Vibration

SDOF Vibration

Single-degree-of-freedom spring–mass–damper analysis: natural frequency, critical damping and damping ratio from the mass (or weight), stiffness (or natural frequency) and damping given as c, ζ, a logarithmic decrement or half-power points; the free response with its envelope and logarithmic decrement; the steady-state amplitude, phase, magnification factor, force transmissibility and isolation efficiency under a harmonic force, a rotating unbalance (MX/me) or harmonic base motion (absolute and relative transmissibility), with Bode-style charts of the magnification and phase over a frequency sweep for a family of damping ratios; and the transient response to a standard shock pulse (half-sine, versed sine, rectangular, terminal-peak or initial-peak sawtooth) or a pasted acceleration or force history by Newmark-β average-acceleration integration, with the maximum response, its time, the dynamic amplification and the maximax shock response spectrum over 0.1 to 10 pulse-duration ratios or an entered frequency range. The default case is Example 3.1 of Rao's Mechanical Vibrations (a 150 lb pump on a 1,200 lb/in plate under 50 cos 62.832t lb, amplitude 0.1504 in out of phase) with a 50 g, 11 ms half-sine shock as the second example.

1 worked example reproduced, listed on the tool page.

Multi-DOF Vibration and Critical Speed

Natural frequencies, mode shapes and critical speeds of lumped systems: a chain of masses, springs and dashpots with fixed or free ends (eigen-solve of K x = ω² M x by Householder reduction and the QL algorithm, Holzer's tabulation for the example mode, Rayleigh's quotient and Dunkerley's bound beside the exact values, modal mass and participation factors, the damped steady-state response to a harmonic force at a chosen mass by the direct complex solve and by mode superposition with a receptance chart over a frequency sweep, and a Newmark-β transient from an initial displacement or a half-sine base shock); a torsional system of discs on shaft segments with gear ratios reduced to the reference shaft (torsional frequencies, Holzer table, node locations, critical speeds of every engine order and the Campbell diagram); and the lateral critical speeds of a stepped shaft with discs on rigid or flexible bearings (Rayleigh with the static deflection curve, Dunkerley, the Myklestad–Prohl transfer-matrix pass beside the lumped-station eigen-solve, influence coefficients, deflection and mode charts, the Campbell line). Gyroscopic effects and shear deflection are not modelled. Defaults: a three-mass chain of 10 lb masses on 100 lb/in springs (the closed-form uniform chain), three discs on 2 in and 1.5 in steel shafts, and a 2 in steel shaft on a 30 in bearing span with two discs.

1 worked example reproduced, listed on the tool page.

Other Mechanical Design sections

Shafts · Gears · Bearings · Springs · Fasteners and Power Screws · Pressure Vessels · Tolerances · Mass Properties · Acoustics · Brakes, Clutches and Belts · Machining and Manufacturing · Lubrication and Wear