On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes

2007 ◽  
Author(s):  
C. H. Pak ◽  
Y. S. Choi
Keyword(s):  
Normal Mode ◽  
Natural Frequencies ◽  
Normal Modes ◽  
New Normal ◽  
Perturbed System ◽  
Saddle Node ◽  
Non Linear ◽  
The Difference ◽  
Generic Bifurcation ◽  
The Stability

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.

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.

Extraction of Normal Modes from Contaminated Measurement With Noise for Highly Coupled Structures

1996 ◽  
Vol 118 (3) ◽  
pp. 430-435 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Normal Mode ◽  
Mode Shape ◽  
Present Method ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Mode Shapes ◽  
Response Functions ◽  
Exact Relation ◽  
Phase Data ◽  
Coupled Structures

A frequency-domain technique to extract the normal mode shapes from the contaminated FRF measurements for highly coupled structures is developed. The relation between the complex FRFs and the normal mode shapes is derived. It is found that the normal mode shape cannot be extracted exactly from the complex mode shape. However, an exact relation between the normal mode shape and the complex FRF does exist. In the present method, only the magnitude and phase data at the undamped natural frequencies are utilized to extract the normal mode shapes. Hence, the effects of measurement noise can be reduced. A numerical example is employed to illustrate the applicability of the technique. The results indicate that this technique can successfully extract the normal modes from the noisy frequency response functions of a highly coupled structure.

Notes on the stability problem for inhomogeneous equilibria

1983 ◽  
Vol 29 (2) ◽  
pp. 275-286 ◽  
Author(s):  
K. R. Symon
Keyword(s):  
Growth Rate ◽  
Normal Mode ◽  
Stability Problem ◽  
Normal Modes ◽  
Electromagnetic Mode ◽  
Spectral Matrix ◽  
First Order ◽  
Real Frequency ◽  
Special Cases ◽  
The Stability

Several conclusions regarding the stability of inhomogeneous Vlasov equilibria are drawn from earlier work. A technique is presented for generating first-order formulae for the change δω in the frequency of any normal mode, when a parameter λ characterizing the equilibrium is changed slightly. Several applications are given, including a first-order calculation of the growth rate or damping of an electromagnetic mode due to the presence of plasma. A condition is derived for the existence of a normal mode with real frequency. When there are ignorable co-ordinates, the normal modes can be written in the form of waves propagating in the ignorable directions. The character of the modes depends on certain symmetries in the dynamic spectral matrix. Special cases arise when the orbits can be approximated in certain ways.

Effect of Fluid Mass on Non-Linear Natural Frequencies of a Rotating Beam

2003 ◽  
Author(s):  
Ahmad A. Al-Qaisia
Keyword(s):  
Harmonic Balance ◽  
Natural Frequencies ◽  
Harmonic Balance Method ◽  
Continuous System ◽  
Assumed Mode Method ◽  
Non Linear ◽  
Mode Method ◽  
The Stability ◽  
The Mathematical Model ◽  
Analytical Expressions

The non-linear natural frequencies of the first three modes of a beam clamped to a rigid rotating hub and carrying a distributed fluid along its span are investigated. The mathematical model is derived using the Lagrangian method and the continuous system is discretized using the assumed mode method. The resulted unimodal nonlinear equation of motion was solved using two methods; harmonic balance (HB) and time transformation (TT), to obtain approximate analytical expressions for the nonlinear natural frequencies. Results have shown that the two terms harmonic balance method (2THB) is more accurate than the time TT method. Results for the effect and type of distribution, i.e. uniform or linearly distributed, on the variation of the nonlinear natural frequency with the rotational speed of the system and how they affect the stability are studied and presented in non-dimensional form.

Nonlinear Normal Modes in the Presence of Internal Resonances: An Energy-Based Approach

1995 ◽  
Author(s):  
Melvin E. King ◽  
Alexander F. Vakakis
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Weakly Nonlinear ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

Abstract In this work, modifications to existing energy-based nonlinear normal mode (NNM) methodologies are developed in order to investigate internal resonances. A formulation for computing resonant NNMs is developed for discrete, or discretized for continuous systems, sets of weakly nonlinear equations with uncoupled linear terms (i.e systems in modal, or canonical, form). By considering a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies. Additionally, the canonical formulation allows for a single (linearized modal) coordinate to parameterize all other (modal) coordinates during a resonant modal response. Energy-based NNM methodologies are then applied to the canonical equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered. Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the application of the resonant NNM methodology. Resonant normal mode solutions are obtained and the stability characteristics of these computed modes are considered. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus the transformation to canonical coordinates is necessary in order to define appropriate NNM relations.

Non-Normal Effects on Salt Finger Growth

2005 ◽  
Vol 35 (5) ◽  
pp. 616-627 ◽  
Author(s):  
Ian Eisenman
Keyword(s):  
Normal Mode ◽  
Stability Theory ◽  
Normal Modes ◽  
Growth Dynamics ◽  
Normal Growth ◽  
Initial Growth ◽  
Transient Effects ◽  
Salt Fingers ◽  
Long Time ◽  
The Difference

Abstract Salt fingers, which occur because of the difference in diffusivities of salt and heat in water, may play an important role in ocean mixing and circulation. Previous studies have suggested the long-time dominance of initially fastest growing finger perturbations. Finger growth has been theoretically derived in terms of the normal modes of the idealized system, which include a growing mode and a pair of decaying internal wave modes. Because these normal modes are not orthogonal, however, transient effects can occur related to the interaction between the modes, as explained by the generalized stability theory of non-normal growth. Initial growth of a perturbation that is not along a normal mode can be faster than the leading normal mode. In this study, the effects of non-normal growth on salt finger formation are investigated. It is shown that some salt finger perturbations that are a superposition of the growing mode and the decaying modes initially grow faster than pure growing normal mode perturbations. These non-normal effects are found to be significant for up to 10 or more e-folding times of the growing normal mode. The generalization of the standard idealized salt finger growth dynamics to include non-normal effects is found to lead to fastest-growing fingers that agree less well with observed fully developed salt fingers than the fastest-growing normal mode previously investigated.

