Vibration Analysis of a Piecewise-Smooth System with Negative Stiffness under Delayed Feedback Control

The principal resonance of a delayed piecewise-smooth (DPWS) system with negative stiffness under narrow-band random excitation is investigated in aspects of multiscale analysis, design methodology of the controller, and response properties. The amplitude-frequency response and steady-state moments together with the corresponding stability conditions of the controlled stochastic system are derived, in which the degradation case is also under consideration. Then, from the perspective of the equivalent damping, the comparisons of the response characteristics of the controlled system to the uncontrolled system, such as the phenomenon of frequency island, are fulfilled. Furthermore, sensitivity of the system response to feedback gain and time delay is studied and interesting dynamic properties are found. Meanwhile, the classification of the steady-state solution is also discussed. To control the maximum amplitude, the feedback parameters are determined by the frequency response together with stability boundaries which must be utilized to exclude the combinations of the unstable parameters. For the case with small noise intensity, mean-square responses present the similar characteristics to what is discussed in the deterministic case.

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STEADY STATE SYMMETRY BREAKING IN PERIODICALLY EXCITED SYSTEMS INVOLVING TIME DELAY BY HARMONIC HOMOTOPY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500910 ◽

2013 ◽

Vol 23 (05) ◽

pp. 1350091

Author(s):

A. Y. T. LEUNG ◽

ZHONGJIN GUO

Keyword(s):

Steady State ◽

Time Delay ◽

Symmetry Breaking ◽

Moving Load ◽

Excitation Frequency ◽

Delayed Feedback Control ◽

Feedback Gain ◽

Homotopy Continuation ◽

External Excitation ◽

Wide Range

Symmetry breaking is a ubiquitous and important phenomenon arising in a wide range of physical systems. We propose the use of the harmonic balance in combination with homotopy continuation to investigate symmetry breaking occurrence in the periodically excited systems involving time delay. Two numerical examples are given to show the details. When the Hopfield neural network is subject to external excitation, we investigate the relation of the magnitude of excitation versus the amplitude of the bias term and analyze the effect of time delay on the steady state response. The second example concerns the delayed feedback control of a nonlinear beam subject to moving load. The relationship of the position feedback gain, external excitation frequency and time delay versus the amplitudes of steady state responses are studied analytically. The symmetry breaking points are accurately predicted. In addition, the Runge–Kutta numerical simulation results are used to cross-check the efficiency and accuracy.

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Jump Avoidance Conditions for Piecewise Linear Vibration Isolator

Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2005-85396 ◽

2005 ◽

Cited By ~ 3

Author(s):

Sagar Deshpande ◽

Sudhir Mehta ◽

G. Nakhaie Jazar

Keyword(s):

Steady State ◽

Frequency Response ◽

Perturbation Methods ◽

Averaging Method ◽

Piecewise Linear ◽

Bilinear System ◽

System Response ◽

Implicit Function ◽

Vibration Isolator ◽

Linear Vibration

Piecewise linear systems being highly non-linear, standard perturbation methods cannot produce an analytical expression for the frequency response. Hence, an adapted averaging method is employed to obtain an implicit function for frequency response of a bilinear system under steady state. This function is examined for jump-avoidance and a condition is derived which when met ensures that the undesirable phenomenon of ‘Jump’ does not occur and the system response is functional and unique.

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Response Prediction and Dynamic Substructuring for Coupled Structures in the Frequency Domain

Journal of Mechanics ◽

10.1017/jmech.2020.49 ◽

2020 ◽

Vol 36 (6) ◽

pp. 867-879

Author(s):

X. H. Liao ◽

W. F. Wu ◽

H. D. Meng ◽

J. B. Zhao

Keyword(s):

Frequency Response ◽

Response Function ◽

Frequency Response Function ◽

Dynamic Properties ◽

Main Idea ◽

Prediction Method ◽

Response Prediction ◽

Synthesis Methods ◽

Dynamic Substructuring ◽

Coupled Structures

ABSTRACTTo evaluate the dynamic properties of a coupled structure based on the dynamic properties of its substructures, this paper investigates the dynamic substructuring issue from the perspective of response prediction. The main idea is that the connecting forces at the interface of substructures can be expressed by the unknown coupled structural responses, and the responses can be solved rather easily. Not only rigidly coupled structures but also resiliently coupled structures are investigated. In order to further comprehend and visualize the nature of coupling problems, the Neumann series expansion for a matrix describing the relation between the coupled and uncoupled substructures is also introduced in this paper. Compared with existing response prediction methods, the proposed method does not have to measure any forces, which makes it easier to apply than the others. Clearly, the frequency response function matrix of coupled structures can be derived directly based on the response prediction method. Compared with existing frequency response function synthesis methods, it is more straightforward and comprehensible. Through demonstration of two examples, it is concluded that the proposed method can deal with structural coupling problems very well.

