Generalization of Seide’s problem by the regulated stochastic linearization technique

In this paper, the nonlinear random vibration of a uniform beam clamped at both of its ends is investigated by the regulated stochastic linearization technique. A closed system of nonlinear algebraic equations for linearization coefficients is obtained using the frequency–response function matrix. Root mean-square response of maximum deflection of the beam that is obtained from the present technique is subsequently compared with that furnished by the conventional linearization. The numerical results show that the regulated stochastic linearization technique constitutes an excellent alternative to the classical linearization scheme for analyzing responses of the clamped–clamped beam.     

Bifurcation analysis for non-local design of a hybrid observer for the impulsive Goodwin’s oscillator

Nonlinear Dynamics - The impulsive Goodwin’s oscillator is a mathematical model capturing the dynamics arising in a closed-loop system, where a third-order linear time-invariant plant is...     

Longitudinal–lateral velocity control design and implementation for a model-scaled unmanned helicopter

In this paper, a nonlinear controller is designed and implemented for longitudinal–lateral motion of a model-scaled helicopter. The underlying principle of controller design is the backstepping technique with slight modifications to accommodate the helicopter model. It is proved theoretically that, under the proposed controller, velocities and yaw angle of the closed-loop system are capable of tracking reference signals. A practical helicopter testbed is constructed to test the performances of the closed-loop system. Experimental results of practical flight tests demonstrate that performances of the closed-loop system are satisfactory.     

Optimal control of the transmission dynamics of tuberculosis

This paper deals with the problem of optimal control for the transmission dynamics of tuberculosis (TB). A tuberculosis model which incorporates the essential biological and epidemiological features of the disease such as exogenous reinfection and chemoprophylaxis of latently infected individuals, and treatment of the infectious is developed and rigorously analyzed. Based on this continuous model, the tuberculosis control is formulated and solved as an optimal control theory problem, indicating how a control term on the chemoprophylaxis should be introduced in the population to reduce the number of individuals with active TB. The feedback control law has been proved to be capable of reducing the number of individuals with active TB. An advantage is that the proposed scheme accounts for the energy wasted by the controller and the closed-loop performance on tracking. Numerical results show the performance of the optimization strategy.     

On Saint-Venant’s principle in the dynamics of elastic beams

In dynamics, Saint-Venant’s principle of exponential decay of stress resulting from a self-equilibrating load is not valid. For a beam type structure, a self-equilibrated load may penetrate well inside the beam. Although this effect has been known for a long time, at least since Lamb’s paper [Proc. Roy. Soc. Lon. Ser. A 93 (1916) 114], it was not clear how to characterize it quantitatively. In this paper we propose a “probabilistic approach” to evaluate the magnitude of the penetrating stress state. The key point is that, in engineering problems, the distribution of the self-equilibrated load is usually not known. By assigning to the self-equilibrated load some probabilistic measure one can find probabilistic characteristics of the penetrating stress state. We develop this reasoning for the simplest case: longitudinal vibrations of a two-dimensional semi-infinite, elastic isotropic homogeneous strip, excited by a periodic load at the end. We show the frequency range where Saint-Venant’s principle can be used with good accuracy, and thus, one-dimensional classical beam theory still can be applied. We characterize also the increase in this range which is achieved in the refined plate theory proposed by Berdichevsky and Le [J. Appl. Math. Mech. (PMM) 42 (1) (1978) 140].     

Optimal control of phytoplankton–fish model with the impulsive feedback control

Nonlinear Dynamics - Considering how to exploit the lake resource reasonably, we propose a phytoplankton–fish model with the impulsive feedback control and investigate the sufficient...     

Kepler’s equation and limit cycles in a class of PWM feedback control systems

The aim of this paper is to point out some new results concerning the ripple instability in the closed-loop control system using pulse width modulators (PWM), with natural sampling, as power amplifier. The presented analysis, based on the dual-input describing function method and the theoretical framework of Kepler’s problem, shows an equivalence between the computation of switching instants of the PWM and the eccentric anomaly of the planet orbit around the sun, giving a simple stability criterion and a sufficient condition for the absence of solutions of the harmonic balance equation and, therefore, the probable absence of limit cycles of a period of a multiple of that characteristic of the modulator. The derived stability criterion, by using the describing function method, is successively compared with the local stability of the closed-loop PWM system for first- and second-order plants. In the first case it has been formally proved that the proposed criterion ensures the local stability of an equilibrium point, while in the second one a Monte Carlo simulation has confirmed that the selection of the modulator parameters, according to the proposed criterion, gives an effective method to avoid limit cycles and to ensure the local stability.     

