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Chapter 3 Performance Characteristics of Road Vehicles Chapter 5 Handling Characteristics of Road Vehicles. Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 5 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 6 43. Chapter 7 Vehicle Ride Characteristics L building, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 7 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 8 43. L building L building, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 9 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 10 43. L building L building, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 11 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 12 43. Stability Stability,Direction of motion, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 13 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 14 43. Stability Stability, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 15 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 16 43.

Tapered wheels Tapered wheels Basic motion,Why is the wheels on a train tapered. Consider a wheelset with tapered wheels on a rail In the steady. motion basic motion the wheels are moving on a straight line in the. longitudinal direction, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 17 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 18 43. Tapered wheels A train taking a turn Tapered wheels Perturbed motion. One reason for using tapered wheels is illustrated in the following figure What will happen if the basic motion is perturbed. showing a wheelset of a train taking a right turn Basic motion is shown to the left and perturbed motion to the right. 2r0 2 y 2r0 2 y Vxl, The longitudinal speed is larger for the outside wheel Vxl than for the w x. inside wheel Vxr but the rotational speed is the same The basic motion. in this case includes a constant drift y in the lateral direction which. compensates for this difference,Vxl r0 y Vxr r0 y, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 19 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 20 43. Tapered wheels Tapered wheels, Lateral drift causes a difference in the longitudinal.

velocity of the wheels in the same way as before,dx Using Vxl r0 y and Vxr r0 y the. dx Vxl r0 y,angular velocity can be written as,Vxl Vxr r0 y. Vxr Vxl 2 y, The longitudinal velocity of the center of gravity is Differentiating y r0 and using the expression. x t y t now given by w,for the angular velocity above the following. Vxl Vxr differential equation for y is obtained,The approximation gives the lateral velocity Vxr.

Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 21 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 22 43. Tapered wheels Harmonic oscillation Tapered wheels Unstable system. For a wheelset with positive taper angle as in the figure the solution. 2r0 2 For a wheelset with negative taper angle the solutions of the differential. y t y t 0 equation,is a harmonic oscillation y t y t 0. y t cos n t,with natural frequency r, 2r0 which means that the a small perturbation would cause an exponential. w growth of the lateral displacement and the system is clearly unstable. If there is friction in the system then the wheelset will return to the basic. motion asymptotically, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 23 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 24 43. Tapered wheels Spatial coordinates Tire,The dynamic equation. can be rewritten by using the relations,Figure 1 1 Tire construction.

d 2y x 2 Figure 1 2 Coordinates forces and moments. x t y t dx 2 r02,and the result is the following,y 00 x y x 0. A model that doesn t depend on speed, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 25 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 26 43. Rolling resistance Hysteresis,Exampel of a hysteresis loop caused by friction. Direction of motion, The rolling resistance of tires is primarily caused by the hysteresis in tire. materials due to the deflection of the carcass while rolling Ffriction. Other less important contributors to the rolling resistance are. Direction of motion, Friction between the tire and the road caused by sliding F Ffriction.

Displacement,Air circulating inside the tire, The energy loss due to hysteresis is equal to the shaded in the figure. 2 d Ffriction, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 27 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 28 43. Rolling resistance Hysteresis Rolling resistance, The center of normal pressure is shifted in the direction of motion due to. the hysteresis The coefficient of rolling resistance fr is defined as the ratio of the rolling. Normal pressure resistance Rr to the normal load W i e fr Rr W. Empirical formulas for calculating the rolling resistance coefficient as a. function of speed V based on experimental data, O Radial ply passenger car tire fr 0 0136 0 40 10 7 V 2. Radial ply truck tire fr 0 006 0 23 10 6 V 2, Fz Other factors that affect the rolling resistance.

Deformation,Surface texture Figure 1 5,Fr Inflation pressure Figure 1 7 and 1 8. The applied wheel torque on free rolling tire is zero Therefore a Internal temperature Figur 1 11 and 1 12. horizontal force Rr at the contact patch must exists to maintain. equilibrium This force is called known as the rolling resistance. Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 29 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 30 43. A Tire Under the Action of a Driving Torque A Tire Under the Action of a Driving Torque. Longitudinal slip,i 1 100 1 100,Limit cases,Free rolling tire i 0. Definitions The tire is not moving i 100 om V 0,Rolling radius of a free rolling tire r V. Effective rolling radius under the action of a driving torque re V. where V is the linear speed of the tire center and is the angular speed. Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 31 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 32 43. Driving Wheel The Brush Model Driving Wheel The Brush Model. The brush model is a very simple physical model of tire The tread of the The contact patch is assumed to rectangular and can be divided into an. tire is modeled as elastic bristles attached to the rim and longitudinal adhesion region 0 x lc and a sliding region lc x lt. force is generated by the deflection of the brush elements. Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 33 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 34 43. Driving Wheel The Brush Model Driving Wheel The Brush Model. The objective is to find the length of the adhesion region lc When does User a linear model for the relation between deflection and longitudinal. the longitudinal force becomes so large that the bristles begins to slide force per unit of length. Consider a bristle in the adhesion region,x kt e kt ix. It is assumed that normal force W is uniformly distributed in the contact. where lt is the length of the contact region, The velocity at the rim is r V The time since the bristle first touch Assumption The bristle will not slide if.

the ground is t x r The deflection at the distance x is. e x r V 1 x ix,r r where p is the coefficient of friction. Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 35 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 36 43. Driving Wheel The Brush Model Driving Wheel The Brush Model. The distribution of the longitudinal force in this case i ic. The condition can be written,First case When is there no sliding region. Answer When x lt fulfills the condition above i e x. kt lt i or i ic,lt kt lt2 1,Fx Area of the shaded region kt lt2 i Ci i. In the limit case i ic k l 2 is,Fx kt lt2 2, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 37 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 38 43. Driving Wheel The Brush Model Driving Wheel The Brush Model. Solution Recall that the bristle will not slide if kt ix p W lt i e. The second case There is a sliding region i ic x lc. The distribution of the longitudinal force in this case dFx. lc lt lc lt, How do we calculate the length of the adhesion region lc The longitudinal force is equal to the shaded area.

1 p W p W 1 lc,Fx lc lt lc p W 1,2 lt lt 2 lt, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 39 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 40 43. The Brush Model Summary Braking Wheel The Brush Model. Critical values if longitudinal slip and force The skid is defined. p W p W p W r r,ic och Fxc Ci ic is 1 100 1 100,kt lt2 2Ci 2 V re. There is no sliding region when i ic eller Fx Fxc and in this case when a braking torque is applied to the wheel. Limit cases,2 Free rolling tire is 0, If i ic eller Fx Fxc then the length of the adhesion region is Locked wheel is 100. p W Relations between i and is,kt lt i is,and the longitudinal force is 1 is. F x p W 1 p W 1 i,2 lt 4Ci i is, Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 41 43 Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 42 43.

Braking Wheel Summary,Critical values of skid and longitudinal force. Cs isc p W,No slide region is isc,With slide region is isc. Jan A slund Linko ping University Vehicle Dynamics and Control Lecture 1 43 43. Course book is Theory of Ground Vehicles 4th edition by J Y Wong You can borrow a copy during the course Chapter 1 Mechanics of Pneumatic Tires Chapter 3 Performance Characteristics of Road Vehicles Chapter 5 Handling Characteristics of Road Vehicles Chapter 7 Vehicle Ride Characteristics Some additional material is taken from the books Vehicle Dynamics Stability and Control 2nd

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