Lift Force from Lagrangian Mechanics

[thrillhouse86]

I am having a lot of trouble conceptually understanding this issue in Lagrangian mechanics:

I have an airfoil which is immersed in incompressible flow: it has two degrees of freedom: a rotation: alpha and a pitching (up down) motion: h. Now the lift is due to both alpha and the first derivative of h.
i.e. Lift = (1/2)*\rho*b*U^{2}(\alpha + \dot{h}/b)

Whenever the equations of motion for the airfoil are derived the inertial forces and the elastic restoring forces of the airfoil are included in the Lagrangian and the aerodynamic forces are included as generalised forces.

What I really want to know is whether the alpha term of the Lift can be included in the Lagrangian, rather than included as part of the generalised force. My rationale for this is that it is a function of alpha, and therefore is not a dissapative term. Also doing an eigen value analysis shows that if the only term that is neglected from the equations is the \dot{h} term, then the eigenvalues have no real component (as I understand it, indicating no energy gain or loss).

But conceptually this lift term is obviously feeding energy into the airfoil, and as this is only a two degree of freedom system, and I'm not including the (infinite) degrees of freedom of the pressure field around the airfoil the energy of my system should not be conserved and thus I shouldn't be able to include it in my Lagrangian.

Can anyone help me resolve this connundrum ?

On an unrelated note - can someone direct me to a page showing how you insert Latex into these pages ?

Cheers,
Thrillhouse

 

[LightHero]

The alpha term of the lift can be included in the Lagrangian, as it is a function of the state variables and not a dissipative term. The energy of the system may not be conserved if the pressure field around the airfoil is neglected. However, this does not mean that the alpha term of the lift cannot be included in the Lagrangian. It can still be included as a function of the state variables, and the energy conservation will simply not hold.Regarding the Latex question, you can find instructions for using Latex on this page: https://stackexchange.com/editing-help

 

[astrokreng]

Dear Thrillhouse,

Thank you for sharing your thoughts and concerns about the lift force in Lagrangian mechanics. Let me try to address your questions and provide some insight into this issue.

In Lagrangian mechanics, the equations of motion are derived from the Lagrangian, which is a function of the generalized coordinates and their time derivatives. In the case of an airfoil in incompressible flow, the Lagrangian will include terms for the inertial forces and the elastic restoring forces of the airfoil, as well as the aerodynamic forces as generalized forces.

The lift force, as you correctly pointed out, is a function of both the angle of attack (alpha) and the time derivative of the pitching motion (h). This means that it cannot be included in the Lagrangian as it is not a function of the generalized coordinates and their time derivatives. Including it in the Lagrangian would lead to incorrect equations of motion.

However, the lift force is still a conservative force, meaning that it does not dissipate energy. This is why you observed that neglecting the time derivative of h in the equations of motion does not result in any energy gain or loss. The lift force is simply a function of the airfoil's orientation and the flow conditions, and it does not take into account the pressure field around the airfoil.

In terms of energy conservation, it is important to understand that the Lagrangian formalism in mechanics is based on the principle of least action, which states that a system will always follow a path that minimizes the action (a quantity related to energy) between two points in time. This means that even though the pressure field may not be explicitly included in the equations, the system will still conserve energy as it follows the path of least action.

I hope this helps to resolve your conundrum and provides some clarity on the issue. As for inserting LaTeX into these pages, there are several online resources that can help with formatting and inserting equations. One option is to use a LaTeX editor and then copy and paste the formatted equation into your post. Alternatively, you can use online LaTeX converters to convert your equations into images that you can then insert into your post.

Best of luck with your research and understanding of Lagrangian mechanics.