KINEMATIC AND KINETIC ANALYSIS OF QUICK

RETURN MECHANISM SCALED CAD DRAWING VECTORIAL ILLUSTRATION POSITION ANALYSIS VELOCITY ANALYSIS ACCELERATION ANALYSIS FREE BODY DIAGRAMS FORCE ANALYSIS ENERGY EQUATION APPENDIX

Table1– Position Analysis Parametric Results

Vector R⃗2 R⃗3 R⃗5 R⃗6 R⃗7 R⃗8 R⃗9 R⃗3-1 R⃗3-2 R⃗5-1 R⃗5-2 R⃗7-1

Definition O⃗2 C C⃗A B⃗D O⃗6 D ¿⃗ H⃗O8 G⃗K B⃗A C⃗B B⃗E E⃗D H⃗E

Length 3001675

525 900 550 750 850 6751000

175 350 450

Angle 135 ϴ3 ϴ5 ϴ6 ϴ7 ϴ8 ϴ9 ϴ3 ϴ3 ϴ5 ϴ5 ϴ7

Vector R⃗7-2 R⃗1 R⃗4 R⃗4-1 R⃗10 R⃗11 R⃗12 R⃗13 R⃗14 R⃗15 R⃗16

Definition G⃗H O⃗6 L O⃗2 A O⃗2 P N⃗K O⃗8 M O⃗ 8 N M⃗A A⃗L O⃗2 L P⃗O8

Length 100 375 R⃗4 1700 175 R⃗11 R⃗12 175 R⃗14 1975 175

Angle ϴ7 90 180 180 90 180 180 270 180 180 90

Loop 1 O2CAO2 R⃗2 + R⃗3 - R⃗4 = 0

Loop 2 O6DBALO6 R⃗6 - R⃗5 + R⃗3−1+ R⃗14 - R⃗1 = 0

Loop 3 ABEHO8MA -R⃗3−1 + R⃗5−1 - R⃗7−1 - R⃗8 + R⃗11+R⃗13 = 0

Loop 4 O8GHKNO8 -R⃗8 - R⃗7−2 + R⃗9- R⃗10 - R⃗12 = 0

Unknown

Parametric Equation Eq. Value

Ө3β1 = (-R2 sin(Ө2 - Ө4))/ R3

Ө3 = Ө4 + sin-1 β1 OR Ө3 = 180 + Ө4 + sin-1 β1 6 187.2760

R4 R2sin(Ө2 - Ө3)/ sin(Ө4 - Ө3) 5 1.8736 mδ1

β2 = [(R12+R3−1

2 - R52 - R6

2 + R142 + 2R1R3-1cos(Ө1–Ө3) – 2R14(sinӨ14(R1sinӨ1

– R3-1sinӨ3) + cosӨ14( R1cosӨ1 – R3-1cosӨ3)))/(-2R5R6)]

δ1 = cos-1 β2 OR δ1 = 180 + cos-1 β2

8 72.810

Ө5

β3 = [[R6cosδ1 – R5 + √{¿¿R6cosδ1-R5)2 + (R6sinδ1)2 - (R1sinӨ1 – R3-1sinӨ5 -R14sinӨ14)2}]/ ( R6sinδ1 + R1sinӨ1 – R3-1sinӨ3 -R14sinӨ14 )]

Ө5 = 2tan-1 β3 OR Ө5 = 180 + 2tan-1 β3

-√ component was considered for the results

8 284.08450

Ө6 δ1+ Ө5 - 356.90

δ2

β4 = [(R3−12 +R5−1

2 - R7−12 - R8

2 + R112 + R13

2 - 2R3-1R5-1cos(Ө5–Ө3) + 2R11R13cos(Ө13–Ө11) + 2((-R3-1sinӨ3 + R5-1sinӨ5)×( R11sinӨ11 +

R13sinӨ13)) + 2((-R3-1cosӨ3 + R5-1cosӨ5)×( R11cosӨ11 + R13cosӨ13)))/(2R7-1R8)]

δ2 = cos-1 β4 OR δ2 = 180 + cos-1 β4

8 52.830

Ө8

β5 = [[R7-1cosδ2 – R8 + √{¿¿R7-1cosδ2-R8)2 + (R7-1sinδ2)2- (-R3-1sinӨ3 + R5sinӨ5 + R11sinӨ11 + R13sinӨ13)2}]/ (R7-1sinδ2 – R3-1sinӨ3 + R5-1sinӨ5 +

R11sinӨ11 + R13sinӨ13)]Ө8 = 2tan-1 β5 OR Ө8 = 180 + 2tan-1 β5

-√ component was considered for the results

8 191.20

Ө7 δ2+ Ө8 - 2440Ө9

β6 = ((R8 sin(Ө8–Ө12) + R7-2 sin(Ө7–Ө12) + R10 sin(Ө10–Ө12))/ R9)Ө9 = Ө12 + sin-1 β6 OR Ө9 = 180 + Ө12 + sin-1 β6

