Mechanisms: count the freedom, close the loop

How many motors does it need?

• A dosing pump turns a motor's rotation into a piston stroke of exactly the right length. Choosing the links, and knowing how many inputs the mechanism needs, comes before any force calculation.
• A mechanism is a chain of links joined by kinematic pairs: lower pairs (pin, slider) allow one relative motion; higher pairs (cam, gear contact) allow two.

Gruebler's count

• Gruebler's equation for planar mechanisms: $M = 3(n - 1) - 2j_1 - j_2$, with $n$ links (ground included), $j_1$ one-freedom joints and $j_2$ two-freedom joints.
• The slider-crank has 4 links and 4 joints: $M = 9 - 8 = \mathbf{1}$, one motor. A five-bar with 5 pins has $M = 2$ and needs two inputs.

Where is the piston?

Can a link turn fully?

• The Grashof condition: shortest plus longest $\le$ the sum of the other two. Links 40, 100, 80 and 110 give $150 \le 180$, so it is Grashof, and with the shortest as crank it is a crank-rocker.
• Ground a different link and the same chain gives a different mechanism: these are its inversions.

The rod pushes sideways too

• The rod meets the piston at angle $\phi$, so an axial force $F$ in the stroke needs a side force $F\tan\phi$ from the cylinder wall. The transmission angle 传动角, here $90° - \phi$, measures how directly the rod drives.
• $\phi$ peaks at $\theta = 90°$, where $\sin\phi = r/l = 0.25$, so $\phi = 14.5°$: a 1 kN piston load presses the wall with $\mathbf{258}$ N. A longer rod lowers the side load and the wear.

Worked example: how far does the rocker swing?

• In the crank-rocker (crank 40, coupler 100, rocker 80, ground 110 mm) the rocker's extremes come when crank and coupler line up: stretched to 140 mm, folded to 60 mm.
• The cosine rule in each triangle with the 110 mm ground and 80 mm rocker gives $93.6°$ and $32.2°$: the rocker swings $\mathbf{61.4°}$ for every turn of the crank.

Three counting slips

Recap