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import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
plt.style.use('fivethirtyeight')
from matplotlib import rc
plt.rc('text', usetex=True)
plt.rc('font', family='sans')
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from IPython.lib.display import YouTubeVideo
YouTubeVideo('fcx9jR0O0rY')
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YouTubeVideo('QcENiGydyF0')
Nonlinear pendulum solution#
The first solution to the pendulum was to linearize the equation of motion using a Taylor series expansion. As you look at engineering systems, this is always the best first estimate for a problem. Linear solutions are much easier to solve and usually its good enough to make a plan.
If you have a linear solution, but want the nonlinear solution then you can approach this in two ways
use a different integration or equation e.g. work-energy
numerically integrate the equations of motion
Using work-energy for the pendulum#
The motion of the pendulum must obey the work-energy. Add potential and kinetic energy you have
\(T_0 + V_0 = T(t) + V(t)\)
The kinetic energy is \(T = 1/2mv^2 = 1/2mL^2\dot{\theta}^2\)
the potential energy is \(V = mgh = mgL(1 - \cos\theta)\)
initial kinetic energy (if starting from rest) is \(T_0=0\)
initial potential energy is \(V_0 = mgL(1 - \cos\theta_0)\)
So you can solve for \(\dot{\theta} = f(\theta)\)
\(\dot{\theta} = \pm \sqrt{\frac{2g}{L}(\cos\theta - \cos\theta_0)}\)
Note: This is the same solution derived in the video by using integration by parts. \(\frac{d\dot{\theta}}{dt} = -\frac{g}{L}\sin\theta \rightarrow \frac{d\dot{\theta}}{d\theta}\frac{d\theta}{dt} = -\frac{g}{L}\sin\theta\). The work-energy formula is a shortcut to integrating an equation of motion by parts.
Below, you can see the angular velocity as a function of angle for a L=0.5 m pendulum released at \(\theta_0 = 90^o = \pi/2~rad\).
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L = 0.5
theta = np.linspace(-np.pi/2, np.pi/2, 200)
dtheta = np.sqrt(2*9.81/L*(np.cos(theta) - np.cos(np.pi/2)))
plt.plot(theta*180/np.pi, dtheta*180/np.pi, 'b-')
plt.plot(theta*180/np.pi, -dtheta*180/np.pi, 'b-')
# plt.axis('equal')
plt.xlabel(r'$\theta$ (deg)')
plt.ylabel(r'$\dot{\theta}$ (deg/s)');
Error in callback <function _draw_all_if_interactive at 0x7f7e7ad44280> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
266 def _draw_all_if_interactive() -> None:
267 if matplotlib.is_interactive():
--> 268 draw_all()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
129 for manager in cls.get_all_fig_managers():
130 if force or manager.canvas.figure.stale:
--> 131 manager.canvas.draw_idle()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1903 if not self._is_idle_drawing:
1904 with self._idle_draw_cntx():
-> 1905 self.draw(*args, **kwargs)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
384 # Acquire a lock on the shared font cache.
385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
386 else nullcontext()):
--> 387 self.figure.draw(self.renderer)
388 # A GUI class may be need to update a window using this draw, so
389 # don't forget to call the superclass.
390 super().draw()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3137, in _AxesBase.draw(self, renderer)
3134 if artists_rasterized:
3135 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3137 mimage._draw_list_compositing_images(
3138 renderer, self, artists, self.figure.suppressComposite)
3140 renderer.close_group('axes')
3141 self.stale = False
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1424, in Axis.draw(self, renderer)
1421 renderer.open_group(__name__, gid=self.get_gid())
1423 ticks_to_draw = self._update_ticks()
-> 1424 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1426 for tick in ticks_to_draw:
1427 tick.draw(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2172 # we do this instead of `self.figure.draw_without_rendering`
2173 # so that we can inject the orientation
2174 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175 self.figure.draw(renderer)
2176 if bbox_inches:
2177 if bbox_inches == "tight":
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3137, in _AxesBase.draw(self, renderer)
3134 if artists_rasterized:
3135 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3137 mimage._draw_list_compositing_images(
3138 renderer, self, artists, self.figure.suppressComposite)
3140 renderer.close_group('axes')
3141 self.stale = False
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1424, in Axis.draw(self, renderer)
1421 renderer.open_group(__name__, gid=self.get_gid())
1423 ticks_to_draw = self._update_ticks()
-> 1424 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1426 for tick in ticks_to_draw:
1427 tick.draw(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Phase plot of the simple pendulum#
This type of plot is called a phase plot. You can use it to compare different initial conditions and stability. Extending this phase plot to account for initial angles, \(\theta_0 = (18^o...180^o)\), you see the concentric circles move outward (from light to dark) with higher speeds.
