Calculos de velocidades y aceleraciones

Cálculos para ɵ=0.

VELOCIDADES

VA = ω2 ∗ rA

VA = 200 mm s

VB = VA + VB/A

100 𝑚𝑚𝑠

sin 94.54=

𝑉𝐵

sin 75.11

𝑉𝐵 = 96.95 𝑚𝑚𝑠

𝑉𝐵𝐴

= 100 𝑚𝑚𝑠 2 + (96.95 𝑚𝑚

𝑠 )2 − 2 100 𝑚𝑚𝑠 96.95 𝑚𝑚

𝑠 ∗ cos 10. 35

𝑉 𝐵𝐴 = 18.02 𝑚𝑚

𝑠

𝜔4 =𝑉𝐵

𝑟𝐵

𝜔4 = 1.43 𝑟𝑎𝑑𝑠

𝜔3 =𝑉 𝐵

𝐴

𝑟 𝐵𝐴

𝜔3 = 0.277 𝑟𝑎𝑑 𝑠

𝑉𝐷 = 𝜔4 ∗ 𝑟𝐷

𝑉𝐷 = 1.43 ∗ 115

𝑉𝐷 = 164.45 𝑚𝑚𝑠

VE = VD + VE/D

164.45 𝑚𝑚𝑠

sin 17.6=

𝑉𝐸

sin 90

𝑉𝐸 = 543.87 𝑚𝑚𝑠

543.87 𝑚𝑚𝑠

sin 17.6=

𝑉𝐸/𝐷

sin 72.4

𝑉𝐸/𝐷 = 518.87 𝑚𝑚𝑠

𝜔5 =𝑉𝐸/𝐷

𝑟𝐸/𝐷

𝜔5 = 4.5 𝑟𝑎𝑑𝑠

𝜔6 =𝑉𝐸

𝑟𝐸

𝜔6 = 2.6 𝑟𝑎𝑑 𝑠

ACELERACIONES

𝑎𝐴 = 𝑎𝐴𝑛+ 𝑎𝐴𝑡

Como ω2 es constante, entonces α2 es 0 y por ende aceleración tangencial de A es cero.

𝑎𝐴 = 𝑎𝐴𝑛= 𝜔2

2𝑟𝐴 = 22 ∗ 50 = 200 𝑚𝑚𝑠2

𝑎𝐵𝑛+ 𝑎𝐵𝑡

= 𝑎𝐴𝑛+ 𝑎𝐵/𝐴𝑛

+ 𝑎𝐵/𝐴𝑡

𝑎𝐵𝑛=𝜔4 𝑋 𝜔4 𝑋 𝑟𝐵

𝑎𝐵𝑛= 1.43 𝑘 𝑋 (1.43 𝐾 𝑋(66.7 𝑖 − 12.18 𝑗)

𝑎𝐵𝑛= −136.39 𝑖 + 26.18 𝑗

𝑎𝐵𝑡= 𝛼4 𝑋 𝑟𝐵 = 𝛼4𝑘 𝑋 66.7 𝑖 − 12.18 𝑗

𝑎𝐵𝑡= 66.70 ∗ 𝛼4𝑖 + 12.18 ∗ 𝛼4 𝑗

𝑎𝐵/𝐴𝑛=𝜔3 𝑋 𝜔3 𝑋 𝑟𝐵/𝐴

𝑎𝐵/𝐴𝑛= 0.277 𝑘 𝑋 4.62 𝑗 − 17.39 𝑖 = −1.27 𝑖 − 4.81𝑗

𝑎𝐵/𝐴𝑡= 𝛼3 𝑋 𝑟𝐵/𝐴

𝑎𝐵𝐴𝑡

= −𝛼3 ∗ 62.81 𝑗 + 𝛼3 ∗ 16.702 𝑖

𝑖 → 𝛼4 ∗ 12.18 + 𝛼362.81 = −64.88

𝑗 → 𝛼4 ∗ 66.70 − 𝛼3 ∗ 16.702 = −31.69

𝛼4 = −0.0699 𝑟𝑎𝑑𝑠2

𝛼3 = 1.01 𝑟𝑎𝑑𝑠2

𝑎𝐵 = 139.01 𝑚𝑚𝑠2

𝑎𝐷𝑛= 42.24 𝑗 − 231.23 𝑖

𝑎𝐷𝑡= 1.46 𝑖 + 8.005 𝑗

𝑎𝐷 = 235.9 𝑚𝑚 𝑠2

𝑎𝐸𝑛+ 𝑎𝐸𝑡

= 𝑎𝐷𝑛+ 𝑎𝐷𝑡

+ 𝑎𝐸/𝐷𝑛+ 𝑎𝐸/𝐷𝑡

𝑎𝐸𝑛 =𝜔6 𝑋 𝜔6 𝑋 𝑟𝐸

𝑎𝐸𝑛= −2.6 𝑘 𝑋 (−2.6 𝐾 𝑋(−63.5 𝑖 − 200.17 𝑗)

