Rod Equation Research Articles

A magnetoelastic theory for Kirchhoff rods having uniformly distributed paramagnetic inclusions and its buckling

We present a theory for finite and spatial elastic deformation of rods under the influence of arbitrary magnetic field and boundary condition. The rod is modeled as a Kirchhoff rod and is assumed to have uniformly distributed array of uniaxial spheroidal paramagnetic inclusions embedded in it all pointing in the same direction in the undeformed state. The governing equations of the magnetoelastic rod are derived which are further non-dimensionalized and linearized to investigate buckling in such rods. Analytical expressions for the onset of buckling from the rod’s trivial state are obtained in terms of loading parameters (applied magnetic field, axial load, torque) as well as geometric (inclusion orientation in the undeformed state) and material (ratio of bending and torsional stiffnesses) parameters for different combinations of boundary conditions. The buckled shape of the rod at the onset of buckling is also examined. The presented work can be useful in simulation and design of magnetic soft continuum robots.

Pull-out tests on bond behavior between timber and near-surface-mounted steel bars

Mounting bars on the surface of timber is a kind of special glued-in rod connection, which has great value of practical application. The near surface mounted method can effectively improve the bearing capacity of timber members (timber columns and beams), and the reliable bond performance between timber and bars is the crucial foundation for their collaborative working. To investigate the bond properties between timber and near surface mounted steel bars, pull-out tests of 48 timber specimens with embedded steel bars were conducted. There were four typical failure modes, which were pull-out failure of steel bar, splitting failure of timber, mixed failure and shear failure of timber. The load-slip curves of each specimen group were obtained, and the distribution curves of steel bar strain along anchorage length under different load level were determined, through the test data acquired by internally attached strain gauges. The steel bar strain decreased gradually from loaded end to free end. Distributions of bond stress between timber and steel bar were fitted, which were uneven and exhibited saddle-shaped distribution along the anchorage length. Besides, this paper investigated and discussed the effects of anchorage length, bar diameter, glue-line thickness and groove depth on bond performance between timber and mounted steel bars. Referring to classical equations of glued-in rod in timber, the design model of pull-out load for specimens with mounted steel bars was provided. By comparison the calculation results with other existing models, the newly proposed model had good prediction effectiveness. Based on classical mBPE and CMR models, the bond stress-slip models were established, which was suitable for timber and mounted steel bars. And the model curves and test results were in good agreement, which verified the reliability of the proposed theoretical models.

New refined model for curved linear anisotropic rods with circular cross section

An asymptotic reduction method is introduced to construct a curved rod theory for a general anisotropic linearized elastic material. For the sake of simplicity, the cross section is assumed to be circular. The starting point is Taylor expansions about the mean-line in curvilinear coordinates, and the goal is to eliminate the two spatial variables in the cross section in a pointwise manner in order to obtain a closed system for the displacement coefficients. We achieve this by using a Fourier series for the lateral traction condition together with the use of cylindrical coordinates in the cross section and by considering exact tridimensional equilibrium equation. We get a closed differential system of ten vector unknowns, and after a reduction process we obtain a differential system of the vector of the mean line displacement and twist angle. Six boundary conditions at each edge are obtained from the edge term in the tridimensional virtual work principle, and a unidimensional virtual work principle is also deduced from the weak forms of the rod equations. Through one example, we show that our theory gives more accurate results than the ones of both classical Euler-Bernoulli rod theory and Timoshenko rod theory. The displacement field is computed for two types of material symmetry : isotropy and transverse isotropy.

Blow-up criteria for modified two-component generalization of hyper-elastic rod equation

We investigate the blow-up phenomena for the two-component generalization of hyper-elastic rod equation on the real line. We establish some blow-up criteria for system of coupled equations with density function that in one case alternates its sign and keeps its sign in another case under certain natural initial profiles. Presented results extend and specify the earlier blow-up criteria for such type system.

Lump-soliton, lump-multisoliton and lump-periodic solutions of a generalized hyperelastic rod equation

In this paper, we will obtain lump-soliton solution for (1[Formula: see text]+[Formula: see text]1)-dimensional generalized hyperelastic rod equation, also known as generalized KdV equation by aid of Hirota bilinear method (HBM). We also obtain lump-multisoliton (which is an interaction of lump with one kink or two kink soliton) and lump-periodic solutions (which is formed by an interaction between lump and periodic waves). The dynamics of these solution are examined graphically by selecting significant parameters.

