Mechanical engineering: power, vibration and mechanism geometry

Mass is kg, lengths m, force N, torque N m, inertia kg m^2 and angular speed rad/s. Gear ratios use positive integer tooth counts; output speed preserves the supplied sign convention and does not automatically flip rotation direction. Output torque includes efficiency in [0,1], so torque*speed equals input power times efficiency. Bearing L10 returns revolutions at 90% survival, not hours; the exponent is typically 3 for balls and 10/3 for rollers.

Vibration

The passive oscillator solves m*x’’+c*x’+k*x=F(t), m>0, c>=0, k>=0. The frequency helpers require k>0. damped_frequency returns zero for critical/ overdamping, indicating no oscillatory frequency rather than a decay rate. Transmissibility is the harmonic force/base-motion amplitude ratio for this linear SDOF model; exact undamped resonance is singular. No nonlinear stiffness, friction, actuator saturation or contact is modeled.

import numpy as np
from engineeringkit.mechanical_engineering import simulate_sdof
result = simulate_sdof(1, 4, 4, x0=1, t=np.linspace(0, 5, 101))
# Critical damping: x=(1+2*t)*exp(-2*t) [m].
fig, ax = result.plot()

Tests check the analytic critical response, undamped energy conservation, monotonic dissipative energy, constant-force equilibrium and gear power balance. Solver time grids must strictly increase; adaptive integration errors are visible.

Mechanisms

Slider-crank displacement uses the positive connecting-rod square-root branch, r>0, l>=r. Four-bar Grashof compares the shortest+longest links against the other pair; it does not determine which grounded mechanism can fully rotate. Square-thread screw torque excludes collar friction; before applying it check that friction angle plus lead angle stays below pi/2. Models assume rigid links and ideal joints unless efficiency is explicitly supplied.