Use of cross-spectrum measurement technique in finding damping and stiffness properties of polymers.

Stoica, Gina Florica ; Iordan, Andrei ; Cananau, Sorin 等

1. INTRODUCTION

The materials possessing both elastic and viscous properties are called viscoelastic, the polymers being the typical example. Poisson's ratio is defined as being the ratio between the lateral contraction and the elongation of an isotropic homogenous material in an infinitesimally small extension. For those viscoelastic materials, Poisson's ratio is a function of time, frequency or temperature (Tschoegl, 2002). In the following, we will present the ameliorations to a method of obtaining the frequency depending Poisson ratio. Due to experimental difficulties, the literature on this subject is rather scarce. Despite this, we have identified to ways to determine the Poisson ratio: an indirect and a direct way (Tschoegl, 2002).

For most frequency dependent measurements, a beam-like specimen is submitted to sinusoidal charges, the ratio being measured with strain gauges. For identification, the least square or the reduction of variables method is used. The method presented here, gathers full-field information by aid of an optical measurement method (deflectometry) and uses the Virtual Fields Method for identification. Also, the specimen used is not a beam, but a plate in vibration.

2. EXPERIMENTAL ARRANGEMENT

The experimental set-up was introduced by Giraudeau and Pierron (Giradeau, 2005). It consists of an arrangement is presented in the Fig. 1. The tested material was a PMMA type polymer cut into thin plates size 200x 160x3 mm and a mirror treatment was applied on one of their faces. This mirror treatment was required by the optical method required for the measurement. The plate is submitted to a sine driving motion, leading to the bending of the plate. For the full description of the sine driving movement two images are taken at different moments in time with a CCD camera and a flash.

As it is shown, the plate is being moved by an electrodynamical shaker and driven by a drivin device introduced to guide the movement and limit it's the amplitude.

[FIGURE 1 OMITTED]

The driving device consists of a series of metal membranes mounted perpendicular to an axle.

After these precautions, we can assure that the movement has small amplitude and that bending vibrations are preponderant. An accelerometer is mounted on the axle holding the plate in order to describe its movement. Facing the mirror treated plate is a grating and behind the grating a CCD camera used for picture taking. Near to the plate we have a flash used to trigger the camera. In order to set the electrodynamic shaker in motion we use a National Instruments signal generating and measurement card. With the help of this card, we generate two signals, a sine one, set to generate the movement and a square one for triggering the flash. The method we use for measurement (deflectometry) requires 2 pictures taken at two precise instants in time, therefore the two signals (sine and square) must be synchronized. We measure the amplitude of the sine movement with the accelerometer and we impose a lag of 0 or pi/2 between the measured signal and the generated square.

3. IMPROVEMENTS-USE OF THE CROSS-SPECTRUM

In order to introduce the improvements we added to this experimental set-up, we will talk about the synchronization of the two signals present in the set-up: the sine and the square. This synchronization is done according to the following graph (Surrel, 2003) (Fig. 2).

Where, the phase lag was measured with a phasemeter (case A.) or by means of the cross-spectrum (case B.).

[FIGURE 2 OMITTED]

The reason we introduced the cross-spectrum measurement technique was that the phasemeter could only allow us to impose a phase lag between the two signals with 0.5 degrees of precision maximum. We obtained [10.sup.-3] degrees of precision. The cross spectrum method of analysis is based on the Fourier transform, which serves in the conversion of information from the time domain to the frequency domain and vice-versa. No information concerning the signal is lost by doing this transformation. For a time-domain signal, a(t), the Fourier transform is: (Herlufsen, 1984)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Likewise, for another signal b(t):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The cross-spectrum is defined like: (/cite)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

So we can see the phase lag clearly:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

By averaging, for we apply a FFT (fast Fourier transform), we can obtain the phase lag between the two signals a(t) and b(t). In the following we will present the steps took for validating this technique for the experimental set-up and the results obtained by doing so.

There have been more stages in the validation of this technique. Firstly, we have used a DAC by DSP Technology--called [SIGLAB.sup.i] to generate two cosines; the same apparatus was used for measurement. The second series of tests consisted in adding a second SIGLAB for measurement to prove the fact that the method works, not indifferent to the apparatus used. To finalize this stage, we have crossed the input and output channels of the two machines following this diagram: Input 1 [right arrow] [right arrow] Output 2; Input 2 [right arrow] Output 1.

