# What is a mode shape and a natural frequency?

## Definition

The dynamic deflection shape of linear structures can be decomposed in a sum of elementary vibration patterns. These vibration patterns are called mode shapes, they are illustrated below with the example of a cantilever beam.

A particular frequency can be associated to each mode shape of the excited structure, independently of the frequency content of the excitation forces. These frequencies are called natural frequencies, or resonance frequencies. A structural mode is therefore characterized by the association of a mode shape and a natural frequency (mathematically, they respectively correspond to an eigenvector and an eigenvalue of the motion equation).

For the cantilever beam, the mode shapes and natural frequencies below are obtained analytically with given parameters E the Young’s modulus, I the quadratic moment of the beam, m the mass and L the length of the beam.

It shows that the modes are inherent to the structure and depends only on its own characteristics. It is also important to note that systems with n degrees of freedom thus have n modes and n possibilities of resonance in case of external solicitation occurring at known natural frequencies.

## Modal Basis

A modal basis is the series of structural modes (mode shape + natural frequency) associated to a linear structure within a given frequency range. One can show that these structural modes are orthogonal to each other, which means that mode shapes form a mathematical basis of linearly independent vectors. The dynamic motion of the structure can therefore be described as a unique, linear combination of the mode shapes. The weight associated to each mode within this modal expansion is called a modal participation factor.

For instance, the motion of the cantilever beam above is decomposed as the weighted sum of its first three structural modes.

## Modal Analysis

To understand how a structure behaves and identify its natural frequencies and modal shapes, a modal analysis technique is performed. It can be done either through numerical simulation (e.g. using Finite Element Analysis) or Experimental Modal Analysis (EMA). The result of this modal analysis can be used in many applications such as vibration response calculation or noise calculation, as in Manatee software.

Although EMA can be used directly to solve vibration problems, it can also be used to check the accuracy of a numerical model and if necessary tune it with experiments. Calibration or model updating of the numerical model is a necessary step before further calculations.

## Examples of modes

In the same way, it is possible to determine the modal shapes and natural frequencies of a stator. 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. Here, the modes are computed from a FEA model, generated and visualized in Manatee Software.

Application to Manatee

Manatee software calculates the vibration response of electrical systems using modal expansion principle. This is particularly useful to split the contribution of structural modes and magnetic load cases in the magnetic noise and vibration generation process. Manatee can run its own modal basis calculation on a simplified stator / rotor geometry, or import a more complex system-level modal basis coming from a 3D FEA software.