Rod Equation Research Articles

Modeling and simulation of non-linear dynamic process of the induction motor system with fluctuating potential loads

The induction motor system with fluctuating potential loads is a non-linear complex electro-mechanical system. With beam pumping motors as an example, this paper proposes a multiple factor non-linear mathematical model to study the operating performance of the induction motor system. This model consists of non-linear time varying electromagnetic field equations, mechanical wave equations of sucker rods and non-linear coupling equation of reducer and four-bar linkage. The equations are numerically solved by combining time-step finite element method (TS-FEM), finite difference method (FDM), a linear dimension reduction method and the Newton-Raphson method. Simulation results, which are validated by experiments, reveal the influence of the fluctuating potential load on magnetic field distributions, stator and rotor currents, the input power and the power factor. The model and simulation results provide theoretical and technical supports for subsequent researches on model simplification and energy saving technologies.

Simulation Analysis of Designing a New Technique KER Warhead

The major technique of an air-defense warhead is to improve the damage after effect. Therefore, a new Kinetic Energy Rod (KER) warhead named profiled rod warhead is proposed in this paper. The detonation process is simulated by ANSYS/LS-DYNA, and the deployment velocity and initial flight attitude of rod are achieved. Rigid body dynamics equations of rod, which accounts for the influence of air resistance, are set up to predict the flight trajectory of long-distance. The results show that profiled rods are conducted to determine a higher penetrability compared with traditional cylindrical rods, and the profiled rods may provide a better penetration angle that still maintains a significant penetrability against projectiles when the rods move off long-distance range.

“Wunderlich, Meet Kirchhoff”: A General and Unified Description of Elastic Ribbons and Thin Rods

The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich's model) and the limit of small width (Sadowksky's model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Fr\'enet-Serret's frame, the present work unifies the description of thin ribbons and thin rods.

In-plane and out-of-plane free vibration and stability of a curved rod in flow

This paper investigates the free vibration and stability of a curved rod in flow. The equations of the three-dimensional motions of the rod are derived by the Newtonian approach. The differential quadrature method (DQM) is introduced to formulate the discrete forms of the governing equations of the inextensible rod with clamped–clamped supports. Based on numerical calculations, the effects of several system parameters, especially the flow velocity, on the natural frequencies and stability of the system are discussed. Buckling and flutter instability are detected as the flow velocity is varied in a certain range. Moreover, a derivation of the generalized slender-body theory for such a deformable curved rod is given in Appendix A.

Dynamic Modeling and Analysis of Seven-Bar Seedling Planting Mechanism of Transplanting Machine

Seven-bar seedling planting mechanism is one kind of developed transplanting mechanism of vegetable seedling transplanting machines. Dynamic equations of the link rods were established and analysis program was compiled so as to analyze the dynamic characteristics of the mechanism; Meanwhile, simulation model of the mechanism was established and corresponding virtual dynamic test was conducted based on Adams. The consistency between calculation results based on kinematic theory model and simulation results achieved from Adams was proved through comparison, which verified the dynamics model was correct and reliable. It can be concluded that the dynamic model was the basis of dynamic parameter optimization model of seven-bar seedling planting mechanism, which facilitates the parameter optimization process of the kind of seedling planting mechanism with various specifications, and it also avoided the tedious process of repeated modeling based on Adams dynamics optimization. With the analysis results of the seven-bar seedlings planting mechanism based on the established kinematic model, it was found out that the force fluctuation at the chassis caused by the reciprocating motion of the mechanism was large, which could cause machine vibration and increase the labor intensity of operators.

Response of infinite length rods and beams with periodically varying area

This paper develops a solution method for the longitudinal motion of a rod or the flexural motion of a beam of infinite length whose area varies periodically. The conventional rod or beam equation of motion is used with the area and moment of inertia expressed using analytical functions of the longitudinal (horizontal) spatial variable. The displacement field is written as a series expansion using a periodic form for the horizontal wavenumber. The area and moment of inertia expressions are each expanded into a Fourier series. These are inserted into the differential equations of motion and the resulting algebraic equations are orthogonalized to produce a matrix equation whose solution provides the unknown wave propagation coefficients, thus yielding the displacement of the system. An example problem of both a rod and beam are analyzed for three different geometrical shapes. The solutions to both problems are compared to results from finite element analysis for validation. Dispersion curves of the systems are shown graphically. Convergence of the series solutions is illustrated and discussed.

