Balance control naturally deteriorates with age, so it comes as no surprise that nearly 30% of the elderly population in the United States report stability problems that lead to difficulty performing daily activities or even falling. Postural stability is an integral task to daily living which is reliant upon the control of the ankle and hip. To this end, the estimation of ankle and hip parameters in quiet standing can be a useful tool when analyzing compensatory actions aimed at maintaining postural stability. Using an analytical approach, this work builds upon the results obtained by the authors and expands it to a two degrees of freedom system where the first two modes of vibration of a standing human are considered. The physiological parameters a second-order Kelvin-Voigt model were estimated for the actuation of the ankle and hip. Estimates were obtained during quiet standing when healthy volunteers were subjected to a step-like perturbation. This paper presents the analysis of a second-order nonlinear system of differential equations representing the control of lumped muscle-tendon units at the ankle and hip. This paper utilizes motion capture measurements to obtain the estimates of the control parameters of the system. The dynamic measurements are utilized to construct a simple time-dependent regression that allows calculating the time-varying estimates of the control and body segment parameters with a single perturbation. This work represents a step forward in estimating the control parameters of human quiet standing where, usually, the analysis is either restricted to the first vibrational mode of an inverted pendulum model or the control parameters are assumed to be time-invariant. The proposed method allows for the analysis of hip related movement in the control of stability and highlights the importance of core muscle training.

Emissions compliance during engine start-up conditions is a major obstacle for current automotive manufacturers across global markets. The challenges to meeting emissions targets are both due to increasingly stringent regulations and the difficulty in developing control strategies for a high degree-of-freedom and highly non-linear system. Online extremum-seeking (ES) methods offer a promising alternative to traditional optimization based on design-of-experiment based automotive calibration. With extremum-seeking methods, results from all prior experiments are used to intelligently and efficiently generate the next iteration of the control parameter(s). In this work, the applicability of the online extremum-seeking method is explored to optimize five performance variables (injection timing for two injection events, the injection fuel mass divided between the first and second injection events, air-fuel equivalence ratio and exhaust cam timing) to minimize brake specific fuel consumption while imposing different constraints on NO x emissions. The experiments were conducted using a production turbocharged four-cylinder gasoline engine with an advanced fuel injection system. The results show the utility of the ES strategy to quickly identify optimal control parameter combinations and the emissions and engine performance improvements during the calibration process. The results also demonstrate the dramatic benefit of the ES calibration strategy in terms of test time required.

The structural health monitoring by piezoelectric wafer active sensor (PWAS) using electromechanical impedance method used for monitoring of structure. In present work impedance method of elasto-plastic beam structure is studied. In order to model the effect of a plastic in beam, the moment-curvature relationship for elasto-plastic region for loading and unloading is used. The finite difference method is used to discretize beam with piezoelectric. The piezoelectric actuator is modeled by equivalent moment. Then output current of piezoelectric sensor is calculated. Firstly, elastic modeling of beam is considered that this is leads to linear system equation. In linear system, time domain system equations are calculated and Fourier transform of current output obtained, and then impedance of PWAS in frequency domain is calculated. Secondly, the elasto-plastic of beam is modeled. This phenomenon leads to the nonlinear system equations. These nonlinear equations are solved using finite difference method for any harmonic voltage applied to actuator. Then impedance of PWAS is calculated. Two methods are used to detect elasto-plastic modeling on PWAS impedance. At the first, frequency response of elastic beam as intact model is compared with elasto-plastic results in a desired frequency range. Second, only frequency response of one harmonic is computed with its super-harmonics. Finally, the detection method of linear is compared with nonlinear model.

