Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Reference texts (subject). R. L. Norton, Design of Machinery, 6th ed. (mobility Ch. 2, synthesis Ch. 3, velocity/acceleration Ch. 6–7, cams Ch. 8, epicyclic trains §9.6–9.9, balancing Ch. 13); Uicker, Pennock & Shigley, Theory of Machines and Mechanisms, 5th ed.; C. E. Wilson & J. P. Sadler, Kinematics and Dynamics of Machinery, 3rd ed.; S. S. Rao, Mechanical Vibrations, 6th ed. (Part B multi-DOF torsional Ch. 6, impact Ch. 1).
Note — figures printed without dimensions. Every figure on this paper is printed without dimensions, and several carry inconsistent numbers (e.g. Q4 shows two conflicting tooth-count sets, and a ring $N_3=20$ cannot mesh a sun $N_1=32$; Q5 prints $AG_3=BG_3=82.5$ mm with $AB=179$ mm, a geometrically impossible triangle). The methods below are exact and complete; where a numerical answer depends on figure geometry (Q2, Q5) or on internally inconsistent data (Q4), the adopted reading is stated explicitly and the value is flagged as representative. Scale-free results (mobility, Grashof class, angular-velocity ratios, transmission-angle location, mode shapes, non-dimensional natural frequencies) are exact regardless of the figure dimensions.
