Nonlinear Response Characteristics Research Articles

A GENERALIZED LOCAL LINEARIZATION PRINCIPLE FOR NON-LINEAR DYNAMICAL SYSTEMS

Tangent spaces in non-linear dynamical systems are state dependent. Hence, it is generally not possible to exactly represent a non-linear dynamical system by a linear one over finite segments of the evolving trajectories in the phase space. It is known from the well-known theorem of Hartman and Grobman that a non-linear oscillator admits a linearization within a neighbourhood of any hyperbolic fixed point in the sense that the linearized flow bears a topological conjugacy to the non-linear flow within the same neighbourhood. This linearization is based on the construction of the tangent space to the non-linear vector field at the hyperbolic fixed point. Even though useful in bifurcation analysis, such a linearization procedure cannot be successfully applied to identically simulate the non-linear flow over the phase space because the functional form of the topological conjugacy is not known. With this in mind, an implicit approach to local linearization is evolved in this paper such that the tangent space of the linearized equation transversally intersects the tangent space to the non-linear dynamical system at that point in the state space where the solution vector is desired. Several numerical schemes for the implementation of this implicit and local linearization procedure are explored and illustrated with several numerical results. It is shown that the locally transversal linearization (LTL) procedure finally reduces the given set of non-linear ordinary differential equations (ODEs) to a set of transcendental algebraic equations with the desired solution vector as the unknown. The variables in the state space appear as unknown quantities in this approach. Finally, one arrives at non-linear Poincare maps for the non-linear oscillator. In this scheme, the characteristic time interval for the map is arbitrarily fixed. The methodology is found to be quite versatile for handling non-linear dynamical systems. In particular, it is verified that these schemes are capable of accurately predicting a wide spectrum of typically non-linear response characteristics, such as limit cycles, multi-periodicity, almost periodicity and chaos. For a limited class of response patterns, the principle proposed has another advantage in that the time step for integrating the non-linear ODEs need not be small. An improved and a more general higher order version of the LTL method is also considered. A distinct advantage of this higher version is in its improved ability for a closer simulation of phase-dependent time histories where a difference in the initial conditions leads to a phase difference in the realized solution history.

An ANN-based smart capacitive pressure sensor in dynamic environment

A multilayer artificial neural network (ANN) is proposed for modeling of a capacitive pressure sensor (CPS). When the ambient temperature changes over a wide range, the nonlinear response characteristics of a CPS change significantly. In many practical conditions, the effect of temperature on the change in the CPS characteristics may be nonlinear. The proposed ANN model can provide correct readout of the applied pressure under such conditions. A novel scheme for estimation of the ambient temperature from the sensor characteristics itself is proposed. A second ANN is utilized to estimate the ambient temperature from the knowledge of the offset capacitance, i.e., the zero-pressure capacitance. A microcontroller unit (MCU)-based implementation scheme for this model is also considered. Simulation results show that this model can estimate the pressure with a maximum error of ±2% over a wide variation of temperature from −50°C to 150°C.

Nonlinear impedance of a microwave-driven Josephson junction with noise

The nonlinear impedance of a point Josephson junction is calculated under various conditions for the resistively shunted junction model in the presence of noise. The calculation proceeds by solving the Langevin equation for the mechanical problem of a Brownian particle in a tilted cosine potential in the presence of a strong ac force ignoring inertial effects. The exact solution of the infinite hierarchy of equations for the moments (expectation values of the Fourier components of the phase angle), which describe the dynamics of the junction, is expressed in terms of a matrix continued fraction. This solution allows one to evaluate the nonlinear response of the junction (nonlinear microwave impedance, for example) to an ac microwave current of arbitrary amplitude. Strong nonlinear effects in the resistance and the reactance are observed for large ac currents as is demonstrated by plotting the nonlinear response characteristics as a function of the model parameters. For weak ac currents and low noise strengths, our results agree closely with previously available linear response and nonlinear response noiseless solutions, respectively. Applications of the model to the interpretation of recent experimental data found in the literature for the nonlinear behavior of microwave impedance of superconducting weak links are discussed.

