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2.3 Lagrangian

The Lagrangian is ℒ = KE − PE, or
(19) ℒ = (1)/(2)Mx1̇2 + (1)/(2)x2̇2⎛⎝m + (I)/(R2)⎞⎠ − (1)/(2)x21(k1 + k2) − (1)/(2)k2x22 + k2x1x2

2.4 Generalized force

The generalized forces are derived from forces acting on each body not already accounted for in the potential energy. In this case, the only force that needs to be included in the generalized forces is the damper force. The equation for generalized forces is
(20) Qk = n⎲⎳i = 1 Fi⋅(∂vi)/(∂qk̇),   k = 1, ..., N
where k is the index of generalized variables. Here, it can easily be seen that there is no additional force acting on body 2 that was not already accounted for in the potential energy, so F2 = 0. Also, (∂v1)/(∂x2̇) = 0, so the generalized force Q2 is also zero. Therefore, the only generalized force is Q1, which is
(21) Q1 =  Fd⋅(∂v1)/(∂x1̇)
or
(22) Q1 =  − cx1̇

2.5 Lagrange’s equation

Lagrange’s equation can now be written for each generalized coordinate. Lagrange’s equation is
(23) (d)/(dt)⎛⎝(∂ℒ)/(∂qk̇)⎞⎠ − (∂ℒ)/(∂qk) = Qk,   k = 1, ..., N
where, in this case, q1 = x1 and q2 = x2. As part of this process, one must evaluate the partial derivatives (∂ℒ)/(∂x1̇), (∂ℒ)/(∂x2̇), (d)/(dt)⎛⎝(∂ℒ)/(∂x1̇)⎞⎠, (d)/(dt)⎛⎝(∂ℒ)/(∂x2̇)⎞⎠, (∂ℒ)/(∂x1), and (∂ℒ)/(∂x2). This process is elementary and so all the terms will not be given here. Applying Lagrange’s equation for the generalized coordinate x1 gives
(24) Mx1̈ + x1(k1 + k2) − k2x2 =  − cx1̇
or, in the same form as was derived in the Newtonian method,
(25) Mx1̈ + cx1̇ + (k1 + k2) − k2x2 = 0
The equation for the second generalized coordinate is indentical to that derived for the second body in the Newtonian method:
(26) ⎛⎝m + (I)/(R2)⎞⎠x2̈ + k2x2 − k2x1 = 0
As expected, the equations of motion derived by Lagrange’s method are identical to those derived by Newton’s method. If this were not the case, then something must be wrong in at least one of the derivations! Therefore, it is a good practice to derive the equations both ways to make sure there are no mistakes.
In this particular problem, Newton’s method is probably easier to apply, because there are no complicated constraints or forces and moments acting between the bodies that are difficult to express. However, in many more complicated problems, energy methods such as Lagrange’s equations are much simpler to apply. Lagrange’s method is also very systematic, making it easier to check the work.
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