Steady-state Response Of System Research Articles

Drag reduction and the Vogel exponent of a flexible beam in transient shear flows

Interactions between a flexible beam and a fluid in a channel are of great relevance to biological hairy surfaces, aquatic vegetation, marine life (e.g., fish gills), and many industrial systems alike. While steady state response of a beam to such flows is fairly well-explored, their behavior in the transient regime is not fully understood. A series of numerical simulations are performed to study the laminar Couette flow of an incompressible viscous fluid past an elastic beam in a two-dimensional channel. The flexible beam is perpendicular to the direction of flow, and its base is fixed to the stationary bottom of the channel. We measure the evolution of the Vogel exponent, drag reduction, and reconfiguration number during the transient and steady-state response of the fluid–structure system for different geometrical and physical properties. Our benchmark shows a good agreement between numerical and experimental observations. Our results show that the system's steady-state response at different bulk-fluid velocities can be reproduced by investigating the shear flow response during the transient regime. We define a new variable that characterizes the evolution of the local velocity profile in the proximity of the free end of the beam and use that to characterize the transient-regime response. The analysis yields insight into the competing effects of elasticity of the beam and non-linear flow response.

A Fast Frequency Domain Method for Steady-State Solution of Forced Vibration of System with Complex Damping

The conventional frequency domain method (CFDM) and dual-force-based time domain method (DTDM) are often used to solve the steady-state response of system with complex damping under an arbitrary force. However, the calculation efficiency of the DTDM is low due to the straightforward summation operation of series even if the solution of the DTDM is the exact real part of the solution. In addition, since the CFDM only can obtain the real part of solution not the complete solution, it gives misleading information that the solution does not have an imaginary part. In this paper, a fast frequency domain method (FFDM) is proposed to calculate the complete response of complex damping system including the imaginary part with a higher accuracy in a much faster manner. The new FFDM uses half of the Fourier series of the discrete Fourier transform of the actual arbitrary force to construct the Fourier series of the dual force, followed by calculating the time history response using the inverse fast Fourier transform. The new developed method is validated through three numerical examples with harmonic and seismic excitations. The numerical results show that the accuracy of the new FFDM is compatible to the DTDM but with much higher computational efficiency.

Automatic Gate System Based on Water Flow Within the Intake of a Micro-Hydro Power Plant

The micro-hydro power plant is an alternative to provide electricity especially in regions that are far from the PLN network. The micro-hydro power plant uses water flow to generate electricity. The micro-hydro power plant uses a gate that is utilised to distribute water to the turbine. However, the watergate is commonly controlled manually which results in drawbacks such as unfavourable stabilization and the need for direct monitoring that requires lots of constant expense as well as labour. The objective of this research is to make equipment that can work automatically in substituting a labourer’s role in opening and closing the watergate according to the required water flow. A water flow sensor is used to measure the water flow, the result of the sensor reading is carried on to microcontroller (set up as Fuzzy Logic Control) and Controler sends the control signal to the actuator (DC motor). The Fuzzy Logic method as the watergate controller is used as the research’s prototype system. Based on the data acquired by the research that has been done, the system is oscillated towards the change of the water flow. However, it’s capable to approach the set value point. It can be seen from the water flow trial with a set point of 3 L/min the steady-state response of system undergoes at range of 2,92 L/min–2,95 L/min and with a set point of 3,5 L/min, the steady-state response of system undergoes at range of 3,44 L/min - 3,47 L/min.

Limit cycle global asymptotic stability of continuous-time switched affine systems

This paper treats global asymptotic stability of a limit cycle, determined by the designer, for continuous-time switched affine systems, taking into account sampled and non-sampled switching rules. More specifically, our goal is to design a state-dependent switching function assuring global asymptotic stability of a limit cycle, which is determined from criteria of interest related to the system steady-state response. The conditions, expressed in terms of differential linear matrix inequalities, are based on a time-varying quadratic Lyapunov function and take into account a guaranteed cost, which assures a suitable performance level for the system transient response. These conditions can be converted into two linear matrix inequalities, making the problem simple-to-solve. An academic example is used for validation and comparison.

