POSITION AND DISPLACEMENT ANALYSIS
(4.2) The magnitude of the angular displacement is the angle
between the initial and final configuration of a link during an interval. This magnitude will be in rotational units (e.g., degrees, radians, and revolutions), and denoting either clockwise or counterclockwise specifies the direction.
¢u3 = u3œ- u3
FIGURE 4.2 Position vector for point .P
3'
FIGURE 4.4 Angular displacement.
74 CHAPTER FOUR
FIGURE 4.5 Typical position analysis.
FIGURE 4.6 Two geometric inversions of a four-bar mechanism.
4.4 DISPLACEMENT ANALYSIS
A common kinematic investigation is locating the position of all links in a mechanism as the driver link(s) is displaced.
As stated in Section 4.2, the degrees of freedom of a mecha-nism determine the number of independent driver links. For the most common mechanisms, those with one degree of freedom, displacement analysis consists of determining the position of all links as one link is displaced. The positions of all links are called the configuration of the mechanism.
Figure 4.5 illustrates this investigation. The mechanism shown has four links, as numbered. Recall that the fixed link, or frame, must always be included as a link. The mechanism also has four revolute, or pin, joints.
From equation (1.1), the degrees of freedom can be calculated as follows:
With one degree of freedom, moving one link precisely posi-tions all other links in the mechanism. Therefore, a typical displacement analysis problem involves determining the position of links 3 and 4 in Figure 4.5 as link 2 moves to a specified displacement. In this example, the driving displacement is angular,¢u2 = 15°clockwise.
M = 3(4 - 1) - 2(4) = 1
disassembling the mechanism or traveling through dead points. Thus, when conducting a displacement analysis, inspecting the original configuration of the mechanism is necessary to determine the geometric inversion of interest.
4.5 DISPLACEMENT: GRAPHICAL ANALYSIS
4.5.1 Displacement of a Single Driving Link
In placing a mechanism in a new configuration, it is neces-sary to relocate the links in their respective new positions.
Simple links that rotate about fixed centers can be relocated by drawing arcs, centered at the fixed pivot, through the moving pivot, at the specified angular displacement. This was illustrated in Figure 4.5 as link 2 was rotated 15° clockwise.
In some analyses, complex links that are attached to the frame also must be rotated. This can be done using several methods. In most cases, the simplest method begins by relocating only one line of the link. The other geometry that describes the link can then be reconstructed, based on the position of the line that has already been relocated.
Figure 4.7 illustrates the process of rotating a complex link. In Figure 4.7a, line AB of the link is displaced to its desired position, clockwise. Notice that the relocated position of point is designated as .B Bœ
¢u2 = 80°
Nearly all linkages exhibit alternate configurations for a given position of the driver link(s). Two configurations for the same crank position of a four-bar mechanism are shown in Figure 4.6. These alternate configurations are called geometric inversions. It is a rare instance when a linkage can move from one geometric inversion to a second without
80°
FIGURE 4.7 Rotating a complex link.
Position and Displacement Analysis 75
Constrained 1 path of point A
Constrained path of point C
Constrained path of point E E
FIGURE 4.8 Constrained paths of points on a link pinned to the frame.
4 D
Constrained path of point C C path of point B
FIGURE 4.9 Constructing the constrained path of .C The next step is to determine the position of the
relocated point , which is designated as . Because the complex link is rigid and does not change shape during movement, the lengths of lines AC and BC do not change.
Therefore, point can be located by measuring the lengths of AC and BC and striking arcs from points and , respec-tively (Figure 4.7b).
A second approach can be employed on a CAD system.
The lines that comprise the link can be duplicated and rotated to yield the relocated link. All CAD systems have a command that can easily rotate and copy geometric entities.
This command can be used to rotate all lines of a link about a designated point, a specified angular displacement. It is convenient to display the rotated link in another color and place it on a different layer.
4.5.2 Displacement of the Remaining Slave Links
Once a driver link is repositioned, the position of all other links must be determined. To accomplish this, the possible paths of all links that are connected to the frame should be constructed. For links that are pinned to the frame, all points on the link can only rotate relative to the frame. Thus, the possible paths of those points are circular arcs, centered at the pin connecting the link to the frame.
Figure 4.8 illustrates a kinematic diagram of a mecha-nism. Links 2, 4, and 6 are all pinned to the frame. Because points , and are located on links 2, 4, and 6, respec-tively, their constrained paths can be readily constructed.
The constrained path of point is a circular arc, centered at point , which is the pin that connects link 2 to the frame.
The constrained paths of and can be determined in a similar manner.
The constrained path of a point on a link that is connected to the frame with a slider joint can also be easily determined. All points on this link move in a straight line, parallel to the direction of the sliding surface.
After the constrained paths of all links joined to the frame are constructed, the positions of the connecting links can be determined. This is a logical process that stems from the fact that all links are rigid. Rigidity means that the links do not change length or shape during motion.
In Figure 4.5, the positions of links 3 and 4 are desired as link 2 rotates 15° clockwise. Using the procedures described
E
in Section 4.5.1, Figure 4.9 shows link 2 relocated to its displaced location, which defines the position of point . The constrained path of point has also been constructed and shown in Figure 4.9.
Because of its rigidity, the length of link 3 does not change during motion. Although link 2 has been reposi-tioned, the length between points and does not change. To summarize the facts of this displacement analy-sis, the following is known:
1. Point has been moved to
2. Point must always lay on its constrained path (length from and
3. The length between and must stay constant (C
must be a length from .
From these facts, the new position of link 3 can be constructed. The length of line BC should be measured.
Because point has been moved to , an arc of length is constructed with its center at . By sweeping this arc, the feasible path of point has been determined. However, point must also lay on its constrained path, as shown in Figure 4.9. Therefore, point must be located at the intersection of the two arcs. This process is illustrated in Figure 4.10. Note that the two arcs will also intersect at a second point. This second point of intersection is a consider-able distance from and represents a second geometric inversion for this linkage. The linkage must be disassembled and reassembled to achieve this alternate configuration, so that intersection can be ignored.
It is possible that the two arcs do not intersect at all.
Cases where the constrained path and feasible path do not intersect indicate that length of the individual links prevents the driver link from achieving the specified displacement.
76 CHAPTER FOUR
4
Constrained path of point C Intersection represents precise location of C′ Feasible path of C′
C
D A
3
1
2 15°
B B′
rBC
rAB
rCD
Second intersection represents another geometric
inversion
1
D C
B