Inverted Pendulum Dynamics Identification

[sandpants]

Homework Statement

I am trying to establish the the dynamics of an inverted pendulum for further analysis. I understand that this is a well known problem, but
the governing equations in various sources are different and make it difficult to keep track of what forces are being applied on the pendulum. I understand there are some assumptions made and hence different models - but in the following source in particular:
http://ctms.engin.umich.edu/CTMS/index.php?example=InvertedPendulum&section=SystemModeling
I am confused and hope someone can clarify the following:

Homework Equations

In the above link the equations I am having trouble with are (2) and (4)

The Attempt at a Solution

In (2) the 2nd and 3rd terms md²x/dt² and mld²θ/dt²cosθ - are these not one and the same? As in the transversal acceleration ld²θ/dt² is related to the horizontal component d²x/dt² by cosθ. If true, why is it counted twice in (2) and if/whether that's an error.

In (4) The last term on the left and the 2 terms on the right; again my intuition is suggesting there are repeating terms:
mld²θ/dt² force vector due to angular acceleration, in transversal direction
mgsinθ - force due to weight
md²x/dt²cosθ - force due to horizontal translation of the point

Aren't the latter 2 just constituents of the first?
if θ =180° mld²θ/dt² = -md²x/dt²cosθ = -md²x/dt²
if θ = 90° then mld²θ/dt² = mgsinθ = mg
Hence shouldn't for any θ
mld²θ/dt² = mgsinθ + md²x/dt²cosθ ?

In the same equation (4) the terms P and N - are these no repeating as well? Especially since the moment is taken about the hinge, why are these even counted?

It also seems like there is a rabbithole going on with the terms md²x/dt²cosθ and mld²θ/dt²cosθ

Sorry, I am just very confused as to which forces are being resolved into what directions and why can they be resolved like that. The last 2 terms just seems like they are taking a cosθ of a cosθ component.

 

[Dr.D]

Rather than try to follow some other person's work at the link, why don't you simply sit down and work the problem for yourself? Then you will not be confused by their notations.

 

[sandpants]

I have; my questions stem from these workings. I would need an external model to confirm my own work. Otherwise I would be blindly do random things and accept them to be true.

 

[OldEngr63]

Otherwise I would be blindly do random things and accept them to be true.

No, you should proceed from knowledge, so that when you have finished your work you can say, "I have reasoned correctly from well established physical principals, therefore, I have confidence that my result is correct."

What will you do for other problems where there is no "external model" to check your work?

 

[sandpants]

In the link posted, in (4) I understand that P and N are reaction forces. But taking the moment around the hinge they shouldn't be counted. All the other forces are taken around the hinge (L dimension is used). Taking it around half that dimension would make it into a couple, P = mg. Hence just taking mlgsinθ should sufficiently account for the forces.

When I resolve the gravitational force I also find the relating transversal acceleration from which I would find the horizontal acceleration of the point mass. But I don't understand how the trig is set up. From geometry you should be able to take mld2θ/dt2=md2x/dt2cosθ, but you can also take mld2θ/dt2cosθ=md2x/dt2, this would switch the hypotenuse and would make either vector smaller. I don't know which one is the correct form.

consulting this link http://www.profjrwhite.com/system_dynamics/sdyn/s7/s7invp1/s7invp1.html I've understood why there are repeating term in (2) of the link in the OP. md2x/dt2 comes from consdering the position of the point mass relative to some arbitrary point, related to the position of the hinge. Whereas my error was taking the position of the point mass only in relation to the hinge.

In the previous link however, fig 7.17 shows a vertical force on the point mass which I do not understand. Why is this force not identical to -mg?
I'm losing track of the forces applied and developed.

Please advise.