Calculation of the Dynamic Moduli of Linear Viscoelastic Materials from Vibrating Reed Measurements

Abstract

The direct relation between the complex longitudinal modulus E* and the experimentally measured quantities—the amplitude ratio A and phase angle φ—has been derived for the forced vibrating reed instrument and is given by: exp (iφ)/A=1+∑K=1∞(−ω2ρ/E*)K[AK+BKρ−1+CKρ−2+….], where terms in ρ with negative exponents are not admitted. E* is equal to E′+iE″, where E′ is the longitudinal storage modulus and E″ is the longitudinal loss modulus. A is the ratio of the amplitude of the free end of the reed—either loaded with a pin or unloaded—to that of the clamped sinusoidally driven end and φ is the angle by which the former lags the latter. In practice, the sample dimensions and frequency range are chosen such that the sum in the above equation is rapidly convergent. The dynamic moduli then can be calculated by including only up to the first two or three terms in the sum. Where phase measurements are relatively inconvenient to make and when all but the first term under the sum can be neglected, the equation can be used to calculate dynamic moduli from two amplitude measurements at the same frequency for reeds of two different lengths; or from amplitude at resonance alone, provided E″≪E′. Illustrative examples are given. The procedures for calculating the dynamic moduli are applicable to the data obtained in all experiments where a relation of the form given above is obtained.