Dynamic model based nonlinear tracking control of a planar parallel manipulator

Based on the dynamic model, a novel nonlinear tracking controller is developed to overcome the nonlinear dynamics and friction of a planar parallel manipulator. The dynamic model is formulated in the active joint space, and the active joint friction is described with the Coulomb + viscous friction model. A nonlinear tracking controller is designed to eliminate the tracking error by using the power function. The nonlinear tracking controller is proven to guarantee asymptotic convergence to zero of both the tracking error and error rate with the Barbalat’s lemma. The trajectory tracking experiment of the proposed controller is implemented on an actual five-bar planar parallel manipulator both at the low-speed and high-speed motion. Moreover, the control performances of the proposed controller are compared with the results of the augmented PD (APD) controller.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Flow and heat transfer of a slightly rarefied gas over a stretching surface

This work deals with the study of the boundary layer flow and mass transfer of a visco-elastic fluid immersed in a porous medium over a stretching surface in the presence of surface slip, chemical reaction and variable viscosity. The partial differential equations governing the flow have been transformed by similarity transformation into a system of coupled nonlinear ordinary differential equations which is solved numerically by means of the fourth order Runge-Kutta integration scheme coupled with the shooting technique. The effects of various involved interesting parameters on the velocity fields and concentration fields are shown graphically and investigated. In addition, tabulated results for the local skin-friction coefficient and the local Sherwood number are presented and discussed.     

Multiple crack propagation along the interface of a non-homogeneous composite subjected to anti-plane shear loading

The present work concerns with the multiple crack propagation along the interface of two bonded dissimilar strips. The crack faces are subjected to anti-plane shear traction. Galilean transformation is employed to reduce the problem to a quasi-static one. Then, using Fourier transforms and asymptotic analysis, the quasi-static problem is reduced to a pair of singular integral equations. That are solved numerically, using Gauss-Chebyshev integration formulae. The values of the dynamic stress intensity factors are obtained and compared with the previous similar works. Further, a parametric study is introduced to investigate the effect of crack growth rate, geometric and elastic characteristics of the composite on the values of dynamic stress intensity factors.     

Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance

Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated motions. The theoretical results are in quantitative agreement with the experimental data.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Detection of coarse-grained unstable states of microscopic/stochastic systems: a timestepper-based iterative protocol

We address an iterative procedure that can be used to detect coarse-grained hyperbolic unstable equilibria (saddle points) of microscopic simulators when no equations at the macroscopic level are available. The scheme is based on the concept of coarse timestepping (Kevrekidis et al. in Commun. Math. Sci. 1(4):715–762, 2003) incorporating an adaptive mechanism based on the chord method allowing the location of coarse-grained saddle points directly. Ultimately, it can be used in a consecutive manner to trace the coarse-grained open-loop saddle-node bifurcation diagrams of complex dynamical systems and large-scale systems of ordinary and/or partial differential equations. We illustrate the procedure through two indicative examples including (i) a kinetic Monte Carlo simulation (kMC) of simple surface catalytic reactions and (ii) a simple agent-based model, a financial caricature which is used to simulate the dynamics of buying and selling of a large population of interacting individuals in the presence of mimesis. Both models exhibit coarse-grained regular turning points which give rise to branches of saddle points.     

Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect

In the present paper, we study the effects of radiation on the thermal boundary layer flow induced by a linearly stretching sheet immersed in an incompressible micropolar fluid with constant surface temperature. Similarity transformation is employed to transform the governing partial differential equations into ordinary ones, which are then solved numerically using the Runge-Kutta-Fehlberg method. Results for the local Nusselt number as well as the temperature profiles are presented for different values of the governing parameters. It is found that the heat transfer rate at the surface decreases in the presence of radiation. Comparison with known results for certain particular cases is excellent.     

