Mechanical systems with transmissions

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1. 1 DOF MECHANICAL SYSTEMS Let us consider a mechanical system composed of an engine, E, a mechanical transmission, T, and motor-driven system, MDS (Figure 1.1). The engine provides a drive torque Md while the angular velocity of the drive (input) shaft is ωi. Conversely, the angular velocity of the driven (output) shaft of the transmission is denoted ωo. The transmission i s characterised by a transmission ratio o iτ = ωω and an efficiency coefficient η. In the end, M r is the resistance tor que exerted by the motor-driven system on the output shaft of the transmission. Figure 1.1 The engine can be composed of some translating, rotating or roto -translating rigid bodies . The angular velocity of the j-th mechanical component , subjected to a rotational motion, is i j ω , while J jd is the corresponding mass moment of inertia evaluated with respect to the barycentre (for the sake of simplicity, let us assume that any component is subjected to a planar motion) . Vector Gik v is the absolute velocity of the centre of gravity, Gik, of the k-th mechanical component while kdm is the corresponding mass. Therefore, the drive part of the system (Figure 1.2) can be mo delled considering the drive torque, M d, provided by the engine, the drive shaft and a “virtual” fly-wheel, mounted on the drive shaft, whose mass moment of inertia is 'Jd . Besides, it is necessary to take into account the equivalen t resistance torque ' 1rM exerted by the remaining system on the drive shaft. For a given time t , and a corresponding angular velocity iω of the drive shaft (or input shaft), the kinetic energy of the fly -wheel must be equal to the global kinetic energy of the drive part of the real system. That is: 12 ' i i i i Gi Gi 11 11 1JJ22 2 jj j kk k nn dd d jk m = = ×= × + × ∑∑ vvωω ω ω…