This is all surprisingly very difficult to think in terms of generalised co-ordinates when you don't know what the mathematical D'Alembert framework is supposed to be doing for you and what you still have to do for yourself!! Clearly, I still don't get that. |

Perhaps that is because dear old D'Alembert did not introduce his principle in the way your textbook probably does.

That is simply because generalised coordinates and Hamilton's Principle were not introduced for another century after D'Alembert.

A good way to study D'Alembert is to do the same problem in the Newtonian and D'Alembert way side by side.

In fact, you can start with the Newtonian equations of motion, motivate then deduce the energy principles, the principle of virual work and D'Alembert's principle in short order. Then you can go on to derive Lagrange and Hamilton from this base and further to Noether, holonomy, symmetry invariants from this base.

But the original D'Alemebert principle was a mathematical transformation to reduce a more difficult dynamic problem to an equivalent easier static one, where the equations of equilibrium can be employed.

This was done by introducing imaginary or fictitious forces, sometimes called the inertial reaction(s) or the kinetic reaction(s).

Translation from old French

"When any particle is acted upon by an external force, the resultant of this external force and the kinetic reaction of the particle is zero"

Applying this should help with signs and trigonometry/geometry.

If you want to know more we should discuss it in a separate thread, rather than mess up this one about a specific problem.