1. Basic Principles and Engineering Characteristics of Belt Drives

A belt drive is a representative flexible transmitter that transmits power by wrapping a flexible belt around two or more pulleys, utilizing friction or meshing force. Compared to gear drives, it can transmit power economically over longer center distances, offers excellent shock absorption, and operates relatively quietly.

• Flat Belt: Suitable for high-speed operation and offers good flexibility, but is highly prone to slip.
• V-Belt: The most widely used for industrial power transmission, as it can transmit relatively large torque without slipping due to increased friction from the wedge action within the pulley groove.
• Timing Belt (Synchronous Belt): Features molded teeth like a gear, enabling synchronous drive with 0% slip.

2. Mathematical Derivation Formulas for Belt Length and Lap Angle

The geometric dimensions and mechanical relational equations that form the basis of belt drive design are derived as follows.

① Open Belt Length Calculation Formula: A standard approximation formula obtained by integrating the geometric trajectory, where D1 and D2 are the diameters of the two pulleys, and C is the center distance.

L ≈ 2C + π(D1 + D2)/2 + (D2 - D1)^2 / (4C)  [mm]

② Lap Angle (Angle of Contact, θ): One of the most critical factors determining power transmission capacity. The smaller the lap angle θ1 on the driver pulley, the more likely slipping is to occur.

θ1 = π - 2 × sin^(-1)((D2 - D1) / 2C)  [rad]

θ2 = π + 2 × sin^(-1)((D2 - D1) / 2C)  [rad]

③ Belt Pitch Line Speed (v): The linear speed of the power-transmitting belt, with the formula based on the driver pulley defined as follows:

v = (π × D1 × N1) / 60,000  [m/s]

3. Euler's Formula for Friction Drives (Slip and Tension Design)

When a belt wraps around a pulley and rotates, the ratio of the tight side tension (T_t) to the slack side tension (T_s) just before slipping is exponentially determined by the coefficient of friction (μ) between the belt and pulley, and the lap angle (θ). This is known as **Euler's Belt Formula**.

T_t / T_s = e^(μ × θ)

This equation demonstrates that as the lap angle θ increases, the tension ratio increases exponentially, allowing for the transmission of a greater effective tension (F_e = T_t - T_s). Therefore, in designs with a large reduction ratio where the lap angle of the smaller pulley drops below 120°, corrective design measures such as installing an idler pulley to forcibly secure the contact angle on the slack side become essential.