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small observation on a problem in the textbook

Nasser M. Abbasi, Feb 2011

January 13, 2016
The following question relates to the derivation of the linearized equation of motion of the center of the cart as given by equation 1.18 in the textbook.

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   ′   2
[𝜃 (t)]   . 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

pict

Eliminate F  from the above, we obtain the following equation

mby ′′ + I 𝜃′′cos𝜃 =  0
         z
(1)

Now, referring to equation 1.16 in the textbook, which is

 ′         ′
y cos𝜃 − b𝜃 − U  sin 𝜃 = 0
(1.16)

Solve for y′ in the above gives

 ′   (b𝜃′ +-U-sin𝜃)
y =      cos 𝜃

Differentiating the above once w.r.t time, gives

            (                              )
y ′′ = --1--- b cos𝜃𝜃′′ + bsin𝜃 [𝜃 ′(t)]2 + U 𝜃′
      cos2𝜃

Substituting y′′ found above into eq. (1) gives

         (                              )
mb --1--- b cos𝜃𝜃′′ + bsin𝜃 [𝜃 ′(t)]2 + U 𝜃′ + Iz𝜃′′cos 𝜃 = 0
   cos2𝜃

Now we can linearize by assuming 𝜃 ≪ 1  , hence cos𝜃 →  1  , sin𝜃 →  𝜃  in the above, which gives

pict

Comparing the above equation to the one given in the textbook, which is 1.18

  (         )
𝜃′′ mb2  + Iz +  mbU  𝜃′ = 0
(1.18)

We see that eq. (2) has an extra term    2   ′   2
mb  𝜃 [𝜃 (t)]   which can be ignored only when  ′
𝜃 (t) ≪ 1