Small Amplitude Forcing

FIGURE 27.22 Morse potential (a Large force, small amplitude, b Small force, large amplitude). 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

This energy has minima at /x = /Xs. where yxs is a saturated dipole value, so that the saturated force has a value Fg = 2KfZg. The frequency of small-amplitude ferroelectric mode, where all dipoles vibrate in phase around fig, is given by twpE = /mY, where is the second derivative of... 

Figure 13.10 shows a representation of the phase plane behaviour appropriate to small-amplitude forcing. There are two basic cycles which make up the full motion first, there is the natural limit cycle, corresponding for example to Fig. 13.9(a) around which the unforced system moves secondly, there is a small cycle, perpendicular to the limit cycle, corresponding to the periodic forcing term. The overall motion, obtained as the small cycle is swept around the large one, gives a torus and the buckled limit cycle oscillations at low rf in Fig. 13.9 draw out a path over the surface of such a torus. 

We begin with an innocuous case. Consider a pendulum suspended in air and consequently subject to damping accompanied by a Langevin force. This force is, of course, the same as the one in equation (1.1) for the Brownian particle, because the collisions of the air molecules are the same. They depend on the instantaneous value of V, but they are insensitive to the fact that there is a mechanical force acting on the particle as well. Hence for small amplitudes the motion is governed by the linear equation (1.10). For larger amplitudes the equation becomes nonlinear ... 

The temporal evolution of y/(x,t) in the laboratory frame, as described by (61) in the ID case, is shown in Fig. 22 for three regimes of the traveling forcing starting with the initial condition y/ = 0 and superimposed with small-amplitude noise. In Fig. 22a the velocity is sufficiently small and belongs to the range where the... 

Figure 10 shows the convolution with the proper normalization factor, and it is immediately apparent from this figure how the use of small amplitudes increases the weight of the short-range atomic forces over the unwanted long-range forces. The amplitude in FM-AFM allows one to tune the sensitivity of the AFM to forces of various ranges. 

Fig. 10. A/ is a convolution of the weight function w with the tip-sample force gradient. For small amplitudes, short range interactions contribute heavily to the frequency shift, while long-range interactions are attenuated.

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

Dynamic plowing lithography (DPL), i.e. the lithography technique in tapping mode in which the force between tip and sample is increased by suddenly increasing the amplitude of the cantilever oscillations, has been developed by Klehn and coworkers [257-260]. The topography of the sample is acquired with a normal, small amplitude of the cantilever oscillations. When the amplitude is increased, the feedback makes the sample approach to the tip, in order to keep constant the oscillation amplitude, and the tip indents the sample. 

Fig. 5 Schematic description of the contact conditions encountered under small amplitude cyclic lateral micro-motions (fretting). S is the applied lateral displacement, Q is the lateral force and P is the applied constant normal load. The elliptic and trapezoidal Q(S) loops correspond to partial slip and gross slip condition respectively...

Next we can consider the small amplitude motions which present a standard GF-eigenvalue problem, but with p as a free parameter. G°-elements corresponding to a basic set of internal valence coordinates, Rt where t= 1,2,... 3N-1, are derived from s°-vectors using Eq. (3.41), and thus they vary with p according to the variation of the first derivatives of Eq. (3.3). Also the force constants, Fn, may be functions of p and contribute to the general functional properties of L- and /-elements as well as of the eigenvalues, Xk (Sect. 4.7). 

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