Sliding a body on snow

Snow is considered as an ideal nonlinear elastoplastic medium. A body performs planeparallel motion on snow. The area of its contact with snow is a part of a rectangular plate. The contact zone changes during the motion of the body. Steady motions are found from the derived equations of motion in the case when the constant external forces and the moment exerted on the body are given. The inverse problem of determining the forces and moments is solved for a given steady motion of a vehicle.     

On the Computer Formulations of the Wheel/Rail Contact Problem

In this investigation, four nonlinear dynamic formulations that can be used in the analysis of the wheel/rail contact are presented, compared and their performance is evaluated. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail (constraint approach), while in the other two formulations the contact force is modeled using a compliant force element (elastic approach). The goal of the four formulations is to provide accurate nonlinear modeling of the contact between the wheel and the rail, which is crucial to the success of any computational algorithm used in the dynamic analysis of railroad vehicle systems. In the formulations based on the elastic approach, the wheel has six degrees of freedom with respect to the rail, and the normal contact forces are defined as function of the penetration using Hertzs contact theory or using assumed stiffness and damping coefficients. The first elastic method is based on a search for the contact locations using discrete nodal points. As previously presented in the literature, this method can lead to impulsive forces due to the abrupt change in the location of the contact point from one time step to the next. This difficulty is avoided in the second elastic approach in which the contact points are determined by solving a set of algebraic equations. In the formulations based on the constraint approach, on the other hand, the case of a non-conformal contact is assumed, and nonlinear kinematic contact constraint equations are used to impose the contact conditions at the position, velocity and acceleration levels. This approach leads to a model, in which the wheel has five degrees of freedom with respect to the rail. In the constraint approach, the wheel penetration and lift are not permitted, and the normal contact forces are calculated using the technique of Lagrange multipliers and the augmented form of the system dynamic equations. Two equivalent constraint formulations that employ two different solution procedures are discussed in this investigation. The first method leads to a larger system of equations by augmenting all the contact constraint equations to the dynamic equations of motion, while in the second method an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated. Numerical results are presented in order to examine the performance of various methods discussed in this study.     

Slip-based experimental studies of a vehicle interacting with natural snowy terrain

As longitudinal slip affects vehicle–pavement interactions on roads and hard surfaces, so too does it play an important role in interactions between vehicles and soft terrains, including snow. Although many slip-based models have been developed recently for tire–snow interactions (e.g., [1] and references cited therein), these models have only been partially validated, due to a lack of relevant experimental data. This paper presents comprehensive data from tests that were performed using a newly-developed test vehicle traversing natural snowy terrain, over a wide range of values for longitudinal slip, vertical load and torque via an effective accelerate/brake maneuver. Drawbar pull, motion resistance, wheel states and tire stiffness were presented as a function of slip; tire sinkage was obtained using a laser profilometer; strength and depth of snow were found using a snow micropenetrometer. The effects of the rear tire going over snow compacted by the front tire were also studied. The maximum traction force normalized by the vertical load is found to be ≈0.47, maximum motion resistance normalized by the vertical load is ≈0.4. Comparison of the trend and order-of-magnitude of test results with those from existing slip-based numerical model [1] shows good comparison in motion resistance, tire sinkage, and longitudinal stiffness, but indicates that a better traction model is needed to improve the comparison.     

轨下支承失效对直线轨道动态响应的影响

建立了基于Timoshenko梁模型的车辆/轨道耦合动力学模型,分析轨下支承失效对直线轨道动态响应的影响. 钢轨被视为连续弹性离散点支承上的无限长Timoshenko梁,通过假设轨道系统刚度沿纵向分布发生突变来模拟轨下支承失效状态. 推导了考虑钢轨横向、垂向和扭转运动的轮轨滚动接触蠕滑率计算公式. 利用Hertz法向接触理论和沈氏蠕滑理论计算轮轨法向力及轮轨滚动接触蠕滑力. 采用移动轨下支承模型的车辆/轨道耦合系统激振模式,考虑轨枕离散支承对系统动力响应的影响. 通过新型显式积分法求解车辆/轨道耦合动力学系统运动方程,由数值分析计算得到不同轨下支承失效状态下直线轨道的动态响应. 结果表明,轨下支承失效对直线轨道变形及加速度有显著的影响,随着失效轨下支承个数的增加,轮轨相互作用力和轨道部件的位移、加速度将会急剧增大,将加速失效区段线路状况的恶化.      

