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Shigley Formula Sheet
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Wesley Botha
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MOW32 MOW323 3 Formulae Formulae sheet - Shigley Shigley
Clutches and Brakes =
Wear
K A A pl Vt
Long shoe dN = =
M N
Cone clutch
pmax br sin θ d θ
F =
sin θ a abr pmax
2 (θ 2 − θ1 ) − sin 2θ 2 4 sin θ a
M N
− M f
c + sin 2θ1
R x
=
+
M f
F
− sin
=
c T
µ b r pmax a 2 −r ( cos θ 2 − cos θ1 ) − ( sin θ 2 M F = 2 sin θ a T
M N
F =
µ pmax br 2 ( cos θ1 − cos θ 2 ) sin θ a
= − Fx +
pmax br
{2 ( sin θ 4 sin θ 2
2
( D − d )
2 π µ c ( D 2 − d 2 ) 4 sin α
T =
θ1 )
2
=
π pmax d
= −F +
R R x
= − Fx +
2 θ
− sin
4 sin θ a
−θ
{2 ( sin θ 4 sin θ 2
2
= − Fy +
pmax br
{2 (θ 4 sin θ
2
θ1 ) − µ 2 (θ 2 − θ1 ) − sin 2θ 2 + sin 2θ1 }
− sin 2θ + sin 2θ + 2
2
sin θ
− sin
2
θ
t 1
− sin
θ1 ) + µ 2 (θ 2 − θ1 ) − sin 2θ 2 + sin 2θ1 }
2
u
)
}
2µ ( sin 2θ2 − sin 2θ1 )
F 2
H loss
µφ
=e
Tmax
=
hCR A ( T
= T∞ +
− T∞
) = ( hr +
∆T
1 − exp ( β t1 )
f v hc ) A ( T
Axial wear t t Axial wear= =f1 ff21Kf 2PKVPV
F
=
T
=
T
=
π pmax d 2
( D − d )
πµ pmax d 8
(D
2
I1 I 2 (ω1 − ω 2 )
F 2
− d
)
T
=
=
π pmax 4
πµ pmax 12
( D 2 − d 2 )
(D
3
3
− d
)
− T ∞
)
T
− rinner
µ pmax ri (θ 2 − θ 1 ) ( ro2
− ri
1 2
r e
=
(r
o
+ r i
− T ∞
T1 − T∞
pmax ri (θ 2 − θ 1 ) ( router
)
2
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
2
)
2
2
∆ T =
2 ( I1 + I 2 )
−hCR A W C p
Disk Brakes
=
=
I1 I 2 ( ω1 − ω 2 )
with β =
Axial Clutch
F
3sin α ( D 2 − d 2 )
I1 + I 2 ω1 − ω 2 − T I1I 2
= T θ = T
E =
a
F 1
µ F ( D3 − d 3 )
I1 I 2
2
− θ1 − sin 2θ 2 + sin 2θ1 −
Band Brakes
4
I + I θ = θ1 − θ2 = ω1 − ω2 − T 1 2 t
a
R y
)
π p0 µ ( D 3 − d 3 ) 12 sin α
1
pmax br
2
− d
Energy
a
pmax br
2
=
T =
4 sin α
(D
=
T
µ F ( D + d )
π p0
F
)
F T
=
=
E WC p
− hCR A W C p
exp
) 12 ( ro2 − ri 2 )
(θ 2 − θ 1 ) 13 ( ro3 − ri3 )
pmax (θ 2
= µ pmax
r e
t
=
− θ 1
3 3 2 ( ro − r i )
3 ( ro2 − r i 2 )
BELTS Flat Belts
F1 − m r 2 ω 2
w = γ bt
θ d
= π − 2 sin
θ D L
θ
= π +
= π +
L
2
F1 − F c F2
2C
−1
2
+
1 2
( Dθ D
T2
=
w
=
H nom K s nd
π 2
=
F1 + ( Fb )2
2
+
1 2
=
dip
( D + d )θ
V 2
( F1 ) a
g
f ' =
=
1
φ
F1
H d
2π n
P
( D − d )
( F1 ) a − F c
=
=
F1 + F1 +
= T ω = T
H a
=
−
d K b
FOS =
1
2
−b
FC
=
Fi
=
FC
2π n 60
≥
N b
− 2( D − d )
F i
F i
d
T d
×
f φ
2e
e f φ + 1 T
F C −
+
2 FC T
e
d 2 f φ
( F1 − FC )
=
F i =
+
+
+
F C +
+
=
F1 − F2
F c
K1 K 2 H tab
π L − D + d) ( p 2
K = N P T1
D
F2
2
4C
K b
F2
8 F i
Fi
=
b Fa C p Cv
ln
2T
=
d F1 + F2 = 2 Fi
L w
=
F1 − ∆F
F1 − F 2
f φ
T =
=
+1
e f φ − 1 e f φ
e
f φ
+1
e
f φ
−1
V FC = K C 2.4
H d H a
2
K + T 2
H a N b
∆F =
=
H d / N b
π nd
−b
H d ∆F =
H nom K s
F1 − F 2
N b
π nd
F1
=
Fi
=
2
F C +
∆F e
f φ
e f φ − 1
F1 + F 2
− F C
2
0.51
=
b
=
σ max =
e
2
d)
( D + d ) +
F1 + ( Fb )1
=
σ b
S f
d θ d )
e
=
F c
2C
( F1 − F2 )V
2C +
=
f typ
+
−
=
ω 2
D + d
π C = 0.25 L p − ( D + d ) + 2
T1
2
− mr
= π d n
V H d
L p
=
2C − 1 D − d
− (D +
m r ω
=
H nom
2 Sin
2
D − d
− (D − d )
