Free vibration methods

The new instrument introduced for inspection of multi-layer structures from polymeric and composite metals and materials in air-space industry and this is acoustic flaw detector AD-64M. The principle of its operation based on impedance and free vibration methods with further spectral processing of the obtained signal. 

Free- Vibration Methods. Free-vibration instmments subject a specimen to a displacement and allow it to vibrate freely. The oscillations are monitored for frequency and damping characteristics as they disappear. The displacement is repeated again and again as the specimen is heated or cooled. The results are used to calculate storage and loss modulus data. The torsional pendulum and torsional braid analy2er (TBA) are examples of free-vibration instmments. 

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. 

Free vibration methods such as the torsion pendulum are covered by ISO 4663 and are limited to cry low strains and frequencies, and are in much less frequent use these days than the forced vibration nonresonant systems on which this chapter will focus. The early Du Pont DMA and German Myrenne used input energy to maintain the resonant oscillation amplitude, but the main limitations were variable frequency according to the sample size (which had to be glassy or plastic) or one frequency only (1 Hz) respectively. 

Free vibration methods suffer from the disadvantage that the frequency of vibration depends on the stiffness of the specimen, which varies with temperature, so that forced vibration methods are to be preferred when the frequency and temperature dependence of viscoelastic behaviour are to be investigated. 

Free vibration and forced vibration methods have been employed to determine the respective data. Free vibration was common earlier in polymer blend studies, but forced vibration data is much more common today due to the availability of more sophisticated testing equipment. Free vibration methods include the torsion pendulum, the vibrating reed and the torsional braid analyzer. The torsion pendulum is comprised of an inertial source (disk or rod), which can freely vibrate and is attached to a specimen, which is rigidly fixed at one end. Upon angular deformation of the inertial source and releasing, a damped sinusoidal curve depicts the resultant deformation of the sample [18,19]. Tan S can be calculated from... 

Table 2 Equations for Dynamic Moduli from Free and Resonance Vibrations Method and specimen Modulus (dyn/cm2)...

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. 

The method of measurement of free vibration frequency gives average... 

Test methods where there is free vibration applied and then allowed to decay. 

E /y and tan 8 were measured using a free-free beam method. The specimens were susp>ended horizontally with an iron piece at both ends by two fine threads at the nodal points. They were excited by an electromagnetic driver at one end, while the vibration was detected by an electromagnetic... 

Much attention has also been devoted to modal identification without measuring the input time history. In particular, a lot of effort has been dedicated to the case of free vibration (or impulse response) and to the case of ambient vibration. In the former case, often time-domain methods based on auto-regressive moving average (ARMA) models are employed, using least squares as the core ingredient in their formulations. However, it was found that the least-squares method yields biased estimates [76], A number of methods have been developed to eliminate this bias, including the instrumental matrix with delayed observations method [76], the correlation fit method [275], the double least-squares method [114,202] and the total least-squares method [92]. A detailed comparison of these methods can be found in Cooper [61],... 

ES, resonance electrostatic method FO, forced oscillation dynamic-mechanical analysis FV, free vibration TP, torsion pendulum TSC, thermally stimulated discharge current measurement D, dielectric VR, vibrating reed. 

All the major peak or resonanee frequencies obtained are also listed and compared with the nominal resonances and the resonances by the FEM analysis in Table 3.1. Generally speaking, the peak frequencies determined by three different methods are in reasonable agreement. In the case of the nominal resonance frequencies, no boundary conditions are assigned. The values of resonances are listed as possible corresponding values. In the cylindrical element, the effect of fixed boundary is minor, because the small area at the bottom is only constraint. It is noted that the resonance frequency lower than the compression mode is obtained in the case of fixed boimdary, which is close to the resonance of the shear mode. In the disk-shaped element, lower resonance frequencies than that of the radial mode is obtained in both the FEM analysis and the experiment in the case of free vibration. These disapper in the case of fixed boimdary, where the resonance frequency corresponds to the compression mode obtained in the FEM analysis. In the conical element, higher resonance frequencies are obtained in the case of fixed boundary. According to Fig. 3.6, the responses of these resonance frequencies are fairly weak. 

The gas phase free energy (AGggs) contribution is computed by quantum mechanical methods. Zero-point (A (0)) and vibrational (AE vib)) energies and entropies should be included in the gas phase free energies to obtain accurate results. The SFE (AAGsqi ) contribution is obtained by the use of free energy methods in aqueous environment. ... 

Nardini, D., Brebbia, C. A., in New Approach for Free Vibration Analysis Using Boundary Elements Boundary Element Methods in Engineering, Springer Verlag (1982)... 

For certain types of non linear dynamical systems subject to stochastic loading, a method has been proposed based on the free-vibration response of the structure (Koo et al. 2005). In several cases, the output of this method will be a rough approximation of the design point which can be used as a starting point (a so-called warm solution) for a specific optimization algorithm. 

Because of the frequency dependence of Tg, the convention adopted for assignment of the glass transition temperature is an important consideration. Traditionally, a frequency of 1 Hz has been used as a standard value. This is based on the historic precedence, since the torsion pendulum was the most widely used DMA technique in the early days of viscoelastic property measurements. The torsion pendulum is a free vibration technique with a natural frequency of approximately 1 Hz. The 1 Hz value also is reasonably close (within 10 °C) to the Tg values determined by other widely used methods such as DSC, dilatometry, and TMA. The relation between DMA and DSC Tg values is considered further at the end of this chapter (Sircar and Drake 1990). Because of the ambiguity inherent in the kinetic nature of Tg, it is most important that the test frequency be reported along with any Tg value determined by a DMA technique. 

Dynamic viscoelastic measurements are useful in studying the structure of polymers, because these mechanical properties are sensitive to glass transition, crystallinity, cross-linking, filling systems (filler or plasticizer), molecular aggregation, and phase separation. To determine dynamic viscoelastic properties, such as storage modulus, loss modulus, and tan, various methods have been proposed, and recently many types of instruments are commercially available. Typical methods to measure the dynamic viscoelasticity are classified into three categories damped free vibration, resonance free vibration, and nonresonance forced vibration. These methods are standardized by the international standard ISO 6721 [3]. 

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