Theory of Machines and Mechanics
Kinematic fundamentals
A mechanism is a constrained chain of links producing a definite motion; a machine is a mechanism that also transmits force and does useful work. Degrees of freedom follow the Kutzbach (Grübler) criterion for planar mechanisms:
F = 3(n − 1) − 2j − h, where n = number of links including the fixed frame, j = number of lower pairs (revolute, prismatic) and h = number of higher pairs (cam, gear, roller contact).
A four-bar chain has n = 4, j = 4, h = 0, so F = 3(3) − 8 = 1. Its inversions are the double-crank, crank-rocker and double-rocker, decided by Grashof's condition s + l ≤ p + q (shortest plus longest ≤ sum of the other two). Equality gives a change-point mechanism.
Other core areas
- Cams: follower motion - uniform velocity, simple harmonic, cycloidal; pressure angle must stay small to avoid jamming.
- Gear trains: simple, compound and epicyclic; velocity ratio from tooth numbers; hunting tooth equalises wear.
- Governors vs flywheel: a governor regulates mean speed against load variation; a flywheel absorbs cyclic energy fluctuation.
- Balancing: static balancing of a rotor needs centre of gravity on the axis; dynamic balancing needs zero resultant force and zero resultant couple.
- Mechanics: Lami's theorem for three concurrent forces, friction angle, screw jack, belt drive ratio T1/T2 = e^(μθ), maximum power when centrifugal tension equals one-third of maximum tension.
Worked example
Slider-crank mechanism: links = crank, connecting rod, slider, frame (n = 4); joints = three revolute plus one sliding (j = 4); no higher pairs (h = 0). F = 3(4 − 1) − 2(4) − 0 = 9 − 8 = 1 degree of freedom - the crank angle fully determines the motion.
Traps
- Counting a higher pair as two constraints; a higher pair removes only one degree of freedom.
- Forgetting the fixed frame in n, which shifts every answer.
- Confusing a structure (F = 0) with a mechanism.