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Mechatronics in Automotive Engineering MEC-C405 2020 - 2021 Part 3 Driveline Dynamics 3. Driveline Dynamics The maximum achievable acceleration of a vehicle is limited by two factors: maximum torque at driving wheels, and maximum traction force at tireprints. The first one depends on engine and transmission performance, and the second one depends on tire-road friction. In this part, we examine engine and transmission performance. 3.1 Engine Dynamics The maximum attainable power Pe of an internal combustion engine is a function of the engine angular velocity ωe. This function must be determined experimentally, however, the function Pe = Pe (ωe), which is called the power performance function, can be estimated by a third-order polynomial If we use ωM to indicate the angular velocity, measured in [ rad/s], at which the engine power reaches the maximum value PM , measured in [Watt = Nm/s], then for spark ignition engines we use: The following figure illustrates a sample for power performance of a spark ignition engine that provides PM = 50kW at ωM = 586 rad/ s ≈ 5600 rpm. The curve begins at an angular velocity at which the engine starts running smoothly. A sample of power and torque performances for a spark ignition engine. For indirect injection Diesel engines we use and for direct injection Diesel engines we use The driving torque of the engine Te is the torque that provides Pe Example 120 Porsche 911T M and Corvette Z06T M engines. A model of Porsche 911 turbo has a flat-6 cylinder, twinturbo engine with 3596 cm3 total displacement. The engine provides a maximum power PM = 353kW ≈ 480 hp at ωM = 6000 rpm ≈ 628 rad/s, and a maximum torque TM = 620Nm at ωe = 5000rpm ≈ 523 rad/s. The car weighs around 1585 kg and can move from 0 to 96 km/ h in 3.7 s. Porsche 911 has a top speed of 310 km/ h. The power performance equation for the Porsche 911 engine has the coefficients and, its power performance function is A model of Corvette Z06 uses a V 8 engine with 6997 cm3 total displacement. The engine provides a maximum power PM = 377 kW ≈ 512 hp at ωM = 6300 rpm ≈ 660 rad/ s, and a maximum torque TM = 637Nm at ωe = 4800 rpm ≈ 502 rad/ s. The Corvette weighs around 1418 kg and can move from 0 to 100km/ h in 3.9 s in first gear. Its top speed is 320 km/ h. The power performance equation for the engine of Corvette Z06 has the coefficients and, its power performance function is The power performance curves for the Porsche 911 and Corvette Z06 are plotted in the following figure. Power performance curves for the Porsche 911 and Corvette Z06. Although there is almost no limit for developing a powerful engine, any engine with power around 100 hp would be enough for street cars with normal applications. It seems that engines with 600 hp reach the limit of application for street cars. However, race cars may have higher power depending on the race regulations. As an example, formula 1 regulations dictates the type of engine permitted. It must be a four-stroke engine, less than 3000 cm3 swept volume, no more than ten cylinders, and no more than five valves per cylinder, but there is no limit for power. 4.2 Driveline and Efficiency We use the word driveline, equivalent to transmission, to call the systems and devices that transfer torque and power from the engine to the drive wheels of a vehicle. Most vehicles use one of two common transmission types: manual gear transmission, and automatic transmission with torque convertor. A driveline includes the engine, clutch, gearbox, propeller shaft, differential, drive shafts, and drive wheels. The following figure illustrates how the driveline for a rear-wheel-drive vehicle is assembled. Driveline components of a rear wheel drive vehicle. 1. The engine is the power source in the driveline. The output from the engine is an engine torque Te, at an associated engine speed ωe. 2. The clutch connects and disconnects the engine to the rest of the driveline when the vehicle is equipped with a manual gearbox. 3. The gearbox can be used to change the transmission ratio between the engine and the drive wheels. 4. The propeller shaft connects the gearbox to the differential. The propeller shaft does not exist in front-engined front-wheel-drive and rear-engined rear-wheel-drive vehicles. In those vehicles, the differential is integrated with the gearbox in a unit that is called the transaxle. 5. The differential is a constant transmission ratio gearbox that allows the drive wheels to have different speeds. So, they can handle the car in a curve. 6. The drive shafts connect the differential to the drive wheels. 7. The drive wheels transform the engine torque to a traction force on the road. The input and output torque and angular velocity for each device in a driveline are indicated in the following figure. The input and output torque and angular velocity of each driveline component. The available power at the drive wheels is where η < 1 indicates the overall efficiency between the engine and the drive wheels ηc < 1 is the convertor efficiency and ηt < 1 is the transmission efficiency. The relationship between the angular velocity of the engine and the velocity of the vehicle is where ng is the transmission ratio of the gearbox, nd is the transmission ratio of the differential, ωe is the engine angular velocity, and Rw is the effective tire radius. Transmission ratio or gear reduction ratio of a gearing device, n, is the ratio of the input velocity to the output velocity while the speed ratio ωr is the ratio of the output velocity to the input velocity. 