vmod_trapθβ, h,()θsmod_trapθβ, h,()dd:=Using MathCAD's Approximate Numerical Differentiationinstead of Equations 8.15-8.19 for v:smod_trapθβ, h,()hCa⋅bπθβ⋅bπ2sinπbθβ⋅⋅−⋅ 0 θ≤b2β⋅<ifhCa⋅12θβ2⋅ b1π12−⋅θβ⋅+ b2181π2−⋅+⋅b2β⋅θ≤1d−2β⋅<ifhCa⋅bπc2+θβ⋅dπ2+ b2181π2−⋅+1d−()28−dπ2cosπdθβ1−2−⋅⋅−⋅hCa⋅12−θβ2bπ1+b2−θβ⋅+ 2d2⋅ b2−()1π218−⋅+14−⋅1d+2β⋅≤ifhCa⋅bπθβ⋅2d2b2−()⋅π2+1b−()2d2−4+bπ2sinπdθβ1−⋅⋅−⋅ 1b2−if:=Ca4.888=Ca4 π2⋅π28−()b2d2−()⋅ 2 π⋅π2−()⋅ b⋅−π2+:=(from Figure 8-18)d .25:=c 0.5:=b 0.25:=Modified Trapezoidal Acceleration Functions (from Equations 8.15-8.19):rise_fall:hrise_fall1.5 in⋅:=βrise_fall130 deg⋅:=θstart_rise_fall200 deg⋅:=βfall50 deg⋅:=θstart_fall120 deg⋅:=fall:hrise1in⋅:=βrise60 deg⋅:=θstart_rise30 deg⋅:=rise:ω 1000π30⋅radsec⋅:=camshaft:Motion Specifications:Multiple Dwell Cam Design Exampleamod_trapθβ, h,()Cahβ2⋅ sinπbθβ⋅⋅ 0 θ≤b2β⋅<ifCahβ2⋅b2β⋅θ≤1d−2β⋅<ifCahβ2⋅ cosπdθβ1d−2−⋅⋅1d−2β⋅θ≤1d+2β⋅<ifCa−hβ2⋅1d+2β⋅θ≤ 1b2−β⋅<ifCahβ2⋅ sinπdθβ1−⋅⋅ 1b2−β⋅θ≤β≤if:=Example Modified Trapezoidal Acceleration s-v-a Diagrams:β 100 deg⋅:= θ 0 2 deg⋅,β..:= h1:=0 50 10000.51smod_trapθβ, h,()θdeg0 50 10000.511.5vmod_trapθβ, h,()θdeg0 50 100202amod_trapθβ, h,()θdeg3-4-5-6 Single Dwell Polynomial Function (Equation 8.26): s3456θβ, h,()h64θβ3⋅ 192θβ4⋅− 192θβ5⋅+ 64θβ6⋅−⋅:=Using MathCAD Symbolic Differentiation (copy and paste expression,put cursor next to θ, select from menu: Symbolics-Variable-Differentiate,copy expression to function definition):h64θβ3⋅ 192θβ4⋅− 192θβ5⋅+ 64θβ6⋅−⋅h 192θ2β3⋅ 768θ3β4⋅− 960θ4β5⋅ 384θ5β6⋅−+⋅v3456θβ, h,()h 192θ2β3⋅ 768θ3β4⋅− 960θ4β5⋅ 384θ5β6⋅−+⋅:=a3456θβ, h,()h 384θβ3⋅ 2304θ2β4⋅− 3840θ3β5⋅ 1920θ4β6⋅−+⋅:=j3456θβ, h,()h384β34608θβ4⋅− 11520θ2β5⋅ 7680θ3β6⋅−+⋅:=Example 3-4-5-6 Polynomial s-v-a Diagrams:β 100 deg⋅:= θ 0 2 deg⋅,β..:= h 1:=0 50 10000.51s3456θβ, h,()θdeg0 50 100202v3456θβ, h,()θdeg0 50 10010010a3456θβ, h,()θdegs-v-a Diagrams for Entire Cam:s θ()0 0 deg⋅θ≤θstart_rise<ifsmod_trapθθstart_rise−βrise, hrise,()θstart_riseθ≤θstart_riseβrise+()<ifhriseθstart_riseβrise+()θ≤θstart_fall<ifhrisesmod_trapθθstart_fall−βfall, hrise,()−θstart_fallθ≤θstart_fallβfall+()<if0 θstart_fallβfall+()θ≤θstart_rise<ifs3456θθstart_rise_fall−βrise_fall, hrise_fall,()θstart_rise_fallθ≤θstart_rise_fallβrise_fall+()<if0 330 deg⋅θ≤if:=θ 0 deg⋅ 1 deg⋅, 360 deg⋅..:=0 50 100 150 200 250 300 35000.010.020.030.04s θ()θdegMaximum acceleration will occur in rise-fall segment:F 264.832 lbf=F ma⋅:=m 0.062 slug=mWg:=g 32.174ftsec2=W 2 lbf⋅:=a 4.26 103×ftsec2=a ω2amax⋅:=amax0.388 ft=amaxa3456θmax_accθstart_rise_fall−βrise_fall, hrise,():=θmax_acc265.001 deg=θmax_accroot j3456θθstart_rise_fall−βrise_fall, hrise,()θ,():=initial guessθ 270 deg⋅:=Peak acceleration amplitude occurs where the jerk is zero:200 250 3000.20.100.1a3456θθstart_rise_fall−βrise_fall, hrise,()θdegθ θstart_rise_fallθstart_rise_fall1