Below, I derive this equation of motion, however, I delay the linearization process to the very end. I obtain an ODE which is different from the one given in the textbook. The ODE found has an extra term involving . This term can be approximated to zero only if the angular speed was very small.
Here is my derivation. Starting with equation 1.13 and 1.14
Eliminate from the above, we obtain the following equation
| (1) |
Now, referring to equation 1.16 in the textbook, which is
| (1.16) |
Solve for in the above gives
Differentiating the above once w.r.t time, gives
Substituting found above into eq. (1) gives
Now we can linearize by assuming , hence , in the above, which gives
Comparing the above equation to the one given in the textbook, which is 1.18
| (1.18) |
We see that eq. (2) has an extra term which can be ignored only when