线性分数阶强迫振动的稳态响应

利用分数阶导数算子-∞D_t~β研究线性分数阶振动系统在谐波激励下的稳态响应.采用复指数函数形式的谐波激励,利用待定函数法得到与激励同频率的稳态响应,以及幅频关系和相频关系.讨论了分数阶导数项对刚度和阻尼的影响.     

Dynamic performance analysis of non-linear tuned vibration absorbers

The main purpose of this study is to evaluate the dynamic characteristics of a damping controlled semi-active tuned vibration absorber (TVA). A base-excited, single-degree-of-freedom structure coupled with a TVA model is adopted as the baseline model for our analysis. A non-model based groundhook control (displacement-based on–off control or “on–off DBG”) was used to control the damping of the TVA. After developing numerical models of a non-linear TVA along with the on–off DBG control and a passive TVA, the optimal tuning parameters of both TVA models were obtained using an optimization routine. Using the two optimally tuned models, parametric studies were preformed by varying the values of both on and off-state damping to evaluate the dynamic performances of the TVAs using the peak transmissibility criteria. The results showed that the peak transmissibility of the semi-active TVA is nearly 21% lower than that of the passive, indicating that the semi-active TVA is more effective in reducing vibration levels. The results further showed that increases in off-state damping decreases the effectiveness of semi-active TVA in reducing maximum vibration levels.     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.     

Brief note: viscoelastic modelling of cantilever vibration absorbers

Internal damping is considered in a cantilever beam supported on a single degree-of-freedom vibrating system. The internal damping mechanism is modelled by a Voigt viscoelastic coefficient in the stress-strain law. The governing equations reduce to a partial differential equation with time-dependent boundary conditions. The solution yields expressions for the amplitude ratio in terms of the tuning and frequency ratios, as well as the viscoelastic damping coefficient. These results are simulated numerically on the computer to obtain three different plots of the frequency response. The first plot has parametric values of the number of cantilevers. In the second plot, three parametric values of the internal damping coefficient are portrayed; the third plot shows the frequency response with various values of the natural frequency of the main system. These results show advantages in using cantilever beams with internal damping as dynamic vibration absorbers.     

GCS of a class of chaotic dynamic systems

This article studies a guaranteed cost synchronization (GCS) problem for a class of chaotic systems. Attention is focused on the design of state feedback controllers such that the resulting closed-loop error system is asymptotically stable and an adequate level of performance is also guaranteed. Using the Lyapunov method and LMI (linear matrix inequality) technique, two criteria for the existence of the controller for GCS are derived in terms of LMIs. To show the effectiveness of the proposed method, GCS problem of Genesio system verified by a numerical example.     

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials.     

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials. (Received: September 27, 2005)     

Adaptive feedback control for a class of chaotic systems

In this paper, the problem of control for a class of chaotic systems is considered. The nonlinear functions of chaotic systems are not necessarily to satisfy the Lipsichtz conditions, but bounded by a polynomial with the gains unknown. Employing adaptive method, the corresponding controller which renders the closed-loop system asymptotically stable is constructed. The designed controller is robust with respect to certain class of disturbances in the chaotic systems. Simulations on unified chaotic systems and Arneodo chaotic system are performed and the results verify the validity of the proposed techniques.     

Solving a class of linear projection equations

Summary. Many interesting and important constrained optimization problems in mathematical programming can be translated into an equivalent linear projection equation Here, is the orthogonal projection on some convex set (e.g. ) and is a positive semidefinite matrix. This paper presents some new methods for solving a class of linear projection equations. The search directions of these methods are straighforward extensions of the directions in traditional methods for unconstrained optimization. Based on the idea of a projection and contraction method [7] we get a simple step length rule and are able to obtain global linear convergence. Received April 19, 1993 / Revised version received February 9, 1994     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.     

