discussed。

Strain Estimation via Displacement  Measurements

Laser vibrometer measurements are Eulerian (since the laser beam reflects variation at the observation point rather than at a point on the structure), whereas contact strain measurements (such as those provided by strain gauges) are for a Lagrangian reference frame。 In order to map the surface strain using any external device (laser vibrometers included) displacements must be re- stricted to small strain where the Lagrangian derivative approaches the Eulerian derivative。

Under the assumption of small deflections the Eulerian and the Lagrange description of the mechani- cal properties are approximately the same。 The relation between displacements uk and strains εkl with k, l  = x, y, z takes the form

εkl  = 2 。uk,l  + ul,k。 , (1)

where uk,l is the partial derivative of uk with respect to

l。 Due to its definition the strain tensor εkl is symmetric。 For plate structures only the in-plane strains

1

εx  = ux,x, εy = uy,y and    γxy  = 2 (ux,y + uy,x) (2)

are relevant。

Measured data usually contain noise, a fact that poses problems to numerical differentiation。   Hence, it is essential to filter measured data prior to the differentiation process。 A spatial filter, introduced by Savitzky and Golay in 1964 [12] for one-dimensional problems and later extended for the use of smoothing and differentiation in arbitrary dimension [14, 15] was utilized in this study。 This filter is a convolution filter that is applicable for structured meshes。 The filter in- volves a spatial convolution, and the weights are de- rived such that the approximated value exactly corre- sponds to a least-squares approximation on a defined interval。

Two parameters define the behaviour of the filter; the order k of the polynomial with which the mea- sured data are approximated and the filter width m which defines the number of points in the convolution interval。

The order of the polynomial should be high enough to approximate the curve well but not too high, so that the efficiency of the smoothing is unaffected。 In most cases a third order polynomial is sufficient, as most smooth curves can be approximated with a quadratic function in the vicinity of a peak and with a cubic function in the vicinity of a shoulder。 However, if higher derivatives shall be evaluated, the value of k must be chosen appropriately high。

When choosing the smoothing width m two trade- off factors have to be taken into account。 On the one hand the noise attenuation increases with the smooth- ing width but on the other hand the distortion of peaks also increases for high m。

Experimental Setup

Polytec PSV-3D Scanning Laser Vibrometer

As described in the introduction, scanning laser vi- brometers employ a sequential method of   measuring

the displacement data at a number of predefined points。 Scan times are equal to the product of the number of points to be scanned and the duration needed at each point to acquire the data as well as locate (tri- angulate) the three laser beams。 Compared to full- field measurement methods, e。g。 speckle interferome- try, measurements made with scanning laser Doppler vibrometers have longer overall measurement (scan) times。 However, unlike the the full-field measurement methods which are limited by the sampling time of the utilized CCD camera (typically less than 100 Hz), the frequency resolution of vibrometer measurements are much higher and is typically restricted by the analog to digital cards employed (often in excess of 100 kHz)。

It should be noted that since laser vibrometry is based on interferometry, then strain estimates can only be made for differential movement over a measure- ment cycle。 This does therefore allow single cycle quasi- static measurements to be made so long as the stress cy- cle can be repeated for each measurement point。 How- ever, the technique is best suited to vibrating surfaces with repetitive loading cycles, which when combined with Fourier analysis, enable improved signal to noise ratios through averaging。