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5.2.1 General Ride Model of the Non-articulated Vehicle A 2-axle model is widely adopted for the non-articulated vehicles in automobile engineering since non-articulated vehicles with more than two axles usually can be transformed into equivalent 2-axle vehicle models in the dynamic analysis (Wong 1993; Gillespie, 1993). In 102 the present study, a 2-axle four-wheel vehicle is modeled as a combination of several rigid bodies connected by several axle mass blocks with springs and damping devices. The suspension system and the elasticity of tires are modeled with springs. The energy dissipation capacities of the suspension as well as the tires are modeled as damping devices with viscous damping assumed. The mass of the suspension system and the tires are assumed to concentrate on idealized mass blocks on each side of the vehicle and no mass in the spring and damping devices exists (Fig. 4.6). The displacements of the rigid body of the qth vehicle are denoted as: vertical displacement Z vr, pitching displacement in x-z plane θ vr and rolling displacement in y-z q q plane φvr. In the subscripts, “vr” refers to the rigid body of the vehicle. Vertical displacements of q
the “L” and “R” represent the left and right mass blocks on the jth axle, respectively. The lower “a” represents the axle suspension. The superscript “qj” represents the jth axle of the qth vehicle.
Same definitions apply hereafter. The longitudinal, lateral, and vertical directions of the bridge are set as x, y, and z axis, respectively.
5.2.2 Coupled Equations of Vehicle-Bridge Model in Modal Coordinates Assuming all displacements remain small, virtual works generated by the inertial forces, damping forces, and elastic forces acting on each vehicle on the bridge at a given time can be obtained. Assuming there are totally nv vehicles running on the bridge, and the initial conditions are the equilibrium conditions of the bridge under the self-weight of the bridge only without vehicles on it. The coupled equations can be finally built from the principle of virtual work as
where subscripts “b” and “v” represent for the bridge and vehicle, respectively; γ v and γ b are the displacement vectors of the vehicles and the bridge, respectively; superscripts of “s” and “v” in the stiffness and damping terms for the bridge refer to the terms of bridge structure itself and those contributed by the vehicles, respectively; subscripts “bv” and “vb” refer to the vehiclesbridge coupled terms; Matrices M, C and K are the mass, damping and stiffness matrices, respectively. F is the external loading terms. Subscripts “r”, “w” and “G” for the F refer to the loadings due to the road roughness, wind forces, and the gravity of the vehicles, respectively;
superscripts of “v” and “b” refer to the forces acting on the vehicles and on the bridge, respectively. Some vectors are shown as follows and details of other terms in Eq. (5.1) can be found in Chapter Four and are omitted here for the sake of brevity.
where the subscripts “st”, “ae” and “b” refer to static wind force component, self-excited wind force component, and buffeting force component, respectively.
With the generated wind velocity time history, the buffeting force time history corresponding to each point along the bridge span can be obtained (Cao et al. 2000). With the mode shapes obtained from finite element analysis, the wind buffeting forces on the whole bridge can be obtained through integrating all the force time histories along the bridge span.
The quasi static wind forces on vehicles are usually adopted (Baker, 1986):
where Fwx, Fwy, Fwz, Mwφ, Mwθ and MwΨ are the drag force, side force, lift force, rolling moment, pitching moment and yawing moment acting on the vehicle, respectively; CD, CS, CL, CR, CP and CY are the coefficients of drag force, side force, lift force, rolling moment, pitching moment and yawing moment for the vehicle, respectively; “A” is the frontal area of the vehicle; hv is the distance from the gravity center of the vehicle to the road surface; Ur is the relative wind speed to
the vehicle, which is defined as:
5.2.4 Numerical Approaches The position of any vehicle running on the bridge changes with time. Correspondingly, the coefficients of the coupled equations are also time dependent. Therefore, the matrices in Eq.
(5.1) should be updated at each time step after a new position of each vehicle is identified.
Rouge-Kutta method is chosen and a computer program based on Matlab is developed to solve the differential equations.
5.3 Accident Analysis Model for Vehicles on Bridges In the previous section, the dynamic interaction model of vehicle-bridge-wind system is briefly introduced. Such model is used to consider the dynamic interaction effects between vehicles and the bridge based on the detailed simulation of vertical stiffness and damping effect from the suspension system as well as from the tires. This interaction analysis model, however, is built based on the assumption that each vehicle wheel has full point contact with the bridge surface all the time and there exists no lateral relative movement between the wheels and the bridge surface. Such model predicts the responses of the bridge in all directions and the responses of vehicles only in several directions such as vertical, rolling and pitching directions.
The dynamic responses of vehicles in the vertical, rolling, and pitching directions from global bridge-vehicle analysis will be carried into the local accident analysis. Relative lateral and yaw responses of vehicles, which are not available in the global analysis, however, will be calculated separately with the local accident model which emphasizes on simulating the lateral relative movement and friction effects. The effects from lateral vibrations of the bridge on vehicle dynamics are considered through treating the lateral acceleration of the bridge as the external base excitation source of the vehicles.
5.3.1 General Model of Vehicle for Accident Analysis In the derivations of section 2, a general case with multiple vehicles is considered and each individual vehicle is generalized as the qth vehicle. In the hereafter accident model derivation, only one typical vehicle is considered. Fig. 5.2 (a) shows the force coordinates for the typical 2axle vehicle and the four wheels are defined for accident analysis. For most vehicles, the driving wheels are usually the rear ones and the steering wheels are the front ones. So the traction forces T only exists in the two rear wheels and the steering angle δ only exists for the two front wheels.
