| 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:
- Assume θ₁ = 1, T₁ = 0 (free end)
- At each station: T_{i+1} = T_i - ω² J_i θ_i
- θ_{i+1} = θ_i - T_i/k_i (deflection through stiffness)
- Natural frequency when T_final = 0 (free end condition)
- 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.