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (t0,x0)∈R+×S1 is the identically zero solution. Such conclusion holds provided the physical parameter γ of the model (related to the Finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa–Holm equation, corresponding to γ=1. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.

Local-in-Space Criteria for Blowup in Shallow Water and Dispersive Rod Equations

We unify a few of the best known results on wave breaking for the Camassa--Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that $u_0'+|u_0|$ is strictly negative in at least one point $x_0$ of the real line. Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean's necessary and sufficient condition for blowup, our approach applies to other equations that are not integrable: we illustrate this fact by establishing new local-in-space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods.

A Numerical Method to Model Dynamic Behavior of Thin Inextensible Elastic Rods in Three Dimensions

Static equations for thin inextensible elastic rods, or elastica as they are sometimes called, have been studied since before the time of Euler. In this paper, we examine how to model the dynamic behavior of elastica. We present a fairly high speed, robust numerical scheme that uses (i) a space discretization that uses cubic splines, and (ii) a time discretization that preserves a discrete version of the Hamiltonian. A good choice of numerical scheme is important because these equations are very stiff; that is, most explicit numerical schemes will become unstable very quickly. The authors conducted this research anticipating describing the dynamic Kirchhoff problem, that is, the behavior of general springs that have natural curvature, and for which the equations take into account torsion of the rod.

On the Cosserat model for thin rods made of thermoelastic materials with voids

In this paper we employ a Cosserat model for rod-like bodies and study the governing equations of thin thermoelastic porous rods. We apply the counterpart of Korn's inequality in the three-dimensional elasticity theory to prove existence and uniqueness results concerning the solutions to boundary value problems for thermoelastic porous rods, both in the dynamical theory and in the equilibrium case.

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.

Dynamic equations of thermoelastic Cosserat rods

A thermoelastic Cosserat rod with a heat flux along its length is modeled after reviewing a simple Cosserat rod model. Extended Kirchhoff constitutive relations that include thermal effects, and the associated heat conduction equation, are derived using the first law of thermodynamics. The rate of internal dissipation of the Cosserat rod is estimated by the Clausius–Duhem inequality. Nonlinear dynamic equations of the thermoelastic Cosserat rod, which extend the simple Cosserat rod model, are obtained. Dynamic equations of a planar thermoelastic Cosserat rod, the Timoshenko thermoelastic beam, and the planar Euler–Bernoulli thermoelastic beam are derived as a special case within the framework of the thermoelastic Cosserat rod.

Nonlinear tension-bending deformation of a shape memory alloy rod

Based on the measured shape memory alloy (SMA) stress–strain curve and the nonlinear large deformation theory of extensible beams (or rods), the first-order nonlinear governing equations of a SMA cantilever straight rod are established. They consist of a boundary-value problem of ordinary differential equations with a strong nonlinearity, in which seven unknown functions are contained and the arc length of the deformed axis is considered as one of the basic unknown functions. The shooting method combining with the Newton–Raphson iteration method is applied to solve the equations numerically. For a SMA cantilever rod subjected to a transverse uniformly distributed force, the deformation characteristics curves, the maximum strain and the maximum stress distribution curves along the longitudinal direction of rod, and the relation curves between deformation characteristic parameters and transverse uniformly force under different slenderness ratios are obtained. The effects of material nonlinearity, geometrical nonlinearity and slenderness ratio on the tension-bending deformation of the SMA cantilever rod are investigated. The numerical simulation results are in good agreement with the experimental data from the literature, verifying the soundness of the entire numerical simulation scheme.

Surface traction and the dynamics of elastic rods at low Reynolds number.

Molecular and cell biological processes often use proteins and structures that are significantly longer in one dimension than they are in the other two, for example, DNA, actin, and bacterial flagella. The dynamics of these structures are the consequence of the balance between the elastic forces from the structure itself and viscous forces from the surrounding fluid. Typically, the motion of these filamentary objects is described using variations of the Kirchhoff rod equations with resistive forces from the fluid treated as body forces acting on the centerline. In reality, though, these forces are applied to the surface of the filament; however, the standard derivation of the Kirchhoff equations ignores surface traction stresses. Here, we rederive the Kirchhoff rod equations in the presence of resistive traction stresses and determine the conditions under which treating the drag forces as body forces is reasonable. We show that in most biologically relevant cases the standard implementation of resistive forces into the Kirchhoff rod equations is applicable; however, we note one particular biological system where the Kirchhoff rod formalism may not apply.

