Velocity & Acceleration Analysis

Transcription

1 Velociy & Acceleraion Analysis Secion 4 Velociy analysis deermines how fas pars of a machine are moving. Linear Velociy (v) Sraigh line, insananeous speed of a poin. ds s v = d Linear velociy is a vecor. v A = 3/s 45 0 Roaional Velociy (w) The rae of roaion of a link. dθ θ ω = d Roaional velociy is a vecor. ω 2 = 300 rad/s ccw Relaing Linear Velociy (v) and Roaional Velociy (w) v For poins on he same link: v = r w Poins have linear velociy (v) A 2 2 r = 18 in Links have roaional velociy (ω) Relaive Velociy Mehod v A =50 f/s r A = 8 in ω Main principle is ha wo poins on a rigid body can only have a relaive velociy: Perpendicular o he line ha connecs hem. A A he moion of, as seen from A v /A = v fi v A Relaive velociy equaion is used o form vecor polygons, and deermine velociies of key poins. v = v A +> v /A 27

2 Acceleraion Analysis deermines he exen ha pars of a machine are speedingup or slowing down. I is imporan because a force is required o produce acceleraions. Linear Acceleraion (a): Acceleraion of a poin, a, is caused by a change in velociy. Velociy can change is: Magniude: angenial acceleraion dv v a = d Direcion: normal acceleraion 2 v = r a n = ω 2 In direcion of velociy if par is acceleraing. Opposie direcion of velociy if par is deceleraing. r direced owards cener of roaion (or relaive roaion). Angular Acceleraion (a): Angular acceleraion of a link, α, is influenced by he angenial acceleraion. a = rα v a n a Relaive Acceleraion Mehod A 2 α 2 ω Main principle is ha wo poins on a rigid body can only have a relaive angenial acceleraion: Perpendicular o he line ha connecs hem. Therefore, he relaive normal acceleraion is: A Along he line ha connecs hem. A he moion of, as seen from A a /A = a fi a A 28

3 Relaive Acceleraion Equaions Relaive acceleraion equaion is used o form vecor polygons, and deermine velociies of key poins. a i = a j +> a i/j reaking each componen ino normal and angenial componens gives: a i n +> a i = a j n +> a j +> a i/j n +> a i/j The relaive moion equaions dicae he creaion of vecor polygons: A 2 α ω Acceleraion Polygon a A n = (v A ) 2 /r A a n = (v ) 2 /r a /A n = (v /A ) 2 /r A a A = (α 2 )r A 3 4 Velociy Polygon v A = ω 2 r A v = v A +> v /A a n +> a = a A n +> a A +> a /A n +> a /A v A v v /A Measure: v v /A Calculae: ω 4 = (v )/r ω 3 = (v /A )/r A a n a a A n a /A a /A n a A Measure: a a /A Calculae: α 4 = (a )/r α 3 = (a /A )/r A Hins for Solving Acceleraion Problems Sar wih deermining he velociy of all in he mechanism Wrie he relaive acceleraion equaion wih he known acceleraion on he righ. Deermine all normal acceleraion componens. Poins on ranslaing links have no normal acceleraion. Poins on links ha roae a consan speed have no angenial acceleraion. Draw acceleraion polygon and measure desired acceleraion componens. 29

4 Problems: Velociy Problems: 4-1. Deermine he velociy of he pison, as he crank roaes a 600 rpm, cw. 4Ans: 101 in/s in 8 in 4-2. For problem 4-1, use a spreadshee o deermine he velociy of he pison for an enire cycle of crank roaion For problem 4-1, use Working Model o deermine he velociy of he pison for an enire cycle of crank roaion Deermine he velociy of he pison, as he crank roaes a 600 rpm, ccw. 4Ans: 90 in/s 1.5 in in 8 in 4-5. For problem 4-4, use a spreadshee o deermine he velociy of he pison for an enire cycle of crank roaion For problem 4-4, use Working Model o deermine he velociy of he pison for an enire cycle of crank roaion Deermine he roaional velociy of he crushing ram, as he crank roaes a 60 rpm, cw. 4Ans: 2.2 rad/s, cw 360 mm 180 mm 60 mm 400 mm 30

