The Temperature Coefficients of Shear and Longitudinal Modes of Vibration

1936 ◽  
Vol 7 (9) ◽  
pp. 339-341 ◽  
Author(s):  
Harold Osterberg
Keyword(s):  
Longitudinal Modes ◽  
Temperature Coefficients ◽  
Modes Of Vibration

Development of 2DOF Ultrasonic Vibration Cutting Device for Ultraprecision Elliptical Vibration Cutting

2010 ◽  
Vol 447-448 ◽  
pp. 164-168 ◽  
Author(s):  
Toshimichi Moriwaki
Keyword(s):  
Ultrasonic Vibration ◽  
Design Principle ◽  
Elliptical Vibration Cutting ◽  
Vibration Cutting ◽  
Longitudinal Modes ◽  
Elliptical Vibration ◽  
Good Surface ◽  
Tool Edge ◽  
Modes Of Vibration ◽  
Ultrasonic Vibration Cutting

A new ultrasonic vibration cutting device for ultraprecision elliptical vibration cutting has been developed. The elliptical vibration cutting device developed utilizes combination of bending and longitudinal modes of vibration of the stepped vibrator to generate circular or elliptical vibration locus at the cutting tool edge. The design principle as well as the structure and the performance of the elliptical vibration cutting device is introduced here. The experimental results of diamond cutting of Co-Cr-Mo Alloy proved that good surface quality with roughness of 20 nmPV can be obtained stably with the elliptical vibration cutting device developed.

DISPERSION RELATIONS FOR PHONONS IN MAGNESIUM FLUORIDE

1967 ◽  
Vol 45 (9) ◽  
pp. 3079-3090 ◽  
Author(s):  
R. S. Katiyar ◽  
R. S. Krishnan
Keyword(s):  
Secular Equation ◽  
Normal Modes ◽  
Theoretical Method ◽  
Dispersion Relations ◽  
Magnesium Fluoride ◽  
Ionic Character ◽  
Axially Symmetric ◽  
Longitudinal Modes ◽  
Modes Of Vibration ◽  
Crystallographic Axes

The lattice dynamics of magnesium fluoride have been investigated theoretically. The group theoretical method is used to decompose the dynamical secular equation at the center of the Brillouin zone. In order to get the expressions for all the infrared-active transverse and longitudinal modes, it is necessary to decompose the secular equation for the wave vector approaching zero along the two nonequivalent crystallographic axes, say y and z, of the crystal. Analysis of the results shows that a rigid ion model of the crystal with axially symmetric forces can provide qualitative agreement with the observed frequencies appearing in Raman and infrared absorption. The ions in the crystal are found to retain 70% of their charges, showing thereby the ionic character of the crystal. The dispersion relations for normal modes of vibration propagating along the (0 0 qz) and (0 qy 0) directions have been evaluated.

Dynamic properties of timber school buildings

1971 ◽  
Vol 61 (4) ◽  
pp. 961-974
Author(s):  
Dixon Rea ◽  
Jack G. Bouwkamp
Keyword(s):  
Dynamic Properties ◽  
Mode Shapes ◽  
School Buildings ◽  
Test Results ◽  
Dynamic Tests ◽  
Resonant Frequencies ◽  
Longitudinal Modes ◽  
Santa Clara County ◽  
Modes Of Vibration ◽  
Mass Type

abstract Dynamic tests were conducted on four buildings of the old McKinley School in Santa Clara County, California. The buildings, single-story timber structures, were vibrated by means of an eccentric-mass type vibration generator to determine their resonant frequencies, associated mode shapes, and damping factors. The test results from two of the buildings are used to illustrate the dynamic behavior of such structures. Typically, the buildings had three basic modes of vibration that have been designated transverse, longitudinal, and flexure of the roof diaphragm. The resonant frequencies of transverse and longitudinal modes ranged from 7 to 10 cps, and the damping factors varied from 3 to 4 per cent. The resonant frequencies of the flexure modes of the roof diaphragms ranged from 6 to 10 cps and the damping factors from 1 to 3 per cent.

