E a V + - -~ r r dr (10-32)

O" r - - A B pco2 E a [ r 2 8 (3 + u)(r2o - r 2) - - f l - - a Tr dr

(10-33)

where A and B are Lam6's constants and can be found from boundary or initial conditions

a = linear coefficient of thermal expansion, mm/°C (in/°F)

T = temperature, °C or K (°F) p = density of rotating cylinder or disk material,

kg/m 3 (lbm/in 3) E = modulus of material of disk or cylinder, GPa

(Mpsi)


Particular Formula

ROTATING LONG H O L L O W CYLINDER WITH U N I F O R M THICKNESS ROTATING AT w rad/s SUBJECT TO THERMAL STRESS

The general expression for the radial stress in the cylinder wall at any radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length.

The general expression for the tangential stress in the cylinder wall at any radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length.

The general expression for the axial stress in the cylinder wall at any radius r when the temperature distribution is symmetrical with respect to the axis and constant along its length.

(31 )[ 22] 2 ri ro _ pod 2 -- 2U r 2 + ro r2 r2 Or - - 8 m Id

r4r2 2Ir° r jr° r r 1 + (1 - u)r ~ - - - ~ [d2o d 2 r i r i

(10-34)

( ) [ 2 2 ( 1 4 - 2 z j ) 1 pw 2 3 -- 2u r 2 4- 2 ri ro r 2 ~rO= ~ 1 " u r o + - - ~ - - 3 2u

oeE I4r2 4-d 2 ~o 4- ( 1 - u)r------------Z i d2o-d2i v i

Tr dr

;r ° ] 4- r r dr - Tr 2 (10-35) i

) 2 - 2r2] pco 2 ly [r 2 + ro ~r°-- ---4-- 1 - - ~

+ i - - lJ a2o - d 2 r i T r d r - r

where do = 2ro and di = 2ri

(10-36)

DEFLECTION OF A ROTATING DISK OF U N I F O R M THICKNESS IN RADIAL DIRECTION WITH A CENTRAL CIRCULAR CUTOUT

The tangential stress within elastic limit, a0, in a rotating disk of uniform thickness (Fig. 10-7)

The expression for the inner deflection (5i, of rotating thin uniform thickness disk with centrally located circular cut-out as per Stodala a (Fig. 10-7)

6 E or° = h (10-37)

n )2( 7.5K2 + 5) 6i = 3.077 x 10 .6 (lo-38)

a Source: Stodala "Turbo-blower and compressor"; Kearton, W. J. and Porter, L. M., Design Engineer, Pratt and Whitney Aircraft; McGraw-Hill Publishing Company, New York, U.S.A. Douglas C. Greenwood, Editor, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

6 . 0 - -

5.0 - -

4 . 0 - -

..,,..,

3 . 0 - -

0 .10-- -

¢ -

( / )

;5 2 . 5 - - 0

"10

L_ ¢_

( 9 ~- 2.0 E

I I

D

1.5

1.0-=

0.9

0.85

Based on Equations:

6~ 6 i = 3.077 x 10-6 (7.5 K 2 + 5) ri

j . %

. . . . , o = x 1 where

~o ro ~ ~ /,,,~ K = E-/

0 . 0 5 - 0 . 0 4 - -

0.001 -

+_ 0 . 0 0 0 5 - - 0 . 0 0 0 4 - - 0 . 0 0 0 3 - -

0 . 0 0 0 2 -

0 . 0 3 - -

~- 0 . 0 2 - - .~- U)

"o 0.01 - - O ~ d l

,4D ~

....---" ~ o .oo5- (9 = 0 . 0 0 4 - - (9 "o 0 . 0 0 3 - -

: 5 0 . O 0 2 - t _

0

I I

~o

0.0001 -

ri

S ~ ,--J

t

I t -

I t - t,,..,

I-

.y-"

n

10.8 CHAPTER TEN

~o 8--j-lo~ ~- . . - " 9 u) 7__-~'- =_,, ._= ;6

E 6 ~ 7 ,'-- L I ( 9

• " 5 _~---B: ' -

~ - " o 4"---'-~ 4 = , ~ " ~ C : O

~ 3 (9 (9 3 - - - ~ - "~

" 0 e -

. ~ _ 2 ~ -

KEY

14

- - 5 , 0 0 0 m

m

- 6 ,000 B

- - 7 , 000 m

- 8 ,000 m

9 ,000

10,000

E- "d

- ( 9 ( 9

- - Q .

- I I

- t::::

m

- - 20 ,000

- -30 ,000 m

n

- 40 ,000 m n

m

50 ,000

m

- - 6 0 , 0 0 0

- - 70 ,000

80 ,000 D

- 90 ,000

FIG URE 10-7 Nomogram for radial deflection of rotating disks with constant thickness with a centrally located circular hole.


Particular Formula

The expression for the outer deflection ~5 o of rotating thin uniform thickness disk with centrally located circular cut-out as per Stodala a (Fig. 10-7)

(ny 5o = 3.077 x 10 -6 ~ (1.5K 2 Jr- 7.5K)

where

(10-39)

K = ro/ri ao = tangential stress, psi

(~ - - ~ i -~- (~o : total deflection of disk, in r i : inner radius of disk, in ro = outer radius of disk, in n = speed, rpm

The Nomogram can be used for steel, magnesium and aluminum since the modulus of elasticity E = 29 x 10 6 psi (200 MPa) for steel and Poisson's ratio u = 1/3. The error involved in using this equation with E and u of steel for aluminum is about 0.5% and for magnesium is 2.5%.

REFERENCES

1. Lingaiah, K., and B. R. Narayana Iyengar, Machine Design Data Handbook, Volume I (SI and Customary Metric Units), Suma Publishers, Bangalore, 1986.

2. Lingaiah, K., Machine Design Data Handbook, McGraw-Hill Publishing Company, New York, 1994. 3. Douglas C. Greenwood, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York,

1961.

a Source: Stodala "Turbo-blower and compressor"; Kearton, W. J. and Proter, L. M., Design Engineer, Pratt and Whitney Aircraft; McGraw-Hill Publishing Company, New York, U.S.A. Douglas C. Greenwood, Editor, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961.

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67071_10.pdf