5 4-8. For problem 4-7, use a spreadshee o deermine he angular velociy of he crushing ram for an enire cycle of crank roaion For problem 4-7, use Working Model o deermine he velociy of he crushing ram for an enire cycle of crank roaion For he embossing press, deermine he required roaional velociy of he handle if he samp is o move downward a a rae of 3 in/s 4Ans: 0.64 rad/s, cw Deermine he linear velociy of poin X, if he shear needs o conac he shee meal a a rae of 10 in/sec. X 10 in Ans: v x = 25.5 in/s in Deermine he angular velociy of he windshield wiper, and he linear velociy of he cener of he wiper, as he moor roaes a 45 rpm, counerclockwise. 4Ans: 2.6 rad/s, cw 2 in 13 in in 1 31

6 4-13. For problem 4-12, use a spreadshee o deermine he angular velociy of he wiper for an enire cycle of crank roaion For problem 4-12, use Working Model o deermine he angular velociy of he wiper for an enire cycle of crank roaion Deermine he velociy of he package-moving ram, as he crank roaes 1200 rpm, clockwise m 4Ans: 12.3 m/s 0.4 m 0.32 m 0.4 m 0.1 m m 0.2 m 0.3 m 0.37 m For problem 4-15, use Working Model o deermine he velociy of he packagemoving ram for an enire cycle of crank roaion For he coin operaed mechanical donkey ride, deermine he roaional velociy of he donkey, if he 5 inch crank roaes a 75 rpm, clockwise. 4Ans: 5.5 rad/s, ccw 3 27 in 5 in For problem 4-17, use a spreadshee o deermine he velociy of he donkey for an enire cycle of crank roaion For problem 4-17, use Working Model o deermine he velociy of he donkey for an enire cycle of crank roaion. 15 in 32

7 4-20. For he sewing machine linkage, deermine he linear velociy of he needle, as he drive wheel is roaing a 200 rpm, counerclockwise. 4Ans: 413 mm/s 15 mm mm 8 mm For problem 4-20, use a spreadshee o deermine he velociy of he needle for an enire cycle of crank roaion For problem 4-20, use Working Model o deermine he velociy of he needle for an enire cycle of crank roaion For he caron urnover machine shown, deermine he angular velociy of he carriage, if he 15 in link is driven clockwise a 5 rad/sec. 4Ans: 2.5 rad/s, cw 7 in 1 15 in in For problem 4-23, use a spreadshee o deermine he angular velociy of he carriage for an enire cycle of crank roaion For problem 4-23, use Working Model o deermine he angular velociy of he carriage for an enire cycle of crank roaion. 33

8 4-26. For he maerial handling mechanism shown, deermine he velociy of he crae if he 4-inch link is driven clockwise a 12 rad/sec. 20 in 3 in 3 in in For problem 4-26, use a spreadshee o deermine he velociy of he needle for an enire cycle of crank roaion For problem 4-26, use Working Model o deermine he velociy of he needle for an enire cycle of crank roaion The horizonal member supporing he scoop of a backhoe is saionary. Deermine he angular velociy of he scoop if he cylinder is exending a a rae of 2 f/s. 2 f 1.5 f 5 f 2.5 f 1 f For he drum pedal mechanism shown, deermine he velociy of he hammer if he pedal roaes 24 rad/sec, clockwise. hammer 105 mm 125 mm pedal 35 mm mm 70 mm 34

9 Acceleraion Problems: Deermine he acceleraion of he pison, as he crank roaes a a consan rae of 600 rpm, cw. 4Ans: 7607 in/s in 8 in For problem 4-31, use a spreadshee o deermine he acceleraion of he pison for an enire cycle of crank roaion For problem 4-31, use Working Model o deermine he acceleraion of he pison for an enire cycle of crank roaion Deermine he acceleraion of he pison, as he crank roaes a a consan rae of 600 rpm, ccw. 1.5 in in 8 in 4Ans: 6120 in/s For problem 4-34, use a spreadshee o deermine he acceleraion of he pison for an enire cycle of crank roaion For problem 4-34, use Working Model o deermine he acceleraion of he pison for an enire cycle of crank roaion. 360 mm Deermine he angular acceleraion of he crushing ram, as he crank roaes a a consan 60 rpm, clockwise. 60 mm Deermine he angular acceleraion of he crushing ram, shown above. The 60 mm crank roaes a 60 rpm, clockwise, and is speeding up a a rae of 30 rad/s 2. 4Ans: 8.42 rad/s 2, cw 180 mm 400 mm For problem 4-37, deermine he acceleraion of he cener of he 400 mm link For problem 4-38, deermine he acceleraion of he cener of he 400 mm link For problem 4-37, use a spreadshee o deermine he angular acceleraion of he crushing ram for an enire cycle of crank roaion. 35

