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### Flywheel and Shaft Torque Analysis

Engineering Analysis

**Gear Engineering and Design **

**Machine Design, Equation and Calculators**

**Flywheel and Shaft Torque Analysis**

A flywheel mounted on a relatively stiff shaft is shown in Fig. a. Considering the equilibrium of torques,

eq1

where,

T_{i} = driving or input torque (N-m)

T_{o} = load or output torque (N-m)

I = mass moment of inertia of flywheel (kg-m^{2})

ω = angular velocity of shaft (rad/s) (dθ/dt)

Fig. 1

Torque on Flywheel Shaft

When the driving torque is more than the load torque, the term (T_{i} - T_{o}) is positive and the flywheel is accelerated. When the driving torque is less than the load torque, the term (T_{i} - T_{o}) is negative indicating retardation of flywheel. A T -θ diagram for a particular application is shown in Fig. 2(a). In this case, the power is supplied to the machine by a constant-torque motor, developing a torque Tm. The work cycle of the driven machine consists of three parts—AB, BC and CD. During the elements AB and CD, the torque required by the machine remains constant at T_{1}. At the point B, it increases to T_{2} and remains constant during the element BC. The cycle is repeated for each revolution.

Fig. 2

(a) Torque Diagram, (b) Speed Diagram

During the first element AB, the torque required by the machine is less than that supplied by the motor (T_{1} < T_{m}). The flywheel is, therefore, accelerated and its angular velocity increases to wmax. at the point B. During the element BC, the torque required by the machine is more than the torque supplied by the motor (T_{2} > T_{m}). The flywheel is retarded and its angular velocity decreases to wmin. at point C. This variation in angular velocity is shown in Fig. 2(b).

During elements AB and CD, the energy is supplied to the flywheel. The input energy U_{i} is given by,

eq 2

During the element BC, the energy is taken from the flywheel and the energy output U_{o} from the flywheel is given by,

eq 3

The change in kinetic energy from point B to C is given by

eq 4

or

eq 5

The difference between the maximum and minimum speeds (ω_{max.} – w_{min.}) during a cycle is called the maximum fl uctuation of speed. The ratio of maximum fl uctuation of speed to the mean speed is called coefficient of fluctuation of speed. Therefore, the coefficient of fluctuation of speed, denoted by C_{s} is given by,

eq 6

where ω is the average or mean angular velocity of the flywheel. It is given by,

eq 7

Substituting Eqs (6) and (7) in expression (a),

The values of the coefficients of fluctuation of speed, used in practice are given in Table 1. The coefficient of fluctuation of speed can be expressed in the following ways:

or

Similarly,

or

or

where n is the speed in rpm.

Sometimes, a term called coefficient of ‘steadiness’ is used. The coefficient of steadiness is defined as the reciprocal of the coefficient of fluctuation of speed. It is denoted by the letter m.

Therefore,

Table 1, Coefficients of Fluctuations of Speed

Type of Equipment |
C_{s} |

Punching, shearing and forming presses | 0.200 |

Compressor (belt driven) | 0.120 |

Compressor (gear driven) | 0.020 |

Machine tools | 0.025 |

Reciprocating pumps | 0.040 |

Geared drives | 0.020 |

Internal combustion engines | 0.030 |

D.C. generators (direct drive) | 0.010 |

A.C. generator (direct drive) | 0.005 |

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