{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Universidade Federal do Rio Grande do Sul (UFRGS)   \n",
    "Programa de Pós-Graduação em Engenharia Civil (PPGEC)   \n",
    "\n",
    "# PEC00025: Introduction to Vibration Theory\n",
    "\n",
    "\n",
    "## Test P1 (2023/1): time and frequency domain analysis of sdof systems\n",
    "\n",
    "---\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Importing Python modules required for this notebook\n",
    "# (this cell must be executed with \"shift+enter\" before any other Python cell)\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from MRPy import *\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Questão 1\n",
    "\n",
    "Uma pessoa com massa $m = 80$kg impacta verticalmente a extremidade de um trampolim com velocidade $v = 3$m/s. O trampolim tem comprimento $L = 2$m, massa por unidade de comprimento $\\mu = 20$kg/m, e amortecimento $\\zeta = 2$% (razão do crítico). A rigidez à flexão, $EI$, é tal que o deslocamento estático sob o peso da pessoa é 0,1m. Pergunta-se:\n",
    "\n",
    "1. Qual o máximo deslocamento na extremidade do trampolim após o impacto?\n",
    "2. Qual o tempo transcorrido até que a pessoa perca o contato com o trampolim ao ser projetada de volta para cima?\n",
    "3. Quantos ciclos de vibração o trampolim completará antes que a pessoa volte a cair impactando o mesmo pela segunda vez?\n",
    "4. Quais as máximas reações de apoio $M_A$ (momento) e $V_A$ (força vertical)? \n",
    "\n",
    "<img src=\"resources/tests/PEC00025A_231_P1_Q1.jpg\" alt=\"Question 1\" width=\"480px\"/>\n",
    "\n",
    "\n",
    "### Cálculo das propriedades do sistema\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rigidez do trampolim:                      7848 N/m\n",
      "Massa total (pessoa + trampolim):         100.0 kg\n",
      "Velocidade inicial (pessoa + trampolim):   2.40 m/s\n",
      "Frequência natural do sistema:             1.41 Hz\n"
     ]
    }
   ],
   "source": [
    "L  =  2.             # comprimento do trampolim (m)\n",
    "vp =  3.             # velocidade da pessoa no contato (m/s)\n",
    "\n",
    "mp =  80.            # massa da pessoa (kg)\n",
    "mt =  20*L/2         # massa do trampolim (metade da massa total, kg)\n",
    "m  =  mp + mt        # massa oscilante total (kg)\n",
    "\n",
    "zt =  0.02           # amortecimento razão do crítico\n",
    "ue =  0.1            # deslocamento estático sob o peso da pessoa (m)\n",
    "g  =  9.81           # gravidade (m/s2)\n",
    "\n",
    "k  =  mp*g/ue        # coeficiente de rigidez\n",
    "Q  =  mp*vp          # quantidade de movimento inicial\n",
    "v0 =  Q/m            # velocidade pessoa+trampolim após contato\n",
    "wn =  np.sqrt(k/m)   # frequência natural (rad/s)\n",
    "fn =  wn/2/np.pi     # frequência natural (Hz)\n",
    "\n",
    "print('Rigidez do trampolim:                     {0:5.0f} N/m'.format(k))\n",
    "print('Massa total (pessoa + trampolim):         {0:5.1f} kg'.format(m))\n",
    "print('Velocidade inicial (pessoa + trampolim):  {0:5.2f} m/s'.format(v0))\n",
    "print('Frequência natural do sistema:            {0:5.2f} Hz'.format(fn))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1) Deslocamento máximo\n",
    "\n",
    "Após o contato, a massa da pessoa se soma à massa (parcial) do trampolim com uma velocidade inicial dada.\n",
    "O deslocamento máximo tem duas parcelas:\n",
    "\n",
    "1. um deslocamento devido a uma velocidade inicial, mais\n",
    "2. um deslocamento devido a uma força impulsiva retangular de duração infinita, com fator de amplificação A = 2. \n",
    "\n",
    "<img src=\"images/dynamic_factors.jpg\" alt=\"Dynamic factors\" width=\"480px\"/>\n",
    "\n",
    "Na verdade esses dois deslocamentos de pico não acontecerão exatamente no mesmo instante e, portanto,\n",
    "simplesmente somá-los é uma simplificação que resulta em um deslocamento total superestimado. \n",
    "Além disso, ao se desprezar o efeito do amortecimento na amplitude observada após 1/4 do período,\n",
    "o deslocamento máximo devido à velocidade inicial também é um pouco superestimado.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Deslocamento devido à velocidade inicial:  0.27 m\n",