𝑎𝐸𝑛= −429.26 𝑖 − 1353.15 𝑗

𝑎𝐸𝑡= 𝛼6 𝑋 𝑟𝐸 = 𝛼6 𝑋 −63.5 𝑖 + 200.17 𝑗

𝑎𝐸𝑡= −200.17 ∗ 𝛼6𝑖 − 63.5 ∗ 𝛼6 𝑗

𝑎𝐸/𝐷𝑛 =𝜔5 𝑋 𝜔5 𝑋 𝑟𝐸/𝐷

𝑎𝐸/𝐷𝑛= −4.6 𝑘 𝑋 84.18 𝑗 + 522.19 𝑖 = 387.22 𝑖 − 2402.047𝑗

𝑎𝐸/𝐷𝑡= 𝛼5 𝑋 𝑟𝐸/𝐷

𝑎𝐸/𝐷𝑡= −𝛼5 ∗ 18.37 𝑗 + 𝛼5 ∗ 113.52 𝑖

𝑖 → −𝛼6 ∗ 200.77 − 𝛼5113.5 = −273.27

𝑗 → −𝛼6 ∗ 63.5 − 𝛼5 ∗ 18.37 = −3712.97

𝛼6 = −38.67 𝑟𝑎𝑑𝑠2

𝛼5 = 70.81 𝑟𝑎𝑑𝑠2

𝑎𝐸 = 8266.85 𝑚𝑚𝑠2

Cálculos para ɵ=45

VELOCIDADES

𝑉𝐴 = 𝜔2 𝑋 𝑟𝑎

𝑉𝐴 = −2𝑘 𝑋 35.36 𝑖 + 35.36 𝑗 = −70.72 𝑗 + 70.72 𝑖 = 100 𝑚𝑚𝑠2

VB = VA + VB/A

𝑉𝐵 = 𝜔4 𝑋 𝑟𝐵

𝑉𝐵 = 𝜔4𝑘 𝑋 65.37 𝑖 + 18 𝑗 = 𝜔4 ∗ 65.3 𝑗 − 𝜔4 ∗ 18 𝑖

𝑉𝐵𝐴

= 𝜔3 𝑋 𝑟𝐵/𝐴

𝑉𝐵𝐴

= 𝜔3𝑘 𝑋 30.02 𝑖 + 57.65 𝑗 = 𝜔3 ∗ 30.02 𝑗 − 𝜔3 ∗ 57.65 𝑖

𝑖 → −𝜔3 ∗ 57.65 + 𝜔4 ∗ 18 = 70.72

𝑗 → 𝜔3 ∗ 30.02 − 𝜔4 ∗ 65.37 = −70.72

𝜔3 = −1.82 𝑟𝑎𝑑 𝑠

𝜔4 = −1.91 𝑟𝑎𝑑 𝑠

𝑉𝐵 = 129.50 𝑚𝑚𝑠

𝑉𝐷 = 𝜔4 𝑋 𝑟𝐷

𝑉𝐷 = −1.92 𝐾 𝑋 110.8 𝑖 + 30.53 𝑗 = −212.73 𝑗 + 58.61 𝑖 = 220.65 𝑚𝑚𝑠

VE = VD + VE/D

VE = ω6 X rE

𝑉𝐸 = 𝜔6 𝑘 𝑋 5.46 𝑖 + 209.9 𝑗 = 𝜔6 ∗ 5.46 𝑗 − 𝜔6 ∗ 209,9 𝑖

𝑉𝐸/𝐷 = 𝜔5 𝑋 𝑟𝐸/𝐷

𝑉𝐸/𝐷 = 𝜔5 𝑘 𝑋 89.59𝑖 + 72.09 𝑗 = 𝜔5 ∗ 85.59 𝑗 − 𝜔5 ∗ 72.09 𝑖

𝑖 → −𝜔6 ∗ 209.9 + 𝜔5 ∗ 72.09 = 55.56

𝑗 → 𝜔6 ∗ 5.46 − 𝜔5 ∗ 89.59 = −201.65

𝜔6 = −1.05 𝑟𝑎𝑑 𝑠

𝜔5 = −2.28 𝑟𝑎𝑑 𝑠

𝑉𝐸 = 220.46 𝑚𝑚𝑠

ACELERACIONES



𝑎𝐴𝑛 = 𝜔2 𝑋(𝜔2 𝑋 𝑟𝐴)

𝑎𝐴𝑛 = −2𝑘 𝑋 −70.72 𝑗 + 70.72 𝑖 = −141.44 𝑖 − 141.44 𝑗 = 200 𝑚𝑚s2



+ 𝑎𝐵/𝐴𝑡


𝑎𝐵𝑛= −1.91 𝑘 𝑋 (−124.85 𝑗 + 34.81 𝑖)

𝑎𝐵𝑛− 238.46 𝑖 − 65.66 𝑗

𝑎𝐵𝑡= 𝛼4 𝑋 𝑟𝐵 = 𝛼4 𝑘 𝑋 65.37 𝑖 + 18 𝑗

𝑎𝐵𝑡= −18 ∗ 𝛼4𝑖 + 65.37 ∗ 𝛼4 𝑗


𝑎𝐵/𝐴𝑛= −1.82 𝑘 𝑋 −54.63 𝑗 + 104.9 𝑖 = −99.4 𝑖 − 190.9𝑗


𝑎𝐵/𝐴 𝑡 = 𝛼3 ∗ 30.02 𝑗 − 𝛼3 ∗ 57.65 𝑖

𝑖 → −𝛼4 ∗ 18 + 𝛼3 ∗ 57.65 = −2.28

𝑗 → 𝛼4 ∗ 65.37 − 𝛼3 ∗ 30.02 = −266.68

𝛼4 = −4.76 𝑟𝑎𝑑𝑠2

𝛼3 = −1.53 𝑟𝑎𝑑𝑠2

𝑎𝐵 = 400.61 𝑚𝑚𝑠2

𝑎𝐷 = 𝑎𝐷𝑛+ 𝑎𝐷𝑡

𝑎𝐷𝑛 = 𝜔4 𝑋 𝜔4 𝑋 𝑟𝐵 = −1.92 𝑘 𝑋 −212.73 𝑗 + 58.61 𝑖 = −408.44 𝑖 − 112.53 𝑗

𝑎𝐷𝑡= 𝛼4 𝑋 𝑟𝐵 = −4.76 𝑘 𝑋 110.8 𝑖 + 30.53 𝑗 = −527.408 𝑖 − 639, .93 𝑗

𝑎𝐷 = 691.91 𝑚𝑚𝑠2





𝑎𝐸𝑛− 1.05 𝑘 𝑋 (−5.73 𝑗 + 220.39 𝑖)

