Four-bar linkage

Four-bar linkage#

The four-bar linkage, or simply four-bar mechanism, is one of the most common mechanisms:

  • Four rigid bodies: #1 (ground), and three moving bars #2, #3, and #4.

  • Two moving points: A and B.

  • Two fixed points: \(O_2\) and \(O_4\).

4barras

It features one degree of freedom: moving one bar fully determines the position of all other bars. The same applies to velocities and accelerations: fixing the velocity or acceleration of one “input” bar, we can compute the corresponding values for the whole mechanism.

Bar names#

  • Crank: A bar connected to the ground that can undergo complete revolutions. We may have 0, 1, or 2 cranks in a four-bar mechanism.

  • Rocker: A bar connected to the ground that can only oscillate back and forth. We may have 0, 1, or 2 rockers.

  • Connecting rod or coupler: The bar that is not connected to the ground.

Grashof’s law#

For at least one of the links connected to the ground to be a crank, the sum of the lengths of the shortest (S) and longest (L) links must not be greater than the sum of the other two (P and Q).

\[\begin{equation*} S + L \leq P + Q \end{equation*}\]

grashof

Types of Grashof mechanisms: Depending on which link is chosen as the ground (fixed link), a Grashof chain can give rise to three different mechanism types:

  1. Crank-rocker: Obtained when the link adjacent to the shortest link (S) is fixed. Link S acts as a crank (turns 360°) and the output link oscillates as a rocker.

  2. Double-crank (or drag-link): Obtained when the shortest link (S) itself is fixed. Both the input and output links can turn 360°.

  3. Double-rocker: Obtained when the link opposite the shortest one (S) is fixed. Both links pivoted to the ground oscillate, and neither can complete a full turn.

If the condition is not met (\(S+L > P+Q\)), the mechanism is non-Grashof, and it will always be a double-rocker, regardless of which link is fixed.

Example of a case with a double crank:

To find out more details on the kind of motion, there is a criterion to find out whether there are one or two cranks. First, we name bar lengths according to their role:

  • \(a\): Input bar (left)

  • \(b\): Output bar (right)

  • \(g\): Ground bar

  • \(h\): Connecting rod (top)

grashof2

Then, we may have 8 different situations (bar names are shown for both the picture above and the interactive demo further below). The following test variables are defined:

\[\begin{eqnarray*} T_1=g+h-a-b = r_1 + r_3 - r_2 - r_4 \\ T_2=b+g-a-h = r_4 + r_1 - r_2 - r_3 \\ T_3=b+h-a-g = r_4 + r_3 - r_2 - r_1 \\ \end{eqnarray*}\]

Possible outcomes:

Linkage type

\(T_1\)

\(T_2\)

\(T_3\)

1

Crank-rocker

+

+

+

2

Rocker-crank

+

-

-

3

Double-crank

-

-

+

4

Double-rocker

-

+

-

5

Double-rocker

-

-

-

6

Double-rocker

+

+

-

7

Double-rocker

+

-

+

8

Double-rocker

-

+

+

4bar

Try it yourself!#

Change the bar lengths in this interactive demo (source) to verify Grashof’s law:

Some tips:

  • Drag the slider \(\theta_2\)

  • The lengths \(a\), \(b\), \(g\), and \(h\) above are named \(r_2\), \(r_4\), \(r_1\), and \(r_3\) in the GeoGebra model, respectively.

  • Example of double-crank: \((r_1,r_2,r_3,r_4)=(5,7,8,9)\). \(T_1=5+8-7-9=-\), \(T_2=9+5-7-8=-\), \(T_3=9+8-5-7=+\) (--+).

Motorbike swingarm#