4C 2
=
FC
2 sin
4C 2
=
−1
F2
F2
f φ
Et
(1 −ν ) D 2
=
(σ b )1 +
97, 70 2 N p−0.407 S y
3
(σ )1 = F 1 bt
σ min
F1
(σ )2 =
bt =
(σ b )2 +
F 2
f ' =
bt
1
φ
ln
F 1 F 2
t (hrs ) =
F 2 bt
f o r 30 1 / 30 2 sta i n less
for other o ther material mate rialss
F1a = S f Fi
MOW 323 323 Formulae Formulae Sheet - Shigley 2011-11-26
=
=
T NP
b
K
N P Lp
3600V
tb = ab − 2 1 υ − D ( ) Et
ab + ab − ∆F
2
=
ab −
∆F
2
POWER SCREWS: l
Tan λ =
P R
=
P L
=
T R
=
T L
=
π d m
F ( l / π d m ) + f 1 − ( f l / π d m ) F f
( l / π d m ) 1 + ( f l / π d m )
PR
PL
−
dm
2 dm
F d m [l + π f d m ]
=
2
F d m [π f d m
=
2
π d m − f l
2
−l
π f d m
Self-l Self-locki ocking ng :
]
e=
Efficiency:
π d m + f l
ACME Threads:
T R
≈
F d m [l + π f d m Secα ]
π d m − f l Secα
2
(Replace f by f Sec α α in Square thread formulae) Collar Torque:
T c
=
F f c d c
2
Power screw stress analysis:
σ B
=
σv
=
σ x
=
τ xy
=
2F
σ b
π d m nt p 1 2 6F
(σ
x
−σ y
σy
π d r nt p 0 τ yz
=
16T
π d r 3
2
) + (σ =
0
τ zx
y
−σz
σ z =
=
=
6 F
π d r nt p
2
) + (σ
z
−σx
)
2
+6
(τ
2 xy
2
2
+ τ yz + τ zx
−4 F
π d r 2
0
Use nt = 1 and 0.38F of the load for calculating σx. Failure criteria:
σ v
=
S y MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
)
>l
T 0 T R
=
⇒f Fl
2π T R
> Tanλ
Couplings and Universal Joints: Tanα
= Tan β Cosδ
Sec 2α
dα
=
dt
Sec 2α ωa
Sec2 β
d β
Cosδ dt
=
Sec 2 β ωb Cosδ
=
(1 + Tan β ) ω Cosδ 2
b
Tan 2α = 1 + ωb Cosδ 2 Cos δ ω b ωa
=
ω b ωa
γ
1 − Sin 2δ Cos 2α
= max
ωb ωa
=
ωb
Cosδ
ω b ω a
1 Cosδ
ω b − ω a
max
d ω b dt
min
min
=
Sin Sinδ Tan Tanδ
⇒ Cosδ
= 1 − Sin
⇒ Tanα
=±
2
δ Cos 2α
Cosδ
2
Cosδ Sin δ Sin2α
(1 − Sin δ Cos α ) 2
2
2
2 − Sin 2δ 2 Sin 2δ 2 − Sin 2δ
= Tan
Cosδ
Tan β
= Tanγ
Cosδ
Tanα
= Tan β
⇒α
=
β
= ω b
ωa Cos 2δ
2 d Sin2α 2 Cosδ Sin δ Si −ω a =0 2 2 2 d α (1 − Sin δ Cos α )
Sin 2δ ( 2 − Cos 2 2α )
Tanα
ωa
2
= −ω a
Cos 2α =
Cos 2α ≈
1
= ωa
= Cosδ
D =
< ω b <
4T
π S s r n
d Sin2α =0 ⇒ 2 2 2 d α (1 − Sin δ Cos α )
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
ω a Cos 2δ
Flywheels: θ ω max
ω max
∫ ( T − T ) dθ = ∫ l
avg
θω min
C f
I m
I m ω dω
=
I m
(ω 2
ω min
=
=
ωmax − ωmin ωavg md 2
8
= ρ
m ( d o2
=
2
2 max
)
2
− ωmin =
ωmax − ωmin ωmax + ω min
Ke
I m
=
K e 2 C f ω avg
π d 2 d 2 2
+ di
I m
=
σθ
= σ θω + σ θ p
t
4
)
8
= ρ
,
(Solid circular cross-section)
8
π ( d o2 − di2 ) 4
σr
t
(d
= σ rω + σ rp
2 o
+
di2 )
(Hollow cross-section)
8
r o2 pi r i 1 + 2 2 2 ν 1 + 3 r r ( ) 3 +ν r r 2 + σ θ = ρω 2 ri 2 + ro2 + i 2o − r ro2 − r i 2 8 3 + ν 2
r o2 pi r i 1 − 2 2
σr
=
2
8
ρω ri
Brittle fracture:
2
2 o
+r −
N =
i
2 2 o 2
r
2
− r
+
ro2
2
− r i
S ut
Yield Failure: N =
σ 1
σv Circumferential strain:
ε θ
=
σ θ E
Stress due to interference : pi =
−
ν σ r
Deflection:
E
Eδ i ( ro2
2
− r i
δ
)
2ri r o2
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
=
S y
σ v σ θ2 + σ r2 − σ θ σ r
= ε θ r
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
MOW 323 Formulae Formulae Sheet - Shigley 2011-11-26
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