Time optimal motions of mechanical system with a prescribed trajectory

The problem of time minimization of a holonomic scleronomic mechanical system on a prescribed trajectory between two specified positions in configuration space is solved. The generalized force with restricted coordinates is taken as the controlling force. The application of the Green theorem (the well-known Miele method in flight mechanics) has shown that at every instant at least one control is at its boundary and possesses controlling functions with interruptions. It is assumed that at least one generalized coordinate exists that is monotonous during the interval of movement. An algorithm for numerical computation is presented for assessing the boundary of the admissible domain in the state space, thus, solving the problem of finding the optimal control as a function of time. Numerical integration is, therefore, carried out forward from the start point and backward from the end point by the use of the Runge-Kutta method. The mentioned procedure is illustrated in the example of time minimization for a manipulator which has its tip moving in a straight line. The application of the presented method simplifies solving of this type of problem compared to other methods, for instance, dynamic programming.     

Almost periodic solution of a non-autonomous model of phytoplankton allelopathy

In this paper, a non-autonomous two species competitive allelopathic phytoplankton model in presence of a discrete time delay is considered. We have obtained the sufficient conditions for permanence along with existence-uniqueness of an almost periodic solution. Sufficient conditions are derived for the existence of unique almost periodic solution. Analytical findings are supported through exhaustive numerical simulations. With the help of the numerical example, we have demonstrated that initial density dependent almost periodic co-existence is possible in some situations when parameter values fail to satisfy all the conditions of permanence.     

Delayed saturation controller for vibration suppression in a stainless-steel beam

This paper considers the effect of time delays on the saturation control of first-mode vibration of a stainless-steel beam. Time delay is commonly caused by measurements of the system states, transport delay, on-line computation, filtering and processing of data, calculating and executing of control forces as required in control processing. The method of multiple scales is employed to obtain the analytical solutions of limit cycles and their stability and to investigate the bifurcations of the system under consideration. All the predictions from analytical solutions are in agreement with the numerical simulation. The analytical results show that a delay can change the range of the saturation control, either widening or shrinking the effective frequency bandwidth. Thus, vibration control of a beam can be achieved using an appropriate choice of the delay in a self-feedback signal. From the examples illustrated, this paper provides a positive example that time delay can also be utilized to suppress vibration in systems when time delay cannot be neglected.     

Analysis of a frictional oblique impact observed in skew bridges

The oblique contact/impact of skew bridges triggers a unique rotational mechanism which earthquake reconnaissance reports correlate with deck unseating of such bridges. Building on the work of other researchers, the present study adopts a fully non-smooth rigid body approach and set-valued force laws, in order to analyze in depth this oblique multi-impact phenomenon. A linear complementarity formulation is proposed which yields a great variety of (multi-) impact states, depending on the initial (pre-impact) conditions, such as “slip” or “stick” at one corner (single-impact) or two corners (double-impact) of the body. The pertinent existential conditions of those impact states reveal a complex dynamic behavior. With respect to the rotational mechanism associated with double-impact, the physically feasible impact states as well as, counter-intuitive exceptions are recognized. The study proves that double oblique impact, both frictionless and frictional, may or may not produce rotation of the body and proposes criteria that distinguish each case. Most importantly, it is shown that the tendency of skew bridges to rotate (and hence unseat) after deck-abutment collisions is not a factor of the skew angle alone, but rather of the overall geometry in-plan, plus the impact parameters (coefficient of restitution and coefficient of friction). The study also provides a theoretical justification of the observed tendency of skew bridges to jam at the obtuse corner and rotate in such a way that the skew angle increases. Finally, counter-intuitive trends hidden in the response are unveiled which indicate that, due to friction, a skew bridge may also rotate so that the skew angle decreases.     

Continuously Dislocated Elastic Bodies with a Neo-Hookean Like Energy Subjected to Anti-plane Shear

In this paper, we consider a materially uniform but inhomogeneous body and we are interested in three particular cases of inhomogeneities corresponding to three distinct distributions of dislocations. The field of defects enters the equilibrium equations through the components of the tensor field describing the relaxation procedure. We examine what form should these components take in order for the material to admit states of anti-plane shear. The results obtained in this paper hold for a class of materials that obey a specific form for the stored energy function. In the special case of no dislocations, this class falls under the well known class of Neo-Hookean materials.      

