Discontinuous and impulsive excitation

In this paper, we study the solution of differential equation with Dirac function and Heaviside function, arising from discontinuous and impulsive excitation. Firstly, according to the theory of differential equation, we suggest, then we derive the equation of x ₁ (t) and x ₂ (t) by terms of property of distribution, and by solving x ₁ (t) and x ₂ (t) we obtain x(t); finally, we make a thorough investigation about periodic impulsive parametric excitation.     

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

Comparison of two methods for stress relaxation data presentation of solid foods

Published exponential relaxation equations, derived from Maxwellian models, were used to generate data for linear representation in the form ofP(0) ·t/(P(0) —P(t)) =k ₁ +k ₂t whereP(t) is the decaying parameter (force, stress or modulus),P(0) its initial value (att = 0) andk ₁ andk ₂ constants. The computer plots indicated that the fit of this normalized and linearized form was excellent for equations containing at least three exponential decay terms. The fit was not as good for some of the two-term exponential equations mainly due to the lack of accurate account for the initial stage of the relaxation process. In all the cases, however, the linear representation could clearly reveal the general rheological character of the analysed materials in terms of the relative degree of solidity.     

ZMP based neural network inspired humanoid robot control

This paper concerns ZMP-based control that is inspired by artificial neural networks for humanoid robot walking on varying sloped surfaces. Humanoid robots are currently one of the most exciting research topics in the field of robotics, and maintaining stability while they are standing, walking or moving is a key concern. To ensure a steady and smooth walking gait of such robots, a feedforward type of neural network architecture, trained by the back-propagation algorithm, is employed. The inputs and outputs of the neural network architecture are the ZMPx and ZMPy errors of the robot, and the x, y positions of the robot, respectively. The neural network developed allows the controller to generate the desired balance of the robot positions, resulting in a steady gait for the robot as it moves around on a flat floor, and when it is descending or ascending slopes. In this paper, experiments of humanoid robot walking are carried out, in which the actual position data from a prototype robot are measured in real-time situations, and fed into a neural network inspired controller designed for stable bipedal walking. In addition, natural walking motions on the different surfaces with varying slopes are obtained and the performance of the resulting controller is shown to be satisfactory.     

The mathematical principles of vibration reductor

In engineering and technology, it is often demanded that self-oscillation.be eliminated.so that the equipment or machinery may not be damaged In this paper, a mathematicalmodel for reducing vibration is given by the following equations:(?)_1+(?)((?)_1) +k_1(x_1-x_2) =0, (?)_2+c(?)_1+k_2(x_2-x_1) =0 (*)We have discussed how to choose suitable parameters c_1, k_1,k_2; of equations (*),so as to make the zero solution to be of global stability. Several theorems on the globalstability of the zero solution of equations (*) are also given.     

Pseudo-fault induction in engineering structures: Multi-input multi-output control

In a previous paper, it was demonstrated that any linear system could be made to respond to harmonic excitation as if a static nonlinearity of specified type and position were present, this response being obtained at a single predetermined point on the structure. The method requires the excitation of the linear structure by an additional or auxiliary input. In the present paper, the theory is extended to allow the possibility of producing a specified nonlinear response at more than one point on a linear structure. It is shown that N responses can be obtained by specifying N or less auxiliary inputs. The theory is also extended to provide for polynomial damping in addition to stiffness nonlinearity. The theory is validated using numerical simulation of MDOF lumped-parameter systems.     

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

Neural networks based self-learning PID control of electronic throttle

An electronic throttle is a low-power DC servo drive which positions the throttle plate. Its application in modern automotive engines leads to improvements in vehicle drivability, fuel economy, and emissions. In this paper, a neural networks based self-learning proportional-integral-derivative (PID) controller is presented for electronic throttle. In the proposed self-learning PID controller, the controller parameters, K _P , K _I , and K _D are treated as neural networks weights and they are adjusted using a neural networks algorithm. The self-learning algorithm is operated iteratively and is developed using the Lyapunov method. Hence, the convergence of the learning algorithm is guaranteed. The neural networks based self-learning PID controller for electronic throttle is verified by computer simulations.     

On an analogue of the Euler-Cauchy polygon method for the numerical solution of ux y = f(x,y, u,ux, uy)

This paper develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x ₀)=y ₀) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation u _xy =f(x, y, u, u _x , y _y ), u(x, y ₀)=(x), u(x ₀, y)=(y), where (x ₀)=(y ₀). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions (x) and (y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.