10 4-42. For problem 4-37, use Working Model o deermine he angular acceleraion of he crushing ram for an enire cycle of crank roaion For he embossing press, deermine he angular acceleraion of he handle if he samp is moving downward a a rae of 3 in/s and deceleraing a 10 in/s For problem 4-43, deermine he acceleraion of he cener of he 6 inch link Deermine he linear velociy and acceleraion of poin X, if he shear conacs he shee meal a a rae of 10 in/sec, and acceleraing a 125 in/s For problem 4-45, deermine he acceleraion of he cener of he connecing link. X 10 in in Deermine he angular acceleraion of he windshield wiper, and he linear acceleraion of he cener of he wiper, as he moor roaes a a consan rae of 45 rpm, counerclockwise. 2 in 13 in Ans: 4.0 rad/s 2, cw in For problem 4-47, use a spreadshee o deermine he angular acceleraion of he wiper for an enire cycle of crank roaion. 36

11 4-49. For problem 4-47, use Working Model o deermine he angular acceleraion of he wiper for an enire cycle of crank roaion Deermine he acceleraion of he package-moving ram, as he crank roaes a a consan rae of 1200 rpm, clockwise. 4Ans: 3306 m/s m 0.4 m 0.32 m 0.4 m 0.1 m m 0.2 m 0.3 m 0.37 m For problem 4-50, use Working Model o deermine he acceleraion of he package-moving ram for an enire cycle of crank roaion For he coin operaed mechanical donkey ride, deermine he roaional acceleraion of he donkey, if he 5 inch crank roaes a a consan 75 rpm, clockwise. 27 in For he coin operaed mechanical donkey ride shown above, deermine he roaional acceleraion of he donkey. The 5 inch crank roaes a 75 rpm, clockwise, and is acceleraing a a rae of 40 rad/sec 2. 4Ans: 38.0 rad/s 2, cw 3 5 in 15 in For problem 4-52, deermine he acceleraion of he cener of he 3ch link For problem 4-53, deermine he acceleraion of he cener of he 3ch link For problem 4-52, use a spreadshee o deermine he angular acceleraion of he donkey for an enire cycle of crank roaion. 37

12 4-57. For problem 4-52, use Working Model o deermine he angular acceleraion of he donkey for an enire cycle of crank roaion For he sewing machine linkage, deermine he acceleraion of he needle, as he drive wheel is roaing a a consan 200 rpm, counerclockwise. 4Ans: 2.9 m/s For he sewing machine linkage, deermine he acceleraion of he needle, as he drive wheel is roaing a 200 rpm, counerclockwise, and acceleraing a a rae of 300 rad/sec For problem 4-58, deermine he acceleraion of he cener of he 30 mm link For problem 4-59, deermine he acceleraion of he cener of he 30 mm link For problem 4-58, use a spreadshee o deermine he acceleraion of he needle for an enire cycle of crank roaion For problem 4-58, use Working Model o deermine he acceleraion of he needle for an enire cycle of crank roaion For he caron urnover machine shown, deermine he angular acceleraion of he carriage, if he 15 in link is driven clockwise a 5 rad/sec, and deceleraing a 40 rad/sec 2. 7 in 1 15 mm mm 8 mm 15 in For he maerial handling mechanism shown, deermine he acceleraion of he crae if he 4-inch link is driven clockwise a a consan rae of 12 rad/sec. 7.3 in 1 20 in 3 in 3 in 8 in

13 4-66. For problem 4-65, deermine he acceleraion of he cener of he 1ch link For problem 4-65, use a spreadshee o deermine he acceleraion of he needle for an enire cycle of crank roaion For problem 4-65, use Working Model spreadshee o deermine he acceleraion of he package-moving ram for an enire cycle of crank roaion The machine shown is used o advance film on a heare qualiy movie projecor. Deermine he acceleraion of he claw as he crank drives a a consan rae of 600 rpm, clockwise. 4Ans: 31.4 m/s The machine shown is used o advance film on a heare qualiy movie projecor. Deermine he acceleraion of he claw as he crank drives a 600 rpm, clockwise, and is acceleraing a 2000 rad/sec mm 45 mm mm 48 mm 18 mm For problem 4-69, use Working Model o deermine he acceleraion of he claw for an enire cycle of crank roaion For he windshield wiper linkage shown, deermine he acceleraion of he cg of he connecing link. The moor is running a 30 rpm clockwise in 2 in 13 in in 3.5 in For he windshield wiper linkage shown, deermine he acceleraion of he cg of he connecing link. The moor is running a 30 rpm clockwise, and acceleraing a 10 rad/sec 2. 4Ans: 6./s