Auto-calibration and -compensation of a capacitive pressure sensor using multilayer perceptrons

Using multilayer perceptrons (MLPs), a smart model for a capacitive pressure sensor (CPS) is proposed. When the ambient temperature changes, the nonlinear response characteristics of a CPS may vary widely. Under such conditions, calibration of the sensor and compensation of the nonlinear sensor characteristics to obtain correct readout becomes a difficult task. The proposed MLP model can provide automatic nonlinear compensation and calibration of the CPS characteristics. A microcontroller unit (MCU)-based implementation scheme for this model is also considered. Simulation results show that this model can estimate the pressure with a maximum full-scale error of ±1% over a variation of temperature from −50 to 150°C.

ANN-based intelligent pressure sensor in noisy environment

A novel artificial neural network (ANN)-based intelligent capacitive pressure sensor (CPS) in noisy environment is proposed in this paper. A switched capacitor circuit (SCC) is used to convert the change in capacitance of the CPS due to applied pressure into a proportional voltage which is then applied to the ANN model to estimate the pressure. Because of the nonlinear response characteristics of the CPS and its temperature dependence, complex signal processing of the SCC output is required to estimate the applied pressure accurately, especially when the room temperature changes with time, place, or both. The situation becomes further complicated when the CPS encounters random noise, as is the case in many practical situations. To alleviate these difficulties in estimation of unknown applied pressure in a CPS, a multilayer perceptron (MLP) has been utilized to model the CPS characteristics over a wide temperature range with noise. By training the MLP model suitably, a direct digital readout of the applied pressure can be obtained. From the simulation studies it was verified that the performance of this model is quite satisfactory for a wide variation of temperature, starting from −20°C to 70°C, and for a signal-to-noise ratio (SNR) of 40 dB and above. This modeling technique provides greater flexibility and accuracy in a changing and noisy environment.

Nonlinear dc response in composites: A percolative study

The dc response, namely the $I\ensuremath{-}V$ and $G\ensuremath{-}V$ characteristics, of a variety of composite materials are in general found to be nonlinear. We attempt to understand the generic nature of the response characteristics and study the peculiarities associated with them. Our approach is based on a simple and minimal bond percolative network model. We do simulate the resistor network with appropriate linear and nonlinear bonds and obtain macroscopic nonlinear-response characteristics. We discuss the associated physics. An effective-medium approximation of the corresponding resistor network is also given.

Characteristics of nonlinear response of deep saturated soil deposits

Abstract Recent EPRI seismic design guidelines call for dynamic soil properties (shear modulus ratio and damping) and liquefaction strength curves to be characterized as a function of the effective vertical stress (or depth). A modified version of the DESRA2 constitutive model for saturated soil has been applied to study the nonlinear seismic response including liquefaction of medium dense soil deposits of various thicknesses. The results of the stress-dependent soil properties model show lower deamplification and higher first-mode (resonant) frequency than that of the stress-independent soil properties model. By using the stress-dependent model with impulse base excitation, the nonlinear behavior of various soil deposits has been investigated under a variety of conditions. The results show that (1) the saturated soil deposit has a smaller surface amplitude and significantly lower resonant frequency than the unsaturated soil deposit of the same thickness; (2) for the saturated soil conditions, the larger the base excitation, the lower the surface amplification and the resonant frequency; (3) the deep soil deposits show lower surface amplification and resonant frequency compared to the response of shallow deposits; (4) when shallow and deep deposits are compared, the shallow deposits develop much higher residual pore-water pressure; and (5) the amplification and residual pore-water-pressure response of deposits deeper than 100 m or so are very similar. The application of the method has also been illustrated using a strong synthetic base excitation applied to the base at a site near Reno. The results in general are consistent with those computed using the impulse loading. The study reveals that the response predicted from the conventionally used stress-independent soil properties model is unconservative for deep deposit.

Finite Element Method for Nonlinear Free Vibrations of Composite Plates

A multimode time-domain modal formulation based on the finite element method for large-amplitude free vibration of thin composite plates is presented. Accurate frequency-maximum deflection relations can be predicted for the fundamental and the higher nonlinear modes. A modal participation is defined, and accurate and convergent frequencies can be determined with minimum number of linear modes. A procedure for the selection of initial conditions for periodic plate response is presented. Convergence of frequency with gridwork refinement and number of linear modes is studied. The classical single-mode elliptic function frequency solutions for simply supported beams and square plates are assessed. Examples of orthotropic and composite plates are given, and the characteristics of nonlinear response are studied.

