Transmission of Rotational Motion

Rotational-motion transmissions transfer motion from an input member to an output member. In kinematics, the main objective is to relate angular velocities, rotational frequencies, and geometric parameters of the members.

No-slip condition

For two wheels in contact or connected by a transmission without slipping, the magnitudes of the corresponding contact-point velocities are equal:

$$\omega_1 r_1=\omega_2 r_2.$$

Thus the transmission ratio by magnitude may be written:

$$i=\frac{\omega_1}{\omega_2}=\frac{r_2}{r_1}.$$

For gears, pitch-circle radii are proportional to tooth numbers, so $i=z_2/z_1$ using the same convention for input and output members.

Equality of tangential velocities at contact is the basis for analyzing simple friction, belt, and gear transmissions under the ideal no-slip assumption.

Friction wheels

For two externally contacting wheels without slip, $|\omega_1|r_1=|\omega_2|r_2$. They rotate in opposite directions. With internal contact, their rotation directions are the same.

Belt drives

For an open belt drive without slip, belt speed is the same at both pulleys, so $\omega_1r_1=\omega_2r_2$ in magnitude. An open belt gives the pulleys the same rotation direction; a crossed belt gives opposite directions.

Gear drives

For an external gear pair, $|\omega_1|/|\omega_2|=z_2/z_1$, where $z_1$ and $z_2$ are tooth numbers. External meshing reverses rotation direction, while internal meshing preserves it.

Transmission ratio

A transmission ratio must always be interpreted with its definition. Here $i=\omega_1/\omega_2$, where member 1 is the input and member 2 is the output. If only the magnitude is required, rotation direction is handled separately.

Multistage transmissions

For successive stages, the overall transmission ratio is the product of the individual stage ratios. Output rotation direction follows from the number of external gear meshes or crossed-belt stages.

Example

An input gear has $z_1=20$ teeth and rotates at $\omega_1=60$ rad/s. The output gear has $z_2=60$. Then $|\omega_2|=60\cdot20/60=20$ rad/s, with opposite direction for external meshing.

Common mistakes

• reversing the radius ratio $r_2/r_1$;
• defining $i$ without stating which member is input;
• ignoring rotation direction in external gear meshing;
• using equal contact speeds when the problem explicitly includes slip;
• considering only one pair in a multistage transmission.