Methods Of Analytical Mechanics Research Articles

Determining the parameters of oscillation dissipation in a column of sucker rods

In order to calculate a column of pumping rods as a mechanical system with concentrated masses, to determine dissipation coefficients, we applied a traditional method of analytical mechanics. A mathematical model is developed of longitudinal oscillations of a three-stage conditionally vertical column of sucker rods using the functions of displacement and load of its separate stages. Dependence for determining the dissipation coefficient of oscillations is derived from the solution of the system of equations. In accordance with the chosen set-up of a three-stage column of sucker rods, we examined a change in the dissipation coefficient of oscillations depending on the relationship between the rigidities of its stages. A change in the relationship between rigidities of the stages was carried out by changing their material. It was found that for the selected set-up of a column of sucker rods, the use of a fiberglass stage instead of that made of steel reduces its rigidity by approximately 4 times, and increases the dissipation of oscillations almost as much. Such approach makes it possible to prevent the phenomenon of resonance in the operation of a SR column during transition modes under the action of alternating load.

Flexoelectricity in low densification materials and its implication

Coupling between the strain gradient and the electric polarization, also named flexoelectricity, is a fundamental but often overlooked property in solid dielectrics. The past ten years has witnessed its great potential in sensing and actuating applications, especially in nanomaterials or nanostructures. The recently discovered giant flexoelectricity in barium strontium titanate (BST) perovskite above its Curie temperature has boosted the research interests on flexoelectricity to a new height. The possible interpretation of the enhanced flexoelectricity in BST was initially attributed to the non-crystalline polar-phases or polar nano-regions existing in the perovskites, then to the inner micro strain generated from the material densification process. In this paper, we analyzed the origin of flexoelectricity in this kind of materials by using both analytical mechanics method and finite element simulation method. Our results suggested the inner micro defect generated during the material densification process would weaken the flexoelectricity of the materials. The observed polarization in the studied materials was likely induced by other mechanisms rather than the pure flexoelectricity.

Methods of analytical mechanics in the constrained adaptive identification problems

The application of analytical mechanics methods to the problem of adaptive identification is discussed. The estimated constant parameters satisfy nonlinear equations considered as parametric constraints. Some modifications of identification algorithms are proposed to satisfy such constraints.

Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod

By Kirchhoff dynamic analogy, the thin elastic rod static equals to rotation of rigid body dynamic. The analytical mechanics methods reflect their advantages in the study of the modeling and equilibrium and stability of elastic rod static, especially for the constrained problems. The Lagrangian structure of the equation of motion for elastic rod is deduced from the integral variational principle. The definition of conformal invariance of Mei symmetry of elastic rod in Lagrangian form is given. The determining equation of conformal invariance of Mei symmetry is obtained based on the Lie point transformation group. The relation between conformal invariance of Mei symmetry and Mei symmetry is discussed. The structure equation and conserved quantity by using the Lagrangian structure along arc coordinate deduced from conformal invariance of Mei symmetry of elastic rod are constructed. Take rod with circular cross section as example to illustrate the application of the results get in this paper. These conserved quantities will be helpful in the study of exact solutions and stability, as well as the numerical simulation of the thin elastic rod nonlinear mechanics.

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems, where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case by applying the Pontryagin maximum principle. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws, a minimum required value of the coefficient of sliding friction is defined, so that the considered system is moving in accordance with nonholonomic bilateral constraints.

10.5937/fmet1403199r = Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

The paper analyzes the problem of brachistochronic motion of a nonholonomic mechanical system, using an example of a simple car model. The system moves between two default positions at an unaltered value of the mechanical energy during motion. Differential equations of motion, containing the reaction of nonholonomic constraints and control forces, are obtained on the basis of general theorems of dynamics. Here, this is more appropriate than some other methods of analytical mechanics applied to nonholonomic systems, where the provision of a subsequent physical interpretation of the multipliers of constraints is required to solve this problem. By the appropriate choice of the parameters of state as simple a task of optimal control as possible is obtained in this case, which is solved by the application of the Pontryagin maximum principle. Numerical solution of the two-point boundary value problem is obtained by the method of shooting. Based on the thus acquired brachistochronic motion, the active control forces are determined as well as the reaction of constraints. Using the Coulomb laws of friction sliding, the minimum value of the coefficient of friction is determined to avoid car skidding at the points of contact with the ground.

