"Ready to Fly" Kinematic Analysis

In my kinematic analysis I will calculate the penguin body position, velocity and acceleration. Additionally, I will calculate the wing angle, velocity of the wing tip, and acceleration of the wing tip. In my analysis I assume that the input angular velocity (w_in) is a constant 6 rad/s. After characterizing the radius as a function of input angle, I found the position of the base of the penguin head. This was simply calculated by adding the length of the follower(connecting the base of the head to the cam) to the radius at each angle. After calculating the position, I moved onto the penguin body velocity (V_body). This was simple, as velocity is a derivative of position. After finding the velocity of the body, I calculated the body acceleration (A_body) by taking the derivative of the velocity. Thus, to find V_body  and A_body I used the following equation:

V_body=(dr/dΘ)*w_in 

_Abody=(dV_body/dΘ)*w_in 

After calculating the position, velocity and acceleration for the penguin body, I moved onto analyzing the penguin wing. The following diagram will be helpful in this analysis.

Figure. Wing-Body Geometry

When the body of the penguin moves up, the wing at the point where it intersects the body has the same velocity and acceleration as the body. Then by simple geometry we can find the angle of the wing as a function of the position of the body. 

Θ_wing=atan((r-y₀)/a)

Then after finding the wing angle, the velocity and acceleration tangent to the wing can be found by more geometric relations.

V_tan,wing=V_body*cos(Θ_wing)

_Atan,wing=A_body*cos(Θ_wing)

After the tangent components are found, the angular velocity and accelerations will be found by the following equations.

w_wing=V_tan,wing/l=V_tan,wing*cos(Θ_wing)/a

_alphawing=A_tan,wing/l=A_tan,wing*cos(Θ_wing)/a

Lastly, the magnitude of the velocity and acceleration of the wing tip can be found by the following relations.

V_wing=w_wing*l_wing

_Awing=alpha_wing*l_wing