Periodic motion induced

In liquid-state NMR spectroscopy, the anisotropic interactions cannot be seen directly due to the rapid tumbling of the molecule. In soHd-state NMR, an ingenious technique called MAS averages out anisotropic interactions, leaving behind mainly isotropic interactions. Unless specified, we will only consider MAS experiments in this contribution. Due to the periodic motions induced by MAS, the NMR interactions can be expended into a Fourier series in units of the rotor frequency, using zeroth-, first-, and second-order terms. Addressing the chemical shift interaction, we can write the Flamiltonian as... 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

All of the above forms of microscopic motion are what we might describe as intrinsic. That is, the motion takes place all by itself, without intervention by any external agent. However, it is possible under certain circumstances to induce particles to engage in additional forms of periodic motion. Still, to achieve resonance, we need to match the frequency of this induced motion with that of the incident radiation [Eq. (1.5)]. 

The concept of the soliton has been introduced in the theoretical treatment by Su et al. (1979, 1980), in which they have employed a model Hamiltonian within the framework of the Hiickel approximation, including both cr-bond compressibility and the kinetic energy term of the CH units. The soliton is an elementary excitation and, in the case of the transient in Fig. 12a, is expected to satisfy a wave equation akin to the if/4 field theory (Krumhansl and Schrieffer, 1975). The estimated energy of the creation of a soliton is 0.4 eV and the periodic-lattice-induced activation energy for the soliton motion is 0.002 eV, the latter being consistent with the result of the ESR observation. The energy of a soliton is most stabilized when its tail (that is, the spatial halfwidth of the kink) extends over seven carbon sites. 

Once a ship is correctly positioned, the issue then becomes maintaining that position. This is typically accomplished in relatively shallow water by positioning anchors at various angles to the vessel. This is a time-consuming process, especially if the vessel has to be moved periodically. To deal with this issue, a dynamically ship positioning system has been developed to hold a vessel on location. In a dynamically positioned ship, thrusters below the waterline apply thrust laterally fore and aft to coxmter the motions induced by wind, wave, and current so that the ship can stay on station without anchors. This technique is used in combination with GPS and shipboard computers (Ingham, 1975). 

T. Sawaragi and M. Kubo, Long-period motions of a moored ship induced by harbor oscillations. Coast. Eng. Japan 25, 261-275 (1982). 

When light is incident on a polymer sample the sample is subg ed to an alternating electric field with frequencies in the ran 10 to 10 Hertz. In response to this field the electrons in the sample move with a periodic motion and become themselves a source of secondary dipole radiation. Hence the induced electric dipole moment, Pj, in the i direction (i = x, y, z) may be expanded in terms of the electric field by the equation ... 

The radius i i of the circular motion induced by the periodic modulation is proportional to the modulation amplitude Gi. It is inversely proportional to the difference of the two frequencies and thus diverges when the condition u/Q = oj of the complete resonance is reached. 

Cabling is an underrated source of excess long-period horizontal noise in vault-type installations. It is recommended to strain-relieve the cable to the pier, sometimes by placing a heavy weight on it close to the sensor. If this is not done, thermal expansion or other motion induced in the cable can be transferred to the seismometer, generating tilt. Sensitivity to cable-induced noise is particularly acute with stiff cables cables designed for flexibility make the whole job much easier. 

Leine RI, Brogliato B, Nijmeijer H (2002) Periodic motion and bifurcations induced by the Painleve paradox. Eur J Mech A Solids 21(5) 869-896... 

Duct flows can be converted into efficient mixing flows (i.e., flows with an exponential stretch of material lines with time) by time- modulation or by spatial changes along the duct axis. One example of the spatially periodic class is the partitioned-pipe mixer (PPM). This flow consists of a pipe partitioned with a sequence of orthogonally placed rectangular plates (Fig. 5a). The cross-sectional motion is induced through rotation of the pipe with respect to the assembly of plates, whereas the axial flow is caused by... 

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