scholarly journals The effects of nonlinear damping on degenerate parametric amplification

2020 ◽  
Vol 102 (4) ◽  
pp. 2433-2452
Author(s):  
Donghao Li ◽  
Steven W. Shaw
Keyword(s):  
Maximum Amplitude ◽  
Nonlinear Effects ◽  
Nonlinear Damping ◽  
Parametric Amplification ◽  
Resonant Peak ◽  
System Response ◽  
Parameter Study ◽  
Response Curves ◽  
Direct Excitation ◽  
General Phase

AbstractThis paper considers the dynamic response of a single degree of freedom system with nonlinear stiffness and nonlinear damping that is subjected to both resonant direct excitation and resonant parametric excitation, with a general phase between the two. This generalizes and expands on previous studies of nonlinear effects on parametric amplification, notably by including the effects of nonlinear damping, which is commonly observed in a large variety of systems, including micro- and nano-scale resonators. Using the method of averaging, a thorough parameter study is carried out that describes the effects of the amplitudes and relative phase of the two forms of excitation. The effects of nonlinear damping on the parametric gain are first derived. The transitions among various topological forms of the frequency response curves, which can include isolae, dual peaks, and loops, are determined, and bifurcation analyses in parameter spaces of interest are carried out. In general, these results provide a complete picture of the system response and allow one to select drive conditions of interest that avoid bistability while providing maximum amplitude gain, maximum phase sensitivity, or a flat resonant peak, in systems with nonlinear damping.

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

scholarly journals Characteristic Analysis of longitudinal vibration Sucker Rod String With Variable Equivalent Stiffness

2018 ◽  
Vol 153 ◽  
pp. 06009 ◽  
Author(s):  
Jian Lv ◽  
Mingming Xing
Keyword(s):  
Longitudinal Vibration ◽  
Step Function ◽  
Load Force ◽  
System Response ◽  
Equivalent Stiffness ◽  
Function Form ◽  
Characteristic Analysis ◽  
Response Curves ◽  
Sucker Rod ◽  
Force Excitation

Considering the influence of variable equivalent stiffness on system response, the equivalent stiffness is defined as a step function, and a mathematical model of nonlinear longitudinal vibration of sucker rod string (SRS) is built. The dynamic response under displacement and load force excitation is solved by fourth-order Runge-Kutta method with zero initial condition. The results show the steady-state responses under the displacement and load force excitation of different function forms are different. The response curves of both displacement and velocity under the displacement and load force excitation of cosine function form have larger fluctuation than it under the displacement and load force excitation of sine function form. Therefore, the characteristic analysis of SRS plays an important role in understanding the influence of the excitation form and sensitive parameters on steady response.

Analysis of Free Nonlinear Vibration Behavior for Curved Embedded Carbon Nanotubes on Elastic Foundation

2010 ◽  
Author(s):  
M. Rezaee ◽  
H. Fekrmandi
Keyword(s):  
Carbon Nanotubes ◽  
Dynamic Behavior ◽  
Continuum Mechanics ◽  
Nonlinear Term ◽  
Nonlinear Oscillations ◽  
Equation Of Motion ◽  
System Response ◽  
Response Curves ◽  
Vibration Behavior ◽  
The Continuum

Carbon nanotubes (CNTs) are expected to have significant impact on several emerging nanoelectromechanical (NEMS) applications. Vigorous understanding of the dynamic behavior of CNTs is essential for designing novel nanodevices. Recent literature show an increased utilization of models based on elastic continuum mechanics theories for studying the vibration behavior of CNTs. The importance of the continuum models stems from two points; (i) continuum simulations consume much less computational effort than the molecular dynamics simulations, and (ii) predicting nanostructures behavior through continuum simulation is much cheaper than studying their behavior through experimental verification. In numerous recent papers, CNTs were assumed to behave as perfectly straight beams or straight cylindrical shells. However, images taken by transmission electron microscopes for CNTs show that these tiny structures are not usually straight, but rather have certain degree of curvature or waviness along the nanotubes length. The curved morphology is due to process-induced waviness during manufacturing processes, in addition to mechanical properties such as low bending stiffness and large aspect ratio. In this study the free nonlinear oscillations of wavy embedded multi-wall carbon nanotubes (MWCNTs) are investigated. The problem is formulated on the basis of the continuum mechanics theory and the waviness of the MWCNTs is modeled as a sinusoidal curve. The governing equation of motion is derived by using the Hamilton’s principle. The Galerkin approach was utilized to reduce the equation of motion to a second order nonlinear differential equation which involves a quadratic nonlinear term due to the curved geometry of the beam, and a cubic nonlinear term due to the stretching effect. The system response has been obtained using the incremental harmonic balanced method (IHBM). Using this method, the iterative relations describing the interaction between the amplitude and the frequency for the single-wall nanotube and double-wall nanotube are obtained. Also, the influence of the waviness, elastic medium and van der Waals forces on frequency-response curves is researched. Results present some useful information to analyze CNT’s nonlinear dynamic behavior.