In this paper, periodic motions in a first-order, time-delayed, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motion. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation tree of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

A nonlinear dynamics investigation is conducted on the frequency-amplitude response of electrostatically actuated micro-electro-mechanical system (MEMS) clamped plate resonators. The Alternating Current (AC) voltage is operating in the realm of superharmonic resonance of second order. This is given by an AC frequency near one-fourth of the natural frequency of the resonator. The magnitude of the AC voltage is large enough to be considered as hard excitation. The external forces acting on the MEMS resonator are viscous air damping and electrostatic force. Two proven mathematical models are utilized to obtain a predicted frequency-amplitude response for the MEMS resonator. Method of Multiple Scales (MMS) allows the transformation of a partial differential equation of motion into zero-order and first-order problems. Hence, MMS can be directly applied to obtain the frequency-amplitude response. Reduced Order Model (ROM), based on the Galerkin procedure, uses mode shapes of vibration for undamped circular plate resonator as a basis of functions. ROM is numerically integrated using MATLAB software package to obtain time responses. Also, ROM is used to conduct a continuation and bifurcation analysis utilizing AUTO 07P software package in order to obtain the frequency-amplitude response. The time responses show the movement of the center of the MEMS circular plate as a function of time. The frequency-amplitude response allows one to observe bifurcation and pull-in instabilities within the nonlinear system over a range of frequencies. The influences of parameters (i.e. damping and voltage) are also included in this investigation.

In this paper, a robust adaptive controller is proposed for a class of uncertain nonlinear systems subject to time-varying input delay, parametric uncertainties and additive bounded disturbances. The desired trajectory based adaptive feedforward technique and a predictor-like robust delay compensating term are integrated via backstepping in the controller design. The proposed controller theoretically ensures semi-global uniformly ultimately bounded tracking performance based on Lyapunov stability analysis by employing Lyapunov-Krasovskii (LK) functionals. Simulation results are obtained to illustrate the effectiveness of the proposed control strategy.

One of the main objectives of the structural health monitoring by piezoelectric wafer active sensor (PWAS) using electromechanical impedance method is continuously damage detection applications. In present work impedance method of beam structure is considered and the effect of early crack using breathing crack modeling is studied. In order to model the effect of a crack in beam, the beam is connected with a rotational spring in crack location. The Rayleigh–Ritz method is used to generate ordinary differential equation of cracked beam. Firstly, only open crack is considered that this is leads to linear system equation. In linear system, time domain system equations are converted to frequency domain, and then impedance of PWAS in frequency domain is calculated. Secondly, the breathing crack is modeled to be fully open or fully closed. This phenomenon leads to the nonlinear system equations. These nonlinear equations are solved using pseudo-arc length continuation scheme and collocation method for any harmonic voltage applied to actuator. Then impedance of PWAS is calculated. Two methods are used to detect early crack using breathing crack modeling on PWAS impedance. At the first, frequency response of breathing crack in the frequency range with its sub-harmonics is calculated. Second, only frequency response of one harmonic is computed with its super-harmonics. Finally, the detection method of linear is compared with nonlinear model.

This paper presents the nonlinear vibration of a simply supported Euler-Bernoulli beam with a mass-spring system subjected to a primary resonance excitation. The nonlinearity is due to the mid-plane stretching and cubic spring stiffness. The equations of motion and the boundary conditions are derived using Hamiltons principle. The nonlinear system of equations are solved using the method of multiple scales. Explicit expressions are obtained for the mode shapes, natural frequencies, nonlinear frequencies, and frequency response curves. The validity of the results is demonstrated via comparison with results in the literature. Exact natural frequencies are obtained for different locations, rotational inertias, and masses.

Parameter estimation and model order reduction (MOR) are important techniques used in the development of mechanical system models. A variety of classical parameter estimation and MOR methods are available for nonlinear systems but performance generally suffers when little is known about the system model a priori. Recent advancements in information theory have yielded a quantity called causation entropy, which is a measure of the influence between multivariate time series. In parameter estimation problems involving dynamic systems, causation entropy can be used to identify which functions in a discrete-time model are important in driving the subsequent state values. This paper extends on previous works’ use of a Causation Entropy Matrix to nonlinear systems modeled from the real world. This work explores the conversion of continuous systems to a discrete model and applies the causation entropy matrix to the system. Results show that model structure can be estimated by the causation entropy matrix. This work extends the previous work by showing that the method can be applied to general nonlinear systems. Previously shown examples were toy, additively separable nonlinear problems. This work shows that the methodology can be extended to any nonlinear system, including time varying systems, which provides a framework to examine parameter estimation for general nonlinear systems.