---

     

Explicit expressions of S(v), H(v) and L(v) for anisotropic elastic materials

The three matrices L(v), S(v) and H(v), appearing frequently in the investigations of the two-dimensional steady state motions of elastic solids, are expressed explicitly in terms of the elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with a plane of symmetry at x₃ = 0, x₁ = 0, and x₂ = 0 are all deduced. Results for orthotropic materials appearing in the literature may be recovered from the present explicit expressions.     

Classifying wakes produced by self-propelled fish-like swimmers using neural networks

We consider the classification of wake structures produced by self-propelled fish-like swimmers based on local measurements of flow variables. This problem is inspired by the extraordinary capability of animal swimmers in perceiving their hydrodynamic environments under dark condition. We train different neural networks to classify wake structures by using the streamwise velocity component, the crosswise velocity component, the vorticity and the combination of three flow variables, respectively. It is found that the neural networks trained using the two velocity components perform well in identifying the wake types, whereas the neural network trained using the vorticity suffers from a high rate of misclassification. When the neural network is trained using the combination of all three flow variables, a remarkably high accuracy in wake classification can be achieved. The results of this study can be helpful to the design of flow sensory systems in robotic underwater vehicles.     

确定非线性隔振装置参数的一种方案

对单自由度刚度可调非线性隔振模型进行数学建模仿真和实验，对于待训练的神经网络模型，将数值仿真结果作为学习样本，将实验结果作为检验样本，训练成功的神经网络模型具有较好的内延能力和一定的外推能力，在设定非线性隔振装置要求的隔振效能参数后，可以给出隔振装置对应的刚度和阻尼参数，从而为设计非线性隔振装置提供了一种方案。     

On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

The dynamic behavior of spiral waves in stochastic Hodgkin–Huxley neuronal networks with ion channel blocks

Chemical blocking is known to affect neural network activity. Here, we quantitatively investigate the dynamic behavior of spiral waves in stochastic Hodgkin–Huxley neuronal networks during sodium- or potassium-ion channel blockages. When the sodium-ion channels are blocked, the spiral waves first become sparse and then break. The critical factor for the transition of spiral waves (x _Na) is sensitive to the channel noise. However, with the potassium-ion channel block, the spiral waves first become intensive and then form other dynamic patterns. The critical factor for the transition of spiral waves (x _K) is insensitive to the channel noise. With the sodium-ion channel block, the spike frequency of a single neuron in the network is reduced, and the collective excitability of the neuronal network weakens. By blocking the potassium ion channels, the spike frequency of a single neuron in the network increases, and the collective excitability of the neuronal network is enhanced. Lastly, we found that the behavior of spiral waves is directly related to the system synchronization. This research will enhance our understanding of the evolution of spiral waves through toxins or drugs and will be helpful to find potential applications for controlling spiral waves in real neural systems.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.     

Linear state representations for identification of bilinear discrete-time models by interaction matrices

Bilinear systems can be viewed as a bridge between linear and nonlinear systems, providing a promising approach to handle various nonlinear identification and control problems. This paper provides a formal justification for the extension of interaction matrices to bilinear systems and uses them to express the bilinear state as a linear function of input–output data. Multiple representations of this kind are derived, making it possible to develop an intersection subspace algorithm for the identification of discrete-time bilinear models. The technique first recovers the bilinear state by intersecting two vector spaces that are defined solely in terms of input–output data. The new input–output-to-state relationships are also used to extend the equivalent linear model method for bilinear system identification. Among the benefits of the proposed approach, it does not require data from multiple experiments, and it does not impose specific restrictions on the form of input excitation.     

Symmetry of Mountain Pass Solutions of Some Vector Field Equations

We prove radial symmetry (or axial symmetry) of the mountain pass solution of variational elliptic systems − AΔu(x) + ∇ F(u(x)) = 0 (or − ∇.(A(r) ∇ u(x)) + ∇ F(r,u(x)) = 0,) u(x) = (u ₁(x),...,u _N (x)), where A (or A(r)) is a symmetric positive definite matrix. The solutions are defined in a domain Ω which can be , a ball, an annulus or the exterior of a ball. The boundary conditions are either Dirichlet or Neumann (or any one which is invariant under rotation). The mountain pass solutions studied here are given by constrained minimization on the Nehari manifold. We prove symmetry using the reflection method introduced in Lopes [(1996), J. Diff. Eq. 124, 378–388; (1996), Eletron. J. Diff. Eq. 3, 1–14].     

