Dynamics - Lagrange's Equations

[Andrew85]

Hi, revising for my christmas exams and am really struggling to get my bearing with Lagrangian Equations. I think its just the initial steps that I can't seem to follow.
This is a tutorial question that I just can't get my head around:

Use Lagrange’s Equations to obtain the equations of motion of the vehicle model shown, about the equilibrium position. The vehicle is symmetrical about the vertical centre line and you may assume small displacements and hence neglect any lateral movements of the masses, spring ends and link pivots.
Note: L1 = (L2 + L3)/2 i.e. the spring is attached half way along the link and the links attaching the wheels to the body are horizontal ( φ1 = φ2 = 0 ) at the equilibriumposition shown. Also, because we are interested in motion about the equilibriumposition it is not necessary to take into account any change in gravitational potential energy associated with the motion of the masses.

There should be a pic of the problem attached. Thanks for any help on how to get started.

 

[AlephZero]

The first thing to do is decide what your generalized variables are. Looking at the picture, they are Y, theta, theta_1, theta_2 with the sign conventions shown by the arrows.

Then work out the kinetic and potential energy of each part of the structure in terms of the generalized variables.

E.g. the right hand wheel has velocity (upwards) Ydot + L3 theta-dot - (L3-L2) theta1-dot. Write the KE = 1/2.Mw.v^2 as a matrix expression like

1/2 [Ydot thetadot theta1dot theta2dot]Transposed . [A symmetric 4x4 matrix with Ms and Ls in it] . [Ydot thetadot theta1dot theta2dot].

Repeat for the other parts of the structire - add all the KE terms for the different parts into the same 4x4 matrix of course.

The PE is very similar except using displacements not velocities.

Lagrangian dynamics is a great tool, provided somebody else has set up the equations of motion already, before you get to work on the problem :-(

Hope that helps you get going.