      "Deslocamento devido à força impulsiva:     0.20 m\n",
      "Deslocamento máximo total:                 0.47 m\n"
     ]
    }
   ],
   "source": [
    "umax_1 = v0/wn      # parcela referente à conservação da quantidade de moviment\n",
    "umax_2 = 2*ue       # parcela referente à força impulsiva \n",
    "\n",
    "umax   = umax_1 + umax_2\n",
    "\n",
    "print('Deslocamento devido à velocidade inicial: {0:5.2f} m'.format(umax_1))\n",
    "print('Deslocamento devido à força impulsiva:    {0:5.2f} m'.format(umax_2))\n",
    "print('Deslocamento máximo total:                {0:5.2f} m'.format(umax))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O cálculo abaixo, obviamente não exigido na prova, apresenta o resultado correto para essa questão utilizado\n",
    "uma integração por Duhamel.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Deslocamento máximo total (por Duhamel):   0.38 m\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 720x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "F = MRPy(-mp*g*np.ones((1,1024)), Td=8)/m         # força constante a partir de t = 0\n",
    "u = F.sdof_Duhamel(fn, zt, V0=-v0)                # resolve por Duhamel, com velocidade inicial dada\n",
    "\n",
    "u.plot_time(fig=2, figsize=[10,4], axis_t=[0, u.Td, -0.6, 0.6]);\n",
    "\n",
    "u_pk = np.max(np.abs(u[0]))\n",
    "print('Deslocamento máximo total (por Duhamel):  {0:5.2f} m'.format(u_pk))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O que confirma que a simples superposição dos dois efeitos superestima o pico de deslocamento.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2) Tempo até nova perda de contato\n",
    "\n",
    "A resposta devida à velocidade inicial reverte o sinal e é maior que a resposta devida à força impulsiva,\n",
    "que não reverte o sinal. Portanto a resposta total terá reversão de sinal, como de fato pode ser observado\n",
    "no gráfico da solução por Duhamel.\n",
    "\n",
    "Pode-se então supor que o contato só será desfeito no momento em que o deslocamento atinge o pico com o\n",
    "sinal revertido, o que ocorre mais ou menos aos 3/4 do período natural de vibração livre do sistema\n",
    "pessoa+trampolim.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A perda de contato ocorrerá aos 0.532 s\n",
      "Pela solução por Duhamel seria aos 0.571 s\n"
     ]
    }
   ],
   "source": [
    "Tn  = 1/fn               # período de vibração (pessoa + trampolim)\n",
    "tk1 = 3*Tn/4             # perda de contato estimada\n",
    "\n",
    "t   = u.t_axis()         # perda de contato pela solução por Duhamel\n",
    "kt  = np.argmax(u[0])\n",
    "tk2 = t[kt]\n",
    "\n",
    "print('A perda de contato ocorrerá aos {0:5.3f} s'.format(tk1))\n",
    "print('Pela solução por Duhamel seria aos {0:5.3f} s'.format(tk2))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Portanto o critério usado para a estimativa é razoável.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "band = [2/u.Td, u.fs/32]\n",
    "a    =  u.differentiate(band=band).differentiate(band=band)\n",
    "\n",
    "a.plot_time(fig=2, figsize=[10,4], axis_t=[0, a.Td/2, -40, 40]);\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3) Trampolim sem a pessoa\n",
    "\n",
    "A partir do momento em que o trampolim está oscilando sem a pessoa sua frequência natural aumenta, \n",
    "porque embora a rigidez permaneça a mesma a massa aderida diminui.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Período de vibração pessoa+trampolin:      0.71 s\n",
      "Período de vibração só trampolim:          0.32 s\n"
     ]
    }
   ],
   "source": [
    "wnt = np.sqrt(k/mt)\n",
    "fnt = wnt/2/np.pi\n",
    "Tnt = 1/fnt\n",
    "\n",
    "print('Período de vibração pessoa+trampolin:     {0:5.2f} s'.format(Tn))\n",
    "print('Período de vibração só trampolim:         {0:5.2f} s'.format(Tnt))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Portanto o período reduz-se para aproximadamente a metade. \n",
    "\n",
    "Considerando que houve uma restituição total da energia cinética, a velocidade da pessoa ao se soltar do trampolim é a mesma do instante de contato inicial. \n",
