Reducing measurement time for a laser Doppler vibrometer using regressive techniques

Abstract

In the last decade the laser Doppler vibrometer (LDV) has become a widely spread instrument for measuring vibrations. It often offers accurate measurements with a high spatial resolution. However, the measurement time of the LDV and especially for the scanning LDV is long. Therefore, reducing the measurement time is an attractive objective. A way to achieve this is to use a single sine excitation (on a resonance frequency). However, this technique has two major drawbacks: the inability to provide information on the damping and a operational deflection shape that can differ from the true mode shape. In this article two methods will be introduced to reduce measurement time without these defaults. In the first method introduced in this article a narrow band multisine is used as excitation signal and the measured vibration signal in the time domain is represented by a model using sines and cosines with these fixed narrow band frequencies. The coefficients of those sines and cosines are then estimated on a global scale by means of a least-squares estimator. An important advantage of this particular technique is that one does not have to measure a full period of the signal, reducing time. The second method accelerates the measurement time for scanning LDV measurements. Using the time domain sequence from each previous scan point and a limited number of time samples from the current scan point, the full time domain sequence of the current scan point can be estimated. Both these methods are a key benefit for in-line quality control, which can have upwards of 1000 spatial measurement locations. The proposed techniques will be validated on both simulations and experiments of varying complexity.

Introduction

Modal analysis has for some time now been an important tool in structural analysis in many different fields of application [1]. The measurements themselves were mostly executed using contact-based measurement systems such as accelerometers. They have, however, some disadvantages such as the cumbersome task of applying the accelerometers to the structure (sometimes several times when working in different patches) and of course there is always the mass loading aspect (important for lightweight structures). Recently however, with the ever ongoing progress in optical techniques some interesting alternatives have been developed. However, this also led to entirely new measurement systems themselves such as the laser vibrometer, or scanning laser Doppler vibrometer (LDV) [2]. The LDV made it possible to increase the number of degrees of freedom (DOF) measured on the surface by two to three orders of magnitude. Where in the past one was limited to a few hundred measurement points it was now feasible to measure tens of thousands of points. However, measuring such high spatial resolution data takes a relatively long time (e.g. a simple plate with 1000 points and a frequency resolution of 1 Hz, will roughly take about 20 min). In short, there are two possibilities to execute a scanning vibrometer measurement with the existing commercial LDV-systems. The first mode is to perform a so-called ‘full scan’ where all frequency lines are excited with which a frequency response function (FRF) over the entire frequency range is obtained. This mode takes a considerable amount of time inversely proportional to the frequency resolution. The second is a fast scan mode where only one frequency line is excited (normally on a resonance frequency). The measurement time of this mode is of course proportional to the inverse of the imposed frequency and is therefore faster. However, it is not possible to estimate damping, for which at least two frequencies are needed, nor will the estimated deflection shape be exact and even in this mode at least one period of the frequency has to be measured.

The only way to reduce the actual measurement time is to simply measure less of the vibration time signal for each individual point on the structure. However, this is, as already stated, limited by the period of that signal. For one always has to measure at least one entire period (or an integer number of periods) of the vibration sequence in order to avoid leakage as is the case with a discrete Fourier transform (DFT). In the first method introduced in this paper, the signal is approximated by a series of sines and cosines with fixed frequency. The frequencies in the measured signal are known a priori for linear systems, because excitation is a multisine with a narrow fixed frequency band, which makes it possible to model an entire signal period using only a portion of the period. In this paper a regressive discrete Fourier series (RDFT) technique [3], [4] was used to estimate the sine and cosine coefficients, yielding an approximation of the vibration signal. Using the proposed technique on a time sequence, it is clearly possible to not only generate an estimate for the entire sequence but maybe even more importantly to reduce the measurement time. Theoretically, one can estimate a single frequency using only two points, however in practice this strongly depends on several parameters in the data.

The second method presented in this article is a spatially accelerated regressive technique used to reduce measurement time specifically for scanning LDV measurements. The technique is based on the fact that the time sequences measured in each sequential scan point are basically the same and only differ by their amplitude. Therefore, it is possible to approximate the vibration signal in a particular scan point by means of a previous scan point by simply estimating the amplitude. This can be done by using a small number of time samples of each adjacent scan point, hence reducing the measurement time.

In the following section the methods will be unveiled. In Section 3 simulations on the regressive Fourier series technique will be shown on different examples showing the capabilities towards high spatial resolution data and multiple frequency signals. In Section 4 both methods will be put to practice on measurements on an aluminium beam and finally some conclusions will be drawn in the last section.