A systematic relaxation method for the determination of the normal modes and the natural frequencies of vibrating systems

The determination of the natural frequencies and normal modes of vibration for continuous panels, representing more or less typical fuselage skin-panel construction for modern airplanes, is discussed in this paper. The time-dependent boundary conditions at the supporting stringers are considered. A numerical example is presented, and analytical results for a particular structural configuration agree favorably with available experimental measurements.

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

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1952.0034 ◽

1952 ◽

Vol 211 (1105) ◽

pp. 204-224 ◽

Cited By ~ 29

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.

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Vibrations of a Rectangular Plate With Concentrated Mass, Spring, and Dashpot

Journal of Applied Mechanics ◽

10.1115/1.3630102 ◽

1963 ◽

Vol 30 (1) ◽

pp. 31-36 ◽

Cited By ~ 25

Author(s):

Y. C. Das ◽

D. R. Navaratna

Keyword(s):

Rectangular Plate ◽

Composite System ◽

Natural Frequencies ◽

Normal Modes ◽

Double Series ◽

Concentrated Mass ◽

Orthogonal Functions ◽

Isotropic Rectangular Plate ◽

Mass Spring

An analytical method, developed by Young [1], is here extended to the determination of the natural frequencies of a composite system which consists of an isotropic rectangular plate with a concentrated mass, spring, and dashpot attached at any point of the plate. This method makes use of a double series expansion in terms of two sets of orthogonal functions which represent the normal modes of the vibration of the plate alone. Numerical examples for a square plate with (a) a combined concentrated mass and spring, (b) two concentrated masses, and (c) an attached dashpot have been presented.

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Statistical determination of normal modes

Journal of Chemical Education ◽

10.1021/ed064p425 ◽

1987 ◽

Vol 64 (5) ◽

pp. 425 ◽

Cited By ~ 6

Author(s):

John F. Geldard ◽

Lawrence R. Pratt

Keyword(s):

Normal Modes ◽

Statistical Determination

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Studies on Determination of Natural Frequencies of Industrial Turbine Blades

Volume 3B: 15th Biennial Conference on Mechanical Vibration and Noise — Acoustics, Vibrations, and Rotating Machines ◽

10.1115/detc1995-0514 ◽

1995 ◽

Author(s):

Mahesh M. Bhat ◽

V. Ramamurti ◽

C. Sujatha

Keyword(s):

Steam Turbine ◽

Dynamic Behavior ◽

Turbine Blade ◽

Natural Frequencies ◽

Complex Structure ◽

Turbine Blades ◽

Blade Root ◽

Steam Turbine Blade ◽

Manufacturing Tolerances

Abstract Steam turbine blade is a very complex structure. It has geometric complexities like variation of twist, taper, width and thickness along its length. Most of the time these variations are not uniform. Apart from these geometric complexities, the blades are coupled by means of lacing wire, lacing rod or shroud. Blades are attached to a flexible disc which contributes to the dynamic behavior of the blade. Root fixity also plays an important role in this behavior. There is a considerable variation in the frequencies of blades of newly assembled turbine and frequencies after some hours of running. Again because of manufacturing tolerances there can be some variation in the blade to blade frequencies. Determination of natural frequencies of the blade is therefore a very critical job. Problems associated with typical industrial turbine bladed discs of a 235 MW steam turbine are highlighted in this paper.

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Effect of the Material Types on the Fundamental Frequencies of Uniaxial Composite Conical Springs

10.1115/imece1999-1185 ◽

1999 ◽

Author(s):

Vebil Yildirim ◽

Erol Sancaktar ◽

Erhan Kiral

Keyword(s):

Transfer Matrix Method ◽

Pitch Angle ◽

Natural Frequencies ◽

Axial Deformation ◽

Helical Springs ◽

Fundamental Frequencies ◽

Helix Pitch ◽

Constant Pitch ◽

Α Helix

Abstract This paper deals with the effect of the material types (Graphite-Epoxies and Kevlar-Epoxy) on the fundamental frequencies of uniaxial constant-pitch composite conical helical springs with solid circle section and fixed-fixed ends. The transfer matrix method is used for the determination of the fundamental natural frequencies. The rotary inertia, the shear and axial deformation effects are taken into account in the solution. The free vibrational charts for each material presented in this study cover the following vibrational parameters: n (number of active turns) = 5–10, α = (helix pitch angle) = 5° and 25°, R2/R1, (minimum to maximum radii of the cylinder) = 0.1 and 0.9, and Dmax/d (maximum cylinder to wire diameters) = 5 and 15. These charts can be used for the design of uniaxial composite conical springs.

