Frequency Domain Techniques for Operational Modal Analysis

Operational Modal Analysis, also known as Output Only Modal Analysis, has in the recent years, been used for determination of modal parameters of civil engineering structures and is now becoming widely used also for mechanical structures. The advantage of the method is that no artificial excitation needs to be applied to the structure, or force signals to be measured. In this paper, the non-parametric technique based Frequency Domain Decomposition (FDD), as well as the more elaborate Enhanced Frequency Domain Decomposition (EFDD) identification technique are discussed. The methods are illustrated by measurements on a wing from a wind turbine acoustically excited by a loudspeaker in the Brüel & Kjær laboratory. INTRODUCTION Operational Modal Analysis, also known as Ambient Modal Analysis and Output Only Modal Analysis is presented here. Traditionally, measuring the input forces and the output responses for a considered linear, time-invariant mechanical system performs a modal test of a structure. The excitation usually used is transient (Impact Hammer testing), or random, burst-random, or sinusoidal (Shaker testing). The advanced signal processing tools used in the Operational Modal Analysis technique allow us now to determine the inherent properties of a mechanical structure (Resonance Frequencies, Damping, Mode Shapes), by measuring only the response of the structure, without using an artificial excitation. The advantage of this technique is that it provides a complete modal model under operating conditions, meaning within true boundary conditions, and actual force and vibration levels. The measurement technique is similar to the “Operating Deflection Shapes” type procedure, where one or more accelerometers are used as reference(s), and a series of roving accelerometers are used for the responses at all the Degrees of Freedom (DOF’s), or all DOF’s are just measured simultaneously. Figure (1) shows a schematic description of an ambient response system. Responses Stationary Zero Mean Gaussian White Noise Loading System Structural system (linear, time-invariant) Unknown Excitation Forces Combined Ambient System Figure 1, Combined Ambient Model The measurements were taken using the Brüel & Kjær PULSE Multi-analyzer system (Type 3560), and the Modal Test Consultant (Type 7753) to create the geometry, assign the measurement points and directions (DOF’s), and capture the data. The analysis was then done using the Brüel & Kjær Operational Modal Analysis software (Type 7760), where all the advanced signal processing and modal extraction procedures were performed. MEASUREMENT PROCEDURE In this paper the use of the Operational Modal Analysis method for a 1:5 scale wind turbine wing is described. The object is a detailed model of one of the blades from a 675 kW wind turbine. The wing has been made for lab investigations of static as well as dynamic parameters. Figure (2) shows a picture of the set-up used for the measurements. The wing itself is supported by a console which is regarded as stiff compared to the wing itself. 24 accelerometers Brüel & Kjær Type 4507 B4 with a sensitivity of 10mV/ms are mounted in two rows along the wing. Two time recordings were taken, one with the accelerometers perpendicular to the surface (Z-direction) and one pointing in the direction of wing rotation (X-direction). To determine the combined modes in this twodimensional model the results from the two recordings are linked together using a tri-axial accelerometer Type 4506 with a sensitivity of 10mV/ms as a reference. The wing is considered as stiff in the length direction, Y, so vibrations in this direction are disregarded. Thus Type 4506 was used as a biaxial accelerometer. The total numbers of DOF’s are 50. The data acquisition system used was a portable PULSE platform Type 3560D composed of a 31 channel front-end for the hardware, and a laptop computer for the software. The wing was exposed to an acoustic excitation by means of a loudspeaker Brüel & Kjær Type 4224 placed beneath the wing. Although the aim is to use Operational Modal Analysis for the test the sound level was measured using a microphone placed in front of the wing and analysed as well. This measurement was done to ensure presence of energy in the frequency range of interest. The spectral distribution does not need to be flat so white noise is not a requirement. The PULSE system was set up with a Time Capture Analyzer with a frequency range of 200 Hz (sampling frequency of 512 Hz, Nyquist frequency of 256 Hz) corresponding to a sampling interval of 1.953 ms. A record length of 60 s (i.e. 30720 samples) was chosen in order to capture 600 cycles of the lowest frequency of interest, 10 Hz. Figure 2, Test object, 1:5 scale wind turbine wing All the raw time data, the geometry, and the series of measurements are then directly exported from PULSE to the Brüel & Kjær Operational Modal Analysis software for advanced signal processing calculations, and modal extraction. SIGNAL PROCESSING AND FREQUENCY DOMAIN DECOMPOSITION The first step of the analysis is to perform a Discrete Fourier Transform (DFT) on the raw time data, to obtain the Power Spectral Density Matrices that will contain all the frequency information. The excitation being broadband and having a continuous type of spectrum, the proper frequency descriptor is the Power Spectral Density (in (m/s)/Hz), that normalizes the measurement with respect to the Noise Bandwidth of the band-pass filters (FFT). Hanning Weighting with 66.6% overlap is used. In general, for all the series of measurements or datasets, we now have the Spectral Density Matrices calculated. The size of the matrix is n*n, n being the number of transducers (26 in our case 26 measured DOF’s). In this example we have two data sets, i.e. two matrices (of a size 26*26) were calculated for each frequency. Each element of those matrices is a Spectral Density function. The diagonal elements of the matrix are the real valued Spectral Densities between a response and itself (Auto Power Spectral Density). The off-diagonal elements are the complex Cross Spectral Densities between two different Responses. All those matrices are Hermitian (they are complex conjugate symmetric, meaning that they have complex conjugate elements around the diagonal). FREQUENCY DECOMPOSITION THEORY BACKGROUND The Frequency Domain Decomposition (FDD) is an extension of the Basic Frequency Domain (BFD) technique, or more often called the Peak-Picking technique. This approach uses the fact that modes can be estimated from the spectral densities calculated, in the condition of a white noise input, and a lightly damped structure. It is a nonparametric technique that estimates the modal parameters directly from signal processing calculations, Refs. [1,2]. The FDD technique estimates the modes using a Singular Value Decomposition (SVD) of each of the data sets. This decomposition corresponds to a Single Degree of Freedom (SDOF) identification of the system for each singular value. The relationship between the input x(t), and the output y(t) can be written in the following form, Refs. [3,4]: [ ] [ ] [ ][ ]T xx * yy ) ( H ) ( G ) ( H ) ( G ω ω ω ω = , (1) where [Gxx(ω)] is the input Power Spectral Density (PSD) matrix. [Gyy(ω)] is the output PSD matrix, and [H(ω)] is the Frequency Response function (FRF) matrix, and * and superscript T denote complex conjugate and transpose, respectively. The FRF matrix can be written in a typical partial fraction form (used in classical Modal analysis), in terms of poles, λ and residues, R [ ] [ ] [ ] [ ] [ ] * k * k m 1 k k k j R j R ) ( X ) ( Y ) ( H λ ω λ ω ω ω ω − + − = = ∑ = , (2) with (3) dk k k jω σ λ + − = , where m being the total number of modes of interest, λk being the pole of the k mode, σk the modal damping (decay constant) and ωdk the damped natural frequency of the k mode. Using the expression (1) for the matrix [Gyy(ω)], and the Heaviside partial fraction theorem for polynomial expansions, we obtain the following expression for the output PSD matrix [Gyy(ω)] assuming the input is random in both time and space and has a zero mean white noise distribution, i.e. its PSD is a constant, i.e. [Gxx(ω)] = [C]: [ ] [ ] [ ] [ ] [ ] * k * k