Structural Modal Multifurcation With International Resonance: Part 1—Deterministic Approach

1993 ◽  
Vol 115 (2) ◽  
pp. 182-192 ◽  
Author(s):  
R. A. Ibrahim ◽  
A. A. Afaneh ◽  
B. H. Lee
Keyword(s):  
Numerical Simulation ◽  
Fourth Order ◽  
Internal Resonance ◽  
Experimental Testing ◽  
Multiple Time Scales ◽  
Small Range ◽  
Multiple Time ◽  
The Third ◽  
Nonstationary Response ◽  
Detuning Parameter

The bifurcation and multifurcation in multimode interaction of nonlinear continuous structural systems is investigated. Under harmonic excitation the nonstationary response of multimode interaction is considered in the neighborhood of fourth-order internal resonance condition. The response dynamic characteristics are examined via three different approaches. These are the multiple scales method, numerical simulation, and experimental testing. The model considered is a clamped-clamped beam with initial static axial load. Under certain values of the static load the first three normal modes are nonlinearly coupled and this coupling results in a fourth-order internal resonance. The method of multiple time scales yields nonstationary response in the neighborhood of 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 bifurcation diagram which represents the boundaries that separate unimodal and mixed mode responses is 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 numerical simulation of the original equations of motion suggested the occurrence of complex response characteristics for certain values of damping ratios and excitation amplitude. Both numerical integration and experimental results reveal the occurrence of multifurcation as reflected by multi-maxima of the response probability density curves.

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.

scholarly journals Critical Parameters and Influence on Dynamic Behaviours of Nonlinear Electrostatic Force in a Micromechanical Vibrating Gyroscope

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Shuying Hao ◽  
Yulun Zhu ◽  
Yuhao Song ◽  
Qichang Zhang ◽  
Jingjing Feng ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Bias Voltage ◽  
Electrostatic Force ◽  
Dynamic Performance ◽  
Great Influence ◽  
Critical Parameters ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Element Analysis ◽  
Dc Bias

The electrostatic force nonlinearity caused by fringe effects of the microscale comb will affect the dynamic performance of the micromechanical vibrating gyroscopes (MVGs). In order to reveal the influence mechanism, a class of four-degree-of-freedom (4-DOF) electrostatically driven MVG is considered. The influence of DC bias voltage and comb spacing on the nonlinearity of electrostatic force and the dynamic response of the MVG by using multiple time scales method and numerical simulation are discussed. The results indicate that the electrostatic force nonlinearity causes the system to show stiffness softening. The softening characteristics of the electrostatic force cause the offset of the resonance frequency and a decrease in sensitivity. Although the electrostatic nonlinearity has a great influence on the dynamic behaviour, its influence can be avoided by the reasonable design of the comb spacing and DC bias voltage. There exists a critical value for comb spacing and DC bias voltage. In this paper, determining the critical values is demonstrated by nonlinear dynamics analysis. The results can be supported by the finite element analysis and numerical simulation.

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.

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.

scholarly journals Internal Resonance and Energy Transfer of a Cable-stayed Beam With a Tuned Mass Damper

2021 ◽  
Author(s):  
Xiaoyang Su ◽  
Hou Jun Kang ◽  
Tieding Guo ◽  
Yunyue Cong
Keyword(s):  
Energy Transfer ◽  
Internal Resonance ◽  
Primary Resonance ◽  
Tuned Mass Damper ◽  
Phase Portraits ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Response Curves ◽  
Mass Damper ◽  
One To One

Abstract This study considers a novel nonlinear system, namely, a cable-stayed beam with a tuned mass damper (cable-beam-TMD model), allowing the description of energy transfer among the beam, cable and TMD. In this system, the vibration of the TMD is involved and one-to-one-to-one internal resonance among the modes of the beam, cable and TMD is investigated when external primary resonance of the beam occurs. Galerkin’s method is utilized to discretize the equations of motion of the beam and cable. In this way, a set of ordinary differential equations (ODEs) are derived, which are solved by the method of multiple time scales (MTS). Then the steady state solutions of the system are obtained by suing Newton-Raphson method and continued by pseudo arclength algorithm. The response curves, time histories and phase portraits are provided to explore the effect of the TMD on the nonlinear behaviours of the model. Meanwhile, a partially coupled system, namely, a cable-beam-TMD model ignoring the vibration of the TMD, is also studied. The nonlinear characteristics of the two cases are compared with each other. The results reveal the occurrence of energy transfer among the beam, cable and TMD.

Model Reduction for Nonlinear Elastic Structures With Inertial Quadratic Nonlinearities

2009 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Model Reduction ◽  
Time Scales ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Nonlinear Response ◽  
Normal Modes ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities

In order to achieve accurate and high fidelity nonlinear response predictions, discrete models usually obtained through Galerkin approximation utilizing linear normal modes of the structure, need to retain a large number of degrees of freedom. This is specially the case if the structural response has the possibility of modal interactions. Then, a possible approach suggested in the literature to decrease the required degrees of freedom while retaining same accuracy is to use nonlinear normal modes of the structure to perform further model reduction. In this work, we discuss model reduction for nonlinear structural systems under harmonic excitations. The analysis needs to carefully consider the possibility of external resonances, parametric resonances, combination parametric resonances (the parametric excitation frequency being near the sum or difference of frequencies of two modes), and internal resonances. A master-slave separation of degrees of freedom is used, and a nonlinear relation between the slave coordinates and the master coordinates is constructed based on the multiple time scales approximation. More specifically, three cases are considered: external resonance of a mode without any internal resonance, and subharmonic as well as superharmonic excitation for systems with 1:2 internal resonance. The steady state periodic responses determined by the method of multiple time scales are compared to exact solutions of the discrete model computed by the bifurcation analysis and parameter continuation software AUTO. It is seen that for systems with essential inertial quadratic nonlinearities, the technique based on nonlinear model reduction through multiple time scales approximation over-predict the softening nonlinear response.

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.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

SOME REMARKS ON THE MULTIPLE TIME SCALES OF DIFFUSION IN MINERALS AND MELTS

2018 ◽  
Author(s):  
Yan Liang ◽  
◽  
Daniele J. Cherniak ◽  
Chenguang Sun
Keyword(s):  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time

Vapour-liquid equilibrium in the isobutyl formate-isobutyl alcohol and n-butyl formate-isobutyl alcohol systems at atmospheric pressure

1979 ◽  
Vol 44 (12) ◽  
pp. 3501-3508 ◽  
Author(s):  
Jan Linek
Keyword(s):  
Experimental Data ◽  
Fourth Order ◽  
Atmospheric Pressure ◽  
Isobutyl Alcohol ◽  
Liquid Equilibrium ◽  
The Third

Isobaric vapour-liquid equilibria in the isobutyl formate-isobutyl alcohol and n-butyl formate-isobutyl alcohol systems have been measured at atmospheric pressure. A modified circulation still of the Gillespie type has been used for the measurements. The experimental data have been correlated by means of the third- and fourth-order Margules equations.

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