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DESIGN OF MACHINERY ! SOLUTION MANUAL 5-8-1 PROBLEM 5-8 Statement: Design a linkage to carry the body in Figure P5-1 through the two positions P1 and P2 at the angles shown in the figure. Use analytical synthesis without regard for the fixed pivots shown. Use the free choices given below. Given: Coordinates of the points P1 and P2 with respect to P1: P 1x 0.0 P 1y 0.0 P 2x 1.236 P 2y 2.138 Angles made by the body in positions 1 and 2: 210 .deg θ P1 θ P2 147.5 .deg β2 27.0 .deg φ 204.4 .deg γ2 40.0 .deg ψ 74.0 .deg Free choices for the WZ dyad : z 1.075 Free choices for the US dyad : s 1.240 Two argument inverse tangent return 0.5 .π if x 0 atan2( x , y ) return atan y x atan y x Solution: if x > 0 π otherwise See Figure P5-1 and Mathcad file P0508. 1. Note that this is a two-position motion generation (MG) problem because the output is specified as a complex motion of the coupler, link 3. Because of the data given in the hint, the second method of Section 5.3 will be used here. 2. Define the position vectors R1 and R2 and the vector P21 using Figure 5-1 and equation 5.1. P 1x R1 R2 P 1y P 2x P 21x P 2y P 21y R2 R1 P 21x = 1.236 P 21y = 2.138 p 21 3. 4. P 21x 2 P 21y 2 p 21 = 2.470 From the trigonometric relationships given in Figure 5-1, determine α2 and δ2. α2 θ P2 θ P1 α 2 = 62.500 deg δ2 atan2 P 21x , P 21y δ 2 = 120.033 deg Solve for the WZ dyad using equations 5.8. Z 1x 2nd Edition, 1999 z .cos( φ ) Z 1x = 0.979 Z 1y z .sin( φ ) Z 1y = 0.444 DESIGN OF MACHINERY A cos β 2 B sin β 2 C cos α 2 W 1x W 1y θ 1 A = 0.109 D sin α 2 D = 0.887 B = 0.454 E p 21 .cos δ 2 E = 1.236 C = 0.538 F p 21 .sin δ 2 F = 2.138 D .Z 1y B . C .Z 1y E D .Z 1x F 2 .A A . C .Z 1y D .Z 1x B . C .Z 1x F D .Z 1y E W 1x = 1.462 W 1y = 3.367 2 .A W 1x 2 w = 3.670 W 1y atan2 W 1x , W 1y θ = 246.528 deg Solve for the US dyad using equations 5.12. s .cos( ψ ) S 1x A cos γ 2 B sin γ 2 C cos α 2 U 1x U 1y σ S 1x = 0.342 1 1 A . C .S 1x s .sin( ψ ) S 1y D sin α 2 D = 0.887 B = 0.643 E p 21 .cos δ 2 E = 1.236 C = 0.538 F p 21 .sin δ 2 F = 2.138 D .S 1y A . C .S 1y E B . C .S 1y D .S 1x F D .S 1x F B . C .S 1x D .S 1y E 2 .A U 1x S 1y = 1.192 A = 0.234 2 .A 2 u 6. 1 A . C .Z 1x 2 w 5. SOLUTION MANUAL 5-8-2 2 U 1x = 3.180 U 1y = 4.439 u = 5.461 U 1y atan2 U 1x , U 1y σ = 234.381 deg Solve for links 3 and 1 using the vector definitions of V and G. Link 3: V 1x z .cos( φ ) s .cos( ψ ) V 1x = 1.321 V 1y z .sin( φ ) s .sin( ψ ) V 1y = 1.636 θ3 Link 1: 2nd Edition, 1999 atan2 V 1x , V 1y 2 v V 1x G 1x w .cos( θ ) θ 3 = 231.086 deg 2 v = 2.103 V 1y v .cos θ 3 u .cos( σ ) G 1x = 0.398 DESIGN OF MACHINERY SOLUTION MANUAL 5-8-3 w .sin( θ ) G 1y θ1 8. u .sin( σ ) atan2 G 1x , G 1y 2 g 7. v .sin θ 3 θ 1 = 54.796 deg 2 G 1x g = 0.690 G 1y Determine the initial and final values of the input crank with respect to the vector G. θ 2i θ θ 2f θ 2i θ 2i = 301.323 deg θ1 θ 2f = 274.323 deg β2 Define the coupler point with respect to point A and the vector V. rp δp z φ θ3 δ p = 26.686 deg r p = 1.075 9. G 1y = 0.564 Locate the fixed pivots in the global frame using the vector definitions in Figure 5-2. ρ1 atan2 P 1x , P 1y 2 R1 P 1x ρ 1 = 90.000 deg 2 R 1 = 0.000 P 1y O 2x R 1 .cos ρ 1 z .cos( φ ) O 2y R 1 .sin ρ 1 z .sin( φ ) O 4x R 1 .cos ρ 1 s .cos( ψ ) O 4y R 1 .sin ρ 1 s .sin( ψ ) w .cos( θ ) w .sin( θ ) u .cos( σ ) u .sin( σ ) O 2x = 2.441 O 2y = 3.811 O 4x = 2.838 O 4y = 3.247 10. Determine the rotation angle of the fourbar frame with respect to the global frame (angle from the global X axis to the line O2O4. θ rot O 2x , O 4y atan2 O 4x O 2y θ rot = 54.796 deg 11. Determine the Grashof condition. Condition( S , L , P , Q ) SL S PQ P L Q return "Grashof" if SL PQ return "non-Grashof" otherwise Condition( g , w , u , v ) = "Grashof" 12. DESIGN SUMMARY 2nd Edition, 1999 Link 2: w = 3.670 θ = 246.528 deg Link 3: v = 2.103 Link 4: u = 5.461 θ 3 = 231.086 deg σ = 234.381 deg DESIGN OF MACHINERY SOLUTION MANUAL 5-8-4 Link 1: g = 0.690 Coupler: r p = 1.075 θ 1 = 54.796 deg δ p = 26.686 deg Crank angles: θ 2i = 301.323 deg θ 2f = 274.323 deg 13. Draw the linkage, using the link lengths, fixed pivot positions, and angles above, to verify the design. O2 1.236 G1 Y O4 U2 P2 S2 B2 V2 2.138 W2 A2 A1 X P1 V1 S1 B1 2nd Edition, 1999 U1 Z2 Z1 62.5° W1