Modelica.Mechanics.Rotational.Components

Components for 1D rotational mechanical drive trains

Information

This package contains basic components 1D mechanical rotational drive trains.

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
Fixed	Flange fixed in housing at a given angle
Inertia	1D-rotational component with inertia
Disc	1-dim. rotational rigid component without inertia, where right flange is rotated by a fixed angle with respect to left flange
Spring	Linear 1D rotational spring
Damper	Linear 1D rotational damper
SpringDamper	Linear 1D rotational spring and damper in parallel
ElastoBacklash	Backlash connected in series to linear spring and damper (backlash is modeled with elasticity)
ElastoBacklash2	Backlash connected in series to linear spring and damper (backlash is modeled with elasticity; at start of contact the flange torque can jump, contrary to the ElastoBacklash model)
BearingFriction	Coulomb friction in bearings
Brake	Brake based on Coulomb friction
Clutch	Clutch based on Coulomb friction
OneWayClutch	Parallel connection of freewheel and clutch
IdealGear	Ideal gear without inertia
LossyGear	Gear with mesh efficiency and bearing friction (stuck/rolling possible)
IdealPlanetary	Ideal planetary gear box
Gearbox	Realistic model of a gearbox (based on LossyGear)
IdealGearR2T	Gearbox transforming rotational into translational motion
IdealRollingWheel	Simple 1-dim. model of an ideal rolling wheel without inertia
InitializeFlange	Initializes a flange with pre-defined angle, speed and angular acceleration (usually, this is reference data from a control bus)
RelativeStates	Definition of relative state variables
AngleToTorqueAdaptor	Signal adaptor for a Rotational flange with torque as output and angle, speed, and optionally acceleration as inputs (especially useful for FMUs)
TorqueToAngleAdaptor	Signal adaptor for a Rotational flange with angle, speed, and acceleration as outputs and torque as input (especially useful for FMUs)
GeneralAngleToTorqueAdaptor	Signal adaptor for a rotational flange with torque as output and angle, speed and acceleration as input (especially useful for FMUs)
GeneralTorqueToAngleAdaptor	Signal adaptor for a rotational flange with angle, speed, and acceleration as outputs and torque as input (especially useful for FMUs)

Modelica.Mechanics.Rotational.Components.Fixed

Flange fixed in housing at a given angle

Information

The flange of a 1D rotational mechanical system is fixed at an angle phi0 in the housing. May be used:

• to connect a compliant element, such as a spring or a damper, between an inertia or gearbox component and the housing.
• to fix a rigid element, such as an inertia, with a specific angle to the housing.

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Inertia

1D-rotational component with inertia

Information

Rotational component with inertia and two rigidly connected flanges.

Extends from Rotational.Interfaces.PartialTwoFlanges (Partial model for a component with two rotational 1-dim. shaft flanges).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Disc

1-dim. rotational rigid component without inertia, where right flange is rotated by a fixed angle with respect to left flange

Information

Rotational component with two rigidly connected flanges without inertia. The right flange is rotated by the fixed angle "deltaPhi" with respect to the left flange.

Extends from Rotational.Interfaces.PartialTwoFlanges (Partial model for a component with two rotational 1-dim. shaft flanges).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Spring

Linear 1D rotational spring

Information

A linear 1D rotational spring. The component can be connected either between two inertias/gears to describe the shaft elasticity, or between a inertia/gear and the housing (component Fixed), to describe a coupling of the element with the housing via a spring.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialCompliant (Partial model for the compliant connection of two rotational 1-dim. shaft flanges).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Damper

Linear 1D rotational damper

Information

Linear, velocity dependent damper element. It can be either connected between an inertia or gear and the housing (component Fixed), or between two inertia/gear elements.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelativeStates (Partial model for the compliant connection of two rotational 1-dim. shaft flanges where the relative angle and speed are used as preferred states), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.SpringDamper