The Lagrange Mechanization and Application of Multi-Machine Power System

The fast development of Hamilton system theory provides a powerful tool for the nonlinear control and stability analysis of power system. The construction of Hamilton function plays an important role in Hamilton realization. Based on Hamilton canonical equations, the energy integration of system is given. This paper expands the odd-order power system to even-order system and gives the responding conditions of Hamilton realization by using the self-adjoint methods of analytical mechanics. Besides, the standard Hamilton form of power system is derived by the combinations of Lagrange mechanization and Hamilton theory, and the form of Hamilton function is also given. Based on non-conservative analytical mechanical principles, the controllers are designed in allusion to the characteristics of power system and Matlab program is realized in IEEE3-9 standard system. The simulation effects are discussed in the transient conditions of three-phase fault by comparison of different approaches, which proves the effectiveness of the control strategy. This approach to construct Hamilton function and design the controllers has an extensive prospect of application

Dynamics and control for an in‐plane morphing wing

Purpose – The purpose of this paper is to clarify the dynamic principle of internal structure of a complex morphing wing and control the wing to change configurations rapidly and smoothly. It includes modeling the dynamics of the morphing wing and designing a rational morphing control system.Design/methodology/approach – The dynamic model of the morphing wing is developed based on Lagrange method of analytical mechanics. The generalized forces are obtained by virtual work principle. Since the morphing wing is a strongly coupled, over‐actuated and nonlinear system with multi‐input and multi‐output, the control system design includes a control allocator, a dynamic inversion controller and two PID controllers. The control allocator is designed based on pseudo inverse method; the dynamic inversion controller is applied to make the original system decoupled into two independent linear systems by proper nonlinear feedback transformation; two classical PID controllers are adopted for the linearlized systems.Find...

Methods of analytical mechanics for exact Cosserat elastic rod dynamics

Thin elastic rod mechanics with background of a kind of single molecule such as DNA and other engineering object has entered into a new developing stage. In this paper the vector method of exact Cosserat elastic rod dynamics is transformed into the form of analytical mechanics with the arc length and time as its independent variables, whose aims are to find new tools for studying rod mechanics and to develop the area of applications of classical analytical mechanics. Based on the plane cross-section assumption, a cross-section of the rod is taken as an object. Basic formulas on deformation and motion of the section are given. After defining virtual displacement of a cross-section and its equivalent variation rule, a differential variational principle such as d’Alembert-Lagrange one is established, from which dynamical equations of thin elastic rod are expressed as Lagrange equations or Nielsen equations under the condition of linear elasticity of the rod. For the rod statics when there exist conserved quantities, Lagrange equation which makes use of these quantities is derived and its first integral is discussed. Finally integral variational principle is derived from differential one, and expressed as Hamilton principle under the condition of linear elasticity. Hamilton canonical equations in phase space with 3×6 dimensions are also derived. All of the results have formed the method of analytical mechanics of dynamics of an exact Cosserat elastic rod, so that the further problems such as symmetry and conserved quantities, and numerical simulation of the rod dynamics may be further studied.

Application of the rigid finite element method to modelling ropes

Two-dimensional motion of a rope fixed at one end is considered. The Rigid Finite Element Method (RFEM) is reviewed and applied to obtain a model of the rope, including its elastic and dissipative properties. Equations of motion are derived without the small displacement assumption, using the Lagrange equations. The resulting model is compared to another one, being derived within the framework of standard analytical mechanics methods and the Lagrange formalism. Advantages of the RFE approach are discussed from a computational point of view. The presented, alternative model can be a basis for e cient numerical simulations, which seem to be useful in further, comparative studies of the rope dynamics.

