shigley formula sheet
TRANSCRIPT
A lWear K A p Vt=
Clutches and Brakes
MOW323 Formulae sheet - Shigley
Long shoe
max
a
p br sin ddN
sin
θ θ
θ=
( )max2 1 2 12 2 2
4N
a
abr pM sin sin
sinθ θ θ θ
θ= − − +
( ) ( )2 2max2 1 2 1
2F
a
br p aM r cos cos sin sin
sin
µθ θ θ θ
θ
= − − − −
N fM MF
c
−=
( )2
max1 2
a
p brT cos cos
sin
µθ θ
θ= −
( ) ( ){ }2 2max2 1 2 1 2 12 2 2 2
4y y
p brR F sin sin sin sin
sinθ θ θ θ µ θ θ
θ= − + − − + + −
( ) ( ){ }2 2max2 1 2 1 2 12 2 2 2
4x x
a
p brR F sin sin sin sin
sinθ θ µ θ θ θ θ
θ= − + − − − − +
N fM MF
c
+=
Cone clutch
( )max
2
p dF D d
π= −
( )4
F D dT
sin
µ
α
+=
( )2 2
4
cT D d
sin
π µ
α= −
( )2 20
4
pF D d
π= −
( )3 30
12
pT D d
sin
π µ
α= −
( )( )
3 3
2 23
F D dT
sin D d
µ
α
−=
−
Energy
1 21 2 1 2
1 2
I IT t
I Iθ θ θ ω ω
+= − = − −
� � �
( )( )
1 2 1 2
1
I It
T I I
ω ω−=
+( ) ( ){ }2 1 2 1 2 12 2 2 2
4y y
a
R F sin sin sin sinsin
θ θ θ θ µ θ θθ
= − + − − + + −
( ) ( ){ }2 2max2 1 2 1 2 12 2 2 2
4x x
a
p brR F sin sin sin sin
sinθ θ µ θ θ θ θ
θ= − + − + − − +
( ) ( ){ }2 2max2 1 2 1 2 12 2 2 2
4y y
a
p brR F sin sin sin sin
sinθ θ θ θ µ θ θ
θ= − + − − + − −
( )1
1 2T I I+
1 21 2
1 2
I Iu T T T t
I Iθ ω ω
+= = − −
�
( )( )
2
1 2 1 2
1 22
I IE
I I
ω ω−=
+p
ET
WC∆ =
1
exp CR
p
h AT T
T T W C
∞
∞
−−= − 1
2
Fe
F
µφ=
Band Brakes
Disk Brakes
( ) ( )3 31max 2 1 3 o iT p r rµ θ θ= − −
( ) ( )2 21max 2 1 2 o iF p r rθ θ= − −
( )( )
3 3
2 2
2
3
o i
e
o i
r rr
r r
−=
−
( )( )max 2 1i outer innerF p r r rθ θ= − −
( ) ( )2 21max 2 12 i o iT p r r rµ θ θ= − −
( )2
o i
e
r rr
+=
( ) ( ) ( )loss CR r v cH h A T T h f h A T T∞ ∞= − = + −
( )max
11 exp
CR
p
h ATT T with
t W Cβ
β∞
−∆= + = −
1 2Axial wear f f K PV t=
Axial Clutch
( )max
2
p dF D d
π= −
( )2 2max
8
p dT D d
πµ= −
( )2 2max
4
pF D d
π= −
( )3 3max
12
pT D d
πµ= −
1 2Axial wear f f K PV t=
MOW 323 Formulae Sheet - Shigley 2011-11-26
Flat Belts
2 2
1
2 2
2
1
2
f
fc
c
F mre
F mr
F Fe
F F
φ
φ
ω
ω
−=
−
−=
−
( )1 2 1
1f
C f
eF F F F
e
φ
φ
−− = −
V d nπ=
w btγ=
22 2
C
VF mr w
gω= =
1 2
1 2
2
2 2i C
TF F
d
F F F F
− =
+ = +
2
dH
Tnπ
=
1
1
f
i f
T eF
d e
φ
φ
+= ×
−
1
2
1
i C
f
C i f
TF F F
d
eF F
e
φ
φ
= + +
= ++
2
2
1
i C
C i f
TF F F
d
F Fe
φ
= + −
= ++
( )1 2nomH F F V= −
d nom s dH H K n=
( )1 a p vaF bF C C=
1
1
22
22
d
D
D dsin
C
D dsin
C
θ π
θ π
−
−
−= −
−= +
( )2 2 14 ( )
2D d
L C D d D dθ θ= − − + +
122
D dSin
Cθ π − +
= +
( )2 2 14 ( )
2L C D d D d θ= − + + +
2 1F F F= − ∆
2
60
nP T T
πω= =
2
8 i
L wdip
F=
( )1
2
1' ln
ca
c
F Ff
F Fφ
−=
−
BELTS
b b
PT N K=
( )3600
P pN L
t hrsV
=( )21b
Et
Dσ
ν=
−( ) ( )1 2
1 2
F F
bt btσ σ= =
( ) 1max 1b
F
btσ σ= + ( ) 2
min 2b
F
btσ σ= +
0.40797, 702 301 / 302
3
f p
y
S N for stainless
Sfor other materials
−=
=
1
2
1' ln
Ff
Fφ=
( )( )
2
22 4
p
D dL C D d
C
π −= + + +
( ) ( ) ( )2
20.25 2
2 2p pC L D d L D d D d
π π = − + + − + − −
db
a
HN
H≥
2
2.4C C
VF K
=
1 2