Some newly explored exact solitary wave solutions to nonlinear inhomogeneous Murnaghan’s rod equation of fractional order

The use of improved generalized Riccati equation mapping method has been demonstrated to find some new exact travelling wave solutions to space-time fractional non-liner double dispersive equation (DDE). The equation is used for modelling wave propagation in elastic inhomogeneous Murnaghan's rod. We have used Caputo's fractional derivative to achieve the fractional version of Murnaghan's rod equation. Improved generalized Riccati equation mapping method proves to be very effective tool to find a variety of soliton solutions. As a result, we found many new and general solutions including dark, combined dark-bright, singular periodic wave, combined singular periodic wave solutions and rational solutions. We have simulated the solitons, to check their types, with the help of graphs and all the solutions obtained in this article have been verified by back substitution in original equation by using Maple 17.

GENERALIZATION OF KIRCHHOFF KINETIC ANALOGY TO THIN ELASTIC SHELLS

The Kirchhoff kinetic analogy is generalized from thin elastic rods to thin elastic shells. The generalization makes thin shell deformations physically correspond and mathematically equivalent to rigid body motions. Hence theories and methods of rigid body dynamics can be applied to investigate deformations of thin elastic shell, and also provide a novel discretization for continuous thin elastic shells. An orthogonal spatial axis system is established along the coordinate lines of the middle surface under the straight normal assumption. The moving of the axis system along the coordinate lines in unit velocity forms its angular velocity, which is the curvature-twist vector with two independent variables. The curvature-twist vector along two coordinate lines expresses the deformation and the configuration of a thin elastic shell. It is demonstrated that curvature-twist vectors are compatible, and curvature-twist vectors and tangential vectors of middle surface are compatible. Nonholonomic constraints and differential equations of middle surfaces are established in the Euler angles and the Lam$\acute{e}$ coefficient form. The strain, the stress and the internal forces are formulated in the curvature-twist vectors and the Lam$\acute{e}$ coefficients. The equilibrium partial differential equations are presented with distributed internal forces intensity of thin elastic shells. The forms of the equations are similar to the Euler equations of rigid body dynamics and Kirchhoff equations of thin elastic rods. The fact means that the Kirchhoff kinetic analogy of thin elastic rods is generalized to thin elastic shells. The analogy relations between thin elastic shells and dynamics of rigid body or thin elastic rods are concluded. Finally, an example is given to show the application of this method. The proposed analogy leads to novel views and approaches to model and to analyze deformation of thin elastic shells. It is possible to generalize further the analogy for dynamics of thin elastic shells.

Design and Modeling of Soft Continuum Manipulators Using Parallel Asymmetric Combination of Fiber-Reinforced Elastomers

Abstract Soft continuum arms (SCAs) have a large workspace, dexterity, and adaptability, but at the cost of complex design construction highlighted by concatenating several serial segments. In this paper, we propose a new design architecture for SCAs composed of a parallel combination of pneumatic actuators. The BR2 SCA featured in this work is asymmetric as it combines one soft bending (B) actuator and two soft rotating (R2) actuators as opposed to state of art symmetric architectures that adopt bending segments. Spatial deformation is obtained by combining the bending and rotating feature of the individual actuators. This paper also formulates an approximate forward analysis method based on Kirchhoff rod equations to predict the spatial deformation under external loads with an accuracy less than 9% of the SCAs length. In addition, the model also takes into account the “coupling effect” inherent to the asymmetric parallel combination, where actuating rotating actuator attenuates the bending performance and vice versa. Consequently, this work also refines the design of the SCA that minimizes the coupling effect. A detailed performance study of the refined BR2 manipulator on a swiveling base demonstrates larger workspace and higher dexterity when compared with state of art single section SCAs. The performance of the design is validated through different tasks like obstacle avoidance, pick and place task, and whole arm grasping. These performance attributes surpass any other single segment soft module and is a potential building block for constructing customized SCAs.

Численный метод расчета нелинейной упругой характеристики продольно-поперечного преобразователя

This paper presents a numerical method for studying the stress-strain state of longitudinal-transverse transducers and obtaining their nonlinear elastic characteristic. The authors propose a mathematical model that uses a direct numerical solution of the boundary value problem based on the plain curved rod equations in MATLAB. The stress-strain state and the nonlinear elastic characteristic of the system are obtained using a method of successive loading based on linearized equations of the curved rod. The proposed model can be considered as an initial approximation to the solution of the spatial problem of the longitudinal-torsional transducer.

