Issuu on Google+

Cálculos para ɵ=0. VELOCIDADES VA = ω2 ∗ rA VA = 200 mm s

VB = VA + VB/A 100 𝑚𝑚 𝑠 𝑉𝐵 = sin 94.54 sin 75.11 𝑉𝐵 = 96.95 𝑚𝑚 𝑠 𝑉𝐵

𝐴

=

100 𝑚𝑚 𝑠

2

+ (96.95 𝑚𝑚 𝑠)2 − 2 100 𝑚𝑚 𝑠 96.95 𝑚𝑚 𝑠 ∗ cos 10. 35 𝑉𝐵

𝐴 = 18.02 𝑚𝑚 𝑠

𝜔4 =

𝑉𝐵 𝑟𝐵

𝜔4 = 1.43 𝑟𝑎𝑑 𝑠 𝜔3 =

𝑉𝐵 𝐴 𝑟𝐵 𝐴

𝜔3 = 0.277 𝑟𝑎𝑑 𝑠 𝑉𝐷 = 𝜔4 ∗ 𝑟𝐷 𝑉𝐷 = 1.43 ∗ 115 𝑉𝐷 = 164.45 𝑚𝑚 𝑠 VE = VD + VE/D 164.45 𝑚𝑚 𝑠 𝑉𝐸 = sin 17.6 sin 90 𝑉𝐸 = 543.87 𝑚𝑚 𝑠 𝑉𝐸/𝐷 543.87 𝑚𝑚 𝑠 = sin 17.6 sin 72.4


𝑉𝐸/𝐷 = 518.87 𝑚𝑚 𝑠 𝜔5 =

𝑉𝐸/𝐷 𝑟𝐸/𝐷

𝜔5 = 4.5 𝑟𝑎𝑑 𝑠 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 2 𝑟𝐴 = 22 ∗ 50 = 200 𝑚𝑚 𝑠 2 𝑎𝐵𝑛 + 𝑎𝐵𝑡 = 𝑎𝐴𝑛 + 𝑎𝐵/𝐴𝑛 + 𝑎𝐵/𝐴𝑡 𝑎𝐵𝑛 =𝜔 4 𝑋

𝜔 4 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 = 𝛼4 𝑘 𝑋 66.7 𝑖 − 12.18 𝑗 𝑎𝐵𝑡 = 66.70 ∗ 𝛼4 𝑖 + 12.18 ∗ 𝛼4 𝑗 𝑎𝐵/𝐴𝑛 =𝜔 3 𝑋

𝜔 3 𝑋 𝑟 𝐵 /𝐴

𝑎𝐵/𝐴𝑛 = 0.277 𝑘 𝑋 4.62 𝑗 − 17.39 𝑖 = −1.27 𝑖 − 4.81𝑗 𝑎𝐵/𝐴𝑡 = 𝛼3 𝑋 𝑟𝐵/𝐴 𝑎𝐵 = −𝛼3 ∗ 62.81 𝑗 + 𝛼3 ∗ 16.702 𝑖 𝐴𝑡

𝑖 → 𝛼4 ∗ 12.18 + 𝛼3 62.81 = −64.88 𝑗 → 𝛼4 ∗ 66.70 − 𝛼3 ∗ 16.702 = −31.69 𝛼4 = −0.0699 𝑟𝑎𝑑 𝑠 2


𝛼3 = 1.01 𝑟𝑎𝑑 𝑠 2 𝑎𝐵 = 139.01 𝑚𝑚 𝑠 2 𝑎𝐷𝑛 = 42.24 𝑗 − 231.23 𝑖 𝑎𝐷𝑡 = 1.46 𝑖 + 8.005 𝑗 𝑎𝐷 = 235.9 𝑚𝑚 𝑠 2 𝑎𝐸𝑛 + 𝑎𝐸𝑡 = 𝑎𝐷𝑛 + 𝑎𝐷𝑡 + 𝑎𝐸/𝐷𝑛 + 𝑎𝐸/𝐷𝑡 𝑎𝐸𝑛 =𝜔 6 𝑋

𝜔 6 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 ∗ 𝛼6 𝑖 − 63.5 ∗ 𝛼6 𝑗 𝑎𝐸/𝐷𝑛 =𝜔 5 𝑋

𝜔 5 𝑋 𝑟 𝐸/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 − 𝛼5 113.5 = −273.27 𝑗 → −𝛼6 ∗ 63.5 − 𝛼5 ∗ 18.37 = −3712.97 𝛼6 = −38.67 𝑟𝑎𝑑

