# Lagrangian, 2 DOF (rotation with torsion, springs)

• Kolodny
To solve this problem, you would need to find the values of \dot{y} and \dot{\theta} and then find the displacements for the potential energy. There are many examples of 2 DOF Lagrangian problems available online, and it is important to understand the general concept of Lagrangians before attempting to solve them.
Kolodny

## Homework Statement

[PLAIN]http://mityaka.com/users/kolodny/img/lagprob.png

## Homework Equations

L = T - V

T = $$\frac{1}{2}$$*m*U2

Vs = $$\frac{1}{2}$$*k*x2

## The Attempt at a Solution

I worked out the equations of motion as:

FL = m*$$\ddot{y}$$+k*y-k*b*$$\theta$$

FL*e = IG*$$\ddot{\theta}$$+(k*b+kT)*$$\theta$$+k*b*yAnd from here I'm not sure where to proceed. I understand in general that I would have to find the $$\dot{y}$$ and $$\dot{\theta}$$, and then the displacement for the potential energy, but given the equations of motion I'm not sure how to go about doing that.

If someone could direct me to an example of a 2 DOF Lagrangian problem, that would be awesome. My textbook (Palm's System Dynamics) doesn't cover Lagrangians at all for some odd reason, and I can't find anything other than general explanations of Lagrangians, as well as some 1 DOF problems, through Google.

Edit: Nevermind, I was making it a lot harder than it should be. For some reason I thought I had to derive the y-dot and theta-dot in terms of position and use that, but I've got it now.

Last edited by a moderator:
The Lagrangian is L = T - V, with T = \frac{1}{2}*m*\dot{y}^2 + \frac{1}{2}*IG*\dot{\theta}^2V = \frac{1}{2}*k*(y-b*\theta)^2 + k*b*y*\thetaThe equations of motion are found by taking the time derivative of the Lagrangian:m*\ddot{y}+k*y-k*b*\theta = 0IG*\ddot{\theta}+(k*b+kT)*\theta+k*b*y = 0

## 1. What is a Lagrangian in the context of a 2 DOF system?

A Lagrangian is a mathematical function that describes the dynamics of a system with two degrees of freedom (DOF), specifically rotation with torsion and the presence of springs. It takes into account the potential and kinetic energies of the system to derive the equations of motion.

## 2. How are the equations of motion derived using the Lagrangian for a 2 DOF system?

The equations of motion for a 2 DOF system can be derived by taking the partial derivatives of the Lagrangian with respect to the two variables, typically the angle of rotation and the displacement of the torsion spring. These equations can then be solved to determine the behavior of the system over time.

## 3. What are some applications of the Lagrangian for 2 DOF systems?

The Lagrangian for 2 DOF systems is commonly used in mechanical engineering and physics to model the behavior of systems such as pendulums, double pendulums, and torsion springs. It can also be applied to study the dynamics of molecules and particles in quantum mechanics.

## 4. How do the springs affect the behavior of a 2 DOF system?

The presence of springs in a 2 DOF system adds an additional force that must be taken into account in the equations of motion. The stiffness of the springs can greatly affect the natural frequencies of the system and how it responds to external forces or disturbances.

## 5. Are there any limitations to using the Lagrangian for 2 DOF systems?

While the Lagrangian is a powerful tool for analyzing the dynamics of 2 DOF systems, it does have some limitations. It assumes that the system is conservative, meaning that energy is conserved and there are no dissipative forces. It also does not take into account any external forces or torques acting on the system.

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