Prediction and Elimination of Subharmonic Resonances in Mechanical Systems With Nonlinear Foundations

1995 ◽  
Author(s):  
Sotirios Natslavas ◽  
Petros Tratskas
Keyword(s):  
Internal Resonance ◽  
Equations Of Motion ◽  
Single Mode ◽  
Harmonic Excitation ◽  
Multiple Time Scales ◽  
Model Parameters ◽  
Multiple Time ◽  
Algebraic Equations ◽  
Existence And Stability ◽  
Mode Response

Abstract In the first part of this work an analysis is presented on the dynamics of a two degree of freedom nonlinear mechanical oscillator. The model consists of a rigid body which rests on a foundation with nonlinear stiffness. This body can exhibit both vertical and rocking motions, which are coupled through the nonlinearities only. In the present study, attention is focused on the response of the system under external harmonic excitation of the vertical translation only, leading to conditions of subharmonic resonance of order three. Also, the model parameters are chosen so that its two linear natural frequencies are almost identical (1:1 internal resonance). For this case, the method of multiple time scales is first applied and a set of four coupled odes is derived, governing the amplitudes and phases of approximate motions of the system. Then, determination of approximate periodic steady state response of the oscillator is reduced to solving a set of four nonlinear algebraic equations. It is shown that besides linear and nonlinear single-mode response, two-mode response is also possible, due to the internal resonance. In addition, the stability of the various single- and two-mode periodic responses of the system is analyzed. In the last part of the work, the analytical findings are verified and complemented by numerical results. The main interest lies on identifying the effect of system parameters on the existence and stability of the predicted motions. The results of this study reveal patterns of appearance of these motions, which provide valuable help in the efforts to eliminate them. Finally, direct integration of the original equations of motion reveals the existence of other more complex motions, which coexist with the analytically predicted motions within the frequency ranges of interest.

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-Linear Modal Interaction in a Clamped-Clamped Beam: Part I — Deterministic Approach

1991 ◽  
Author(s):  
R. A. Ibrahim ◽  
A. Afaneh ◽  
B. Lee
Keyword(s):  
Internal Resonance ◽  
Resonance Condition ◽  
Harmonic Excitation ◽  
Multiple Time Scales ◽  
Small Range ◽  
Multiple Time ◽  
Transfer Energy ◽  
Response Characteristics ◽  
The Third ◽  
Detuning Parameter

Abstract The nonstationary response characteristics of multimode interaction in a clamped beam subjected to harmonic excitation is investigated. The nonlinear coupling of the first three modes is considered and resulted in a fourth order internal resonance condition for certain values of initial static axial load. The method of multiple time scales is employed to derive five equations in amplitudes and phase angles. It is found that the beam cannot reach any stationary solution in the neighborhood of the combination internal resonance. Within a small range of internal detuning parameter the third mode, which is externally excited, is found to transfer energy to the first two modes. Outside this region, the response is governed by a unimodal response of the third mode which follows the Duffing oscillator characteristics. The boundaries that separate unimodal and mixed mode responses are obtained in terms of the excitation level, damping ratios and internal resonance detuning parameter. The domains of attraction of the two response regimes are also obtained. The experimental results and response characteristics to random excitation will be reported in parts II and III, respectively.

Nonlinear Dynamic Response of a Thin Plate in a Fractional Viscoelastic Medium under Internal Resonance 1:1:2

2014 ◽  
Vol 518 ◽  
pp. 60-65 ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Nonlinear Equations ◽  
Time Scales ◽  
Viscoelastic Medium ◽  
Equations Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Nonlinear Dynamic Response ◽  
New Approach ◽  
New Type

Dynamic behaviour of a nonlinear plate embedded in a fractional derivative viscoelastic medium and subjected to the conditions of the internal resonances two-to-one has been studied by Rossikhin and Shitikova in [1]. Nonlinear equations, the linear parts of which occur to be coupled, were solved by the method of multiple time scales. A new approach proposed in this paper allows one to uncouple the linear parts of equations of motion of the plate, while the same method, the method of multiple time scales, has been utilized for solving nonlinear equations. The new approach enables one to find a new type of the internal resonanse, i.e., one-to-one-to-two, as well as to solve the problems of vibrations of thin bodies more efficiently.

