Acceleration

Handle Acceleration

In the constant condition, α₂ = 0, but follows a sinusoidal curve in the variable condition (Figure 19).  

Figure 19: The input angular acceleration α₂ over time for the constant (left) and variable (right) conditions.

The handle acceleration (Figure 20) can be found by taking the time-derivative of its velocity:

Figure 20: The magnitude and angle of the acceleration of the handle for the constant (left) and variable (right) input conditions as a function of input angle θ₂.

Cam Acceleration

The horizontal linear acceleration of the cam (Figure 21) can be found by taking the time-derivative of its velocity:

Figure 21: Horizontal linear acceleration of the cam as a function of input angle θ₂ for constant (left) and variable (right) conditions.

Gear Acceleration

The angular acceleration of the intermediate gear can be found by taking the time-derivative of its angular velocity (See Velocity, Figure 22):

Figure 22: Angular acceleration of the intermediate gear that spins the bicycle wheels as a function of time for the constant (left) and variable (right) conditions.

Four-bar Linkage Acceleration

For the leg links, an acceleration analysis can be performed by taking the derivative of the velocity vector loop equation and solving for α_lleg and α_uleg (Figure 23).

Figure 23: Angular acceleration of the upper and lower leg segments as a function of input angle θ₂ for the constant (left) and variable (right) input conditions.

Then, the linear acceleration of the foot (Figure 24) and knee (Figure 25) can be found by taking the time-derivative of their velocity vectors.

 and 

Figure 24: Magnitude and angle of the acceleration of the foot as a function of input angle θ₂ for the constant (left) and variable (right) conditions.

Figure 25: Magnitude and angle of the acceleration of the knee as a function of input angle θ₂ for the constant (left) and variable (right) conditions.