### Definition

If an infinite beam is hit in the middle, bending vibration waves will propagate on the left and right sides of the impact point. If the beam has a finite length, these vibration waves will reflect on the edges, creating two additional vibration waves which will interfere wwith previous waves, creating a particular vibration pattern. This vibration pattern is called a **modal shape or a structural mode**, it occurs at a specific frequency called a **natural frequency**, because it is the frequency at which the beam will naturally vibrate just after the external perturbation.

In fact, a real mechanical system has generally several degrees of freedom (bending, torsion, etc) so it exhibits several structural modes.
One can demonstrate that** these modal shapes represent a mathematical basis** when the mechanical structure behave linearly, which means that any deflection shape can be represented has a weighted sum of modal shapes (e.g. 90% of the first bending mode, 10% of the first torsional mode).

### Application to e-NVH

Large acoustic noise and vibrations can be due to **resonances between electromagnetic excitations and structural modes of the electric powertrain**.
It is therefore important to characterize the modal basis of the structures excited by Maxwell forces (in particular rotor and stator components).
The stator structural modes are generally characterized using the analogy with a cylindrical shell, whose structural modes can be labelled (m,n) where m is the rank of the circumferential deflection and n is the rank of the longitudinal deflection:

**Examples of structural modes of a cylindrical shell (numerical calculation)**

**Examples of structural modes of a stator stack (from experimental modal analysis))**

One must make the difference between the circumferential order *m* of a cylindrical mode and the circumferential wavenumber *r* of a electromagnetic excitation. The first one characterizes the structural modal behaviour, while the second one characterizes the electromagnetic excitation.

One can show that **EV HEV NVH ** is driven by pulsating forces when pole pair number is above 4. This means that the **breathing mode** (0,0) of the stator / housing assembly is the one that can respond the most to magnetic forces, resulting in airborne noise. When taken into account geometrical and magnetic asymmetries, other structural modes can be excited and structure borne noise can be generated, in particular through excitation of rotor bending mode by UMP.

### Application to MANATEE

MANATEE software can calculate the structural modes of the stator and rotor orthotropic structures, either by analytic methods or using FEA (e.g. coupling with Ansys, Optistruct). Alternatively the modal parameters can also be enforced.

The structural validation cases show the accuracy of MANATEE structural models compared to experiments.