Linear 1D rotational spring and damper in parallel

Information

A spring and damper element connected in parallel. The component can be connected either between two inertias/gears to describe the shaft elasticity and damping, or between an inertia/gear and the housing (component Fixed), to describe a coupling of the element with the housing via a spring/damper.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelativeStates (Partial model for the compliant connection of two rotational 1-dim. shaft flanges where the relative angle and speed are used as preferred states), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.ElastoBacklash

Backlash connected in series to linear spring and damper (backlash is modeled with elasticity)

Information

This element consists of a backlash element connected in series to a spring and damper element which are connected in parallel. The spring constant shall be non-zero, otherwise the component cannot be used.

In combination with components IdealGear, the ElastoBacklash model can be used to model a gear box with backlash, elasticity and damping.

During initialization, the backlash characteristic is replaced by a continuous approximation in the backlash region, in order to reduce problems during initialization, especially for inverse models.

If the backlash b is smaller as 1e-10 rad (especially, if b=0), then the backlash is ignored and the component reduces to a spring/damper element in parallel.

In the backlash region (-b/2 ≤ flange_b.phi - flange_a.phi - phi_rel0 ≤ b/2) no torque is exerted (flange_b.tau = 0). Outside of this region, contact is present and the contact torque is basically computed with a linear spring/damper characteristic:

desiredContactTorque = c*phi_contact + d*der(phi_contact)

         phi_contact = phi_rel - phi_rel0 - b/2 if phi_rel - phi_rel0 >  b/2
                     = phi_rel - phi_rel0 + b/2 if phi_rel - phi_rel0 < -b/2

         phi_rel     = flange_b.phi - flange_a.phi;

This torque characteristic leads to the following difficulties:

1. If the damper torque becomes larger as the spring torque and with opposite sign, the contact torque would be "pulling/sticking" which is unphysical, since during contact only pushing torques can occur.
2. When contact occurs with a non-zero relative speed (which is the usual situation), the damping torque has a non-zero value and therefore the contact torque changes discontinuously at phi_rel = phi_rel0. Again, this is not physical because the torque can only change continuously. (Note, this component is not an idealized model where a steep characteristic is approximated by a discontinuity, but it shall model the steep characteristic.)

In the literature there are several proposals to fix problem (2). However, there seems to be no proposal to avoid sticking. For this reason, the most simple approach is used in the ElastoBacklash model, to fix both problems by slight changes to the linear spring/damper characteristic:

// Torque characteristic when phi_rel > phi_rel0
if phi_rel - phi_rel0 < b/2 then
   tau_c = 0;          // spring torque
   tau_d = 0;          // damper torque
   flange_b.tau = 0;
else
   tau_c = c*(phi_rel - phi_rel0);    // spring torque
   tau_d = d*der(phi_rel);            // damper torque
   flange_b.tau = if tau_c + tau_d ≤ 0 then 0 else tau_c + min( tau_c, tau_d );
end if;

Note, when sticking would occur (tau_c + tau_d ≤ 0), then the contact torque is explicitly set to zero. The "min(tau_c, tau_d)" part in the if-expression, limits the damping torque when it is pushing. This means that at the start of the contact (phi_rel - phi_rel0 = b/2), the damping torque is zero and is continuous. The effect of both modifications is that the absolute value of the damping torque is always limited by the absolute value of the spring torque: |tau_d| ≤ |tau_c|.