Identification of robot arm’s joints’ time-varying stiffness under loads

This paper is concerned with the identification problem of two degree of freedom robot arm’s joints’ time-varying stiffness under time-varying torque loads. The industrial robot arms can be simplified into series hinged linkage mechanism. The unrestraint dynamic equation of two degrees of freedom robot arm can be obtained by using analytical mechanics method. The model which was established in Pro/E had been imported into ADAMS for simulation. And the simulation consisted of free vibration and forced vibration. Then by choosing limited memory least square method and using the post-processing measured variables, time-varying stiffness could have been identified. Finally, the calculative stiffness was compared to the “real” stiffness which was simulated in ADAMS. The whole process shows that the robot arm’s dynamic model and the method of identification are both effective. DOI: http://dx.doi.org/10.11591/telkomnika.v10i8.1644 Full Text: PDF

Identification of Two Degree of Freedom Robot Arm’s Joints’ Time-Varying Stiffness

This paper is concerned with the identification problem of two degree of freedom robot arm’s joints’ time-varying stiffness. The dynamic equation of two degrees of freedom robot arm can be obtained by using analytical mechanics method. Then by choosing limited memory least square method, time-varying stiffness can be identified. Finally, the calculative stiffness is compared to the “real” stiffness which is simulated in ADAMS. The whole process shows that the robot arm’s dynamic model and the method of identification are both effective.

Asymptotic Hamiltonian reduction for geodesics on deformed spheres and the Funk-Minkowski transform

asymbol. In problems of finding geodesics on two-dimensional surfaces, exact analytical solutions are known only for a limited circle of special cases [1]. Thus, it seems reasonable to solve these problems by using the asymptotic methods of analytical mechanics. We consider geodesics on surfaces obtained by a small deformation of the standard two-dimensional sphere. By means of averaging, we asymptotically reduce equations for geodesics to a Hamiltonian system with one degree of freedom. For a special class of deformations of the sphere, such a reduction was performed earlier in [2]. In this paper, we perform reduction for any smooth small deformation of the sphere and prove that the Hamiltonian of the averaged system can be obtained from the function determining the deformation of the sphere by using the Funk –Minkowski integral transform (known also as the Radon transform for the group SO(3)); see [3]. A function whose principal part coincides with that of an averaged Hamiltonian was used by Poincar ´

FREE TRANSLATIONAL OSCILLATIONS OF ICY BODIES WITH A SUBSURFACE OCEAN USING A VARIATIONAL APPROACH

We analyze the influence of the interior structure of an icy body with an internal ocean on the relative translational motions of its solid constituents. We consider an isolated body differentiated into three homogeneous layers with spherical symmetry: an external ice-I layer, a subsurface ammonia‐water ocean, and a rocky inner core. This composition represents icy bodies such as Europa, Titania, Oberon, and Triton, as well as Pluto, Eris, Sedna, and 2004 DW. We construct the equations of motion by assuming that the solid constituents are rigid and that the ocean is an ideal fluid, the internal motion being characterized by the relative translations of the solids and the induced flow in the fluid. Then we determine the dynamics of the icy body using the methods of analytical mechanics, that is, we compute the kinetic energy and the gravitational potential energy, and obtain the Lagrangian function. The resulting solution of the Lagrange equations shows that the solid layers perform translational oscillations of different amplitudes with respect to the barycenter of the body. We derive the dependence of the frequency of the free oscillations of the system on the characteristics of each layer, expressing the period of the oscillations as a function of the densities and masses of the ocean and the rocky inner core, and the mass of the icy body. We apply these results to previously developed subsurface models and obtain numerical values for the period and the ratio between the amplitudes of the translational oscillations of the solid components. The features obtained are quite different from the cases of Earth and Mercury. Our analytical formulas satisfactorily explain the source of these differences. When models of the same icy body, compatible with the existence of an internal ocean, differ in the thickness of the ice-I layer, their associated periods experience a relative variation of at least 10%. In particular, the different models for Titania and Oberon exhibit a larger variation of about 37% and 30%. This indicates an absolute difference of the order of three and two hours, respectively. This suggests that the free period of the internal oscillations might provide a new procedure to constrain the internal structure of icy bodies with a subsurface ocean.

