﻿ General Design Information > Basic Kinematics > Kinematics - General > Kinematics: Grashof Criterion

# 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 LIf 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 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 Therefore: 2.Q ≤ P  + (L – S) (Grashof)Also: 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 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.