LossyGear

Gear with mesh efficiency and bearing friction (stuck/rolling possible)

LossyGear

Library

Mechanics/Rotational/Components

Description

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:

\|w_a\|	eta_mf1	eta_mf2	\|tau_bf1\|	\|tau_bf2\|
...	...	...	...	...
...	...	...	...	...

with

\|w_a\|	Absolute value of angular velocity of input shaft flange_a
eta_mf1	Mesh efficiency in case that flange_a is driving
eta_mf2	Mesh efficiency in case that flange_b is driving
\|tau_bf1\|	Absolute resultant bearing friction torque with respect to flange_a in case that flange_a is driving (= \|tau_bf_a*eta_mf1 + tau_bf_b/i\|)
\|tau_bf2\|	Absolute resultant bearing friction torque with respect to flange_a in case that flange_b is driving (= \|tau_bf_a/eta_mf2 + tau_bf_b/i\|)

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.

Parameters

LossyGear_0

Name	Label	Description	Data Type	Valid Values
mo_useSupport	useSupport	= true, if support flange enabled, otherwise implicitly grounded	Number	0 1
mo_ratio	ratio	Transmission ratio (flange_a.phi/flange_b.phi)	Scalar
mo_lossTable	lossTable	Array for mesh efficiencies and bearing friction depending on speed	Matrix of size Mx5
mo_useHeatPort	useHeatPort	=true, if heatPort is enabled	Number	0 1
mo_Unknown	Unknown	Value of mode is not known	Scalar
mo_Free	Free	Element is not active	Scalar
mo_Forward	Forward	w_a > 0 (forward rolling)	Scalar
mo_Stuck	Stuck	w_a = 0 (forward rolling, locked or backward rolling)	Scalar
mo_Backward	Backward	w_a < 0 (backward rolling)	Scalar
mo_unitAngularAcceleration	unitAngularAcceleration		Scalar
mo_unitTorque	unitTorque		Scalar
mo_eta_mf1_0	eta_mf1_0		Scalar
mo_eta_mf2_0	eta_mf2_0		Scalar
mo_tau_bf1_0	tau_bf1_0		Scalar
mo_tau_bf2_0	tau_bf2_0		Scalar
mo_tau_bf_a_0	tau_bf_a_0		Scalar

