![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\" "\\"Open"){kind=icon}
![\"Download\"](\"https://img.shields.io/badge/Github-Download-blue.svg\" "\\"Download"){kind=icon}
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Manometer Models and Dynamics"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Summary\n",
"\n",
"This notebook demonstrates the modeling and interactive simulation of a u-tube manometer. This device demonstrates a variety of behaviors exhibited by a linear second order system. An interesting aspect of the problem is the opportunity for passive design of dynamics for a measurement device."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Learning Goals\n",
"\n",
"* Develop linear differential equations models for mechanical systems from momentum/force balances. \n",
"* Describe role of position and velocity as state variables in a dynamic model.\n",
"* Describe undamped, underdamped, overdamped, and critically damped responses.\n",
"* Represent a second order system in standard form with natural frequency and damping factor.\n",
"* Describe second order response to sinusoidal input, and resonance.\n",
"* Construct a state space representation of a second order linear differential equation."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Initializations"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from scipy.integrate import odeint\n",
"from scipy import linalg as la\n",
"from ipywidgets import interact,interactive\n",
"from control.matlab import *\n",
"\n",
"# scales for all subsequent plots\n",
"tmax = 20\n",
"ymin = -0.02\n",
"ymax = +0.02\n",
"axis = [0.0,tmax,ymin,ymax]\n",
"t = np.linspace(0.0,tmax,1000)\n",
"\n",
"# physical properties\n",
"g = 9.8 # m/s\n",
"rho = 1000.0 # density of water kg/m^3\n",
"nu = 1.0e-6 # kinematic viscosity of water m/s^2\n",
"\n",
"# system dimensions\n",
"L = 7 # meters\n",
"d = 0.08 # meters"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Model 1. Steady State Response to a Pressure Differential\n",
"\n",
"For this first model we will that the ends of the u-tube are exposed to a pressure differential $\\Delta P$. How does the level in the tubes change?\n",
"\n",
"The u-tube manometer of cross-sectional area $A$, filled with a liquid of density $\\rho$, the total length of the liquid column $L$. When the ends are open and exposed to the same environmental pressure $P$ the liquid levels in the two the legs of the device will reach the same level. We'll measure the levels in the tubes as a deviation $y$ from this equilibrium position.\n",
"\n",
"At steady state the difference in the levels of the tubes will be $h$. The static pressure difference \n",
"\n",
"$$\\Delta P = \\rho g h$$\n",
"\n",
"or \n",
"\n",
"$$y = \\frac{\\Delta P}{\\rho g}$$\n",
"\n",
"This is simple statics. Notice that neither the cross-sectional area or the length of the liquid column matter. This is the rationale behind the common water level.\n",
"\n",
"\n",
"\n",
"(By [Bd](https://de.wikipedia.org/wiki/User:Bd) at the [German language Wikipedia](https://de.wikipedia.org/wiki/), [CC BY-SA 3.0](http://creativecommons.org/licenses/by-sa/3.0/), [Link](https://commons.wikimedia.org/w/index.php?curid=46342405))"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"jupyter": {
"outputs_hidden": false
}
},
"outputs": [
{
"data": {
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"text/plain": [
""
]
}
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