Symbolic-numerical Solution of Multibody Systems with Closed Loops

For multibody systems with closed kinematic Loops a set of ordinary differential equations and decoupled algebraic equations is formulated which can be solved with explicit multistep integration algorithms. This is achieved by introducing a minimal set of generalized coordinates being specified during numerical integration. For avoiding restart of the integration algorithm after changing these variables transformation relationships are given. Velocity and acceleration constraints are satisfied exactly, position constraints are fulfilled approximately by a dynamic invariant projection onto the constraint manifold. The method is demonstrated by an application to a five-point wheel suspension.     

Simulation of Multibody Vehicles Moving over Elastic Guideways1

SUMMARY

This paper describes the present state of a general purpose computer program for calculating the dynamic response of vehicles travelling over guideways which may be elastic.

The linearized state-equations of motion for general multibody vehicles are constructed automatically by the program, these equations are supplemented by the equations for the active subsystems. Finally, the vehicle system equations are combined with the modal equations for elastic guideways and the complete set of coupled equations is solved simultaneously by numerical integration.     

On Inverse Equations for Vehicle Dynamics

Abstract

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

Exact Equations for Tractrix Curves Associated with Vehicle Offtracking

Abstract

Vehicle offtracking behavior at low speeds is closely approximated by a geometric entity called a tractrix. This paper presents differential equations for generalized coordinates of a planar multibody vehicle model based on tractrix behavior. The equations are exact, can be used with any type of input path, are valid for forward and backward movements, and are much simpler than previously published formulations used to compute transient offtracking. The differential equations can be integrated using conventional numerical integration algorithms to obtain plots of the low-speed tracking performance of articulated vehicles. The equations were formulated symbolically by a computer program used to analyze the kinematic and dynamic behavior of multibody systems. Example numerical results are plotted.     

Tractor-Semitrailer Dynamics: Design of the Fifth Wheel∗

SUMMARY

The nonlinear equations of motion are derived for a tractor-semitrailer truck where both the itractor and the semitrailer yaw, pitch, roll, and translate. Special emphasis is placed on the constraints imposed by the fifth wheel on the vehicle motion. In particular, the effects of two proposed fifth wheel design changes on the jackknifing behavior of a vehicle in a turning, braking maneuver are studied. The results demonstrate that the tendency of the vehicle to jackknife can be reduced with a geometric modification of the fifth wheel.     

Modeling and Analysis of Nonlinear Multibody Systems

SUMMARY

The method of multibody systems is introduced for the modeling of nonlinear vehicle systems. The equations of motion and the equations of reaction are found. For the dynamical analysis a classification of nonlinear phenomena is presented using objective and subjective criteria. As an illustrative example the dynamical behaviour of a four-body pendulum is investigated.     

Flexible Bodies in Multibody System Codes

SUMMARY

Multibody codes are'efficient tools to simulate nonlinear dynamic behaviour of rigid and flexible multibody systems undergoing large overall motions overlaid by small elastic deformation. This paper gives an overview of common approaches for the equations of motion of flexible body models and presents a general way to prepare the required data including geometric stiffening terms. In particularly, for the nodal approach the data are derived using standard results of a finite element analysis of the body. The computation of coefficient matrices describing the equations of motion is done outside the finite element code by matrix manipulations only. The data are stored in a standardised object-oriented structure. Consequently, the data set is independent of the formulation of the multibody system code.     

Dynamic Parameter Estimation of a MacPherson Strut Suspension

SUMMARY

The dynamic parameters of a MacPherson strut suspension were estimated from the kinematic responses and measured external forces on the system. First, Lagrange equation was used to develop the equations of motion of the system, and then equations of motion were written as linear with respect to the dynamic parameters being estimated. The generalized coordinate partitioning technique was applied to create a reduced set of equations of motion of the constrained dynamic system. In this method only the measurements of positions and kinematic responses of the independent coordinates, and the external forces applied on the system are required as inputs. The rank deficient linear system was solved by QR decomposition. Good correlation was obtained between the actual parameters and the estimated parameters, by comparison between the simulated system response from the actual parameters and from the estimated parameters.     

Robust hierarchical controller with conditional integrator based on small gain theorem for reference trajectory tracking of autonomous vehicles

ABSTRACT

A robust trajectory tracking controller is designed for autonomous vehicles based on a hierarchical architecture to make the autonomous vehicle track a given reference trajectory. The controller consists of two sub controllers: kinematic controller and dynamic controller. Based on the kinematics of tracking reference trajectory, a desired yaw rate is calculated by kinematic controller to make the lateral deviation global asymptotic stable. Then, steering wheel angle is calculated by a vehicle dynamic controller to make the vehicle yaw rate converge to the desired value and make the vehicle dynamic stable. Conditional integration method is used in the sub controllers. This method guarantees global asymptotic stability of tracking reference values and considers the uncertainty of parameters and constraints of desired yaw rate and actuators. Then based on small-gain theorem, the condition of the finite-gain L stability is given to the hierarchical controller to ensure the interconnected sub systems stable and prevent the amplification of system disturbance. Finally, the effectiveness and robustness of the controller are validated by real vehicle experiments.     

Effect of Frame Compliance on the Lateral Dynamics of Motorcycles

SUMMARY

The lateral dynamics of an uncontrolled motorcycle, running on a straight, level road surface, is investigated in this paper. The structural compliances in the front and the rear frames of the motorcycle are taken into account by introducing additional degrees of freedom in the analysis. The kinematics of the tires is represented by linear differential equations which are based on the taut-string model of pneumatic tires. The linear differential equations of motion are solved to yield the eigensolutions of the system. Numerical results, obtained for parameters corresponding to a Honda CB750 motorcycle, are presented and discussed.     

