A static force–displacement curve can describe seating and stiffness. It cannot by itself determine how much energy a contact dissipates during vibration. That requires the force and displacement timing, measured under the relevant preload and dynamic conditions.
For an approximately sinusoidal, locally linear response, define complex stiffness K* = F/X = K′ + iK″ using the exp(iωt) convention. F and X are force and displacement phasors, giving stiffness in newtons per metre. K′ describes the in-phase elastic component; K″ describes the loss component under this convention.
Measure the phase with the mechanics
With displacement x = X cos ωt and peak amplitude X in metres, the corresponding force is X[K′ cos ωt − K″ sin ωt]. The energy dissipated per cycle is πK″X² joules for a passive response with positive K″. This result follows from integrating force against displacement around the cycle.
The loss angle satisfies tan δ = K″/K′ when that description is meaningful. It is not a universal material constant for a clamped lamination interface.
| Test variable | Record | Reason |
|---|---|---|
| Preload and contact area | Applied condition and restraint | Sets seating and slip behavior |
| Frequency | Excitation and response frequency | Contact and fixture dynamics can vary |
| Displacement amplitude | Measured interface motion | Equivalent damping may depend on amplitude |
| Force–displacement phase | Synchronized calibrated channels | Separates storage from loss |
| Higher harmonics and loop shape | Full time records | Challenges the single-harmonic assumption |
Check sensor phase and fixture dynamics before assigning all observed lag to the contact. A transducer, support or acquisition channel can introduce its own phase error. Measure relative displacement across the intended interface rather than an unrelated fixture motion.
Look for the transition from sticking to slip
Repeat a controlled amplitude sweep within an approved range. A change in the loop shape, harmonic content or equivalent K* can indicate that the contact response is no longer adequately linear. Repeat on decreasing amplitude to identify history dependence.
Do not reduce a strongly distorted loop to one loss angle without showing the limitation. Integrating the measured closed loop gives cycle energy for that particular motion, while its relationship to a general dynamic model still requires care.
Compare preload conditions separately. Changing clamp force can alter both stiffness and dissipation. A model fitted at one preload should not claim to describe a different assembled core automatically.
Use the measured amplitude–frequency map to choose either local equivalent properties or a nonlinear contact law suited to the analysis. Validate the resulting prediction at an additional condition. Adding a convenient damping ratio until one resonance matches may conceal the wrong interface mechanics and fail as soon as the excitation amplitude changes.
Convert phase-resolved contact stiffness into dissipated work
Use catalogue 0.23 mm nominal sheet thickness [1]. Assume 100 sheets of 0.50×0.10 m give a steel volume of 0.00115 m³ and mass 8.7975 kg at the catalogue assumed density. At an assumed 100 Hz small-amplitude contact state, let K′ = 10 MN/m, K″ = 1 MN/m and displacement peak X = 1.0 μm. Cycle loss is πK″X² = 3.142 μJ; average dissipated power is f times that loss, 0.314 mW. Specific cycle loss is 0.357 μJ/kg, stored elastic energy is K′X²/2 = 5.0 μJ and tan δ = 0.10. Retain the phase measurement and preload: a static stiffness alone supplies none of the loss term [2]. Phase-resolved cycle energy is the required basis for a dissipative stiffness description [2].
Burr contact, coating condition, packet registration and clamp pressure determine the interfaces that carry tangential motion. Associate the measured force–displacement loop with those controlled assembly variables. An equivalent damping factor from one preload and amplitude should not be transferred to another core build without verification.
References
- Nippon Steel Corporation, Grain-Oriented Electrical Steel Sheets, catalogue D004je_03_202004, 2020, Specifications; Size and Form Tolerances.
- Sönnerlind, H., Damping in Structural Dynamics: Theory and Sources, COMSOL, 14 March 2019.