Dispersion of characteristics of optical nonlinear response used for phase conjugation

The frequency dispersion and decay time dependence for the third-order nonlinear optical response in a substance and the dependence of its value on material parameters are analyzed on the basis of an anharmonic-oscillator approach for a variety of mechanisms of optical nonlinear response.

Calibration of VISSR on board GMS-5

Operational Calibration method of visible and IR channels of GMS-5 VISSR is as follows. In the visible channel, albedo is computed by the following equation. A={ (C−b 0) b 1} 2 a − V 0 a , where C and V 0 are digital counts from VISSR and V 0 is bias voltage corresponding to 0 albedo respectively, other coefficients are determined from prelaunch test data. A principle of IR channel calibration depends upon a linear relation between incoming IR radiance and output voltage of IR channel. Taking the space as a cold source and a calibration shutter in VISSR as a high source, a calibration regression equation to convert output voltage of IR channel into temperature is determined. A little correction is necessary to compensate non-linear response characteristics of IR detector at low temperature. To cope with a malfunction of calibration shutter (not frequent occurrence) a method of calibration without a shutter information is developed. The monitoring of solar radiance for one year indicates a small degradation in visible channel detectors. The degradation among 4 detectors is not uniform.

Wave propagation in nonlinear multilayer structures.

We investigate the adequacy of the Kronig-Penney \ensuremath{\delta}-function model in describing the electromagnetic wave propagation in periodic structures consisting of thin layers of materials with an intensity-dependent dielectric constant. We find that the model captures the most essential features of nonlinear response to radiation. Excellent agreement is found between the results from the \ensuremath{\delta}-function model and the exact solutions of nonlinear wave equations. However, discrepancies do exist below the bottom of transmission bands due to the rigidness of the band edge in the \ensuremath{\delta}-function model. Consequently, gap solitons cannot form in the \ensuremath{\delta}-function model when the nonlinear Kerr coefficient is positive. \textcopyright{} 1996 The American Physical Society.

A bound for the magnitude characteristics of nonlinear output frequency response functions Part 2: Practical computation of the bound for systems described by the nonlinear autoregressive model with exogenous input

In Part 1 of this paper the concept of a bound for the output frequency response magnitude characteristics of nonlinear systems was proposed, and general calculation and analysis procedures were developed. In this, Part 2 of the paper, a new recursive algorithm for the computation of the gain bounds for the generalized frequency response functions of the polynomial nonlinear autoregressive model with exogenous input is proposed, and effective procedures for the practical computation of the new bound are developed. Simulated examples are included to verify the effectiveness of the proposed procedures.

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.

A bound for the magnitude characteristics of nonlinear output frequency response functions Part 1. Analysis and computation

A bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs. It is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included. The paper is divided into two parts. In Part 1, an expression for the output magnitude bound is derived, properties of the result are discussed and general procedures for the practical computation of the bound are developed. In Part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed.

Effect of Vertical Seismic Load on Shear-Bending Buckling Strength of Thin Cylindrical Shells

In order to investigate various nonlinear response characteristics, including buckling, of thin cylindrical shells under vertical and horizontal seismic motion, pseudodynamic experiments and nonlinear response simulation analysis are performed. It is confirmed that buckling is caused mainly by horizontal seismic loads, and that vertical seismic loads reduce the lateral load-carrying capacity of cylinders and amplify the response displacement for a given horizontal seismic load. To evaluate the amplification of nonlinear horizontal responses due to vertical input motions, we define a response amplification factor, which is calculated from the floor response spectra of seismic waves.

Ionic current model of the retinal photoreceptor and simulation analysis of the photoresponses

AbstractThe fact that the reception of optical information by the retinal photoreceptor and the process of its transformation to the membrane voltage response are due to an interaction between the photocurrent from the outer segment and each ionic current in the inner segment has been confirmed by many physiological experiments. However, the properties of each ionic current in the photoreceptor have not been clear since no physiologically reliable model of the retinal photoreceptor has been constructed.This paper proposes a model of a retinal photoreceptor, which is a combination of the outer segment model (devised by Torre et al.) and the inner segment model (devised by the present authors). It has been confirmed that this model can accurately represent the electron physiological properties of a retinal photoreceptor. Using this model simulation, the relationship between the photocurrent and the photoresponse dynamics, and an oscillatory phenomena caused by a strong light flash have been analyzed. The results show that the dynamic and nonlinear response characteristics of a retinal photoreceptor are due to the interactions between the nonlinear ionic current in the inner segment and the mechanism of the intracellular calcium.

