The crank-rocker is the most commonly built form of the four-bar linkage: a revolving crank drives a link that rocks back and forth. This page states when one arises, which quantities describe its travel, and what of it can be designed in closed form.
When a crank-rocker arises
Two conditions have to come together. First, Grashof’s condition must hold - shortest plus longest link shorter than the other two combined. Second, the shortest link must sit at the frame. It then revolves fully as the crank, while the opposite link at the frame rocks between two limit positions. If the shortest link sits elsewhere, the same four lengths give a double crank or a double rocker (Kerle/Pittschellis/Corves, Einführung in die Getriebelehre, 3rd edition 2007, page 38). The type is therefore not in the lengths alone but in which link is held still.
Limit positions, swing angle, time ratio
The travel of the rocker is bounded by two positions in which crank and coupler lie on one line: extended in one, folded in the other. These are the dead-centre or limit positions; Kerle covers them in chapter 6.1, pages 151 to 158. Both quantities used to judge a crank-rocker fall out of them:
- The swing angle ψ₀ is the full travel of the rocker between its two limit positions - the figure a required swivel motion is checked against.
- The time ratio relates the two parts of the crank revolution in between, the longer to the shorter. For the centric crank-rocker it is 1, otherwise larger.
A time ratio above 1 is the actual reason for this type: quick-return mechanisms on shaping and planing machines work slowly under load and travel back fast, while the crank keeps turning uniformly. Which of the two parts is the working stroke is decided by the application, not by the geometry.
The transmission angle decides the quality
Two crank-rockers with the same swing angle can run very differently. The difference lies in the transmission angle μ, the angle between coupler and rocker: at 90° the entire coupler force goes into the motion, towards 0° it jams. Its two smallest values occur in the positions where crank and frame are aligned and can be computed in closed form from the four lengths (Kerle, page 161, equation 6.5). As an empirical value for slow-running mechanisms Kerle gives 40 to 50 degrees. Enlarging the swing angle without changing the lengths usually costs a smaller μ - that is the trade-off of this type.
Designing instead of checking: VDI 2130
The inverse task - obtaining the dimensions from a required swing angle - can be solved in closed form for the centric case. Centric means equal times for forward and return stroke, that is a time ratio of 1. From swing angle, crank length and one further length the remaining dimensions then follow without any iteration. The method is in Kerle, exercise 6.2 b) (solution page 289, the laying unit of a textile machine) and, independently, in exercise 14 of FH Jena; both compute the same two relations from different directions.
For unequal stroke times this does not work without an additional input. In VDI 2130 the selection angle β comes from a chart over crank angle and swing angle (Kerle, figure 6.11) and exists as a formula only for the centric case. Kinematics Studio therefore designs the centric crank-rocker and not the general one - a number picked out of the air in a verification document would be worse than a missing function. What is designed in closed form alongside it is the slider-crank with unequal stroke times, which VDI 2130 does give as a formula (Kerle, exercise 6.1).
What you can compute it with
Kinematics Studio builds the chain from links and joints and solves it across the whole revolution. It reports the Grashof type, finds the limit positions and outputs swing angle and time ratio, carries the transmission angle as the smallest value over the revolution, draws coupler curves and returns drive torque and joint forces. Every design recomputes from its own dimensions what it was meant to achieve, and that check is printed in the verification document - a designed mechanism without its calculation path is an assertion.
Further: the four-bar linkage and its types as the general case, the slider-crank mechanism as its nearest relative, and the drive sizing calculator once the drive torque is known.
Frequently asked questions
When does a four-bar become a crank-rocker?
When Grashof's condition holds and the shortest link sits at the frame. That link then revolves fully as the crank while the opposite one rocks. The same four lengths give a double crank if the frame itself is the shortest link, and a double rocker if the coupler is - so the type does not depend on the lengths alone, but on which link is held still.
What does the time ratio tell you?
It relates the two parts of the revolution between the limit positions, always the longer one to the shorter. k = 1 means forward and return stroke take equally long. k = 2 means one takes twice as long as the other. Quick-return mechanisms are built for exactly that - slow under load, fast on the way back. Which of the two parts is the working stroke is decided by the application, not by the geometry.
Can a crank-rocker be designed for a required swing angle?
For equal stroke times yes, in closed form and without iteration: from swing angle, crank length and one further length the remaining dimensions of the centric crank-rocker follow. For unequal stroke times it does not work without an additional input: in VDI 2130 the selection angle comes from a chart over crank and swing angle and exists as a formula only for the centric case. Inventing a number for it would mean writing something into a verification document that has no source.