Complex Analysis: Solve Last Question in Section 1.3

[skilambi]

Hi All,

I am trying to learn complex analysis on my own and for this I have chosen Fundamentals of Complex Analysis by Saff and Snider. I am stuck at the last question in section 1.3 which is as follows.

For the linkage illustrated in the figure, use complex variables to outline a scheme for expressing the angular position, velocity and acceleration of arm c in terms of those of arm a.

As an attempt to the solution, this is what I have thought of so far.

Let the arm a be dictated by the complex variable Za, similarly Zb for b and Zc for c. Also since the distance between the bottom of a and c is fixed (a + b - c), we can say

Za + Zb = Zc + (a+b-c). (Parallelogram law)

However I am not sure of what comes next as I am not sure what Zb is. How can i proceed? To express Zc with respect to Za, I will somehow need to know what Zb is in terms of Za. But how do I do that?

The figure is in this thread

https://www.physicsforums.com/showthread.php?t=547863

SMK.

 

[lippyka]

Solution:The angular position of each arm can be expressed in terms of the complex variable Za. Since arm a and b are connected, we can use the law of cosines to express Zb as a function of Za: Zb = Za + 2ab cos(theta) where theta is the angle between arm a and b. Then, using the parallelogram law, we can express Zc in terms of Za and theta: Zc = Za + 2ab cos(theta) + (a+b-c) For the velocity and acceleration of arm c, we can take the derivatives of the above equation with respect to time. This will give us an expression for the velocity and acceleration of arm c in terms of the velocity and acceleration of arm a.