Rapid deployment of marine structures is of growing importance to U.S. naval forces. Surface-based inflatable structures including Rigid Inflatable Boats (RIBs), inflatable causeways and bridging, and launch and recovery systems provide unique solutions for temporary structures during sea-based missions. When performance specifications demand minimal weight and stowage, rapid deployability and temporary rigidity, solutions are limited to inflatable structures constructed of flexible materials. Driven by air pressure, today’s inflatables provide significant load-carrying capacities per unit weight (or stowed volume) utilizing technical textiles, elastomers or “soft” composites. Overloading of inflatable structures produces unique fail-safe behaviors (reversible wrinkling) that allow the structures to assume rigidity and load-carrying capacity upon load removal. Design standards are virtually nonexistent for inflatable structures involving shapes constructed of spheres, beams, arches and most recently flat panels using 3D woven drop-stitch panels. Predictive performance tools (analytical and numerical) for static applications lag significantly behind those for conventional structures. Nonlinear system behaviors (material and geometric), thermo-mechanical coupling and fluid-structure interactions (FSI’s) pose significant challenges when applying existing design tools to inflatable structures. This gap is further exacerbated for dynamic applications as inflatable structures exhibit pressure-dependent natural frequencies and mode shapes. Surface-based structures must be designed with consideration given to operational sea state frequencies and wave periods so that the onset of structural instabilities (wrinkling, buckling) and loss of load-carrying capacities can be prevented. The present research establishes the validity of physics based models using the Ideal Gas Law as an equation of state (EOS) to predict the natural frequencies and corresponding mode shapes of air-inflated drop-stitch fabric panels as functions of inflation pressure. Particular concern is given to the breathing modes for inflation pressures ranging from 5.0 to 30.0 psig. The presence of breathing modes can negatively impact the riding performance of RIBs vessels constructed with drop-stitch fabric hulls by amplification of the panel’s skin separation displacements and vertical accelerations, and are not seen in this material system for the pressures considered. Both numerical and experimental methods are pursued; the results of laboratory modal experiments are used to validate the numerical models. Predicted and experimental natural frequencies and mode shapes are compared and excellent correlation was observed. Increasing inflation pressures produced increasing in-plane and through-thickness normal stresses and modal frequencies of the drop-stitch fabric panels.

Input shaping suppresses residual vibration by destructive interference of the impulse responses. Because proper destructive interference requires superposition property of the linear system, traditional input shaper only applies to the linear flexible system. In this paper, the work and energy principle is used to derive input shaper for flexible system having nonlinear spring and damper. It was shown via simulation and experiment that this type of shaper performs well with nonlinear systems. Positive, robust, and negative input shapers are discussed.

Metamaterials demonstrate unique frequency dependent responses due to the presence of internal resonators; hence, it can be used to filter, absorb, cloak, or otherwise manipulate waves in unique ways. However, its applicability is normally limited to a very narrow frequency range (bandwidth) due to a dependency on linear resonance. The applications of these linear metamaterials are limited when used under the broadband excitation spectra that are common in real life applications. This paper numerically investigates the effect of introducing the two main classes of Duffing type cubic nonlinearities, namely monostable and bistable, on the attenuation bandwidth of an elasto-dynamic metamaterial. From the analysis, it is found that the attenuation bandwidth of a bistable nonlinear system is two to three times wider than that of an equivalent linear system; whereas, in case of a monostable system the bandwidth is remained same. In both cases, the attenuation bandwidth shifts towards the higher end of the frequency spectra and for higher nonlinearity and excitation amplitude, second transmission zone completely vanishes.

