Lift Force from Lagrangian Mechanics

In summary, the alpha term of the lift can be included in the Lagrangian as it is a function of the state variables, and the neglect of the pressure field does not affect this inclusion. Instructions for using Latex can be found on this page: https://stackexchange.com/editing-help#latex
  • #1
thrillhouse86
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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
 
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  • #2
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
 
  • #3
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.
 

1. What is lift force?

Lift force is the upward force exerted on an object, such as an airplane or a bird, as it moves through a fluid, typically air. It is a result of the difference in pressure between the upper and lower surfaces of the object.

2. How is lift force calculated using Lagrangian mechanics?

In Lagrangian mechanics, lift force is calculated through the use of Bernoulli's equation, which relates the fluid's pressure, density, and velocity at different points along the object's surface. This equation is then incorporated into the Lagrangian, which is the difference between the kinetic and potential energy of the object in motion.

3. What factors affect the magnitude of lift force?

The magnitude of lift force is affected by several factors, including the shape and size of the object, the speed of the object through the fluid, the density and viscosity of the fluid, and the angle of attack (the angle between the object's direction of motion and its orientation in the fluid).

4. Can Lagrangian mechanics accurately predict lift force?

Yes, Lagrangian mechanics can accurately predict lift force as long as the assumptions and equations used in the calculations are appropriate for the given situation. However, it is important to note that there are certain scenarios, such as turbulent flow, where Lagrangian mechanics may not be the most accurate method for predicting lift force.

5. How is lift force important in the design of aircraft?

Lift force is a crucial factor in the design of aircraft as it allows them to generate enough upward force to stay in flight. Engineers and scientists use the principles of lift force to design the shape and size of the wings and other aerodynamic features of an aircraft to optimize its lift and overall performance.

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