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.

Belts and lubrication

belt_tension_ratio is the Euler-Eytelwein limit exp(mu*theta/sin(beta/2)) at the point of slip, for a massless, inextensible belt; centrifugal tension and belt creep are ignored. Omit groove_angle for a flat belt or a rope on a drum. petroff_friction_torque assumes a centred journal and a linear shear profile across a gap c much smaller than r; it overestimates the exact concentric Couette torque by about 1.5*c/r. sommerfeld_number uses N = omega/(2*pi) in rev/s and the projected pressure P = W/(2*r*L). long_bearing_eccentricity inverts the full Sommerfeld solution for an infinitely long bearing. That solution keeps the negative film pressures a real film cavitates away, so it puts the journal at 90 degrees to the load and underestimates the eccentricity of short bearings. No finite-length, short-bearing or thermal correction is applied.

Linkages and robot arms

fourbar_output_angle grounds link d along the x axis, with the input crank a at the origin and the output link c at (d, 0). It raises ValueError if the loop cannot close at any requested angle, so trying a full turn tests whether the crank can rotate fully. crossed=False is the open circuit. freudenstein_coefficients fits K1..K3 exactly for three precision points and by least squares for more. freudenstein_link_lengths rejects nonpositive K1 or K2 rather than reinterpreting a negative length as a link turned through pi. dh_transform uses the original (distal) Denavit-Hartenberg convention Rz(theta) Tz(d) Tx(a) Rx(alpha); the modified (Craig) convention is not provided.

Absorbers, torsion and rotors

tuned_mass_damper follows Den Hartog: the absorber damping ratio is c2/(2*m2*omega1), with omega1 the primary natural frequency, not the absorber’s. The default tuning f = 1/(1+mu) and damping sqrt(3*mu/(8*(1+mu)^3)) make the fixed points equal and put the peaks within about 1% of sqrt(1+2/mu); this is the force-excited, undamped-primary optimum. holzer_frequencies treats a free-free chain of rigid disks on massless shafts, excludes the zero rigid-body mode, and finds only roots that the trial grid separates. jeffcott_response gives the steady synchronous whirl of a disk midway on a massless shaft with external viscous damping; gyroscopic moments, internal damping, bearing flexibility and run-up transients are not modeled.

Forced vibration and several degrees of freedom

magnification_factor and phase_angle give the steady state of m*x’’+c*x’+k*x=F0*cos(w*t): amplitude (F0/k)*M and lag phi in [0, pi], with r=w/wn and zeta=c/(2*sqrt(k*m)). The transient is not included; use simulate_sdof for run-up. normal_modes solves K*phi=w^2*M*phi with scipy.linalg.eigh for symmetric M (positive definite) and K (positive semidefinite), returns every mode including zero-frequency rigid-body modes, and scales the shapes so phi^T*M*phi=I. modal_response superposes the modes in closed form with modal (proportional) damping ratios in [0, 1); a general, non-proportional damping matrix couples the modes and is not handled there. harmonic_response solves (K-w^2*M+i*w*C)*X=F directly and raises ValueError at an undamped resonance. rayleigh_frequency is an upper bound on the lowest frequency for any trial shape; dunkerley_frequency, from the masses and the diagonal of the flexibility (influence-coefficient) matrix, is a lower bound. Neither includes the shaft’s own mass unless it is lumped into the masses.

Lagrange’s equations

lagrange_accelerations and simulate_lagrangian take a Python function L(q, qdot) = T - V in joules and the generalised non-conservative forces Q(t, q, qdot), and form H*q’’=dL/dq-J*qdot+Q by central differences (H the Hessian of L in qdot, J the mixed derivative). L must be smooth and its Hessian in qdot nonsingular, i.e. the coordinates independent; time-dependent Lagrangians and constraints with multipliers are not supported. The finite differences cost O(n^2) evaluations of L per step and carry a relative error of about step^2 (1e-8 by default); this is a teaching tool, not a multibody code.

