Use Only the Angular Quantities in Analysis? Three Sample Problems to Consider...

A recent discussion at iMechanica following my last post here [^] leads to this post. The context of that discussion is assumed here.

I present here three sample problems, thought of almost at random, just to see how the suggestions made by Jaydeep in the above post work out.

Problem 1. For simplicity of analysis, assume a gravity-free, frictionless, airless, field-free, etc. kind of a physical universe, and set up a suitable Cartesian reference frame in it such that the xy-plane forms the ground and the z-axis is vertical. (Assume a right-handed coordinate system.)

Consider a straight rigid rod of zero thickness and length 2a, initially positioned parallel to the x-axis and at a height of h, translating with a uniform linear velocity, v, in the positive y-direction.

Suppose that the translating rod runs into a rigid vertical pole of infinite strength, zero thickness and a height > h so that a collision between the two is certain.

If the collision were to occur at the midpoint of the rod (i.e. at its CM), it would simply begin translating back with a -v velocity.

However, assume that the collision occurs at a distance d (0 < d < a) from the CM. This will impart an angular motion to the rod after the collision.

Derive the equations of motion for the rod before and after the impact, using (a) the usual method of analysis (having both the linear and angular quantities in it), and (b) the method / lines of analysis suggested by Jaydeep.

Comment on the linear and angular quantities before and after the impact in both the cases.

Make any additional assumptions as necessary.

Problem 2. Repeat the Problem 1., using both the usual analysis and Jaydeep's method, now assuming that the rod is linear elastic with infinite strength. All other assumptions and data remain the same; in particular, the pole continues to remain rigid.

Problem 3. Provide a detailed derivation for an FE model of the beam element, replacing the three rotations and three translations by six rotations so as to conform to Jaydeep's method.

Asides: BTW, I plan to solve none. But I will make sure to have a look at any solution(s) that are offered, but without promising in advance that I will also comment on them.

With that said, solutions and comments are most welcome!

--Ajit

PS: Also posted at my blog here [^]

Update on May 29: Corrected a typo. Now the Problem 1 reads: "...a uniform linear velocity, v, in the positive y-direction. ..." This is the way it was intended. Earlier, it incorrectly read: "...a uniform linear velocity, u, in the positive x-direction. ..." 

[E&OE]