Pendulum resonance

The convenience and value of the concept of resonance in discussing the problems of chemistry are so great as to make the disadvantage of the element of arbitrariness of little significance. This element occutb in the classical resonance phenomenon also—it is arbitrary to discuss the behavior of a system of pendulums with connecting springs in terms of the motion of independent pendulums, since the motion can be described in a way that is mathematically simpler by use of the normal coordinates of the system—but the convenience and usefulness of the concept have nevertheless caused it to be widely applied. 

A few words should be said about the difference between resonance and molecular vibrations. Although vibrations take place, they are oscillations about an equilibrium position determined by the structure of the resonance hybrid, and they should not be confused with the resonance among the contributing forms. The molecule does not resonate or vibrate" from one canonical structure to another. In this sense the term resonance is unfortunate because it has caused unnecessary confusion by invoking a picture of vibration. The term arises from a mathematical analogy between the molecule and the classical phenomenon of resonance between coupled pendulums, or other mechanical systems. 

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. 

Pn is sometimes said to represent the coupling of q with term resonance integral has similar roots (Coulson C. A., Valence, Oxford University Press, Oxford, 2nd edn, p. 79). 

Various methods (1-1) have used to determine the dynamic mechanical properties of polymers. Many of the instruments described are well known and are widely used (torsional pendulum, rheovibron, vibrating reed, and Oberst beam ASTM D4065-82). Newer instruments like the torqued cylinder apparatus (4), resonant bar apparatus (5) and Polymer Laboratories Dynamic Mechanical Thermal Analyzer (6) are becoming more popular in recent times. 

Except for the negative sign in front of the kinetic energy, which can be interpreted as a negative mass , the Hamiltonian (6.4.6) is the (autonomous) Hamiltonian of a pendulum. The resonance width Wm can immediately be calculated. Using the results derived in Section 5.2 and the approximation (6.1.56) for the derivatives of the Bessel functions we obtain ... 

A circular rod of a viscoelastic material of length h and radius R located between the two clamps of a torsion pendulum is rotated slightly from its equilibrium position by a deflecting torque. The torque is released, and the system begins to oscillate. Calculate the resonance frequency of the system. 

Hence, G can be obtained from the resonance frequency of the torsion pendulum by means of the expression... 

According to Chirikov [23J, the onset of chaos is associated with the overlap of neighboring nonlinear resonances. The overlap criterion, which bears the qualitative significance, uses the model of isolated resonances. Each resonance is characterized by its width, the maximum distance (in the action variable) from the elliptic fixed point The overlap means that the sum of the widths of two neighboring resonances is equal to the distance between two fixed points of these isolated resonances. We start with the pendulum Hamiltonian, which describes an isolated 1 N resonance under the periodic perturbation of frequency Q ... 

The functional principle of these kind of sensors is based on the effect that was discovered by G. H. Bryan in 1890 [8]. The sensor element of this gyroscope consists of a hollow, axial symmetrical resonator, such as a ring or a cylinder. It is possible to stimulate radial oscillations, which form a kind of standing wave on the wall of the cylinder. If the cylinder rotates around its axis, the nodes of the oscillation waves remain at rest, similar to Foucaults pendulum. 

The results of the characterization of cured phthalonitrile resins are presented. This includes their thermal properties from thermal gravimetric analysis and differential scanning calorimetry, spectroscopic properties from the infra-red and nuclear magnetic resonance techniques, and mechanical properties from torsional pendulum analysis and fracture mechanics evaluation. 

Pendulums were more accurate, followed by quartz oscillators. In 1880, Pierre and Jacques Curie discovered that sending a current through a quartz crystal could result in a resonance situation with cyclic behavior, making a quartz oscillator that could be used to mark time. Because crystals grow in miniature, quartz crystal watches became the standard in the 1960s. But the periodic nature of resonant quartz excitations depends on the shape of each crystal as well as the ambient temperature and humidity. Every crystal behaves differently, and none can constitute a reputable standard for the accuracy needed by global positioning systems, for example. 

---