The Determination of Vibration Modes

1949 ◽  
Vol 53 (468) ◽  
pp. 1095-1099
Author(s):  
N. F. Harpur
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Dynamic Loads ◽  
Vibration Modes ◽  
Resonance Modes ◽  
Modes Of Vibration

At some stage in the design of every aeroplane it is necessary to estimate or to measure the resonance modes of vibration. This has not always been the case, but the problems of flutter, control reversal and dynamic loads have increased in importance as speeds have risen. Nowadays, it is an airworthiness requirement that these effects be considered and the aircraft made safe for all conditions of flight. A knowledge of the normal modes of vibration is essential for all accurate estimates of these aeroelastic effects.Taking flutter as an example, the technique of flutter investigations consists of first determining which combinations of the various possible degrees of freedom are liable to excite dangerous oscillations. Typical degrees of freedom for a wing are bending and twist in each normal mode, aileron deflection and tab deflection; for a tailplane and elevator we might consider tailplane bending or twist, elevator deflection, tab deflection, fuselage bending and twist, and pitching of the whole aeroplane.

A Method of Calculating Space Free Resonance Modes of an Aircraft

1953 ◽  
Vol 57 (510) ◽  
pp. 420-422 ◽  
Author(s):  
B. A. Hunn
Keyword(s):  
Free Space ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Characteristic Root ◽  
Frequency Range ◽  
Resonance Modes ◽  
Characteristic Roots ◽  
Modes Of Vibration ◽  
4 Degrees Of Freedom

The problem of deducing resonance modes of vibration of an aircraft in free space is a concomitant of flutter calculations if the number of degrees of freedom used is to be small. When the structure is complex in that it involves wings, fuselage and tailplane, each of which possesses infinitely many normal modes, it becomes apparent that the number of point masses which must be considered, in constructing a dynamical equivalent to give a sufficient coverage of the frequency range in which flutter is likely, is very large. For example, it may be necessary to use four rods per wing and per tailplane and four point masses on the fuselage. This would involve 8 + 8 + 4 degrees of freedom, and if the usual technique of characteristic roots is used it would be necessary to consider a characteristic root matrix of order 20 × 20.

Applications of Infrared Methods in the Structural Examination of Synthetic Rubber

1946 ◽  
Vol 19 (4) ◽  
pp. 1113-1123 ◽  
Author(s):  
J. E. Field ◽  
D. E. Woodford ◽  
S. D. Gehman
Keyword(s):  
Absorption Spectra ◽  
Organic Molecules ◽  
Normal Modes ◽  
Structural Variations ◽  
Structural Examination ◽  
Absorption Bands ◽  
Synthetic Rubber ◽  
Modes Of Vibration ◽  
Long Chain

Abstract Infrared absorption spectra have been long recognized as a convenient means for studying the structure of organic molecules. The interpretations of the spectra are based on the energy interactions of the molecule and the radiations which arise from the vibration of the constituent atoms and molecular rotations. For simple or highly symmetrical molecules, the determination of the normal modes of vibration and the calculation of the absorbing frequencies are relatively simple and straightforward. For more complicated organic molecules, this becomes increasingly difficult because with each additional atom, the number of degrees of freedom is increased by three and the determination of the normal modes of vibration becomes practically impossible. However, interpretations can be made to a useful extent through empirical comparisons with the absorption spectra of simpler known structures. The data that have been accumulated by investigators in this field have made it possible to assign rather definite absorption frequencies to some of the chemical linkages and functional groups. These correlations which have appeared in numerous places in the literature are partially reproduced in Table I. Organic compounds generally have strong absorption bands below 1300 cm−1, to which few definite assignments can be made with certainty because the vibrations of many of the atoms of the molecule may be involved rather than a specific part of it. It is clear that such empirical relationships must be relied upon in studying the structural variations of the long chain, complex molecules which occur in butadiene and isoprene polymers and copolymers and other synthetic rubbers. This procedure has been applied to determine the effects of oxidation and of variations in monomers and polymerizing conditions on the structure of synthetic rubber. It is practically certain that physical deficiencies of synthetic rubber are due principally to the structure of the long chain molecules rather than to the chemical nature of the monomers used.

Free Vibration of Continuous Skin-Stringer Panels

1960 ◽  
Vol 27 (4) ◽  
pp. 669-676 ◽  
Author(s):  
Y. K. Lin
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Time Dependent ◽  
Experimental Measurements ◽  
Structural Configuration ◽  
Modes Of Vibration ◽  
Fuselage Skin

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.

