Euler Lagrange equations with viscous dissipation

In summary, the conversation discusses determining the Lagrangian for a system that can pivot at point O, using small angle approximations and the E-L equation with dissipation. The potential and kinetic energies for the system, including a circular mass and a spring, are also mentioned. There is some concern about the potential energy signs.
  • #1
Dustinsfl
2,281
5

Homework Statement


The system can pivot at point O and I am taking small angle approximations.
I am trying to determine the Lagrangian, ##\mathcal{L} = T - U## for the following system:
2nh6RzK.png


Homework Equations


E-L equation with dissipation: ##\frac{\partial\mathcal{L}}{\partial q_i} - \frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{q}_i} + \frac{\partial D}{\partial\dot{q}_j} = 0##

The Attempt at a Solution


I am going to use the generalized coordinate theta.

For the circular mass, I have the potential energy to be ##mg(1-\cos(\theta)) = \frac{mg\theta^2}{2}## and the kinetic energy is ##\frac{1}{2}J\dot{\theta}^2## where J is the mass moment of inertia. The potential energy of the spring is ##\frac{1}{2}kx^2##, and the dissipative energy is ##D = c\frac{\dot{x}^2}{2}##.

Before I convert the xs to thetas am I missing a kinetic or potential energy?
 
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  • #2
Don't think you are missing anything. I do worry about the potential energy signs.
 

1. What are Euler-Lagrange equations with viscous dissipation?

Euler-Lagrange equations with viscous dissipation are a set of equations used to describe the motion of a fluid with viscosity. They take into account the effects of both inertia and friction on the fluid's motion.

2. How are Euler-Lagrange equations with viscous dissipation derived?

These equations are derived from the Navier-Stokes equations, which describe the motion of a viscous fluid. By applying the principle of virtual work, the Euler-Lagrange equations are obtained.

3. What is the significance of viscous dissipation in these equations?

Viscous dissipation represents the energy lost due to the frictional forces within the fluid. It is an important factor to consider in fluid dynamics, as it affects the overall behavior and stability of the system.

4. How do Euler-Lagrange equations with viscous dissipation differ from the standard Euler-Lagrange equations?

The addition of viscous dissipation introduces an extra term in the equations, known as the viscous term. This term accounts for the effects of friction on the fluid's motion, making the equations more complex and difficult to solve analytically.

5. What applications do Euler-Lagrange equations with viscous dissipation have?

These equations have various applications in fluid dynamics, such as in the study of turbulence, boundary layers, and flow around objects. They are also important in the design and analysis of fluid systems, such as in the fields of aerodynamics and hydrodynamics.

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