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1 Notations 2
2 Flywheel 3
3 Flywheel Performance 33.1 Coefficient of Fluctuation of Speed Cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 coefficient of steadiness M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 Maximum Fluctuation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 Coefficient of Fluctuation of Energy CE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.4.1 Work Done per Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.5 Energy Stored in a Flywheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Stresses in a Flywheel Rim 44.1 Tensile(Hoop) Stress due to The Centrifugal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Tensile Bending Stress Caused by Restraint of the Arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
5 Stresses in Flywheel Arms 65.1 Tensile Stress due to The Centrifugal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65.2 Bending Stress due to The Torque Transmitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6 Flywheel Design 76.1 Design of Flywheel Arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.2 Design of Shaft, Hub and Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6.2.1 Shaft Diameter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.2.2 Hub Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76.2.3 Key Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
7 Examples 87.1 Flywheel Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87.2 Stresses in a Flywheel Rim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.3 Flywheel Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8 References 20
9 Contacts 20
1 Notations
• N1 = Maximum speed in r.p.m. during the cycle
• N2 = Minimum speed in r.p.m. during the cycle
• N = Mean speed in r.p.m. = N1+N2
2
• w1 = Maximum angular speed in rad. during the cycle
• w2 = Minimum angular speed in rad. during the cycle
• w = Mean angular speed in rad. = w1+w2
2
• v1 = Maximum linear speeds during the cycle
• v2 = Minimum linear speeds during the cycle
• v = Mean linear speeds = v1+v22
• Cs = Coefficient of fluctuation of speed
• CE = Coefficient of fluctuation of energy
• Tmean = Mean torque
• θ = Angle turned in radians per revolution = 2π, in case of steam engines and two stroke internal combustion engines.= 4π, in case of four stroke internal combustion engines
• P = Power transmitted in watts
• m = Mass of the flywheel in kg
• k = Radius of gyration of the flywheel in meters
• I = Mass moment of inertia of the flywheel about the axis of rotation in kg −m2 = m.k2
• b = Width of the rim
• t = Thickness of the rim
• A = Cross-section area of the rim
• ρ = Flywheel material density
• D = Mean diameter of flywheel
• R = Mean radius of flywheel
• σt = Tensile or hoop stress
• l = Length between fixed ends of arms
• Z = section modulus
• σ = Total stress
• σt1 = Tensile stress in the arms
• T = Maximum torque transmitted by the shaft
• r = Radius of the hub
• n = Number of arms
• σb1 = Bending stress in arms
• σb2 = Bending stress due to the belt tensions if a flywheel used as a belt pulley
• a1 = Major axis of flywheel arm
• b1 = Minor axis of flywheel arm
• d1 = Inner diameter of hub or diameter of shaft (Diameter of the shaft)
• d = Outer diameter of hub
• τ = Allowable shear stress for the material
• L = Length of the key
2 Flywheel
A flywheel used in machines serves as a reservoir which stores energy during the period when the supply of energy is more thanthe requirement and releases it during the period when the requirement of energy is more than supply.A little consideration will show that when the flywheel absorbs energy, its speed increases and when it releases, the speed decreases.Hence a flywheel does not maintain a constant speed, it simply reduces the fluctuation of speed.
Figure 1: Flywheel.
3 Flywheel Performance
3.1 Coefficient of Fluctuation of Speed Cs
The ratio of the maximum fluctuation of speed to the mean speed.
Cs =N1 −N2
N=
2 (N1 −N2)
N1 +N2
=w1 − w2
w=
2 (w1 − w2)
w1 + w2
=v1 − v2v
=2 (v1 − v2)
v1 + v2
3.2 coefficient of steadiness M
M =1
Cs=
N
N1 −N2
=w
w1 − w2
=v
v1 − v2
3.3 Maximum Fluctuation of Energy
Figure 2: Turning moment diagram for a multi-cylinder engine.
Let the energy in the flywheel at A = EEnergy at B = E + a1Energy at C = E + a1 − a2Energy at D = E + a1 − a2 + a3Energy at E = E + a1 − a2 + a3 − a4Energy at F = E + a1 − a2 + a3 − a4 + a5Energy at G = E + a1 − a2 + a3 − a4 + a5 − a6 = Energy at ALet us now suppose that the maximum of these energies is at B and minimum at E.
∴ Maximum energy in the flywheel = E + a1and minimum energy in the flywheel = E + a1 − a2 + a3 − a4∴ Maximum fluctuation of energy (∆E) = Maximum energy Minimum energy
∆E = (E + a1) − (E + a1 − a2 + a3 − a4) = a2 − a3 + a4
3.4 Coefficient of Fluctuation of Energy CE
CE =Maximum fluctuation of energy
Work done per cycle
3.4.1 Work Done per Cycle
1. Work done per cycle = Tmean x θ
Tmean =P x 60
2πN=P
w
2. Work done per cycle = P x 60n
n = Number of working strokes per minute.= N, in case of steam engines and two stroke internal combustion engines.= N/2, in case of four stroke internal combustion engines.