𝑎𝐸𝑛= − − 6.01 𝑖 − 231.4 𝑗

𝑎𝐸𝑡= 𝛼6 𝑋 𝑟𝐸 = 𝛼6𝑘 𝑋 5.46 𝑖 + 209.9 𝑗

𝑎𝐸𝑡= −209.9 ∗ 𝛼6𝑖 + 5.46 ∗ 𝛼6 𝑗


𝑎𝐸/𝐷𝑛= −2.28𝑘 𝑋 −204.2 𝑗 + 164.36 𝑖 = −465.71 𝑖 − 374.74 𝑗


𝑎𝐸/𝐷𝑡= 𝛼5 ∗ 89.59 𝑗 − 𝛼5 ∗ 72.09 𝑖

𝑖 → −𝛼6 ∗ 209.9 + 𝛼572.09 = −722.82

𝑗 → 𝛼6 ∗ 5.46 − 𝛼5 ∗ 81.59 = −783.27

𝛼6 = −3.6 𝑟𝑎𝑑𝑠2

𝛼5 = 0.45 𝑟𝑎𝑑𝑠2

𝑎𝐸 = 6770.24 𝑚𝑚𝑠2

Cálculos para ɵ=105

VELOCIDADES

𝑉𝐴 = 𝜔2 𝑋 𝑟𝑎

𝑉𝐴 = −2𝑘 𝑋 12.94 𝑖 + 98.29 𝑗 = −25.88 𝑗 + 96.59 𝑖 = 100 𝑚𝑚𝑠2

VB = VA + VB/A

𝑉𝐵 = 𝜔4 𝑋 𝑟𝐵

𝑉𝐵 = 𝜔4𝑘 𝑋 48.77 𝑖 − 47.11 𝑗 = 𝜔4 ∗ 48.77 𝑗 + 𝜔4 ∗ 47.11 𝑖

𝑉𝐵𝐴

= 𝜔3 𝑋 𝑟𝐵/𝐴

𝑉𝐵𝐴

= 𝜔3𝑘 𝑋 61.71 𝑖 − 20.40 𝑗 = 𝜔3 ∗ 61.71 𝑗 + 𝜔3 ∗ 20.40 𝑖

𝑖 → −𝜔3 ∗ 20.9 + 𝜔4 ∗ 47.11 = 96.59

𝑗 → −𝜔3 ∗ 61.71 + 𝜔4 ∗ 48.77 = −25.88

𝜔3 = 3.13 𝑟𝑎𝑑 𝑠

𝜔4 = 3.44 𝑟𝑎𝑑 𝑠

𝑉𝐵 = 233.25 𝑚𝑚𝑠

𝑉𝐷 = 𝜔4 𝑋 𝑟𝐷

𝑉𝐷 = 3.44𝐾 𝑋 82.71 𝑖 − 79.9 𝑗 = 284.52 𝑗 + 274.85 𝑖 = 295.544 𝑚𝑚𝑠

VE = VD + VE/D

VE = ω6 X rE

𝑉𝐸 = 𝜔6 𝑘 𝑋 −165.39 𝑖 + 129.4𝑗 = −𝜔6 ∗ 165.39 𝑗 − 𝜔6 ∗ 129.4 𝑖

𝑉𝐸/𝐷 = 𝜔5 𝑋 𝑟𝐸/𝐷

𝑉𝐸/𝐷 = 𝜔5 𝑘 𝑋 −102 𝑖 − 53.1 𝑗 = −𝜔5 ∗ 102 𝑗 + 𝜔5 ∗ 53.1 𝑖

𝑖 → −𝜔6 ∗ 129.4 − 𝜔5 ∗ 53.1 = 274.85

𝑗 → −𝜔6 ∗ 165.39 + 𝜔5 ∗ 102 = 284.85

𝜔6 = −1.96 𝑟𝑎𝑑 𝑠

𝜔5 = −0.39 𝑟𝑎𝑑 𝑠

𝑉𝐸 = 411.59 𝑚𝑚𝑠

ACELERACIONES



𝑎𝐴𝑛 = 𝜔2 𝑋(𝜔2 𝑋 𝑟𝐴)