scholarly journals ANALYSIS OF NONLINEAR FORCED VIBRATIONS OF FRACTIONALLY DAMPED SUSPENSION BRIDGES SUBJECTED TO THE ONE-TO-ONE INTERNAL RESONANCE

2020 ◽  
Vol 16 (2) ◽  
pp. 113-131
Author(s):  
Marina Shitikova ◽  
Aleks Katembo
Keyword(s):  
Internal Resonance ◽  
Fractional Power ◽  
Suspension Bridge ◽  
Multiple Time Scales ◽  
Suspension Bridges ◽  
Multiple Time ◽  
Method Of Solution ◽  
Nonlinear Force ◽  
One To One ◽  
The One

Nonlinear force driven coupled vertical and torsional vibrations of suspension bridges, when the frequency of an external force is approaching one of the natural frequencies of the suspension system, which, in its turn, undergoes the conditions of the one-to-one internal resonance, are investigated. The method of multiple time scales is used as the method of solution. The damping features are described by the fractional derivative, which is interpreted as the fractional power of the differentiation operator. The influence of the fractional parameters (orders of fractional derivatives) on the motion of the suspension bridge is investigated.

scholarly journals Nonlinear Dynamics of a Single-Mode Semiconductor Laser with Long Delayed Optical Feedback: A Modern Experimental Characterization Approach

Photonics ◽  
2022 ◽  
Vol 9 (1) ◽  
pp. 47
Author(s):  
Xavier Porte ◽  
Daniel Brunner ◽  
Ingo Fischer ◽  
Miguel C. Soriano
Keyword(s):  
Semiconductor Laser ◽  
Semiconductor Lasers ◽  
Optical Feedback ◽  
Dynamical Behavior ◽  
Laser Cavity ◽  
Single Mode ◽  
Point Of View ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Delayed Optical Feedback

Semiconductor lasers can exhibit complex dynamical behavior in the presence of external perturbations. Delayed optical feedback, re-injecting part of the emitted light back into the laser cavity, in particular, can destabilize the laser’s emission. We focus on the emission properties of a semiconductor laser subject to such optical feedback, where the delay of the light re-injection is large compared to the relaxation oscillations period. We present an overview of the main dynamical features that emerge in semiconductor lasers subject to delayed optical feedback, emphasizing how to experimentally characterize these features using intensity and high-resolution optical spectra measurements. The characterization of the system requires the experimentalist to be able to simultaneously measure multiple time scales that can be up to six orders of magnitude apart, from the picosecond to the microsecond range. We highlight some experimental observations that are particularly interesting from the fundamental point of view and, moreover, provide opportunities for future photonic applications.

An Experimental Study of Parametrically Excited Cantilever Beam and System Identification of Nonlinear Modes

2003 ◽  
Author(s):  
Timothy A. Doughty ◽  
Patricia Davies ◽  
Anil Bajaj
Keyword(s):  
Steady State ◽  
System Identification ◽  
Nonlinear System Identification ◽  
Single Mode ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Parametrically Excited ◽  
The Third ◽  
Mode Behavior ◽  
Second Mode

The nonlinear response of a parametrically excited cantilevered beam is experimentally investigated and nonlinear system identification techniques are used to generate nonlinear modal models to explain the observed behavior. Three techniques are applied to data from simulation of a nonlinear single-mode model as well as from experiments, for a beam which is excited with stationary harmonic input at nearly twice the frequency of the beam’s second mode. The first technique is based on the continuous-time differential equation model of the system, the second uses relationships generated by the method of harmonic balance, and the third is based on fitting steady-state response data to steady-state amplitude and phase predictions resulting from a multiple time scales analysis. Each approach is successful when applied to identify models from simulation data. For the experimental data obtained from a beam under nominally identical conditions, difficulties with using higher harmonic information lead to the incorporation of nonlinear damping terms and an investigation of two-mode behavior. Simulated two-mode behavior demonstrates how the beam’s third mode, with natural frequency nearly three times the frequency of the second mode, is excited in the physical structure, thus explaining the mismatch between the previous model and experiment at the third harmonic in the beam’s response.

