Free vibration amplitude

Fig. 1.12 Tip-sample distance d and free vibration amplitude A0. When the tip approaches the surface, the value of A0 will decrease. In tapping, the tip touches the surface at each cycle, which changes the oscillation amplitude to A. The setpoint at which images is done is defined as A/A0...

The amplitudes of oscillations will depend upon weight, stiffness and configuration. The record of these oscillations is known as free vibration record. The rate of oscillations will determine the natural frequency of the object. Figure 14.20 shows one such free vibration record. 

Here A is the amplitude, cp the initial phase, and coo the frequency of free vibrations. Thus, in the absence of attenuation free vibrations are sinusoidal functions and this result can be easily predicted since mass is subjected to the action of the elastic force only. In other words, the sum of the kinetic and potential energy of the system remains the same at all times and the mass performs a periodic motion with respect to the origin that is accompanied by periodic expansion and compression of the spring. As follows from Equation (3.105) the period of free vibrations is... 

Before considering particular test methods, it is useful to survey the principles and terms used in dynamic testing. There are basically two classes of dynamic motion, free vibration in which the test piece is set into oscillation and the amplitude allowed to decay due to damping in the system, and forced vibration in which the oscillation is maintained by external means. These are illustrated in Figure 9.1 together with a subdivision of forced vibration in which the test piece is subjected to a series of half-cycles. The two classes could be sub-divided in a number of ways, for example forced vibration machines may operate at resonance or away from resonance. Wave propagation (e.g. ultrasonics) is a form of forced vibration method and rebound resilience is a simple unforced method consisting of one half-cycle. The most common type of free vibration apparatus is the torsion pendulum. 

In free vibration methods, the rubber test piece, with or without an added mass, is allowed to oscillate at the natural frequency determined by the dimensions and viscoelastic properties of the rubber and by the total inertia. Due to damping in the rubber, the amplitude of oscillations will decay with time and, from the rate of decay and the frequency of oscillation, the dynamic properties of the test piece can be deduced. 

ISO 4663 gives no advice as to the relative merits of the three methods it specifies. Method C, which is not strictly a free vibration method, removes the difficulties associated with changing amplitude through the course of the test but at the expense of a rather more complex apparatus. When the inertia member is supported by a torsion wire, as in method B, the tensile strain in the test piece can be controlled to a low level by means of counterweights. 

The recoil-free fraction /, while strictly speaking not the result of a chemical interaction, can indirectly provide useful chemical, as well as structural, information. As shown earlier, / is related to , the mean square vibrational amplitude of the resonant atom in the direction of the y ray. The temperature dependence of is often approximated using the... 

Whatever the nanotube production method, understanding the properties of materials filled with nanotubes requires the knowledge of the nanotube properties. As a consequence, many efforts were made to experimentally measure the nanotube Young modulus and intrinsic conductivity. Fortunately in TEM, it was observed that the nanotubes were vibrating when clamped at one end and free at the other one (see Figure 3.6). Thus, the measurement of the mean-square vibration amplitude in function of the temperature allowed the determination of the Young modulus (higher than 1 TPa for bundles of SWNTs) (51). 

The above discussion implies steady-state response in time. An equivalent reciprocal view of steady-state resonance response is that in the vicinity of resonance there is a dip in the force required to maintain a constant level of response. The force-reduction ratio is Q, and the fractional bandwidth of the force reduction is T. In contrast, a truly force-free response of a resonant system (once excited) would involve the exponential decay of vibration amplitude with time. As we have mentioned earlier, decay is also controlled by the system loss factor as follows ... 

From an entropic viewpoint, imposition of even a fairly mild restriction on the vibrational amplitude reduces the entropy associated with that vibration essentially to zero, and little significant additional loss is possible on further restriction. Although a bond with a i/ of 1000 cm has a substantial vibrational amplitude owing to its zero point energy, it has only 0.1 eu of entropy because only a small number of quantum states are effectively occupied. Further restriction on the magnitude of this vibrational amplitude will further reduce this small residual entropy but will not significantly increase the much larger difference in entropy between these restricted states and that of a free rotation. 

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