Writing Equations of Motion for an Airplane Model

[Whitebread]

Homework Statement

I was tasked with finding the equations of motion for an the airplane model pictured below. Mass 1 and Mass 3 represent the wings. K1 and K2 are linear springs that represent the stiffness of the wings. M2 is the fuselage and M4 is the landing gear. K3 is the linear spring that represents the stiffness of the landing gear structures. The circle marked landing gear is not the landing gear. It is actually the ground. And the linear spring between them, K4, is a representation of the stiffness of the tires. This was a mistake. My apologies.

Homework Equations

N/A

The Attempt at a Solution

Here is my solution:

My professor marked them wrong, but did not say why. Can one of you enlighten me please? This is all.

Thank you.

 

[Filip Larsen]

You should probably explain what your state variables represent. I assume they are the (initial) vertical displacement from the "unloaded" configuration of each element, but your diagram suggests that they are heights.

Depending on the assumptions given in the problem text (I can't see if the basic structure was given, or you made that too), your model should probably also include the effect of gravity, aerodynamic lift, wheel/ground separation, and spring dampening. For modeling realistic dynamics of an airplane structure during landing most of these aspects would be essential to include.

Also, it is also common to model simple airplane structures as symmetric in the vertical plane, which in your case will result in one less state variable.

 

[Whitebread]

You should probably explain what your state variables represent. I assume they are the (initial) vertical displacement from the "unloaded" configuration of each element, but your diagram suggests that they are heights.

Depending on the assumptions given in the problem text (I can't see if the basic structure was given, or you made that too), your model should probably also include the effect of gravity, aerodynamic lift, wheel/ground separation, and spring dampening. For modeling realistic dynamics of an airplane structure during landing most of these aspects would be essential to include.

Also, it is also common to model simple airplane structures as symmetric in the vertical plane, which in your case will result in one less state variable.

The question as presented to us was extremely vague. It did not clarify if the state variables were initial displacements, or if they were relative displacements from Y=0. I took them to be displacements from their rest positions and wrote the state equations with that assumption. Sorry for not including that in my initial post. I forgot I even had to make that distinction.

I do agree that the factors you mentioned should be included, but the point of this question was actually to perform modal analysis on a lumped parameter mass-spring system and not to accurately model an airplane. I did not include that information in order to simplify this thread. Respondents need not know what the actual goal is as it has nothing to do with writing the equations of motion correctly (at least in my limited experience).

As for the symmetry about the vertical plane, I do believe modal analysis would make the fourth state variable necessary, no? I do not see how one can account for modes with asymmetric wing movement with only 3 variables. Am I right, or wrong here?

Thank you.

 

[Filip Larsen]

... the point of this question was actually to perform modal analysis on a lumped parameter mass-spring system and not to accurately model an airplane. I did not include that information in order to simplify this thread. Respondents need not know what the actual goal is as it has nothing to do with writing the equations of motion correctly (at least in my limited experience).

As you hopefully know, there is not a single "correct" model that can be used for all purposes, especially not if the model has to be as simple as possible. The purpose and and any simplifying assumptions are in fact very decisive for how the model will end up, so such information is vital if you want relevant comments to your model. As it were, my comments ended up being more or less irrelevant for you because I assumed a different purpose than you had.

Since its all so vague and "secret" as it is, I suggest that you simply discuss it with your professor (he should know) or one of your co-students.

As for the symmetry about the vertical plane, I do believe modal analysis would make the fourth state variable necessary, no? I do not see how one can account for modes with asymmetric wing movement with only 3 variables. Am I right, or wrong here?

You are correct that you cannot model asymmetric dynamics using a symmetric model, so if that is one of your goals then you should of course have a model that allow asymmetries. Again, its all about purpose and assumptions.

 

[Whitebread]

As you hopefully know, there is not a single "correct" model that can be used for all purposes, especially not if the model has to be as simple as possible. The purpose and and any simplifying assumptions are in fact very decisive for how the model will end up, so such information is vital if you want relevant comments to your model. As it were, my comments ended up being more or less irrelevant for you because I assumed a different purpose than you had.

Since its all so vague and "secret" as it is, I suggest that you simply discuss it with your professor (he should know) or one of your co-students.You are correct that you cannot model asymmetric dynamics using a symmetric model, so if that is one of your goals then you should of course have a model that allow asymmetries. Again, its all about purpose and assumptions.

I do, indeed, know this. Although, I have no experience building airplane models. My only experience is with cars. The incite is interesting. Thanks!

I would discuss this with my professor and/or one of my peers, but I'm taking an internet class and not one single peer responds to class forum posts. The professor is also difficult to communicate with. He will mark incorrect responses incorrect, but never gives an explanation. I've actually asked him about this twice already, via email, and each time he has not provided the correct equations, but taken up some other, peripheral, issue. I'd like for him to provide me with the correct equations so that I may sharpen my technique, but I have yet to see them.

The question as presented to the class is vague, but nothing here is "secret". This isn't anything special. I just simplified the question as presented to the physics forums so that I could reduce the amount of time individuals spent looking over this thread. This is all. Modal analysis takes quite a few steps, and so I decided not to bother individuals with all the subsequent steps when I'm only concerned with the first. That was my intention. I am not trying to be patronizing, I do not take anyone here to be obtuse, and I do not mean to come off as arrogant. I'm just looking for some help with these equations of motion so that I may improve myself. That is all.