Derivatives of BP(t)
Problem 16.151
The gear has the angular motion shown. Omega and
BP(t) denotes the
variable distance BP(t)
The numbers used here for BP deviate from the problem statement.
Here is a general analysis, valid for any angle Theta. The time derivatives, cumbersome to derive by hand,
are computed with ease using the Mathcad symbolics features. See work at left.
The geometry (variable length AB = L(t),
angle of swing arm BC = phi) is found as:
Time derivatives of phi(t):
position cursor at 't' in the
expression at right, select
Variable --> Differentiate
in the symbolics menu.
Textbook illustration
Symbolics--> Simplify removes the double fraction. We get for p_dot(t):
Second Deriv, raw
Second Deriv. of phi, cleaned up
We now differentiate phi(t) symbolically and obtain angular vel. and accel of arm AB.
The results are shown below as p_dot(t) and p_ddot(t), respectively.