Harmonic oscillatory measure

The viscoelastic parameters are generally measured by dynamic oscillatory measurements. Apparatus of three different configurations can be used cone and plate, parallel plates, or concentric cylinders. In the case of cone and plate geometry, the test material is contained between a cone and a plate with the angle between cone and plate being small (<4°). The bottom member undergoes forced harmonic oscillations about its axis and this motion is transmitted through the test material to the top member, the motion of which is constrained by a torsion bar. The relevant measurements are the amplitude ratio of the motions of the two members and the associated phase lag. From this information it is relatively simple to determine G and G". 

Rich and reliable viscoelastic data can be obtained from measuring the dynamic mechanical response of the polymeric liquid to a rate-of-strain X t) which is harmonically oscillatory ... 

A spectrum is the distribution of physical characteristics in a system. In this sense, the Power Spectrum Density (PSD) provides information about fundamental frequencies (and their harmonics) in dynamical systems with oscillatory behavior. PSD can be used to study periodic-quasiperiodic-chaotic routes [27]. The filtered temperature measurements y t) were obtained as discrete-time functions, then PSD s were computed from Fast Fourier Transform (FFT) in order to compute the fundamental frequencies. 

Besides the phase of the fundamental mode, strictly speaking, the preferred phase, many other characteristics have been studied in [226]. Because a large mismatch was chosen, they have lacked any trend, but an interesting oscillatory behavior has been discovered for the initial two-mode coherent state. Within each period, the phase-matched second-harmonic and second-subharmonic generation processes can be prepared. The model of an ideal Kerr-like medium [223] have been considered for a comparison with cascaded quadratic non-linearities. It follows that these nonlinearities exhibit not only self-phase modulation in the fundamental mode but also a cross-phase modulation of the modes that can be considered for a nondemolition measurement. 

Fig. 2 Schematic depiction of the procedure for conducting LAOS experiments and analysis using FT-rheology. a Measurement of the oscillatory shear strain and shear stress response in the time domain, b Normalized frequency spectra after the Fourier transformation of the shear stress exhibit the fundamental peak at the angular frequency (d. Higher harmonics / /i with n being a positive odd integer are detected for a periodic nonlinear shear stress, c By variation of yo the transition from linear to nonlinear mechanical behavior can be observed in the increase of/ /i...

As an extension of the dynamic test in the linear regime, a large-amplitude oscillatory shear (LAOS), y(t) = /o sin or, is often applied to a material to measure the oscillating shear stress (Ts (and sometimes the first normal stress difference, too). In the stationary state, cTs under LAOS is contributed not only from the fundamental harmonics (oscillating at the angular frequency of strain, cu) but also from the higher order (mostly odd) harmonics and can be expressed as... 

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