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.     

Analysis of Mars Exploration Rover wheel mobility processes and the limitations of classical terramechanics models using discrete element method simulations

A previous three-dimensional discrete element method (DEM) model of Mars Exploration Rovers (MERs) wheel mobility demonstrated agreement with test data for wheel drawbar pull and sinkage for wheel slips from 0.0 to 0.7. Here, results from the previous model are compared with wheel mobility data for non-MER wheels that cover the range of wheel slip from 0.0 to 1.0. Wheel slips near 1.0 are of interest for assessing rover mobility hazards. DEM MER wheel model predictions show close agreement with weight-normalized wheel drawbar pull data from 0.0 to 0.99 wheel slip and show a similar trend for wheel sinkage. The nonlinear increase in MER wheel drawbar pull and sinkage for wheel slips greater that 0.7 is caused by development of a tailings pile behind the wheel as it digs into the regolith.Classical terramechanics wheel mobility equations used in the ARTEMIS MER mobility model are inaccurate above wheel slips of 0.6 as they do not account for the regolith tailings pile behind the wheel. To improve ARTEMIS accuracy at wheel slips greater that 0.6 a lookup table of drawbar pull, wheel torque, and sinkage derived from DEM mobility simulations can be substituted for terramechanics equation calculations.     

A spatial motion analysis model of tracked vehicles with torsion bar type suspension

A spatial motion analysis model for high-mobility tracked vehicles was constructed for evaluation of ride performance, steerability, and stability on rough terrain. Ordinary high-mobility tracked vehicles are equipped with independent torsion bar type suspension system, which consists of road arms and road wheels. The road arm rotates about the axis of torsion bar, and rigidity of the torsion bar and cohesion of damper absorb sudden force change exerted by interaction with the ground. The motion of the road arms should be considered for the evaluation of off-road vehicle performance in numerical analysis model. In order to obtain equations of motion for the tracked vehicles, the equations of motion for the vehicle body and for the assembly of a road wheel and a road arm were constructed separately at first. Two sets of equations were reduced with the constraint equations, which the road arms are mechanically connected to the vehicle body. The equations of motion for the vehicle have been expressed with minimal set of variables of the same number as the degrees of freedom for the vehicle motion. We also included the effect of track tension in the equations without constructing equations of motion for the tracks. Numerical simulation based on the vehicle model and experiment of a scale model passing over a trapezoidal speed bump were performed in order to examine the numerical model. It was found that the numerical results reasonably predict the vehicle motion.     

Finite element formulation of slender structures with shear deformation based on the Cosserat theory

This paper addresses the derivation of finite element modelling for nonlinear dynamics of Cosserat rods with general deformation of flexure, extension, torsion, and shear. A deformed configuration of the Cosserat rod is described by the displacement vector of the deformed centroid curve and an orthogonal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parameterised by a rotational vector. The shape functions with up to third order nonlinear terms of generic nodal displacements are obtained by solving the nonlinear partial differential equations of motion in a quasi-static sense. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A cantilever is presented as a simple example to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures are presented through numerical simulations using the MATLAB software. In addition, a MicroElectroMechanical System (MEMS) device is presented. The developed equations of motion have furthermore been implemented in a VHDL-AMS beam model. Together with available models of the other components, a netlist of the device is formed and simulated within an electrical circuit simulator. Simulation results are verified against Finite Element Analysis (FEA) results for this device.     

Upwinded finite difference schemes for dense snow avalanche modeling

Avalanche dynamics models are used by engineers and land‐use planners to predict the reach and destructive force of snow avalanches. These models compute the motion of the flowing granular core of dense snow avalanches from initiation to runout. The governing differential equations for the flow height and velocity can be approximated by a hyperbolic system of equations of first‐order with respect to time, formally equivalent to the Euler equations of a one‐dimensional isentropic gas. In avalanche practice these equations are presently solved analytically by making restrictive assumptions regarding mountain topography and avalanche flow behaviour. In this article the one‐dimensional dense snow avalanche equations are numerically solved using the conservative variables and stable upwinded and total variation diminishing finite difference schemes. The numerical model is applied to simulate avalanche motion in general terrain. The proposed discretization schemes do not use artificial damping, an important requirement for the application of numerical models in practice. In addition, non‐physical M‐wave solutions are not encountered as in previous attempts to solve this problem using Eulerian finite difference methods and non‐conservative variables. The simulation of both laboratory experiments and a field case study are presented to demonstrate the newly developed discretization schemes. Copyright © 2000 John Wiley & Sons, Ltd.     

