A monolithic approach toward coupled electrodynamic–thermomechanical problems with regard to weak formulations

Weak formulations of boundary value problems are the basis of various numerical discretization schemes. They are classically derived applying the method of weighted residuals or a variational principle. For electrodynamical and caloric problems, variational approaches are not straightforwardly obtained from physical principles like in mechanics. Weak formulations of Maxwell’s equations and of energy or charge balances thus are frequently derived from the method of weighted residuals or tailored variational approaches. Related formulations of multiphysical problems, combining mechanical balance equations and the axioms of electrodynamics with those of heat conduction, however, raise the additional issue of lacking consistency of physical units, since fluxes of charge and heat intrinsically involve time rates and temperature is only included in the heat balance. In this paper, an energy-based approach toward combined electrodynamic–thermomechanical problems is presented within a classical framework, merging Hamilton’s and Jourdain’s variational principles, originally established in analytical mechanics, to obtain an appropriate basis for a multiphysical formulation. Complementing the Lagrange function by additional potentials of heat flux and electric current and appropriately defining generalized virtual powers of external fields including dissipative processes, a consistent formulation is obtained for the four-field problem and compared to a weighted residuals approach.

Optimal Robust Control With Cooperative Game Theory For Lower Limb Exoskeleton Robot

For achieving trajectory tracking issue of the lower limb exoskeleton robot, a novel optimal robust control with cooperative game theory is proposed. The uncertainties are considered (possible time-varying, bounded and fast) and the fuzzy set theory is creatively adopted to describe the boundary. From the view of analytical mechanics, the trajectory tracking is treated as the constraints control problem, including holonomic and nonholonomic constraints. A robust control is designed with two adjustable parameters to guarantee the uniform boundedness and uniform ultimate boundedness. For obtaining the optimal selection of two adjustable parameters, a novel optimal control employed cooperative game theory is proposed. Combining the robust control and optimal design, optimal robust control is formulated. The Pareto optimal solution is obtained to guarantee the minimum control cost. In the simulation, the adaptive robust control of Sun is chosen as a comparison. The existence of Pareto optimality and the effectiveness of optimal robust control have been verified via simulation results.

Prediction of Frequency Response Function for Cylindrical Thin-Walled Workpiece with Fixture Support Constraints

Auxiliary fixtures are widely used to enhance the rigidity of cylindrical thin-walled workpieces (CTWWs) in the machining process. Nevertheless, the accurate and efficient prediction of frequency response function (FRF) for the workpiece-fixture system remains challenging due to the complicated contact constraints between workpiece and fixture. This paper proposes an analytical solution for the comprehensive FRF analysis of the CTWW-fixture system. Firstly, based on the vector mechanics, the mode shape functions of the workpiece are presented using the classical theory of thin shell. The variable separation method is utilized to deal with the inter-mode coupling of the workpiece. Secondly, the motion equation of the CTWW with fixture constraints is established using analytical mechanics from the viewpoint of energy balance. Finally, the FRFs of the CTWW-fixture system are derived by means of modal superposition. Experimental modal tests verify that the predicted FRFs are in good agreement with the measured curves.

THE MAIN HAMILTON FUNCTION AND THE NECESSARY CONDITIONS FOR THE EXISTENCE OF A SITUATION IN THE FORM OF THE HAMILTON-JACOBI EQUATIONS

Interdisciplinary application of learning outcomes is essential to the field of mathematics, attainable at intersection of math with other subjects – which includes applied tasks When studying a number of technical disciplines, as well as solving applied problems, it is possible to use certain aspects of the theory of optimal control - which is an example of interdisciplinary link. Analytical mechanics, among other disciplines, enables leverage of certain aspects of the theory of differential games, namely, equilibrium conditions in non-coalitional differential games of several players. This article provides studies of the necessary and sufficient conditions for the existence of equilibrium situations, using some concepts of analytical mechanics. In line with Hamilton’s definition, necessary conditions were obtained in the form of Hamilton-Jacobi equations. This form of necessary conditions in differential games of N persons is of interest to students of natural and technical fields. The main goal of the article is to demonstrate interdisciplinary link, an important component of the process of training future engineers for various sectors of the economy. It is necessary for the holistic understanding of the material, so that students of technical specialties of various fields can use it. Proposed work can aid in study of this area of analytical mechanics by university students and young scientists alike.

