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An analytical model is developed to describe the torsional responses of the powertrain system. The model is used to analyze system equilibrium, free vibration, forced and self-excited vibrations. The equations of motion are linearized about the equilibrium to determine natural frequencies and mode shapes of the torsional modes. The forced responses of the system are investigated by including the excitations of gas combustion forces and inertia torques induced by the reciprocating motions of the piston and connecting rod. The self-excited vibration induced by negative damping behavior of clutch torque capacity is studied. For an example rear-wheel drive powertrain considered, the free vibration analyses show the natural frequencies and the associated mode shapes. The forced and the self-excited vibrations for the transmission gearset and the driveline components are examined. Experimental measurements from a test powertrain are used to confirm the theoretical predictions.

An analytical model is developed to describe the torsional responses of the chain drive and crankshaft. The model is used to analyze system equilibrium, free vibration and forced responses. The equations of motion are linearized about the equilibrium to determine natural frequencies and mode shapes of the torsional modes. The forced responses of the system are investigated by including the excitations of valvetrain torque, gas combustion forces and inertia torques induced by the reciprocating motions of the piston and connecting rod. For an example engine considered, the free vibration analyses show the natural frequencies and the associated mode shapes. The forced responses display the resonant amplitudes of the system excited by the torques/forces. Experimental measurements from a test engine are used to confirm the theoretical predictions.

An analytical model is developed to describe the longitudinal response of the timing chain and the associated torsional response of all the sprockets and tensioner. A closed form equilibrium analysis reveals that equilibrium tensions are functions of tensioner stiffness, chain preload, steady cam torques and engine speed. The equations of motion are linearized about the equilibrium position to determine natural frequencies, mode shapes of the torsional modes and the forced response due to cam torque harmonics. Experimental measurements of the system natural frequencies and the forced amplitudes are in good agreement with the theoretical predictions.