硅微机械振动轮陀螺仪的机理研究

本文针对Ｄｒａｐｅｒ 实验室公布的硅微机械振动轮陀螺仪的设计方案,给出了它的非线性 动力学方程和运动规律,分析了该陀螺仪壳体绕空间任意轴匀速旋转时干扰力矩所引起的漂 移率,给出了比通常线性化处理后得到的角速率测量关系式更为精确的测量表达式。     

DYNAMICAL ANALYSIS OF AUTOMOBILE HYSTERESIS NONLINEAR SYSTEM

This paper adopts free interface modal synthesis method to divide the whole automobile model into many sub-structures and establish dynamical equations of automobile nonlinear coupled system. The Monte Carlo method is used to simulate the spectrum of the random excitation of the road and the engine. Based on the automobile dynamical equations, a simulation is carried out within time domain and frequency domain on the characteristic of vibration due to the excitation of automobile wheel and the engine. The results are verified by bench experiment to make the research more practicable. In order to do research of rubber hysteresis’ influence on automobile dynamic property, Poincare diagrams and amplitude frequency characteristic curves were drawn with automobile linear and nonlinear models. The results show that the nonlinear dynamical model concerning rubber hysteresis not only can improve the simulation accuracy, but also is beneficial to find some complex nonlinear dynamical behaviors of vehicles.     

Nonlinear dynamic analysis on rigid-flexible coupling system of an elastic beam

Previous work examined the effect of the attached stiffness matrix terms on stability of an elastic beam undergoing prescribed large overall motion. The aim of the present work is to extend the nonlinear formulations to an elastic beam with free large overall motion. Based on initial stress method, the nonlinear coupling equations of elastic beams are obtained with free large overall motion and the attached stiffness matrix is derived by solving sub-static formulation. The angular velocity and the tip deformation of the elastic pendulum are calculated. The analytical results show that the simulation results of the present model are tabled and coincide with the one-order approximate model. It is shown that the simulation results accord with energy conservation principle.     

Nonlinear dispersive waves in a relaxing medium

The dispersion of nonlinear waves in a relaxing medium is analysed by making use of the evolution equations for longitudinal waves. The dispersion relations are obtained and the behaviour of the waves compared to those that arise when they are governed by the well-known Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations that describe unidirectional motion and also by the time regularized long wave (TRLW) equation that describes bi-directional motion. The nonlinear steady wave solutions are obtained. The general mathematical model used throughout this paper is obtained by the theory of nonlinear elasticity with weak relaxation effects (standard viscoelasticity). A further generalization using a four-element model is also discussed briefly.     

Reduced order nonlinear aeroelasticity of swept composite wings using compressible indicial unsteady aerodynamics

Nonlinear dynamic aeroelasticity of composite wings in compressible flows is investigated. To provide a reasonable model for the problem, the composite wing is modeled as a thin walled beam (TWB) with circumferentially asymmetric stiffness layup configuration. The structural model considers nonlinear strain displacement relations and a number of non-classical effects, such as transverse shear and warping inhibition. Geometrically nonlinear terms of up to third order are retained in the formulation. Unsteady aerodynamic loads are calculated according to a compressible model, described by indicial function approximations in the time domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to derive the final explicit form of the coupled fluid-structure equations of motion. The final nonlinear governing aeroelastic system of equations is solved using the eigenvectors of the linear structural equations of motion to approximate the spatial variation of the corresponding degrees of freedom in the Ritz solution method. Direct time integrations of the nonlinear equations of motion representing the full aeroelastic system are conducted using the well-known Runge–Kutta method. A comprehensive insight is provided over the effect of parameters such as the lamination fiber angle and the sweep angle on the stability margins and the limit cycle oscillation behavior of the system. Integration of the interpolation method employed for the evaluation of compressible indicial functions at any Mach number in the subsonic compressible range to the derivation process of the third order nonlinear aeroelastic system of equations based on TWB theory is done for the first time. Results show that flutter speeds obtained by the incompressible unsteady aerodynamics are not conservative and as the backward sweep angle of the wing is increased, post-flutter aeroelastic response of the wing becomes more well-behaved.     

