Advances in the Modelling of Motorcycle Dynamics - ResearchGate

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272 R.S. SHARP ET AL.(PID) feedback of motorcycle lean angle error to steering torque, with the leanangle target being set by an initial value and a constant rate of change. The targetlean angle must therefore be a ramp function of time. This would be easy to alterif it were considered restrictive.The steering control gains need to be speed adaptive and they need choosingwith considerable care to achieve effective stabilisation. Especially difficult arecases involving very low or very high speed and high lean angles. Each of the threePID gains is linearly related to speed, as indicated by the relations:G p = spg0 + spg1 · u; G i = sig0 + sig1 · u; G d = sdg0 + sdg1 · u;corresponding to the control law:∫ tτ = G p (φ − φ ref ) + G i (φ − φ ref )dt + G d ˙φ;0where u is the forward speed, τ is the steering control torque, φ is the lean angleand φ ref is the target lean angle.7. Equilibrium State Checking and Power BalancingWith suitable stabilisation, the motorcycle can be run to equilibrium at any feasiblespeed and lean angle. To describe such an equilibrium state, force and momentbalance equations can be set up, as was done in [5, 30]. As described in [30], thechecking process includes a power balance, whereby the engine power is shown toaccount precisely for the aerodynamic and tyre losses. In steady turning, the forcebalance check is to ensure that the sum of the external forces is equal to the sum ofthe inertial and gravitational forces. The force error calculated is:F error = ∑ iF i + ∑ jm j (g − ω j × v j ),the first sum containing all the external forces, while the second deals with gravitationaland centripetal effects. The external forces include: (i) aerodynamic liftand drag forces, (ii) the front and rear wheel normal loads, (iii) the tyre side forcesand (iv) the tyre longitudinal forces, including the driving force at the rear tyresufficient to maintain the steady speed. In the second term, m j represents the massof the jth body, ν j is the velocity of the body’s mass centre, w j is the body’s angularvelocity vector and g is the gravitational acceleration vector. Invariably, in a fullyestablished steady turn, |F error |〈0.02 N.In much the same way, the following moment error should be zero:M error = ∑ il i × F i + ∑ j{l j × m j (g − v j × ω j ) − ω j × H j }+ ∑ kM k ,
ADVANCES IN THE MODELLING OF MOTORCYCLE DYNAMICS 273where l i and l j are moment arm vectors referred to the rear wheel contact pointand H j is the moment of momentum of body j about its mass centre. The firstsum accounts for the moments generated by the external forces listed above, whilethe second contains a part treating gravitational moments and moments of inertialforces on the body mass centres and a part accounting for the rate of change ofmoment of momentum of each body, with respect to its mass centre. The thirdsummation deals with aerodynamic pitching and tyre aligning moments.Each of the terms w × H is calculated as w × (H x i + H y j + H z k), with Hhaving components H x , H y and H z in directions denoted by the unit vectors i, jand k, which must be chosen so that the moment of momentum components areinvariant, when the motorcycle is in a steady turn. For all the non-spinning bodies,the body reference axes satisfy this requirement. For the wheels, the parent body’sreference system needs to be used and a moment of momentum term for the spinadded on. For the most general case applicable here in which I xy and I yz are zerobut I xz is non-zero [9], noting a change of sign of products of inertia, as comparedwith the reference, because [9] and Autosim use opposite definitions:H x = I xx ω x + I xz ω z ; H y = I yy ω y ; and H z = I xz ω x + I zz ω z ,in which ω x , ω y and ω z are the components of ω in the i, j and k directions.Thus, for the main body, this second component of rate of change of momentof momentum, that about the mass centre, is of the form:ω main ×{(I mainx ω x + I mainxz ω z ) · i main+ I mainy ω y · j main+(I mainxz ω x + I mainz ω z ) · k main}.Here, i main, j mainand k maindenote the unit vectors i, j and k for the main body.To deal with the rear wheel, its diametral inertia is added to the corresponding termsbelonging to the swing arm, its parent body, as if it were part of the swing arm.The spin is accounted for by a term: ω swingarm × I rwy ω y · [rwy] and similarly forthe front wheel, for which the lower fork body is the parent. For any steady turn,|M error |〈0.02 Nm.The power error is given by:P error = τ · ω spin + ∑ iF i · v i + ∑ kM k · ω k ,in which τ is the rear wheel driving torque and ω spin is its spin velocity relativeto the swing arm. v i is the velocity of the point of application of force, F i , andω k is the absolute angular velocity of the body to which moment, M k ,isapplied.Describing the power associated with tyre forces requires care also. The velocityinvolved is that of the tyre tread base material [30], already calculated in connectionwith finding the slip ratio and the slip angle. For any steady turn, indicating amazingprecision, P error 〈0.3 mW.
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Advances in the Modelling of Motorcycle Dynamics - ResearchGate

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