Sinusoidal oscillations

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

Consider a deformation consisting of repeated sinusoidal oscillations of shear strain. The relation between stress and strain is an ellipse, provided that the strain amplitude is small, and the slope of the line joining points where tangents to the ellipse are vertical represents an effective elastic modulus, termed the storage modulus /r. The area of the ellipse represents energy dissipated in unit volume per cycle of deformation, expressed by the equation... 

Intracellular calcium oscillations generally fall into one of two categories involving different mechanisms baseline transients, or spikes, and sinusoidal oscillations. Figure 22-4 illustrates these two oscillatory patterns. 

Na+ concentrations confirms their common marine origin. However, the concentration variations with depth are much less clear than the sinusoidal oscillations at J-9. At Station Q-13, only 70 km... 

If the system is initially at rest (all derivatives equal zero) and we start to force it with a sine wave the output x, will go through some transient period as shown in Fig. 12,3 and then settle down to a steady sinusoidal oscillation. In the Laplace domain, the output is by definition... 

Figure 14. Simple model demonstrating how adsorption and surface diffusion can co-Urnit overall reaction kinetics, as explained in the text, (a) A semi-infinite surface establishes a uniform surface coverage Cao of adsorbate A via equilibrium of surface diffusion and adsorption/desorption of A from/to the surrounding gas. (b) Concentration profile of adsorbed species following a step (drop) in surface coverage at the origin, (c) Surface flux of species at the origin (A 4i(t)) as a function of time. Points marked with a solid circle ( ) correspond to the concentration profiles in b. (d) Surface flux of species at the origin (A 4i(ft>)) resulting from a steady periodic sinusoidal oscillation at frequency 0) of the concentration at the origin.

If one assumes a simplified sinusoidal oscillation, the frequency of the oscillation, V, is represented by... 

Molerus (M7) and Hjelmfelt and Mockros (H6) have developed complete solutions to Eq. (11-48). Velocities can be expressed as Fourier integrals. It therefore suffices to consider pure sinusoidal oscillations ... 

Fig. 11.16 Ratio of mean terminal velocity to terminal velocity in absence of oscillations for particles in sinusoidally oscillating fluids. Unbroken lines are predictions from Eq. (11-58) broken lines are numerical predictions (M8) for 2 mm spheres in water with y = 2.5 and values as follows curve A-0.28 B -0.42 C—0.56 D -1.11 E -1.67.

Levitation is defined as a stable condition in which a particle responds to vertically oscillating fluid so that net gravity forces are completely neutralized and the particle merely oscillates about a fixed position (H12). In terms of the preceding analysis, this means 14 = 0 (cf. Fig. 11.16). Contrary to the predictions of the numerical solutions, Feinman (FI) found that levitation can be caused by sinusoidal oscillations. Equation (11-58) predicts that 14 becomes zero if ... 

Sinusoidal oscillations of the continuous phase cause levitation or countergravity motion much more readily for gas bubbles, due to changes in bubble volume which cause a steady component in the pressure gradient drag term (Jl, J2). If the fluid motion is given by Eq. (11-49), the pressure in the vicinity of the bubble also varies sinusoidally. For normal experimental conditions, the resulting volume oscillations are isothermal (P2), and given by (Jl) ... 

Dynamic melt viscosity studies on the star blocks and a similar triblock were carried out using a Rheometric Mechanical Spectrometer (RMS) (Rheometrics 800). Circular molded samples with -1.5 mm thickness and 2 cm diameter were subjected to forced sinusoidal oscillations (2% strain) between two parallel plates. The experiment was set in the frequency sweep mode. Data were collected at 180 and 210 °C. 

The dynamic melt viscosity measurements of select star blocks and a similar triblock were carried out on a rheometric mechanical spectrometer, RMS. Circular molded samples of 2 cm diameter and -1.5 mm thickness were subjected to forced sinusoidal oscillations. Dynamic viscosities were recorded in the frequency range of 0.01-100 rad/s at 180 °C. Figure 10 shows the complex viscosities of two select star blocks and a similar linear triblock. The plots showed characteristic behavior of thermoplastic elastomers, i.e., absence of Newtonian behavior even in the low frequency region. The complex viscosity of the star block... 

The liquid jet formed by a pressure nozzle is inherently unstable. The breakup of the laminar jet occurs by symmetrical oscillation, sinusoidal oscillation, and atomization. 

If, in a dynamic test with forced sinusoidal oscillation, force is plotted against deflection a hysteresis loop is obtained as shown in Figure 9.4. 

Fig. 3. Experimental traces of bromide ion concentration in closed system studies of the Belousov-Zhabotinski reaction, showing (a) quasiharmonic (i.e., sinusoidal) oscillations, (A>) and (c) increasingly nonlinear oscillations, and (

Moore, 1977b] Moore, F. R. (1977b). Table Lookup Noise for Sinusoidal Oscillators. Computer Music Journal, l(2) 26-29. pages 326-334. 

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).

Signal amplitude maximum vertical displacement from equilibrium for a sinusoidally oscillating waveform. Signal amplitude is thus half of the peak-to-peak vertical displacement. 

In the classical framework of single scattering (SS), the EXAFS signal, %(k), is related to the sum of sinusoidal oscillations for each shell of neighbours around the absorbing atom ... 

Because x (t) must always be positive, a comparison of equations (12) and (13) indicates that the rate of removal of R will be enhanced by these small amplitude oscillations of undefined waveform. Similarly, for the case of sinusoidal oscillations in X... 

---