Frequencies of Longitudinal Vibration for a Slender Rod of Variable Section

1953 ◽  
Vol 20 (2) ◽  
pp. 173-177
Author(s):  
James L. Lubkin ◽  
Yudell L. Luke
Keyword(s):  
Asymptotic Expansions ◽  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Sectional Area ◽  
Dimensional Theory ◽  
Cross Sectional ◽  
One Dimensional ◽  
Variable Section ◽  
Fixed End

Abstract The natural frequencies of a slender, homogeneous, fixed-free rod of variable section are studied in a one-dimensional theory. The rod is uniform at the fixed end for a certain distance L1, and then tapers for a distance L2 so that the cross-sectional area varies linearly. The frequency equation is derived and its first five zeros are tabulated for several ratios of L1 to L2, and various tapers. The first few terms of asymptotic expansions for the roots of the frequency equation are presented and discussed.

A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures With Arbitrary Nonuniformities

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Numerical Approach ◽  
Mode Shapes ◽  
Cross Sectional ◽  
One Dimensional ◽  
Dirac Delta ◽  
Mass Distributions ◽  
Jump Discontinuities

Abstract One-dimensional continuous structures include longitudinal vibration of bars, torsional vibration of circular shafts, and transverse vibration of beams. Using the linear time-varying system theory, algorithms are developed in this paper to compute natural frequencies and mode shapes of these structures with nonuniform spatial parameters (mass distributions, material properties and cross-sectional areas) which can have jump discontinuities. A general numerical approach has been presented to include Dirac-delta functions and their spatial derivatives due to jump discontinuities. Numerical results are presented to illustrate the application of these techniques to the solution of different types of spatial variations of parameters and boundary conditions.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

A New Approach to Compute Natural Frequencies and Mode Shapes of Non-Uniform Continuous Bar, Circular Shaft and Beam Vibration

2019 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Linear Time ◽  
Longitudinal Vibration ◽  
Mode Shapes ◽  
Engineering Structures ◽  
Cross Sectional ◽  
Closed Form Solutions ◽  
Continuous Structure ◽  
Circular Shaft ◽  
Spatial Parameters

Abstract The wave equation governing longitudinal vibration of a bar and torsional vibration of a circular shaft, and the Euler-Bernoulli equation governing transverse vibration of a beam were developed in the eighteenth century. Natural frequencies and mode shapes are easily obtained for uniform or constant spatial parameters (cross sectional area, material property and mass distribution). But, real engineering structures seldom have constant parameters. For non-uniform continuous structure, a large number of papers have been written for more than 100 years since the publication of Kirchhoff’s memoir in 1882. There are analytical solutions only in few cases, and there are approximate numerical methods to deal with other (almost all) cases, most notably Stodola, Holzer and Myklestad methods in addition to Rayleigh-Ritz and finite element methods. This paper presents a novel approach to compute natural frequencies and mode shapes for arbitrary variations of spatial parameters on the basis of linear time-varying system theory. The advantage of this approach is that now it can be claimed that “almost” closed-form solutions are available to find natural frequencies and mode shapes of any non-uniform, linear and one-dimensional continuous structure.

Generalized Hypergeometric Function Solutions on the Transverse Vibration of a Class of Nonuniform Beams

1967 ◽  
Vol 34 (3) ◽  
pp. 702-708 ◽  
Author(s):  
Han-Chung Wang
Keyword(s):  
Transverse Vibration ◽  
Hypergeometric Functions ◽  
Beam Theory ◽  
Frequency Equation ◽  
Cantilever Beams ◽  
Sectional Area ◽  
Cross Sectional ◽  
Generalized Hypergeometric Functions ◽  
Normal Functions ◽  
Longitudinal Coordinate

With simple beam theory, solutions of normal functions for transverse vibration of a tapered beam are obtained in terms of generalized hypergeometric functions by the method of Frobenius. For the beams considered, the cross-sectional area and the area moment of inertia vary along the beam according to any two arbitrary powers of the longitudinal coordinate. The frequency equation is formulated, and the numerical results for many different tapered cantilever beams are presented.