Stabilization of High-Speed Railway Vehicles

SUMMARY

The problems of critical speeds of railway vehicles with dry friction units determination are discussed. A new approach is used which extends the field of application of dynamic response linear analysis methods to vital nonlinear multibody systems. The special features concerning the influence of dry friction forces in the body supports on the trucks and parameters of horizontal constraints of wheelsets and truck frames on critical speed are indicated. It is shown that a significant rising of railway vehicle critical speeds can be reached by changing the structure of constraints between the body and the trucks.     

Damping Performance of Power Transmission System with Hydraulic Torque Converter

Abstract

In this work three first-order nonlinear, nonstationary differential equations describing a hydraulic torque converter in a power transmission system are used.

A linear model is obtained on the basis of these equations. This model is verified both theoretically and by an experimental test. On the basis of the calculations the torsional damping was determined. This permitted the damping performance of a power transmission system with a hydraulic torque converter to be defined and established.     

In-plane Dynamics of Tires on the Road Based on an Experimentally Verified Rolling Ring Model

SUMMARY

The dynamic behavior of tires rolling on the road is predicted based on a ring model that is experimentally verified to well represent a real tire. The road contact is kinematically represented by two displacement constraints, one in radial and one in tangential direction. The displacement constraints are enforced via the receptance theory. The receptance matrix developed for the rolling tire is found to be Hermitian. Numerical results show that the natural (or eigen-) frequencies of the rolling tire with road contact are, as expected, higher than those of the tire with no contact. The influence of the rotating speed to the eigen-frquencies of the tire is also shown. Analytic expressions for the mode shapes of a rolling tire with ground contact are derived.     

Numerical Simulation and Experimental Verification of the German Roller Rig for Rail Vehicles*

SUMMARY

A numerical simulation model of the roller test stand located at Munich and loaded by a bogie is discussed including its technical structure, the governing physical equations of motion and the structure of the simulation program.

Both, the set up of the mathematical and numerical models time and the computation time of simulation runs have been considerably reduced (by a factor of 20) using formula manipulation programs.

Simulation results concerning the

• starting behaviour of a bogie,

• stationary limit cycle behaviour of bogies with ideal and wear profile and

• influence of gauge changes and spring/damper modifications on limit cycle behaviour of a bogie are presented, some of which are compared with experimental results gained from test facility measurements. The simulation results are in good agreement with the experimental results and provide an experimental verification of the roller rig simulation model presented.

     

Simulation of Multibody Vehicles Moving over Elastic Guideways1

This paper describes the present state of a general purpose computer program for calculating the dynamic response of vehicles travelling over guideways which may be elastic.

The linearized state-equations of motion for general multibody vehicles are constructed automatically by the program, these equations are supplemented by the equations for the active subsystems. Finally, the vehicle system equations are combined with the modal equations for elastic guideways and the complete set of coupled equations is solved simultaneously by numerical integration.     

Self-Excited Vibrations of Railway Vehicle with Dry Friction Units

SUMMARY

Some results obtained during developing a new approach to analysis of conditions of self-excitation of lateral vibrations of railway vehicles as systems of variable structure are given in this article. Structure variability is conditioned by the state of constraints with dry friction. An idea of the replacement of a nonlinear multibody system by a number of subsystems, which are linearized in a certain way, is worked out. The efficiency of the method suggested for the determination of the characteristics of self-excited vibrations of active multibody systems with dry friction is illustrated by way of an example of the investigation of the dynamic behaviour of a freight car.     

Directional Stability of Truck-Dolly-Trailer System

SUMMARY

The present state of knowledge on the handling behaviour of truck-dolly-trailer system is based on parametric studies made through simulation or eigen value analysis. Any convincing study using the equations to the stability boundaries have not been reported so far due to algebraic complexities. This paper fills the gap. Effects of system parameters are assessed using equations to the stability boundaries obtained from Routh-Hurwitz criterion. The system equations of motion are derived from a bondgraph model of a linearised system.     

车速和路面附着系数的滚动时域估计

基于Dugoff轮胎模型建立了车辆的非线性动力学方程,并给出了路面附着系数的约束条件.针对车速和路面附着系数约束的非线性估计,提出了一种基于滚动优化原理的滚动时域估计法(MHE),并给出了MHE法的具体步骤.在不同路面上对MHE法进行了多种工况的实验验证,并在同样条件下与扩展Kalman滤波法进行了比较.实验结果表明,MHE法的估计性能优于扩展Kalman滤波法.     

A book review of “Dynamik der Kraftfahrzeuge (German)”, Dynamics of Motor Vehicles by M. Mitschke,Springer-Verlag, 1972, 529 pp., 382 figs.

SUMMARY

The demands upon the dynamic behaviour of a vehicle can be stated as an optimization problem. The optimum solution is calculated using an iterative optimization algorithm. Scaling the problem by non-linear transformations reduces the number of iterations. Lagrange multipliers provide useful information about the sensitivity of the optimum with respect to changes of the constraints. The analysis of the dynamic behaviour is performed in the frequency domain. New structural variants are calculated using system synthesis. As an example, the engine, cab and wheel suspension systems of a tractor/semi-trailer have been optimized all together.     

Natural Frequencies of Bernoulli–Euler Beams Resting on Two Elastic Supports: Application to Railway Vehicles

SUMMARY

The purpose of the present study is to investigate the effect of the different parameters of the elastic supports upon which a deformable beam lies, upon the natural frequencies of the system.

The influence of support parameters is obtained by exploiting characteristic transcendental equations. The natural flexural vibrations of the beam are only affected by the stiffness of the supports over a very restricted range; outside this range the frequencies are those of a completely free beam or of a beam supported by two simple rigid supports (infinite stiffness).