6 1840

R12 (-R8sin(Ө8 – Ө9) - R7-2sin(Ө7 – Ө12) – R10sin(Ө10 – Ө9))/ sin(Ө12 – Ө9) 5 0.0682 mPosition Analysis of Joints and Centroids Parametric Results (r⃗i = ri∠Өi; m∠degree)

r⃗c R2∠Ө2 0.3∠135 r⃗h r⃗e - R7-1∠Ө7 1.02∠161.64r⃗a r⃗c + R3∠Ө3 1.8736∠180 r⃗ g r⃗h - R7-2∠Ө7 1∠156

r⃗ g 3 r⃗c + 12R3∠Ө3 1.0483∠174.19 r⃗ g 9 r⃗ g +

12 R9∠Ө9 1.39∠164.26

r⃗b r⃗c + R3-2∠Ө3 1.2∠175.95 r⃗k r⃗ g + R9∠Ө9 1.8∠168.86

r⃗e r⃗b + R5-1∠Ө5 1.1648∠184.15 r⃗o6 r⃗d - R6∠Ө6 2.0103∠190.75

r⃗d r⃗e + R5-2∠Ө5 1.157∠201.5 r⃗o8

r⃗4−1 + R16

∠Ө161.71∠174.12

r⃗ g 6 r⃗d - 12 R6∠Ө6 1.58∠194.67

Table 3– Acceleration Analysis Parametric Results

Loop 1 R⃗2 + R⃗3 - R⃗4 = 0Loop 2 R⃗6 - R⃗5 + R⃗3−1+ R⃗14 - R⃗1 = 0Loop 3 -R⃗3−1 + R⃗5−1 - R⃗7−1 - R⃗8 + R⃗11+R⃗13 = 0Loop 4 -R⃗8 - R⃗7−2 + R⃗9- R⃗10 - R⃗12 = 0

Unknown Parametric Equation Eq Valueα 2 given - 0α 3 (R2ω2

2sin(Ө2 – Ө4)+ R3ω32sin(Ө3 – Ө4))/ R3cos(Ө3 – Ө4) 19 -11.156 rad/s2

R̈4 (-R2ω22sin(Ө2 – Ө3)- R3ω3

2)/cos(Ө4 – Ө3) 20 -18.8821 rad/s2α 5

(R3-1α3sin(Ө3 – Ө6) + R3-1ω32cos(Ө3 – Ө6 )- R5ω5

2sin(Ө5 – Ө6)+ R6ω62 -

R̈14 cos(Ө14 – Ө6) )/ R5sin(Ө5 – Ө6)20 -33.1 rad/s2

α 6

(-R3-1α3sin(Ө3 – Ө5 )- R3-1ω32cos(Ө3 – Ө5)+ R5ω5

2- R6ω62 cos(Ө6 – Ө5)

+R̈14 cos(Ө14 – Ө5) )/ R6sin(Ө6 – Ө5)21 -3.18 rad/s2

α 8

(R3-1α3sin(Ө3 – Ө7)- R5-1α5sin(Ө5 – Ө7 )+ R3-1ω32cos(Ө3 – Ө7)

-R5-1ω52cos(Ө5 – Ө7) + R7-1ω7

2+ R8ω8

2 cos(Ө8 – Ө7) +R̈11 cos(Ө11 – Ө7)

)/ -R8sin(Ө8 – Ө7)

20 25.052 rad/s2

α 7

(R3-1α3sin(Ө3 – Ө8)- R5-1α5sin(Ө5 – Ө8 )+ R3-1ω32cos(Ө3 – Ө8)

-R5-1ω52cos(Ө5 – Ө8) + R7-1ω7

2cos(Ө7 – Ө8) + R8ω8

2 +R̈11 cos(Ө11 – Ө8)

)/ -R7-1sin(Ө7 – Ө8)

21 25.44 rad/s2

R̈12(R8ω8

2cos(Ө8 – Ө9)+ R7-2ω72cos(Ө7 – Ө9)- R9ω9

2+ R8α8sin(Ө8 – Ө9)+R7-2α7sin(Ө7 – Ө9 ))/ cos(Ө12 – Ө9)

20 5.1786 m/s2

α 9(R8ω8

2sin(Ө8 – Ө12)+ R7-2ω72sin(Ө7 – Ө12)- R9ω9

2sin(Ө9 – Ө12)- R7-2α7cos(Ө7 – Ө12)- R8α8cos(Ө8 – Ө12))/ -R9cos(Ө9 – Ө12)

