Technology / Technical guide
Two-Inertia Torsional Vibration in Wind Generators
Two rotating masses joined by an elastic torsional element can share an overall rotation while oscillating relative to one another. The relative twist and the common rotation are different coordinates.
A free system has both a rigid mode and an elastic mode.
A torsional model must preserve that distinction. Its natural frequency cannot be obtained from a gear ratio or an electrical power rating alone.
01 / Model definition
Two inertias do not imply two fixed supports.
Consider a hypothetical free system with positive inertias J1 and J2, joined only by a torsional spring of stiffness k.
Consider a hypothetical free system with positive inertias J1 and J2, joined only by a torsional spring of stiffness k. Both angles θ1 and θ2 are measured in the same positive rotation direction. There is no elastic attachment to ground in this starting model.
The relative twist is δ = θ1 −θ2. Equal angular increments applied to both sides leave δ unchanged. This fact is central: a coupling can resist differential motion without resisting common rotation. A drawing that fixes one side describes a different boundary condition.
02 / Torque balance
The coupling acts internally on both sides.
For a linear spring, its torque on inertia 1 is −kδ and on inertia 2 is +kδ.
For a linear spring, its torque on inertia 1 is −kδ and on inertia 2 is +kδ. With no external torques or damping, J1θ1¨ = −kδ and J2θ2¨ = +kδ. The opposite signs preserve angular momentum of this isolated system.
Torsional stiffness k is expressed as N·m/rad and inertia as kg·m². Their ratio has the dimensions of angular frequency squared when radians are treated as dimensionless. A translational spring value in N/m cannot be substituted without a defined geometric transformation.
03 / Relative mode
Both inertias participate in the oscillation.
Subtracting the two angular-acceleration equations gives δ¨ + k(1/J1 +1/J2)δ = 0.
Subtracting the two angular-acceleration equations gives δ¨ + k(1/J1 +1/J2)δ = 0. The relative-mode frequency is ωrel = √[k(1/J1 +1/J2)]. Using only k/J1 would incorrectly treat the second inertia as fixed.
For an entirely hypothetical J1 = 1 kg·m², J2 = 3 kg·m² and k = 12 N·m/rad, ωrel is 4 rad/s, or about 0.637 Hz. These values explain the relation and are not turbine shaft properties, an operating speed or a resonance limit.
04 / Rigid mode
Zero natural frequency does not mean zero rotation.
Define the inertia-weighted common angle Θ = (J1θ1 +J2θ2)/(J1 +J2).
Define the inertia-weighted common angle Θ = (J1θ1 +J2θ2)/(J1 +J2). In the free unforced model, Θ¨ = 0. The system can therefore retain constant common angular velocity while having no relative twist and no spring torque.
This is the rigid-rotation mode with zero restoring natural frequency. Zero frequency describes the absence of a spring restoring term for common motion; it does not require the rotor to be stationary. In the other mode, the two angles oscillate oppositely about the common coordinate, weighted by their inertias.
05 / Relative damping
Internal dissipation removes differential energy.
Adding a viscous coupling gives torque −cδ̇ on side 1 and +cδ̇ on side 2.
Adding a viscous coupling gives torque −cδ̇ on side 1 and +cδ̇ on side 2. For positive c, the coupling removes energy at rate cδ̇². It responds to relative angular velocity, not either absolute speed by itself.
Pure common rotation has δ̇ = 0, so this internal damper alone does not slow that rigid motion. The elastic-mode kinetic energy can be written using Jeq = J1J2/(J1 +J2). This reduced inertia and the coupling describe differential motion, while total inertia belongs to the common coordinate.
06 / Physical boundary
Torsion differs from lateral shaft motion.
Torsional vibration changes relative angular position about the shaft axis.
Torsional vibration changes relative angular position about the shaft axis. Lateral whirl concerns motion of the shaft centreline. They may interact in a larger system, but a two-angle model cannot replace a lateral shaft-dynamics model or identify bearing reactions.
External generator torque, rotor loading, additional elastic elements and grounded restraints can change these equations and their modes. A direct-drive description does not imply infinite stiffness. The FD14-50KW is confirmed as PMSG direct drive, but no actual coupling stiffness, inertia pair or modal frequency is inferred from that fact.
Related technical guides: Campbell Diagrams: Modal Branches, Rotor Orders & Crossings · Bearing Stiffness and Damping Coefficients · Magnetic Energy and Coenergy: Fixed Variables in Torque.
General background: MathWorks — Torsional compliance between rotating inertias. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Relative twist | δ = θ1 −θ2 | Same angular sign convention |
| Spring torque | −kδ and +kδ | Equal and opposite internal torques |
| Reduced inertia | J1J2/(J1 +J2) | Relative-mode kinetic energy |
| Relative frequency | √[k(1/J1 +1/J2)] | Free, undamped, two-inertia model |
| Rigid mode | Zero restoring natural frequency | Common rotation need not be stationary |
Engineering questions.
Does a free two-inertia model have a zero mode?
Yes. Common rotation causes no spring twist, so there is no restoring stiffness for that coordinate.
Does zero natural frequency mean the shaft is stopped?
No. The free common coordinate can have constant angular velocity.
Why does the frequency involve both inertias?
Both sides accelerate under equal and opposite spring torques; neither side is fixed in this model.
Can internal relative damping slow pure common rotation?
Not by itself. Its torque vanishes when relative angular velocity is zero.
Is torsional vibration the same as shaft whirl?
No. Torsion concerns angular twist, whereas whirl concerns lateral centreline motion.
Does direct drive eliminate torsional dynamics?
No. Direct drive identifies a drive arrangement, not infinite stiffness or a measured modal spectrum.
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