modelLossyGear

Information

This component models the gear ratio and the losses of a standard gear box in a reliable way including the stuck phases that may occur at zero speed. The gear boxes that can be handled are fixed in the ground or on a moving support, have one input and one output shaft, and are essentially described by the equations:

             flange_a.phi  = i*flange_b.phi;
-(flange_b.tau - tau_bf_b) = i*eta_mf*(flange_a.tau - tau_bf_a);

// or        -flange_b.tau = i*eta_mf*(flange_a.tau - tau_bf_a - tau_bf_b/(i*eta_mf));

where

• i is the constant gear ratio,
• eta_mf = eta_mf(w_a) is the mesh efficiency due to the friction between the teeth of the gear wheels,
• tau_bf_a = tau_bf_a(w_a) is the bearing friction torque on the flange_a side,
• tau_bf_b = tau_bf_b(w_a) is the bearing friction torque on the flange_b side, and
• w_a = der(flange_a.phi) is the speed of flange_a

The loss terms "eta_mf", "tau_bf_a" and "tau_bf_b" are functions of the absolute value of the input shaft speed w_a and of the energy flow direction. They are defined by parameter lossTable[:,5] where the columns of this table have the following meaning:

with

With these variables, the mesh efficiency and the bearing friction are formally defined as:

if (flange_a.tau - tau_bf_a)*w_a > 0 or
   (flange_a.tau - tau_bf_a) == 0 and w_a > 0 then
   eta_mf := eta_mf1
   tau_bf := tau_bf1
elseif (flange_a.tau - tau_bf_a)*w_a < 0 or
       (flange_a.tau - tau_bf_a) == 0 and w_a < 0 then
   eta_mf := 1/eta_mf2
   tau_bf := tau_bf2
else // w_a == 0
   eta_mf and tau_bf are computed such that der(w_a) = 0
end if;
-flange_b.tau = i*(eta_mf*flange_a.tau - tau_bf);

Note, that the losses are modeled in a physically meaningful way taking into account that at zero speed the movement may be locked due to the friction in the gear teeth and/or in the bearings. Due to this important property, this component can be used in situations where the combination of the components Modelica.Mechanics.Rotational.IdealGear and Modelica.Mechanics.Rotational.GearEfficiency will fail because, e.g., chattering occurs when using the Modelica.Mechanics.Rotational.GearEfficiency model.

Acknowledgement

• The essential idea to model efficiency in this way is from Christoph Pelchen, ZF Friedrichshafen.
• The article (Pelchen et.al. 2002), see Literature below, and the first implementation of LossyGear (up to version 3.1 of package Modelica) contained a bug leading to a non-converging solution in cases where the driving side is not obvious. This was pointed out by Christian Bertsch and Max Westenkirchner, Bosch, and Christian Bertsch proposed a concrete solution how to fix this bug, see Literature below.

Literature

• Pelchen C., Schweiger C., and Otter M.: "Modeling and Simulating the Efficiency of Gearboxes and of Planetary Gearboxes," in Proceedings of the 2nd International Modelica Conference, Oberpfaffenhofen, Germany, pp. 257-266, The Modelica Association and Institute of Robotics and Mechatronics, Deutsches Zentrum für Luft- und Raumfahrt e. V., March 18-19, 2002.
• Bertsch C. (2009): "Problem with model LossyGear and a proposed solution", Ticket #108, Sept. 11, 2009.

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