DESIGN OF MACHINERY SOLUTION MANUAL 8-7-1! PROBLEM 8-7Statement:Design a double-dwell cam to move a follower from 0 to 2.5 in in 60 deg, dwell for 120 deg, fall 2.5 in in 30 deg and dwell for the remainder. The total cycle must take 4 sec. Choose suitable programs for rise and fall to minimize accelerations. Plot the SVAJ diagrams.Given:RISE DWELL FALL DWELLβ160deg.β2120deg.β330deg.β4150deg.h12.5in.h20in.h32.5in.h40in.Cycle time:τ4sec.Solution:See Mathcad file P0807.1. The camshaft turns 2π rad during the time for one cycle. Thus, its speed isω2π.rad.τω1.571radsec=2. From Table 9-2, the motion program with lowest acceleration that does not have infinite jerk is the modified trapezoidal. The modified trapezoidal motion is defined in local coordinates by equations 8.13. The numerical constants in these equations are first defined below in constants C1 through C11.c10.38898448c20.0309544c34.888124c461.425769c52.44406184c60.22203097c70.00723407c81.6110154c90.3055077c104.6660917c111.22926483. The equations for the rise or fall interval (β) are divided into 5 subintervals. These are:for 0 <= θ <= β/8 where, for these equations, θ is a local coordinate that ranges from 0 to β,s1θβ,h,()hc1θβ.c2sin4π.θβ...v1θβ,h,()c1hβ.1cos4π.θβ..a1θβ,h,()c3hβ2.sin4π.θβ..j1θβ,h,()c4hβ3.cos4π.θβ..for β/8 <= θ <= 3β/8s2θβ,h,()hc5θβ2.c6θβ.c7.v2θβ,h,()hβc3θβ.c6.a2θβ,h,()c3hβ2.j2θβ,h,()0in.for 3β/8 <= θ <= 5β/8s3θβ,h,()hc8θβ.c2sin4π.θβ.π.c9.v3θβ,h,()hβc8c1cos4π.θβ.π..2nd Edition, 1999DESIGN OF MACHINERY SOLUTION MANUAL 8-7-2a3θβ,h,()c3hβ2.sin4π.θβ.π.j3θβ,h,()c4hβ3.cos4π.θβ.π.for 5β/8 <= θ <= 7β/8s4θβ,h,()hc5θβ2.c10θβ.c11.v4θβ,h,()hβc3θβ.c10.a4θβ,h,()c3hβ2.j4θβ,h,()0in.for 7β/8 <= θ <= βs5θβ,h,()hc1θβ.c2sin4π.θβ.3π..c81.v5θβ,h,()c1hβ.1cos4π.θβ.3π..a5θβ,h,()c3hβ2.sin4π.θβ.3π..j5θβ,h,()c4hβ3.cos4π.θβ.3π..4. The above equations can be used for a rise or fall by inserting the proper values of θ, β, and h. To plot the SVAJ curves, first define a range function that has a value of one between the values of a and b and zero elsewhere. Rθa,b,()ifθa>()θb().1,0,()5. The global SVAJ equations are composed of four intervals (rise, dwell, fall, and dwell). The local equations above must be assembled into a single equation each for S, V, A, and J that applies over the range 0 <= θ <= 360 deg.6. Write the global SVAJ equations for the first interval, 0 <= θ <= β1. For this interval, the local and global frames are coincident so the local equations can be used as written, substituting only for h1 for h and β1 for β. Note that each subinterval function is multiplied by the range function so that it will have nonzero values only over its subinterval.For 0 <= θ <= β1S1θ()Rθ0deg.,β18,s1θβ1,h1,.Rθβ18,38β1.,s2θβ1,h1,.Rθ38β1.,58β1.,s3θβ1,h1,.Rθ58β1.,78β1.,s4θβ1,h1,.+...Rθ78β1.,β1,s5θβ1,h1,.+...V1θ()Rθ0deg.,β18,v1θβ1,h1,.Rθβ18,38β1.,v2θβ1,h1,.Rθ38β1.,58β1.,v3θβ1,h1,.Rθ58β1.,78β1.,v4θβ1,h1,.+...Rθ78β1.,β1,v5θβ1,h1,.+...2nd Edition, 1999DESIGN OF MACHINERY SOLUTION MANUAL 8-7-3A1θ()Rθ0deg.,β18,a1θβ1,h1,.Rθβ18,38β1.,a2θβ1,h1,.Rθ38β1.,58β1.,a3θβ1,h1,.Rθ58β1.,78β1.,a4θβ1,h1,.