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum-Arrays Subjected to Horizontal Excitation

2013 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
System Response ◽  
Stable Steady State ◽  
Frequency Range ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol’s method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant after which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

scholarly journals Center/surround organization of retinal bipolar cells: High correlation of fundamental responses of center and surround to sinusoidal contrasts

2011 ◽  
Vol 28 (3) ◽  
pp. 183-192 ◽  
Author(s):  
DWIGHT A. BURKHARDT ◽  
THEODORE M. BARTOLETTI ◽  
WALLACE B. THORESON
Keyword(s):  
Receptive Field ◽  
Maximum Amplitude ◽  
High Gain ◽  
Bipolar Cells ◽  
Ambystoma Tigrinum ◽  
Response Curves ◽  
On And Off Cells ◽  
Field Organization ◽  
Receptive Field Organization ◽  
Contrast Response

AbstractReceptive field organization of cone-driven bipolar cells was investigated by intracellular recording in the intact light-adapted retina of the tiger salamander (Ambystoma tigrinum). Centered spots and concentric annuli of optimum dimensions were used to selectively stimulate the receptive field center and surround with sinusoidal modulations of contrast at 3 Hz. At low contrasts, responses of both the center and surround of both ON and OFF bipolar cells were linear, showing high gain and thus contrast enhancement relative to cones. The contrast/response curves for the fundamental response, measured by a Fast Fourier Transform, reached half maximum amplitude quickly at 13% contrast followed by saturation at high contrasts. The variation of the normalized amplitude of the center and surround responses was remarkably similar, showing linear regression over the entire response range with very high correlations, r2 = 0.97 for both ON and OFF cells. The contrast/response curves of both center and surround for both ON and OFF cells were well fit (r2 = 0.98) by an equation for single-site binding. In about half the cells studied, the nonlinear waveforms of center and surround could be brought into coincidence by scaling and shifting the surround response in time. This implies that a nonlinearity, common to both center and surround, occurs after polarity inversion at the cone feedback synapse. Evidence from paired whole-cell recordings between single cones and OFF bipolar cells suggests that substantial nonlinearity is not due to transmission at the cone synapse but instead arises from intrinsic bipolar cell and network mechanisms. When sinusoidal contrast modulations were applied to the center and surround simultaneously, clear additivity was observed for small responses in both ON and OFF cells, whereas the interaction was strikingly nonadditive for large responses. The contribution of the surround was then greatly reduced, suggesting attenuation at the cone feedback synapse.

Nonlinear Dynamic Analysis and Experimental Verification of an Unbalanced Rotor Supported by Ball Bearings

2011 ◽  
Author(s):  
S. H. Upadhyay ◽  
Satish C. Sharma ◽  
S. P. Harsha
Keyword(s):  
Dynamic Properties ◽  
Nonlinear Dynamic Analysis ◽  
Peak Frequency ◽  
Nonlinear Damping ◽  
Rotor System ◽  
System Stability ◽  
System Response ◽  
Ball Bearings ◽  
Unbalanced Rotor ◽  
Radial Forces

In this paper, a dynamic model is presented for studying the dynamic properties of unbalanced rotor system supported by ball bearings under the effects of radial internal clearance and unbalanced rotor effect. The Newmark-β method is used to solve the nonlinear equations. The dynamics behaviors of a rigid rotor system are studied through frequency responses of the system. Clearances, nonlinear stiffness & nonlinear damping, radial forces and unbalanced forces—all these bring a significant influence to bear on the system stability. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The peak-to-peak frequency response of the system for each speed is obtained.

OTEC Plant Response and Control Analysis

1982 ◽  
Vol 104 (3) ◽  
pp. 208-215 ◽  
Author(s):  
W. L. Owens
Keyword(s):  
Nonlinear Differential Equations ◽  
System Response ◽  
Small Scale ◽  
Control Analysis ◽  
Plant Design ◽  
Operational Control ◽  
Response Curves ◽  
Control Schemes ◽  
Mass Flow Rates ◽  
And Control

An analysis is presented which allows prediction of closed-cycle OTEC power plant system response and control. Two basic operational control schemes are presented, which are primarily related to the type of seawater pumps employed. Variable flow seawater pumps allow optimization of the OTEC thermal-cycle state points for maximization of net generated power. Constant flow pumps are cheaper and simpler, but do not allow direct control over the evaporator and condenser operating temperatures. A system of nonlinear differential equations representing the basic elements of a constant seawater flow OTEC plant with turbine bypass flow control has been formulated for computer solution. Typical normalized response curves are presented for pressures, temperatures, mass flow rates, and generator speed for a small-scale, 50-kW OTEC plant design.

