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Drive Shaft Model

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Drive Shaft Model VEHICLE DYNAMICS PROJECT DRIVELINE AND ENGINE CONTROL GROUP: ME10B014 E.KARTHIK ME10B016 GONA UDAY KUMAR ME10B021 M NAVYA TEJ ME10B037 TADI CHAITANYA VIKAS ME10B039 V SHARATH CHANDRA ME10B040 V SAI MUKESH CHANDRA ME10B041 V CHIRANJEEVI INTRODUCTION TO DRIVELINE • A driveline is the part of a motorized vehicle which connects the engine and transmission to the wheel axles. • In order to transmit this torque in an efficient way, a proper model of the driveline is needed for controller design purposes, with the aim of lowering emissions, reducing fuel consumption and increasing comfort. • It can be rear drive, front drive or four wheel drive. • Schematic of driveline: CAD model CAD model of car(Dodge Challenger SRT8) CAD model of driveline CAR model- car(Dodge Challenger SRT8) A COMPONENTS OF DRIVELINE The components of driveline are • Engine • Clutch • Transmission • Shafts • Wheels Simplest model : Flexible drive shaft model This Picture shows the driveline of heavy truck driveline. Fundamentals equation of driveline will be derived by using the generalized Newton’s Second law of motion. Relations between inputs and outputs will be described for each part in the given figure. Schematic of Driveline Parameters used in mathematical model Driving torque: Mm External load from Clutch: Mc Moment of Inertia of the engine: Jm Angle of flywheel: theta m Conversion ratio of transmission: i t internal friction torque of transmission: Mf r:t • Engine :The Output torque of the engine characterized by the driving torque (Mm) resulting from the combustion, the internal friction form the engine (Mfr:m ) and the external load from the clutch (Mc).Newtons’s second law of motion gives the following model where Jm is the mass moment of interia of the engine and the is θmthe angle of the flywheel. Jm휃푚 = Mm - Mfr:m - Mc • Clutch: A friction clutch found in vehicles ins equipped with a manual transmission consists of a clutch disk connecting the flywheel of the engine and the transmission input shaft .When clutch is engaged and no internal friction is assumed ,Mc=Mt is obtained .The transmission torque is a function of the angular difference (Θm- Θc)and the angular velocity difference over the clutch Mc = Mt = fc(θm-θc , θm - θc) • Transmission: A transmission has a set of fears ,each with a conversion ratio it . This gives the following relation between the input and the output torque of the transmission where the internal frcition torque of the transmission is labeled Mfr:t . The reason for considering the angle difference Θc-Θt it is the possibility of having the torsional effects in the transmission. Mp=ft(Mt,Mfr:t,Θc-Θt it, Θc-Θt it ,it ) • Propeller Shaft: The Propeller shaft connects the transmission’s output shaft with the final drive .No friction is assumed (Mp =Mf) giving the following model of the torque input to the final drive Mp =Mf =fp(Θt-Θp,Θt-Θp) • Final