DESIGN OF MACHINERY SOLUTION MANUAL 4-21-1! PROBLEM 4-21Statement:For the linkage in Figure P4-8, find its limit (toggle) positions in terms of the angle of link O2A referenced to the line of centers O2O4 when driven from link O2A. Then calculate and plot the xy coordinates of coupler point P between those limits, referenced to the line of centers O2O4.y31O22A14.000°4YBPO4XxGiven:Link lengths:Input (O2A) a 5.00 in.Coupler (AB) b 4.40 in.Rocker (O4B) c 5.00 in.Ground link d 9.50 in.Coupler point data:p 8.90 in.δ56 deg.Coordinate transformation angle:α14 deg.Two argument inverse tangentatan2 x y,( ) 0.5π.return x 0ifatanyxreturn x 0>ifatanyxπotherwiseSolution:See Figure P4-8 and Mathcad file P0421.1. Define the coordinate systems. The local frame has origin at O2 with the positive x axis going through O4. Let the global frame also have its origin at O2 with the positive X axis to the right.2. Check the Grashof condition of the linkage.Condition S L,P,Q,()SL S LPQ P Q"Grashof"return SL PQif"non-Grashof"return otherwiseCondition b d,a,c,( ) "non-Grashof"=3. Using equations 4.33, determine the crank angles (relative to the line AD) at which links 3 and 4 are in toggle.arg1a()2d()2b()2c()22 a.d.()bc.ad.()arg11.209=arg2a()2d()2b()2c()22 a.d.()bc.ad.()arg20.283=θ2toggleacos arg2θ2toggle73.6 deg=The other toggle angle is the negative of this.2nd Edition, 1999DESIGN OF MACHINERY SOLUTION MANUAL 4-21-24. Define one cycle of the input crank between limit positions:θ2θ2toggleθ2toggle2 deg.,θ2toggle..5. Determine the values of the constants needed for finding θ3 from equations 4.11b and 4.12.K1daK4dbK5c2d2a2b22 a.b.()K11.9000=K42.1591=K52.4911=Dθ2cosθ2K1K4cosθ2.K5Eθ22 sinθ2.Fθ2K1K41 cosθ2.K56. Use equation 4.13 to find values of θ3 for the open circuit.θ3θ22 atan2 2 Dθ2.Eθ2Eθ224 Dθ2.Fθ2.,.7. Use equations 4.27 to define the x- and y-components of the vector RP.RPRARPARAa cosθ2j sinθ2..RPAp cosθ3δj sinθ3δ..RPxθ2a cosθ2.p cosθ3θ2δ.RPyθ2a sinθ2.p sinθ3θ2δ.8. Transform the coupler point coordinates in the local frame to the global frame using coordinate transformation equations.XPθ2RPxθ2cosα().RPyθ2sinα().YPθ2RPxθ2sinα().RPyθ2cosα().9. Plot the coordinates of the coupler point in the global system. 75 50 25 0 25 50 751001020COUPLER POINT COORDINATESCrank Angle - degCoupler Point CoordinatesXPθ2inYPθ2inθ2deg2nd Edition,