    "\n",
    "O tempo para retornar ao contato é dado pela equação da posição de um corpo acelerado:\n",
    "\n",
    "$$ u = u_0 + v_0 t + \\frac{a t^2}{2} $$\n",
    "\n",
    "com $u = u_0$ (retorno à mesma posição inicial) e $a = g$. Isolando-se $t$ tem-se:\n",
    "\n",
    "$$ t = \\frac{2v_0}{g}$$\n",
    "\n",
    "onde $v_0$ é a velocidade da pessoa no momento do contato inicial (3m/s).\n",
    "Dividindo-se o tempo transcorrido pelo período do trampolim vazio tem-se então:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tempo até novo contato:                    0.61 s\n",
      "Número aprox. de oscilações do trampolim:       2\n"
     ]
    }
   ],
   "source": [
    "T1 = 2*vp/g\n",
    "N  = T1/Tnt\n",
    "\n",
    "print('Tempo até novo contato:                   {0:5.2f} s'.format(T1))\n",
    "print('Número aprox. de oscilações do trampolim: {0:7.0f}'.format(np.round(N)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4) Máximas reações de apoio\n",
    "\n",
    "As máximas reações podem ser estimadas a partir da força estática equivalente. \n",
    "Para isso, vamos usar o deslocamento máximo (superestimado) do item (1).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Momento reativo no engaste:  7391 Nm\n",
      "Reação vertical no engaste:  3696 N\n"
     ]
    }
   ],
   "source": [
    "Feq = umax*k           # força estática equivalente\n",
    "\n",
    "MA  = L*Feq            # momento reativo\n",
    "VA  = Feq              # reação vertical\n",
    "\n",
    "print('Momento reativo no engaste: {0:5.0f} Nm'.format(MA))\n",
    "print('Reação vertical no engaste: {0:5.0f} N'.format(VA))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Contudo, o valor correto calculado a partir da solução por Duhamel (não exigida na prova) seria:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Momento reativo no engaste:  5949 Nm\n",
      "Reação vertical no engaste:  2974 N\n"
     ]
    }
   ],
   "source": [
    "Feq = u_pk*k           # força estática equivalente\n",
    "\n",
    "MA  = L*Feq            # momento reativo\n",
    "VA  = Feq              # reação vertical\n",
    "\n",
    "print('Momento reativo no engaste: {0:5.0f} Nm'.format(MA))\n",
    "print('Reação vertical no engaste: {0:5.0f} N'.format(VA))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Questão 2\n",
    "\n",
    "Um carrinho se desloca com velocidade estacionária $v$ sobre uma superfície com relevo senoidal, cujo comprimento de onda (igual à distância entre as rodas) é $\\lambda_0 = 2$m e amplitude (zero-a-pico) $h = 0,1$m. Sobre o carrinho é transportada uma massa $m = 200$kg, que está suportada por um console elástico cujo amortecimento é $\\zeta = 1$% (razão do crítico). Considere que a massa do carrinho é desprezável. \n",
    "\n",
    "5. Admitindo inicialmente um console rígido, qual a máxima velocidade $v_{\\rm MAX}$ que o carrinho pode atingir antes que as rodas percam contato com a superfície. \n",
    "6. Para $v = v_{\\rm MAX}/2$, determine a rigidez da mola k de modo que a amplitude total de vibração da massa (zero-a-pico) seja no máximo igual a $2h$.\n",
    "7. Para a rigidez $k$ calculada, determine a máxima e a mínima forças de reação vertical $R$ em cada roda durante o movimento.\n",
    "\n",
    "<img src=\"resources/tests/PEC00025A_231_P1_Q2.jpg\" alt=\"Question 2\" width=\"480px\"/>  \n",
    "\n",
    "\n",
    "### Cálculo das propriedades do sistema\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "h  =  0.1            # amplitude do deslocamento na base\n",
    "l0 =  2.             # distância entre rodas (m)\n",
    "m  =  200.           # massa oscilante total (kg)\n",
    "zt =  0.01           # amortecimento razão do crítico\n",
    "g  =  9.81           # gravidade (m/s2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.1 Velocidade máxima\n",
    "\n",
    "Com a mola considerada rígida, a massa acompanha o relevo do terreno. Isso implica em um deslocamento da base\n",
    "igual ao deslocamento da massa com uma frequência igual a $f_0 = v_{\\rm max}/\\lambda_0$.\n",
    "O deslocamento vertical é uma função harmônica:\n",
    "\n",
