## Operational Modal Analysis

by Nikolay Donets – March 1, 2012 · 4 min read

In this article, we will review the basic concepts of Operational Modal Analysis **OMA** and we will present the most common **OMA** techniques, namely: the Frequency Domain Decomposition **FDD** method, and the Stochastic Subspace Identification **SSI** method. We will also introduce the basics of the system identification method, which is used in **OMA** to estimate the modal parameters of a structure. **OMA** does not require special excitation of a structure and this is the main advantage of this method.

## What is Operational Modal Analysis

Operational modal analysis is used in the field of experimental modal analysis to identify the modal parameters of a structure using only response measurements. The structure is assumed to be excited by unknown random forces and the response is measured at several locations on the structure. The unknown excitation is modeled as white noise and the response is used to estimate the modal parameters of the structure.

## Modal Parameters

A structure will vibrate at a certain *natural frequency* when it is subjected to an excitation at that frequency. *Natural frequency* is the frequency at which a structure will vibrate when it is not subjected to *damping*.

*Damping* is the dissipation of energy that occurs when a structure is subjected to vibration. The amount of damping in a structure can be represented by the damping ratio.

*Mode shape* is a plot of the amplitude of vibration at each measurement point at a particular natural frequency. A mode shape is a plot of the vibration pattern of a structure at a particular natural frequency.

## Theoretical Background

Operational modal analysis **OMA** is a method for extracting modal parameters of a structure from output-only measurements, i.e. without measuring the excitation forces. In **OMA**, the input force to the structure is not measured but rather assumed to be a zero-mean white-noise process. The main assumptions in **OMA** are the following:

- The input force to the structure is a zero-mean white-noise process: the input force is a random process with constant Power Spectral Density
**PSD**over all frequencies. This assumption is required to ensure that the Cross-Spectral Density**CSD**matrix of the response is proportional to the Frequency Response Function**FRF**matrix. - The input force is spatially uncorrelated with itself: the force at one point on the structure is uncorrelated with the force at any other point. The assumption is important because the spatial correlation of the input force will affect the
**CSD**matrix of the response - The input force is uncorrelated with the response of the structure: the input force is not correlated with the response of the structure since the correlation between the input force and the response will affect the
**CSD**matrix of the response as well - The structure is lightly damped: the damping ratio of the structure is small

The last three assumptions are generally satisfied for many structures. For example, bridges usually have low damping ratios. The first assumption is satisfied in the case of ambient vibration, in which the input force to the structure is the wind and the traffic loads. The ambient vibration can be assumed to be a zero-mean white-noise process if the wind speed is constant over the duration of the measurement and if the traffic loads are uniformly distributed over the bridge deck.

The main advantage of **OMA** is that the modal parameters can be identified without measuring the input forces. **OMA** is especially useful for structures that are difficult to excite, e.g. offshore structures, or for structures that are excited by ambient forces, e.g. bridges.

The main disadvantage of **OMA** is that the modal parameters can only be identified for the modes that are excited by the unknown input forces. The modal parameters of the unexcited modes cannot be identified.

## Techniques

### Frequency Domain Decomposition Method

The Frequency Domain Decomposition **FDD** method is a simple **OMA** method that can be used to identify the natural frequencies, the damping ratios, and the mode shapes of a structure.

The **FDD** method is based on the on the fact that the Singular Value Decomposition **SVD** of the output Power Spectral Density **PSD** matrix can be used to estimate the natural frequencies, the damping ratios, and the mode shapes of a structure. The **FDD** method involves the following steps:

- Compute the output
**PSD**matrix - Compute the Singular Value Decomposition
**SVD**of the**PSD**matrix - Plot the singular values of the
**PSD**matrix versus the frequency - The natural frequencies and damping ratios are estimated from the peaks of the singular values
- The mode shapes are estimated from the first singular vector at the natural frequencies

### Stochastic Subspace Identification Method

The Stochastic Subspace Identification **SSI** method is a more advanced **OMA** method than the **FDD** method. The **SSI** method can be used to estimate the natural frequencies, the damping ratios, and the mode shapes of a structure.

The **SSI** method involves the following steps:

- Construct the block Hankel matrix from the measured data
- Perform the singular value decomposition of the Hankel matrix
- Perform a projection to obtain the Extended Observability matrix
- Perform a projection to obtain the state sequence
- Perform a singular value decomposition to obtain the system matrices and calculate modal parameters
- An additional step is to vary the model order and plot a stabilisation diagram: grouping together identified modal parameters helps to remove noisy estimations