The wheel rolling resistance forces Fi (moving along the x axis direction) are related to the
vertical forces as:
Fi f = n re N i f Fi r = n re N ir (i=1 to 2) (5.18a, b) where nre is a coefficient of rolling friction (with negative value) and can be estimated as a constant or with some simple formulas at elementary level (Gillespie, 1993); superscripts “f” and “r” denote the front wheels and rear wheels; and Ni is the reaction forces of the ith wheel.
Aligning moment arising from the tire lateral frictions is omitted here.
105 When the side slipping angle is small, the tire side slipping forces Hi (along the y axis direction) can be related, approximately in a linear way, to the vertical reactions as (Gillespie, 1993):
H i f = mlaα f N i f H ir = mlaα r N ir ( i=1 to 2) (5.19a, b) where mla is a cornering stiffness coefficient and α f and α r are the side slipping angles for the front wheels and rear wheels, respectively (Fig. 5.2 (b)).
The side slipping angles for the front wheels and rear wheels can be expressed as (Gillespie, 1993; Chen and Ulsoy, 2001; Shin et al. 2002)
where ψ is the yaw angle of the vehicle at the center of the gravity around axis z; δi is the steering angle of the ith wheel (it is zero for rear wheels and the same for the two front wheels here); L1 and L2 are the horizontal distance between the center of gravity to the front wheels and rear wheels, respectively (Fig. 5.5); γ is the vehicle body side slipping angle at the gravity center and defined by γ = − arc tan ( v / V ) ≈ −v / V (5.21) where v is the side slipping (lateral) velocity of the vehicle body relative to the road surface; and V(t) is the longitudinal driving speed of vehicle at time t. In Ref. (Baker, 1986, 1991a, b, 1999), γ is chosen to approximate side slipping angles for both front and rear wheels as shown in Eqs.
For vehicles considering driver behavior as shown in Fig. 5.1, the following force and moment equilibrium equations should be satisfied.
(a) Force equilibrium in the x axis:
where Fwx is the aerodynamic wind force on the vehicle in the x direction as defined in Eq. (5.9);
M v is the total mass of the vehicle; the derivative of V enables the acceleration/decelerations to be considered; θ g is the grade and M v g sin θ g is also called grade resistance; g is the gravity acceleration; Ti is the traction force (be zero for non-drive wheels) or braking force of the ith tire and is believed approximately in proportion to the vertical reaction forces N i as
where Fwy is the aerodynamic wind force on the vehicle in y direction as defined in Eq. (5.10);
Ybr ( t ) is the lateral acceleration of the supporting surface under the vehicle, e. g. the bridge, v
where Fwz is the aerodynamic wind force on the vehicle in the z direction as defined in Eq. (5.11);
Z vr is the vertical displacement of the vehicle obtained from the previous section in Eq. (5.4).
where Mwψ is the aerodynamic yawing moment of the vehicle about the z axis (Eq. (5.14); Θvr is the polar moment inertia about the z axis; ψ is the yaw displacement of the vehicle about the z axis that is to be calculated. Please be noted that aligning moment of the tire is omitted here.
(g) Compatibility conditions 108 Vertical tire displacements for the non-articulated vehicle or any rigid body of the articulated vehicle should remain coplanar. If it is assumed that the tire reaction forces Ni are proportional to the tire displacements, then for conventional 2-axle vehicle, the following equation is satisfied (Baker, 196):
It should be noted that for articulated vehicle, compatibility equations exist for each rigid body of the whole vehicle.
Assuming the steering angle is small, approximations of cos δ ≈ 1 and sin δ ≈ δ can be made. Equations (5.22-23, 25-27 and 29) can be rewritten as following,
With the assumed initial conditions about ξ and δ, Eq. (5.54) can be solved at time step t and the reaction forces of four wheels can also be quantified with Eqs. (5.36-39). The new steering angle in time t+∆t can be predicted with the steering angle model (discussed next) based on the obtained response in time t. Then this calculation procedure continues to time t+∆t.
111 Repeat such procedure and the whole time history of ξ, reaction forces Ni and steering angle δ can be derived. The vehicle accident can be identified based on suitable accident criteria.
5.3.2 Driver Behavior Model The driver behavior is considered as the way that a driver will steer his/her vehicle being blown laterally and rotationally across the road. While each driver’s behavior may be different, it is expected that a driver would set the steering angle in according to the lateral and yaw displacements, velocities and accelerations in order to keep the vehicle in position (Baker 1991b, 1999). In automobile engineering, steering angle is mostly studied about how to negotiate the cornering other than how to correct the driving to avoid accident in a straight route (Wong, 1993;
Gillespie, 1992). It was also pointed out in (Chen and Ulsoy, 2001) that the driver behavior model is still a topic with a lot of uncertainties. Hence most existent models cannot be easily used in the current problem. Among very few people working on the driver behavior model serving current problems, Baker (1991b and 1999) once proposed the steering angle model, which is related to the lateral response and the lateral velocity. After some trial-and-error investigations of the driver behavior model, one similar to that proposed by Baker is suggested as follows.
The present driver behavior model is developed based on a simple idea that the steering angle should be adjusted to correct any lateral displacement of the front (steering) wheels. The adoption of the lateral responses of the front wheel other than that of the vehicle body (at C. G.) enables the yawing response can be taken care of as well. As mentioned earlier, each driver may react differently, and therefore, the present model is for demonstration only even though it gives very reasonable results. It is out of the scope of the current work in determining how driver
behaviors in actual driving in a windy environment. The model is suggested as:
L1 + L2 − λ1 (Y +ψ L1 ) − λ2 (Y + ψ L1 ) δ= (5.59) R where R is the radius of turn; λ1 and λ2 are related to the driver behavior and assumed to be constants for the same driver.
5.3.3 Accident Criteria
For vehicles, it is usually believed that three types of typical accidents may happen: