Question 3.3.5: Belt drives and chain drives, like those used on bicycles, h......
Belt drives and chain drives, like those used on bicycles, have similar characteristics and can be analyzed in a similar way. A belt drive is shown in Figure 3.3.4. The input shaft (shaft 1) is connected to a device (such as a bicycle crank) that produces a torque T_{1} at a speed ω_{1}, and drives the output shaft (shaft 2). The mean sprocket radii are {r}_{1} and {r}_{2}, and their inertias are I_{1} and I_{2}. The belt mass is m.
Derive the expression for the equivalent inertia I_{e} felt on the input shaft.
The kinetic energy of the system is
\mathrm{KE}={\frac{1}{2}}I_{1}\omega_{1}^{2}+{\frac{1}{2}}I_{2}\omega_{2}^{2}+{\frac{1}{2}}m v^{2}If the belt does not stretch, the translational speed of the belt is v=r_{1}\omega_{1}=r_{2}\omega_{2}. Thus we can express KE as
{\mathrm{KE}}={\frac{1}{2}}I_{1}\omega_{1}^{2}+{\frac{1}{2}}I_{2}{\bigg(}{\frac{r_{1}\omega_{1}}{r_{2}}}{\bigg)}^{2}+{\frac{1}{2}}m{\big(}r_{1}\omega_{1}{\big)}^{2}={\frac{1}{2}}{\bigg[}I_{1}+I_{2}{\bigg(}{\frac{r_{1}}{r_{2}}}{\bigg)}^{2}+m r_{1}^{2}{\bigg]}\omega_{1}^{2}Therefore, the equivalent inertia felt on the input shaft is
I_{e}=I_{1}+I_{2}\left({\frac{r_{1}}{r_{2}}}\right)^{2}+m r_{1}^{2} (1)
This means that the dynamics of the system can be described by the model I_{e}{\dot{\omega}}_{1}=T_{1}.