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Nonlinear filter properties of the thick ascending limb

AJP Renal Physiology ◽

10.1152/ajprenal.1997.273.4.f625 ◽

1997 ◽

Vol 273 (4) ◽

pp. F625-F634 ◽

Cited By ~ 20

Author(s):

H. E. Layton ◽

E. Bruce Pitman ◽

Leon C. Moore

Keyword(s):

Steady State ◽

Frequency Response ◽

Transit Time ◽

Concentration Profile ◽

Tubuloglomerular Feedback ◽

Thick Ascending Limb ◽

Transport Parameters ◽

Nodal Structure ◽

Nacl Concentration ◽

Fluid Transit Time

A mathematical model was used to investigate the filter properties of the thick ascending limb (TAL), that is, the response of TAL luminal NaCl concentration to oscillations in tubular fluid flow. For the special case of no transtubular NaCl backleak and for spatially homogeneous transport parameters, the model predicts that NaCl concentration in intratubular fluid at each location along the TAL depends only on the fluid transit time up the TAL to that location. This exact mathematical result has four important consequences: 1) when a sinusoidal component is added to steady-state TAL flow, the NaCl concentration at the macula densa (MD) undergoes oscillations that are bounded by a range interval envelope with magnitude that decreases as a function of oscillatory frequency; 2) the frequency response within the range envelope exhibits nodes at those frequencies where the oscillatory flow has a transit time to the MD that equals the steady-state fluid transit time (this nodal structure arises from the establishment of standing waves in luminal concentration, relative to the steady-state concentration profile, along the length of the TAL); 3) for any dynamically changing but positive TAL flow rate, the luminal TAL NaCl concentration profile along the TAL decreases monotonically as a function of TAL length; and 4) sinusoidal oscillations in TAL flow, except at nodal frequencies, result in nonsinusoidal oscillations in NaCl concentration at the MD. Numerical calculations that include NaCl backleak exhibit solutions with these same four properties. For parameters in the physiological range, the first few nodes in the frequency response curve are separated by antinodes of significant amplitude, and the nodes arise at frequencies well below the frequency of respiration in rat. Therefore, the nodal structure and nonsinusoidal oscillations should be detectable in experiments, and they may influence the dynamic behavior of the tubuloglomerular feedback system.

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Steady-State and Dynamic Properties of the Floating-Ring Journal Bearing

Journal of Lubrication Technology ◽

10.1115/1.3601543 ◽

1968 ◽

Vol 90 (1) ◽

pp. 243-253 ◽

Cited By ~ 26

Author(s):

F. K. Orcutt ◽

C. W. Ng

Keyword(s):

Steady State ◽

Journal Bearing ◽

Dynamic Properties ◽

Calculated Data ◽

Experimental Results ◽

Supply Pressure ◽

Pressure Parameter

Calculated data on steady-state and dynamic properties of the plain cylindrical floating-ring bearing with pressurized lubricant supply are given. The data are for a bearing with L/D of 1, and values of the ratio of inner to outer film clearances of 0.7 and 1.3. One value of dimensionless supply pressure parameter is covered. Experimental results are presented which verify the calculated results and which supplement them, particularly with respect to stability characteristics of the bearing.

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Dynamic control of maximal ventricular elastance via the baroreflex and force-frequency relation in awake dogs before and after pacing-induced heart failure

AJP Heart and Circulatory Physiology ◽

10.1152/ajpheart.00922.2009 ◽

2010 ◽

Vol 299 (1) ◽

pp. H62-H69 ◽

Cited By ~ 5

Author(s):

Xiaoxiao Chen ◽

Javier A. Sala-Mercado ◽

Robert L. Hammond ◽

Masashi Ichinose ◽

Soroor Soltani ◽

...

Keyword(s):

Heart Failure ◽

Transfer Function ◽

Transfer Functions ◽

Dynamic Properties ◽

System Response ◽

Force Frequency ◽

Arterial Baroreflex ◽

Ventricular Elastance ◽

Arterial Blood ◽

Frequency Relation

We investigated to what extent maximal ventricular elastance ( Emax) is dynamically controlled by the arterial baroreflex and force-frequency relation in conscious dogs and to what extent these mechanisms are attenuated after the induction of heart failure (HF). We mathematically analyzed spontaneous beat-to-beat hemodynamic variability. First, we estimated Emax for each beat during a baseline period using the ventricular unstressed volume determined with the traditional multiple beat method during vena cava occlusion. We then jointly identified the transfer functions (system gain value and time delay per frequency) relating beat-to-beat fluctuations in arterial blood pressure (ABP) to Emax (ABP→ Emax) and beat-to-beat fluctuations in heart rate (HR) to Emax (HR→ Emax) to characterize the dynamic properties of the arterial baroreflex and force-frequency relation, respectively. During the control condition, the ABP→ Emax transfer function revealed that ABP perturbations caused opposite direction Emax changes with a gain value of −0.023 ± 0.012 ml−1, whereas the HR→ Emax transfer function indicated that HR alterations caused same direction Emax changes with a gain value of 0.013 ± 0.005 mmHg·ml−1·(beats/min)−1. Both transfer functions behaved as low-pass filters. However, the ABP→ Emax transfer function was more sluggish than the HR→ Emax transfer function with overall time constants (indicator of full system response time to a sudden input change) of 11.2 ± 2.8 and 1.7 ± 0.5 s ( P < 0.05), respectively. During the HF condition, the ABP→ Emax and HR→ Emax transfer functions were markedly depressed with gain values reduced to −0.0002 ± 0.007 ml−1 and −0.001 ± 0.004 mmHg·ml−1·(beats/min)−1 ( P < 0.1). Emax is rapidly and significantly controlled at rest, but this modulation is virtually abolished in HF.

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An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

Journal of Vibration and Control ◽

10.1177/1077546315575471 ◽

2016 ◽

Vol 23 (1) ◽

pp. 119-130 ◽

Cited By ~ 2

Author(s):

Yaping Zhao

Keyword(s):

Steady State ◽

Probability Density Function ◽

Probability Density ◽

Averaging Method ◽

Stochastic Averaging ◽

Stochastic Averaging Method ◽

System Response ◽

Mean Square ◽

Fpk Equation ◽

The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

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