MATHEMATICAL PENDULUM MODEL WITH MOBILE SUSPENSION POINT

Background. The new dynamic problem, which is posed and solved in this article, is a theoretical generalization of the well-known classical problem of free oscillations of a mathematical pendulum. In the proposed setting, it is new and relevant, and can be successfully used in such fields of technology as vibration protection, vibration isolation and seismic protection of high-rise flexible structures, long power lines, long-span bridges and other large-sized supporting objects. Objective. The aim of the work is to derive the equations of own oscillations of a new mathematical pendulum-absorber, to find a formula for determining the frequency of its small own oscillations and to establish those control parameters that allow you to tune the single-mass pendulum absorber to the frequency of the fundamental tone of the carrier object. Methods. To achieve this goal, we used the methods of analytical mechanics, namely, the Appel’s formalism, as well as the linearization of the obtained differential equations. Results. A mathematical model is constructed in the work that describes the own oscillations of a new-design mathematical pendulum with a movable (spring-loaded) suspension point with length L. The model is a system of differential equations obtained using the Appel’s formalism. Based on them, after linearization of nonlinear equations, a formula is established for the frequency of small own oscillations of a pendulum with a mobile suspension point. Conclusions. It is shown that the frequency of own oscillations of the new mathematical pendulum coincides with the frequency of own oscillations of the classical mathematical pendulum with an equivalent suspension length, which is equal to . In the case where the suspension point is fixed (k ® ¥), the frequency formula turns into a well-known formula for the frequency of small own oscillations of a classical mathematical pendulum . If the stiffness coefficient of elastic elements tends to zero (k ® 0), then the frequency w of the damper also tends to zero. An important structural feature of the proposed pendulum is noted, consisting in the fact that due to the appropriate choice of the three control parameters of the pendulum (k, L, m), its frequency in magnitude can be made any in the range from zero to .

Investigation of fiber slippage in the chopping process of carbon fiber tows under rigid support

As an important application form of carbon fiber (CF), short CFs and their production process have self-evident research value. In this work, the chopping process of CF tows under rigid support and the essential cause of high cutting forces were explored. Large-tow CFs containing 1–3000 single filaments were chopped, and the fracture processes were observed and described. It was found that obvious fiber slippage phenomena and intermediate fracture behaviors occurred during the chopping process. These factors not only increased the cutting force but also caused an uneven distribution of the cutting force along the width. A mechanical model was established to explain the fiber slippage and intermediate fracture. Based on material mechanics and analytical mechanics, the real process of fiber slippage and intermediate fracture was described by Hamilton’s principle. Moreover, a width constraint experiment was designed to indirectly verify fiber slippage phenomena and intermediate fracture behaviors. Through the analysis of the stress curve, it was proven that a reasonable width constraint could effectively limit fiber slippage and improve the uniformity of the distribution of the cutting force along the width of the tool, thus reducing the cutting force. This work can be used as an excellent guide for the chopping process in CF production.

Cross-country ability of tracked vehicles

Research aims to determine conditions and parameters under which the tracked vehicle overcomes stone fields and separate blocks (boulders). Research methodology included the methods of analytical mechanics and mathematical analysis. Results. Mining and geological conditions of deposits gradually get more complicated, while mining moves to remote northern regions, which is a characteristic feature of the Holocene epoch. Road, weather, and natural conditions in arctic and northern territories differ markedly from other regions. Ridges, mounds, lake plains, and marshland are the predominant landscape. Stone fields, marshes, permafrost, marshy hollows, and deep ravines impede the traffic. Due to these factors it is particularly difficult to access and explore in arctic and northern territories. There are particular technical specifications for the exploration equipment, namely, increased cross-country ability for weak soil (snow and swamp), ability to overcome stone fields, increased haulage and towing capacity with the account of maximum possible environmental regulatory compliance, etc. The conditions of ascent and overturning are derived from geometry, while the sliding condition is derived from statics. Summary. A mathematical model has been created, which makes it possible to describe the tracked vehicle’s travel across the uneven ground and, under known design values of the vehicle, determine the limitary obstacle clearance.

Analysis of the motion and stability of the holonomic mechanical system in the arbitrary force field

In order to give an insight into the work of the machine before the production and assembly and to obtain good analysis, this paper presents detailed solutions to the specific problem occured in the field of analytical mechanics. In addition to numerical procedures in the paper, a review of the theoretical foundations was made.Various types of analysis are very common in mechanical engineering, due to the possibility of an approximation of complex machines. For the proposed system, Lagrange's equations of the first kind, covariant and contravariant equations, Hamiltons equations and the generalized coordinates, as well as insight in Coulumb friction force are provided.Also, the conditions of static equilibrium are solved numerically and using intersection of the two curves. Finally, stability of motion for the disturbed and undisturbed system was investigated.

Kinematics of the Maneuver of the Multifunctional Machine for Metallurgical Furnaces

The article proposes a simulation model of a loading machine for metallurgical furnaces and builds a mathematical model that makes it possible to find optimal trajectories of motion and obtain laws of change in kinematic, dynamic and inertial characteristics for a full operation cycle. The description of the machine motion is made taking into account the nonholonomy of the imposed connections and provides for the possibility of considering arbitrary laws of motion along optimal trajectories. Compliance with optimal working conditions is also ensured by the choice of a special law of motion between the nodal points on the trajectory of movement, corresponding to the change in the direction of movement of the loading machine. Based on the research results, a design scheme and a simulation model with an analytical description of the mechanism operation, including kinematic graphs, are proposed. A simulation model of a loading machine for metallurgical furnaces and a description of its operation modes are made by methods of analytical mechanics.