Show code cell source
opacity = 0
for theta0 in np.linspace(np.pi/10, np.pi, 6):
theta = np.linspace(-theta0, theta0, 100)
dtheta = np.sqrt(2*9.81/L*(np.cos(theta) - np.cos(theta0)))
plt.plot(theta*180/np.pi, dtheta*180/np.pi, '-', color = (0, 0, 1, 0.2+opacity))
plt.plot(theta*180/np.pi, -dtheta*180/np.pi, '-', color = (0, 0, 1, 0.2+opacity))
opacity += 0.1
plt.title('Pendulum speed vs angle for\n'+
r'$\theta_0=(18^o...180^o)$')
plt.xlabel(r'$\theta$ (deg)')
plt.ylabel(r'$\dot{\theta}$ (deg/s)');
Error in callback <function _draw_all_if_interactive at 0x7f7e7ad44280> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
266 def _draw_all_if_interactive() -> None:
267 if matplotlib.is_interactive():
--> 268 draw_all()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
129 for manager in cls.get_all_fig_managers():
130 if force or manager.canvas.figure.stale:
--> 131 manager.canvas.draw_idle()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1903 if not self._is_idle_drawing:
1904 with self._idle_draw_cntx():
-> 1905 self.draw(*args, **kwargs)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
384 # Acquire a lock on the shared font cache.
385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
386 else nullcontext()):
--> 387 self.figure.draw(self.renderer)
388 # A GUI class may be need to update a window using this draw, so
389 # don't forget to call the superclass.
390 super().draw()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3101, in _AxesBase.draw(self, renderer)
3098 for spine in self.spines.values():
3099 artists.remove(spine)
-> 3101 self._update_title_position(renderer)
3103 if not self.axison:
3104 for _axis in self._axis_map.values():
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3045, in _AxesBase._update_title_position(self, renderer)
3043 top = max(top, bb.ymax)
3044 if title.get_text():
-> 3045 ax.yaxis.get_tightbbox(renderer) # update offsetText
3046 if ax.yaxis.offsetText.get_text():
3047 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1372, in Axis.get_tightbbox(self, renderer, for_layout_only)
1369 renderer = self.figure._get_renderer()
1370 ticks_to_draw = self._update_ticks()
-> 1372 self._update_label_position(renderer)
1374 # go back to just this axis's tick labels
1375 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2654, in YAxis._update_label_position(self, renderer)
2650 return
2652 # get bounding boxes for this axis and any siblings
2653 # that have been set by `fig.align_ylabels()`
-> 2654 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2655 x, y = self.label.get_position()
2656 if self.label_position == 'left':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2206, in Axis._get_tick_boxes_siblings(self, renderer)
2204 axis = ax._axis_map[name]
2205 ticks_to_draw = axis._update_ticks()
-> 2206 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2207 bboxes.extend(tlb)
2208 bboxes2.extend(tlb2)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2172 # we do this instead of `self.figure.draw_without_rendering`
2173 # so that we can inject the orientation
2174 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175 self.figure.draw(renderer)
2176 if bbox_inches:
2177 if bbox_inches == "tight":
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3101, in _AxesBase.draw(self, renderer)
3098 for spine in self.spines.values():
3099 artists.remove(spine)
-> 3101 self._update_title_position(renderer)
3103 if not self.axison:
3104 for _axis in self._axis_map.values():
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3045, in _AxesBase._update_title_position(self, renderer)
3043 top = max(top, bb.ymax)
3044 if title.get_text():
-> 3045 ax.yaxis.get_tightbbox(renderer) # update offsetText
3046 if ax.yaxis.offsetText.get_text():
3047 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1372, in Axis.get_tightbbox(self, renderer, for_layout_only)
1369 renderer = self.figure._get_renderer()
1370 ticks_to_draw = self._update_ticks()
-> 1372 self._update_label_position(renderer)
1374 # go back to just this axis's tick labels
1375 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2654, in YAxis._update_label_position(self, renderer)
2650 return
2652 # get bounding boxes for this axis and any siblings
2653 # that have been set by `fig.align_ylabels()`
-> 2654 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2655 x, y = self.label.get_position()
2656 if self.label_position == 'left':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2206, in Axis._get_tick_boxes_siblings(self, renderer)
2204 axis = ax._axis_map[name]
2205 ticks_to_draw = axis._update_ticks()
-> 2206 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2207 bboxes.extend(tlb)
2208 bboxes2.extend(tlb2)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Numerical integration#
The phase plot can be used for a number of engineering analyses. It tells you the maximum and minimum speeds and displacements for the pendulum, how fast the system is moving at given location, etc. It does not tell you what \(\theta(t)\) is. If you have a nonlinear equation of motion and you need the solution for \(\theta(t)\), you have to use a numerical integration.
