Free Vibration Method Research Articles

Modal Method for Free Vibration of Oval Cylindrical Shells With Simply Supported or Clamped Ends

The free vibration of an oval cylindrical shell of finite length was investigated with the aid of the kinematic relations of the first-order shell theory of Sanders. Transverse and in-plane inertia terms were retained throughout. A method incorporating a type of eigen-function expansion into Hamilton’s principle, was developed and found to be far more convenient than a parallel Fourier analysis. In addition to the determination of the natural frequencies and deformation characteristics, attention was focused on the influence of various types of simple support and clamped end conditions. Two modes of deformation corresponding to a “higher” and a “lower” frequency were observed for every pair of axial and circumferential wave numbers, depending upon the degree of circumferential symmetry in the deformation pattern.

Viscoelastic behavior of deformed polytetrafluoroethylene

The effect of pressure and heat treatment on the viscoelastic properties of polytetrafluoroethylene has been studied by the free torsional vibration method. It is shown that the deformation of PTFE produces an important change in the principal relaxation processes associated with the molecular mobility in the amorphous regions. On the basis of the measurements made it is concluded that the mechanical strength of the PTFE increases, which is in good agreement with the results of static tests. A conclusion is drawn concerning the molecular mechanism of the processes associated with the deformation of PTFE.

Molecular mobility in polyvinylfluoride and polyvinylidenefluoride

The method of free torsional vibrations has been used to study molecular mobility in polyvinylfluoride. The α-, ß- and γ-relaxation processes have been observed and investigated. An analogy has been established between relaxation processes and molecular mobility in polyvinylfluoride and the similar processes in polyethylene at low temperatures and polytetrafluorethylene at high temperatures. Unusual melting characteristics have been found, that are characterized by unusually high values of the tangent of the mechanical loss angle and by a minimum on the temperature dependence of the low frequency velocity of sound. The rate of propagation of longitudinal and shear waves in polyvinylidenefluoride has been measured at frequencies of one and five MHz. Low temperature relaxation processes have been discovered and investigated close to liquid helium temperature. The principal viscoelastic characteristics of polyvinylidenefluoride at low temperatures have been calculated.

Finite element displacement method for large amplitude free flexural vibrations of beams and plates

A finite element method to determine the nonlinear frequency of beams and plates for large amplitude free vibrations is presented. The equation of motion is characterized by the basic stiffness, mass, geometrical stiffness and the associated inplane force matrices. The procedures for solving the system equations of motion are discussed, and the explicit formulations of the geometrical stiffness and inplane force matrices of a rectangular plate element are given. Examples of large amplitude free oscillations for rectangular plates and beams with various boundary conditions are given. Characteristics of convergence are investigated. In all the cases where comparisons with previous investigations are made, good agreement has been obtained. It indicates that the present method will give results entirely adequate for engineering purposes.

Effect of low molecular weight compounds on the relaxation behavior of poly(2‐hydroxyethyl methacrylate) in the glassy state and in the transition region from the glassy to the rubberlike state

AbstractThe method of free torsional vibrations was used to determine the temperature dependence of the storage and loss shear modulus of poly(2‐hydroxyethyl methacrylate) samples swollen with ethylene glycol, formamide, n‐propanol, and water. The measurements included the glassy region (starting with temperatures from −130 to −190°C) and the main (α) transition from the glassy to the rubberlike state. At a volume fraction of the low molecular weight compound vd > 0.2, the above systems exhibit, besides the α dispersion, only the secondary (βSW) dispersion, which is generally attributed to the relaxation motions of the hydroxyethyl groups of the side chains interacting with molecules of the diluent. If no separation of the diluent in a second phase occurs at the measurement temperatures, the temperatures of both dispersions decrease with increasing vd and approach the glass transition temperature of the low molecular weight compound. The concentration dependences of the dispersion temperatures were described by an equation derived elsewhere for the concentration dependence of the glass transition temperature. The results indicate that molecules of the diluent contribute significantly to the intensity of the βSW transition and simultaneously affect its limiting temperatures (for vd = 1 and vd = 0). Specific differences among the systems described above appear only at those temperatures where same of the low molecular weight compound separates into a crystalline or glassy phase.

Measurement of the Attenuation Constant of Sound Waves in Rock

A laboratory measurement of the attenuation constant of sound waves travelling through rock was carried out. The so-called free vibration method was employed for this purpose. The travelling and oscillating longitudinal sound waves were excited in a long boring-core specimen of rock by means of an impact of small steel ball tied at one end of the specimen which was horizontally suspended by strings. The fundamental frequency of the oscillating waves was decided by the length of the bar. Observation was also enacted of the damped free oscillation of the sound waves.The reasonable attenuation constants and Q values have been obtained from several rock specimens. The Q's which have been directly calculated from the attenuation constants are substantially independent of frequency, but the Q of the Kuzuu (in Tochigi Prefecture) limestone is evidently dependent on the wave frequency.

Dynamic-elastic investigation of the chemical denaturation of collagen fibers.

AbstractThe dynamic elastic behavior of collagen fibers treated by LiBr solutions was studied by the method of free longitudinal vibrations. The frequency response functions and the stress–strain relationship were evaluated for fibers denatured to different extents by various concentrations of the salt solution. The James and Guth model for rubber elasticity was applied to the experimental data. The elastometric parameter β, which is a measure of the degree of folding of the macromolecular chains, was found to decrease on increasing the salt concentration. It might thus serve as a characteristic of the degree of denaturation of fibrillar proteins.

Secondary (β) Relaxation Process of Alkaline Polycaprolactam Swollen by Low Molecular Weight Substances

AbstractBy using the method of free torsion vibrations, a detailed dependence was determined of the temperature position and the height of the secondary (β) loss maximum of alkali‐polymerized polycaprolactam upon the concentration of water in the range of 0 < PA < 1.3, where PA is the number of moles of water per mole of monomer units of the portions accessible to the sorption of water. The increase in the secondary loss maximum and the decrease in the height of the low‐temperature loss maximum due to the effect of water in the concentration range 0 < PA < 0.5 is interpreted as consequence of a transformation of the corresponding relaxation processes; subsequent increase in the water content then exhibits only little influence on the height of both the maxima. The β relaxation process is attributed to kinetic units containing amide groups which, owing to interaction with molecules of water, are more firmly bonded to neighboring molecules than the amide groups of a dry polymer. The secondary loss maximum arises also because of the effect of caprolactam monomer and of some low molecular weight amides, being markedly shifted to higher temperatures with the decreasing molecular weight of the amide.

Beziehungen zwischen den dynamischen Eigenschaften und der Schlagzähigkeit der Mischungen des Polyvinylchlorids mit Kautschuk

AbstractMeasurements of dynamic properties of a series of polyvinylchloride‐rubber blends at temperatures of −60 to +100°C and frequencies of 0.3–3.0 Hz have been carried out by means of the free vibration method. The dynamic data have been compared to the results of impact resistance Charpy‐experiments. The linear correlation obtained between the dynamic shear modulus at −10°C (G′−10) and the impact resistance at 20°C indicated that the value G′−10 has to be decreased compared to that of unmodified PVC to increase the resistance of compatible as well as incompatible PVC‐rubber blends. The impact resistance increases with higher decrease of G′−10. The correlation given and the breaking characteristics of the samples investigated show that, ductile fracture, relieving the deformational stress mainly by developing cracks, occurs in blends with G′−10 in the range of 12.0–7.5 × 108 N/m2. This is valid for compatible as well as incompatible PVC rubber blends.