end of fallvβ0inrad⋅:=sβ0in⋅:=maximum lift of riseshh:=beginning of risev00inrad⋅:=s00in⋅:=Critical Extreme Positions (CEPs):jrfθ C3, C4,()1β36C3⋅ 24 C4⋅θβ⋅+⋅:=arfθ C2, C3, C4,()1β22C2⋅ 6C3⋅θβ⋅+ 12 C4⋅θβ2⋅+⋅:=vrfθ C1, C2, C3, C4,()1βC12C2⋅θβ⋅+ 3C3⋅θβ2⋅+ 4C4⋅θβ3⋅+⋅:=srfθ C0, C1, C2, C3, C4,()C0C1θβ⋅+ C2θβ2⋅+ C3θβ3⋅+ C4θβ4⋅+:=Polynomial functions for rise-fall segment:polynomial angle at max liftθhβ2:=lift heighth2in⋅:=rise-fall angle durationβ 240 deg=βθcam_endθcam_start−:=camshaft end angle of fallθcam_end300 deg⋅:=camshaft start angle of riseθcam_start60 deg⋅:=Rise-fall specs:'Dwell - Rise-Fall - Dwell' Cam Polynomial Function DesignC2C3C43264−32in=C2C3C4Find C2C3, C4,():=vrfβ C1, C2, C3, C4,()vβ=srfβ C0, C1, C2, C3, C4,()sβ=srfθhC0, C1, C2, C3, C4,()sh=Giveninitial guessesC410 in⋅:=C310 in⋅:=C210 in⋅:=Alternative Solution Method:0 100 2000.0500.050.1srfθ C0, C1, C2, C3, C4,()θdegθ 0 deg⋅ 1 deg⋅,β..:=C2C3C43264−32in=C2C3C4A1−B⋅:=Aθhβ212βθhβ313βθhβ414β:=Bh0in⋅0in⋅:=C10in⋅:=C00in⋅:=Resulting Equations and Solution:s-v-a-j diagrams:θ 0 deg⋅ 1 deg⋅, 360 deg⋅..:=s θ()0 θθcam_start≤ifsrfθθcam_start− C0, C1, C2, C3, C4,()θcam_startθ<θcam_end<ifsrfβ C0, C1, C2, C3, C4,()θθcam_end≥if:=v θ()0 θθcam_start≤ifvrfθθcam_start− C1, C2, C3, C4,()θcam_startθ<θcam_end<if0 θθcam_end≥if:=a θ()0 θθcam_start≤ifarfθθcam_start− C2, C3, C4,()θcam_startθ<θcam_end<if0 θθcam_end≥if:=0 50 100 150 200 250 300 350 4000.0100.010.020.030.040.050.06s θ()θdeg0 50 100 150 200 250 300 350 4000.040.0200.020.04v θ()θdeg0 50 100 150 200 250 300 350 4000.0500.050.1a θ()θdegNOTE: discontinuities in a(θ) at start and end of rise-fall segment result in non-finite