LossyGear_1

Name	Label	Description	Data Type	Valid Values
mo_lossPower	lossPower	lossPower	Structure
mo_lossPower/fixed	fixed		Cell of scalars	true false
mo_lossPower/start	start		Cell of scalars
mo_phi_a	phi_a	phi_a	Structure
mo_phi_a/fixed	fixed		Cell of scalars	true false
mo_phi_a/start	start		Cell of scalars
mo_phi_b	phi_b	phi_b	Structure
mo_phi_b/fixed	fixed		Cell of scalars	true false
mo_phi_b/start	start		Cell of scalars
mo_sa	sa	sa	Structure
mo_sa/fixed	fixed		Cell of scalars	true false
mo_sa/start	start		Cell of scalars
mo_w_a	w_a	w_a	Structure
mo_w_a/fixed	fixed		Cell of scalars	true false
mo_w_a/start	start		Cell of scalars
mo_a_a	a_a	a_a	Structure
mo_a_a/fixed	fixed		Cell of scalars	true false
mo_a_a/start	start		Cell of scalars
mo_interpolation_result	interpolation_result	interpolation_result	Structure
mo_interpolation_result/fixed	fixed		cell of matrices of size 1x4	true false
mo_interpolation_result/start	start		cell of matrices of size 1x4
mo_eta_mf1	eta_mf1	eta_mf1	Structure
mo_eta_mf1/fixed	fixed		Cell of scalars	true false
mo_eta_mf1/start	start		Cell of scalars
mo_eta_mf2	eta_mf2	eta_mf2	Structure
mo_eta_mf2/fixed	fixed		Cell of scalars	true false
mo_eta_mf2/start	start		Cell of scalars
mo_tau_bf_a	tau_bf_a	tau_bf_a	Structure
mo_tau_bf_a/fixed	fixed		Cell of scalars	true false
mo_tau_bf_a/start	start		Cell of scalars
mo_tau_eta	tau_eta	tau_eta	Structure
mo_tau_eta/fixed	fixed		Cell of scalars	true false
mo_tau_eta/start	start		Cell of scalars
mo_tau_bf1	tau_bf1	tau_bf1	Structure
mo_tau_bf1/fixed	fixed		Cell of scalars	true false
mo_tau_bf1/start	start		Cell of scalars
mo_tau_bf2	tau_bf2	tau_bf2	Structure
mo_tau_bf2/fixed	fixed		Cell of scalars	true false
mo_tau_bf2/start	start		Cell of scalars
mo_quadrant1	quadrant1	quadrant1	Structure
mo_quadrant1/fixed	fixed		Cell of scalars	true false
mo_quadrant1/start	start		Cell of scalars
mo_quadrant2	quadrant2	quadrant2	Structure
mo_quadrant2/fixed	fixed		Cell of scalars	true false
mo_quadrant2/start	start		Cell of scalars
mo_quadrant3	quadrant3	quadrant3	Structure
mo_quadrant3/fixed	fixed		Cell of scalars	true false
mo_quadrant3/start	start		Cell of scalars
mo_quadrant4	quadrant4	quadrant4	Structure
mo_quadrant4/fixed	fixed		Cell of scalars	true false
mo_quadrant4/start	start		Cell of scalars
mo_quadrant1_p	quadrant1_p	quadrant1_p	Structure
mo_quadrant1_p/fixed	fixed		Cell of scalars	true false
mo_quadrant1_p/start	start		Cell of scalars
mo_quadrant2_p	quadrant2_p	quadrant2_p	Structure
mo_quadrant2_p/fixed	fixed		Cell of scalars	true false
mo_quadrant2_p/start	start		Cell of scalars
mo_quadrant3_m	quadrant3_m	quadrant3_m	Structure
mo_quadrant3_m/fixed	fixed		Cell of scalars	true false
mo_quadrant3_m/start	start		Cell of scalars
mo_quadrant4_m	quadrant4_m	quadrant4_m	Structure
mo_quadrant4_m/fixed	fixed		Cell of scalars	true false
mo_quadrant4_m/start	start		Cell of scalars
mo_tauLoss	tauLoss	tauLoss	Structure
mo_tauLoss/fixed	fixed		Cell of scalars	true false
mo_tauLoss/start	start		Cell of scalars
mo_tauLossMax	tauLossMax	tauLossMax	Structure
mo_tauLossMax/fixed	fixed		Cell of scalars	true false
mo_tauLossMax/start	start		Cell of scalars
mo_tauLossMin	tauLossMin	tauLossMin	Structure
mo_tauLossMin/fixed	fixed		Cell of scalars	true false
mo_tauLossMin/start	start		Cell of scalars
mo_tauLossMax_p	tauLossMax_p	tauLossMax_p	Structure
mo_tauLossMax_p/fixed	fixed		Cell of scalars	true false
mo_tauLossMax_p/start	start		Cell of scalars
mo_tauLossMin_m	tauLossMin_m	tauLossMin_m	Structure
mo_tauLossMin_m/fixed	fixed		Cell of scalars	true false
mo_tauLossMin_m/start	start		Cell of scalars
mo_tau_aPos	tau_aPos	tau_aPos	Structure
mo_tau_aPos/fixed	fixed		Cell of scalars	true false
mo_tau_aPos/start	start		Cell of scalars
mo_tau_etaPos	tau_etaPos	tau_etaPos	Structure
mo_tau_etaPos/fixed	fixed		Cell of scalars	true false
mo_tau_etaPos/start	start		Cell of scalars
mo_startForward	startForward	startForward	Structure
mo_startForward/fixed	fixed		Cell of scalars	true false
mo_startForward/start	start		Cell of scalars
mo_startBackward	startBackward	startBackward	Structure
mo_startBackward/fixed	fixed		Cell of scalars	true false
mo_startBackward/start	start		Cell of scalars
mo_locked	locked	locked	Structure
mo_locked/fixed	fixed		Cell of scalars	true false
mo_locked/start	start		Cell of scalars
mo_ideal	ideal	ideal	Structure
mo_ideal/fixed	fixed		Cell of scalars	true false
mo_ideal/start	start		Cell of scalars
mo_mode	mode	mode	Structure
mo_mode/fixed	fixed		Cell of scalars	true false
mo_mode/start	start		Cell of scalars
mo_tau_eta_p	tau_eta_p	tau_eta_p	Structure
mo_tau_eta_p/fixed	fixed		Cell of scalars	true false
mo_tau_eta_p/start	start		Cell of scalars
mo_tau_eta_m	tau_eta_m	tau_eta_m	Structure
mo_tau_eta_m/fixed	fixed		Cell of scalars	true false
mo_tau_eta_m/start	start		Cell of scalars

Ports

Name	Type	Description	IO Type	Number
flange_a	implicit	Flange of left shaft	input	1
flange_b	implicit	Flange of right shaft	output	1
Port 3	implicit	Support/housing of component	input	mo_useSupport
Port 4	implicit	Optional port to which dissipated losses are transported in form of heat	input	mo_useHeatPort

LossyGear

Library

Description

Acknowledgement:

Literature

Parameters

Ports

See Also