A 3D Model of One-Legged Hopping Machine

The paper models the spatial motion of a hopping machine with allowance for the inertial properties of its leg. A control system is synthesized and a program trajectory is determined in view of the collision processes accompanying touchdowns. The synthesis procedure is based on optimization of periodic systems. The results from numerical simulation of the hopper's nonstationary motion are demonstrative of the high efficiency of the control system     

A Control Algorithm for a 3D Hopping Machine

The paper addresses several control problems for a three-dimensional hopping machine. The machine consists of a ponderable torso and an imponderable leg. The leg is a kinematic chain with joints allowing rotation and translation. The leg is attached at the torso's center of mass. The force in the leg is developed by a linear spring. A control system is synthesized within the framework of a periodic linear quadratic problem. The efficiency of the controller is exemplified by a case where the hopper changes the direction of its motion     

High-accuracy algorithms for solving the discrete-time periodic Riccati equation

The paper proposes an algorithm for solving the discrete-time periodic Riccati equation, which arises, for instance, in designing a stabilization system for a hopping robot. This algorithm does not have a number of constraints on the matrix coefficients peculiar to other algorithms. It is significant that variable-precision arithmetic can be used in numerical implementation of the algorithm. This makes it possible to solve discrete-time periodic Riccati equations with high accuracy. Some examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 98–105, September 2007.     

A Note on a Walking Machine Model

The paper continues studies intended to find out whether it is possible to create a prototype walking machine with relatively simple components. In this connection, the control problem is solved for a two-dimensional model of biped machine. It has a torso and two telescopic legs. Each leg includes a ponderable section of constant length and an imponderable section of variable length. The machine, regarded as a system with variable constraints, implements a single-stance gait (one stance leg at a time) with a step of constant duration. The contact of the swing leg with the ground is analyzed within the framework of Carnot's theorem (perfectly inelastic impact). It is assumed that the force developed in the stance leg is due to the deformation of the leg's spring and that this deformation can be controlled. An algorithm is proposed to synthesize a control system that takes into account collisions occurring at reverse of the roles of the legs. This algorithm is based on methods of optimizing periodic systems. The algorithm is compared with approaches used by other authors     

Stable Periodic Motion of a System Using Echo for Position Control

We study a system of equations which is based on Newton's law and models automatic position control by echo. Rewritten as a delay differential equation with state-dependent delay the model defines a semiflow on a submanifold in the space of continuously differentiable maps [-h, 0] ². All time-t maps and the restriction of the semiflow given by t > h are continuously differentiable. For simple nonlinearities and suitable parameters we find a set of initial data to which the solution curves return, after an excursion into the ambient manifold. The associated return map is semiconjugate to an interval map. Estimates of derivatives yield a unique, attracting fixed point of the latter, which can be lifted to the return map. The proof that the resulting periodic orbit is stable and exponentially attracting with asymptotic phase involves a Poincaré return map and a discussion of derivatives of its iterates.     

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。     

Stability of Nonstationary Automatic-Control Systems of Variable Structure in Forced Motion

Conditions of stability in sliding mode are obtained for nonstationary dynamic automatic-control processes of variable structure. The nonstationary parameters of a control process subjected to external perturbations vary in given ranges, whereas the parameters of the switching plane remain constant for the appropriately chosen parameters of discontinuous logic control laws. The corresponding system of differential equations contains coefficients that vary stepwise in time together with the parameters of the controlled process. The general stability criteria for the solutions of variable-structure systems are used to analyze the stability of motion in sliding mode of an automatic variable-structure control system of the second order     

结构试验机的神经网络内模自适应跟踪控制系统

针对结构试验系统的非线性和不确定性特性，研究了一种基于神经网络的非线性内模自适应加载控制方法。引入的神经网络内模可跟踪学习对象的时变动力学，控制器的设计较少依赖于对象的先验知识，控制参数的调整是基于被控过程的测量信息，利用导出的神经网络算法来实现的。实验结果证明该系统具有良好的控制效果。     

Technical Stability of Forced Motion in Nonstationary Automatic-Control Systems of Variable Structure

Sufficient conditions of technical stability are established for nonstationary automatic-control systems of variable structure. The case is considered where for appropriate controls and for all admissible initial perturbations from a predefined measurable set of initial states, the nonstationary parameters of a process under external actions vary within given ranges. The control process depends on the mismatch signal and its derivatives up to an order that is less by one than the order of the equations of motion. The associated system of differential equations contains discontinuous time-dependent coefficients. A relationship is established between the eigenvalues of the quadratic forms of the given Lyapunov functions and the technical-stability criteria. It is shown that the technical-stability conditions do not depend on the sliding-mode conditions for variable-structure control processes     

The solution for the generalized riccati algebraic equations of linear equality constraint system

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper. Project supported by the National Natural Science Youth Foundation of China     

Damping the vibrations of a rectangular plate with piezoelectric actuators

The paper addresses the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators. The problem is solved by two methods: (i) the classical method of balancing the fundamental vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to an algebraic Riccati equation. The results produced by both approaches are presented and compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 78–84, February 2008.     