The imposed phase lags, frequencies and sample rates of the signals are presented below (Table 1 to Table 3).

In order to fully emulate real life conditions, we finished the set of experiments by simulating noise into our signals. The results were also very impressing, a precision of 10-2 degrees exists at high levels of noise, and only at 30% level of noise precision sadly diminishes. The table 4 present the results obtained during the study of the influence on noise levels.

4. CONCLUSION

A technique for measurement of the Poisson coefficient was discussed here and the improvements to the existing method (Giraudeau, 2006) were underlined. The cross-spectrum technique for phase lag measurement was presented and also its results. Sadly, due to the lack of stability of the acquired signals, this technique did not produce the break-through it was expected, because the results of the measurements were practically the same as in the previous work (Giraudeau, 2006) when a phasemeter was used to measure phase lag.

The reasons this technique did not worked as expected are, in our opinion, due to the fact that the plate's driving movement is assured by an electrodynamical shaker, therefore its amplitude is not constant in time. Since the cross-spectrum calculus of the phase lag is an average of the signal, it is obvious that on the experimental set-up we didn't obtained the same results as we did in the presented tests. Due to this factor and of course of random errors, the precision of measurement of the phase lag diminished to 0.4 degrees, comparable with that of a phasemeter.

5. REFERENCES

Giradeau, A. (2005), Identification of Stifness and Damping Properties of Thin Isotropic Vibrating Plates Using the Virtual Fields Method: Theory and Simulations, Journal of Sound and Vibration, 284, 757-781

Giradeau, A. (2006), Stiffness and Damping Identification from Full Field Measurements on Vibrating Plates, Experimental Mechanics

Herlufsen, H. (1984), Dual Channel FFT Analysis (part I), Technical Review

Surrel, Y. (cours CNAM 2003-2004), Images optiques; mesures 2D et 3D

Tschoegl, N.W. (2002), Poissons Ratio in Linear Viscoelasticity. A Critical Review, Mechanics of Time Dependent Materials, 6, 3-51.

Table 1. Same SIGLAB for generation and measurement;
frequency 200Hz, sample rate 1024

Imposed lag 0 [pi]/2 [pi]/4 [pi]/6 [pi]/10

Measured lag [10.sub.-4] 89,998 44,998 29,997 17,997

Table 2.2 SIGLABS; bandwidth 500Hz, frequency
200Hz, sample rate 1024

Imposed lag [pi]/2 [pi]/4 [pi]/6

Measured lag 89,99994 44,99977 29,99910

Imposed lag [pi]/10 [pi]/100 [pi]/1000

Measured lag 18,02220 1,798277 0,178782

Table 3. For 50 Hz

Imposed lag [pi]/2 [pi]/4 [pi]/6

Measured lag 89,99872 45,00014 30,00054

Imposed lag [pi]/10 [pi]/100 [pi]/1000

Measured lag 18,00128 1,798127 0,178444

Table 4.

 100 Hz 150 Hz

 Measured Measured
Noise Imposed lag Imposed lag
level lag (degrees) lag (degrees)

3% [pi]/4 44,995 n/4 44,997
 [pi]/6 30,006 n/6 30,002
 [pi]/10 17,999 n/10 17,997
 [pi]/100 1,8039 n/100 1,8008
 [pi]/1000 0,1799 n/1000 0,1809

5% [pi]/4 45,004 n/4 44,996
 [pi]/6 30,004 n/6 29,996
 [pi]/10 18,001 n/10 18,007
 [pi]/100 1,8050 n/100 1,7952
 [pi]/1000 0,1838 n/1000 0,1727

10% [pi]/4 45,002 n/4 44,998
 [pi]/6 30,026 n/6 29,993
 [pi]/10 18,005 n/10 18,001
 [pi]/100 1,7958 n/100 1,7973
 [pi]/1000 0,1723 n/1000 0,1876

30% [pi]/4 45,014 n/4 44,984
 [pi]/6 30,015 n/6 30,021
 [pi]/10 17,968 n/10 17,988
 [pi]/100 1,8536 n/100 1,8233
 [pi]/1000 0,1837 n/1000 0,2093