Vibrações Forçadas

Sistemas de 1 GDL\nVibração Forçada 27/10/10 (prática)\nVibração forçada:\n\nm = 87.6 kg\nk = 3.5x10^4 N/m\nf(x) = F senωt ; F = 44.5N\n\nWrms = ?\n\nXrms = ? ; se C = 1048.6 Ns/m X(t) = X0 sen(ωt + δ)\nX = √((k-w²m)² + (ωc)²)\n\nω = F/K\nX = F/√(k/2) + (2ξβ²)\n\nWrms = Wrms = √((3.5x10^4)/87.6) = 20 rad/s\n\nXrms = F/(ωc)\nXrms = 14.3 mm\n\nξ = ? x/ d ?\n\nS = ln [(xi/x0) = 2ln(4.2) = (2π/ξ) = 2\n\n = 0.223\n\n\nF = ta √ (ξ) \n\nTa = 2πj ; juse = Unm √ 1-ξ2 = Unum = 3.58 rad\n\nβ = U = 3 = 0.84\n\n\nw = √ (K/m)\n\nX = 8.1/5.533\n\n^{1/2} = - 0.035\n\n {} \n\n\nz = ty = (2x0.23x0.84)\n\n \n\n0.9 rad\n\nα = t/gy = (2x0.23x0.84)\n\n\n\n19 N; F = 24.5 N; Tm = 0.2 s; x max = 0.0127 m\n\na) C = ?\n\n0.0127 = 24.5 \n\n∴ C = (24.5*0.2)/0.0127 = 61.91 N/m\n\nb) f = 4 Hz;\n \n\n\n(x(t) = (x(0) - x) x 100%)\n R = 2∫(0.6) \nM = 3.85 \nF = R = A\n\n\n\nAssuming C = 0\nR = 0.0127 = 0.022\n\nX = \n\n= 0.0145m\n\nx' =\n = 24.5 \n\nX' = 0.0357\n\n(0.0357 - 0.045) x 100 = 46%.\n\n0.0145\n\n\n m x \" + Cx + kx = -a0/2\nx0 = -a0 dt\nx0 = 0 ; x0 = 0\nP.47\n\n\nk1\nC \n \n \n \n \n \n\nk2\nx(t)\n\n \n \n \n\ny(t)\n\n \n \n\ny(t) = A T\n\n\na0 = 2/T ∫ (A dt + t 2 = A)\n\nT2 T2\n\n\na0 = A/2\n\n\nam = 2/T2 ∫ (A cos a dt = A)\n\nT \n T\n\n= 2A/T2 { [t 1 summation to T0] - ∫ 0 to T1 summation dt =\n\na = 2A/T2 . 1/(mw) - (1) 1/(mw) \n\n2A/T2 - e[-2*cos 0] = 0 b_n = \\frac{2}{T} \\int_{0}^{T} A + A \\sin(\\omega t) dt = 2A \\left[ -\\frac{t}{T} \\sin(\\omega t) \\right]_{0}^{T} + \\int_{0}^{T} \\frac{1}{\\omega} \\sin(\\omega t) dt\n= \\frac{2A}{T} \\left[ \\frac{T}{\\omega} \\cos(\\omega t) \\right]_{0}^{T} - 0 + 1 \\cdot \\left( \\frac{A}{\\omega^2} \\right) \n\n= \\frac{2A}{T} \\left[ 0 - 1 \\right] + \\frac{A}{\\omega^2} \\therefore \n\ny(t) = \\frac{A}{2} + \\sum_{m=1}^{\\infty} \\left( \\frac{-A}{m\\pi} \\sin(m\\omega t) \\right)\n\nT = \\frac{1}{2} m x^2\nv = \\frac{1}{2} k x^2 + \\frac{1}{2} k (x-y)^{2}\ng = \\frac{1}{2} C x^2\n\nmx^{\\cdot \\cdot} + C + k x = k_2 y(t)\n\\text{Is: } \\frac{km}{k}\n\\therefore x(t) = \\frac{1}{k} \\sum \\frac{V m}{\\sin(\\omega t - \\alpha)} \n\\sqrt{\\frac{1}{(1-\\beta^{2}) + (2\\xi\\beta)^{2}}}\n\nb_n = tg^{-1} \\left( \\frac{\\sum_{i=1}^{n} B_n}{\\alpha - \\beta_n} \\right) ; B_n = \\frac{A_n}{\\omega_n}\n\\omega_n = \\sqrt{\\frac{k}{m}} \n\\zeta = \\frac{C}{2m}\n\\text{Respeito a uma solu\\u00e7\\u00e3o fis\\u00eddica: }\nf(t)\n\\text{Grafico: } (3/11/10) (pratica):\n\n\\beta = 13.3Hg\nfa\\cdot Fa\\cdot St = 