Formula Sheet: Flywheel | Theory of Machines (TOM) - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Flywheel
Introduction to Flywheels
• Definition : A flywheel is a rotating mechanical device used to store rota-
tional energy and s mooth out fluctuations in speed or torque in machines
lik e engines an d presses.
• Applications : Internal combustion engines, punch presses, and power gen-
er ation systems .
• K ey Function : Reduces speed fluctuations b y storing and releasing kinetic
energy .
Energy Stor age in Flywheel
• Kinetic Ener gy :
E =
1
2
I?
2
whereE = kinetic energy (J),I = mass moment of inertia (kg m
2
),? = angular
velocity (r ad/s).
• Energy Fluct uation :
?E =E
max
-E
min
=
1
2
I(?
2
max
-?
2
min
)
where?
max
,?
min
= maximum and minimum angu lar velocities.
• Energy Fluct uation (Simplified) :
?E =I?
2
avg
C
s
where?
avg
=
? max+?
min
2
,C
s
= coefficient of speed fluctuation.
Mass Moment of Inertia
• Solid Disk Fly wheel :
I =
1
2
mR
2
wherem = mass,R = r adius.
• Rim-Type Fly wheel (Thin Ring Approximation) :
I =mR
2
whereR = mean r adius of the rim.
• Gener al F orm :
I =
?
r
2
dm
wherer = r adial distanc e,dm = differential mass.
1
Page 2


F ormula Sheet: Flywheel
Introduction to Flywheels
• Definition : A flywheel is a rotating mechanical device used to store rota-
tional energy and s mooth out fluctuations in speed or torque in machines
lik e engines an d presses.
• Applications : Internal combustion engines, punch presses, and power gen-
er ation systems .
• K ey Function : Reduces speed fluctuations b y storing and releasing kinetic
energy .
Energy Stor age in Flywheel
• Kinetic Ener gy :
E =
1
2
I?
2
whereE = kinetic energy (J),I = mass moment of inertia (kg m
2
),? = angular
velocity (r ad/s).
• Energy Fluct uation :
?E =E
max
-E
min
=
1
2
I(?
2
max
-?
2
min
)
where?
max
,?
min
= maximum and minimum angu lar velocities.
• Energy Fluct uation (Simplified) :
?E =I?
2
avg
C
s
where?
avg
=
? max+?
min
2
,C
s
= coefficient of speed fluctuation.
Mass Moment of Inertia
• Solid Disk Fly wheel :
I =
1
2
mR
2
wherem = mass,R = r adius.
• Rim-Type Fly wheel (Thin Ring Approximation) :
I =mR
2
whereR = mean r adius of the rim.
• Gener al F orm :
I =
?
r
2
dm
wherer = r adial distanc e,dm = differential mass.
1
Coefficient of Speed Fluctuation
• Definition :
C
s
=
?
max
-?
min
?
avg
or
C
s
=
N
max
-N
min
N
avg
whereN
max
,N
min
,N
avg
= maximum, minimum, and aver age rotational speeds
(RPM).
• Typical V alues : C
s
˜ 0.01-0.05 for engines, lower for smoother oper ation.
Flywheel Design
• Energy Req uirement :
?E =
?
(T
load
-T
engine
)d?
whereT
load
,T
engine
= load and engine torques,? = angular displacement.
• Moment o f Inertia from Energy :
I =
?E
C
s
?
2
avg
• Rim Stre ss (Centrifugal Stress) :
s =?R
2
?
2
where? = material density ,s = hoop stress in the rim.
Power and T orque
• Power :
P =T?
whereP = power (W),T = torque (N·m),? = angular velocity (r ad/s).
• T orque Fluct uation :
T =Ia
wherea =
d?
dt
= angular acceler ation.
2
Page 3


F ormula Sheet: Flywheel
Introduction to Flywheels
• Definition : A flywheel is a rotating mechanical device used to store rota-
tional energy and s mooth out fluctuations in speed or torque in machines
lik e engines an d presses.
• Applications : Internal combustion engines, punch presses, and power gen-
er ation systems .
• K ey Function : Reduces speed fluctuations b y storing and releasing kinetic
energy .
Energy Stor age in Flywheel
• Kinetic Ener gy :
E =
1
2
I?
2
whereE = kinetic energy (J),I = mass moment of inertia (kg m
2
),? = angular
velocity (r ad/s).
• Energy Fluct uation :
?E =E
max
-E
min
=
1
2
I(?
2
max
-?
2
min
)
where?
max
,?
min
= maximum and minimum angu lar velocities.
• Energy Fluct uation (Simplified) :
?E =I?
2
avg
C
s
where?
avg
=
? max+?
min
2
,C
s
= coefficient of speed fluctuation.
Mass Moment of Inertia
• Solid Disk Fly wheel :
I =
1
2
mR
2
wherem = mass,R = r adius.
• Rim-Type Fly wheel (Thin Ring Approximation) :
I =mR
2
whereR = mean r adius of the rim.
• Gener al F orm :
I =
?
r
2
dm
wherer = r adial distanc e,dm = differential mass.
1
Coefficient of Speed Fluctuation
• Definition :
C
s
=
?
max
-?
min
?
avg
or
C
s
=
N
max
-N
min
N
avg
whereN
max
,N
min
,N
avg
= maximum, minimum, and aver age rotational speeds
(RPM).
• Typical V alues : C
s
˜ 0.01-0.05 for engines, lower for smoother oper ation.
Flywheel Design
• Energy Req uirement :
?E =
?
(T
load
-T
engine
)d?
whereT
load
,T
engine
= load and engine torques,? = angular displacement.
• Moment o f Inertia from Energy :
I =
?E
C
s
?
2
avg
• Rim Stre ss (Centrifugal Stress) :
s =?R
2
?
2
where? = material density ,s = hoop stress in the rim.
Power and T orque
• Power :
P =T?
whereP = power (W),T = torque (N·m),? = angular velocity (r ad/s).
• T orque Fluct uation :
T =Ia
wherea =
d?
dt
= angular acceler ation.
2
Applications
• Used in engin es to smooth out torque variations during the power cycle.
• Essential in punch presses to provide energy for sudden load demands.
• Critical for GA TE problems on flywheel sizing, energy stor age, and speed
fluctuation.
3