Electric vehicle’s energy consumption of car-following models

Nonlinear Dynamics - In this paper, we use the optimal velocity model, full velocity difference model, full velocity and acceleration difference model, and the car-following model with...     

Modelling of the dynamics of Euler’s beam by ϕ6 potential

We derive in this paper the mathematical model, describing the single mode dynamics of a beam subjected to external loads. Taking into account the geometrical nonlinearities and considering the second order approximation of the deflected length of the beam, we show that the single mode dynamics of a simply supported (hinged–hinged) beam can be described by a ϕ⁶ potential with various configurations.     

T–S fuzzy control design for a class of nonlinear networked control systems

This paper is devoted to controlling a class of nonlinear systems over a communication network in the presence of packet transmission delays, packet losses, quantization errors, and sampling-related phenomena. Specifically, based on the Takagi–Sugeno (T–S) fuzzy approach, this paper presents a way of designing a quantized controller that allows all the states of nonlinear NCSs to converge exponentially to a bounded ellipsoid. In particular, this paper provides a method capable of exploiting fully the time-varying delay term d(t), induced by Jensen inequality, for nonlinear NCSs.     

Stability for the Equilibrium State of Chaplygin’s Systems

ＳＴＡＢＩＬＩＴＹＦＯＲＴＨＥＥＱＵＩＬＩＢＲＩＵＭＳＴＡＴＥＯＦＣＨＡＰＬＹＧＩＮ’ＳＳＹＳＴＥＭＳＺｈｕＨａｉｐｉｎｇ（朱海平）ＳｈｉＲｏｎｇｃｈａｎｇ（史荣昌）ＭｅｉＦｅｎｇｘｉａｎｇ（梅凤翔）（ｐｅｋｉｎｇＩｎｓｔｉｔｒｔｅｏｆＴｅｃｈｎｏｌ...     

Carlo Cercignani’s interests for the foundations of physics

Carlo Cercignani was known all over the world for his works on the Boltzmann equation and on kinetic theory. There was however another aspect of his scientific life, which is not much known. Namely, his interest for the foundations of physics, in particular for the possibility of understanding quantum mechanics through classical mechanics, which he shared with several people in Milan. A review of such researches is given here, together with some personal recollections of him.     

First-round design of the flight scenario for Chang’e-2’s extended mission:take off from lunar orbit

Chang'e-2, Chinese second lunar probe, was inserted into a 100 km altitude low lunar orbit on October 9th, 2010, its purpose is to continuously photograph the lunar surface and possibly chosen landing sites for future lunar missions. The probe will still carry considerable amount of propellant after completing all prescribed tasks in about six months. After the successful launch of Chang'e-2, we began designing the probe's subsequent flight scenario, considering a total impulse of 1 100 m/s for takeoff from low lunar orbit and a maximum 3×10 6 km distance for Earth-probe telecommunication. Our first-round effort proposed a preliminary flight scenario that involves consecutive arrivals at the halo orbits around the Earth-Moon L1/L2 and Sun-Earth L1/L2 points, near-Earth asteroid flyby, Earth return, and lunar im- pact. The designed solution of Chang'e-2's subsequent flight scenario is a multi-segment flight trajectory that serves as a reference for making the final decision on Chang'e-2's extended mission, which is a flight to the Sun-Earth L2 point, and a possible scheme of lunar impact via Earth flyby after remaining at the Sun-Earth L2 point was also presented. The proposed flight trajectory, which possesses acceptable solution accuracy for mission analysis, is a novel design that effectively exploits the invariant manifolds in the circular restricted three-body problem and the patched-manifold-conic method.     

A revisit to synchronization of Lurie systems with time-delay feedback control

This paper revisits the problem of synchronization for general Lurie systems with time-delay feedback control. Differently from most of existing results, the more restrictively slope restrictions on the nonlinearities of Lurie systems are considered in view of the fact that the slope restrictions may improve synchronization conditions compared with the sector ones. The Kalman–Yakubovich–Popov (KYP) lemma and the Schur complement formula are applied to get novel and less conservative synchronization criteria, which have the forms of linear matrix inequalities (LMIs). Numerical examples are presented to illustrate the efficiency of the proposed results.     

A Simple Proof of the Generalized Cauchy’s Theorem

The Cauchy’s theorem for balance laws is proved in a general context using a simpler and more natural method in comparison to the one recently presented in Segev (Arch. Ration. Mech. Anal. 154:183–198, 2000). By “generality” we mean that the ambient space is considered to be an orientable smooth manifold, and not only the Euclidean space.