4.3 Gearbox and Clutch Dynamics The internal combustion engine cannot operate below a minimum engine speed ωmin. Consequently, the vehicle cannot move slower than a minimum speed vmin while the engine is connected to the drive wheels. At starting and stopping stages of motion, the vehicle needs to have speeds less than vmin. A clutch or a torque converter must be used for starting, stopping, and gear shifting. Consider a vehicle with only one drive wheel. Then, the forward velocity vx of the vehicle is proportional to the angular velocity of the engine ωe, and the tire traction force Fx is proportional to the engine torque Te where Rw is the effective tire radius, nd is the differential transmission ratio, ni is the gearbox transmission ratio in gear number i, and η is the overall driveline efficiency. These two equations are called the speed equation, and the traction equation. Proof. The froward velocity vx of a driving wheel with radius Rw is: and the traction force Fx on the driving wheel is: Tw is the applied spin torque on the wheel, and ωw is the wheel angular velocity. The wheel inputs Tw and ωw are the output torque and angular velocity of differential. The differential input torque Td and angular velocity ωd are: where nd is the differential transmission ratio and ηd is the differential efficiency. The differential inputs Td and ωd are the output torque and angular velocity of the vehicle’s gearbox. The engine’s torque Te and angular velocity ωe are the inputs of the gearbox. The input-output relationships for a gearbox depend on the engaged gear ratio ni. ηg is the gearbox efficiency, and ni is the gear reduction ratio in the gear number i. Therefore, the forward velocity of a driving wheel vx, is proportional to the engine angular velocity ωe, and the tire traction force Fx is proportional to the engine torque Te, when the driveline is engaged to the engine. Having the torque performance function Te = Te (ωe) enables us to determine the wheel torque Tw as a function of vehicle speed vx at each gear ratio ni. Using the approximate equation for Te: Then, Example A six-gear gearbox. Consider an inefficient passenger car with the following specifications: m = 1550 kg Rw = 0.326 m η = 0.24 torque = 392Nm at 4400 rpm ≈ 460.7 rad/ s power = 206000W at 6800 rpm ≈ 712.1 rad/ s 1st gear ratio = n1 = 3.827 2nd gear ratio = n2 = 2.36 3rd gear ratio = n3 = 1.685 4th gear ratio = n4 = 1.312 5th gear ratio = n5 = 1 6th gear ratio = n6 = 0.793 reverse gear ratio = nr = 3.28 final drive ratio = nd = 3.5451 Based on the speed equation, The angular velocities associated to maximum power and maximum torque are indicated by dashed lines. The power and torque performance equations for the engine can be approximated by because Using the above torque and traction equations, we can plot the wheel torque as a function of vehicle speed at different gears. The following figure shows the wheel torque-speed at each gear ni. The envelope curve for the series of torque-speed equations is similar to the torque curve of a constant power ideal engine. Wheel torque-speed at each gear ni of a gearbox, and the envelope curve simulating an ideal engine behavior. 4.4 Gearbox Design The speed and traction equations: are used to calculate the gear ratios of a gearbox as well as vehicle performance. Theoretically the engine should work at its maximum power to have the best performance. However, to control the speed of the vehicle, we need to vary the engine’s angular velocity. Hence, we pick an angular velocity range (ω1, ω2) around ωM , which is associated to the maximum power PM , and sweep the range repeatedly at different gears. The range (ω1, ω2) is called the engine’s working range. As a general guideline, we may use the following recommendations to design the transmission ratios of a vehicle gearbox: 1. We may design the differential transmission ratio nd and the final gear nn such that the final gear nn is a direct gear, nn = 1, when the vehicle is moving at the moderate highway speed. Using nn = 1 implies that the input and output of the gearbox are directly connected with each other. Direct engagement maximizes the mechanical efficiency of the gearbox. 2. We may design the differential transmission ratio nd and the final gear nn such that the final gear nn is a direct gear, nn = 1, when the vehicle is moving at the maximum attainable speed. 3. The first gear n1 may be designed by the maximum desired torque at driving wheels. Maximum torque is determined by the slope of a desired climbing road. 4. We can find the intermediate gears using the gear stability condition. Stability condition provides that the engine speed must not exceed the maximum permissible speed if we gear down from ni to ni−1, when the engine is working at the maximum torque in ni. 5. The value of cg for relative gear ratios can be chosen in the range. 4.4.1 Geometric Ratio Gearbox Design When the jump of engine speed in any two successive gears is constant at a vehicle speed, we call the gearbox geometric. The design condition for a geometric gearbox is where cg is the constant relative gear ratio and is called step jump. Proof. A geometric gearbox has constant engine speed jump in any gear shift. So, a geometric gearbox must have a gear-speed plot such as that shown in the following figure. A gear-speed plot for a geometric gearbox design. The engine working range is defined by two speeds (ω 1, ω2) When the engine reaches the maximum speed ω2 in the gear number i with ratio ni, we gear up to ni+1 to jump the engine speed down to ω1. The engine’s speed jump is kept constant for any gear change from ni to ni+1. Employing the speed equation, , we have and therefore, Let’s indicate the maximum vehicle speed in gear ni by vi and in gear ni−1 by vi−1, then, and therefore, the maximum speed in gear i to the maximum speed in gear i − 1 is inverse of the gear ratios The change in vehicle speed between gear ni−1 and ni is indicated by: and is called speed span. Having the step jump cg , and knowing the maximum speed vi of the vehicle in gear ni, are enough to find the maximum velocity of the car in the other gears

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