Nonlinear dynamics of a resonant gas sensor

We develop a mathematical model for a resonant gas sensor made up of an microplate electrostatically actuated and attached to the end of a cantilever microbeam. The model considers the microbeam as a continuous medium, the plate as a rigid body, and the electrostatic force as a nonlinear function of the displacement and the voltage applied underneath the microplate. We derive closed-form solutions to the static and eigenvalue problems associated with the microsystem. The Galerkin method is used to discretize the distributed-parameter model and, thus, approximate it by a set of nonlinear ordinary-differential equations that describe the microsystem dynamics. By comparing the exact solution to that associated with the reduced-order model, we show that using the first mode shape alone is sufficient to approximate the static behavior. We employ the Finite Difference Method (FDM) to discretize the orbits of motion and solve the resulting nonlinear algebraic equations for the limit cycles. The stability of these cycles is determined by combining the FDM discretization with Floquet theory. We investigate the basin of attraction of bounded motion for two cases: unforced and damped, and forced and damped systems. In order to detect the lower limit of the forcing at which homoclinic points appear, we conduct a Melnikov analysis. We show the presence of a homoclinic point for a loading case and hence entanglement of the stable and unstable manifolds and non-smoothness of the boundary of the basin of attraction of bounded motion.     

Controlled pendulum on a movable base

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.     

Large amplitude free vibration of symmetrically laminated magneto-electro-elastic rectangular plates on Pasternak type foundation

Nonlinear free vibration of symmetrically laminated magneto-electro-elastic rectangular plate resting on an elastic foundation is studied analytically. The plate is considered to be simply supported on all edges. It is also assumed that the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces. To model the motion of the plate, the first order shear deformation theory along with the Gauss's equations for electrostatics and magnetostatics are used. Then equations of motion are reduced to a single nonlinear ordinary differential equation which is solved analytically by multiple scales method. The results are compared with the published results and good agreement is found. Some numerical examples are presented to investigate the effects of several parameters on the linear and nonlinear behavior of these plates.     

An efficient method for nonlinear aeroelasticy of slender wings

This paper aims the nonlinear aeroelastic analysis of slender wings using a nonlinear structural model coupled with the linear unsteady aerodynamic model. High aspect ratio and flexibility are the specific characteristic of this type of wings. Wing flexibility, coupled with long wingspan can lead to large deflections during normal flight operation of an aircraft; therefore, a wing in vertical/forward-afterward/torsional motion using a third-order form of nonlinear general flexible Euler–Bernoulli beam equations is used for structural modeling. Unsteady linear aerodynamic strip theory based on the Wagner function is used for determination of aerodynamic loading on the wing. Combining these two types of formulation yields nonlinear integro-differentials aeroelastic equations. Using the Galerkin’s method and a mode summation technique, the governing equations will be solved by introducing a numerical method without the need to adding any aerodynamic state space variables and the corresponding equations related to these variables of the problem. The obtained equations are solved to predict the aeroelastic response of the problem. The obtained results for a test case are compared with those of some other works and show a good agreement between results.     

Constraint implementation in wheeled apparatus rolling problems

The models obtained from equations of rolling of an apparatus with small slips of wheels with respect to the supporting plane are considered by passing to the limit of infinite values of rigidity of contact forces (zero values of slip speeds). The conditions under which these equations become the classical nonholonomic wheel no-slip model are discussed. It is shown that, for small angles of turn of the apparatus front wheels about the vertical axis, neglect of slips in the transverse direction is not correct; namely, the limit model is determined by the wheel no-slip conditions in the longitudinal direction and by the primary Dirac constraints arising owing to the degeneration of the Lagrangian of the system. The methods used to decrease the order of equations, where small slips of wheels are taken into account, can be useful for qualitative analysis of motion of wheeled systems and for solving the problems of estimation and control in real time.     

Magnetohydrodynamic and heat transfer effects on the stagnation-point flow of an electrically conducting nanofluid past a porous vertical shrinking/stretching sheet in the presence of variable stream conditions

A steady two-dimensional magnetohydrodynamic stagnation-point flow of an electrically conducting fluid and heat transfer with thermal radiation of a nanofluid past a shrinking and stretching sheet is investigated numerically. The model used for the nanofluid incorporates the effects of the Brownian motion and thermophoresis. A similarity transformation is used to convert the governing nonlinear boundary-layer equations into coupled higher-order nonlinear ordinary differential equations. The result shows that the velocity, temperature, and concentration profiles are significantly influenced by the Brownian motion, heat radiation, and thermophoresis particle deposition.