torsional-vibration

name	torsional-vibration
description	Torsional vibration analysis — MDOF lumped model, natural frequencies, mode shapes, holzer method, excitation orders, resonance avoidance, damping, ISO 1940 balance.
metadata	{"priority":7,"promptSignals":{"phrases":["torsional vibration","torsional resonance","shaft torsional","critical speed torsional","Holzer method","coupling torsional"],"minScore":3}}

Torsional Vibration — Complete Skill

System Model

Lumped model: inertias J_i connected by shaft stiffnesses k_i Equation of motion: [J]{θ̈} + [K]{θ} = {T(t)}

J_i = polar moment of inertia at station i [kg·m²] k_i = torsional stiffness of shaft segment i = GJ_p/L [N·m/rad] G = shear modulus; J_p = π d⁴/32 (solid shaft); L = length

Two-Degree-of-Freedom System

Two inertias J₁ and J₂ connected by shaft k:

Natural frequencies: ω₁ = 0 (rigid body rotation — zero frequency mode) ω₂ = √(k(J₁ + J₂)/(J₁J₂)) = √(k/J_eff) where J_eff = J₁J₂/(J₁+J₂)

Mode shapes: Mode 1 (ω=0): rigid body, both inertias move in phase Mode 2 (ω₂): node between inertias; amplitudes θ₂/θ₁ = -J₁/J₂

Holzer Method (Tabular — MDOF)

Iterative calculation of frequency response for assumed frequency ω:

1. Assume θ₁ = 1, T₁ = 0 (free end)
2. At each station: T_{i+1} = T_i - ω² J_i θ_i
3. θ_{i+1} = θ_i - T_i/k_i (deflection through stiffness)
4. Natural frequency when T_final = 0 (free end condition)
5. Sweep ω and plot T_final vs. ω → zeros are natural frequencies

Excitation Sources

Reciprocating engines (IC engines, compressors): Excitation orders = multiples of shaft speed (for 4-stroke: 0.5, 1, 1.5, 2, 2.5, 3... × n) Critical excitations: major orders (number of cylinders × 0.5 for 4-stroke) 4-cylinder 4-stroke: major orders at 2, 4, 6... × engine frequency

Gears: excitation at gear mesh frequency = (n × z) [Hz; z = number of teeth]

Electric motors: 2× line frequency excitation (120 Hz for 60 Hz supply); VFD adds harmonics

Propellers/fans: blade pass frequency = n × N_blades

Resonance Avoidance

Campbell diagram: plot all excitation orders × speed on same graph with natural frequencies Resonance = intersection of order line with horizontal natural frequency line

Speed range avoidance: If resonance within operating speed range:

• Avoid running at resonance speed (short transit only)
• Add damping (coupling, damper)
• Detune (change k or J to shift natural frequency) ±20% separation rule: natural frequency should be ≥ 20% away from critical order at operating speed

Torsional Stiffness

Solid shaft: k = GJ_p/L = Gπd⁴/(32L) [N·m/rad] Hollow shaft: k = Gπ(D⁴-d⁴)/(32L) Series shafts: 1/k_total = 1/k₁ + 1/k₂ + ... Stepped shaft equivalent: use J_p1/L1 + J_p2/L2... for distributed shaft

Coupling Torsional Stiffness

Flexible couplings add known torsional stiffness:

• Jaw coupling: k = 5,000–100,000 N·m/rad (size-dependent)
• Elastomeric disc: k = 500–50,000 N·m/rad
• Gear coupling (rigid): k = very high (rigid body, add to shaft stiffness)
• Quill shaft coupling: low k → isolates torsional vibration

Torsional Damping

System damping typically low (ζ < 2% for steel shafts) Response amplification at resonance: Q = 1/(2ζ) Resonant torque amplitude: T_res = T_excitation × Q

Dampers: Rubber coupling: adds damping + reduces stiffness (shifts frequency down) Viscous torsional damper: adds viscous damping c; optimal c = 2√(J k) (critical) Geislinger damper: spring + viscous; used in large diesel engines

Stress Calculation

Torque amplitude at resonance → stress: τ = T_max × r / J_p = 16 T_max / (πd³) Safety factor: τ_allow / τ_max ≥ 1.5 (minimum); ≥ 2.0 (preferred)

Standards

ISO 10814: torsional vibration of reciprocating compressors API 619, API 671: torsional analysis requirements for rotating equipment ISO 1940: balance quality grades (G1 to G6.3 for precision to industrial)

Output

Provide: natural frequencies ω_n [Hz and rpm], critical orders, resonance speeds [rpm], Campbell diagram interpretation, resonance margin %, damping recommendation, max dynamic torque [N·m] vs. allowable.