Presentation and modeling of complex modulus
Authors: Rogers Lynn 
Published: October 2005

One of the historical challenges in the field of passive vibration damping technology is the modeling of experimental complex modulus data as a function of temperature and frequency. Several models exist for the temperature shift function (TSF); i.e., Arrhenius, WLF, exponential, etc. Several models of complex modulus as a function of reduced frequency also exist. All existing TSF and CM models fail to represent at least some sets of data with desired accuracy and efficiency. Consider a linear, constant coefficient, stable system, and its frequency response function. It is well known that if the real component of the complex-valued frequency response function is given over the infinite frequency range, then the imaginary component may be obtained. The complex modulus of vibration damping materials is such a system. Extensive work with fractional calculus based models for complex modulus has established their viability and potential attractiveness. A ratio of factored polynomials of one-half order is proposed to model the complex modulus. This CM model is attractive from a number of viewpoints: The proper interrelationship of the real and imaginary components is guaranteed; an adequately large number of terms may be used in order to accurately model the complex modulus; an expression may be developed for the real component that lends itself to fitting data by collocating through a number of points; closed-form expressions may be developed for compliance, relaxation modulus, and creep compliance which also lend themselves to collection fitting of experimental data, etc. With modern computational power, this model becomes both accurate and efficient. Previous work has established the slope of the TSF as the characteristic which causes complex modulus data to be properly shifted; therefore, a new approach to modeling the TSF is proposed. The new model is based on determining values of slopes at equally spaced temperatures, fitting a cubic spline through these points (i.e., knots), storing the coefficients, and integrating the cubic spline analytically. The concept of reduced temperature is introduced, used as a convenience for the present effort and proposed as an additional method of presenting data in a form useful to the damping industry. The core of the revolutionary concept is using the simultaneous modeling of both real and imaginary components as the criteria to enable the set of data to establish its TSF. Previous techniques have used real modulus, imaginary modulus, and loss factor as a function or reduced frequency, sometimes in a least-squares sense, and sometimes visually, as the criteria. The above CM and TSF models are essential to the iteration strategy required to determine parameter values for both models. The iteration scheme is conceptually straight-forward. Approximations to the CM and to the TSF are obtained. For each TSF knot, the reduced temperature is used to determine the associated reduced frequency, the current loss factor curve is compared to the corresponding experimental value and the value of the slope adjusted accordingly, the real component collocated for the updated TSF, etc. Examples are given and discussed.