OF DESIGN MACHINERY
SOLUTION MANUAL 3-66-1
PROBLEM 3-66 Statement: Design a fourbar mechanism to give the three positions shown in Figure P3-18 using the fixed pivots O2 and O4 shown. (See Example 3-7.) Build a cardboard model and add a driver dyad to limit its motion to the range of positions designed, making it a sixbar. See Figure P3-18 and Mathcad file P0366. Length of link 5: L5 4.000 Length of link 2b: L2b 2.000
Solution:
Design choices: 1. 2. 3.
Draw link CD in its three design positions C 1D1, C 2D2, C 3D3 in the plane as shown. Draw the ground link O2O4 in its desired position in the plane with respect to the first coupler position C 1D1. Draw construction arcs from point C 2 to O2 and from point D2 to O2 whose radii define the sides of triangle C2O2D2. This defines the relationship of the fixed pivot O2 to the coupler line CD in the second coupler position. Draw construction arcs from point C 2 to O4 and from point D2 to O4 whose radii define the sides of triangle C2O4D2. This defines the relationship of the fixed pivot O4 to the coupler line CD in the second coupler position. Transfer this relationship back to the first coupler position C 1D1 so that the ground plane position O2'O4' bears the same relationship to C1D1 as O2O4 bore to the second coupler position C 2D2. Repeat the process for the third coupler position and transfer the third relative ground link position to the first, or reference, position. The three inverted positions of the ground link that correspond to the three desired coupler positions are labeled O2O4, O2'O4', and O2"O4" in the first layout below and are renamed E1F1, E2F2, and E3F3, respectively, in the second layout, which is used to find the points G and H.
4.
5. 6. 7.
D1 D2 C1 C2 O2''
D3
C3 O2 O2'
8. 9.
O4'
O4
O4''
Draw construction lines from point E1 to E2 and from point E2 to E3. Bisect line E1E2 and line E2E3 and extend their perpendicular bisectors until they intersect. Label their intersection G.
10. Repeat steps 2 and 3 for lines F1F2 and F2F3. Label the intersection H.
DESIGN OF MACHINERY
SOLUTION MANUAL 3-66-2
11. Connect E1 with G and label it link 2. Connect F1 with H and it label link 4. Reinverting, E1 and F1 are the original fixed pivots O2 and O4, respectively. 12. Line GH is link 3. Line O2O4 is link 1a (ground link for the fourbar). The fourbar is now defined as O2GHO4 and has link lengths of Ground link 1a Link 3 L1a 4.000 L3 6.002 Link 2 Link 4 L2 2.000 L4 7.002
H
3
E3
4
G 2 O2 E1 O4 E2 F 1 F3
F2
13. Check the Grashof condition. Note that any Grashof condition is potentially acceptable in this case. Condition( a ) S min ( a ) b c d b c d L max (a ) b c d SL S L PQ a b c d SL return "Grashof" if SL PQ return "Special Grashof" if SL = PQ return "non-Grashof" otherwise Condition1a 2 34 "Grashof" L L L L The fourbar that will provide the desired motion is now defined as a non-Grashof crank rocker in the open configuration. It now remains to add the original points C 1 and D1 to the coupler GH and to define the driving dyad, which in this case will drive link 4 rather than link 2. 14. Select a point on link 2 (O2G) at a suitable distance from O2 as the pivot point to which the driver dyad will be
DESIGN OF MACHINERY
SOLUTION MANUAL 3-66-3
connected and label it B. (Note that link 2 is now a ternary link with nodes at O2, B, and G.) In the solution below, the distance O2B was selected to be L2b 2.000 . Thus, in this case B and G coincide. 15. Draw a construction line through B1B3 and extend it up to the left. 16. Layout the length of link 5 (design choice) along the extended line. Label the other end A. 17. Draw a circle about O6 with a radius of one-half the length B1B3 and label the intersections of the circle with the extended line as A1 and A3. In the solution below the radius was measured as L6 1.399. 18. The driver fourbar is now defined as O2BAO6 with link lengths Link 6 (crank) L6 1.399
Link 5 (coupler) L5 4.000 Link 1b (ground) L1b 4.257 Link 2b (rocker) L2b 2.000 19. Use the link lengths in step 18 to find the Grashoff condition of the driving fourbar (it must be Grashoff and the shortest link must be link 6). Condition6 1b 2b 5 "Grashof" L L L L
H2 H1
3 3 H3 D2 D3 C1 C2 4 4 G2 2 3 C3 2 O2 2 G1 1a 4 O4 D1
1b A1 6 O6 5
G3
A3