Thomas Industrial Library

SIZING THE FLYWHEEL We now must determine how large a flywheel is needed to absorb this energy with an acceptable change in speed. The change in shaft speed during a cycle is called its fluctuation (Fl) and is equal to:

We can normalize this to a dimensionless ratio by dividing it by the average shaft speed. This ratio is called the coefficient of fluctuation (k) .

This coefficient of fluctuation is a design parameter to be chosen by the designer. It typically is set to a value between 0.01 and 0.05, which correspond to a 1 to 5% fluctuation in shaft speed. The smaller this chosen value, the larger the flywheel will have to be. This presents a design trade-off. A larger flywheel will add more cost and weight to the system, which factors have to be weighed against the smoothness of operation desired.

We found the required change in energy E by integrating the torque curve

and can now set it equal to the right side of equation 9.14c (p. 254):

Factoring this expression:

If the torque-time function were a pure harmonic, then its average value could be expressed exactly as:

Our torque functions will seldom be pure harmonics, but the error introduced by using this expression as an approximation of the average is acceptably small. We can now substitute equations 9.15b and 9.17 into equation 9.16c to get an expression for the mass moment of inertia Is of the system flywheel needed.

Equation 9.18 can be used to design the physical flywheel by choosing a desired coefficient of fluctuation k , and using the value of E from the numerical integration of the torque curve (see Table 9-3) and the average shaft to compute the needed system Is. The physical flywheel’s mass moment of inertia If is then set equal to the required system Is. But if the moments of inertia of the other rotating elements on the same driveshaft (such as the motor) are known, the physical flywheel’s required If can be reduced by those amounts.

The most efficient flywheel design in terms of maximizing If for minimum material used is one in which the mass is concentrated in its rim and its hub is supported on spokes, like a carriage wheel. This puts the majority of the mass at the largest radius possible and minimizes the weight for a given If . Even if a flat, solid circular disk flywheel design is chosen, either for simplicity of manufacture or to obtain a flat surface for other functions (such as an automobile clutch), the design should be done with an eye to reducing weight and thus cost. Since in general I = mr 2, a thin disk of large diameter will need fewer pounds of material to obtain a given I than will a thicker disk of smaller diameter. Dense materials such as cast iron and steel are the obvious choices for a flywheel. Aluminum is not used. Though many metals (lead, gold, silver, platinum) are more dense than iron and steel, their high cost prohibits use for a flywheel.

Figure 9-26 shows the change in the input torque T 12 for the cam-follower system of Figure 9-25 after the addition of a flywheel sized to provide a coefficient of fluctuation of 0.05. The oscillation in torque about the unchanged average value is now 5%, much less than what it was without the flywheel. The angular velocity of the shaft will also vary 5% as the flywheel must change velocity to deliver and absorb energy as can be seen in equation 9.16b. A much lower power (non-servo) motor can now be used because the flywheel is available to absorb the energy returned from the follower during its cycle, though the added inertia of the flywheel will require sufficient motor torque to accelerate it up to speed in a reasonably short time.