Upper and Lower Bounds of Frequencies for Cantilever Bars of Variable Cross Sections

1966 ◽  
Vol 33 (4) ◽  
pp. 948-950 ◽  
Author(s):  
J. H. Gaines ◽  
Enrico Volterra
Keyword(s):  
Lower Bounds ◽  
Cross Sections ◽  
Transverse Shear ◽  
Natural Frequencies ◽  
Numerical Results ◽  
Upper And Lower Bounds ◽  
Variable Cross ◽  
Tabular Form ◽  
Rotatory Inertia ◽  
Transverse Vibrations

Upper and lower bounds of frequencies of transverse vibrations of cantilever bars of variable cross sections are presented, taking into account the effects of transverse shear and of rotatory inertia. Numerical results for the first four natural frequencies are presented in tabular form for different inertia characteristics of the bars.

The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

1952 ◽  
Vol 211 (1105) ◽  
pp. 204-224 ◽  
Keyword(s):  
Lower Bounds ◽  
Natural Frequencies ◽  
Hermitian Operator ◽  
Normal Modes ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Subsequent Paper ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Vibrating Systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.

Bandgap enhancement of periodic nonuniform metamaterial beams with inertial amplification mechanisms

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1309-1318 ◽  
Author(s):  
Shoaib Muhammad ◽  
Shuai Wang ◽  
Fengming Li ◽  
Chuanzeng Zhang
Keyword(s):  
Cross Sections ◽  
Vibration Reduction ◽  
Flexural Vibration ◽  
Experimental Methods ◽  
Numerical Results ◽  
Variable Cross ◽  
Bragg Scattering ◽  
Elastic Beams ◽  
Lower Frequencies

The aim of this study was to obtain bandgaps that are much better, that is at lower frequencies and in broader frequency ranges. Novel nonuniform metamaterial beams with periodically variable cross sections and inertial amplification mechanisms are designed and investigated by numerical and experimental methods. Flexural vibration equations of the nonuniform metamaterial beams are established, and the enhanced bandgap and vibration reduction properties are achieved by combining Bragg scattering and the inertial amplification mechanisms. Numerical results of the bandgaps for the periodic elastic beams with and without the inertial amplification mechanisms are validated by comparing them with the results of vibration experiments. Effects of the amplification mass and angle on the bandgap properties are investigated. Larger amplification mass and angle lead to much enhanced bandgap performances of the nonuniform metamaterial beams in lower to higher frequency ranges.

On the Influence of Uniform Stress States on the Natural Frequencies of Spherical Shells

1962 ◽  
Vol 29 (3) ◽  
pp. 502-505 ◽  
Author(s):  
R. R. Archer
Keyword(s):  
Spherical Shell ◽  
Compressive Stress ◽  
Shallow Shell ◽  
Natural Frequencies ◽  
Numerical Results ◽  
Uniform Stress ◽  
Spherical Shells ◽  
Transverse Vibrations ◽  
Stress States ◽  
General Frequency

The influence of uniform tensile and compressive stress states on the natural frequencies of transverse vibrations for shallow spherical shell segments is determined from the general equations for elastokinetic shallow shell problems with longitudinal inertia terms neglected. General frequency determinants are derived and detailed numerical results obtained for a range of shell geometries and stress states.

Bounds for the Natural Frequencies of a Simply Supported beam Carrying a Uniformly Distributed Axial Load

1967 ◽  
Vol 9 (2) ◽  
pp. 149-156 ◽  
Author(s):  
G. Fauconneau ◽  
W. M. Laird
Keyword(s):  
Lower Bounds ◽  
Axial Load ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Distributed Load ◽  
Made In

Upper and lower bounds for the eigenvalues of uniform simply supported beams carrying uniformly distributed axial load and constant end load are obtained. The upper bounds were calculated by the Rayleigh-Ritz method, and the lower bounds by a method due to Bazley and Fox. Some results are given in terms of two loading parameters. In most cases the gap between the bounds over their average is less than 1 per cent, except for values of the loading parameters corresponding to the beam near buckling. The results are compared with the eigenvalues of the same beam carrying half of the distributed load lumped at each end. The errors made in the lumping process are very large when the distributed load and the end load are of opposite signs. The results also indicate that the Rayleigh-Ritz upper bounds computed with the eigenfunctions of the unloaded beam as co-ordinate functions are quite accurate.

Vibration Analysis of a Cracked Beam on an Elastic Foundation

2016 ◽  
Vol 16 (05) ◽  
pp. 1550006 ◽  
Author(s):  
Ali Çağri Batihan ◽  
Fevzi Suat Kadioğlu
Keyword(s):  
Elastic Foundation ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Crack Depth ◽  
Cracked Beam ◽  
Crack Location ◽  
Transverse Vibrations ◽  
Foundation Type ◽  
Foundation Parameters ◽  
Open Edge

The transverse vibrations of cracked beams with rectangular cross sections resting on Pasternak and generalized elastic foundations are considered. Both the Euler–Bernoulli (EB) and Timoshenko beam (TB) theories are used. The open edge crack is represented as a rotational spring whose compliance is obtained by the fracture mechanics. By applying the compatibility conditions between the beam segments at the crack location and the boundary conditions, the characteristic equations are derived, from which the nondimensional natural frequencies are solved as the roots. Sample numerical results showing the effects of crack depth, crack location, foundation type and foundation parameters on the natural frequencies of the beam are presented. It is observed that the existence of crack reduces the natural frequencies, whereas the elasticity of the foundation increases the stiffness of the system and thus the natural frequencies. It is also observed that the type of elastic foundation has a significant effect on the natural frequencies of the cracked beam.

scholarly journals A classical approach to eigenvalue problems associated with a pair of mixed regular Sturm-Liouville equations I

2001 ◽  
Vol 14 (2) ◽  
pp. 205-214 ◽  
Author(s):  
M. Venkatesulu ◽  
Pallav Kumar Baruah
Keyword(s):  
Boundary Value Problems ◽  
Cross Sections ◽  
Eigenvalue Problems ◽  
Fundamental System ◽  
Variable Cross ◽  
Classical Approach ◽  
New Class ◽  
Transverse Vibrations ◽  
Acoustic Waveguides ◽  
Sturm Liouville

In the studies of acoustic waveguides in ocean, buckling of columns with variable cross sections in applied elasticity, transverse vibrations in non homogeneous strings, etc., we encounter a new class of problems of the type L1y1=−d2y1dx2+q1(x)y1=λy1 defined on an interval [d1,d2] and L2y2=−d2y2dx2+q2(x)y2=λy2 on the adjacent interval [d2,d3] satisfying certain matching conditions at the interface point x=d2.Here in Part I, we constructed a fundamental system for (L1,L2) and derive certain estimates for the same. Later, in Part II, we shall consider four types of boundary value problems associated with (L1,L2) and study the corresponding spectra.

Stability of a dislocation : Discrete model

1998 ◽  
Vol 9 (4) ◽  
pp. 373-396 ◽  
Author(s):  
A. B. MOVCHAN ◽  
R. BULLOUGH ◽  
J. R. WILLIS
Keyword(s):  
Shear Stress ◽  
Lower Bounds ◽  
Nonlinear Model ◽  
Discrete Model ◽  
Critical Shear Stress ◽  
Numerical Results ◽  
Upper And Lower Bounds ◽  
Evolution Problem ◽  
Stability Of Solutions ◽  
The Stability

An algorithm, based on a discrete nonlinear model, is presented for evaluation of the critical shear stress required to move a dislocation through a lattice. The stability of solutions of the corresponding evolution problem is analysed. Numerical results provide upper and lower bounds for the critical shear stress.

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