LossyGear

Description

This component models the gear ratio and the losses ofa standard gear box in a reliable way including the stuck phasesthat may occur at zero speed. The gear boxes that canbe handled are fixed in the ground or on a moving support, have one input and oneoutput 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 theabsolute value of the input shaft speed w_a and of the energyflow direction. They are defined by parameter lossTable[:,5] where the columns of this table have the followingmeaning:

with

With these variables, the mesh efficiency and the bearing frictionare 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_bf1elseif (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_bf2else // w_a == 0   eta_mf and tau_bf are computed such that der(w_a) = 0end if;-flange_b.tau = i*(eta_mf*flange_a.tau - tau_bf);

Note, that the losses are modeled in a physically meaningful way takinginto account that at zero speed the movement may be locked dueto the friction in the gear teeth and/or in the bearings.Due to this important property, this component can be used insituations where the combination of the componentsModelica.Mechanics.Rotational.IdealGear andModelica.Mechanics.Rotational.GearEfficiency will fail because,e.g., chattering occurs when using theModelica.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.:"Modelingand Simulating the Efficiency of Gearboxes and of Planetary Gearboxes," inProceedings 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):"Problemwith model LossyGear and a proposed solution",Ticket #108,Sept. 11, 2009.