Simple harmonic motion amplitude

Imagine a Maxwell liquid placed between two parallel plates and sheared by moving the upper plate in its own plane. However, instead of moving the plate at a constant velocity as discussed in Chapter 1, let the displacement of the plate vary sinusoidally with time, ie the plate undergoes simple harmonic motion. If the maximum displacement of the upper plate is X and the distance between the plates is h, then the amplitude A of the shear strain in the liquid is given by... 

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

The potential energy of such an oscillator can be plotted as a function of the separation r, or, for a normal mode in a polyatomic molecule, as a function of a parameter characterizing the phase of the oscillation. For a simple harmonic oscillator, the potential energy function is parabolic, but for a molecule its shape is that indicated in Figure 2.6. The true curve is close to a parabola at the bottom, and it is for this reason that the assumption of simple harmonic motion is justified for vibrations of low amplitude. 

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... 

Figure 1-7 indicates the normal modes of vibration in CO2 and H2O molecules. In each normal vibration, the individual nuclei carry out a simple harmonic motion in the direction indicated by the arrow, and all the nuclei have the same frequency of oscillation (i.e., the frequency of the normal vibration) and are moving in the same phase. Furthermore, the relative lengths of the arrows indicate the relative velocities and the amplitudes for each nucleus. The 2 vibrations in CO2 are worth comment, since they differ from the others in that two vibrations (i>2a and p2 ) I ave exactly the same frequency. Apparently, there are an infinite number of normal vibrations of this type, which differ only in their directions perpendicular to the molecular axis. Any of them, however, can be resolved into two vibrations such as P2o and p2fe>... 

As the molecule vibrates (undergoes atom displacements)) the electronic charge distribution and, hence, the polarizability (a) varies in time. The polarizability is related to the electron density of the molecule and is often visualized in three dimensions as an ellipsoid and represented mathematically as a symmetric second-rank tensor. The time-dependent amplitude (Q ) of a normal vibrational mode executing simple harmonic motion is written in terms of the equilibrium amplitude Q , the normal mode frequency o), and time t). 

Properties. It is of considerable importance to examine the nature of die solutions obtained above. It is evident from Eq. (9), Sec. 2-2, that each atom is oscillating about its equilibrium position with a simple harmonic motion of amplitude Aik — Kkhk, frequency x /27t, and phase e. Ihirthermore, corresponding to a given solution X of the secular equation, i he frequency and phase of the motion of each coordinate is the same, but I lie amplitudes may be, and usually are, different for each coordinate. On account of the equality of phase and frequency, each atom reaches its position of maximum displacement at the same time, and each atom pa.sscs through its equilibrium position at the same time. A mode of ibration having all these characteristics is called a normal mode of vibra-iion, and its frequency is known as a normal, or fundamental, frequency of (he molecule. 

If the journal is excited into a simple harmonic motion of small amplitude, the instantaneous eccentricity ratio and attitude angle may be expressed respectively as ... 

The travel of the vanes can be adjusted so that at one end of their swing they either overlap, just meet, or fail to meet. The resulting waveform is a pure sinusoid only if the vanes just meet. If the resulting irradiance must be known, the total swing 2L determines the maximum open aperture area. To determine 2L, adjust the amplitude until the pattern in Figure 9.9a is seen, and then measure the rest width Lg. The total swing will be 2Lg if the chopper obeys simple harmonic motion. 

Doppler width of this transition at 300 K is AwDoppier 5-71 x 10 s . If, instead of moving freely, the radiating atoms are constrained to oscillate about their mean positions with simple harmonic motion of frequency f/2ir and amplitude L, show that the emitted wave is frequency modulated. By qualitative arguments show that the line width of the radiation is given approximately by equation (17.17). Assuming a gas kinetic collision cross-section of... 

Consider a 5 kg mass attached to a spring with a spring constant / = 400Nm undergoing simple harmonic motion with amplitude A = 10 cm. Assume that the energy this mass can attain is quantized according to Eq. 3.97. What is the quantum number nl The potential energy is... 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

But this is just the energy function for a linear oscillator with the linear amplitude q — l. The motion is therefore a simple harmonic vibration (j> — coscot, and we have therefore... 

