APPLICATION OF ANALYTICAL SOLUTIONS FOR BENDING BEAMS IN THE METHOD OF MOVEMENT

Introduction: Usually, to analyze statically indeterminate rod systems, the classical displacement method and preprepared tables for two types of rods of the main system are used. A mathematically correct representation of local loads with the use of generalized functions makes it possible to find an accurate solution of the differential equation for the equilibrium of a beam exposed to an arbitrary transverse load. Purpose of the study: We aimed to obtain analytical expressions for functions of deflection, rotation angles, transverse forces, and bending moments depending on four types of local loads for beams with different boundary conditions, so as to apply accurate solutions in the displacement method. Methods: We propose an analytical form of the displacement method to analyze rod structural models. For beams exposed to different types of transverse load (uniformly distributed force, concentrated force, or a couple of forces), accurate analytical solutions were obtained for functions of deflection, bending moments, and transverse forces at different types of beam ends’ restraint. This is possible due to the fact that concentrated load and load in the form of the moment of force can be specified by using unit column functions. By transforming Mohr’s integrals, using integration by parts, we show that the system of canonical equations of the displacement method was obtained based on the Lagrange principle. Results: Based on the analysis of a statically indeterminate frame, the effectiveness of the proposed analytical method is shown as compared with the classical displacement method.

Stability and synchronous characteristics of a two exciters vibration system considering material motion

Nowadays, two exciters vibration system played an indispensable role in a majority of machinery and devices, such as vibratory feeder, vibrating screen, vibration conveyer, vibrating crusher, and so on. The stability of the system and the synchronous characteristics of two exciters are affected by material motion. However, those effects of material on two exciters vibration system were studied very little. Based on the special background, a mechanical model that two exciters vibration system considering material motion is proposed. Firstly, the system's dynamic equations are solved by using Lagrange principle and Newton's second law. Then, the motion stability of the system when material with different mass move on the vibrating body is analyzed by [Formula: see text] mapping and numerical simulation methods, and the motion forms of the material are also studied. Meanwhile, the frequency responses of the vibrating body are analyzed. Finally, the influence of material on the phase difference of the two exciters is revealed. It can be concluded that with the mass ratio of the material to the vibrating body increasing, the system's motion evolves from stable periodic motion to chaotic state, the synchronization ability of two exciters decline, and the unpredictability of abrupt change about the phase difference increases. Further, the uncertainties of both the abrupt change of phase difference and the collision location affect each other and eventually lead to the instability of the system.

Research on Active Control of Rotating Motion and Vibration of Flexible Manipulator

In this study, PZT (piezoelectric) actuators and PD control (PDs’ command line tool) method is selected to control the vibration of the flexible manipulator. The dynamic equations of the flexible manipulator system are established based on Lagrange principle. The control strategy of PZT actuators and joint control torque are designed. It is investigated by a Lyapunov approach that a combined scheme of PD feedback and command voltages applies to segmented PZT actuators. By comparison, only PD feedback control is also considered to control the flexible manipulator. The numerical simulations prove that the method of the designed PZT actuators’ control strategy and PD control is effective to compress the vibration of the flexible manipulator.

Event-triggered adaptive control for upper-limb robot-assisted passive rehabilitation exercises with input delay

In this article, an adaptive tracking control strategy is designed for uncertain electrically driven end-effector type upper-limb rehabilitation robots subject to an input delay and a limited bandwidth channel. This control scheme is implemented to perform upper-limb passive rehabilitation training for different subjects. Primarily, dynamic analysis of the rehabilitation robot is carried out using the Euler–Lagrange principle, which incorporates motor dynamics to allow the voltage-based control commands as desirable in practical implementations. Thereafter, an adaptive backstepping control law with input delay compensation is designed to estimate the unknown dynamical parameters of the rehabilitation robot during the training sessions. Furthermore, a Lyapunov-based triggering mechanism is developed to deal with the limited bandwidth challenge and reduce the transmissions over the network. The experimental validation is conducted for different scenarios, and a comparison study is carried out with two time-triggered control schemes to investigate the potential of the proposed approach. From the experimental runs and the comparative analysis, the proposed control scheme is found to achieve a promising tracking performance with input delay compensation. Moreover, a significant saving in the network resources is attained during the passive rehabilitation training of the subjects.

Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systems

We consider the regularization of the classical optimality conditions (COCs) — the Lagrange principle and the Pontryagin maximum principle — in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space $L^m_2$, the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They “overcome” the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.

