Kinematics: Grashof Criterion

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Kinematics: Grashof Criterion

Grashof Criterion

Frequently, a designer would like to predict that the input to a four-bar can rotate completely, or it cannot rotate completely.

Grashof's Criterion (sometimes called Grashof's Law, Grashof' Rule) can predict if one Part can rotate completely, or not. We apply the criterion to four-bar kinematic-chains that are joined with Pin-Joints. A Grashof mechanism has at least one part that can a rotate fully. A simple Crank-Rocker is the best known Grashof mechanism.

Analysis of the Grashof Criterion - predicting whether a Part can rotate continuously.

We start by making the frame as the shortest Part, designated 'S'. The part we choose as the shortest part does not change the following analysis.

Review the schematics images below.

They show a four-bar in two positions - the 'Stretched' and the 'Overlapping' positions. Each Part is designated a letter. S and L, are the Shortest (Frame) and Longest links, respectively. The other two Parts are designated P and Q.

If L can reach the Stretched and the Overlapping positions (see schematics below), then the mechanism, can also rotate fully, because (sort of proof!)

the distance between the two joints that are 'furthest apart' must be less at all other angles of L .

the distance between the two joints that are 'nearest together' must be greater at all other angles of L

If L can reach these positions it can also move to all of the intermediate positions(needs proof).

If this is agreed, then we want to know the lengths of P and Q that do not prevent L moving the two extreme positions.

Stretched Condition - Grashof Proof

Stretched Condition - Grashof Proof 1

Stretched - top image - a triangle gives one inequality:

1.S + L ≤ P + Q  (Grashof)

If P+Q < L+S, (or L+S > P+Q) then the joints would break.

Overlapping - bottom image - gives two more inequalities, and can exist when:

When Q = P  + (L – S) (Grashof, theoretically) , P is Horizontal and Q is also horizontal, stretching from the fixed-joint on the right to the end of P.

Q ≥ P  + (L – S) (Non-Grashof) - if Q becomes longer, then the end of Q must break from the end of P


2.Q ≤ P  + (L – S) (Grashof)


3.P ≤ Q + (L – S) (Grashof)

If P were to increase in length to become equal to the inequality, then Q rotates counter-clockwise to become horizontal, and if greater, it would break the mechanism.

The Grashof Conditions:

From 1, 2 and 3 above, we can say 'L' can rotate fully when these conditions, call them 'Conditions {A}', are met:

Conditions {A} :

S + L ≤ P + Q:   Condition 1

S + P ≤ Q + L:   Condition 2

S + Q ≤ L + P:   Condition 3

{We can derive the same conditions if we move Q that is adjacent to S to the other side of the mechanism (S adjacent to P). We can also prove - not here - that the coupler can also rotate fully - see Motion Geometry of Mechanism, Dijksman}.

Overalpping Condition - Grashof Proof

Overalpping Condition - Grashof Proof 2

If S, is the shortest Part, one of the remaining Parts must be the longest Part.

Eventually, we can prove - but we have not proved fully in this analysis - if the total lengths of the shortest and longest Part is equal or shorter than the sum of the lengths of the other two parts, then the shortest link can make a complete revolution, with respect to the others, and vice versa. This is the Grashof Criterion (also called the Grashof Rule, or even the Grashof Classification) by Franz Grashof in 1883.

So, stated as an inequality:

(L + S) < (P + Q) : Grashof

(L + S) = (P + Q) : then it is a special condition - see below - Parallelogram and Kite.

(L + S) > (P + Q) : Non- Grashof

Generally, if the the Grashof condition is met, the Parts other than the shortest can only oscillate with respect to the others.

If two parts can make a complete rotation, the two parts must be the shortest at the same time, and therefore have equal length. This is true if two of the three Grashof Conditions are met we can get a Kite, or a Parallelogram - see below.

Grashof Criterion Mechanisms

GRASHOF : Double-Crank

The Shortest Part is the Frame (Base-Part) = then the mechanism is a Double-Crank

Also called a Drag-Line.

Both Parts that are joined to the Base-Part will rotate fully.

GRASHOF: Crank-rocker

The Shortest Part is joined to the Frame (Base-Part) = then the mechanism is a Crank-Rocker

This is the most common four-bar maechanism.

Grashof- Double Rocker Rotating Coupler

GRASHOF:  Double-Rocker with Rotating Coupler

The Shortest Part is the Coupler = then the linkage is a Double-Rocker, but the Coupler can rotate fully.

To model this: Add a Motion-Path FB to a Circle at the end of one of the Parts.

Then add a Linear-Motion FB to the Motion-Path FB to rotate the Motion-Point. Joint the Motion-Point to, one of the rocking Parts.


Then join the Motion-Point to the other Rocking Link.

Grashof- Double Rocker Rotating Coupler


SPECIAL CASE 1 'Parallelogram' / 'Anti-parallelogram'.

Here, the Length of the:

Shortest links are equal

Two longest links are equal

Shortest + Longest = Other two Links (Shortest + Longest!!)

The Shortest and Longest parts are opposite to each other.

It is not possible to predict kinematically what the position of the Dyad will be after the links become 'flat', or in-line.

The inertia of a part may make the mechanism move in one way, but it is not advisable to operate in the flat position, unless you transmit motion from the input to the output with gears or a belt.

If you want to model, and build this mechanism and to make the rotations predictable, then it is best to add two Cranks - with a belt between them, or a Gear-Pair with idler gear between the input and the output Parts.

Grashof- Double Rocker Rotating Coupler


SPECIAL CASE 2 'Kite', or 'Deltoid'.

Here, the Length of the:

Shortest links are equal

Two longest links are equal

The Shortest and Longest parts are adjacent to each other.

You will notice that the crank must rotate two times while the output rotates one time.