.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/analyses/transient/example_sdof_transient.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_gallery_analyses_transient_example_sdof_transient.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_gallery_analyses_transient_example_sdof_transient.py:


SDOF transient — step-load response
===================================

A single-DOF mass-spring-damper subjected to a step load shows off the
Newmark-β transient integrator in
:func:`~femorph_solver.solvers.transient.solve_transient`.  We integrate
for long enough to watch the underdamped oscillation settle toward the
new static equilibrium, then compare the result to the textbook
second-order response.

.. GENERATED FROM PYTHON SOURCE LINES 12-21

.. code-block:: Python


    from __future__ import annotations

    import matplotlib.pyplot as plt
    import numpy as np
    import scipy.sparse as sp

    from femorph_solver.solvers.transient import solve_transient








.. GENERATED FROM PYTHON SOURCE LINES 22-24

Set up a 1-DOF oscillator
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-33

.. code-block:: Python

    m = 1.0  # kg
    k = 4.0 * np.pi**2  # N / m — gives ωn = 2π rad/s (fn = 1 Hz)
    zeta = 0.05
    c = 2.0 * zeta * np.sqrt(k * m)

    M = sp.csr_array(np.array([[m]], dtype=float))
    K = sp.csr_array(np.array([[k]], dtype=float))
    C = sp.csr_array(np.array([[c]], dtype=float))








.. GENERATED FROM PYTHON SOURCE LINES 34-36

Step load at t = 0
------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-39

.. code-block:: Python

    F_step = 1.0  # N
    F = np.array([F_step])








.. GENERATED FROM PYTHON SOURCE LINES 40-42

Newmark-β with the default unconditionally-stable parameters
------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-47

.. code-block:: Python

    dt = 1e-3
    n_steps = 5000
    result = solve_transient(K, M, F, dt=dt, n_steps=n_steps, C=C)
    u_fs = result.displacement[:, 0]








.. GENERATED FROM PYTHON SOURCE LINES 48-50

Analytical comparison
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 50-61

.. code-block:: Python

    omega_n = np.sqrt(k / m)
    omega_d = omega_n * np.sqrt(1 - zeta**2)
    u_static = F_step / k
    phi = np.arctan(zeta / np.sqrt(1 - zeta**2))
    u_exact = u_static * (
        1
        - np.exp(-zeta * omega_n * result.time)
        / np.sqrt(1 - zeta**2)
        * np.cos(omega_d * result.time - phi)
    )








.. GENERATED FROM PYTHON SOURCE LINES 62-64

Plot Newmark vs analytic
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 64-83

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(6, 3))
    ax.plot(result.time, u_fs, label="Newmark-β", color="#1f77b4")
    ax.plot(result.time, u_exact, "--", label="analytic", color="black", linewidth=0.8)
    ax.axhline(
        u_static,
        color="red",
        linestyle=":",
        linewidth=0.8,
        label=f"static limit u_s = {u_static:.4f}",
    )
    ax.set_xlabel("time [s]")
    ax.set_ylabel("displacement [m]")
    ax.set_title("SDOF step-load response (ζ = 0.05)")
    ax.legend(loc="lower right")
    ax.grid(True, alpha=0.3)
    fig.tight_layout()

    err = np.max(np.abs(u_fs - u_exact)) / u_static
    print(f"max relative error vs analytic: {err:.3e}")



.. image-sg:: /gallery/analyses/transient/images/sphx_glr_example_sdof_transient_001.png
   :alt: SDOF step-load response (ζ = 0.05)
   :srcset: /gallery/analyses/transient/images/sphx_glr_example_sdof_transient_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    max relative error vs analytic: 2.423e-05





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.270 seconds)


.. _sphx_glr_download_gallery_analyses_transient_example_sdof_transient.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: example_sdof_transient.ipynb <example_sdof_transient.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: example_sdof_transient.py <example_sdof_transient.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: example_sdof_transient.zip <example_sdof_transient.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_