Grashof’s Law For A Four-Bar Linkage

As is clear, the motion of links in a system must satisfy the constraints imposed by their connections. However, even for the same chain, and thus the same constraints, different motion transformations can be obtained. This is demonstrated in Figure 1.15, where the motions in the inversions of the four-bar linkage are shown. In Figure 1.15, s identifies the smallest link, l is the longest link, and p, q are two other links.

From a practical point of view, it is of interest to know if for a given chain at least one of the links will be able to make a complete revolution. In this case, a motor can drive such a link. The answer to this question is given by Grashof’s law, which states

that for a four-bar linkage, if the sum of the shortest and longest links is not greater than the sum of the remaining two links, at least one of the links will be revolving. For the notations in Figure 1.15 Grashof’s law (condition) is expressed in the form:

Since in Figure 1.15 Grashof’s law is satisfied, in each of the inversions there is at least one revolving link: in Figure 1.15a and b it is the shortest link s; in Figure 1.15c there are two revolving links, l and q; and in Figure 1.15d the revolving link is again the shortest link s.

PROBLEMS

1. What is the difference between a linkage and a mechanism?

2. What is the difference between a mechanism and a structure?

3. Assume that a linkage has N DOF. If one of the links is made a frame, how will it affect the number of DOF of the mechanism?

4. How many DOF would three links connected by revolute joints at point B (Figure P1.1) have? Prove.

FIGURE P1.1

5. A fork joint connects two links (Figure P1.2). What is the number of DOF of this system? Prove.

6. An adjustable slider drive mechanism consists of a crank-slider with an adjustable pivot, which can be moved up and down (see Figure P1.3). a. How many bodies (links) can be identified in this mechanism? b. Identify the type (and corresponding number) of all kinematic joints. c. What is the function of this mechanism and how will it be affected by moving the pivot point up and down?

7. In Figure P.1.4 a pair of locking toggle pliers is shown. a. Identify the type of linkage (four-bar, slider-crank, etc.). b. What link is used as a driving link? c. What is the function of this mechanism? d. How does the adjusting screw affect this function?