On a thermodynamic theory of rods with two temperature fields

This paper presents a thermodynamic theory for elastic rods using the model of directed curves. In this model, the thin rod-like bodies are described as deformable curves with a triad of rigidly rotating vectors attached to each point. To account for the thermal effects in rods, we introduce two independent temperature fields: the absolute temperature field and the temperature deviation field. We present a complete derivation of the non-linear equations of thermoelastic rods, starting from the principles of thermodynamics. Finally, we prove the uniqueness of solution to the linearized equations of thermoelastodynamics for rods.

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An internal-variable rod model for stress-induced phase transitions in a slender SMA layer. I: Asymptotic equations and a two-phase solution

In part I of this series of two papers, an internal-variable rod model is proposed to study the stress-induced phase transitions in a slender shape memory alloy (SMA) layer. To study the mechanical responses of SMAs, two independent energy functions are adopted: the Helmholtz free energy and the rate of mechanical dissipation. A phase state variable is introduced to describe the phase transition process. Starting from the 2-D governing system and by using the coupled series-asymptotic expansion method, one single equilibrium equation is derived, which involves the leading order term of the axial strain and the phase state functions. Further by using the phase transition criteria, the evolution laws of the phase state functions corresponding to the outer loop of the stress–strain response are derived. As a result, the governing ODEs for the purely loading and purely unloading processes are obtained, which are called the asymptotic rod equations. The two-phase solution of the asymptotic rod equations in an infinitely long layer is then constructed. An explicit solution for the phase volume fraction of the corresponding inhomogeneous deformation is deduced, which appears to be the first analytical expression for this important quantity in stress-induced phase transitions. The key parameters in terms of original material constants for the phase transitions and deformations are also identified.

An internal-variable rod model for stress-induced phase transitions in a slender SMA layer. II. Analytical solutions for the outer loop and inner loops

In this sequel paper, the analytical solutions to the asymptotic rod equations (derived in part I of this series of papers) are constructed. More specifically, it is considered that the two ends of the layer satisfy the natural boundary conditions. By conducting a phase plane analysis, both a force-controlled problem and a displacement-controlled problem are studied. It is found that under certain conditions, both the constant solutions and the nontrivial solutions exist. The explicit expressions for the possible solutions are derived and the corresponding stress–strain curves are plotted. By comparing the total pseudo-elastic energies, the preferred solution in the displacement-controlled problem is determined. With the preferred solution obtained, the stress–strain curve and the profiles of the layer at the different stages are plotted, which can capture the key experimental features. Some further analysis has been given on the inner loops, where the reverse loading processes start just after the phase transformations are partially completed. By using the WKB method, the asymptotic solutions corresponding to the inner loops are constructed, which appear to provide the first analytical descriptions for the inner loops. The corresponding stress–strain curves and the graphes of the axial strain distribution of the layer are plotted. It is shown that the analytical solutions for the inner loops are also consistent with the experimental results.

On the linearized disturbance dynamic equations for buckling and free vibration of cylindrical helical coil springs under combined compression and torsion

In this study a set of twelve linearized disturbance dynamic equations in canonical form is derived systematically and in a comprehensive manner based on the first order shear deformation theory to study the buckling and vibration analysis of helical coil springs made of isotropic linear materials. Those complete equations comprise the axial and shear deformation effects together with rotatory inertia effects. The special case of these equations corresponds also to the equations for straight and circular rods. The main differences among the existing formulations based on the same approach are discussed briefly. The resulting equations are used for numerical buckling and free vibration analyses to show its soundness.

Elasticity and geometry: from hair curls to the non-linear response of shells

1. Introduction 2. Three-dimensional elasticity I: RODS 3. Equations for elastic rods 4. Mechanics of the human hair 5. Rippled leaves, uncoiled springs II: PLATES 6. The equations for elastic plates 7. End effects in plate buckling 8. Finite amplitude buckling of a strip 9. Crumpled paper 10. Fractal buckling near edges III: SHELLS 11. Geometric rigidity of surfaces 12. Shells of revolution 13. The elastic torus 14. Spherical shell pushed by a wall Appendix A: Calculus of variations: a worked example Appendix B: Boundary and interior layers Appendix C: The geometry of helices Appendix D: Derivation of the plate equations by formal expansion from 3D elasticity