Formulation of Statistical Linearization for M-D-O-F Systems Subject to Combined Periodic and Stochastic Excitations

A formulation of statistical linearization for multi-degree-of-freedom (M-D-O-F) systems subject to combined mono-frequency periodic and stochastic excitations is presented. The proposed technique is based on coupling the statistical linearization and the harmonic balance concepts. The steady-state system response is expressed as the sum of a periodic (deterministic) component and of a zero-mean stochastic component. Next, the equation of motion leads to a nonlinear vector stochastic ordinary differential equation (ODE) for the zero-mean component of the response. The nonlinear term contains both the zero-mean component and the periodic component, and they are further equivalent to linear elements. Furthermore, due to the presence of the periodic component, these linear elements are approximated by averaging over one period of the excitation. This procedure leads to an equivalent system whose elements depend both on the statistical moments of the zero-mean stochastic component and on the amplitudes of the periodic component of the response. Next, input–output random vibration analysis leads to a set of nonlinear equations involving the preceded amplitudes and statistical moments. This set of equations is supplemented by another set of equations derived by ensuring, in a harmonic balance sense, that the equation of motion of the M-D-O-F system is satisfied after ensemble averaging. Numerical examples of a 2-D-O-F nonlinear system are considered to demonstrate the reliability of the proposed technique by juxtaposing the semi-analytical results with pertinent Monte Carlo simulation data.

Frequency domain response under arbitrary excitation for fading memory nonlinear systems

For dynamic systems, the steady-state system response to periodic excitation is well understood for both linear and certain nonlinear system classes. When the excitation is not periodic, however, the measured response will contain both transient and steady-state contributions. For linear systems, these transient contributions have been thoroughly explored, while no equivalent analysis exists for the nonlinear case. In this paper, we derive an expression in the frequency domain for the system response of all discrete time-invariant nonlinear systems which have fading memory, using the Volterra series representation. The expression contains both steady-state and transient contributions at each nonlinear order, revealing a highly structured view of nonlinear system response. For the nonlinear case, the transient expressions at higher nonlinear orders have a more complex structure than those generated by linear systems, which provides valuable insight for systems theory and identification purposes.

Optimized Control Based on Nonlinear Modelling and Parameter Estimation of the Electro-Hydraulic Servo System

In this paper, a nonlinear physical model of the electro-hydraulic system is built and described in Bond Graph in order to show its physical structure and flow of energy. The nonlinear model is simplified based on response bandwidth of different parts of system and nonlinear degree. A series of typical parameters of the model are chosen based on the effect on nonlinear response of system and estimated with the experiment response. A simple and useful model based nonlinear feedforward PID control methods is proposed based on the steady state response of system and dynamic stribeck friction force. The simulation and experiment results of parameter estimation and feedforward control are presented, which shows that the model response with estimated parameters can fit the experimental data very well, and the simulation and experiment response of the optimized control can eliminate the nonlinear response very well. Therefore, the nonlinear model and control method used in this paper have a guiding significance and application value, and have theory value for the further research of hydraulic servo system with the similar structure.

Traverse Vibration of Axially Moving Laminated SMA Beam considering Random Perturbation

The nonlinear random vibration of axially moving shape memory alloy (SMA) laminated beam under transverse loads is investigated. Considering the effects of axial movement and random perturbation, the dynamic equation of the SMA laminated beam is established by means of physical equation, force balance conditions, deformation compatibility equation, and the constitutive relation by polynomial function. The transverse vibration differential equation of axially moving SMA laminated beam with two simply‐supported edges is obtained through the Galerkin method. The amplitude‐frequency response equation under primary resonance and the probability density function under random perturbation are derived by using the averaging method and stochastic averaging method, respectively, and the theoretical results are numerically validated. In addition, under the combination of the harmonic excitation and random perturbation, the effects of axial movement velocity and random perturbation intensity on system steady‐state response are analyzed.

Steady-state response of systems with fractional dampers

In this paper, steady state vibrations of systems with built-up viscoelastic dampers are considered. The dampers are modeled using the fractional-derivative rheological models. The Caputo type fractional derivative definition is used. In particular, the steady state vibrations of systems are analysed. The solution to the steady state vibrations is written using real quantities. The effects induced by changes of environmental temperature are also considered and, in this context, the time-temperature superposition principle is adopted. The results of several parametric studies are also described and discussed in detail.

Generalized Parametric Resonance

We consider the dynamic response of systems subjected to principal parametric resonant excitation in which the full potential, including nonlinearities, is modulated in a time-periodic manner. This work was motivated by the model of Rhoads et al. [J. Sound Vibration, 296 (2006), pp. 797--829], which was used to describe an interesting bifurcation structure that was experimentally observed in a micro-electro-mechanical system. The goal of the present investigation is to more fully explore this class of systems, described by a generalized nonlinear Mathieu equation, and ascertain general features of their response. The method of averaging is used to derive equations governing the slowly varying amplitude and phase for parametrically excited systems with weak nonlinearity, small damping, and near resonant excitation, allowing for a detailed analysis of the system steady-state response. Results for a general class of models are presented first, followed by details for two examples: (i) generalized parametric ...