𝑠2

𝛼5 = 70.81 𝑟𝑎𝑑 𝑠 2 𝑎𝐸 = 8266.85 𝑚𝑚 𝑠 2


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 𝑚𝑚 𝑠 2 VB = VA + VB/A 𝑉𝐵 = 𝜔4 𝑋 𝑟𝐵 𝑉𝐵 = 𝜔4 𝑘 𝑋 65.37 𝑖 + 18 𝑗 = 𝜔4 ∗ 65.3 𝑗 − 𝜔4 ∗ 18 𝑖 𝑉𝐵 𝑉𝐵

𝐴

𝐴

= 𝜔3 𝑋 𝑟𝐵/𝐴

= 𝜔3 𝑘 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 𝑚𝑚 𝑠 𝑉𝐷 = 𝜔4 𝑋 𝑟𝐷

𝑉𝐷 = −1.92 𝐾 𝑋 110.8 𝑖 + 30.53 𝑗 = −212.73 𝑗 + 58.61 𝑖 = 220.65 𝑚𝑚 𝑠 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 𝑚𝑚 𝑠 ACELERACIONES 𝑎𝐴 = 𝑎𝐴𝑛 + 𝑎𝐴𝑡 Como ω2 es constante, entonces α2 es 0 y por ende aceleración tangencial de A es cero. 𝑎𝐴𝑛 = 𝜔2 𝑋(𝜔2 𝑋 𝑟𝐴) 𝑎𝐴𝑛 = −2𝑘 𝑋 −70.72 𝑗 + 70.72 𝑖 = −141.44 𝑖 − 141.44 𝑗 = 200 𝑚𝑚 2 s 𝑎𝐵𝑛 + 𝑎𝐵𝑡 = 𝑎𝐴𝑛 + 𝑎𝐵/𝐴𝑛 + 𝑎𝐵/𝐴𝑡 𝑎𝐵𝑛 =𝜔 4 𝑋

𝜔 4 𝑋 𝑟𝐵

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 ∗ 𝛼4 𝑖 + 65.37 ∗ 𝛼4 𝑗 𝑎𝐵/𝐴𝑛 =𝜔 3 𝑋

𝜔 3 𝑋 𝑟 𝐵 /𝐴

𝑎𝐵/𝐴𝑛 = −1.82 𝑘 𝑋 −54.63 𝑗 + 104.9 𝑖 = −99.4 𝑖 − 190.9𝑗 𝑎𝐵/𝐴𝑡 = 𝛼3 𝑋 𝑟𝐵/𝐴 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 𝑟𝑎𝑑

𝑠2

𝛼3 = −1.53 𝑟𝑎𝑑 𝑠 2 𝑎𝐵 = 400.61 𝑚𝑚 𝑠 2 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 𝑚𝑚

𝑠2


𝑎𝐸𝑛 + 𝑎𝐸𝑡 = 𝑎𝐷𝑛 + 𝑎𝐷𝑡 + 𝑎𝐸/𝐷𝑛 + 𝑎𝐸/𝐷𝑡 𝑎𝐸𝑛 =𝜔 6 𝑋

𝜔 6 𝑋 𝑟𝐸

𝑎𝐸𝑛 − 1.05 𝑘 𝑋 (−5.73 𝑗 + 220.39 𝑖) 𝑎𝐸𝑛 = − − 6.01 𝑖 − 231.4 𝑗 𝑎𝐸𝑡 = 𝛼6 𝑋 𝑟𝐸 = 𝛼6 𝑘 𝑋 5.46 𝑖 + 209.9 𝑗 𝑎𝐸𝑡 = −209.9 ∗ 𝛼6 𝑖 + 5.46 ∗ 𝛼6 𝑗 𝑎𝐸/𝐷𝑛 =𝜔 5 𝑋

𝜔 5 𝑋 𝑟 𝐸/𝐷

𝑎𝐸/𝐷𝑛 = −2.28𝑘 𝑋 −204.2 𝑗 + 164.36 𝑖 = −465.71 𝑖 − 374.74 𝑗 𝑎𝐸/𝐷𝑡 = 𝛼5 𝑋 𝑟𝐸/𝐷 𝑎𝐸/𝐷𝑡 = 𝛼5 ∗ 89.59 𝑗 − 𝛼5 ∗ 72.09 𝑖 𝑖 → −𝛼6 ∗ 209.9 + 𝛼5 72.09 = −722.82 𝑗 → 𝛼6 ∗ 5.46 − 𝛼5 ∗ 81.59 = −783.27 𝛼6 = −3.6 𝑟𝑎𝑑 𝑠 2 𝛼5 = 0.45 𝑟𝑎𝑑

