POSITION AND DISPLACEMENT ANALYSIS
EXAMPLE PROBLEM 4.6
The mechanism shown in Figure 4.26 is the driving linkage for a reciprocating saber saw. Determine the configura-tions of the mechanism that places the saw blade in its limiting posiconfigura-tions.
1.0"
1.75"
.5"
FIGURE 4.26 Saber saw mechanism for Example Problem 4.6.
SOLUTION: 1. Draw a Kinematic Diagram
The kinematic diagram for the reciprocating saw mechanism is given in Figure 4.27a. Notice that this is a slider-crank mechanism as defined in Chapter 1. The slider-slider-crank has one degree of freedom.
2. Construct the Extended Limiting Position
The saw blade, link 4, reaches its extreme downward position as links 2 and 3 move into a collinear alignment.
This configuration provides the maximum distance between points and . To determine this maximum distance, the lengths of links 2 and 3 must be combined. Adding these lengths,
Once the combined length of lines 2 and 3 is determined, an arc should be constructed of this length, centered at point . As shown in Figure 4.29b, the intersection of this arc and the possible path of point A
L2 + L3 = 0.5 in. + 1.75 in. = 2.25 in.
C A (a) Slider-crank
Stroke, |ΔRC|max
B'
A B''
C' C C''
B β
(b) Four-bar B'
A β
D Throw, (θ4)max B''
C' C
B C''
FIGURE 4.25 Limiting positions.
Position and Displacement Analysis 89
A B
C
(b) (a)
2
4 1 3
A B
C L3– L2
C′ C″ 2
4 3 1
L2+
L1 L3
L1 L2
L3
(c)
FIGURE 4.27 Extreme positions for Example Problem 4.6.
This retracted limiting position can be determined using a technique similar to determining the extended position. Recall that the distance between and in Figure 4.27b represents the combined length of links 2 and 3. Similarly, the distance between points and represents the difference between links 3 and 2.
Using the distance, the position of point at its extreme upward position, denoted as , can be determined (Figure 4.27b). Finally, links 2 and 3 can be drawn and the position of point is located.
4. Measure the Stroke of the Follower Link
As shown in Figure 4.27c, the stroke of the saw blade can be measured as the extreme displacement of point . Scaling this from the kinematic diagram yields the following result:
ƒ ¢RCƒmax = 1.27 in.
C Bœœ
Cœœ C
L3 - L2
Cœœ A
Cœ A
determines the limiting extended position of C, denoted C. Links 2 and 3 can be drawn, and point can be determined. This is shown in Figure 4.29c.
3. Construct the Retracted Limiting Position
Next, the configuration that places the saw blade, link 4, in its extreme upper position must be determined. In this configuration, links 2 and 3 are again collinear but overlapped. This provides the minimum distance between points and . Thus, this minimum distance is the difference between the lengths of links 3 and 2.
Subtracting the link lengths gives
L3 - L2= 1.75 in. - 0.5 in. = 1.25 in.
C A
Bœ
90 CHAPTER FOUR
A
D
B
L2 + L3 L2
L4
L3 L3 – L2
C
(b) (a)
2
4 3 1
A
D
B
C
C C
(c)
FIGURE 4.29 Extreme positions for Example Problem 4.7.
Motor
Water inlet
Nozzle 2.0"
2.0"
2.0"
1.0" .75"
FIGURE 4.28 Water nozzle linkage for Example Problem 4.7.
SOLUTION: 1. Draw the Kinematic Diagram
The kinematic diagram for the water nozzle linkage is given in Figure 4.29. Notice that this is a four-bar mechanism with one degree of freedom.
2. Construct the Extended Limiting Position
The analysis in this example is very similar to Example Problem 4.6. The nozzle, link 4, reaches its extreme downward position as links 2 and 3 become collinear. This configuration provides the maximum distance between points and . To determine this maximum distance, the lengths of links 2 and 3 must be combined.
Adding these lengths gives
L2 + L3 = 0.75 in. + 2.00 in. = 2.75 in.
C A EXAMPLE PROBLEM 4.7
Figure 4.28 illustrates a linkage that operates a water nozzle at an automatic car wash. Determine the limiting positions of the mechanism that places the nozzle in its extreme positions.
Position and Displacement Analysis 91 Once the combined length of lines 2 and 3 is determined, an arc should be constructed of this length, centered at point A. As shown in Figure. 4.28b, the intersection of this arc and the possible path of point C determines the extreme downward position of C, denoted . Links 2 and 3 can be drawn, and point can be determined. This is shown in Figure 4.29c.
3. Construct the Retracted Limiting Position
Next, the configuration that places the nozzle, link 4, in its extreme upper position must be determined.
Similar to the slider-crank discussed in Example Problem 4.6, the retracted configuration occurs when links 2 and 3 are collinear but overlapped. This produces the minimum distance between points A and C.
Thus, this minimum distance is the difference between the lengths of links 3 and 2. Subtracting the link lengths gives
This minimum distance can be constructed similar to the technique for the maximum distance. Recall that the distance between A and in Figure 4.29c represents the combined length of links 2 and 3. Similarly, the distance between points A and represents the difference between links 3 and 2.
Using the distance, the position of point at its extreme upward position, denoted as , can be determined. This is shown in Figure 4.29b. Finally, links 2 and 3 can be drawn and the position of point is located.
4. Measure the Stroke of the Follower Link
As shown in Figure 4.29c, the stroke of the nozzle can be measured as the extreme angular displacement of link 4. Measuring this from the graphical layout yields the following:
ƒ ¢u4ƒmax = 47.0°
Bœœ Cœœ C
L3 - L2
Cœœ Cœ
L3 - L2 = 2.00 in. - .75 in. = 1.25 in.
Bœ Cœ
4.8 LIMITING POSITIONS: ANALYTICAL ANALYSIS
Analytical determination of the limiting positions for a mechanism is a combination of two concepts presented earlier in this chapter:
I. The logic of configuring the mechanism into a limiting configuration. This was incorporated in the graphical method for determining the limiting positions, as presented in Section 4.7.
II. The method of breaking a mechanism into convenient triangles and using the laws of trigonometry to deter-mine all mechanism angles and lengths, as presented in Section 4.6.
Combining these two concepts to determine the position of all links in a mechanism at a limiting position is illustrated through Example Problem 4.8.