Non-parametric and parametric identification of a non-linear system is often performed by estimating instantaneous amplitude and frequency using the Hilbert transform. However, the Hilbert transform cannot be used for the accurate analysis of asymmetric signals and the reliable estimation of intra-wave frequency modulation. This paper proposes two alternatives to the Hilbert transform which not only avoid some of its mathematical and numerical issues, but also allow the above mentioned analyses. The first method, based on zero-crossing, allows the backbone and damping curves as well as the elastic and damping force characteristics of an asymmetric free decay to be identified. The application and accuracy of this method are demonstrated on the free decay of the system with off-centre clearance. The second method, based on direct quadrature, estimates intrawave frequency modulation frequency with sufficient resolution for characterization of non-linear systems which have stiffness non-linearities. The use of this method is shown on a system with cubic hardening stiffness.

A novel time-frequency nonlinear scheme demonstrated to be feasible for the control of dynamic instability including bifurcation, non-autonomous time-delay feedback oscillators, and route-to-chaos in many nonlinear systems is applied to the control of a time-delayed system. The control scheme features wavelet adaptive filters for simultaneous time-frequency resolution. Specifically Discrete Wavelet transform (DWT) is used to address the nonstationary nature of a chaotic system. The concept of active noise control is also adopted. The scheme applied the filter-x least mean square (FXLMS) algorithm which promotes convergence speed and increases performance. In the time-frequency control scheme, the FXLMS algorithm is modified by adding an adaptive filter to identify the system in real-time in order to construct a wavelet-based time-frequency controller capable of parallel on-line modeling. The scheme of such a construct, which possesses joint time-frequency resolution and embodies on-line FXLMS, is able to control non-autonomous, nonstationary system responses. Although the controller design is shown to successfully moderate the dynamic instability of the time-delay feedback oscillator and unconditionally warrant a limit cycle, parameters are required to be optimized. In this paper, the setting of the control parameters such as control time step, sampling rate, wavelet filter vector, and step size are studied and optimized to control a time-delay feedback oscillators of a nonautonomous type. The time-delayed oscillators have been applied in a broad set of fields including sensor design, manufacturing, and machine dynamics, but they can be easily perturbed to exhibit complex dynamical responses even with a small perturbation from the time-delay feedback. These responses for the system have a very negative impact on the stability, and thus output quality. Through employingfrequency-time control technique, the time responses of the time-delay feedback system to external disturbances are properly mitigated and the frequency responses are also suppressed, thus rendering the controlled responses quasi-periodic.

In recent decades, nonlinear time-delay systems were often applied in controlling nonlinear systems, and the stability of such time-delay systems was very actively discussed. Recently, one was very interested in periodic motions in nonlinear time-delay systems. Especially, the semi-analytical solutions of periodic motions in time-delay systems are of great interest. From the semi-analytical solutions, the nonlinearity and complexity of periodic motions in the time-delay systems can be discussed. In this paper, time-delay effects on periodic motions of a periodically forced, damped, hardening, Duffing oscillator are analytically discussed through a semi-analytical method. The semi-analytical method is based on discretization of the differential equation of such a Duffing oscillator to obtain the corresponding implicit discrete mappings. Through such implicit mappings and mapping structures of periodic motions, period-1 motions varying with time-delay are discussed, and the corresponding stability and bifurcation analysis of periodic motions are carried out through eigenvalue analysis. Numerical results of periodic motions are illustrated to verify analytical predictions. The corresponding harmonic amplitude spectrums and harmonic phases are presented for a better understanding of periodic motions in such a nonlinear oscillator.

In this paper, a quadratic nonlinear dynamical system with two periodic excitation forces is discussed. Analytical period-1 motions of such dynamical system are obtained by using generalized harmonic balance method. Stability analysis is carried out via eigenvalues analysis. To verify approximate analytical solutions, numerical simulations are completed to compare analytic and numerical solutions of the dynamical system, the approximate precision is guaranteed with appropriate harmonic balance terms. More harmonic terms should be employed to guarantee good approximation of periodic motions if excitation frequency is small. Furthermore, infinite harmonic balance terms must be introduced for chaotic systems.