𝑎𝐴𝑛 = −2𝑘 𝑋 −25.88 𝑗 + 70.72 𝑖 = −51.76 𝑖 − 193.18 𝑗 = 200 𝑚𝑚s2



+ 𝑎𝐵/𝐴𝑡


𝑎𝐵𝑛= 3.44 𝑘 𝑋 (167.76 𝑗 + 162.05 𝑖)

𝑎𝐵𝑛− 577.09 𝑖 + 557.45 𝑗

𝑎𝐵𝑡= 𝛼4 𝑋 𝑟𝐵 = 𝛼4 𝑘 𝑋 48.77 𝑖 − 47.11 𝑗

𝑎𝐵𝑡= 48.77 ∗ 𝛼4𝑖 + 47.11 ∗ 𝛼4 𝑗


𝑎𝐵/𝐴𝑛= 3.13 𝑘 𝑋 193.1 𝑗 + 63.85 𝑖 = −604.4 𝑖 + 199.85𝑗


𝑎𝐵/𝐴 𝑡 = 𝛼3 ∗ 61.71 𝑗 − 𝛼3 ∗ 20.4 𝑖

𝑖 → −𝛼4 ∗ 18 + 𝛼3 ∗ 57.65 = −2.28

𝑗 → 𝛼4 ∗ 65.37 − 𝛼3 ∗ 30.02 = −266.68

𝛼4 = 11.90 𝑟𝑎𝑑𝑠2

𝛼3 = 31.36 𝑟𝑎𝑑𝑠2

𝑎𝐵 = 580.36 𝑚𝑚𝑠2

𝑎𝐷 = 𝑎𝐷𝑛+ 𝑎𝐷𝑡

𝑎𝐷𝑛 = 𝜔4 𝑋 𝜔4 𝑋 𝑟𝐵 = 3.44 𝑘 𝑋 284.5 𝑗 + 274.85 𝑖 = −978.68 𝑖 + 945.48 𝑗

𝑎𝐷𝑡= 𝛼4 𝑋 𝑟𝐵 = 11.90 𝑘 𝑋 82.71 𝑖 − 79.9 𝑗 = 984.24 𝑗 + 950.81 𝑖

𝑎𝐷 = 1929.92 𝑚𝑚𝑠2





𝑎𝐸𝑛= −1.96 𝑘 𝑋 (324.16 𝑗 + 253.62 𝑖)

𝑎𝐸𝑛= 635.35 𝑖 − 497.09 𝑗

𝑎𝐸𝑡= 𝛼6 𝑋 𝑟𝐸 = 𝛼6𝑘 𝑋 −165.39 𝑖 + 129.4 𝑗

𝑎𝐸𝑡= −165.39 ∗ 𝛼6𝑖 − 129.4 ∗ 𝛼6 𝑗


𝑎𝐸/𝐷𝑛= −0.39𝑘 𝑋 39.78 𝑗 − 20.7 𝑖 = 15.51 𝑖 + 8.073 𝑗


𝑎𝐸/𝐷𝑡= −𝛼5 ∗ 102 𝑗 + 𝛼5 ∗ 53.1 𝑖

𝑖 → −𝛼6 ∗ 129.4 − 𝛼5 ∗ 53.1 = −147.71

𝑗 → −𝛼6 ∗ 165.39 + 𝛼5 ∗ 102 = 2068.737

𝛼6 = −4.31 𝑟𝑎𝑑𝑠2

𝛼5 = 13.28 𝑟𝑎𝑑𝑠2

𝑎𝐸 = 2902.97 𝑚𝑚𝑠2

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Calculos de velocidades y aceleraciones