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Longitudinal dynamics and performance Traction-Limited Drive Prof. R.G. Longoria Spring 2016 ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Overview These slides review the performance model in steady-state longitudinal motion. This gives a basis for understanding the system (plant) that needs to be controlled either in open loop or closed loop. These slides review traction-limited performance and demonstrate how performance can be over-predicted when only looking at ideal traction, motivating need to include power limited source models. ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Longitudinal dynamics Recall the model for longitudinal dynamics of a vehicle, • • • • Fg = grade pɺ x = mvɺx = ∑ Fx Fr = rollingresistance ∑F Fd = drawbar x • Ftx = traction/braking Fa = aerodynamic = Ftx − Fg − Fr − Fa − Fd Maintaining a set speed requires adjusting the traction force to counteract external forces (road loads) and/or changes in mass. If you want to apply basic linear control design concepts, then a linear model is required. However, the forces depend on the vehicle velocity in a nonlinear manner. So, this equation would need to be linearized with respect to the forward velocity if you wanted to used linear control methods. Let’s first review the open-loop model. ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Two-axle vehicle on an incline (ideal traction forces) Fa Along the longitudinal (x) axis: pɺ x = m dvx = m ⋅ ax = ∑ Fx dt cf. Wong, Chapter 3, Fig. 3.1 Frf vɺx Fg Rolling resistance forces placed at wheel centers. Note, RR is often ignored in braking since the effect in that case may not be significant. ha r A h W Ftf B Frr Fd hd θs Frf l1 L l2 ∑F x ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Note: For passenger cars you can assume that: h ≈ h ≈ h a d = Tractive force − Road Loads Department of Mechanical Engineering The University of Texas at Austin Longitudinal tractive force (effort) In the longitudinal (x) direction, need to estimate forces, ∑F x = Ftf + Ftr − Fa − Frf − Frr − Fd − Fg Rolling resistance forces These are the tractive forces (effort). ∑F x The tractive forces should be modeled to estimate the force propelling a vehicle for given road/terrain conditions. It is necessary to make assumptions or gather information about the tire and the road/terrain conditions. = Ftf + Ftr − Fa − Frf − Frr − Fd − Fg Ftf ,r = tractive effort on front and rear Fa = aerodynamic resistance force Frf ,rr = rolling resistance on front and rear Fd = drawbar load Fg = grade resistance = W sin θ s NOTE: See Wong Chapter 3 and/or the appended slides for some details on modeling these forces. ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Dynamic model For vehicle on incline, general relations – for 2D motion: pɺ x = mvɺx = ∑ Fx NOTE: hr = h − r l1 y l2 l1 + l2 = L l1 − hr f r 1 − f r pɺ z = mvɺz = ∑ Fz hɺ = I ωɺ = M −l2 − hr f r 1 − fr ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control y ∑ y One way to solve for the unknowns is to formulate as 3 simultaneous equations: 0 W f −hFt 0 Wr = W cos θ −m vɺx Fg + Fa − Ft Department of Mechanical Engineering The University of Texas at Austin Normal forces and acceleration Solution of the three unknowns yields the two normal loads and an expression for the vehicle acceleration in terms of ‘known’ quantities, (l2 + hr f r ) h W cos θ − L Ft + L W f h (l1 − hr f r ) W = θ F + W cos t r L L vɺx 1 [ Ft − W sin θ − Fa − f rW cos θ ] m The traction force depends on the condition of the front and rear tires: rolling or slipping? Ft = Ftf + Ftr front ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control rear Department of Mechanical Engineering The University of Texas at Austin Shouldn’t the front to rear axle loads show the effect of acceleration? Only if you re-derive the equations, assuming you are given acceleration (or deceleration). From previous case, rewrite x-direction Newton’ law to solve for traction forces in terms of known quantities: Ft = max + f rW f + f rWr + Fg + Fa Then insert into the other two equation to solve for the axle loads: l1 −l2 W f −h ( max + Fg + Fa ) 1 1 W = W cos θ r l2 h W f = W cos θ − max + Fg + Fa L L l1 h Wr = W cos θ + max + Fg + Fa L L ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Consider this problem from Wong, Chapter 3 This problem seeks a steady-state solution for vehicle speed. ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Finding solutions for velocity 1. Steady state: assume maximum traction, e.g., Ft max f = µW f Solve for the front and rear forces: Ft max r l1 − hr f r (v) = µW cos θ L − h µ Ft max f l2 + hr f r (v) = µW cos θ µ L + h Plot these forces and all loads as functions of velocity. Look for intersection between traction force and load curve. 2. Dynamic: need to solve for velocity as a function of time by integrating the 1st order ODE, considering all the loads, which are also nonlinear functions of velocity. Need a model for the traction force – for now assume maximum traction as above. These solutions estimate the ‘traction-limited’ performance of the vehicle. It is necessary to also look at ‘power-limited’ performance. ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Solving source load problems: source and load characteristics intersect, a steady-state speed is achieved The source and load lines are just equations of effort (torque or force) versus flow (angular velocity or velocity), so when you plot them together the intersection corresponds to a solution of the two simultaneous equations. Basically, you are solving a set of algebraic equations for a steady-state (operating) condition. Graphical solution e (T or F ) Source Load eo ( eo , fo ) P = eo ⋅ f o fo Area = power delivered ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control f (ω or v) Next slide shows how this approach is used by Wong to solve Problem 3.1. Department of Mechanical Engineering The University of Texas at Austin Solution to 3.1 from Wong Make note of how the total road loads are plotted here on the same graph as the traction forces (the source). ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin My solution to 3.1 from Wong ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin Summary of vehicle on incline • Plane motion analysis of rigid vehicle, an assumption valid for many practical applications. • Typical of what is required for basic vehicle weight transfer calculations, or simple force analysis, etc. • Analysis not useful for simulation of braking or traction, or for control design • Good for ‘go/no-go’ type assessment, design analysis, etc. • This model assumes best case scenario with maximum traction. • To be more realistic, we need to consider power-limited performance by the including a model of the ‘source’ ME 360/390 – Prof. R.G. Longoria Vehicle System Dynamics and Control Department of Mechanical Engineering The University of Texas at Austin