19 21.732 rad/s2Acceleration Analysis of Joints and Centroids (a⃗ i = αi∠γ i ; m/sec2∠degree)

a⃗c -R2ω22∠Ө2 -26.65∠135

a⃗aR̈4∠Ө4 -18.88∠180

a⃗g 3 a⃗c - 12 R3ω3

2∠Ө3+12 R3α3∠Ө3

' 21.1∠333.46

a⃗b a⃗c – R3-2ω32∠Ө3+ R3-2α3∠Ө3

' 20.3∠338a⃗e a⃗b – R5-1ω5

2∠Ө5+ R5-1α5∠Ө5' 13.83∠334.2

a⃗d a⃗e – R5-2ω52∠Ө5+ R5-2α5∠Ө5

' 2.854∠264.4

a⃗g 6 a⃗d - 12 R6ω6

2∠Ө6+12 R6α6∠Ө6

' 1.415∠263.5

a⃗h a⃗e + R7-1ω72∠Ө7 - R7-1α7∠Ө7

' 19.553∠309.26a⃗g a⃗h + R7-2ω7

2∠Ө7 - R7-2α7∠Ө7' 19.63∠303.34

6

a⃗g 9 a⃗g - 12 R9ω9

2∠Ө9+12 R9α9∠Ө9

' 28.045∠298.35

a⃗kR̈12∠Ө12 5.0124∠180

a⃗06, a⃗o 8 - 0

REE BODY DIAGRAMS (FBD) FOR FORCE ANALYSIS

Table 4 – Kinetic-Analysis Parametric Resultsf⃗ 3 -m3a⃗g 3 168∠333.457 Nf⃗ 6 -m6a⃗g 6

-2.834∠263.47 Nf⃗ 9 -m9a⃗g 9

-84.135∠298.35 Nq3 -J3α3 16.734 Nmq6 -J6α6 3.179 Nmq9 -J9α9 -26.1 Nm

Force FBD Equation Definition Value (N)

Output A F⃗ -F∠Фk -100∠180

F10 A (2R9Fsin(Фk - Ө9)-R9f9sin(γg9 - Ө9) – q9)/2R9cosӨ9 - -59.58

F⃗10 A F10∠ λF10 -59.58 ∠90

G⃗ C K⃗+ f⃗ 9 G∠ λg 61.78∠13.595K⃗ B F⃗+ F⃗10 K∠ λk 116.37∠329.34H C R7Gsin(λg – Ө7)/R7-1sin(Ө8 – Ө7) - -99.611H⃗ C - H∠ λh -99.611∠191.2E⃗ C H⃗ + G⃗ E∠ λe 161.36∠12.117F⃗8 D H⃗ F8∠ λF 8 -99.611∠191.2D5 E (R6f6sin(γg6 – Ө6) + 2q6)/2R6sin(Ө5 - Ө6) - -5.1756D⃗5 E - D5∠ Ө5 -5.1756∠284D6 F -R5-1Esin(λe – Ө5)/R5sin(Ө6 – Ө5) - -56.244D⃗6 F - D6∠ Ө6 -56.244∠356.9D⃗ F D⃗5 +D⃗6 D∠ λd 57.977∠172F⃗6 G D⃗ - f⃗ 6 F6∠ λF 6 57.974∠174.81B⃗ H E⃗ - D⃗ B∠ λb 108.76∠22.67

F4 I (2R3-2Bsin(λb – Ө3) + R3f3sin(γg3 – Ө3) + q3)/-2R3sinӨ3

- -54.444

F⃗4 I F4∠90 F4∠ λF 4 -54.444∠90A⃗ J F⃗4 A∠ λa -54.444∠90C⃗ K A⃗ + B⃗ + f⃗ 3 C∠ λc 80.04∠308.6F⃗2 L -C⃗ F2∠ λF2 80.04∠308.6

Torque T L -R2Csin(λc – Ө2) - -2.676 Nm

∑ F M F⃗2 + F⃗4 + F⃗6 + F⃗8 + F⃗10 + F⃗ + f⃗ 3 + f⃗ 6 + f⃗ 9

- 0.61271∠343.92

∑ M o 2 Mr⃗o 4× F⃗4+ r⃗o 6× F⃗6+ r⃗o 8× F⃗8+ r⃗k× (F⃗10+ F⃗) + r⃗ g 3× f⃗ 3

+ r⃗ g 6× f⃗ 6 + r⃗ g 9× f⃗ 9

- -63.835 Nm

∑ P M Tω2 + ⌊ F⃗ . V⃗ K+ƒ⃗3. V⃗ g3+ƒ⃗6 . V⃗ g 6+ƒ⃗9. V⃗ g9 ⌋ + q3 × ω3

+q6 ×ω6 + q9 ×ω9

- 30.812 Pa