On a class of linear differential games

Saddlepoint strategies are deduced for a class of linear, single-input differential games. Conditions necessary for a saddle-point, as well as sufficient conditions, are discussed. The results are illustrated with a simple example. For the case of one control subject to a norm bound and the other to a quadratic penalty, the results are extended to the vector case and applied to the stabilization of a system subject to norm-bounded input disturbance.     

Control of chaotic vibration in automotive wiper systems

Chaotic vibration has been identified in the automotive wiper system at certain wiping speeds. This irregular vibration not only decreases the wiping efficiency, but also degrades the driving comfort. The purpose of this paper is to propose a new approach to stabilize the chaotic vibration in the wiper system. The key is to employ the extended time-delay feedback control in such a way that the applied voltage of the wiper motor is online adjusted according to its armature current feedback. Based on a practical wiper system, it is verified that the proposed approach can successfully stabilize the chaotic vibration, and provide a wide range of wiping speeds.     

Development of a chaotic nonlinear tuned mass damper for optimal vibration response

In this article an efficient method is developed for optimal design of a nonlinear tuned mass damper (N-TMD) system. Using several horizontal linear springs coming into action sequentially, system nonlinearity can be achieved with ease as a novel method. Friction force between tuned mass and the structure is variably produced by a vertical linear spring that follows a specified controlling curved path allowing reduction of the desired tuned mass. Chaotic behavior of the introduced tuned mass is investigated in terms of the existing parameters in the system. Lyapunov characteristic exponents are determined to demonstrate the chaotic behavior of the system. It is confirmed by comparison that the proposed scheme is able to retrofit structures in a superior way than some other devices such as multiple tuned mass damper systems. The optimization procedure is performed by sequential Simplex algorithm, while Newmark’s beta method for step by step integration is used to find the dynamic response of the structure.     

Solvability of a class of impulsive damped vibration problems

In this paper, a class of impulsive damped vibration problems are considered. Existence results are obtained by using variational method and critical point theorem. The obtained results are also valid and new even if the impulsive damped vibration problem is reduced to impulsive Hamiltonian system and Hamiltonian system. Examples are presented to illustrate the feasibility and effectiveness of the results. Copyright © 2013 John Wiley & Sons, Ltd.     

On a class of forced vibration problems with obstacles

The main purpose of this paper is to study the following damped vibration problems

(1.1)     

On a class of damped vibration problems with obstacles

The main purpose of this paper is to study the following damped vibration problem (1.1)$?\stackrel{?}{x}=g\left(t\right)\stackrel{?}{x}+f(t,x)$ satisfying (1.2)$x\left(0\right)?x\left(2\pi \right)=\stackrel{?}{x}\left(0\right)?\stackrel{?}{x}\left(2\pi \right)=0\text{,}x\left(t\right)\ge 0\text{,}?t\in R$(1.3)$\stackrel{?}{x}\left(t_0^{?}\right)=?\stackrel{?}{x}\left(t_0^{+}\right)\text{,}\text{if}x\left(t_0\right)=0\text{.}$ The variational principles are given and some existence and multiplicity results of nonzero periodic solutions satisfying (1.1), (1.2), (1.3) are obtained.     

Asymptotic properties of a class of linear operators

Siberian Mathematical Journal -     

Robust synchronization of a class of chaotic systems with disturbance estimation

This paper investigates the robust synchronization problem for a class of chaotic systems with external disturbances. By using disturbance-observer-based control (DOBC) and LMI approach, the disturbance observers are developed to ensure the boundedness of the disturbance error dynamical. Then, by employing the sliding mode control technique, an adaptive control law is established to eliminate the effect of disturbance error to realize synchronization between the master and slave systems. Finally, the corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.     

Approximation properties of a class of linear operators

This work focuses on a class of linear positive operators of discrete type. We present the relationship between the local smoothness of functions and the local approximation. Also, the degree of approximation in terms of the moduli of smoothness is established, and the statistical convergence of the sequence is studied. Copyright © 2013 John Wiley & Sons, Ltd.