On a consistent rod theory for a linearized anisotropic elastic material: I. Asymptotic reduction method

An asymptotic reduction method is introduced to construct a rod theory for a linearized general anisotropic elastic material for space deformation. The starting point is Taylor expansions about the central line in rectangular coordinates, and the goal is to eliminate the two cross-section spatial variables in order to obtain a closed system for displacement coefficients. This is first achieved, in an ‘asymptotically inconsistent’ way, by deducing the relations between stress coefficients from a Fourier series for the lateral traction condition and the three-dimensional (3D) field equation in a pointwise manner. The closed system consists of 10 vector unknowns, and further refinements through elaborated calculations are performed to extract bending and torsion terms and to obtain recursive relations for the first- and second-order displacement coefficients. Eventually, a system of four asymptotically consistent rod equations for four unknowns (the three components of the central-line displacement and the twist angle) are obtained. Six physically meaningful boundary conditions at each edge are obtained from the edge term in the 3D virtual work principle, and a one-dimensional rod virtual work principle is also deduced from the weak forms of the rod equations.

A new solution of thin elastic rod by dynamic analogy

Inspired by Kirchhoff dynamic analogy, we write the Kirchhoff equation of thin elastic rod in form of curvature expression. Compared it with nonlinear Schrödinger equation, we extend a Jacobi elliptic function analogy solution to elastic rod equation and give a new alternative way to solve the Kirchhoff equation.

A Dynamic Model for Concentric Tube Robots

Existing static and kinematic models of concentric tube robots are based on the ordinary differential equations of a static Cosserat rod. In this paper, we provide the first dynamic model for concentric tube continuum robots by adapting the partial differential equations of a dynamic Cosserat rod to describe the coupled inertial dynamics of precurved concentric tubes. This generates an initial-boundary-value problem that can capture robot vibrations over time. We solve this model numerically at high time resolutions using implicit finite differences in time and arc length. This approach is capable of resolving the high-frequency torsional dynamics that occur during unstable "snapping" motions and provides a simulation tool that can track the true robot configuration through such transitions. Further, it can track slower oscillations associated with bending and torsion as a robot interacts with tissue at real-time speeds. Experimental verification of the model shows that this wide range of effects is captured efficiently and accurately.

Influence of force load on the stability of ultrasonic longitudinal–torsional composite drilling system

This study aimed to investigate the problems of system detuning, amplitude attenuation, and even vibration stop after the load was applied to the acoustic system during ultrasonic-assisted vibration processing. In the theoretical part, the overall theoretical model of the ultrasonic longitudinal–torsional composite drilling system (ULTCDS), including the longitudinal–torsional composite horn, transducer, and load was established based on the four-terminal network, combined with the longitudinal and torsional vibration equations of the variable cross-section rod. Also, the influence of the load on the input impedance and output amplitude of the acoustic system was preliminarily analyzed. In the simulation part, the static and dynamic drilling force was simulated by the pre-stressed modal and transient dynamic analyses, and the influence of the force load on the resonant frequency and output amplitude of the system was further explored. In the experiment part, the static force loading test and the ultrasonic drilling test of titanium alloy were designed. The micro-morphology of drilling holes under different processing parameters was observed, and the influence of force load on the system characteristics and stability was explored. The results showed that when the system was subjected to a force load greater than 400 N, the system current and frequency were greatly offset, the electromechanical conversion efficiency reduced, and the amplitude was attenuated to below the amplitude at empty load. In the ultrasonic-assisted drilling of titanium alloy processing, the acoustic system undergoes an obvious detuning phenomenon and the drilling quality is attenuated when the current increased by more than 60% and the frequency changed by more than 350 Hz, affecting the stability of the system and the ultrasonic-assisted processing effect.