Nonlinear dynamic behaviour of a rotor–foundation system coupled through passive magnetic bearings with magnetic anisotropy – Theory and experiment

In this work, the nonlinear dynamic behaviour of a vertical rigid rotor interacting with a flexible foundation by means of two passive magnetic bearings is quantified and evaluated. The quantification is based on theoretical and experimental investigation of the non-uniformity (anisotropy) of the magnetic field and the weak nonlinearity of the magnetic forces. Through mathematical modelling the nonlinear equations of motion are established for describing the shaft and bearing housing lateral dynamics coupled via the nonlinear and non-uniform magnetic forces. The equations of motion are solved in the frequency domain by the methods of Finite Difference and pseudo-arclength continuation. The theoretical findings are validated against experiments carried out using a dedicated test-rig and a special device for characterisation of the magnetic anisotropy.The characterisation of the magnetic anisotropy shows that it can be quantified as magnetic eccentricities having an amplitude and a phase, which result in linear and parametric excitation. The magnetic eccentricities are also determined using the steady-state response of the rotor–bearing system due to forcing from the magnetic anisotropies and several levels of mass imbalance. Discrepancies in the results from the two methods in terms of magnetic eccentricity magnitude are due to additional geometric eccentricities in the shaft. The steady-state system response shows clear nonlinear phenomena, e.g. bent resonance peaks, jump phenomena and nonlinear cross-coupling between the two orthogonal directions, especially during counter-phase motion between shaft and bearings. The clear nonlinear behaviour is facilitated by the lack of damping resulting in relatively large vibrations. The overall nonlinear dynamic behaviour is well captured by the theoretical model, thereby validating the modelling approach.

Study of nonlinear dynamics and chaos in MEMS/NEMS resonators

With the successes in numerous applications from signal filtering to chemical and mass sensing, micro- and nano-electro-mechanical resonators continue to be one of the most widely studied topics of the micro-electro-mechanical systems community. Nonlinearities arising out of different sources such as mid-plane stretching and electrostatic force lead to a rich nonlinear dynamics in the time response of these systems which should be investigated for appropriate design and fabrication of them. Motivated by this need, present study is devoted to analyzing the nonlinear dynamics and chaotic behavior of nano resonators with electrostatic forces on both sides. Based on the potential function and phase portrait of the unperturbed system, the resonator dynamics is categorized to four physical situations and it is shown that the system undergoes homoclinic and heteroclinic orbits which are responsible for the appearance of chaos in the resonator response. Bifurcation diagram of nano resonator is plotted by variation of applied AC actuation voltage and it is shown that the system possess rich dynamic behavior such as periodic doubling, quasi-periodic, bifurcation and chaotic motion which are classified and studied in more details by plotting time response and phase plane of the each category.The main result of this paper indicates that the necessary condition for the creation of chaos in the resonator is intersection of the system steady state response with the homoclinic orbit. This occurs when the system steady state velocity or amplitude reaches to the homoclinic orbit maximum speed or amplitude. The critical oscillating amplitudes corresponding to these situations are derived based on the system parameters which can be used to propose the new analytical criteria for chaos detection in resonators.

ANALYSIS OF THE STEADY STATE UNBALANCE RESPONSE OF RIGID ROTORS ON MAGNETORHEOLOGICAL DAMPERS: STABILITY, FORCE TRANSMISSION AND ENERGY DISSIPATION

The rotors working in real technological devices are always slightly imbalanced. This excites their lateral vibrations and generates forces that are transmitted to the rotor casing. These effects can be significantly reduced if damping devices are added to the support elements. The possibility of controlling the damping, in order to achieve their optimum performance, is offered by magnetorheological squeeze film dampers. In this paper, a computational modeling method is used to investigate the dynamical behavior of a rigid flexibly supported rotor loaded by its unbalance and equipped with two short magnetorheological dampers. The equations of motion of the rotor are nonlinear due to the damping forces. Computational procedures were developed to verify the applicability of such dampers by simulating their behavior and analyzing their effect on the amplitude of the rotor vibration, on the magnitude of the forces transmitted to the rotor casing and on the amount of the power dissipated in the magnetorheological films. The proposed approach to study the optimum performance of semiactive magnetorheological dampers applied in rotor systems, in terms of vibration amplitudes and transmitted forces, together with the developed efficient computational methods to calculate the system steady state response and to evaluate its stability represent the new contributions of this paper.