Drive : The final drive is characterized by a conversion ratio If in the same way as for the transmission .The following relation for the input and output torque holds Md =푓푓(푀푓, 푀푓푟:푓, 휃푝 − 휃푓푖푓, 휃푝 − 휃푓푖푓, 푖푓) • Drive Shafts : The drive shafts connect the wheels to the final drive .Here it is assumed that the wheel speed is the same for the 2 wheels. Therefore, the drive shafts are modeled as one shaft .When the vehicle is turning and the speed differs between the wheels ,both drive shafts have to be modeled .No friction ()gives the model equation Mw=Md=fd(Θf-Θw, Θf-Θw ) Wheels: • For vehicle with mass, m and velocity Frictional force 퐹푤 is given by 퐹푤 = 푚푣 + 퐹푎 + 퐹푟 + 푚푔푠푖푛(α) Where air drag force(퐹푎) is 1 퐹 = ∗ 푐 퐴 휌 푣2 푎 2 푤 훼 훼 Rolling resistance(퐹푟) is 퐹푟 = 푚(푐푟1 + 푐푟2푣) Gravitational force is 푚푔푠푖푛(α) where α is slope of the road. • Resulting torque 퐽푤휃 = 푀푤 − 퐹푤푟푤 − 푀푓푟:푤 Where 퐽푤 is the mass moment of inertia of the wheel,푀푓푟:푤 is the friction torque. By using above equations 1 퐽 + 푚푟2 휃 = 푀 − 푀 − ∗ 퐶 퐴 휌 푟3휃 2 − 푟 푚(푐 + 푐 푟 휃 ) 푤 푤 푤 푓푟:푤 2 푤 훼 훼 푤 푤 푤 푟1 푟2 푤 푤 −푟푤푚(푐푟1 + 푐푟2푟푤휃 푤)-푟푤푚gsin 훼 Final equation of torque for wheel Mathematical model Two types of mathematical models are analyzed: 1. Drive shaft model Flexible drive shaft, all other components are rigid 2. Linear Clutch drive shaft model Flexible drive shaft and Clutch, all other components are rigid Flexible drive shaft model Engine: Driving torque from combustion (Mm) and internal friction from the engine (Mf r:m) and the external load from clutch(Mc). Force and moment balance gives: Clutch : The clutch is assumed to be stiff which gives the following equations for the torque and the angle. Mc=Mt, θm= θc ----------------------- (1) Transmission: The transmission is described by one rotating inertia. The friction torque is assumed to be described by a viscous damping coefficient. The model of the transmission, 휃푐 = 휃푡푖푡 ---------(2) 퐽푡휃푡 = 푀푡푖푡 − 푏푡휃 푡 − 푀푝 ----------(3) From equations 1, 2, 3, 2 퐽푡휃푚 = 푀푐푖푡 − 푏푡휃푚 − 푀푝푖푡 -----------(4) Propeller Shaft: The propeller shaft is assumed to be stiff. Hence following equations can be written as, 푀푝 = 푀푓 , 휃푡 = 휃푡---------(5) Final drive: In the same way as for the transmission, the final drive is modelled by one rotating inertia 퐽푓. The friction torque is assumed to be described by a viscous damping coefficient 푏푓. The model of the final drive, is 휃푝 = 휃푓푖푓 ------------ (6) 퐽푓휃푡 = 푀푓푖푓 − 푏푓휃푓 − 푀푑 ----------(7) From equations 5 and 6, 2 퐽푓휃푡 = 푀푝푖푓 − 푏푓휃푓 − 푀푑푖푓 ----------(8) From equations 8 and 2, 2 퐽푓휃푚 = 푀푝푖푓 푖푡 − 푏푓휃푚 − 푀푑푖푓 푖푡------------(9) From equations 9 and 4, 2 2 2 (퐽푡푖푓 +퐽푓)휃푚 = 푀푝푖푡 푖푓 − 푏푡휃푚 푖푓 − 푀푑푖푓 푖푡 -- 푏푓휃푚 • Drive shaft: It is modelled as damped torsional flexibility having stiffness k and internal damping c 휃푚 휃 푚 푀푤 = 푀푑 = k(휃푓 − 휃푤) + c(휃푓 − 휃 푤) = k( − 휃푤) + c( − 휃 푤) 푖푓 푖푡 푖푓 푖푡 By replacing 푀푑 in final drive equation, 2 2 2 2 (퐽푡푖푓 +퐽푓) 휃푚 = 푀푐푖푡 푖푓 − 푏푡휃푚 푖푓 − 푏푓휃푚 − 푘 휃푚 − 휃푤푖푓 푖푡 − 푐(휃푚 − 휃푤 푖푓 푖푡) • Wheel: As discussed earlier , the dynamics of wheel is given by 2 휃푚 휃푚 3 2 (m푟푤+퐽푤) 휃푤 = 푘 − 휃푤 + c( − 휃 푤) − 푏푤휃 푤 − 0.5푐푤퐴푎휌푎푟푤휃푤 − 푟푤m(푐푟1 + 푔푠푖푛훼) 푖푓 푖푡 푖푓 푖푡 The drive shaft model: 퐽 푏 휃 휃 푡 2 2 푡 2 2 푚 푚 (퐽푚 + 2 + 퐽푓/푖푡 푖푓 )휃푚 = 푀푚 − 푀푓푟:푚 − ( 2 + 푏푓/푖푡 푖푓 )휃푚 − 푘( − 휃푤)/푖푓 푖푡 − c( − 휃푤)/푖푓 푖푡 푖푡 푖푡 푖푓 푖푡 푖푓 푖푡 2 휃푚 휃푚 2 3 2 (m푟푤+퐽푤) 휃푤 = 푘 − 휃푤 + c( − 휃 푤) −(푏푤+푚푐푟2푟푤)휃 푤 − 0.5푐푤퐴푎휌푎푟푤휃푤 − 푟푤m(푐푟1 + 푔푠푖푛훼) 푖푓 푖푡 푖푓 푖푡 State space formulation • Input to open-loop drive system is 푢 = 푀푚 − 푀푓푟:푚 (Difference between the driving torque and the fiction torque). • Possible physical state variables are torques, angle differences, and the angular velocity of any inertia. • The state space equation is 푥 = 퐴푥 + 퐵푢 + 퐻푙 Where A, B, H, x, and l are defined for drive shaft model and for the clutch and drive-shaft model. State space formulation for linear drive shaft model 휃푚 • 푥1 = − 휃휔 푖푡푖푓 • 푥2 = 휃푚 • 푥3 = 휃휔 • 푙 = 푟휔푚(푐푟1 + 푔푠푖푛(훼)) This gives A= 0 1/i -1 2 −푘/푖퐽1 −(푏 + 푐/푖 )/퐽1 c/푖퐽1 푘/퐽2 c/푖퐽2 −(푐 + 푏2)/퐽1 T • B=[0 1/퐽1 0] T • H=[0 0 −1/퐽2] Where 푖 = 푖푡푖푓 퐽푡 퐽푓 퐽1 = 퐽푚 + 2 + 2 2 푖푡 푖푡 푖푓 2 퐽2 = 퐽푤 + 푚푟푤 푏푡 푏푓 푏1 = 2 + 2 2 푖푡 푖푡 푖푓 2 푏2 = 푏푤 + 푚푐푟2푟푤 SIMULINK flexible drive shaft model Ramp response INPUT TORQUE,WHEEL SPEED ENGINE SPEED,WHEEL ACCELERATION CONTROL USING PD Linear flexible clutch and drive shaft A model with linear clutch flexibility and torsional flexibility in drive shaft is derived by repeating the process as in flexible drive shaft model with the difference that the model for clutch is a flexibility with stiffness Kc and damping coefficient Cc. 푀푐 = 푀푡 = 퐾푐 θ푚 − θ푐 + 퐶푐 θ푚 − θ푐 = 퐾푐 θ푚 − θ푡푖푡 + 퐶푐 θ푚 − θ푡 푖푡 Engine inertia is given by 퐽푚θ 푚 = 푀푚 − 푀푓푟:푚 − (퐾푐 θ푚 − θ푡푖푡 + 퐶푐 θ푚 − θ푡 푖푡 ) Equation describing the transmission is given by 퐽푡θ 푡 = 푖푡 퐾푐 θ푚 − θ푡푖푡 + 퐶푐 θ푚 − θ푡 푖푡 − 푏푡θ푡 − 푀푝 Substituting Mp 퐽푓 푏푓 퐽푡 + 2 θ푡 = 푖푡 퐾푐 θ푚 − θ푡푖푡 + 퐶푐 θ푚 − θ푡푖푡 − (푏푡+ 2) θ푡 − 푀푑/푖푓 푖푓 푖푓 Drive shaft equation is same as previous case. The Clutch and driveshaft model • 퐽푚휃 푚 = 푀푚 − 푀푓푟:푚 − (푘푐 휃푚 − 휃푡푖푡 + 퐶푐(휃푚 − 휃 푡푖푡)) 퐽푓 푏푓 휃 푡 • 퐽푡 + 2 휃푡 = 푖푡 푘푐 휃푚 − 휃푡푖푡 + 푐푐 휃푚 − 휃푡푖푡 − 푏푡 + 2 휃푡 − 1/푖푓(퐾푑 − 휃푤 + 푖푓 푖푓 푖푓 푐푑(−휃 푤 + 휃 푡푖푡)) 2 휃푡 휃푡 1 3 2 • 퐽푤 + 푚푟푤 휃 푤 = 퐾푑 − 휃푤 + 푐푑 − 휃 푤 − 푏푤 + 퐶푟2푟푤 휃 푤 − ∗ 푐푤퐴푎푟푤휌푎휃 푤 − 푖푓 푖푓 2 푟푤푚(퐶푟1 + 푔푠푖푛(훼)) State space formulation of the linear Clutch and drive-shaft model • 푥1 = 휃푚 − 휃푡푖푡 휃푡 • 푥2 = − 휃휔 푖푓 • 푥3 = 휃푚 • 푥4 = 휃 푡 • 푥5 = 휃휔 • A is given by the matrix: 0 0 1 -푖푡 0 0 0 0 1/푖푓 -1 −푘푐/퐽1 0 −퐶푐/퐽1 퐶푐푖푡/퐽1 0 2 푐푑 푘푐푖푡/퐽2 −푘푑/푖푓퐽2 퐶푐푖푡/퐽2 −(퐶푐푖푡 + 푏2 + 2 )/퐽2 퐶푑/푖푓퐽2 푖푓 T • B=[0 0 1/퐽1 0 0] T • H=[0 0 0 0 -1/퐽2] Where 푖 = 푖푡푖푓 퐽1 = 퐽푚 2 퐽2 = 퐽푡 + 퐽푓/푖푓 2 퐽2 = 퐽푤 + 푚푟푤 푏푓 푏2 = 푏푡 + 2 푖푓 푏3 = 푏푤 + 푐푟2푟푤 • 푥1 = 휃푚 − 휃푡푖푡 휃푡 • 푥2 = − 휃푤 푖푓 • 푥3 = 휃푚 • 푥4 = 휃 푡 • 푥5 = 휃푤 SIMULINK flexible linear clutch and drive shaft model RAMP RESPONSE INPUT TORQUE,WHEEL ENGINE SPEED,ANG ACC OF SPEED WHEEL Control Disturbance description: • Disturbance from the road is assumed to be described by slow varying load l and disturbance v.
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