/d bH NF F F
ndπ∆ = − =
11
f
C f
FeF F
e
φ
φ
∆= +
−
1 2
2i C
F FF F
+= −a b
nom s
H NFOS
H K=
( )1 1 11
b
b
KT F F F
d= + = +
( )2 1 12
b
b
KT F F F
D= + = +
1 2
1b b
P
K K
N T T
− −
= +
d
b
HN
Fndπ
∆ =
1 2a tabH K K H=
( )1 21a f
EtF S tb ab
Dυ
= − =
−
i2 2
ab ab F FF ab
+ − ∆ ∆= = −
0.51typf =
MOW 323 Formulae Sheet - Shigley 2011-11-26
m
lTan
dλ
π=
( )( )
/
1 /
m
R
m
F l d fP
f l d
π
π
+ =−
( )( )
/
1 /
m
L
m
F f l dP
f l d
π
π
− =+
[ ]2 2
mm mR R
m
l f dd F dT P
d f l
π
π
+= =
−
[ ]2 2
mm mL L
m
f d ld FdT P
d f l
π
π
−= =
+
mf d l f Tanπ λ> ⇒ >
0
2R R
T Fle
T Tπ= =
[ ]2
mmR
m
l f d SecF dT
d f lSec
π α
π α
+≈
−
Self-locking :
Efficiency:
ACME Threads:
POWER SCREWS:
2
c c
c
F f dT =
( ) ( ) ( ) ( )2 2 2 2 2 2
2
3
2 6
16
2
6 40
160 0
B b
m t r t
v x y y z z x xy yz zx
x y z
r t r
xy yz zx
r
F F
d n p d n p
F F
d n p d
T
d
σ σπ π
σ σ σ σ σ σ σ τ τ τ
σ σ σπ π
τ τ τπ
= =
= − + − + − + + +
−= = =
= = =
v ySσ =
(Replace f by f Secα in Square thread formulae)
Collar Torque:
Use nt = 1 and 0.38F of the load for calculating σx.
Failure criteria:
MOW 323 Formulae Sheet - Shigley 2011-11-26
Power screw stress analysis:
Tan Tan Cosα β δ=
( )
2 2
2 2
2
2
2
1
1
a b
b
b
d dSec Sec Cos
dt dt
Sec Sec Cos
Tan Cos
TanCos
Cos
α βα β δ
α ω β ω δ
β ω δ
αω δ
δ
=
=
= +
= +
2 21
b
a
Cos
Sin Cos
ω δ
ω δ α=
−
max
1b
aCos
ω
ω δ=
min
b
a
Cosω
δω
=
max min
1
b b
a aSin Tan
ω ω
ω ωγ δ δ
−
= =
( )2 2
2
2 22
2
Sin CosCos
Sin
δ αα
δ
−=
−
2
2
22
2
SinCos
Sin
δα
δ≈
−
Tan Tan Cosα γ δ=
Couplings and Universal Joints:
1
2 21b a
Cos Sin Cos
Tan Cos
ω ω δ δ α
α δ
= ⇒ = −
⇒ = ±
( )
22
22 2
2
1
b
a
d Cos Sin Sin
dt Sin Cos
ω δ δ αω
δ α= −
−
( )
( )
22
22 2
22 2
20
1
20
1
a
d Cos Sin Sin
d Sin Cos
d Sin
d Sin Cos
δ δ αω
α δ α
α
α δ α
− = −
⇒ = −
Tan Tan Cos
Tan Tan Cos
Tan Tan
α γ δ
β γ δ
α β α β
=
=
= ⇒ =
a bω ω=
2
2
aa bCos
Cos
ωω δ ω
δ< <
4
s
TD
S rnπ=
MOW 323 Formulae Sheet - Shigley 2011-11-26
( ) ( )max max
minmin
2 2
max min2
ml avg m e
IT T d I d K
ω
ω
θ ω
θ ω
θ ω ω ω ω− = = − =∫ ∫
max min max min
max min
2f
avg
Cω ω ω ω
ω ω ω
− −= =
+ 2
e
m
f avg
KI
C ω=
2 2 2
8 4 8m
md d dI t
πρ= =
( ) ( ) ( )2 2 2 2 2 2
8 4 8
o i o i o i
m
m d d d d d dI t
πρ
+ − += =
,p r r rpθ θω θ ωσ σ σ σ σ σ= + = +
( )
22
22 22 2 2 2
2 2 2
11 33
8 3
oi i
i oi o
o i
rp r
rr rr r r
r r rθ
ννσ ρω
ν
+
+ + = + + − + + −
22
22 22 2 2 2
13
oi i
i o
rp r
rr rr r r
νσ ρω
−
+ = + − − +
Flywheels:
(Solid circular cross-section)
(Hollow cross-section)
MOW 323 Formulae Sheet - Shigley 2011-11-26
2 2 2 2
2 2 2
3
8
i o
r i o
o i
rr rr r r
r r r
νσ ρω
+ = + − − + −
1
utSN
σ=
2 2
y
v
v r r
SN
θ θ
σ
σ σ σ σ σ
=
= + −
r
E E
θθ
σ νσε = − rθδ ε=
( )2 2
22
i o i
i
i o
E r rp
r r
δ −=
Brittle fracture:
Circumferential strain: Deflection:
Stress due to interference :
Yield Failure:
MOW 323 Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Sheet - Shigley 2011-11-26