On a One-Dimensional Theory of Finite Torsion and Flexure of Anisotropic Elastic Plates

1981 ◽  
Vol 48 (3) ◽  
pp. 601-605 ◽  
Author(s):  
E. Reissner
Keyword(s):  
Linear Theory ◽  
Beam Theory ◽  
Bending Moment ◽  
Elastic Plates ◽  
Equilibrium Equations ◽  
Dimensional Theory ◽  
Cross Sectional ◽  
One Dimensional ◽  
Slender Beams ◽  
Anisotropic Elastic

Equations for small finite displacements of shear-deformable plates are used to derive a one-dimensional theory of finite deformations of straight slender beams with one cross-sectional axis of symmetry. The equations of this beam theory are compared with the corresponding case of Kirchhoff’s equations, and with a generalization of Kirchhoff’s equations which accounts for the deformational effects of cross-sectional forces. Results of principal interest are: 1. The equilibrium equations are seven rather than six, in such a way as to account for cross-sectional warping. 2. In addition to the usual six force and moment components of beam theory, there are two further stress measures, (i) a differential plate bending moment, as in the corresponding linear theory, and (ii) a differential sheet bending moment which does not occur in linear theory. The general results are illustrated by the two specific problems of finite torsion of orthotropic beams, and of the buckling of an axially loaded cantilever, as a problem of bending-twisting instability caused by material anisotropy.

Exact Solutions for Longitudinal Vibration of Multi-Step Bars with Varying Cross-Section

1999 ◽  
Vol 122 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Q. S. Li
Keyword(s):  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Transmission Tower ◽  
Power Functions ◽  
Closed Form Solutions ◽  
High Rise ◽  
One Step ◽  
Area Variation

Using appropriate transformations, the equation of motion for free longitudinal vibration of a nonuniform one-step bar is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the area variation. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step nonuniform bar are derived and used to obtain the frequency equation of multi-step bars. The new exact approach is presented which combines the transfer matrix method and closed form solutions of one step bars. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a television transmission tower are in good agreement with the corresponding experimental data, and the selected expressions are suitable for describing the area variation of typical high-rise structures. [S0739-3717(00)00302-0]

Damping of modal perturbations in solid rocket motors

2016 ◽  
Vol 120 (1231) ◽  
pp. 1425-1445 ◽  
Author(s):  
A.S. Iyer ◽  
V.K. Chakravarthy ◽  
S. Saha ◽  
D. Chakraborty
Keyword(s):  
Computational Fluid Dynamic ◽  
Fluid Dynamic ◽  
Sectional Area ◽  
Dynamic Simulations ◽  
Cross Sectional ◽  
Solid Rocket Motors ◽  
Rocket Motors ◽  
One Dimensional ◽  
Solid Rocket ◽  
High Length

ABSTRACTQuasi-one-dimensional (quasi-1D) tools developed for capturing flow and acoustic dynamics in non-segmented solid rocket motors are evaluated using multi-dimensional computational fluid dynamic simulations and used to characterise damping of modal perturbations. For motors with high length-to-diameter ratios (of the order of 10), remarkably accurate estimates of frequencies and damping rates of lower modes can be obtained using the the quasi-1D approximation. Various grain configurations are considered to study the effect of internal geometry on damping rates. Analysis shows that lower cross-sectional area at the nozzle entry plane is found to increase damping rates of all the modes. The flow-turning loss for a mode increases if the more mass addition due to combustion is added at pressure nodes. For the fundamental mode, this loss is, therefore, maximum if burning area is maximum at the centre. The insights from this study in addition to recommendations made by Blomshield(1)based on combustion considerations would be very helpful in realizing rocket motors free from combustion instability.

Paper 4: An Investigation of Shock Wave Behaviour in Ducts with a Gradual or Sudden Enlargement in Cross-Sectional Area

1969 ◽  
Vol 184 (7) ◽  
pp. 17-27 ◽  
Author(s):  
B. E. L. Deckker ◽  
J. Gururaja
Keyword(s):  
Cross Section ◽  
Sudden Change ◽  
Stationary Wave ◽  
Finite Amplitude ◽  
Sectional Area ◽  
Wave Interactions ◽  
Cross Sectional ◽  
One Dimensional ◽  
Sudden Enlargement ◽  
Net Pressure

This paper details some investigations relating to non-stationary wave interactions and their effects on the propagation of finite amplitude waves in ducts with either a gradual or sudden change in cross-section. The influence of the angle of divergence on the wave-front and the perturbed flow behind it has been examined in different diffuser configurations using Schlieren spark photography. The variation with incident pressure of the net pressure (superimposition pressure) and the transmitted pressure are presented for incident pressure ratios up to 3·4. The results have been compared with a quasi-steady one-dimensional analysis and with the theory of Chisnell. Differences between measured and predicted results are attributed to the failure of the theories to account for the wave interactions. A complete analytical solution does not appear possible in view of the complexities of the flow and wave action, and numerical methods have to be employed.

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