Numerical integration only works on first order differential equations. The idea behind the integration is that if you know the current value and you know how quickly it changes, then you can estimate the next value
\(next~value = current~value + \frac{d~value}{dt}*\Delta t\)
Here, you will separate the second order differential equation,
\(\ddot{\theta} = -\frac{g}{L}\sin\theta\)
into 2 first-order differential equations
\(\frac{d\theta}{dt} = \dot{\theta}\)
\(\frac{d\dot{\theta}}{dt} = -\frac{g}{L}\sin\theta\)
Now, you have 2 differential equations and one state varible, \(\mathbf{x} = [\theta,~\dot{\theta}]\).
\(\frac{dx_1}{dt} = x_2\)
\(\frac{d x_2}{dt} = -\frac{g}{L}\sin x_1\)
g = 9.81
L = 0.5
def pendulum(t, x):
'''pendulum equations of motion for theta and dtheta/dt
arguments
---------
t: current time
x: current state variable [theta, dtheta/dt]
outputs
-------
dx: current derivative of state variable [dtheta/dt, ddtheta/ddt]'''
dx = np.zeros(len(x))
dx[0] = x[1]
dx[1] = -g/L*np.sin(x[0])
return dx
The SciPy function solve_ivp can now be used to integrate pendulum and plot the solutions below. You can use a for-loop to generate multiple solutions on one graph to compare results. Below, the solutions are plotted for \(\theta_0 = (18^o...180^o)\).
Show code cell source
from scipy.integrate import solve_ivp
for theta0 in np.linspace(np.pi/10, np.pi, 6):
sol = solve_ivp(pendulum, [0, 4], [theta0, 0], t_eval=np.linspace(0, 4))
plt.plot(sol.t, sol.y[0]*180/np.pi, label = r'$\theta_0$={:1.0f}$^o$'.format(theta0*180/np.pi))
plt.legend(loc = 'lower right')
<matplotlib.legend.Legend at 0x7f7e52b478e0>
Error in callback <function _draw_all_if_interactive at 0x7f7e7ad44280> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
266 def _draw_all_if_interactive() -> None:
267 if matplotlib.is_interactive():
--> 268 draw_all()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
129 for manager in cls.get_all_fig_managers():
130 if force or manager.canvas.figure.stale:
--> 131 manager.canvas.draw_idle()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1903 if not self._is_idle_drawing:
1904 with self._idle_draw_cntx():
-> 1905 self.draw(*args, **kwargs)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
384 # Acquire a lock on the shared font cache.
385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
386 else nullcontext()):
--> 387 self.figure.draw(self.renderer)
388 # A GUI class may be need to update a window using this draw, so
389 # don't forget to call the superclass.
390 super().draw()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3137, in _AxesBase.draw(self, renderer)
3134 if artists_rasterized:
3135 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3137 mimage._draw_list_compositing_images(
3138 renderer, self, artists, self.figure.suppressComposite)
3140 renderer.close_group('axes')
3141 self.stale = False
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1424, in Axis.draw(self, renderer)
1421 renderer.open_group(__name__, gid=self.get_gid())
1423 ticks_to_draw = self._update_ticks()
-> 1424 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1426 for tick in ticks_to_draw:
1427 tick.draw(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2172 # we do this instead of `self.figure.draw_without_rendering`
2173 # so that we can inject the orientation
2174 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175 self.figure.draw(renderer)
2176 if bbox_inches:
2177 if bbox_inches == "tight":
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3137, in _AxesBase.draw(self, renderer)
3134 if artists_rasterized:
3135 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3137 mimage._draw_list_compositing_images(
3138 renderer, self, artists, self.figure.suppressComposite)
3140 renderer.close_group('axes')
3141 self.stale = False
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1424, in Axis.draw(self, renderer)
1421 renderer.open_group(__name__, gid=self.get_gid())
1423 ticks_to_draw = self._update_ticks()
-> 1424 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1426 for tick in ticks_to_draw:
1427 tick.draw(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Notice in the graphs how the time period changes depending upon initial angle, \(\theta_0\). You may have experienced this first hand if you have played on a swingset. Initially, you need to pump your legs quickly to get started, but then as you swing higher, you spend more time between pumps.
Note: What happens at the \(\theta_0 = 180^o\)? Why is it constant?
Wrapping up#
In this notebook, you used two techniques to solve the nonlinear pendulum equation of motion
work-energy formula which is equivalent to integrating by parts
numerical integration using
solve_ivp
You can use either of these techniques for any engineering system. Similarly, you can always linearize an equation of motion so you don’t need to use nonlinear solutions. Just keep in mind, a linearized solution will introduce error in your model results.