Analysis of a Class of Coupled Nonlinear Oscillators With an Application to Flow Induced Vibrations

2001 ◽  
Author(s):  
T. I. Haaker
Keyword(s):  
Normal Mode ◽  
Mixed Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Original System ◽  
Harmonic Oscillators ◽  
Resonant System ◽  
Coupled Equations ◽  
Averaged Equations ◽  
Internal Resonant

Abstract We consider in this paper the following system of coupled nonlinear oscillatorsx..+x-k(y-x)=εf(x,x.),y..+(1+δ)y-k(x-y)=εf(y,y.). In this system we assume ε to be a small parameter, i.e. 0 < ε ≪ 1. A coupling between the two oscillators is established through the terms involving the positive parameter k. The coupling may be interpreted as a mutual force depending on the relative positions of the two oscillators. For both ε and k equal to zero the two oscillators are decoupled and behave as harmonic oscillators with frequencies 1 and 1+δ, respectively. The parameter δ may therefore be viewed as a detuning parameter. Finally, the term ε f represents a small force acting upon each oscillator. Note that this force depends only on the position and velocity of the oscillator upon which the force is acting. To analyse the system’s dynamic behaviour we use the method of averaging. When k and δ are choosen such that no internal resonance occurs, one typically observes the following behaviour. If the trivial solution is unstable, solutions asymptotically tend to one of the two normal modes or to a mixed mode solution. For the special case with δ = 0 a system of two identical oscillators is found. If in addition k is O(ε) we obtain a 1 : 1 internal resonant system. The averaged equations may then be reduced to a system of three coupled equations — two for the amplitudes and one for the phase difference. Due to the fact that we consider identical oscillators there is a symmetry in the averaged equations. The normal mode solutions, as found for the non-resonant case, are still present. New mixed mode solutions appear. Moreover, Hopf bifurcations in the averaged system lead to limit cycles that correspond to oscillations in the original system with periodically modulated amplitudes and phases. We also consider the case with δ = O(ε), i.e. the case with nearly identical oscillators. If k = O(ε) again a 1 : 1 internal resonant system is found. Contrary to the previous cases the normal mode solutions no longer exist. Moreover, different bifurcations are observed due to the disappearance of the symmetry present in the system for s = 0. We apply some of the results obtained to a model describing aeroelastic oscillations of a structure with two-degrees-of-freedom.

scholarly journals The normal vibrations of carbonate and nitrate ions

1931 ◽  
Vol 134 (823) ◽  
pp. 265-277 ◽  
Keyword(s):  
Raman Spectrum ◽  
Raman Spectra ◽  
Raman Effect ◽  
Normal Modes ◽  
Potassium Nitrate ◽  
Nitrate Ions ◽  
Carbonate Ion ◽  
Infra Red ◽  
Modes Of Vibration

When the Raman effect was first discovered, it was believed that every line in the Raman spectrum referred to some characteristic vibration of the scatter­ing molecule. Later the tendency was to regard the lines as due to transitions between states of vibration of the molecule, so that the energies corresponded not to energies of vibration directly, but to differences in the energy of vibra­tion of two different modes. It is now realised that the infra-red spectrum of a substance and the Raman spectrum which it scatters give complementary information. Certain modes of vibration are represented solely in the infra­red spectrum, others are found only in the Raman spectrum, while others may appear in both spectra. Quite early a rough criterion on the basis of symmetry was put forward by Schaefer, for the determination of whether or not a particular vibration was to be expected in the Raman effect. Recently a selection rule has been formulated by Placzek; no vibration will appear as a fundamental in the Raman effect if it is such that any symmetrical operation upon it can change the signs of the displacements of the normal co-ordinates, without altering the energy. It is clear that a knowledge of the normal modes of vibration of the molecule under discussion must precede the application of any such rule, and it is the purpose of the present communication to discuss the normal modes of vibration of the carbonate and nitrate ions. In 1929 the writer showed that it was possible to obtain Raman spectra from powdered crystals, and the discovery was made when using powdered crystals of potassium nitrate. The method was applied first to carbonates and nitrates, so it became of interest to attempt to fix the structure of the anions of these salts by means of the Raman spectra combined with the infra-red data. In what follows the carbonate ion will first be dealt with in some detail, and then the nitrate ion can be treated summarily owing to the similarity of structure of the two ions.

