Converting complex modes to real modes

Signed amplitude conversion

Starting from complex mode shapes {ΨC}r, we can use signed amplitude conversion to obtain the real mode shapes {Ψ}r. This conversion method works well for lightly damped modes. The modulus of each element of the complex mode shape vector is multiplied by the sign of the cosine of its phase angle NOTE.

Let ΨCj,r denote the jth element of the complex mode shape vector {ΨC}r:

	Eq. 1

where ρj,r and φj,r are, respectively, the modulus and the phase angle of ΨCj,r.

The jth element of the converted real mode shape vector {Ψ}r is then given by:

	Eq. 2

Undamped natural frequency

Assuming the proportional viscous damping model, the natural frequency ωr of the rth mode can be calculated from the complex eigenvalue sr. The complex eigenvalue sr of the rth mode can be expressed as:

	Eq. 3

where ζr is the critical damping ratio for the rth mode.

From Eq. 3, the critical damping ratio is calculated as follows:

	Eq. 4

and the undamped natural frequency is calculated as follows:

	Eq. 5

where:

• Re(sr) is the real part of the eigenvalue sr.

• Im(sr) is the imaginary part of the eigenvalue sr.

In Eq. 5, we can divide ωr by 2π to have the undamped natural frequency fr in Hz.

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Modal Analysis

Accounting for repeated modes

Modal Scale Factor (MSF)

Modal Assurance Criteria (MAC)

Coordinate MAC (COMAC)

Cross-orthogonality (X-Ortho)

Frequency Response Assurance Criterion (FRAC)

Min-MAC algorithm

MODMAC algorithm

Normal Mode Indicator Function (NMIF) algorithm

Driving Point Residue algorithm

Scientific literature references for correlation

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Converting complex modes to real modes, Simcenter 3D 2021.1 Series

© 2020 Siemens

Friswell, M.I. and Mottershead, J.E., “Finite Element Model Updating in Structural Dynamics”, Kluwer Academic Publishers, 1995.

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