Wind-driven nonlinear oscillations of cables

The objective of this paper is the study of the dynamics of damped cable systems, which are suspended in space, and their resonance characteristics. Of interest is the study of the nonlinear behavior of large amplitude forced vibrations in three dimensions. As a first-order nonlinear problem the forced oscillations of a system having three-degrees-of-freedom with quadratic nonlinearities is developed in order to consider the resonance characteristics of the cable and the possibility of dynamic instability. The cables are acted upon by their own weight in the perpendicular direction and a steady horizontal wind. The vibrations take place about the static position of the cables as determined by the nonlinear equilibrium equations. Preliminary to the nonlinear analysis the linear mode shapes and frequencies are determined. These mode shapes are used as coordinate functions to form weak solutions of the nonlinear autonomous partial differential equations.In order to investigate the behavior of the cable motion in detail, the linear and the nonlinear analyses are discussed separately. The first part of this paper deals with the solution to the self adjoint boundary-value problem for small-amplitude vibrations and the determination of mode shapes and natural frequencies. The second problem dealt with in this paper is the determination of the phenomena produced by the primary resonance of the system. The method of multiple time scales is used to develop solutions for the resulting multi-dimensional dynamical system with quadratic nonlinearity.Numerical results for the steady state response amplitude, and their variation with external excitation and external detuning for various values of internal detuning parameters are obtained. Saturation and jump phenomena are also observed. The jump phenomenon occurs when there are multi-valued solutions and there exists a variation of kinetic energy among solutions.Notation A=diag(a _i ,i=1, 2, 3) amplitude matrix (diagonal) - A _n,A undeformed area, deformed area - B span of hanging cables - D sag for static conditions - E Young's modulus - vector of external force - diagonal matrix - symmetric coefficient matrix - H ^* =HR I unit matrix - diagonal matrix - L original length of cables before hanging - M the symmetric stiffness matrix - N integer - P damping constant matrix (diagonal) - R linear mode shape matrix (diagonal) - S sway of hanging cables - T tension of cables - T _o tension of cables for static conditions - T _o(0) tension of the lowest point for static conditions - V eigenfunction matrix - b=y ^T R coefficient vector - b - c,c ₁,c ₂,c ₃ vector, and the components in thex ₁,x ₂,x ₃ directions respectively, in terms of cosine functions. - e, e _o strain, and static strain of elongation - e ₁ time-dependent perturbation ine - f wind force in the sway direction - f, f ₀,f ₁ vector of external force - g gravity constant - h time-dependent amplitude vector - m mass density per unit length of the undeformed cable - r=(R ₁,R ₂,R ₃) ^T vector of modal shapes - s undeformed arc length - t time - u ₁ linear scalar in z - u ₂ quadratic scalar in z - v ₁,v ₂,v ₃ eigenfunctions inx ₁,x ₂, andx ₃ directions, respectively - x=(x ₁,x ₂,x ₃) ^T Cartesian position vector and components - y=(y ₁,y ₂,y ₃) ^T static position vector and components - error vector - matrix operator - =diag[₁, ₂, ₃] internal frequency matrix and components - excitation frequency - global matrix of coordinate functions - T _o(0)/mgL - mgL/EA _o - yy ^T - s/L - = diag[₁, ₂, ₃] phase angle matrix and components of characteristic modes - phase angle of excitation force - ₁, ₂ time-dependent amplitude vectors in timet _o and timet ₁ - _ij,i=1, 2...N,j=1, 2, 3 theith coordinate function of thejth component - _i = diag[_i1, _i2, _i3] theith matrix of coordinate functions - global vector of modal amplitudes - ₁ external detuning parameter - _i,i=2, 3 internal detuning parameter - _i,i=1, 2, 3 phase angles     

Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x _t + f (t, x _t , λ), assuming that the linear equation x′ = L(t)x _t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.     

Neural network reconstruction of fluid flows from tracer-particle displacements

We demonstrate some of the advantages of using artificial neural networks for the post-processing of particle-tracking velocimetry (PTV) data. This study is concerned with the data obtained after particle images have been matched and the obvious outliers have been removed. We show that it is easy to produce simple back-propagation neural networks that can filter the remaining random noise and interpolate between the measurements. They do so by performing a particular form of non-linear global regression that allows them to reconstruct the fluid flow for the entire field covered by the photographs. This is obtained by training these neural networks to learn the fluid dynamics function f that maps the position x of a fluid particle at time t to its position X at time t + Δt. They can do so with a high degree of precision when provided with pairs of matching particle positions (x, X) from only about 2 to 4 pairs of PTV photographs as exemplars. We show that whether they are trained on exact or on noisy data, they learn to interpolate with such a precision that their output is within one pixel of the theoretical output. We demonstrate their accuracy by using them to draw whole streamlines or flow profiles, by iteration from a single starting point. Received: 23 November 1998/Accepted: 14 July 2000