+...Rθ78β1.,β1,a5θβ1,h1,.+...J1θ()Rθ0deg.,β18,j1θβ1,h1,.Rθβ18,38β1.,j2θβ1,h1,.Rθ38β1.,58β1.,j3θβ1,h1,.Rθ58β1.,78β1.,j4θβ1,h1,.+...Rθ78β1.,β1,j5θβ1,h1,.+...7. Write the global SVAJ equations for the second interval, β1 <= θ <= β1+ β2. For this interval, the value of S is the value of S at the end of the previous interval and the values of V, A, and J are zero because of the dwell.For β1 <= θ <= β1+ β2S2θ()Rθβ1,β1β2,S1β1.V2θ()Rθβ1,β1β2,0.in.A2θ()Rθβ1,β1β2,0.in.J2θ()Rθβ1,β1β2,0.in.8. Write the global SVAJ equations for the third interval, β1 + β2 <= θ <= β1.+ β2 + β3. For this interval, the local and global frames are not coincident so we must transform the local θ to the global θ by subtracting the first two intervals from it. Note that, for a fall, the local equation for s is subtracted from the value of S at the end of the previous interval while V, A, and J are simply negated.For β1 + β2 <= θ <= β1.+ β2 + β3Letaβ1β2S3θ()Rθa,aβ38,s1θaβ3,h3,.Rθaβ38,a38β3.,s2θaβ3,h3,.+...Rθa38β3.,a58β3.,s3θaβ3,h3,.+...Rθa58β3.,a78β3.,s4θaβ3,h3,.+...Rθa78β3.,aβ3,s5θaβ3,h3,.+...Rθa,aβ3,S2a().+...2nd Edition, 1999DESIGN OF MACHINERY SOLUTION MANUAL 8-7-4V3θ()Rθa,aβ38,v1θaβ3,h3,.Rθaβ38,a38β3.,v2θaβ3,h3,.+...Rθa38β3.,a58β3.,v3θaβ3,h3,.+...Rθa58β3.,a78β3.,v4θaβ3,h3,.+...Rθa78β3.,aβ3,v5θaβ3,h3,.+...A3θ()Rθa,aβ38,a1θaβ3,h3,.Rθaβ38,a38β3.,a2θaβ3,h3,.+...Rθa38β3.,a58β3.,a3θaβ3,h3,.+...Rθa58β3.,a78β3.,a4θaβ3,h3,.+...Rθa78β3.,aβ3,a5θaβ3,h3,.+...J3θ()Rθa,aβ38,j1θaβ3,h3,.Rθaβ38,a38β3.,j2θaβ3,h3,.+...Rθa38β3.,a58β3.,j3θaβ3,h3,.+...Rθa58β3.,a78β3.,j4θaβ3,h3,.+...Rθa78β3.,aβ3,j5θaβ3,h3,.+...9. Write the global SVAJ equations for the fourth interval, β1 + β2 + β3 <= θ <= β1.+ β2 + β3 + β4. For this interval, the values of S, V, A, and J are zero because of the dwell.For β1 + β2 + β3 <= θ <= β1.+ β2 + β3 + β4Letaβ1β2β3S4θ()Rθa,aβ4,0.in.V4θ()Rθa,aβ4,0.in.A4θ()Rθa,aβ4,0.in.J4θ()Rθa,aβ4,0.in.10. Write the complete global equation for the displacement and plot it over one rotation of the cam, which is the sum of the four intervals defined above.Sθ()S1θ()S2θ()S3θ()S4θ()θ0deg.1deg.,360deg...2nd Edition, 1999DESIGN OF MACHINERY SOLUTION MANUAL 8-7-50 60 120 180 240 300 3600123DISPLACEMENT, SCam Rotation Angle, degDisplacement, inSθ()inθdeg11. Write the complete global equation for the velocity and plot it over one rotation of the cam, which is the sum of the four intervals defined above.Vθ()V1θ()V2θ()V3θ()V4θ()0 60 120 180 240 300 36010505VELOCITY, VCam Rotation Angle, degVelocity, inVθ()inθdeg12. Write the complete global equation for the acceleration and plot it over one rotation of the cam, which is the sum of the four intervals defined above.Aθ()A1θ()A2θ()A3θ()A4θ()2nd Edition, 1999DESIGN OF MACHINERY SOLUTION MANUAL 8-7-60 60 120 180 240 300 360502502550ACCELERATION, ACam Rotation Angle, degAcceleration, inAθ()inθdeg13. Write the complete global equation for the jerk and plot it over one rotation of the cam, which is the sum of the four