Abstract
Geared wheels in a planetary arrangement have been used in automotive pro-pulsion systems from the very early days, e. g., as the primary transmission in the T-Ford and as the differential. The first applications were fairly simple, but already exhibited two basi-cally different types of functionality. Three different shafts were interconnected by a planetary arrangement of gears to form a two degree of freedom mechanism. In the first type of func-tionality, used in primary transmissions, one degree of freedom was suppressed by locking one shaft or interlocking two shafts, resulting in one power input and another power output shaft. In a differential the two rotational degrees of freedom are maintained and a unique torque distribution between the three rotating shafts is enabled.
The abovementioned basic types of planetary arrangements of geared wheels have over the years been successfully treated by well established equations, where the set of wheels has been assigned a basic transmission ratio given by the numbers of teeth of the involved wheels, and a - mostly empirical - basic efficiency. These two quantities are the only constants in the linear governing equations for three connected shafts: one for speeds and two for torques.
When automatic transmissions with more than two gears were introduced as automotive pri-mary transmissions, the planetary gear systems became more complex with internally more than three shafts but still with two rotational degrees of freedom. Such larger systems were for long time treated as being composed by the abovementioned basic subsystems with given characteristics and governing equations.
How to decompose a complex planetary gear arrangement into basic planetary gear sets be-came an art. In addition, some more systematic approaches for treating complex planetary gear trains have been proposed, e. g., based upon network theory.
In 1999 the present author published a novel approach for the analysis, which was inspired of the theoretical background to the MultiBody System software. There the concepts of dynam-ics of constrained systems are applied. In a planetary system originally all shafts are assumed to rotate freely. Gear meshes constitute motion constraints, which constitute the constraint Jacobian matrix, facilitating considerably the formulation of torque equilibrium equations. The theoretical background was given in the paper: A General New Approach for Characteri-zation and Analysis of Planetary Gear Trains, VDI Berichte 1460, 1999. Equations for both stationary and transient operating conditions may be systematically formulated in vector-matrix notation and solved by standard software (e. g., Matlab).
To make the novel concept, which was published in English in a German VDI Proceeding, more visible and attractive to apply for a broader audience, it is used in the present paper to analyze in detail the power distribution and losses in a contemporary commercial Aisin-Warner four speed automatic automobile transmission.
Keywords:idling losses, gear mesh losses, geometrical loss interpretation, constraint Jacobian matrix, automatic transmission.