7.6g\n\\xi = 0\n\\text{( } K)$\n\\text{F_R} = \\frac{F_u}{F_ap} = \\sqrt{\\frac{1 + (2\\xi \\beta)^{2}}{(1 - \\beta^{2})^{2} + (\\xi \\beta)^{2}}}\\text{TR} = \\frac{1 + 0}{(1 - \\beta^{2})^{2} + (\\xi \\beta)^{2}}\\omega_n = \\sqrt{\\frac{k}{m}}\n\\beta = \\frac{2\\pi \\cdot 13.3}{9.8} = 2.33\n\\sqrt{7.6 \\cdot 10^{-3}}\n\\text{Reduz: } = 271\n\\text{TR} = 0.23\n\na)$\n\\text{m}=91kg\nk=7\\times10^{5}N/m\nF=353N\nm=3000rpm\n\\xi=0.2\n\\text{b) TR=} ?\n\\text{c) F_R} = ? \\text{WM} = \\sqrt{\\frac{k}{m}} = 88.7 rad/s\nW = 2\\pi\\cdot3000/60 = 314.16 rad/s\n\\beta = 3.58 \n= \\frac{X}{\\sqrt{(1 - \\beta^{2})^{2} + (2\\beta)(2.0)^{2}}}\n\\text{TR} = 0.147\nF_{TR} = TR\\cdotF_{ap} = 0.147 x 353 = 51.9N\nm = 29.5 kg\nm = 580rpm\nk = ?\n\\text{TR} \\leq 0.1 \\implies 1 > 0.01 \\rightarrow (1 - \\beta^{2})^{2} \n\n\\therefore (1-\\beta^{2}) \\leq 0.1 \\implies \\beta = 3.31 \\text{(para ma\\u00edor)} \n\\beta ^{2} > \\left( \\frac{80}{60} \\right)^{2} > 11 \n\\therefore \\text{(TR < 0.1 conditions)} K < (\\frac{27 \\times 580}{60})^{2} \\times 29.5 = 3291 N/mm^{2} \n40\n\n\\zeta = 0\nT R = \\frac{1}{\\sqrt{1-\\beta^{2}}}\nX(v) = ?\n\\uparrow\nV = \\frac{L}{T}\n\\omega = \\frac{2\\pi}{L} \\cdot V \nX = \\frac{1}{1-\\beta^{2}} \\cdot y\nsituación crítica (\\rightarrow z = 1)\n\\to V_{crit} = \\frac{L}{2 \\pi} \\cdot \\omega_n \n\\begin{tikzpicture}\n\\draw[domain=0:6.28,smooth,variable=\\x] plot ({\\x},{1-1*\\x^2}) ;\n\\end{tikzpicture}\nTPC -> (41) \\begin{bmatrix} |x(t)| = \\begin{bmatrix} [M] \\end{bmatrix} \\begin{bmatrix} [P] \\end{bmatrix} \\end{bmatrix} \n\\begin{bmatrix} 0 \\cdots 1 \\end{bmatrix} \\begin{bmatrix} K_{x1} & K_{x2} \\end{bmatrix} \\begin{bmatrix} K \\end{bmatrix} \\begin{bmatrix} 0 \\end{bmatrix} = 0 \n\n\\begin{array}{l} m \\ddot{x} + \\beta x = 0 \\ m \\ddot{x_2} + \\beta x_1 + \\zeta Kx_2 = 0 \\end{array}\n\\text{Ejemplo:} \\begin{bmatrix} x_{1} \\end{bmatrix} \\to \\frac{1}{2} m \\cdot x_{1}^{2} + \\frac{1}{2} m_{2} x_{2}^{2} \\end{bmatrix} \\begin{align*} d \\end{align*} \n\\frac{\\partial}{\\partial t} - \\frac{1}{z} K_{ij} \\cdot \\frac{\\partial}{\\partial x} = 0\\end{bmatrix}\nq_1 = x_1, j=1,2\n\\begin{bmatrix} x_{3} \\end{bmatrix} = x_{1} + \\theta a\n\\begin{bmatrix} x_{4} \\end{bmatrix} = x_{2} - a\\end{bmatrix}\nx_2 = x_1 + 3 \\omega\\end{bmatrix}\n\\theta = x_2 - x_1 \\leads to \n\\begin{bmatrix} x_{3} = x_{1} + \\frac{1}{3}(x_{2}-x_{1}) \\end{bmatrix} \n\\begin{bmatrix} x_{4} = \\frac{2}{3}x_{2} + \\frac{1}{3}x_{1}\\end{bmatrix}