On the derivation of an admissibility condition for phase boundary propagation in an SMA bar based on a 3-D formulation

There is some considerable difficulty in determining the solution uniquely for a propagating phase boundary in shape memory alloy (SMA) bar. In this paper, we establish an admissibility condition starting from a three-dimensional (3-D) internal-variable formulation to resolve this issue. We adopt a 3-D formulation in literature which is based on a constitutive model with specific forms of the Helmholtz free energy and dissipation rate. Then the 3-D dynamical equations are reduced to the 1-D rod equations for three phase regions (coupled with the radial effect and surface condition) by using two small parameters. Connection conditions at the phase interfaces are determined. By considering the traveling-wave solution for the rod system, we eventually derive three conditions across a sharp phase boundary corresponding to the 1-D sharp-interface model, including the two usual jump conditions and an additional condition. The third condition is then used to supplement the 1-D sharp-interface model to study an impact problem. The unique solution is constructed analytically for all possible impact velocity, including three kinds of wave patterns according to different levels of the impact velocity. The results are compared with those obtained by the maximal dissipation rate criterion.

On solitonic solutions for the hyperelastic rod equation

Formation of solitary wave trains is studied within the scope of the hyperelastic rod equation. The model equation is integrated numerically for the harmonic initial conditions using the discrete Fourier transform-based pseudospectral method. Numerical simulations are carried out for a wide range of material parameters. The analysis of the temporal and spatial behaviour of numerical solutions shows that (i) under a certain condition on material parameters trains of interacting solitary waves form from initial sinusoidal wave and (ii) the behaviour of emerging solitary waves is very close to that of solitons.

Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations

This paper considers the dynamical behavior of solutions near explicit self-similar singularity for a class of nonlinear shallow water models including the Dullin–Gottwald–Holm equation, the Korteweg–de Vires equation, the Camassa–Holm equation, the dispersive rod equation and the Benjamin–Bona–Mahony equation. We first show there are explicit self-similar solutions for those five nonlinear shallow water equations, then we find that those explicit self-similar solutions for the Dullin–Gottwald–Holm equation, the Camassa–Holm equation and the dispersive rod equation are asymptotic stable, but for the Korteweg–de Vries equation and the Benjamin–Bona–Mahony equation being unstable.

Numerical analysis of Volterra integro-differential equations for viscoelastic rods and membranes

We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain Ω. The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank–Nicolson method for the time discretization. Then we derive the uniform long time error estimates in the norm Lt1(0,∞;Lx2) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds.

Experimental statics calibration of a multi-constraint parallel continuum robot

A multi-constraint parallel continuum robot (MPCR) is composed of parallel flexible legs under multiple constraints. Compared with a rigid parallel robot, the model of MPCR, which is formulated based on Cosserat rod equations and coupled boundary conditions, is a boundary value problem (BVP) with only numerical solutions instead of analytical solutions, and the Jacobian matrix in calibration is ill-conditioned due to the parallel structure. Therefore, traditional calibration techniques cannot be used directly on MPCR. In this paper, we present a framework to calibrate this model with tip position data, including identifiability analysis and an optimization method that combines Levenberg-Marquadt method and interior point method. Simulation results of the method show the numerical solvability of the error model and its performance under noise disturbance. The calibration method is then validated experimentally on a prototype. The results show that the model matches experimental tip positions with an average and maximum error of 0.8% and 1.8% respectively of nominal total length, which verifies the effectiveness of the proposed calibration method.

Quasi hyperbolic function expansion method and tanh-function method for solving vibrating string equation and elastic rod equation

This paper applies the quasi hyperbolic function expansion method and the modified extended tanh-function method to exactly solve the nonlinear vibrating string equation and the elastic rod equation. The obtained exact solutions can be expressed by hyperbolic functions, which can outline various vibration properties.

Experimental investigation of the axial strength of glued-in rods in cross laminated timber

This paper presents results from an experimental assessment of glued-in rods in cross laminated timber (CLT). For the purposes of the study more than 60 pull–pull tests were performed, where the specimens varied in terms of bonded-in length (from 80 to 400 mm), rod diameter (16–24 mm) and rod-to-grain angle (parallel and perpendicular). Several different failure modes that are not common for other applications of glued-in rods (e.g., a failure between CLT layers) were obtained for the analysed CLT specimens. It was found that these failure mechanisms can substantially influence the obtained ultimate tension loads. At the end, the experimental results were compared with empirical and semi-empirical equations for estimating the pull-out strength of glued-in rods in structural timber and glulam. The comparison showed that most of the existing equations overestimate the ultimate tension loads for specimens with the rod parallel to the grain and underestimate the ultimate tension load for specimens with the rod perpendicular to the grain. The results vary because the possible CLT failure modes were not included in previous studies. Further studies are proposed to improve the equations for glued-in rods in CLT.