Theory of Machines

The mobility of a planar linkage is found from Gruebler/Kutzbach: F = 3(n − 1) − 2j₁ − j₂, counting links n, lower pairs j₁ and higher pairs j₂. A mechanism needs F = 1 for constrained motion, as SS Rattan explains.

Point velocities in a linkage follow the relative-velocity equation v_B = v_A + v_B/A, and accelerations split into tangential a_t = αr and normal a_n = ω²r components. Sliding pairs add the Coriolis term 2ωv_r, as detailed by SS Rattan.

The follower's motion is derived from the displacement diagram: velocity v = (dS/dθ)·ω and acceleration a = (d²S/dθ²)·ω². The pressure angle φ = arctan((dS/dθ − e)/(S + s₀ + e)) must stay small to avoid jamming, per SS Rattan.

For a simple pair the speed ratio is N_A/N_B = T_B/T_A; a compound train multiplies the ratios of each mesh. Epicyclic trains are solved by the tabular method because the arm rotation superposes on gear rotation, as SS Rattan shows.

A Porter governor balances centrifugal and gravity/spring effects to set sleeve height h; sensitiveness = (N₂ − N₁)/N_mean. A spinning rotor resists tilting with a gyroscopic couple C = I·ω·ω_p, both from SS Rattan.