In the next figure, a typical simulation with the ElastoBacklash model is shown (Examples.Backlash) where the different effects are visualized:

1. Curve 1 (elastoBacklash1.tau) is the unmodified contact torque, i.e., the linear spring/damper characteristic. A pulling/sticking torque is present at the end of the contact.
2. Curve 2 (elastoBacklash2.tau) is the contact torque, where the torque is explicitly set to zero when pulling/sticking occurs. The contact torque is discontinuous at begin of contact.
3. Curve 3 (elastoBacklash3.tau) is the ElastoBacklash model of this library. No discontinuity and no pulling/sticking occurs.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelativeStates (Partial model for the compliant connection of two rotational 1-dim. shaft flanges where the relative angle and speed are used as preferred states), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.ElastoBacklash2

Backlash connected in series to linear spring and damper (backlash is modeled with elasticity; at start of contact the flange torque can jump, contrary to the ElastoBacklash model)

Information

This element consists of a backlash element connected in series to a spring and damper element which are connected in parallel. The spring constant shall be non-zero, otherwise the component cannot be used.

In combination with components IdealGear, the ElastoBacklash2 model can be used to model a gear box with backlash, elasticity and damping.

During initialization, the backlash characteristic is replaced by a continuous approximation in the backlash region, in order to reduce problems during initialization, especially for inverse models.

If the backlash b is smaller as 1e-10 rad (especially, if b=0), then the backlash is ignored and the component reduces to a spring/damper element in parallel.

In the backlash region (-b/2 ≤ flange_b.phi - flange_a.phi - phi_rel0 ≤ b/2) no torque is exerted (flange_b.tau = 0). Outside of this region, contact is present and the contact torque is basically computed with a linear spring/damper characteristic:

desiredContactTorque = c*phi_contact + d*der(phi_contact)

         phi_contact = phi_rel - phi_rel0 - b/2 if phi_rel - phi_rel0 >  b/2
                     = phi_rel - phi_rel0 + b/2 if phi_rel - phi_rel0 < -b/2

         phi_rel     = flange_b.phi - flange_a.phi;

This torque characteristic leads to the following difficulty:

• If the damper torque becomes larger as the spring torque and with opposite sign, the contact torque would be "pulling/sticking" which is unphysical, since during contact only pushing torques can occur.

In the literature this issue seems to be not discussed. For this reason, the most simple approach is used in the ElastoBacklash2 model, by slightly changing the linear spring/damper characteristic to:

// Torque characteristic when phi_rel > phi_rel0
if phi_rel - phi_rel0 < b/2 then
   tau_c = 0;          // spring torque
   tau_d = 0;          // damper torque
   flange_b.tau = 0;
else
   tau_c = c*(phi_rel - phi_rel0);    // spring torque
   tau_d = d*der(phi_rel);            // damper torque
   flange_b.tau = if tau_c + tau_d ≤ 0 then 0 else tau_c + tau_d;
end if;

Note, when sticking would occur (tau_c + tau_d ≤ 0), then the contact torque is explicitly set to zero.

This model of backlash is slightly different to the ElastoBacklash component:

• An event occurs when contact occurs or when contact is released (contrary to the ElastoBacklash component).
• When contact occurs, the torque changes discontinuously, due to the damping. The damping is larger as for the ElastoBacklash component (for the same damping coefficient), because the ElastoBacklash component has a heuristic to avoid the discontinuity of the torque when contact occurs.
• For some models, the ElastoBacklash2 component leads to faster simulations (as compared when using the ElastBacklash component).

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelativeStates (Partial model for the compliant connection of two rotational 1-dim. shaft flanges where the relative angle and speed are used as preferred states), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.BearingFriction

Coulomb friction in bearings

Information

This element describes Coulomb friction in bearings, i.e., a frictional torque acting between a flange and the housing. The positive sliding friction torque "tau" has to be defined by table "tau_pos" as function of the absolute angular velocity "w". E.g.

 w | tau
---+-----
 0 |   0
 1 |   2
 2 |   5
 3 |   8

gives the following table:

tau_pos = [0, 0; 1, 2; 2, 5; 3, 8];

Currently, only linear interpolation in the table is supported. Outside of the table, extrapolation through the last two table entries is used. It is assumed that the negative sliding friction force has the same characteristic with negative values. Friction is modelled in the following way:

When the absolute angular velocity "w" is not zero, the friction torque is a function of w and of a constant normal force. This dependency is defined via table tau_pos and can be determined by measurements, e.g., by driving the gear with constant velocity and measuring the needed motor torque (= friction torque).