Vibration Control of Horizontally Excited Structures Utilizing Internal Resonance of Liquid Sloshing in Nearly Square Tanks

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Passive Control ◽  
Harmonic Excitation ◽  
Liquid Sloshing ◽  
Maximum Efficiency ◽  
System Response ◽  
Response Curves ◽  
System Parameters ◽  
Theoretical Results

Passive control of vibrations in an elastic structure subjected to horizontal, harmonic excitation by utilizing a nearly square liquid tank is investigated. When the natural frequency ratio 1:1:1 is satisfied among the natural frequencies of the structure and the two predominant sloshing modes (1,0) and (0,1), the performance of a nearly square tank as a tuned liquid damper (TLD) is expected to be superior to rectangular TLDs due to internal resonance. In the theoretical analysis, Galerkin's method is used to determine the modal equations of motion for liquid sloshing considering the nonlinearity of sloshing. Then, van der Pol's method is used to obtain the expressions for the frequency response curves for the structure and sloshing modes. Frequency response curves and bifurcation set diagrams are shown to investigate the influences of the aspect ratio of the tank cross section and the tank installation angle on the system response. From the theoretical results, the optimal values of the system parameters can be determined in order to achieve maximum efficiency of vibration suppression for the structure. Hopf bifurcations occur and amplitude modulated motions (AMMs) may appear depending on the values of the system parameters. Experiments were also conducted, and the theoretical results agreed well with the experimental data.

LQG Vibration Control of Aircraft Panel Using Automated State Weighting Procedure

2005 ◽  
Author(s):  
T. C. Waite ◽  
Christopher E. Whitmer ◽  
Atul G. Kelkar
Keyword(s):  
Maximum Amplitude ◽  
Linear Quadratic Regulator ◽  
Initial Guess ◽  
Performance Comparison ◽  
System Response ◽  
Linear Quadratic ◽  
Cabin Noise ◽  
Automated Method ◽  
And Control ◽  
Error State

Optimal control theory has long been plagued by its inability to optimize the state and control weighting matrices specified in the LQR (Linear Quadratic Regulator) cost function. Although this control is optimal for a given set of user-defined state and control weighting matrices, the performance of the controller varies widely based on the selection of these weights. Engineers have been left with the task of choosing appropriate weighting sequences by iterating each weight until the controller performs to their satisfaction. This procedure gets increasingly more frustrating and time consuming as the size of the controller increases. The work in this paper outlines an effective strategy which reduces the engineer’s effort in finding these weights. It is shown that the introduction of an additional performance index along with the repeated perturbation of an initial guess quickly leads to a controller with excellent performance. Comparison of this automated method with a controller exhaustively designed using a standard selection method reveals a closed loop system response which has more energy reduction and more maximum amplitude reduction, all in a fraction of the time it takes to guess and check the weights. The controller will be applied to the model of an aircraft panel in an effort to reduce vibrations caused by wind. The goal is to achieve a reduction in cabin noise by controlling the vibration of such panels. The system identification procedure uses a modification of the SOCIT toolbox to achieve extremely accurate frequency domain system models. The model obtained using this method will then be used in the design and simulation of both the trial-and-error state weighted controller and the automated state weighted controller.

Forced Response of Turbine Engine Bladed Disks and Sensitivity to Harmonic Mistuning

2002 ◽  
Vol 125 (1) ◽  
pp. 113-120 ◽  
Author(s):  
J. A. Kenyon ◽  
J. H. Griffin
Keyword(s):  
Maximum Amplitude ◽  
Mathieu Equation ◽  
Forced Response ◽  
System Response ◽  
Analytical Prediction ◽  
Bladed Disks ◽  
Turbine Engine ◽  
Root Cause ◽  
Random Mistuning ◽  
Shear Spring

The mistuned forced response of turbine engine bladed disks is treated using harmonic perturbations in the properties of a continuous ring. A continuous shear spring is attached to the ring in which the stiffness is allowed to vary along the ring annulus. The modes of such a structure with a single harmonic mistuning pattern are shown to obey the Mathieu equation, which is solved to obtain the natural frequencies and modes of the mistuned system. The forced response of the system is then examined to determine the sensitivity of the system to small mistuning. The model is extended to include multiple harmonics, allowing for the possibility of general mistuning. An expression for the maximum amplitude magnification due to small mistuning is developed by showing that high response is caused by distortion of the structural modes. A method to intentionally mistune systems for maximum forced response is demonstrated, and numerical results demonstrate the accuracy of the analytical prediction. The intentionally mistuned system response is shown to be robust with respect to small random mistuning. Such a result might be useful for designing a test rotor for screening new bladed disk designs or for establishing the root cause of fatigue problems.

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