Sinusoidal mechanical oscillation

In the nano-DMA, a sinusoidal mechanical oscillation is applied to the sample by the transducer shown in Fig.l. The tip of an atomic force microscope is placed in contact with the sample, and the amplitude and phase of the cantilever s response is recorded as a function of position on the sample s surface, forming images related to the sample s elastic modulus and damping. Lateral resolution is of the order of a few tens of nanometers. The microscope may also be operated in spectroscopy mode, where the tip remains nominally on the same point of the surface, and the temperature of the sample is ramped. 

R.F. Boyer Coupling between the alternating electric field and the electric dipole is direct that between the sinusoidal mechanical field and the "mechanical dipole" may be diffuse. One example of this which we studied (unpublished) is the glassy state response of poly(vinyl acetate) to an electric field which senses high frequency rotation of the dipole about the oxygen-carbonyl carbon bond in the side chain, and low frequency mechanical motion which probably detects local motion of the entire side group about the backbone. The latter occurs at 0.75 Tg (K) at 1 Hz and should be classified as a p relaxation the former occurs 0.75 Tg and is a y relaxation. The dipole oscillates about the backbone above Tg. 

If the applied force varies sinusoidally with time, the period of the oscillation defines the time scale. Quite different mechanical responses are expected at different frequencies. This type of experiment will be described in Secs. 3.10 and 3.11. 

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 A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

The inlet monomer concentration was varied sinusoidally to determine the effect of these changes on Dp, the time-averaged polydispersity, when compared with the steady-state case. For the unsteady state CSTR, the pseudo steady-state assumption for active centres was used to simplify computations. In both of the mechanisms considered, D increases with respect to the steady-state value (for constant conversion and number average chain length y ) as the frequency of the oscillation in the monomer feed concentration is decreased. The maximum deviation in D thus occurs as lo 0. However, it was predicted that the value of D could only be increased by 10-325S with respect to the steady state depending on reaction mechanism and the amplitude of the oscillating feed. Laurence and Vasudevan (12) considered a reaction with combination termination and no chain transfer. 

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. 

The motion in the classical domain corresponds to a harmonic oscillator of frequency v, with the displacement from equilibrium varying sinusoidally with time. The transcription of this problem into quantum mechanics is simple and straightforward it is a standard problem in introductory quantum mechanics texts. The energy levels of the quantum system are given by... 

The TA Instruments CSL2 rheometer can perform low frequency oscillatory measurements as well as steady-state viscosity determinations, even though it has a simple mechanical system. The sinusoidal wave form is generated mathematically in the computer rather than with an electromechanical drive system. The stress is controlled, and the resulting strain is determined and stored in memory. The computer analyzes the wave form and calculates the viscosity and elasticity of the specimen at the frequency of the test. As of this writing (1996), the oscillation software covers a frequency range of 10-4 -40 Hz. This range could be increased as faster software and computers become available. 

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... 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

Simple harmonic motion, such as the (undamped by frictional forces) sinusoidal oscillation of a weight suspended by a spring can also be thought of in terms of the projection of a vector traveling in a circular path. This is something you should have covered in your elementary mechanics classes, of course, but we will reexamine it here, first because it is important in infrared spectroscopy, and second because it provides some illumination concerning resonance. 

Finally, one of the most useful ways of measuring viscoelastic properties is dynamic mechanical analysis, or DMA. In this type of experiment, an oscillating stress is applied to the sample and the response is measured as a function of the frequency of the oscillation. By using different instruments this frequency can be varied over an enormous range. Actually, the sample is usually stretched a little bit and oscillated about this strain also, the stress necessary to produce an oscillatory strain of a given magnitude is the quantity usually measured. If the sample being oscillated happens to be perfectly elastic, so that its response is instantaneous, then the stress and strain would be completely in-phase. If a sinusoidal shear strain is imposed on the sample we have (Equation 13-72) ... 

Such a model describes an underdamped oscillatory behavior with a sinusoidal period of 11 minutes and damping coefficient of 0.44. This period of oscillation corresponds to the "hunting swings of the sporangiophore therefore, it appears that our model accounts for the gross mechanical behavior of Phycomyces and its interaction with our measurement system. 

In a typical case of dynamic mechanical analysis, a small stress oscillates periodically in a sinusoidal mode with amplitude cr and frequency co, and the small strain e follows the modulation with a certain phase lag The sinusoidal stress is the imposed stimulation, and in a complex form. 

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