碰摩裂纹转子轴承系统的周期运动稳定性及实验研究

根据碰摩裂纹耦合故障转子轴承系统的非线性动力学方程，利用求解非线性非自治系统周期解的延拓打靶法，研究了系统周期运动的稳定性。研究发现，小偏心量下系统周期运动发生Hopf分岔，大偏心量下系统周期运动发生倍周期分岔，偏心量的加大使周期解的稳定性明显降低；系统碰摩间隙变小，碰摩影响了油膜涡动的形成，使失稳转速有所提高；裂纹深度的加大降低了系统周期运动的稳定性。本文的研究为转子轴承系统的安全稳定运行提供了理论参考。     

Unsteady Motion of a Maxwellian Fluid Droplet in a Maxwellian Medium under the Action of Monotonic and Periodic Forces

Unsteady motion of a Maxwellian fluid droplet, which arises in a quiescent Maxwellian medium under the action of monotonic and periodic forces, is considered. In the initial period of time smaller than the relaxation time, the droplet is affected by elastic forces on the part of the fluid; moreover, the droplet itself is a viscoelastic material. A solution of the problem in the first approximation is found. The dependence of the amplitude of droplet velocity and the shift of the phase of oscillations on the relaxation time of the external and internal media and also on the frequency of oscillations of the driving force is analyzed. The passage to the limit in terms of density and viscosity of the internal medium is performed.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 55–65, July–August, 2005.     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.     

Nonstationary Vibrations of a String with Time-Varying Length and a Mass-Spring Attached at the Lower End

The purpose of this paper is to study the nonstationary vibration of a string with time-varying length and a mass-spring system attached at the lower end. The string is hung vertically and excited sinusoidally by a horizontal displacement at its upper end. The mass is supported by a guide spring horizontally and has two-degrees-of-freedom, vertical and horizontal. It is shown analytically that axial velocity of the string influences the peak amplitude of the string vibration at the passage through resonances. Moreover, it is shown numerically that the amplitudes of both the string and the mass vibrations depend on the sign of the axial velocity, when the natural frequency of the mass-spring system is close to the frequency of the excitation. The above two theoretical results are confirmed experimentally with a simple experimental setup.     

Nonstationary (transient) radiative-conductive heat transfer in a plane layer of gray absorbing medium

The kinetics of the heating of a plane layer of gray absorbing medium by radiative-conductive heat transfer are considered. The nonstationary energy equation is reduced to a nonlinear integral equation by means of a Green's function, and this is solved numerically by the Newton method. The results of the solution are presented in the form of the temperature fields in the layer for various values of the defining parameters (optical thickness, radiative-conductive heat-transfer criterion, heat-transfer criterion at the boundaries).Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 156–159, January–February, 1972.     

Numerical modeling of the acoustic wave dynamics on the basis of a kinetic approach

The results of the numerical modeling of a viscous compressible flow on a finite one-dimensional interval with periodic boundary conditions on the basis of a model kinetic equation are presented. The flow is excited by a small time-dependent force, periodic in space and time. It is shown that when the periodicity interval length L ≥ 10³ nonlinear steady oscillations with sharp temporal and spatial variations of the parameters arise. The system dynamics are investigated by means a pseudospectral method using the standard fast Fourier transformation procedure.     

On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations

For periodic solutions to the autonomous delay differential equation

Magnetic torque attitude control of a satellite using the state-dependent Riccati equation technique

A non-linear attitude control method for a satellite with magnetic torque rods using the state-dependent Riccati equation (SDRE) technique has been developed. The magnetic torque caused by the interaction with the Earth's magnetic field and the magnetic moment of torque rods plays a role of the control torque. The detailed equations of motion for this system are presented using angular velocity and quaternions. The SDRE controller is developed for the non-linear systems which can be formed in pseudo-linear representations using the state-dependent coefficient (SDC) method without linearization procedure. The aim of this control system is to achieve a stable attitude within 5°, and minimize the control effort. The stability regions for the SDRE controlled satellite system are estimated through the investigation of the stability conditions developed for pseudo-linear systems and the application of Lyapunov's theorem. For comparisons, the Linear Quadratic Regulator (LQR) method using the solution of the algebraic Riccati equation (ARE) is also applied to this non-linear system. The performance of the SDRE non-linear control system demonstrates more robustness and stability than the LQR control system when subjected to a wide range of initial conditions.