tutorial3_2.ipynb

{"metadata":{"language_info":{"name":"python","version":"3.7.12","mimetype":"text/x-python","codemirror_mode":{"name":"ipython","version":3},"pygments_lexer":"ipython3","nbconvert_exporter":"python","file_extension":".py"},"kernelspec":{"name":"python3","display_name":"Python 3 (ipykernel)","language":"python"}},"nbformat_minor":5,"nbformat":4,"cells":[{"cell_type":"markdown","source":"# Tutorial 3.2: Solving a cantilever beam with Finite Differences\n\nIn this tutorial we will learn how to obtain the motion of a cantilever beam using Finite Differences\n\nThe equation of motion of the beam is:\n$$\\rho A \\ddot{w} + EI w'''' = q(x,t)$$\n\nwhere $w(t)$ is the deflection of the beam, $\\rho$ its density, $A$ its cross-sectional area, $E$ its Young's modulus, $I$ its second moment of area and $q(x,t)$ a distributed force applied to the beam. \n\nThe boundary conditions for a cantilever beam are:\n$$ w(0) = w'(0) = \\dot{w}(0) = \\ddot{w}(0) = 0 $$\n$$EI\\ w^{'''}(L) = F_{\\mathrm{ext}}(t) $$\n$$EI\\ w^{''}(L) = M_{\\mathrm{ext}}(t) $$\n\n## The roadmap to finde the beam deflection  using Finite Differences\n\nThe steps needed to discretize the beam are listed next. \n1. Discretize space into $N + 1$ points.\n2. Discretize the EOM of the beam. To do this, you will have to approximate the 4th order derivative with FD. Use a second order approximation. You can find the coefficients [here](https://en.wikipedia.org/wiki/Finite_difference_coefficient) (or google `finite difference coefficients' and click the Wikipedia page)}.\n3. Apply the boundary conditions. Use the definition of discrete derivatives and EOM to find the relation between the ghost points and the interior points.\n4. Combine all equations into matrices. \n5. Solve the resulting ODE system","metadata":{},"id":"a59fb708-b557-4f17-9e70-0703026a598e"},{"cell_type":"code","source":"pip install ipympl;","metadata":{"trusted":true},"execution_count":1,"outputs":[{"name":"stdout","text":"Requirement already satisfied: ipympl in /srv/conda/envs/notebook/lib/python3.7/site-packages (0.9.2)\nRequirement already satisfied: ipython-genutils in /srv/conda/envs/notebook/lib/python3.7/site-packages (from ipympl) (0.2.0)\nRequirement already satisfied: traitlets<6 in /srv/conda/envs/notebook/lib/python3.7/site-packages (from ipympl) (5.3.0)\nRequirement already satisfied: ipython<9 in /srv/conda/envs/notebook/lib/python3.7/site-packages (from ipympl) (7.33.0)\nRequirement already satisfied: pillow in /srv/conda/envs/notebook/lib/python3.7/site-packages (from ipympl) (9.2.0)\nRequirement already satisfied: ipywidgets<9,>=7.6.0 in /srv/conda/envs/notebook/lib/python3.7/site-packages (from ipympl) (7.7.0)\nRequirement already satisfied: matplotlib<4,>=3.4.0 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cffi>=1.0.1->argon2-cffi-bindings->argon2-cffi->notebook>=4.4.1->widgetsnbextension~=3.6.0->ipywidgets<9,>=7.6.0->ipympl) (2.21)\nNote: you may need to restart the kernel to use updated packages.\n","output_type":"stream"}],"id":"ccceb60e-a0fb-449b-891c-a6d59ab71b98"},{"cell_type":"code","source":"pip install scipy;","metadata":{"trusted":true},"execution_count":2,"outputs":[{"name":"stdout","text":"Requirement already satisfied: scipy in /srv/conda/envs/notebook/lib/python3.7/site-packages (1.7.3)\nRequirement already satisfied: numpy<1.23.0,>=1.16.5 in /srv/conda/envs/notebook/lib/python3.7/site-packages (from scipy) (1.21.6)\nNote: you may need to restart the kernel to use updated packages.