Velocity

Handle Velocity

Input angular velocity ω₂ changes depending on the condition (Figure 11). 

Figure 11: Input angular velocity ω₂ as a function of time for the constant condition (left) and variable condition (right).

The velocity of the handle itself can be calculated by taking the time derivative of its position for each condition (Figure 12):

Figure 12: The magnitude and angle of the handle velocity as a function of input angle θ₂ for the constant (left) and variable (right) conditions.

Cam Velocity

The angular velocity of the cam disk is the same as the input angular velocity because it spins on the same shaft. Therefore, ω₃ = ω₂. Because we have a time-history of the cam's linear position (see Position), the linear velocity of the rod can be found by taking the time-derivative of its position (Figure 13):

Figure 13: Linear horizontal velocity of the cam as a function of input angle θ₂, for the constant condition (left) and variable condition (right).

Gear Velocity

Input gear 4 rotates on the input shaft, and therefore ω₄ = ω₂. To find the angular velocity of the large gear that turns the wheels, ω₅, use the gear ratio: ω₅/ω₄ = N₄/N₅ (Figure 14).

Figure 14: The angular velocity of the intermediate gear (ω₅) that turns the bicycle wheels as a function of input angle θ₂ for the constant (left) and variable (right) conditions (left).

Figure 15: Even in the variable condition, ω₅ changes with ω₂ at a constant rate, which is the gear ratio.

Four-bar Linkage Velocity

Velocity analysis of the four-bar linkage can be done by taking the time-derivative of the position vector loop:

The input angular velocity ω₆ is the same as ω₅, and therefore can be used to solve for ω_lleg and ω_uleg (Figure 16).

Figure 16: Angular velocities of the leg segments with respect to input angle θ₂ for the constant (left) and variable (right) conditions.

The velocity of the foot (Figure 17) and knee (Figure 18) can be found by taking the time derivative of their position vectors:

and  

Figure 17: Magnitude and angle of the foot velocity as a function of input angle θ₂ for the constant (left) and variable (right) conditions.

Figure 18: Magnitude and angle of the knee velocity as a function of input angle θ₂ for the constant (left) and variable (right) conditions.