Application of Lagrange Equations in Heat Conduction

Based on the new concept of thermal mass which refers to the equivalent mass of the thermal energy in an object according to Einstein's mass-energy relation, the kinetic energy and potential energy of thermal mass as well as its dissipation were introduced to establish the Lagrange equations in heat conduction. The results show that Fourier's law is equivalent to the Lagrange equations with negligible inertia forces. The application of Lagrange equations in heat conduction makes it possible to unify the treatment of heat transfer and mechanics as well as electrics through the method of analytical mechanics.

Finite deformation of thin shells in the context of analytical mechanics of material surfaces

In the framework of the direct approach shells are considered as deformable surfaces consisting of particles, and the relations of the theory are obtained with the methods of analytical mechanics. In the present work we assign to each particle five degrees of freedom, namely three translations and two in-plane rotations. The principle of virtual work produces all the relations of the theory of shells: equations of equilibrium, boundary conditions, definition of the force factors and the general form of constitutive equations. Remarkable consistency and clarity is achieved both in the relations of the theory and in the derivation process. A new formulation of the Piola tensors for a shell is suggested in order to transform the equations to the reference configuration. To analyze the effects of buckling or geometric stiffening, we linearize these equations in the vicinity of a pre-deformed configuration. Some new semi-analytical results on buckling and supercritical behavior of an axially compressed cylindrical shell are presented. The correspondence between the equations and the variational formulation is discussed in view of development of efficient numerical procedures for modeling nonlinear deformations of shells. Results of finite element modeling of the nonlinear deformation of a shell structure are discussed in comparison with the fully three-dimensional solution of the problem.

The Okhotsimskii-Pontryagin method in control theory and analytical mechanics. Part I

The Okhotsimskii method for the differentiation of functionals in control theory is discussed on the basis of the Pontryagin formalism. The relation of this method to other approaches to solving control problems and to other methods of analytical mechanics is studied. Some typical cases of solving optimal control problems are considered. The brachistochrone problem is solved with consideration of viscous friction.

Analytical mechanics methods for solving Whittaker equations

The purpose of this paper is to study the solution of the celebrated Whittaker equations by using analytical mechanics methods, including the Lagrange?Noether method, Hamilton?Poisson method and potential integral method.

Methods of analytical mechanics for integrating the Emden-Fowler equation

The methods of analytical mechanics for solving the Emden-Fowler equation are presented in this paper, including the Hamilton-Poisson method, the Lagrange-Noether method, the Lie-Hojman method and the Hamilton-Jacobi method.

Methods of analytical mechanics for dynamics of the Kirchhoff elastic rod

A cross section of the rod is taken as object of investigation. The freedom of the section in free or constraint case is analyzed and the definition of virtual displacement of the section is given, which can be expressed by a variational operation. Assuming the variational and partial differential operations has commutativity, based on the hypothesis about surface constraint subjected to the rod, the freedom of the section on constraint surface is discussed and the equations satisfied by virtual displacements of the section are given. Combining D'Alembert principle and the principle of virtual work, D'Alembert-Lagrange principle is established. When constitutive equation of material of the rod is linear, the principle can be transformed to Euler-Lagrange form. From the principle, a dynamical equation in various forms such as Kirchhoff, Lagrange, Nielsen and Appell equation can be derived. For the case when a rod is subjected to a surface or a nonholonomic constraint, Lagrange equation with undetermined multipliers is obtained. Integral variational principle of dynamics of a super-thin elastic rod is also established, from which Hamilton principle formulation is obtained when the material of the rod is linear. Finally, canonical variables to describe the state of the section and Hamilton function are defined, and Hamilton canonical equation is derived. The analytical methods of dynamical modeling of a super-thin elastic rod have been constructed, which can serve as a theoretical framework of analytical dynamics of a super-thin elastic rod with two independent variables.