If the mechanical vibration of the simple harmonic oscillator by which we first represented the nuclear motion of the diatomic molecule in the preceding section is accompanied by an oscillation of the dipole moment of the molecule, then, according to classical physics, radiation will be emitted with the frequency of the oscillator. For small amplitudes of vibration we can take the oscillating part of the dipole moment as being proportional to the elongation cc of the molecule introduced in the preceding section, let us say equal to qx. The amount of radiation emitted by the oscillator in unit time is then given by ... 

Note that Eq. (24.18) is still nonlinear in the motion amplitude due to the third term. A simple iterative technique is chosen to solve the above linearized equation. In the first step, a linear harmonic solution for x o =0 I = 1,...,6) on the left-hand side is obtained. This first iteration value is introduced on the left-hand side of solution and the process repeated. Generally, between two and three iterations produce convergence in the results. 

A vibration is a periodic motion or one that repeats itself after a certain interval of time. This time interval is referred to as the period of the vibration, T. A plot, or profile, of a vibration is shown in Figure 43.1, which shows the period, T, and the maximum displacement or amplitude, X - The inverse of the period, j, is called the frequency, f, of the vibration, which can be expressed in units of cycles per second (cps) or Hertz (Hz). A harmonic function is the simplest type of periodic motion and is shown in Figure 43.2, which is the harmonic function for the small oscillations of a simple pendulum. Such a relationship can be expressed by the equation ... 

In practice, this model is oversimplified since the exciting wake shedding is by no means harmonic and is itself coupled with the shape oscillations and since Eq. (7-30) is strictly valid only for small oscillations and stationary fluid particles. However, this simple model provides a conceptual basis to explain certain features of the oscillatory motion. For example, the period of oscillation, after an initial transient (El), becomes quite regular while the amplitude is highly irregular (E3, S4, S5). Beats have also been observed in drop oscillations (D4). If /w and are of equal magnitude, one would expect resonance to occur, and this is one proposed mechanism for breakage of drops and bubbles (Chapter 12). 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

GAMESS . The vibrational data obtained in this way should, however, be used with care, since many weakly bonded systems exhibit large-amplitude motions (especially when simple hydrides are involved) which cannot be accurately modelled at the harmonic level. Indeed, the very concept of the equilibrium structure of such complexes, while formally valid, loses much of its significance, and a much larger section of the surface must be sampled than is customary for ordinary molecules. Appropriate techniques for evaluating vibrational wavefunctions beyond the harmonic level have been reviewed by Le Roy et and very recently by Briels et a/. and the interested reader is referred to these. 

The viscoelastic parameters are generally measured by dynamic oscillatory measurements. Apparatus of three different configurations can be used cone and plate, parallel plates, or concentric cylinders. In the case of cone and plate geometry, the test material is contained between a cone and a plate with the angle between cone and plate being small (<4°). The bottom member undergoes forced harmonic oscillations about its axis and this motion is transmitted through the test material to the top member, the motion of which is constrained by a torsion bar. The relevant measurements are the amplitude ratio of the motions of the two members and the associated phase lag. From this information it is relatively simple to determine G and G". 

Here Ef is the amplitude, t the duration, and co the frequency of the ith pulse. This scheme has been applied in Ref [46] to a generic two-dimensional HT model which incorporated a H-atom reaction coordinate as well as a low-frequency H-bond mode. In a subsequent work [47] the approach has been specified to a simple model of HT in thioacetylacetone. The Hamiltonian was tailored to the form of Eq. (4.1) based on the information available for the stationary points, that is, the energetics as well as the normal modes of vibration. From these data an effective two-dimensional potential was constructed including the H-atom coordinate as well as a coupled harmonic oscillator, which describes the 0-S H-bond motion. Although perhaps oversimplified, this model allowed the study of some principle aspects of laser-driven H-bond motion in an asymmetric low-barrier system. 

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