Generalized stochastic resonance of power function type single-well system

To generalize the harmonic potential of the linear random vibration system, a more general power type potential is presented, and the corresponding power function type nonlinear single-well random vibration system is obtained. The first moment of the system steady-state response and the stationary variance of the system response, which are influenced by noise strength, parameters of the potential and the periodic excitation, are studied by using the second order stochastic Runge-Kutta algorithm. The parameter b, which determines the shape of the potential, goes through b b > 2 and b=2 (harmonic potential), and it is shown that varying the noise strength, if b b=2 (harmonic potential) or b > 2, this phenomenon does not occur; varying the parameters of the potential, the first moment of the system steady-state response and the stationary variance of the system response can also be non-monotonic.

A Computational Investigation of the Steady State Vibrations of Unbalanced Flexibly Supported Rigid Rotors Damped by Short Magnetorheological Squeeze Film Dampers

Unbalance is the principal cause of excitation of lateral vibrations of rotors and generation of the forces transmitted through the rotor supports to the foundations. These effects can be significantly reduced if damping devices are added to the constraint elements. To achieve their optimum performance, their damping effect must be controllable. The possibility of controlling the damping force is offered by magnetorheological squeeze film dampers. This article presents an original investigation of the dynamical behavior of a rigid flexibly supported rotor loaded by its unbalance and equipped with two short magnetorheological squeeze film dampers. In the computational model, the rotor is considered as absolutely rigid and the dampers are represented by force couplings. The pressure distribution in the lubricating layer is governed by a modified Reynolds equation adapted for Bingham material, which is used to model the magnetorheological fluid. To obtain the steady state solution of the equations of motion, a collocation method is employed. Stability of the periodic vibrations is evaluated by means of the Floquet theory. The proposed approach to study the behavior of rigid rotors damped by semi-active squeeze film magnetorheological dampers and the developed efficient computational methods to calculate the system steady state response and to evaluate its stability represent new contributions of this article.

A Chance-Constrained Optimization Approach for Control of Transmission Voltages

In this paper, a chance-constrained optimization (CCO) is presented to handle uncertainty in control of transmission voltages. A control scheme is proposed using a steady-state system model to achieve the goal of online voltage control and preventing long-term voltage instability. In order to model steady-state system response, the long-term model of governors and Automatic Voltage Regulators are employed in the control scheme. The Nordic32 test system is selected to show the simulation results of the proposed technique.

Stochastic resonance in an over-damped linear oscillator driven by multiplicative quadratic noise

For an over-damped linear oscillator driven by multiplicative quadratic noise and periodic modulated noise, the exact analytical expressions of the first two moments, the amplitude and variance of the system steady-state response are obtained. We find that this system has richer dynamic behaviors than the traditional linear systems driven by linear noise; when the coefficients of the quadratic noise satisfy certain conditions, both the amplitude and the variance of the system steady-state response present stochastic resonance.

Dynamics of a rotating shaft—disc under a periodic axial force

Under a periodic axial force, a rotating Timoshenko shaft with a rigid unsymmetrical disc was modelled as a parametrically excited system using the finite-element method. Using a harmonic balance method, dynamic stability of the system was checked, steady-state response, undamped resonance condition, and disc centre orbit were solved analytically and discussed numerically. Furthermore, the time history response was calculated to verify the efficiency of the solutions. The discussion shows that the fluctuating part of the axial force results in system dynamic instability, and the parameter regions of dynamic instability are enlarged with increasing amplitude of the fluctuation; the disc centre orbit of the system steady-state response is limited to an annular region, and the orbit width is increased by the axial force fluctuating amplitude; besides the neighbourhood of the system critical speed, the shaft—disc system can undergo some additional resonances due to the fluctuating axial force.

Steady-state Response of a Hysteretic Nonlinear System with Clearance

A dry friction damped vibration system with elastic stops model is established.By using harmonic linearization method the nonlinear hysteretic restoring force is expressed as equivalent damping and equivalent stiffness.The first order approximation of steady-state response of system subjected to a sinnsoidal excitation is investigated by means of the Kryov-Bogoliubov method.The analysis shows that the amplitude-frequency characteristic of system could be classified into two kinds which represent hysteretic nonlinearity,piecewise linear nonlinearity or the combination of the both in the different conditions of various parameters. Furthermore,the phenomena ofjump could be observed.The influence of changing each parameter upon the amplitode-frequency curve is discussed.

Vehicle-trailer stability and handling control: a working system

In this paper, a combined feedforward and feedback control design approach is proposed with the aim of improving vehicle-trailer overall system stability while maintaining vehicle-like handling performance. While the feedforward control is designed mainly to improve system steady-state response, the feedback control is primarily designed to improve system transient response by 'decoupling' both vehicle and trailer such that each tracks a well-defined second-order reference model. Vehicle testing has been conducted to verify the proposed approach with results indicating that the proposed control method is very effective in achieving required stability while maintaining vehicle-like handling performance.