Parameter identification of electronic throttle using a hybrid optimization algorithm

Aiming at the problems in parameter identification of an electronic throttle, this paper proposes a novel hybrid optimization algorithm to search the optimal parameter values of the plant. The parameter identification of an electronic throttle is considered as an optimization process with an objective function minimizing the errors between the measurement and identification, and the optimal parameter values of the plant are searched by using a hybrid optimization algorithm. The proposed hybrid optimization algorithm, effective combination of parallel chaos optimization algorithm (PCOA) and simplex search method, preserves both the global optimization capability of PCOA and the accurate search ability of simplex search method. Simulation and experiment results have shown the good performance of the proposed approach.     

Control problems of a mathematical model for schistosomiasis transmission dynamics

Drug treatment, snail control, cercariae control, improved sanitation and health education are the effective strategies which are used to control the schistosomiasis. In this paper, we formulate a deterministic model for schistosomiasis transmission dynamics in order to explore the role of the several control strategies. The basic reproductive number is computed. Sufficient conditions for the global asymptotic stability of the disease-free equilibrium are obtained. By using the Center Manifold Theory, we analyze the local stability of endemic equilibrium. Finally, numerical simulations support our analytical conclusions and the sensitive analysis on the basic reproductive number to the changes of control parameters are shown. Our results imply that snail-killing is the most effective way to control the transmission of schistosomiasis.     

The Theory of Thin-Walled Rods of Open Profile as a Result of Asymptotic Splitting in the Problem of Deformation of a Noncircular Cylindrical Shell

A new asymptotic approach to the theory of thin-walled rods of open profile is suggested. For the problem of linear static deformation of a noncircular cylindrical shell we consider solutions, which are slowly varying along the axial coordinate. A small parameter is introduced in the equations of the modern theory of shells. Conditions of compatibility for the shell strain measures are employed. The principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. We conclude the procedure with the subsequent solution for the field of displacements. The analysis shows that the known equations of thin-walled rods, which were previously obtained with some approximate methods using hypotheses and approximations of displacements, are asymptotically exact. The presented semi-numerical analysis of the shell equations allows us to estimate the accuracy of the obtained solution. The results of the paper constitute a sound basis to the equations of the theory of thin-walled rods and provide trustworthy information concerning the distribution of stresses in the cross-section.     

Stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises

The stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises is investigated by using the theory of signal-to-noise (SNR) in the adiabatic limit. The analytic expression of the SNR is obtained for arbitrary signal amplitude without being restricted to small amplitudes. The SNR is a nonmonotonic function of intensities of multiplicative and additive noises and the noise intensity ratio R=D/Q, so stochastic resonance exhibits in the bias monostable system. We investigate the effect of any system parameter (such as D,Q,R,r) on the SNR. It is shown that the SNR is a nonmonotonic function of the static asymmetry r, also; the SNR is decreased when |r| is increased. Moreover, the SNR is increased when the noise intensity ratio R=D/Q is increased.     

Second-order approximation of primary resonance of a disk-type piezoelectric stator for traveling wave vibration

Considering the quadratic nonlinear constitutive relations of piezoelectric materials, a traveling wave dynamic model for a lead zirconate titanate stator of a traveling wave ultrasonic motor is established using Hamilton’s principle and the Rayleigh–Ritz method. Applying the method of multiple scales, the second-order approximation of the primary resonance for traveling wave vibration of the stator is investigated. The second harmonic component is found in the primary response of the stator, which arises from the quadratic stiffness in the condition of weak excitation. In the region of the resonance, the two coupled modals are split and the lower-order peak bends to the left, hence a jump and delay exist in the response. In this way numerical results are given to verify the feasibility of the analytical approach. The results provide a theoretical foundation for further nonlinear dynamic analysis and design of the traveling wave ultrasonic motor.     

Bifurcation of periodic solutions and invariant tori for a four-dimensional system

In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der Pol–Duffing oscillator system is given.     

Nonlinear dynamics of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam’s immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.