Grashof's Criterion: Four-bar Mechanism Double Crank

One of the most commonly used linkages is the four-bar linkage. It consists of 3 moving links and 1 ground link (also called a frame). There are 4 pin joints connected between those links. And from Gruebler's Equation, it's the mechanism with 1 degree of freedom (1DOF) which requires only 1 actuator to drive and control position of all linkages. The four-bar mechanism consists of the following components.

Four-bar mechanism components

The link that connects to the driver or power source is called the input link. The other link connected to the fixed pivot is called the output link. The remaining moving link connected between input and output links is called the coupler. It couples the motion of the input link to the output link.

There are different configurations of lengths of the four-bar mechanism and it results in different movements of the mechanism. Grashof's Criterion helps classifies into the following categories:

1. Double Crank --- also called a drag link mechanism
2. Crank-Rocker
3. Double Rocker
4. Change Point
5. Triple Rocker

In this post, we're going to explore the case of Double Crank which both input and output links are able to rotate through a full revolution. We use Autodesk ForceEffect app on Google chrome to illustrate and demonstrate how the mechanism moves. The links can be setup easily by just dragging. The exact length dimension can also be specified. Once we setup all required items i.e. links, pivots and drive, we can play and see the animation of the linkages with the path of desired tracing points.

Autodesk ForceEffect app on Google chrome

Grashof's criterion states that a four-bar mechanism has at least one revolving link if:

S + L ≤ P + Q

where:
S = length of the shortest link
L = length of the longest link
P = length of one of the intermediate length links
Q = length of the other intermediate length link

For the Double Crank category, the following criteria must be satisfied:

Double Crank:

• S + L < P + Q
• S is the length of the frame (ground link)

So we setup the links in ForceEffect app as follows.

Grashof's Criteria for Double Crank 4-bar Mechanism

The shortest link (S) is the frame which is 400 mm long. The length of the longest link (L) is 1300 mm. The remaining 2 intermediate links (P and Q) have length 700 mm and 1200 mm.

This satisfies Grashof's criteria since (S)400 + (L)1300 < (P)700 + (Q)1200. And this is how this mechanism moves.

Paths of double crank four-bar mechanism

Both input and output links can make a full revolution as desired. Let's find more details of other categories in later post.

The following video shows how to use ForceEffect app to simulate the motion of double crank four-bar linkage.

Reference:

• Machines & Mechanisms 3rd edition, David H. Myszka
• Autodesk ForceEffect