STABILITY ANALYSIS IN PATIENTS WITH NEUROLOGICAL AND MUSCULOSKELETAL DISORDERS USING LINEAR AND NON-LINEAR APPROACHES

2015 ◽  
Vol 15 (04) ◽  
pp. 1530004
Author(s):  
MOHAMMAD KARIMI ◽  
MEISSAM SADEGHISANI ◽  
ABDUL HAFIDZ HAJI OMAR ◽  
EHSAN KOUCHAKI ◽  
MINA MIRAHMADI ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Musculoskeletal Disorders ◽  
Center Of Pressure ◽  
Linear Method ◽  
Approximate Entropy ◽  
Normal Subjects ◽  
Stability Parameters ◽  
Non Linear ◽  
The Difference ◽  
The Stability

Standing stability is controlled by musculoskeletal and neurological systems. Various methods have been used to evaluate the performance of subjects during standing including linear and non-linear methods. It is not clear which method has more sensitivity to represent the stability of subjects with various musculoskeletal disorders. Therefore, the aim of this study was to investigate the stability of the subjects with various musculoskeletal disorders by use of linear and non-linear methods. About 65 subjects including, normal and those with flatfoot, Parkinson and Perthes were recruited into this study. A Kistler forceplate was used to evaluate the stability. The difference between the linear (center of pressure excursion, velocity and path length) and non-linear (approximate entropy) parameters were evaluated using the independent t-test. The mean values of stability parameters (linear and non-linear) of flat arch subjects were more than that of normal subjects. Although there was no difference between linear stability parameters of normal and those with Parkinson disease, their mean value of non-linear parameter was less than that of normal subject (p-< 0.05). The results of stability analysis based on both linear and non-linear approaches showed that the subjects with Perthes disease were more unstable than normal subjects. It seems that non-linear method is more sensitive to represent the difference between stability of subjects with flatfoot, Parkinson and Perthes. However, if a combination of various parameters, based on linear method, is used to measure stability, the difference between stability can be enhanced. Depending on the disease condition increasing and decreasing the value of approximate entropy represent the unstability situations.

On the Stability Behavor of Bifurcated Normal Modes in Coupled Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 155-161 ◽  
Author(s):  
C. H. Pak
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Normal Modes ◽  
Nonlinear Oscillators ◽  
Coupled Nonlinear Oscillators ◽  
Floquet’S Theory ◽  
Generic Bifurcation ◽  
The Stability ◽  
Floquet's Theory

The stability of bifurcated normal modes in coupled nonlinear oscillators is investigated, based on Synge’s stability in the kinematico-statical sense, utilizing the calculus of variations and Floquet’s theory. It is found, in general, that in a generic bifurcation, the stabilities of two bifurcated modes are opposite, and in a nongeneric bifurcation, the stability of continuing modes is opposite to that of the existing mode, and the stabilities of the two bifurcated modes are equal but opposite to that of the continuing mode. Some examples are illustrated.

Use of a Lyophilized Reference Plasma to Compare Coagulation Test Procedures

1975 ◽  
Vol 34 (02) ◽  
pp. 426-444 ◽  
Author(s):  
J Kahan ◽  
I Nohén
Keyword(s):  
Oral Anticoagulant Therapy ◽  
Repeated Measurements ◽  
Test Procedures ◽  
Coagulation Activity ◽  
Close Relationship ◽  
Measured Activity ◽  
Test Materials ◽  
The Difference ◽  
The Stability ◽  
The One

SummaryIn 4 collaborative trials, involving a varying number of hospital laboratories in the Stockholm area, the coagulation activity of different test materials was estimated with the one-stage prothrombin tests routinely used in the laboratories, viz. Normotest, Simplastin-A and Thrombotest. The test materials included different batches of a lyophilized reference plasma, deep-frozen specimens of diluted and undiluted normal plasmas, and fresh and deep-frozen specimens from patients on long-term oral anticoagulant therapy.Although a close relationship was found between different methods, Simplastin-A gave consistently lower values than Normotest, the difference being proportional to the estimated activity. The discrepancy was of about the same magnitude on all the test materials, and was probably due to a divergence between the manufacturers’ procedures used to set “normal percentage activity”, as well as to a varying ratio of measured activity to plasma concentration. The extent of discrepancy may vary with the batch-to-batch variation of thromboplastin reagents.The close agreement between results obtained on different test materials suggests that the investigated reference plasma could be used to calibrate the examined thromboplastin reagents, and to compare the degree of hypocoagulability estimated by the examined PIVKA-insensitive thromboplastin reagents.The assigned coagulation activity of different batches of the reference plasma agreed closely with experimentally obtained values. The stability of supplied batches was satisfactory as judged from the reproducibility of repeated measurements. The variability of test procedures was approximately the same on different test materials.

Sign in / Sign up

Export Citation Format

Share Document