scholarly journals On Kaufmann's theory of the impact of the pianoforte hammer

1920 ◽  
Vol 97 (682) ◽  
pp. 99-110 ◽  
Keyword(s):  
Initial Conditions ◽  
Practical Importance ◽  
Prima Facie ◽  
Point Of View ◽  
Infinite Length ◽  
Modes Of Vibration ◽  
Rigorous Treatment ◽  
The Subject ◽  
Set Up ◽  
The Impact

The theory of the vibrations of the pianoforte string put forward by Kaufmann in a well-known paper has figured prominently in recent discussions on the acoustics of this instrument. It proceeds on lines radically different from those adopted by Helmholtz in his classical treatment of the subject. While recognising that the elasticity of the pianoforte hammer is not a negligible factor, Kaufmann set out to simplify the mathematical analysis by ignoring its effect altogether, and treating the hammer as a particle possessing only inertia without spring. The motion of the string following the impact of the hammer is found from the initial conditions and from the functional solutions of the equation of wave-propagation on the string. On this basis he gave a rigorous treatment of two cases: (1) a particle impinging on a stretched string of infinite length, and (2) a particle impinging on the centre of a finite string, neither of which cases is of much interest from an acoustical point of view. The case of practical importance treated by him is that in which a particle impinges on the string near one end. For this case, he gave only an approximate theory from which the duration of contact, the motion of the point struck, and the form of the vibration-curves for various points of the string could be found. There can be no doubt of the importance of Kaufmann’s work, and it naturally becomes necessary to extend and revise his theory in various directions. In several respects, the theory awaits fuller development, especially as regards the harmonic analysis of the modes of vibration set up by impact, and the detailed discussion of the influence of the elasticity of the hammer and of varying velocities of impact. Apart from these points, the question arises whether the approximate method used by Kaufmann is sufficiently accurate for practical purposes, and whether it may be regarded as applicable when, as in the pianoforte, the point struck is distant one-eighth or one-ninth of the length of the string from one end. Kaufmann’s treatment is practically based on the assumption that the part of the string between the end and the point struck remains straight as long as the hammer and string remain in contact. Primâ facie , it is clear that this assumption would introduce error when the part of the string under reference is an appreciable fraction of the whole. For the effect of the impact would obviously be to excite the vibrations of this portion of the string, which continue so long as the hammer is in contact, and would also influence the mode of vibration of the string as a whole when the hammer loses contact. A mathematical theory which is not subject to this error, and which is applicable for any position of the striking point, thus seems called for.

Axial guided wave characteristics in functionally graded one-dimensional hexagonal piezoelectric quasi-crystal cylinders

2021 ◽  
pp. 108128652110134
Author(s):  
B. Zhang ◽  
X.H. Wang ◽  
L. Elmaimouni ◽  
J.G. Yu ◽  
X.M. Zhang
Keyword(s):  
Coupling Effect ◽  
Energy Conversion Efficiency ◽  
Guided Wave ◽  
Functionally Graded ◽  
Phonon Modes ◽  
One Dimensional ◽  
Longitudinal Modes ◽  
Wave Characteristics ◽  
Piezoelectric Effects ◽  
Hollow Cylinders

In one-dimensional hexagonal piezoelectric quasi-crystals, there exist the phonon–phason, electro–phonon, and electro–phason couplings. Therefore, the phonon–phason coupling and piezoelectric effects on axial guided wave characteristics in one-dimensional hexagonal functionally graded piezoelectric quasi-crystal (FGPQC) cylinders are investigated by utilizing the Legendre polynomial series method. The dispersion curves and cut-off frequencies are illustrated. Wave characteristics in three hollow cylinders with different quasi-periodic directions are comparatively studied. Some new wave phenomena are revealed: the phonon–phason coupling and piezoelectric effects on the longitudinal and torsional phonon modes ( N = 0) vary as the quasi-periodic direction changes; the phonon–phason coupling effect on flexural–torsional modes in the r-, z-FGPQC hollow cylinders, and on flexural–longitudinal modes in ϑ-FGPQC hollow cylinders increases as N increases. The corresponding results obtained in this work lay the theoretical foundation for the design and manufacture of piezoelectric transducers with high resolution and energy-conversion efficiency.

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