A new technique to determine the complex modulus in accordance with ISO 6721

Polymer Testing 13 (1994) 189-194 © 1994 Elsevier Science Limited Printed in Malta. All rights reserved 0142-9418/94/$ 7.00 ELSEVIER

A New Technique to Determine the Complex Modulus in Accordance with ISO 6721 Michael Schulthes Friedenstr. 15, D 25451 Quickborn, Germany (Received 1 July 1993; accepted 23 July 1993)

ABSTRACT To measure the complex modulus according to ISO 6721 is a tedious process, especially when it is necessary to determine the values versus temperature. When the measurement is performed automatically faulty test pieces may cause uncertain results. A new mathematical procedure improves the results with all test pieces. Furthermore, it can compensate for the errors created by faulty specimens to a certain degree, gives warnings when the faults are too severe and automatically calculates the correction factors necessary when dealing with highly damped materials.

NOTATION A d D f fo (mode) k Ko m P V

Vibration amplitude Loss factor Spring constant Frequency Resonance frequency of the mode Friction constant Exciting force Mass Proportionality constant Vibration velocity

Phase angle between force and amplitude f~ 2xnxf f~o (mode) 2 x x xfo (mode) 189

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M. Schulthes

INTRODUCTION ISO 6721 covers the determination of loss factor and storage modulus of test strips excited by a sinusoidal force. The test strip is fixed on one end and actuated using a constant force of variable frequency. The velocity of the forced vibration is measured and the complex modulus is calculated from the results. The most widely used is the 3 dB method: Around the resonances the frequency response is measured and the frequency values are determined where the amplitude has reduced by 3 dB referred to the resonance value. The characteristics of the tested material are calculated using the resonance frequency and the distance of the respective 3 dB points. When the damping value is very small, after determining the resonance frequency, the reverberation method is used: The test piece is excited using the resonance frequency until the amplitude is constant. Then the exciting signal is switched off. The decay of the vibration amplitude is measured and the loss factor calculated from the decay time. When the measurement is made automatically the whole procedure is controlled by a computer. In this case the test is made in a climatic chamber and the test pieces are normally tested at different temperatures. Figure 1 shows the test equipment schematically.

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Determining the complex modulus

191

PROBLEMS Unfortunately, faults which occurred during production of the test specimen are frequently not detected. When strips are made by covering steel with a damping material the adhesion may be incomplete. Furthermore, gas bubbles or mixing faults in the damping material may occur. These, and other faults, may lead to deviations between the theoretical and measured frequency response of the test strips at certain temperatures. Therefore, it is recommended in ISO 6721 to measure the frequency response of the test specimen not only at certain frequency values but over the whole range of interest and to take into account the complete frequency response. Another problem of the measurement appears when dealing with highly damped materials. The 3 dB points are influenced by adjacent resonances and the true loss factor is not exactly the same as the one determined by the 3 dB method, or possibly the 3 dB points cannot be located.

SOLUTION The method described here has large advantages compared to the techniques used hitherto with respect to both problems. With this method the resonances and loss factors of the test specimen are not determined directly using the 3 dB method but the frequency response is additionally synthesized around the resonance frequencies. The parameters of the synthesis function are optimized by a desk-top computer in such a way that the measured and the synthesized frequency responses fit best. Resonance frequency and loss factor are then taken from the synthesized frequency response curve.

PROCEDURE Initially, the first resonance of the test specimen is determined, for all following resonance frequencies the following model is used: The test strip is treated as if it were a system of three independant single mass oscillators. The dampings of these three oscillators are assumed to be equal. The resonance frequency of the left adjacent mode is known. The resonance frequency of the next adjacent mode is calculated using the resonance frequency of the processed mode and the formula according to ISO 6721 part 3. The amplitudes of the two adjacent modes are estimated empirically. Errors in this estimation only minimally influence the cal-

192

M. Schulthes

culated values for resonance frequency and loss factor. The amplitude around the resonance under consideration is the vector sum of the three single amplitudes, whereas the adjacent modes only have a significant influence on the resulting amplitude at medium and high damping. For each resonance of a single mass oscillator the following relations are valid:

A = Ko/x/(m2(Do(mode) 2 - ~2)2 + k 2~2) f~o(mode) = x / ~ / m tan(a) = f~k/(m(t)o (mode) 2 - f~2))

The form of the resonance curve is a function of resonance frequency and loss factor only. Furthermore, neither the effective masses nor the effective spring constants are known, therefore we normalize to k = 1. Having transformed and rewritten the equations above we find for each resonance:

V ( f ) = P/x/(1/(dfl))2((f~o(m°de) 2 -- ~2)2 + D2) tan(a) = fl 2 x d/(f~o(mode)2 - D2) Hence, the frequency-dependent velocity of vibration may be synthesized as follows: V-total ( f ) = ~ P ( i ) x Vi(f)

(i from m o d e - 1 to m o d e + 1)

This algorithm automatically follows the recommendation of ISO 6721 to curve fit the frequency response when dealing with highly damped materials. The values for loss factor and resonance frequency are varied during an optimization process in the calculator until the measured and the synthesized frequency response function fit best. Figure 2 shows the deviations between the measured curve and the synthesized function of a test strip with medium damping. The plot was made with a modified measurement programme that, having calculated resonance frequency and damping, repeats the measurement with a widened frequency range to show the amount of conformity between

Determining the complex modulus 28 °

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Fig. 3. Faulty test strip. Measured curve, solid line; synthesized curve, dotted line. The synthesis error is calculated from the differences between the curves. t h e o r y a n d m e a s u r e m e n t • T h e m e a s u r e d c u r v e is p l o t t e d with a fat, solid line, the s y n t h e s i z e d c u r v e w i t h a d o t t e d line, the u p p e r 3 dB r a n g e is marked• T h e d e v i a t i o n s b e t w e e n m e a s u r e d a n d synthesized c u r v e d e p e n d u p o n

194

M. Schulthes

the quality of the test specimen. They therefore may be used as a warning criteria for the user. Figure 3 shows the calculation with a faulty test strip. It is evident that the programme has mostly corrected the influence of an incidental resonance and therefore there is satisfactory overall coincidence of the curves. Nevertheless, the technician may decide, using the plot, whether or not the point should be accepted. The measurement system described above has been developed using Briiel & K j a e r equipment. It supports climate chambers of various manufacturers and can be modified to customers requirements. Additions to the technique described here have been developed to calculate the damping of sandwich metal sheets and to repair data files with faulty measurements by fitting adjacent, valid measurements in the missing or erroneous parts.