Understanding Speed Ratio in a Gear Train

A gear train is a system formed by meshing two or more gears to transmit power or motion from one shaft to another. In a gear train, the first gear that receives the input motion is called the driver, and the last gear that delivers the output motion is called the driven or follower.

Defining Speed Ratio

The speed ratio of a gear train is a fundamental characteristic that tells us how much the speed of rotation changes from the input shaft to the output shaft. It is conventionally defined as the ratio of the angular velocity (or speed) of the first driver to the angular velocity (or speed) of the last driven gear. Let:

• $\mathop N\nolimits_1$ be the rotational speed (in revolutions per minute or radians per second) of the first wheel (driver).
• $\mathop N\nolimits_n$ be the rotational speed (in revolutions per minute or radians per second) of the $n$-th wheel (the last driven wheel) in a train of $n$ wheels.

The speed ratio is given by the formula: $$ \text{Speed Ratio} = \frac{\text{Speed of the first driver}}{\text{Speed of the last driven}} = \frac{{\mathop N\nolimits_1 }}{{\mathop N\nolimits_n }} $$

Analyzing the Given Options

Let's look at the provided options in the context of the definition of speed ratio for a gear train with $n$ wheels:

1. \(\frac{{\mathop N\nolimits_1 }}{{\mathop N\nolimits_n }}\): This option represents the ratio of the speed of the first wheel ($\mathop N\nolimits_1$) to the speed of the $n$-th (last) wheel ($\mathop N\nolimits_n$). This aligns perfectly with the standard definition of the speed ratio in a gear train.
2. \(\frac{{\mathop N\nolimits_n }}{{\mathop N\nolimits_1 }}\): This represents the ratio of the speed of the last wheel ($\mathop N\nolimits_n$) to the speed of the first wheel ($\mathop N\nolimits_1$). This is the inverse of the speed ratio and is often referred to as the train value or gear ratio, depending on the convention used. However, the speed ratio is defined as input speed over output speed.
3. \(\frac{{\mathop T\nolimits_1 }}{{\mathop T\nolimits_n }}\): This represents the ratio of the number of teeth on the first wheel ($\mathop T\nolimits_1$) to the number of teeth on the last wheel ($\mathop T\nolimits_n$). While the speed ratio is related to the number of teeth (inversely proportional for simple gear trains), the speed ratio itself is defined in terms of speeds, not teeth counts directly.
4. \(\frac{{\mathop \omega\nolimits_n }}{{\mathop \omega\nolimits_1 }}\): This uses angular velocity ($\omega$) instead of rotational speed ($\mathop N\nolimits_1$ or $\mathop N\nolimits_n$), which is acceptable as they are proportional. However, this option shows the ratio of the angular velocity of the last wheel ($\mathop \omega\nolimits_n$) to the angular velocity of the first wheel ($\mathop \omega\nolimits_1$). This is equivalent to option 2 ($\frac{{\mathop N\nolimits_n }}{{\mathop N\nolimits_1 }}$) and represents the inverse of the speed ratio.

Based on the standard definition, the speed ratio of a gear train is the ratio of the speed of the first driver to the speed of the last driven wheel. Therefore, the correct representation is $\frac{{\mathop N\nolimits_1 }}{{\mathop N\nolimits_n }}$.