\n","output_type":"stream"}],"id":"e2f462a8-8399-4a81-a6ef-b6274540087b"},{"cell_type":"code","source":"# Load packages\n%matplotlib inline\nimport matplotlib.pyplot as plt\nfrom matplotlib import cm\nimport numpy as np\nimport scipy as sp\nfrom scipy.integrate import solve_ivp","metadata":{"trusted":true},"execution_count":3,"outputs":[],"id":"85e93fef-de7f-4f7e-a535-72d2b773605b"},{"cell_type":"markdown","source":"### Step 1: Discretize space","metadata":{},"id":"8fb656d2-f8ab-4cf0-8174-ebb5a0a9b929"},{"cell_type":"code","source":"# Your code goes here","metadata":{"trusted":true},"execution_count":4,"outputs":[],"id":"aa557cb7-d425-41f0-a141-8d68c59ea804"},{"cell_type":"markdown","source":"### Step 2: Discretize the EOM of the beam\n\nUsing the coefficients for a centered FD scheme for 4th order derivatives with 2nd order accuracy we have:\n\n$$ w_i^{''''} = \\frac{w_{i-2}-4w_{i-1}+6w_i-4w_{i+1}+w_{i+2}}{\\Delta x^4}$$\n\nThe equivalent scheme for the 3rd order derivative is:\n\n$$ w_i^{'''} = \\frac{-0.5w_{i-2}+w_{i-1}-w_{i+1}+0.5w_{i+2}}{\\Delta x^3}$$\n\nFor the 2nd order derivative:\n\n$$ w_i^{''} = \\frac{w_{i-1}-2w_i+w_{i+1}}{\\Delta x^2}$$\n\nAnd for the 1st order derivative:\n\n$$ w_i^{'} = \\frac{-0.5w_{i-1}+0.5w_{i+1}}{\\Delta x}$$\n\nReplacing these expressions into the equation of motion we get the discrete system:\n\n> Your derivations go here\n\n### Step 3: Apply boundary conditions\n\n> Your derivations go here\n\n","metadata":{"tags":[]},"id":"ce1ba23e-9989-4ec2-8da9-82468e993ba5"},{"cell_type":"markdown","source":"### Step 4: Matrix form\n\nSummarizing we have the following discrete (in space) equations:\n\n- For $i=1$:\n> Your derivations go here\n- For $i=2$:\n> Your derivations go here\n- For $i=3,...,N-2$:\n> Your derivations go here\n- For $i=N-1$:\n> Your derivations go here\n- For $i=N$:\n> Your derivations go here\n\nThis is equivalent to the following system:\n> Your derivations go here\n\nAnd in a compact form:\n$$\\boldsymbol{M}\\ddot{\\boldsymbol{w}}+\\boldsymbol{K}\\boldsymbol{w}=\\boldsymbol{F}$$","metadata":{},"id":"0f603293-8b8e-4f80-b8a0-df5043d99c9b"},{"cell_type":"code","source":"# Construct the matrices and external force vector","metadata":{"trusted":true},"execution_count":5,"outputs":[],"id":"27bafd61-9ca0-4796-9b18-ebc07f52b360"},{"cell_type":"markdown","source":"# Step 5: Solve the ODE system\n\nNow, we just need to apply what we learned in the previous session to solve the dynamic problem. The main novelty is that here we work with matrices.","metadata":{},"id":"706565f3-49b8-40b5-ad20-91eebccf3f96"},{"cell_type":"code","source":"# Define the ODE function \n\n# Define initial state\n\n# Define time interval and time evaluation points\n\n# Solve\n\n# Plot\n","metadata":{"trusted":true},"execution_count":6,"outputs":[],"id":"b38d7035-abf8-4dba-97ea-071227439416"},{"cell_type":"markdown","source":"## Exercise: Add a point mass at the extreme of the beam","metadata":{},"id":"f7bec530-b838-41d1-91f2-70549ef59917"},{"cell_type":"code","source":"","metadata":{},"execution_count":null,"outputs":[],"id":"6251014d-a853-4810-b0f3-668dd3bb60ff"}]}