# Flywheel design

**Flywheel design**

**(a) DESIGN OF RIM**

**Stresses in a rim**

- Tensile stress due to centrifugal force

σ_{t} = ρv^{2}

- Tensile stresses due to bending of arms

σ_{b} = M/Z

**BENDING MOMENT**

M = wL^{2}/12

w = UDL =Centrifugal force between two arms per unit length

w = mRω^{2 }=ρbtLR ω^{2}/L

w = mRω^{2 }=ρbtR ω^{2}

L = length between two arms

= Behaves as a fixed beam

L =2πR/n

**NUMBER OF ARMS**

n = number of arms (normally 4,6 and8)

M = wL^{2}/12= ρbtLR ω^{2}L^{2}/12

M = (ρbtLR ω^{2}/12)( 2πR/n)^{2}

M = π^{2}ρbt v^{2}R/3n^{2}

Z = bt^{2}/6

σ_{b} = M/Z =2 π^{2}ρv^{2}R/n^{2} t

NOTE: Arms stretch by ¾ of free expansion due to the centrifugal force

Total stress in the rim

σ_{total} = (3/4) σ_{t} + σ_{b}/4

σ_{total} = (3/4) ρv^{2} + π^{2}ρv^{2}R/2n^{2} t

σ_{total} = ρv^{2}(0.75 + 4.935R/n^{2} t )

For a safe design for a cast iron flywheel, total stress should be less than the allowable stress of 38 MPa.

CHECK σ_{total} <38 MPa.

- Shrinkage stresses due to uneven rate of of cooling of the casting, it is difficult to calculate but can be accounted for a certain factor say 10 %
**.**

**(b) DESIGN OF ARMS**

**Stresses in arms**

- Tensile stress due to centrifugal force acting on the rim
- Bending stresses due to torque transmitted from the rim to the shaft
- Shrinkage stresses—Cannot be determined. To account for this, 10% stress is added.

Tensile stress due to centrifugal force σ_{t} = 0.75 ρv^{2}

**BENDING MOMENT**

M is due to the torque ‘T’ transmitted, M = = T(R-r)/nR

Where n is the number of arms

σ_{bt1} =Due to torque and hence due to M,

σ_{bt1 }=M/Z

σ_{bt} _{2}=Due to belt tensions =(F_{1}—F_{2})(R–r)/(n/2)Z

Only half the arms are considered for transmitting power.

σ_{total} = σ_{t} +σ_{bt1 + } σ_{bt2}

CHECK σ_{total} < 38 MPa

The cross section of the arms is normally elliptical.

Let a = Major axis near the hub

B= Minor axis near the hub

Z_{y} = section modulus = πba^{2}/32

Taking b = a/2

a= (64Z/π)^{1/3}

Dimension near the rim are taken 2/3 rd of dimension near the hub**.**