3.5 Energy Stored in a Flywheel
Mean kinetic energy of the flywheel,
E =1
2Iw2 =
1
2mk2w2 (in N-m or joules)
Maximum fluctuation of energy,
∆E = Maximum K.E. Minimum K.E. =1
2Iw2
1 −1
2Iw2
2 =1
2I[w2
1 − w22
]=
1
2I (w1 − w2) (w1 + w2)
= Iw (w1 − w2) = Iw2
[w1 − w2
w
]= Iw2Cs = mk2w2Cs = 2ECs
k = R (The thickness of rim is very small as compared to the diameter of rim)
∆E = mR2w2Cs = mv2Cs
m = 2πRAρA = bt (Assuming the cross-section of the rim to be rectangular)
4 Stresses in a Flywheel Rim
4.1 Tensile(Hoop) Stress due to The Centrifugal Force
Volume of the small element = ARδθdm = ARδθρ = ρARδθ
Vertical component ofdF = dF sin θ = ρAR2w2δθ sin θ
∴ Total vertical bursting force across the rim diameter X-Y, = ρAR2w2
∫ π
0
sin θdθ = 2ρAR2w2
This vertical force is resisted by a force of (2P ) = 2σtA
∴ 2ρAR2w2 = 2σtA
∴ σt = ρR2w2 = ρv2
4.2 Tensile Bending Stress Caused by Restraint of the Arms
l =πD
n=
2πR
nThe uniformly distributed load per meter length(w) = btρw2R [N/m]
Maximum bending moment(M) =wl2
12=btρw2R
12
[2πR
n
]2Section modulus(Z) =
1
6bt2
Bending stress(σb) =M
Z=btρw2R
12
[2πR
n
]26
bt2
=19.74 ρw2R3
n2t=
19.74 ρv2R
n2tTotal stress in the rim(σ) = σt + σb
It has been shown by G. Lanza that the arms of a flywheel stretch about 34
th of the amount necessary for free expansion. Thereforethe total stress in the rim,
=3
4σt +
1
4σb =
3
4ρv2 +
1
4
19.74 ρv2R
n2t= ρv2
(0.75 +
4.935R
n2t
)
5 Stresses in Flywheel Arms
5.1 Tensile Stress due to The Centrifugal Force
Tensile stress in the arms,
σt1 =3
4σt =
3
4ρv2
5.2 Bending Stress due to The Torque Transmitted
The load at the mean radius of the rim(F ) =T
R
∴ Load on each arm =T
Rn
,Maximum bending moment which lies on the arm at the hub(M) =T
Rn(R− r)
∴ Bending stress in arms(σb1) =M
Z=
T
RnZ(R− r)
∴ Total tensile stress in the arms at the hub end(σ) = σt1 + σb1
If the flywheel is used as a belt pulley, then the arms are also subjected to bending due to net belt tension (T1−T2), where T1 andT2 are the tensions in the tight side and slack side of the belt respectively. Therefore the bending stress due to the belt tensions,
σb2 =(T1 − T2)(R− r)
n2Z
Only half the number of arms are considered to be effective in transmitting the belt tensions)∴Total bending stress in the arms at the hub end
σb = σb1 + σb2
and the total tensile stress in the arms at the hub end,
σ = σt1 + σb1 + σb2
6 Flywheel Design
6.1 Design of Flywheel Arms
Section modulus,
Z =π
32b1a
21
Maximum bending moment,
M =T
Rn(R− r)
Maximum bending stress,
σb =M
Z=
T
RnZ(R− r)
NOTES
1. The arms of the flywheel have a taper from the hub to the rim. The taper is about 20 mm per metre length of the arm forthe major axis and 10 mm per metre length for the minor axis.
2. The number of arms are usually 6. Sometimes the arms may be 8, 10 or 12 for very large size flywheels.
3. The arms may be curved or straight. But straight arms are easy to cast and are lighter.
4. Since arms are subjected to reversal of stresses, therefore a minimum factor of safety 8 should be used. In some cases likepunching machines amd machines subjected to severe shock, a factor of safety 15 may be used.
5. The smaller flywheels (less than 600 mm diameter) are not provided with arms. They are made web type with holes in theweb to facilitate handling.
6.2 Design of Shaft, Hub and Key
6.2.1 Shaft Diameter Design
Tmax =π
16τd31
6.2.2 Hub Design
Tmax =π
16τ
(d4 − d41d
)6.2.3 Key Design
Tmax = Lwτd12
7 Examples
7.1 Flywheel Performance
7.2 Stresses in a Flywheel Rim
7.3 Flywheel Design