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Erratum: Plasma‐Filled Waveguide with Axial Magnetization. I. Variational Determination of Normal Modes

Journal of Applied Physics ◽

10.1063/1.1657342 ◽

1969 ◽

Vol 40 (12) ◽

pp. 4999-4999

Author(s):

K. S. Champlin ◽

G. H. Glover ◽

D. E. O'Connor

Keyword(s):

Normal Modes

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The Natural Frequencies of Free and Constrained Non-Uniform Beams

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100073703 ◽

1960 ◽

Vol 64 (599) ◽

pp. 697-699 ◽

Cited By ~ 2

Author(s):

R. P. N. Jones ◽

S. Mahalingam

Keyword(s):

Degrees Of Freedom ◽

Natural Frequencies ◽

Ritz Method ◽

Normal Modes ◽

Iteration Process ◽

Conservative System ◽

Basic System ◽

Orthogonal Functions ◽

Added Masses ◽

The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

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High-Fidelity Sensitivity Analysis of Modal Properties of Mistuned Bladed Disks Regarding Material Anisotropy

Volume 7C: Structures and Dynamics ◽

10.1115/gt2018-76572 ◽

2018 ◽

Cited By ~ 1

Author(s):

Adam Koscso ◽

Guido Dhondt ◽

E. P. Petrov

Keyword(s):

Sensitivity Analysis ◽

Finite Element ◽

Natural Frequencies ◽

Mode Shapes ◽

Finite Element Models ◽

Coordinate Systems ◽

Bladed Disks ◽

Bladed Disk ◽

Material Orientation

A new method has been developed for sensitivity calculations of modal characteristics of bladed disks made of anisotropic materials. The method allows the determination of the sensitivity of the natural frequencies and mode shapes of mistuned bladed disks with respect to anisotropy angles that define the crystal orientation of the monocrystalline blades using full-scale finite element models. An enhanced method is proposed to provide high accuracy for the sensitivity analysis of mode shapes. An approach has also been developed for transforming the modal sensitivities to coordinate systems used in industry for description of the blade anisotropy orientations. The capabilities of the developed methods are demonstrated on examples of a single blade and a mistuned realistic bladed disk finite element models. The modal sensitivity of mistuned bladed disks to anisotropic material orientation is thoroughly studied.

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Determination of the complex frequencies for the normal modes below 1mHz after the 2010 Maule and 2011 Tohoku earthquakes

Annals of Geophysics ◽

10.4401/ag-6400 ◽

2014 ◽

Vol 56 (5) ◽

Author(s):

Hao Ding ◽

Wen-Bin Shen

Keyword(s):

Time Series ◽

Normal Modes ◽

Power Spectra ◽

Tohoku Earthquake ◽

Superconducting Gravimeter ◽

Width Ratio ◽

Maule Earthquake ◽

Attenuation Models ◽

First Time

Based upon SG (superconducting gravimeter) records, the autoregressive method proposed by Chao and Gilbert [1980] is used to determine the frequencies of the singlets of seven spheroidal modes (0S2, 2S1, 0S3, 0S4, 1S2, 0S0, and 3S1) and the degenerate frequencies of three toroidal modes (0T2, 0T3, and 0T4) below 1 mHz after two recent huge earthquakes, the 2010 Mw8.8 Maule earthquake and the 2011 Mw9.1 Tohoku earthquake. The corresponding quality factor Qs are also determined for those modes, of which the Qs of the five singlets of 1S2 and the five singlets (m=0, m=±2, and m=±3) of 0S4 are estimated for the first time using the SG observations. The singlet m=0 of 3S1 is clearly observed from the power spectra of the SG time series without using other special spectral analysis methods or special time series from pole station records. In addition, the splitting width ratio R of 3S1 is 0.99, and consequently we conclude that 3S1 is normally split. The frequencies and Qs of the modes below 1mHz may contribute to refining the 3D density and attenuation models of the Earth.

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