Vibration of Rectangular and Skew Cantilever Plates

1951 ◽  
Vol 18 (2) ◽  
pp. 129-134 ◽  
Author(s):  
M. V. Barton
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
Equal Length ◽  
Uniform Beam ◽  
Analytical Results ◽  
Modes Of Vibration ◽  
Experimental Values ◽  
Skew Angles ◽  
Plate Deflection

Abstract The Ritz method is used to determine approximate solutions for the frequencies and modes of vibration of uniform rectangular and skew cantilever plates. The functions used to represent the plate deflection correspond to those which define the normal modes of vibration of a uniform beam. Solutions are obtained for (a) uniform rectangular cantilever plates with cantilever span-to-breadth ratios of 1/2, 1, 2, and 5; (b) uniform skew cantilever plates with sides of equal length and skew angles of 15 deg, 30 deg, and 45 deg. Experimental values are presented and the correlation with analytical results discussed.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

The Normal Modes of Vibrations of Beams Having Noncollinear Elastic and Mass Axes

1940 ◽  
Vol 7 (3) ◽  
pp. A97-A105
Author(s):  
Clyne F. Garland
Keyword(s):  
Physical Properties ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Longitudinal Axis ◽  
Cantilever Beams ◽  
Amplitude Ratios ◽  
Numerical Example ◽  
Modes Of Vibration ◽  
Axis Passing

Abstract This analysis deals with vibration characteristics of cantilever beams in which the longitudinal axis, passing through the mass centers of the elementary sections, is not collinear with the longitudinal axis about which the beam tends to twist under the influence of an applied torsional couple. Expressions are derived from which the natural frequencies and normal modes of vibration of such a beam can be determined. The Rayleigh-Ritz method is employed to determine the frequencies and amplitude ratios. Following the development of the general expressions, more specific equations are derived which express the natural frequencies and relative amplitudes of motion in each of two normal modes of vibration. The theoretical relationships of the several physical properties of the beam to the natural frequencies of vibration are shown graphically. Finally a numerical example is presented for a particular beam, and the computed natural frequencies and normal modes are compared with those determined experimentally.

scholarly journals Phonon Dispersion Relations for Chromium and Tantalum

1975 ◽  
Vol 28 (1) ◽  
pp. 57 ◽  
Author(s):  
Jyoti Prakash ◽  
LP Pathak ◽  
MP Hemkar
Keyword(s):  
Force Constant ◽  
Phonon Dispersion ◽  
Normal Modes ◽  
Inelastic Neutron ◽  
Neutron Spectroscopy ◽  
Phonon Dispersion Curves ◽  
Modes Of Vibration ◽  
Experimental Values ◽  
Good Agreement ◽  
Phonon Dispersion Relations

Phonon dispersion curves for the normal modes of vibration in chromium and tantalum are calculated along the symmetry directions [100], [110] and [111] using the five force-constant model of Behari and Tripathi (1970a). The results are compared with experimental values obtained from inelastic neutron spectroscopy and reasonably good agreement is found.

scholarly journals Transverse Vibrations of Clamped and Simply-Supported Circular Plates with Two Dimensional Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 273-285 ◽  
Author(s):  
N. Bhardwaj ◽  
A.P. Gupta ◽  
K.K. Choong ◽  
C.M. Wang ◽  
Hiroshi Ohmori
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Two Dimensional ◽  
Circular Plates ◽  
Simply Supported ◽  
Modes Of Vibration ◽  
Dimensional Boundary ◽  
Special Case

Two dimensional boundary characteristic orthonormal polynomials are used in the Ritz method for the vibration analysis of clamped and simply-supported circular plates of varying thickness. The thickness variation in the radial direction is linear whereas in the circumferential direction the thickness varies according to coskθ, wherekis an integer. In order to verify the validity, convergence and accuracy of the results, comparison studies are made against existing results for the special case of linearly tapered thickness plates. Variations in frequencies for the first six normal modes of vibration and mode shapes for various taper parameters are presented.

scholarly journals An Approximate Analytical Solution of Sloshing Frequencies for a Liquid in Various Shape Aqueducts