Preliminary design and development of a low-cost lower-limb exoskeleton system for paediatric rehabilitation

In this work, the design, modeling, and development of a low-cost lower limb exoskeleton (LLES) system are presented for paediatric rehabilitation (age: 8–12 years, mass: 25–40 kg, height: 115–125 cm). The exoskeleton system, having three degrees-of-freedom (DOFs) for each limb, is designed in the SolidWorks software. A wheel support module is introduced in the design to ensure the user’s stability and safety. The finite element analysis of the hip joint connector along with the wheel support module is realized for maximum loading conditions. The holding torque capacity of exoskeleton joints is estimated using an affordable spring-based experimental setup. A working prototype of the LLES is developed with holding torque rated actuators. Thereafter, the dynamic analysis for the human-exoskeleton coupled system is carried out using the Euler-Lagrange principle and SimMechanics model. The simulation results of estimating joint actuator torques are obtained for two paraplegic subjects (Case I: 10 years age, 30 kg mass, 120 cm height and Case II: 12 years age, 40 kg mass, 125 cm height). The details of input parameters such as body mass, link lengths, joint angles, and contact forces are discussed. The simulation results of dynamic analysis have shown the potential of estimating the torques of joint actuators for the developed prototype during motion assistance and gait rehabilitation.

Lagrange principle and its regularization as a theoretical basis of stable solving optimal control and inverse problems

The paper is devoted to the regularization of the classical optimality conditions (COC) — the Lagrange principle and the Pontryagin maximum principle in a convex optimal control problem for a parabolic equation with an operator (pointwise state) equality-constraint at the final time. The problem contains distributed, initial and boundary controls, and the set of its admissible controls is not assumed to be bounded. In the case of a specific form of the quadratic quality functional, it is natural to interpret the problem as the inverse problem of the final observation to find the perturbing effect that caused this observation. The main purpose of regularized COCs is stable generation of minimizing approximate solutions (MAS) in the sense of J. Warga. Regularized COCs are: 1) formulated as existence theorems of the MASs in the original problem with a simultaneous constructive representation of specific MASs; 2) expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) are sequential generalizations of the COCs and retain the general structure of the latter; 4) “overcome” the ill-posedness of the COCs, are regularizing algorithms for solving optimization problems, and form the theoretical basis for the stable solving modern meaningful ill-posed optimization and inverse problems.

Nonlinear analysis of a crossing-beams system on an elastic with usage of the Mathematica software

In this paper, the authors consider the method of calculating an infinite system of cross beams on an elastic base by the variationaldifference method. The system of cross beams on an elastic base is most often modeled as shallow strip foundations for buildings of various functional purposes. The variation-difference method is one of the numerical and analytical methods for calculating building structures, it is based on the variational principles of the Ritz-Timoshenko method and on the minimum of the total potential energy of the entire system according to the Lagrange principle, and is also close to the real operating conditions of the foundation – base. A single-layer isotropic artificial base was used as an elastic base in the work, as an elastic layer limited in thickness. The algorithm of nonlinear calculation is based on the use of the iterative method of elastic solutions. The physical nonlinearity of the material of reinforced concrete beams is taken into account through the asymptotic dependence “Moment-curvature”. Numerical approbation of the results of elastic and nonlinear calculations of the system of cross beams on an elastic base was carried out using the MATHEMATICA software package.

On the regularization of the Lagrange principle and on the construction of the generalized minimizing sequences in convex constrained optimization problems

Рассматривается регуляризация принципа Лагранжа (ПЛ) в выпуклой задаче условной оптимизации с операторным ограничением-равенством в гильбертовом пространстве и конечным числом функциональных ограничений-неравенств. Целевой функционал задачи не является, вообще говоря, сильно выпуклым, а на множество ее допустимых элементов, которое также принадлежит гильбертову пространству, не накладывается условие ограниченности. Получение регуляризованного ПЛ основано на методе двойственной регуляризации и предполагает использование двух параметров регуляризации и двух соответствующих условий согласования одновременно. Один из регуляризирующих параметров «отвечает» за регуляризацию двойственной задачи, другой же содержится в сильно выпуклом регуляризирующем добавке к целевому функционалу исходной задачи. Основное предназначение регуляризованного ПЛ — устойчивое генерирование обобщенных минимизирующих последовательностей,аппроксимирующих точное решение задачи по функции и по ограничениям, для целей ее непосредственного практического устойчивого решения