Velocity Ratio of Pulleys Explained

In a belt drive system connecting two pulleys, the velocity ratio is an important concept. It relates the speed of the driving pulley to the speed of the driven pulley.

The velocity ratio is defined as the ratio of the speed of the driven pulley to the speed of the driving pulley. Mathematically, if $\text{N}_1$ is the speed (in revolutions per minute, RPM) of the driving pulley and $\text{N}_2$ is the speed of the driven pulley, the velocity ratio (VR) is:

$\text{VR} = \frac{\text{N}_2}{\text{N}_1}$

Assuming there is no slip between the belt and the pulleys, the linear speed of the belt is the same for both pulleys. The linear speed of a point on the circumference of a pulley is given by the product of its angular velocity and radius, or $\pi \times \text{Diameter} \times \text{Speed (in RPM)}$.

Let $\text{D}_1$ be the diameter of the driving pulley and $\text{D}_2$ be the diameter of the driven pulley.

Linear speed of belt at driving pulley = $\pi \times \text{D}_1 \times \text{N}_1$

Linear speed of belt at driven pulley = $\pi \times \text{D}_2 \times \text{N}_2$

Since the linear speed is the same (assuming no slip):

$\pi \times \text{D}_1 \times \text{N}_1 = \pi \times \text{D}_2 \times \text{N}_2$

We can cancel $\pi$ from both sides:

$\text{D}_1 \times \text{N}_1 = \text{D}_2 \times \text{N}_2$

Rearranging the equation to find the velocity ratio $\frac{\text{N}_2}{\text{N}_1}$:

$\frac{\text{N}_2}{\text{N}_1} = \frac{\text{D}_1}{\text{D}_2}$

So, the velocity ratio is the ratio of the diameter of the driving pulley to the diameter of the driven pulley. This formula shows that the velocity ratio is inversely proportional to the diameters of the pulleys.

If the diameter of the driven pulley ($\text{D}_2$) is larger than the driving pulley ($\text{D}_1$), the speed of the driven pulley ($\text{N}_2$) will be lower than the driving pulley ($\text{N}_1$), resulting in a velocity ratio less than 1. Conversely, if the driven pulley's diameter is smaller, its speed will be higher, and the velocity ratio will be greater than 1.

Relationship to Diameters

Based on the derivation $\frac{\text{N}_2}{\text{N}_1} = \frac{\text{D}_1}{\text{D}_2}$, we can state the relationship between the velocity ratio and the pulley diameters:

• The velocity ratio ($\frac{\text{N}_2}{\text{N}_1}$) is inversely proportional to the diameter of the driven pulley ($\text{D}_2$).
• The velocity ratio ($\frac{\text{N}_2}{\text{N}_1}$) is directly proportional to the diameter of the driving pulley ($\text{D}_1$).

However, the question asks about the relationship of the velocity ratio of two pulleys to their diameters. The formula $\frac{\text{N}_2}{\text{N}_1} = \frac{\text{D}_1}{\text{D}_2}$ can be rewritten as $\text{N}_2 \propto \frac{1}{\text{D}_2}$ (keeping $\text{N}_1$ and $\text{D}_1$ constant) or $\text{N}_1 \propto \frac{1}{\text{D}_1}$ (keeping $\text{N}_2$ and $\text{D}_2$ constant). More importantly, the ratio $\frac{\text{N}_2}{\text{N}_1}$ depends on the ratio of diameters $\frac{\text{D}_1}{\text{D}_2}$. This means the ratio of speeds is inversely related to the ratio of diameters ($\frac{\text{N}_2}{\text{N}_1}$ is proportional to $\frac{1}{\text{D}_2}$ and $\frac{1}{\text{D}_1}$ in an inverse relationship sense when comparing the driven and driver speeds respectively to their diameters). Therefore, the velocity ratio is inversely proportional to their diameters when considering the speed of one pulley relative to the diameter of the other, and vice versa, such that their product is constant ($\text{N} \times \text{D} = \text{constant}$).

So, the velocity ratio of two pulleys is inversely proportional to their diameters.