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yuchun Li ◽  
Zhuang Wang
Keyword(s):  
Analytical Solution ◽  
Approximate Method ◽  
Approximate Analytical Solution ◽  
Ritz Method ◽  
Approximate Solutions ◽  
The Other ◽  
Engineering Applications ◽  
Tuned Liquid Dampers ◽  
Liquid Dampers ◽  
Experimental Values

An approximate analytical solution of sloshing frequencies for a liquid in the various shape aqueducts is formulated by using the Ritz method. The present approximate method is, respectively, applied to find the sloshing frequencies of the liquid in rectangular, trapezoid, oval, circular, U-shaped tanks (aqueducts), and various shape tuned liquid dampers (TLD). The first three antisymmetric and symmetric frequencies by the present approach are within 5% accuracy compared to the other analytical, numerical, and experimental values. The approximate solutions of this paper for the various shape aqueducts are acceptable to the engineering applications.

Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

1968 ◽  
Vol 72 (688) ◽  
pp. 341-344 ◽  
Author(s):  
B. Dawson
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Rotary Inertia ◽  
Characteristic Functions ◽  
Cantilever Beams ◽  
Dynamic Displacement ◽  
Modes Of Vibration ◽  
Good Agreement

Summary The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

scholarly journals Preparation and the Microwave and Infrared Spectra of Vinyl Ketene

1979 ◽  
Vol 34 (11) ◽  
pp. 1269-1274 ◽  
Author(s):  
Erik Bjarnov
Keyword(s):  
Dipole Moment ◽  
Gas Phase ◽  
Infrared Spectra ◽  
Room Temperature ◽  
Normal Modes ◽  
Spectroscopic Constants ◽  
Electrical Dipole ◽  
Electrical Dipole Moment ◽  
Vacuum Pyrolysis ◽  
Modes Of Vibration

Vinyl ketene (1,3-butadiene-1-one) has been synthesized by vacuum pyrolysis of 3-butenoic 2-butenoic anhydride. The microwave and infrared spectra of vinyl ketene in the gas phase at room temperature have been studied. The trans-rotamer has been identified, and the spectroscopic constants were found to be Ã= 39571(48) MHz, B̃ = 2392.9252(28) MHz, C̃ = 2256.0089(28) MHz, ⊿j = 0.414(31) kHz, and ⊿JK = - 34.694(92) kHz. The electrical dipole moment was found to be 0.987(23) D with μa = 0.865(14) D and μb = 0.475(41) D. A tentative assignment has been made for 17 of the 21 normal modes of vibration

Normal modes of vibration of a model peptide adopting the C7 C5 structure

2009 ◽  
Vol 24 (6) ◽  
pp. 543-552 ◽  
Author(s):  
P. LAGANT ◽  
G. VERGOTEN ◽  
G. FLEURY ◽  
M.H. LOUCHEUX-LEFEBVRE
Keyword(s):  
Normal Modes ◽  
Model Peptide ◽  
Modes Of Vibration

Localization of Vibration in Disordered Multi-Span Beams With Damping

1993 ◽  
Author(s):  
Djamel Bouzit ◽  
Christophe Pierre
Keyword(s):  
Weak Coupling ◽  
Normal Modes ◽  
Structural Damping ◽  
Coupling Case ◽  
Effects Of Disorder ◽  
Transfer Matrix Approach ◽  
Modes Of Vibration ◽  
Comparable Magnitude ◽  
Localization Of Vibration ◽  
Statistical Perturbation

Abstract The combined effects of disorder and structural damping on the dynamics of a multi-span beam with slight randomness in the spacing between supports are investigated. A wave transfer matrix approach is chosen to calculate the free and forced harmonic responses of this nearly periodic structure. It is shown that both harmonic waves and normal modes of vibration that extend throughout the ordered, undamped beam become spatially attenuated if either small damping or small disorder is present in the system. The physical mechanism which causes this attenuation, however, is one of energy dissipation in the case of damping but one of energy confinement in the case of disorder. The corresponding rates of spatial exponential decay are estimated by applying statistical perturbation methods. It is found that the effects of damping and disorder simply superpose for a multi-span beam with strong interspan coupling, but interact less trivially in the weak coupling case. Furthermore, the effect of disorder is found to be small relative to that of damping in the case of strong interspan coupling, but of comparable magnitude for weak coupling between spans. The adequacy of the statistical analysis to predict accurately localization in finite disordered beams with boundary conditions is also examined.

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