On normal mode vibrations

1964 ◽  
Vol 60 (3) ◽  
pp. 595-611 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Finite Number ◽  
Linear Systems ◽  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Free Vibrations ◽  
Non Linear ◽  
Non Linear Systems

1. Introduction. In linear systems, the concept of ‘free vibrations in normal modes’ is well defined and fully understood. The meaning of this phrase is far less clear when it is applied to non-linear systems. It is the purpose here to define and examine the free vibrations in normal modes (and their stability) in certain non-linear systems composed of masses and springs and having a finite number of degrees of freedom. Of necessity, such a paper is in some degree conceptual in nature.

scholarly journals Nonsimilar Normal Mode Vibrations of Nonlinear Systems Having Two Degrees of Freedom

1964 ◽  
Vol 31 (2) ◽  
pp. 283-290 ◽  
Author(s):  
R. M. Rosenberg ◽  
J. K. Kuo
Keyword(s):  
Experimental Study ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Wave Shape ◽  
Time Histories ◽  
Mode Vibration ◽  
Wave Shapes

When a nonlinear system having several masses vibrates in normal modes, the time histories of the motion of these masses are, in general, different in wave shape (although in certain special nonlinear systems they differ at most in amplitude, but not in shape). When the wave shapes differ, the normal mode vibration is called nonsimilar. In this paper, nonsimilar normal mode vibrations are analyzed with respect to wave shape and stability. The systems considered are those lying close to systems having similar normal mode vibrations. An example is worked out in detail, and a comparison with an experimental study is reported.

Nonlinear Normal Modes for Vibratory Systems Under Periodic Excitation

2003 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Beam Model ◽  
Periodic Excitation ◽  
Nonlinear Normal Mode ◽  
Nonlinear Normal

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to periodic excitation. The approach is an extension of the nonlinear normal mode formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree of freedom, whose response is known. A reduced order model for the forced system is then determined by the usual nonlinear normal mode procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple two-degree-off-reedom mass-spring system with cubic nonlinearities, and for a discretized beam model with 12 degrees of freedom. The results show that this method provides very accurate responses over a range of frequencies near resonances.

Thermal Stresses in Rectangular Plates

1959 ◽  
Vol 10 (1) ◽  
pp. 65-78 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Thermal Stresses ◽  
Plate Thickness ◽  
Normal Modes ◽  
Thermoelastic Stress ◽  
Characteristic Functions ◽  
Vibration Modes ◽  
Rectangular Plates ◽  
Practical Application ◽  
Flat Plates ◽  
Modes Of Vibration

SummaryThe characteristic functions for beam vibration modes are used to derive an approximate solution for the calculation of thermal stresses in rectangular isotropic flat plates subjected to arbitrary temperature distributions in the plane of the plate and constant temperatures through the plate thickness. The thermal stresses are obtained in the form of generalised Fourier expansions in terms of the characteristic functions, and their derivatives, representing normal modes of vibration of a clamped-clamped beam. Since these functions have recently been tabulated, the practical application of this new method to the thermoelastic stress analysis of plates presents no difficulty.

Champ de force de valence et modes normaux de vibration de la glycylglycine et du monochlorhydrate monohydrate de glycylglycine

1974 ◽  
Vol 52 (14) ◽  
pp. 2590-2602 ◽  
Author(s):  
Christian Destrade ◽  
Eliane Dupart ◽  
Monique Joussot-Dubien ◽  
Chantal Garrigou-Lagrange
Keyword(s):  
Solid State ◽  
Force Field ◽  
Vibrational Spectra ◽  
Molecular Conformation ◽  
Normal Modes ◽  
Valence Force ◽  
Valence Force Field ◽  
Modes Of Vibration

The valence force field and normal modes of vibration of α-glycylglycine have been calculated using the vibrational spectra of ten isotopic derivatives, selectively labeled with deuterium or nitrogen-15. It is shown that this force field permits the determination of the frequencies of glycylglycine-hydrochloride in the solid state, despite a structure very different from that of the zwitterion. The agreement is particularly satisfying for those modes of vibration most sensitive to molecular conformation. It is thus possible to use this type of calculation to determine the conformation of an oligopeptide.

Sign in / Sign up

Export Citation Format

Share Document