Measurements non-linearity

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

Fig. 12. Measured non-linear transmission of a 1-cm-path solution of compound6 (of Fig. 2) in benzene of dg = 0.045 mol L as a function of the input intensity of the -800 nm laser beam. From [9]...

In all relaxation measurement experiments, spectra are recorded as a function of t, a time during which relaxation is allowed to occur. The variation of the intensity of a particular signal in the spectrum as a function of t then constitutes the relaxation decay curve, which we must analyze to retrieve motional data. As this curve will generally have an exponential form (or at least approximately), in order to define this curve properly, it is desirable to arrange to spread the t points of measurement non-linearly and to concentrate them near the t — 0 end of the curve where the signal intensity varies most rapidly. A logarithmic distribution of t points is usually optimal. It is essential that the spin system is fully relaxed before acquiring the next scan or spectrum and also that the t data measurement is continued until the complete decay curve is measured, otherwise errors will inevitably result in its analysis. 

Fig. 7.20 Containment vessel response spectrum (-, measured —, non-linear linear)...

SOLVES FOR THE PARAMETERS IN NON-LINEAR MEASURED VARIABLES ARE SUBJECT TC ERROR ONE OR TWO CONSTRAINTS. 

Density is the most commonly measured property of a gas, and is obtained experimentally by measuring the specific gravity of the gas (density of the gas relative to air = 1). As pressure increases, so does gas density, but the relationship is non-linear since the dimensionless gas compressibility (z-factor) also varies with pressure. The gas density (pg) can be calculated at any pressure and temperature using the real gas law ... 

Figure B2.4.6. Results of an offset-saturation expermient for measuring the spin-spin relaxation time, T. In this experiment, the signal is irradiated at some offset from resonance until a steady state is achieved. The partially saturated z magnetization is then measured with a kH pulse. This figure shows a plot of the z magnetization as a fiinction of the offset of the saturating field from resonance. Circles represent measured data the line is a non-linear least-squares fit. The signal is nonnal when the saturation is far away, and dips to a minimum on resonance. The width of this dip gives T, independent of magnetic field inliomogeneity.

I When the system voltage is linear (an ideal condition that would seldom exist) but the load is non-linear The current will be distorted and become non-sinusoidal. The actual current /, (r.m.s.) (equation (23.2)) will become higher than could be measured by an ammeter or any other measuring instrument, at the fundamental frequency. Figure 23.13 illustrates the difference between the apparent current, measured by an instrument, and the actual current, where / = active component of the current... 

From the above it can be inferred that for an accurate analysis of a system, particularly where the loads are of varying nature or have non-linear characteristics it is necessary to conduct a harmonic analysis. The above corrective measures will provide a reasonably stable network, operat-ing at high p.f. with the harmonics greatly suppressed. The improved actual line loading, eliminating the fifth harmonic component, which is compensated,... 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

BS 7118 Part 2 1989. Measurement of flaiid Flow Assessment of Uncertainty in the Calibration and Use of Flow Measurement Devices Non-linear Calibration Relationships. British Standards Institution, 1989. 

The combination of non-ideal phase behaviour of solutions, the non-linearity of particle formation kinetics, the multi-dimensionality of crystals, their interactions and difficulties of modelling, instrumentation and measurement have conspired to make crystallizer control a formidable engineering challenge. Various aspects of achieving control of crystallizers have been reviewed by Rawlings etal. (1993) and Rohani (2001), respectively. 

The direction of rotation depends on the direction of the current in the coil, and thus the instrument is only suitable for D.C. It is, however, possible to incorporate a full-wave rectifier arranged as shown in Figure 17.11 in order to allow the instrument to measure A.C. quantities. The quantity measured is the RMS value only if the waveform of the current is truly sinusoidal. In other cases, a considerable error may result. In principle, the scale is linear but, if required, it can be made non-linear by suitably shaping the poles of the permanent magnet. The instrument reading is affected by the performance of the rectifier, which is a non-linear device, and this results in the scale also being non-linear. The error when measuring D.C. quantities can be as low as 0.1 per cent of full-scale deflection and instruments are available for currents between microamperes and up to 600 A. 

We employed a comparative Z-scan procedure, wherein we perform a reduced-aperture Z-scan on CS2 immediately followed by reduced and open-aperture Z-scans on the polymer solution at a particular wavelength. The peak on-axis intensity, /0, is then calculated from the CS2 peak to valley transmission change using previously measured values for its non-linear axis of refraction. Based on the work of Sheik-Bahae et al the equation for /0is given by 37]... 

Conjugated polymers are centrosymmetric systems where excited states have definite parity of even (A,) or odd (B ) and electric dipole transitions are allowed only between states of opposite parity. The ground state of conjugated polymers is an even parity singlet state, written as the 1A... PM spectroscopy is a linear technique probing dipole allowed one-photon transitions. Non linear spectroscopies complement these measurements as they can couple to dipole-forbidden trail-... 

A) The use of a calibration graph. This overcomes any problems created due to non-linear absorbance/concentration features and means that any unknown concentration run under the same conditions as the series of standards can be determined directly from the graph. The procedure requires that all standards and samples are measured in the same fixed-path-length cell, although the dimensions of the cell and the molar absorption coefficient for the chosen absorption band are not needed as these are constant throughout all the measurements. 

Conductivity measurements yield molar conductivities A (Scm2 mol-1) at salt concentration c (mol L-1). A set of data pairs (Af, c,), is evaluated with the help of non linear fits of equations [89,93,94] consisting of the conductivity equation, Eq. (7), the expression for the association constant, Eq. (3), and an equation for the activity coefficient of the free ions in the solution, Eq.(8) the activity coefficient of the ion pair is neglected at low concentrations. 

It is seen that, for a truly linear detector, the response index (a) will be unity and the experimentally determined value of (a) will be an accurate measure of the proximity of the response to strict linearity. It is also clear that (a) could be used to correct for any non-linearity that might occur in the detector and thus improve the accuracy of an analysis. 

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