Nonlinear equations frequently arise in engineering analysis, especially when modeling real-world systems that cannot be described by simple linear relationships. In previous chapters, you explored how numerical approaches can be used to solve equations that do not have straightforward analytical solutions. For example, when analyzing the deflection of beams under load, the governing equations often become nonlinear due to the presence of trigonometric or other nonlinear terms. In these cases, root-finding algorithms such as those provided by `scipy.optimize` become essential tools for engineers seeking practical solutions.


import unittest
import user_code
import ast
import re   
import importlib
import csv
import unittest
import importlib
import numpy as np

class TestTask(unittest.TestCase):
    def test_solution_accuracy_P1200(self):
        import user_code
        importlib.reload(user_code)
        P = 1200
        y = user_code.find_beam_deflection(P)
        _dynamic_test(
            self,
            isinstance(y, float),
            "Returned value is a float",
            f"Returned value is not a float: {type(y)}"
        )
        diff = abs(P * np.sin(y) - 1000)
        _dynamic_test(
            self,
            diff < 1e-6,
            "Solution for P=1200 satisfies equation within tolerance",
            f"abs(P*sin(y) - 1000)={diff} is not less than 1e-6 for y={y}"
        )

    def test_solution_accuracy_P1500(self):
        import user_code
        importlib.reload(user_code)
        P = 1500
        y = user_code.find_beam_deflection(P)
        _dynamic_test(
            self,
            isinstance(y, float),
            "Returned value is a float for P=1500",
            f"Returned value is not a float: {type(y)}"
        )
        diff = abs(P * np.sin(y) - 1000)
        _dynamic_test(
            self,
            diff < 1e-6,
            "Solution for P=1500 satisfies equation within tolerance",
            f"abs(P*sin(y) - 1000)={diff} is not less than 1e-6 for y={y}"
        )

    def test_return_type(self):
        import user_code
        importlib.reload(user_code)
        P = 1000
        y = user_code.find_beam_deflection(P)
        _dynamic_test(
            self,
            isinstance(y, float),
            "Return type is float",
            f"Return type is not float: {type(y)}"
        )

def _dynamic_test(test_case, condition, success_message, failure_message):
    if condition:
        test_case._testMethodName = success_message
        test_case.assertTrue(True, success_message)
    else:
        test_case._testMethodName = failure_message
        test_case.fail(failure_message)

def normalize_text(text):
    text = text.lower()
    text = re.sub(r"\\s{2,}", " ", text)
    text = re.sub(r"\\s*([,:?])\\s*", r"\\1 ", text)
    return text.strip()

def change_var(code: str, var_name: str, value: str) -> str:
    tree = ast.parse(code)
    lines = code.splitlines()
    changed = False
    # Collect all assignment nodes to modify
    assign_nodes = [
        (i, node)
        for i, node in enumerate(tree.body)
        if isinstance(node, ast.Assign)
        and any(isinstance(target, ast.Name) and target.id == var_name for target in node.targets)
    ]

    # If nothing to change, return unmodified code
    if not assign_nodes:
        return code

    # Perform replacements for all matching assignments (from last to first to not break line offsets)
    for i, node in reversed(assign_nodes):
        start_line = node.lineno - 1
        line = lines[start_line]
        indent = ' ' * (len(line) - len(line.lstrip()))
        lines[start_line] = f"{indent}{var_name} = {value}"
        next_line = len(lines)
        for next_node in tree.body[i+1:]:
            if hasattr(next_node, 'lineno'):
                next_line = next_node.lineno - 1
                break
        if next_line > start_line + 1:
            lines[start_line+1:next_line] = []
        changed = True

    return '\\n'.join(lines) if changed else code

if __name__ == "__main__":
    unittest.main()


test_main.py

A practical course designed for engineers who want to leverage Python to solve real-world engineering problems. Explore data analysis, mathematical modeling, and simulation using Python's powerful libraries and tools, all through engaging, hands-on examples and challenges.

Learn how to use Python to analyze, summarize, and visualize engineering data. This section covers essential data handling and visualization techniques tailored for engineering applications.

Dive into mathematical modeling and simulation techniques for engineering systems using Python. Learn to model physical processes, solve equations, and simulate system behavior.

Apply data science techniques to engineering problems, including classification, clustering, and predictive modeling using Python's scikit-learn and seaborn libraries.

Challenge: Solve for Beam Deflection

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