Non-sinusoidal oscillation

If the material response in AC magnetisation measurements is nonlinear, the magnetisation oscillations do not follow the fundamental susceptibility perfectly. The perfect sinusoidal AC field induces non-sinusoidal oscillations of the magnetisation which can be described as a sum of several sinusoidal components oscillating at the harmonics of the driving frequency [4] (Fig. 2.4)... 

X. N. Meng, et al., Mechanism of Eiqilaining Liquid Friction and Flux Consumption during Non-sinusoidal Oscillation in Slab Continuous Casting Mould , Canadan Metallurgical... 

Figure 1. Fundamentals of ICR excitation. The applied magnetic field direction is perpendicular to the page, and a sinusoidally oscillating radiofrequency electric field is applied to two opposed plates (see upper diagrams). Ions with cyclotron frequency equal to ("resonant" with) that of the applied rf electric field will be excited spirally outward (top right), whereas "off-resonant" ions of other mass-to-charge ratio (and thus other cyclotron frequencies) are excited non-coherently and are left with almost no net displacement after many cycles (top left). After the excitation period (lower diagrams), the final ICR orbital radius is proportional to the amplitude of the rf electric field during the excitation period, to leave ions undetected (A), excited to a detectable orbital radius (B), or ejected (C).

A single capillary oscillator has also been used for oscillation in case of non-electrolytes [15]. Interesting results have been obtained with four types of membranes, viz. Pyrex, millipore filter, silver coated and inorganic membranes for 0.1N NaCl/KCl 0.01N NaCl/KCl systems. Relaxation oscillations are obtained for Pyrex membrane, whereas high frequency sinusoidal oscillations are observed for the other membranes. Bistability is also observed in some cases as discussed in earlier Chapter 8. Typical results on oscillatory features are recorded in Fig. 11.2. 

This mode is used for accurate determination of the frequency dependence of materials and prediction of end-use product performance. In the fixed frequency mode applied stress (i.e., a force per unit area that tends to deform the body, usually expressed in Pa (N/m)) forces the sample to undergo sinusoidal oscillation at a frequency and amplitude (strain), i.e., the deformation from a specified reference state, measured as the ratio of the deformation to the total value of the dimension in which the strain occurs. Strain is non-dimensional, but is frequently expressed in reference values (such as %strain) selected by the operator. Energy dissipation in the sample causes the sample strain to be out of phase with the applied stress (Figure 15.2(a)). In other words, since the sample is viscoelastic, the maximum strain does not occur at the same instant as maximum stress. This phase shift or lag, defined as phase angle (6), is measured and used with known sample geometry and driver energy to calculate the viscoelastic properties of the sample. 

Most adsorbed surfactant and polymer coils at the oil-water (0/W) interface show non-Newtonian rheological behavior. The surface shear viscosity Pg depends on the applied shear rate, showing shear thinning at high shear rates. Some films also show Bingham plastic behavior with a measurable yield stress. Many adsorbed polymers and proteins show viscoelastic behavior and one can measure viscous and elastic components using sinusoidally oscillating surface dilation. For example the complex dilational modulus c obtained can be split into an in-phase (the elastic component e ) and an out-of-phase (the viscous component e") components. Creep and stress relaxation methods can be applied to study viscoelasticity. 

The factor f reduces the oscillation amplitude symmetrically about R - R0, facilitating straightforward calculation of polymer refractive index from quantities measured directly from the waveform (3,). When r12 is not small, as in the plasma etching of thin polymer films, the first order power series approximation is inadequate. For example, for a plasma/poly(methyl-methacrylate)/silicon system, r12 = -0.196 and r23 = -0.442. The waveform for a uniformly etching film is no longer purely sinusoidal in time but contains other harmonic components. In addition, amplitude reduction through the f factor does not preserve the vertical median R0 making the film refractive index calculation non-trivial. 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% < pi < pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

The oscillating bubble method proves to be very convenient and precise for the evaluation of the non-equilibrium elasticity of surfaces in a wide range of frequencies of external disturbances and surface coverage (adsorption of surfactant) [103-105]. It is based on registration of the sinusoidal variation of bubble volume. The bubble is situated in a capillary containing surfactant solution in which oscillations of different frequencies and amplitudes are created. The treatment of the U = f(ft)) curves (where U is the tension needed to initiate oscillations of constant amplitude) allows the determination of Marangoni elasticities [105]. 

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