Nonlinear Identification of a Capacitive Dual-Backplate MEMS Microphone

2005 ◽  
Author(s):  
Jian Liu ◽  
David T. Martin ◽  
Karthik Kadirvel ◽  
Toshikazu Nishida ◽  
Mark Sheplak ◽  
...  
Keyword(s):  
Nonlinear System Identification ◽  
Nonlinear Least Squares ◽  
Order System ◽  
Multiple Time Scales ◽  
Model Parameters ◽  
Multiple Time ◽  
Element Analysis ◽  
System Parameters ◽  
Lumped Element ◽  
Mems Microphone

This paper presents the nonlinear system identification of model parameters for a capacitive dual-backplate MEMS microphone. System parameters of the microphone are developed by lumped element modeling (LEM) and a governing nonlinear equation is thereafter obtained with coupled mechanical and electrostatic nonlinearities. The approximate solution for a general damped second order system with both quadratic and cubic nonlinearities and a non-zero external step loading is explored by the multiple time scales method. Then nonlinear finite element analysis (FEA) is performed to verify the accuracy of the lumped stiffnesses of the diaphragm. The microphone is characterized and nonlinear least-squares technique is implemented to identify system parameters from experimental data. Finally uncertainty analysis is performed. The experimentally identified natural frequency and nonlinear stiffness parameter fall into their theoretical ranges for a 95% confidence level respectively.

Nonlinear Dynamic Response of a Thin Plate Embedded in a Fractional Viscoelastic Medium under Combinational Internal Resonances

2014 ◽  
Vol 595 ◽  
pp. 105-110 ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Nonlinear Equations ◽  
Time Scales ◽  
Viscoelastic Medium ◽  
Equations Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Nonlinear Dynamic Response ◽  
New Approach ◽  
Difference Type

Dynamic behaviour of a nonlinear plate embedded in a fractional derivative viscoelastic medium and subjected to the conditions of the combinational internal resonances of the additive and difference types has been studied by Rossikhin and Shitikova in [1]. Nonlinear equations, the linear parts of which occur to be coupled, were solved by the method of multiple time scales. A new approach proposed in [2] allows one to uncouple the linear parts of equations of motion of the plate, while the same method, the method of multiple time scales, has been utilized for solving nonlinear equations. The new approach enables one to find an additional combinational resonance of the additive-difference type, as well as to solve the problems of vibrations of thin bodies more efficiently.

Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation

2016 ◽  
Vol 23 (7) ◽  
pp. 1106-1114 ◽  
Author(s):  
Saim Kural ◽  
Erdoğan Özkaya
Keyword(s):  
Elastic Foundation ◽  
Couple Stress Theory ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Approximate Solutions ◽  
Modified Couple Stress Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Classical Beam Theory

In this study, fluid conveying continuous media was considered as micro beam. Unlike the classical beam theory, the effects of shear stress on micro-structure's dynamic behavior not negligible. Therefore, modified couple stress theory (MCST) were used to see the effects of being micro-sized. By using Hamilton's principle, the nonlinear equations of motion for the fluid conveying micro beam were obtained. Micro beam was considered as resting on an elastic foundation. The obtained equations of motion were became independence from material and geometric structure by nondimensionalization. Approximate solutions of the system were achieved with using the multiple time scales method (a perturbation method). The effects of micro-structure, spring constant, the occupancy rate of micro beam, the fluid velocity on natural frequency and solutions were researched. MCST compared with classical beam theory and showed that beam models that based on classical beam theory are not capable of describing the size effects. Comparisons of classical beam theory and MCST were showed in graphics and these graphics also proved that obtained mathematical model suitable for describe the behavior of normal sized beams.

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