Solving for Distributed Spring Work

[hubbat]

Homework Statement

I need to find the distributed spring work over a beam connected to a ceiling. The position on any point on the beam is given by Z = h + σ*(x-x_ea). The spring constant is K. This is a wing model, where h is the plunge, alpha is the pitch, and x is a position on the chord.

Homework Equations

Z = h + σ*(x-x_ea)
Z_dot = h_dot + σ_dot*(x-x_ea)
F = K*Z
dZ = dh + dσ*(x-xea)

The Attempt at a Solution

This seems like a simple question, but, is the potential energy differential for the wing
dW = ∫K*Z*dz = ∫K*Z*(dh + dσ*(x-xea))dx
or can it simply be
dW = 1/2*K*Z^2dx
?

 

[mighty2000]

Hello,

Thank you for your question. To find the distributed spring work over the beam, you will need to use the second equation you provided: dW = 1/2*K*Z^2dx. This equation takes into account the changing position of the beam (Z) as well as the spring constant (K). The first equation you provided does not take into account the position of the beam, so it will not accurately calculate the distributed spring work.

Additionally, since this is a wing model and the beam is connected to a ceiling, you may also need to consider the gravitational potential energy of the wing as it moves. This can be added to the distributed spring work using the equation dW = mgh, where m is the mass of the wing, g is the gravitational constant, and h is the height of the wing.

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!