𝑠2

𝑎𝐸 = 6770.24 𝑚𝑚

𝑠2


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 𝑚𝑚 𝑠 2 VB = VA + VB/A 𝑉𝐵 = 𝜔4 𝑋 𝑟𝐵 𝑉𝐵 = 𝜔4 𝑘 𝑋 48.77 𝑖 − 47.11 𝑗 = 𝜔4 ∗ 48.77 𝑗 + 𝜔4 ∗ 47.11 𝑖 𝑉𝐵 𝑉𝐵

𝐴

𝐴

= 𝜔3 𝑋 𝑟𝐵/𝐴

= 𝜔3 𝑘 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 𝑚𝑚 𝑠 𝑉𝐷 = 𝜔4 𝑋 𝑟𝐷

𝑉𝐷 = 3.44𝐾 𝑋 82.71 𝑖 − 79.9 𝑗 = 284.52 𝑗 + 274.85 𝑖 = 295.544 𝑚𝑚 𝑠 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 𝑚𝑚 𝑠 ACELERACIONES 𝑎𝐴 = 𝑎𝐴𝑛 + 𝑎𝐴𝑡 Como ω2 es constante, entonces α2 es 0 y por ende aceleración tangencial de A es cero. 𝑎𝐴𝑛 = 𝜔2 𝑋(𝜔2 𝑋 𝑟𝐴) 𝑎𝐴𝑛 = −2𝑘 𝑋 −25.88 𝑗 + 70.72 𝑖 = −51.76 𝑖 − 193.18 𝑗 = 200 𝑚𝑚 2 s 𝑎𝐵𝑛 + 𝑎𝐵𝑡 = 𝑎𝐴𝑛 + 𝑎𝐵/𝐴𝑛 + 𝑎𝐵/𝐴𝑡 𝑎𝐵𝑛 =𝜔 4 𝑋

𝜔 4 𝑋 𝑟𝐵

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 ∗ 𝛼4 𝑖 + 47.11 ∗ 𝛼4 𝑗 𝑎𝐵/𝐴𝑛 =𝜔 3 𝑋

𝜔 3 𝑋 𝑟 𝐵 /𝐴

𝑎𝐵/𝐴𝑛 = 3.13 𝑘 𝑋 193.1 𝑗 + 63.85 𝑖 = −604.4 𝑖 + 199.85𝑗 𝑎𝐵/𝐴𝑡 = 𝛼3 𝑋 𝑟𝐵/𝐴 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 𝑟𝑎𝑑

𝑠2

𝛼3 = 31.36 𝑟𝑎𝑑 𝑠 2 𝑎𝐵 = 580.36 𝑚𝑚 𝑠 2 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 𝑚𝑚

𝑠2


𝑎𝐸𝑛 + 𝑎𝐸𝑡 = 𝑎𝐷𝑛 + 𝑎𝐷𝑡 + 𝑎𝐸/𝐷𝑛 + 𝑎𝐸/𝐷𝑡 𝑎𝐸𝑛 =𝜔 6 𝑋

𝜔 6 𝑋 𝑟𝐸

𝑎𝐸𝑛 = −1.96 𝑘 𝑋 (324.16 𝑗 + 253.62 𝑖) 𝑎𝐸𝑛 = 635.35 𝑖 − 497.09 𝑗 𝑎𝐸𝑡 = 𝛼6 𝑋 𝑟𝐸 = 𝛼6 𝑘 𝑋 −165.39 𝑖 + 129.4 𝑗 𝑎𝐸𝑡 = −165.39 ∗ 𝛼6 𝑖 − 129.4 ∗ 𝛼6 𝑗 𝑎𝐸/𝐷𝑛 =𝜔 5 𝑋

𝜔 5 𝑋 𝑟 𝐸/𝐷

𝑎𝐸/𝐷𝑛 = −0.39𝑘 𝑋 39.78 𝑗 − 20.7 𝑖 = 15.51 𝑖 + 8.073 𝑗 𝑎𝐸/𝐷𝑡 = 𝛼5 𝑋 𝑟𝐸/𝐷 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 𝑟𝑎𝑑 𝑠 2 𝛼5 = 13.28 𝑟𝑎𝑑

𝑠2

𝑎𝐸 = 2902.97 𝑚𝑚

𝑠2


Calculos de velocidades y aceleraciones