Non-linear parametric vibrations of a composite column under uniform compression

2012 ◽  
Vol 226 (8) ◽  
pp. 1921-1938 ◽  
Author(s):  
Jerzy Warmiński ◽  
Andrzej Teter
Keyword(s):  
Order Approximation ◽  
Linear Equations ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Parametric Vibrations ◽  
Uniform Compression ◽  
Composite Column ◽  
Approximate Analytical Solutions ◽  
Non Linear ◽  
First Order Approximation

Parametric oscillations of a prismatic, thin-walled composite column with a channel section are considered in the article. The simply supported column is made of a seven-layer composite with a symmetric ply alignment. The non-linear problem of buckling is solved with Koiter’s asymptotic theory within the first-order approximation by adoption of a plate model. The asymptotic approximation leads to non-linear equations allowing evaluation of the two mode buckling effect. Parametric vibrations, produced by a periodically changing load component, are investigated near the principal parametric resonances. The approximate analytical solutions are determined by the multiple time scales of method. The influence of amplitude and frequency of parametric excitation on the structure response is investigated. An example bifurcation scenario and a possible transition to chaotic oscillations are also presented.

scholarly journals Active Vibration Control of a Nonlinear Beam with Self- and External Excitations

2013 ◽  
Vol 20 (6) ◽  
pp. 1033-1047 ◽  
Author(s):  
J. Warminski ◽  
M. P. Cartmell ◽  
A. Mitura ◽  
M. Bochenski
Keyword(s):  
Analytical Solutions ◽  
Order Approximation ◽  
Equations Of Motion ◽  
Active Vibration Control ◽  
Harmonic Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Beam ◽  
Approximate Analytical Solutions ◽  
Full Structure

An application of the nonlinear saturation control (NSC) algorithm for a self-excited strongly nonlinear beam structure driven by an external force is presented in the paper. The mathematical model accounts for an Euler-Bernoulli beam with nonlinear curvature, reduced to first mode oscillations. It is assumed that the beam vibrates in the presence of a harmonic excitation close to the first natural frequency of the beam, and additionally the beam is self-excited by fluid flow, which is modelled by a nonlinear Rayleigh term for self-excitation. The self- and externally excited vibrations have been reduced by the application of an active, saturation-based controller. The approximate analytical solutions for a full structure have been found by the multiple time scales method, up to the first-order approximation. The analytical solutions have been compared with numerical results obtained from direct integration of the ordinary differential equations of motion. Finally, the influence of a negative damping term and the controller's parameters for effective vibrations suppression are presented.

Non-Stationary Oscillations at Slow Transitions Across Instabilities

2007 ◽  
Author(s):  
Rudolf R. Pusˇenjak ◽  
Maks M. Oblak ◽  
Jurij Avsec
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Time Scales ◽  
Primary Resonance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Excitation Parameter ◽  
Linear Dynamical Systems ◽  
Weakly Nonlinear ◽  
Non Linear

The paper presents the study of non-stationary oscillations, which is based on extension of Lindstedt-Poincare (EL-P) method with multiple time scales for non-linear dynamical systems with cubic non-linearities. The generalization of the method is presented to discover the passage of weakly nonlinear systems through the resonance as a control or excitation parameter varies slowly across points of instabilities corresponding to the appearance of bifurcations. The method is applied to obtain non-stationary resonance curves of transition across points of instabilities during the passage through primary resonance of harmonically excited oscillators of Duffing type.

On the non-linear Lamb-Taylor instability

1969 ◽  
Vol 38 (3) ◽  
pp. 619-631 ◽  
Author(s):  
Ali Hasan Nayfeh
Keyword(s):  
Layer Thickness ◽  
Travelling Waves ◽  
Finite Thickness ◽  
Standing Waves ◽  
Linear Case ◽  
Wave Number ◽  
Multiple Time Scales ◽  
Solid Boundary ◽  
Multiple Time ◽  
Non Linear

A non-linear analysis of the inviscid stability of the common surface of two superposed fluids is presented. One of the fluids is a liquid layer with finite thickness having one surface adjacent to a solid boundary whereas the second surface is in contact with a semi-infinite gas of negligible density. The system is accelerated by a force normal to the interface and directed from the liquid to the gas. A second-order expansion is obtained using the method of multiple time scales. It is found that standing as well as travelling disturbances with wave-numbers greater than$K^{\prime}_c = k_c[1+\frac{3}{8}a^2k^2_c + \frac{51}{512}a^4k^4_c]^{\frac{1}{2}}$where a is the disturbance amplitude and kc is the linear cut-off wave-number, oscillate and are stable. However, the frequency in the case of standing waves and the wave velocity in the case of travelling waves are amplitude dependent. Below this cut-off wave-number disturbances grow in amplitude. The cut-off wave-number is independent of the layer thickness although decreasing the layer thickness decreases the growth rate. Although standing waves can be obtained by the superposition of travelling waves in the linear case, this is not true in the non-linear case because the amplitude dependences of the wave speed and frequency are different. A mechanism is proposed to explain the overstability behaviour observed by Emmons, Chang & Watson (1960).

scholarly journals Large-scale Subspace Clustering by Fast Regression Coding

2017 ◽  
Author(s):  
Jun Li ◽  
Handong Zhao ◽  
Zhiqiang Tao ◽  
Yun Fu
Keyword(s):  
Linear Function ◽  
Spectral Clustering ◽  
Large Scale ◽  
Order Approximation ◽  
Subspace Clustering ◽  
Low Rank ◽  
First Order ◽  
Non Linear ◽  
Data Points ◽  
First Order Approximation

Large-Scale Subspace Clustering (LSSC) is an interesting and important problem in big data era. However, most existing methods (i.e., sparse or low-rank subspace clustering) cannot be directly used for solving LSSC because they suffer from the high time complexity-quadratic or cubic in n (the number of data points). To overcome this limitation, we propose a Fast Regression Coding (FRC) to optimize regression codes, and simultaneously train a non-linear function to approximate the codes. By using FRC, we develop an efficient Regression Coding Clustering (RCC) framework to solve the LSSC problem. It consists of sampling, FRC and clustering. RCC randomly samples a small number of data points, quickly calculates the codes of all data points by using the non-linear function learned from FRC, and employs a large-scale spectral clustering method to cluster the codes. Besides, we provide a theorem guarantee that the non-linear function has a first-order approximation ability and a group effect. The theorem manifests that the codes are easily used to construct a dividable similarity graph. Compared with the state-of-the-art LSSC methods, our model achieves better clustering results in large-scale datasets.

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