Mechanical engineering exercises
================================

.. include:: /_generated/nav/mechanical_engineering.rst

Three problems built on the :doc:`mechanical engineering gallery </api/gallery/mechanical_engineering/index>`.
Try each one before opening its solution. The solutions run when the
documentation is built, and the test suite checks every number they print.

1. Damping from a free decay
----------------------------

Damping is hard to calculate but easy to measure: displace the system,
let go, and compare successive peaks. Simulate a 1 kg mass on a 4 N/m
spring with a 0.4 N s/m damper, released from 0.1 m. Estimate the damping
ratio from the logarithmic decrement of the simulated response, and
compare it with :math:`c / (2\sqrt{km})`.

*Builds on:* :doc:`/api/gallery/mechanical_engineering/index`.

.. dropdown:: Solution

   .. plot::
      :context: reset

      import numpy as np

      from engineeringkit.mechanical_engineering import damped_frequency, damping_ratio, simulate_sdof

      m, c, k = 1.0, 0.4, 4.0
      result = simulate_sdof(m, c, k, x0=0.1, t=np.linspace(0, 20, 20001))
      x = result.displacement
      peaks = np.flatnonzero((x[1:-1] > x[:-2]) & (x[1:-1] > x[2:])) + 1
      delta = np.mean(np.log(x[peaks[:-1]] / x[peaks[1:]]))
      zeta = delta / np.sqrt(4 * np.pi**2 + delta**2)
      period = np.mean(np.diff(result.t[peaks]))
      print(f"log decrement {delta:.4f}: zeta {zeta:.4f} (exact {damping_ratio(c, k, m):.4f})")
      print(f"damped period {period:.4f} s (exact {2 * np.pi / damped_frequency(k, m, c):.4f} s)")
      # log decrement 0.6315: zeta 0.1000 (exact 0.1000)
      # damped period 3.1576 s (exact 3.1574 s)
      assert np.isclose(zeta, damping_ratio(c, k, m), rtol=1e-3)

   The log decrement uses only peak ratios, so it needs no calibration of
   the displacement sensor's gain.

2. Isolating a machine
----------------------

A 200 kg compressor runs at 1500 rpm. Choose the total stiffness of its
rubber mounts (damping ratio 0.05) so that only 10% of its unbalanced force
reaches the floor. How far do the mounts deflect under the machine's
weight?

*Builds on:* :doc:`/api/gallery/mechanical_engineering/index`.

.. dropdown:: Solution

   .. plot::
      :context: reset

      import matplotlib.pyplot as plt
      import numpy as np
      from scipy.optimize import brentq

      from engineeringkit.mechanical_engineering import natural_frequency, transmissibility

      mass, zeta, omega = 200.0, 0.05, 1500 * 2 * np.pi / 60
      r = brentq(lambda r: transmissibility(r, zeta) - 0.1, 1.5, 10)
      k = mass * (omega / r) ** 2
      print(f"frequency ratio {r:.3f}, stiffness {k / 1e3:.0f} kN/m, static deflection {1e3 * mass * 9.81 / k:.1f} mm")
      # frequency ratio 3.399, stiffness 427 kN/m, static deflection 4.6 mm
      assert np.isclose(transmissibility(omega / natural_frequency(k, mass), zeta), 0.1)

      ratios = np.linspace(0.01, 5, 500)
      fig, ax = plt.subplots(figsize=(5, 3.2))
      ax.semilogy(ratios, transmissibility(ratios, zeta))
      ax.axhline(0.1, color="k", ls=":")
      ax.axvline(np.sqrt(2), color="0.6", ls="--")
      ax.set(xlabel="frequency ratio r", ylabel="transmissibility")
      fig.tight_layout()

   Isolation needs a soft mount: the machine must run at more than
   :math:`\sqrt 2` times the natural frequency, here 3.4 times. Every soft
   mount has the drawback that the machine passes through resonance as it
   starts and stops.

3. A gearbox and its bearing
----------------------------

A motor delivers 10 N m at 1450 rpm into a two-stage gearbox with a
20-tooth pinion driving a 60-tooth gear, then an 18-tooth pinion driving a
54-tooth gear. Each mesh is 98% efficient. Find the output speed and
torque. The output shaft's bearing has a dynamic load rating of 30 kN and
carries 4 kN; find its :math:`L_{10}` life in hours.

*Builds on:* :doc:`/api/gallery/mechanical_engineering/index`.

.. dropdown:: Solution

   .. plot::
      :context: reset

      import numpy as np

      from engineeringkit.mechanical_engineering import bearing_l10_life, gear_output_speed, gear_output_torque, gear_ratio, power_torque

      speed, torque = 1450.0, 10.0
      for driver, driven in ((20, 60), (18, 54)):
          speed = gear_output_speed(speed, driver, driven)
          torque = gear_output_torque(torque, driver, driven, efficiency=0.98)
      print(f"overall ratio {gear_ratio(60, 20) * gear_ratio(54, 18):.0f}: {speed:.1f} rpm, {torque:.1f} N m")
      # overall ratio 9: 161.1 rpm, 86.4 N m
      power_in = power_torque(10.0, 1450 * 2 * np.pi / 60)
      power_out = power_torque(torque, speed * 2 * np.pi / 60)
      assert np.isclose(power_out / power_in, 0.98**2)

      L10 = bearing_l10_life(30e3, 4e3)
      print(f"L10 = {L10:.3e} revolutions = {L10 / (speed * 60):.0f} hours")
      # L10 = 4.219e+08 revolutions = 43642 hours

   Ball-bearing life goes as the cube of the load ratio: doubling the load
   cuts the life by eight. The slow output shaft turns few revolutions, so
   the same bearing lasts nine times longer there than on the motor shaft.