In this paper, analytical solutions of periodic motions in the first-order nonlinear dynamical system are discussed from the finite Fourier series expression. The first-order nonlinear dynamical system is transformed to the dynamical system of coefficients in the Fourier series. From investigation of such dynamical system of coefficients, the analytical solutions of periodic motions are obtained, and the corresponding stability and bifurcation of periodic motions will be determined. In fact, this method provides a frequency-response analysis of periodic motions in nonlinear dynamical systems, which is alike the Laplace transformation of periodic motions for nonlinear dynamical systems. The harmonic frequency-amplitude curves are obtained for different-order harmonic terms in the Fourier series. Through such frequency-amplitude curves, the nonlinear characteristics of periodic motions in the first-order nonlinear system can be determined. From analytical solutions, the initial conditions are obtained for numerical simulations. From such initial conditions, numerical simulations are completed in comparison of the analytical solutions of periodic motions.

This paper addresses the control problem for trajectory tracking of a class of robotic manipulators presenting uncertainties and switching constraints using a biomimetic approach. Uncertainties, system-inherent as well as environmental disturbances deteriorate the performance of the system. A change in constraints between the robot’s end-effector and the environment resulting in a switched nonlinear system, undermines the stable system performance. In this work, a robust adaptive controller combining sliding mode control and BELBIC (Brain Emotional Learning-Based Intelligent Control) is suggested to remediate the expected impacts on the overall system tracking performance and stability. The controller is based on an interplay of inputs relating to environmental information through error-signals of position and sliding surfaces and of emotional signals regulating the learning rate and adapting the future behaviour based on prior experiences. The proposed control algorithm is designed to be applicable to discontinuous freeform geometries. Its stability is proven theoretically and a simulation, performed on a two-link manipulator verifies its efficacy.

Piping system in thermal power plants usually include nonlinear boundary condition (e.g. Single acting and gap supports) to eliminate the high thermal stresses since those systems are operating in high temperature range with large thermal displacement. Using this kind of non-linear supports may include error due to the linear approximation when it comes to study the response of the piping system in the dynamic analysis. Most of the common pipe stress analysis programs such as CAESAR II [1].cannot perform a dynamic analysis for nonlinear systems. This paper is investigating the effect on maximum reaction forces when modelling nonlinear boundary condition using the dynamic analysis module in ANSYS program [2]. The aim of of this paper is to estimate the impact of gaps and to study the effect of different pipe support stiffness. The dynamic analysis is performed for a range of different gap and stiffness support arrangements. The results from the nonlinear analysis are compared with regular linear analysis (supports with zero gaps) and a correlation is established indicating that large spikes in dynamic force are expected when large gaps are included in dynamic analysis. The results of the paper shows that the design engineer must be very caution in using linear dynamic analysis when large gaps are included and that the nonlinear analysis is required when dynamic amplification is significant.

Various vibration problems occur in rotating machinery. Specifically, the amplitude of shaft motion can be large in the case of vertical pump using a journal bearing. In such case, rotor motion such as self-excited vibration is not able to be predicted accurately without considering both rotordynamic fluid force and contact phenomena to the bearing surfaces or boundary lubrication condition. Moreover, in an actual vertical pump, the stationary side structure that supports the pump shaft may be relatively flexible. However, the vibration characteristics in such situations have not been so far explained simply. The purpose of this paper is a development of the simple and quantitative method for evaluating vibration characteristics of a vertical rotating shaft with journal bearing. In this study, the rotor system flexibly supported by a journal bearing at the axial center is considered. Eigenvalue analysis is demonstrated for the linearized system, and steady-state vibration analysis of the self-excited vibration is carried out for the nonlinear system. As a result, the effect on the vibration characteristics of the system by changing fluid viscosity, radial clearance, stiffness and damping of the support is clarified. Moreover, the obtained experimental results correspond to the theoretical results of the self-excited vibration qualitatively.