Semi-smooth Newton methods for mixed FEM discretizations of higher-order for frictional, elasto-plastic two-body contact problems

International audienceIn this article a semi-smooth Newton method for frictional two-body contact problems and a solution algorithm for the resulting sequence of linear systems are presented. It is based on a mixed variational formulation of the problem and a discretization by finite elements of higher-order. General friction laws depending on the normal stresses and elasto-plastic material behavior with linear isotropic hardening are considered. Numerical results show the efficiency of the presented algorithm

Optimal slip control for tractors with feedback of drive torque

Traction efficiency of tractors barely reaches 50 % in field operations. On the other hand, modern trends in agriculture show growth of the global tractor markets and at the same time increased demands for greenhouse gas emission reduction as well as energy efficiency due to increasing fuel costs. Engine power of farm tractors is growing at 1.8 kW per year reaching today about 500 kW for the highest traction class machines. The problem of effective use of energy has become crucial. Existing slip control approaches for tractors do not fulfil this requirement due to fixed reference set-point. The present work suggests an optimal control scheme based on set-point optimization and on assessment of soil conditions, namely, wheel-ground parameter identification using fuzzy-logic-assisted adaptive unscented Kalman filter.:List of figures VIII List of tables IX Keywords XI List of abbreviations XII List of mathematical symbols XIII Indices XV 1 Introduction 1 1.1 Problem description and challenges 1 1.1.1 Development of agricultural industry 1 1.1.2 Power flows and energy efficiency of a farm tractor 2 1.2 Motivation 9 1.3 Purpose and approach 12 1.3.1 Purpose and goals 12 1.3.2 Brief description of methodology 14 1.3.2.1 Drive torque feedback 14 1.3.2.2 Measurement signals 15 1.3.2.3 Identification of traction parameters 15 1.3.2.4 Definition of optimal slip 15 1.4 Outline 16 2 State of the art in traction management and parameter estimation 17 2.1 Slip control for farm tractors 17 2.2 Acquisition of drive torque feedback 23 2.3 Tire-ground parameter estimation 25 2.3.1 Kalman filter 25 2.3.2 Extended Kalman filter 27 2.3.3 Unscented Kalman filter 27 2.3.4 Adaptation algorithms for Kalman filter 29 3 Modelling vehicle dynamics for traction control 31 3.1 Tire-soil interaction 31 3.1.1 Forces in wheel-ground contact 32 3.1.1.1 Vertical force 32 3.1.1.2 Tire-ground surface geometry 34 3.1.2 Longitudinal force 36 3.1.3 Zero-slip condition 37 3.1.3.1 Soil shear stress 38 3.1.3.2 Rolling resistance 39 3.2 Vehicle body and wheels 40 3.2.1 Short description of Multi-Body-Simulation 40 3.2.2 Vehicle body and wheel models 42 3.2.3 Wheel structure 43 3.3 Stochastic input signals 45 3.3.1 Influence of trend and low-frequency components 47 3.3.2 Modelling stochastic signals 49 3.4 Further components and general view of tractor model 53 3.4.1 Generator, intermediate circuit, electrical motors and braking resistor 53 3.4.2 Diesel engine 55 4 Identification of traction parameters 56 4.1 Description of identification approaches 56 4.2 Vehicle model 58 4.2.1 Vehicle longitudinal dynamics 58 4.2.2 Wheel rotational dynamics 59 4.2.3 Tire dynamic rolling radius and inner rolling resistance coefficient 60 4.2.4 Whole model 61 4.3 Static methods of parameter identification 63 4.4 Adaptation mechanism of the unscented Kalman filter 63 4.5 Fuzzy supervisor for the adaptive unscented Kalman filter 66 4.5.1 Structure of the fuzzy supervisor 67 4.5.2 Stability analysis of the adaptive unscented Kalman filter with the fuzzy supervisor 69 5 Optimal slip control 73 5.1 Approaches for slip control by means of traction control system 73 5.1.1 Feedback compensation law 73 5.1.2 Sliding mode control 74 5.1.3 Funnel control 77 5.1.4 Lyapunov-Candidate-Function-based control, other approaches and choice of algorithm 78 5.2 General description of optimal slip control algorithm 79 5.3 Estimation of traction force characteristic curves 82 5.4 Optimal slip set-point computation 85 6 Verification of identification and optimal slip control systems 91 6.1 Simulation results 91 6.1.1 Identification of traction parameters 91 6.1.1.1 Comparison of extended Kalman filter and unscented Kalman filter 92 6.1.1.2 Comparison of ordinary and adaptive unscented Kalman filters 96 6.1.1.3 Comparison of the adaptive unscented Kalman filter with the fuzzy supervisor and static methods 99 6.1.1.4 Description of soil conditions 100 6.1.1.5 Identification of traction parameters under changing soil conditions 101 6.1.2 Approximation of characteristic curves 102 6.1.3 Slip control with reference of 10% 103 6.1.4 Comparison of operating with fixed and optimal slip reference 104 6.2 Experimental verification 108 6.2.1 Setup and description of the experiments 108 6.2.2 Virtual slip control without load machine 109 6.2.3 Virtual slip control with load machine 113 7 Summary, conclusions and future challenges 122 7.1 Summary of results and discussion 122 7.2 Contributions of the dissertation 123 7.3 Future challenges 123 Bibliography 125 A Measurement systems 137 A.1 Measurement of vehicle velocity 137 A.2 Measurement of wheel speed 138 A.3 Measurement or estimation of wheel vertical load 139 A.4 Measurement of draft force 140 A.5 Further possible measurement systems 141 B Basic probability theoretical notions 142 B.1 Brief description of the theory of stochastic processes 142 B.2 Properties of stochastic signals 144 B.3 Bayesian filtering 145 C Modelling stochastic draft force and field microprofile 147 D Approximation of kappa-curves 152 E Simulation parameters 15

An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams

The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees-of-freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes