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 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.

scholarly journals On Estimating Survival and Hazard Ratios Using One Time-Scale When Cohort Data Have Multiple Time-Scales: A Simulation Study

2021 ◽  
Author(s):  
Nurgul Batyrbekova ◽  
Hannah Bower ◽  
Paul Dickman ◽  
Robert Szulkin ◽  
Paul C. Lambert ◽  
...  
Keyword(s):  
Time Scales ◽  
Simulation Study ◽  
Time Scale ◽  
Proportional Hazards ◽  
Survival Models ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Cohort Data ◽  
Hazard Ratios ◽  
The One

Abstract Background: When estimating survival functions and hazard ratios during theanalysis of cohort data, we often choose one time-scale, such as time-on-study, asthe primary time-scale, and include a xed covariate, such as age at entry, in themodel. However, we rarely consider the possibility of simultaneous effects ofmultiple time-scales on the hazard function. Methods: In a simulation study, within the framework of exible parametricmodels, we investigate whether relying on one time-scale and xed covariate asproxy for the second time-scale is sucient in capturing the true survivalfunctions and hazard ratios when there are actually two underlying time-scales. Result: We demonstrate that the one-time-scale survival models appeared toapproximate well the survival proportions, however, large bias was observed in thelog hazard ratios if the covariate of interest had interactions with the secondtime-scale or with both time-scales. Conclusion: We recommend to exercise caution and encourage tting modelswith multiple time-scales if it is suspected that the cohort data have underlyingnon-proportional hazards on the second time-scale or both time-scales.

scholarly journals New Approach for the Analysis of Damped Vibrations of Fractional Oscillators

2009 ◽  
Vol 16 (4) ◽  
pp. 365-387 ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Fractional Power ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Derivative Term ◽  
Mechanical Oscillators ◽  
Definition Of ◽  
Nonlinear Elastic

The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

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.

Resonant Excitation of a Spinning, Nutating Plate

1979 ◽  
Vol 46 (1) ◽  
pp. 132-138 ◽  
Author(s):  
J. W. Klahs ◽  
J. H. Ginsberg
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Resonant Excitation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Principal Coordinates ◽  
Method Of Solution ◽  
The Galerkin Method ◽  
Plane Displacement

The equations of motion governing the three-dimensional finite-amplitude response of a plate in arbitrary space motion are derived and shown to lead to dynamic coupling between the transverse and in-plane displacement. A general method of solution for such problems is demonstrated in an example involving a simply supported rectangular plate spinning about an axis parallel to an edge and nutating through a small angle. The method involves an asymptotic expansion using the derivative expansion version of the method of multiple time scales, in conjunction with the Galerkin method. A critical spin rate leading to the loss of stability in divergence is determined. Then, a numerical example of resonant excitation of one principal coordinate demonstrates that the nonlinear response resembling the one obtained from linear theory may lose stability in favor of a second response in which several principal coordinates are mutually excited. Consideration of the interaction between in-plane and transverse displacements is shown to be crucial to the prediction of this “unusual” response.

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.

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.

Control of Vibrations due to Moving Loads on Suspension Bridges

2006 ◽  
Vol 11 (3) ◽  
pp. 293-318 ◽  
Author(s):  
M. Zribi ◽  
N. B. Almutairi ◽  
M. Abdel-Rohman
Keyword(s):  
Bridge Deck ◽  
Suspension Bridge ◽  
Lyapunov Theory ◽  
Dynamic Responses ◽  
Suspension Bridges ◽  
Control Force ◽  
Moving Loads ◽  
Hydraulic Actuator ◽  
Long Span ◽  
Suspended Cables

The flexibility and low damping of the long span suspended cables in suspension bridges makes them prone to vibrations due to wind and moving loads which affect the dynamic responses of the suspended cables and the bridge deck. This paper investigates the control of vibrations of a suspension bridge due to a vertical load moving on the bridge deck with a constant speed. A vertical cable between the bridge deck and the suspended cables is used to install a hydraulic actuator able to generate an active control force on the bridge deck. Two control schemes are proposed to generate the control force needed to reduce the vertical vibrations in the suspended cables and in the bridge deck. The proposed controllers, whose design is based on Lyapunov theory, guarantee the asymptotic stability of the system. The MATLAB software is used to simulate the performance of the controlled system. The simulation results indicate that the proposed controllers work well. In addition, the performance of the system with the proposed controllers is compared to the performance of the system controlled with a velocity feedback controller.

Sign in / Sign up

Export Citation Format

Share Document