Gears, shafts, springs and bolts

lewis_bending_stress is W_t*K_v/(F*m*Y) in SI form (module m in metres), with the whole load at the tip of one tooth; lewis_form_factor interpolates Shigley’s Table 14-2 for 20-degree full-depth teeth from 12 to 400 teeth. barth_velocity_factor uses the SI constants of Shigley’s equation 14-4. The AGMA geometry, overload, size, load-distribution and rim factors are not modelled. Shaft functions assume a solid round section and evaluate the surface fibre: shaft_diameter uses von Mises (default) or Tresca with the tensile yield strength; shaft_diameter_goodman is the DE-Goodman equation with a fully corrected endurance limit and fatigue stress-concentration factors supplied by the caller, and does not check first-cycle yield. Spring functions are for round-wire, close-coiled compression springs: spring_rate counts only wire torsion and spring_shear_stress’s corrections are Wahl (default), Bergsträsser, direct shear only or none. Buckling, surge and end coil effects are not modelled. Bolted joints: bolt_stiffness puts the shank and threads in series; member_stiffness is the frustum model for two identical members of one material (washer face 1.5*d and 30 degrees by default); bolted_joint shares the load linearly until separation, after which the bolt carries the whole load. preload_torque’s coefficient K scatters by about 25%.

Linkage motion, cams and balancing

fourbar_kinematics differentiates the loop closure analytically for the coupler and output angular velocities and accelerations of a crank with angular velocity omega2 and acceleration alpha2; at a change point, where coupler and output are collinear, the solution is infinite. slider_crank_kinematics is exact for an in-line slider-crank with l>r; the rod angle is measured from the line of stroke, positive with the crank above it. cam_follower_motion returns the rise from 0 to L over beta for the uniform, parabolic, harmonic, cycloidal, 3-4-5 and 4-5-6-7 laws, as derivatives with respect to cam angle (multiply by omega^k for time derivatives); the return is L - s, and the cam profile, pressure angle and follower dynamics are not modelled. balance_rotating_masses balances a rigid rotor statically in one plane, or force and couple in two planes; flexible rotors above a bending critical speed need modal balancing, which is not provided.

Tests and gallery examples check the involute’s normals, the capstan limit, the Reynolds-equation load and torque against Sommerfeld, Petroff against exact Couette flow, four-bar loop closure on both circuits, Freudenstein synthesis round trips, DH products against hand trigonometry, Den Hartog’s fixed points, Holzer against the matrix eigenvalue problem and the Jeffcott whirl against a simulated orbit. The curriculum additions are checked against the simulated forced steady state, mass and stiffness orthogonality, direct integration of the MDOF equations, the hand-derived double pendulum and its energy, the Rayleigh-Dunkerley bracket, the inscribed-parabola cantilever, the Goodman line, Castigliano’s spring deflection, the integrated pressure cone, joint compatibility, the residual force and couple after balancing, finite differences of the linkage positions and of the cam laws, and the instant centres.

Textbook reference problems

tests/test_reference_problems.py checks Budynas and Nisbett, Shigley’s Mechanical Engineering Design (10th ed.), Example 14-1 and the worked solutions to Problems 7-1, 10-3, 10-5, 11-8, 11-9 and 11-10, and Rao, Mechanical Vibrations (5th ed.), Example 6.11, against the printed answers. All agree; where they differ in the last digit, the cause is the method, not the theory. Shigley rounds intermediates before going on: in Example 14-1 he takes K_v = 1.52 and prints W_t = 365 lbf, where the unrounded barth_velocity_factor gives 364 lbf, and in Problem 10-5 the unrounded safety factor 2.0450 sits on the rounding boundary of the printed 2.04. In Rao’s Example 6.11, three equal masses on a fixed-free spring chain, normal_modes reproduces the first frequency, 0.44504 sqrt(k/m), to every printed digit; Rao’s roots of the cubic for the other two are low-precision, 1.2471 and 1.8025 where the exact values are 1.24698 and 1.80194.