Developments in industrial applications of self-tuning control

Control strategies based on auto-tuning and self-tuning principles are now emerging to provide a realistic option for a wide range of industrial processes and plant. This is particularly the case when considering batch and fed-batch process control systems in which assumptions on slowly varying process dynamics permit the use of such adaptive schemes. However, there are many industrial systems where commercial ‘off the shelf’ controllers do not provide an adequate solution. Consequently there is much on-going research activity in the refining and tailoring of algorithms in order to provide effective implementation on specific industrial applications. With consideration given to two industrial case-study applications, the paper describes extensions and refinements that have been introduced in order to accommodate for improved estimation and control law design algorithms when dealing with systems exhibiting fast dynamic behaviour and non-linear response characteristics respectively. The requirement for improved estimation schemes arises when one is compelled to make use of low-order linear model structures whereas when dealing with non-linear systems, which exhibit slowly varying dynamic characteristics, one can afford to invest in improvements in control law design.

The non-linear analysis of tapered bonded joints

The highly non-linear characteristics of adhesives for bonding aerospace structures have not been accounted for in most cases due to the complexities associated with non-linear analysis, and the linear solutions generally yield overly conservative results for a reasonable assessment of joint strength. Tapered joints are inherently efficient because of the reduction in peeling stresses at the adherent tips. In this paper, a non-linear analysis technique is developed for these joints. A Voltera integral equation of the second kind is formulated for an arbitrary adherend tapered geometry and adhesive non-linear material properties. A numerical solution technique for the integral equation is developed using the successive approximation method. The accuracy and stability of the resulting numerical solutions are typical of the integral equation approach. A more reliable assessment of joint strength is introduced from the non-linear response characteristics.

Modeling nonlinear features of V tail aircraft using MNN

The nonlinear stability and control modeling involves a very popular high performance general aviation aircraft that uses a tail assembly instead of the traditional inverted T tail. The nonlinear response features of this aircraft result from a well developed Dutch roll mode and are caused by a dynamic stall phenomenon that occurs on the V tail during the maneuver. A new approach using multilayered neural network (MNN) for modeling the nonlinear features of this aircraft is suggested here. Both the conventional backpropagation (BP) and the extended Kalman filter (EKF)-based learning algorithm are used for training the neural network. Simulation results that confirm the efficacy of the method are given. Further, performance comparison of the EKF-based and the conventional BP algorithm is made to highlight the effectiveness of the EKF-based learning algorithm. >

Investigation of non‐linear site amplification at two downhole strong ground motion arrays in Taiwan

AbstractNon‐linear seismic response of soil is studied by comparing the spectral ratios of surface to downhole horizontal accelerations on weak and strong motion. Data from two boreholes are analysed. One is drilled in the alluvial deposits in the south–west quadrant of the SMART 1 array. The second one penetrates Pleistocene terrace deposits in the northern part of the SMART2 array. Observed weak and strong motion spectral ratios are compared with the theoretical ones predicted by the geotechnical soil model which postulates a hysteretic constitutive law. A significant non‐linear response is found at the first site for the events with surface peak acceleration exceeding roughly 0–15g. Deamplification of the strong motion occurred in the frequency range from approximately 1 to 10 Hz. The maximum observed difference between the average weak and strong motion amplification functions of an 11 m‐thick near‐surface stratum is a factor of 2–3. Nonlinear response characteristics are in qualitative agreement with the model. An additional corollary is that the amplification function calculated from the shear wave coda is equivalent to the average amplification calculated over the ensemble of small earthquakes. No statistically significant non‐linear response is detected on the second array, that is tentatively accounted for by the stiffer soil conditions and weaker accelerations achieved at the SMART2 site. The results indicate that the non‐linear amplification can be detectable at certain soil conditions above a threshold acceleration level.