    "$$ u(t) = h \\sin (2\\pi f_0 t) = h \\sin \\left( \\frac{2\\pi v_{\\rm max} t}{\\lambda_0} \\right) $$\n",
    "\n",
    "\n",
    "O critério para que o carrinho não descole do terreno é a aceleração vertical não superar a gravidade.\n",
    "A aceleração é a segunda derivada do deslocamento.\n",
    "\n",
    "$$ a = \\frac{d^2 u}{dt^2}  = \n",
    "- h \\left(\\frac{2\\pi v_{\\rm max}}{\\lambda_0} \\right)^2 \\sin \\left( \\frac{2\\pi v_{\\rm max} t}{\\lambda_0} \\right)  > -g $$\n",
    "\n",
    "Portanto o limite de velocidade é dado por:\n",
    "\n",
    "$$ h \\left(\\frac{2\\pi v_{\\rm max}}{\\lambda_0} \\right)^2 = g $$\n",
    "\n",
    "Que resulta em:\n",
    "\n",
    "$$ v_{\\rm max} = \\frac{\\lambda_0}{2\\pi} \\sqrt{\\frac{g}{h}}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Máxima velocidade para manter contato:     3.15 m/s\n"
     ]
    }
   ],
   "source": [
    "vmax = (l0/2/np.pi)*np.sqrt(g/h)\n",
    "\n",
    "print('Máxima velocidade para manter contato:    {0:5.2f} m/s'.format(vmax))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 Amplitude de vibração para velocidade $v_{\\rm max}/2$\n",
    "\n",
    "A aceleração na base é dada pela mesma expressão anterior, mas \n",
    "com a velocidade agora reduzida à metade:\n",
    "\n",
    "$$ a = - h \\left(\\frac{\\pi v_{\\rm max}}{\\lambda_0} \\right)^2 \\sin \\left( \\frac{\\pi v_{\\rm max} t}{\\lambda_0} \\right)$$\n",
    "\n",
    "onde a frequência de excitação é dada por:\n",
    "\n",
    "$$ f_0 = \\frac{\\pi v_{\\rm max}}{2\\pi \\cdot \\lambda_0} = \\frac{v_{\\rm max}}{2\\lambda_0} $$\n",
    "\n",
    "Calculando amplitude e frequência da excitação:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Amplitude da aceleração:                   2.45 m/s2\n",
      "Frequência da excitação harmônica:         0.79 Hz\n"
     ]
    }
   ],
   "source": [
    "aG = h*((np.pi*vmax/l0)**2)\n",
    "f0 = vmax/(2*l0)\n",
    "\n",
    "print('Amplitude da aceleração:                  {0:5.2f} m/s2'.format(aG))\n",
    "print('Frequência da excitação harmônica:        {0:5.2f} Hz'.format(f0))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para que a amplitude total seja no máximo $2h$, a amplitude do sistema deve ser\n",
    "no máximo $h$, visto que já ocorre uma amplitude $h$ inerente ao próprio relevo.\n",
    "\n",
    "No cálculo a seguir, vamos variando a frequência $f_{\\rm n}$ até ter a amplitude \n",
    "de deslocamento desejada:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Frequência natural de vibração livre:     1.114 Hz\n",
      "Resposta em deslocamento estático:        0.050 m\n",
      "AMPLITUDE DA RESPOSTA EM DESLOCAMENTO:    0.100 m\n",
      "Fator de amplificação dinâmica:           2.000\n",
      "Rigidez calibrada:                        9.807 kN/m\n"
     ]
    }
   ],
   "source": [
    "fn =   1.414*f0              # valor obtido por tentativa e erro\n",
    "k  = ((2*np.pi*fn)**2)*m\n",
    "ue =   aG*m/k\n",
    "\n",
    "bt =   f0/fn\n",
    "A  =   np.sqrt(1/((1 - bt**2)**2 + (2*zt*bt)**2))\n",
    "ud =   A*ue\n",
    "\n",
    "print('Frequência natural de vibração livre:     {0:5.3f} Hz'.format(fn))\n",
    "print('Resposta em deslocamento estático:        {0:5.3f} m'.format(ue))\n",
    "print('AMPLITUDE DA RESPOSTA EM DESLOCAMENTO:    {0:5.3f} m'.format(ud))\n",
    "print('Fator de amplificação dinâmica:           {0:5.3f}'.format(A))\n",
    "print('Rigidez calibrada:                        {0:5.3f} kN/m'.format(k/1000))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3 Máxima reação vertical\n",
    "\n",
    "A máxima amplitude de deslocamento é portanto $h$, que multiplicada pela rigidez\n",
    "dá a força da mola sobre a massa. Essa força deve ser ainda ser acrescida do peso\n",
    "da massa (quando a roda passa pelo ponto mais baixo) e dividida entre os\n",
    "dois eixos. Portanto:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Máxima reação em cada eixo:               149.98 kgf\n"
     ]
    }
   ],
   "source": [
    "R = (h*k + m*g)/2/g\n",
    "\n",
    "print('Máxima reação em cada eixo:               {0:5.2f} kgf'.format(R))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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