# Constrained Lagrangian Formulation for Modelling and Analysis of Transmission Synchronizers Rapport, 2015

An engineering model of a generic synchronizer is considered. The generic synchronizer is modeled consisting of the selector sleeve with engaging teeth and frictional cone, the synchronizing ring with blocker and the gearwheel with engaging teeth. It is assumed that for the sleeve and synchronizing ring translational and rotational motions are feasible, for the gearwheel only rotational motion is feasible. The motion of the generic synchronizer is performed under the external and internal loads. The external loads comprise the vehicle resistance torque and the control force applied to the sleeve and the drag torque due to the clutch. The internal loads comprise the forces and torques arises due to interaction between the sleeve, the synchronizing ring, the gearwheel as well as oil splash during the synchronization process. A feasible synchronization process of a generic synchronizer is divided into four main phases: pre-synchronization, main synchronization, blocker transition (indexing for the ring), and engagement. Pre-synchronization and blocker transition are further divided into several sub-phases. The generic synchronizer is fully parametrized and together with description of all phases constitutes kinematics of synchronization process in question. Due to the nature of synchronization process in transmission system a motion of a synchronizer mechanism is subjected to unilateral or/and bilateral constraints imposed on generalized coordinates or/and generalized velocities of the synchronizer. Constrained Lagrangian Formalism (CLF) has been proposed as an approach for developing mathematical and computational models of a generic synchronizer dynamics. The CLF gives possibility to formulate direct dynamics, semi-inverse dynamics and inverse dynamics problems for a generic synchronizer in unified and efficient way. It is shown that CLF can be promising and useful for advanced dynamic analysis of the existing synchronizer mechanisms and to design novel optimized ones. In the present report mathematical and computational models of a generic synchronizer with five degrees of freedom have been developed using CLF. Application of CLF on the modelling of the synchronization process gives system of differential-algebraic equations (DAE). Analysis of the generic synchronizer performance diagrams has shown that the CLF based model is able to predict the synchronization process characteristics reasonably well. The developed model has been validated by using measurement data from test rig at CV AB Scania. Performance of the generic synchronizer is also analyzed in case of driveline vibrations. Sensitivity analysis of the synchronizer mechanism performance with respect to cone angle, coefficient of friction, maximum value of shift force and rate of applying shift force has been done. Next, the developed models will be used to get further insight into synchronizer design and its performance diagrams with aid of global sensitivity analysis and Pareto optimization.

Generic Synchronizer

Force Control and Kinematic Control

Constrained Lagrangian Formulation

Dynamik

Dynamik

Dynamik

Teknisk mekanik