When the absolute angular velocity becomes zero, the elements connected by the friction element become stuck, i.e., the absolute angle remains constant. In this phase the friction torque is calculated from a torque balance due to the requirement, that the absolute acceleration shall be zero. The elements begin to slide when the friction torque exceeds a threshold value, called the maximum static friction torque, computed via:

maximum_static_friction = peak * sliding_friction(w=0)  (peak >= 1)

This procedure is implemented in a "clean" way by state events and leads to continuous/discrete systems of equations if friction elements are dynamically coupled which have to be solved by appropriate numerical methods. The method is described in (see also a short sketch in UsersGuide.ModelingOfFriction):

Otter M., Elmqvist H., and Mattsson S.E. (1999):

Hybrid Modeling in Modelica based on the Synchronous Data Flow Principle. CACSD'99, Aug. 22.-26, Hawaii.

More precise friction models take into account the elasticity of the material when the two elements are "stuck", as well as other effects, like hysteresis. This has the advantage that the friction element can be completely described by a differential equation without events. The drawback is that the system becomes stiff (about 10-20 times slower simulation) and that more material constants have to be supplied which requires more sophisticated identification. For more details, see the following references, especially (Armstrong and Canudas de Wit 1996):

Armstrong B. (1991):

Control of Machines with Friction. Kluwer Academic Press, Boston MA.

Armstrong B., and Canudas de Wit C. (1996):

Friction Modeling and Compensation. The Control Handbook, edited by W.S.Levine, CRC Press, pp. 1369-1382.

Canudas de Wit C., Olsson H., Åström K.J., and Lischinsky P. (1995):

A new model for control of systems with friction. IEEE Transactions on Automatic Control, Vol. 40, No. 3, pp. 419-425.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlangesAndSupport2 (Partial model for a component with two rotational 1-dim. shaft flanges and a support used for textual modeling, i.e., for elementary models), Rotational.Interfaces.PartialFriction (Partial model of Coulomb friction elements), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Brake

Brake based on Coulomb friction

Information

This component models a brake, i.e., a component where a frictional torque is acting between the housing and a flange and a controlled normal force presses the flange to the housing in order to increase friction. The normal force fn has to be provided as input signal f_normalized in a normalized form (0 ≤ f_normalized ≤ 1), fn = fn_max*f_normalized, where fn_max has to be provided as parameter. Friction in the brake is modelled in the following way:

When the absolute angular velocity "w" is not zero, the friction torque is a function of the velocity dependent friction coefficient mu(w), of the normal force "fn", and of a geometry constant "cgeo" which takes into account the geometry of the device and the assumptions on the friction distributions:

frictional_torque = cgeo * mu(w) * fn

Typical values of coefficients of friction mu:

• 0.2 … 0.4 for dry operation,
• 0.05 … 0.1 when operating in oil.

When plates are pressed together, where ri is the inner radius, ro is the outer radius and N is the number of friction interfaces, the geometry constant is calculated in the following way under the assumption of a uniform rate of wear at the interfaces:

cgeo = N*(r0 + ri)/2

The positive part of the friction characteristic mu(w), w >= 0, is defined via table mu_pos (first column = w, second column = mu). Currently, only linear interpolation in the table is supported.

When the absolute angular velocity becomes zero, the elements connected by the friction element become stuck, i.e., the absolute angle remains constant. In this phase the friction torque is calculated from a torque balance due to the requirement, that the absolute acceleration shall be zero. The elements begin to slide when the friction torque exceeds a threshold value, called the maximum static friction torque, computed via:

frictional_torque = peak * cgeo * mu(w=0) * fn   (peak >= 1)

This procedure is implemented in a "clean" way by state events and leads to continuous/discrete systems of equations if friction elements are dynamically coupled. The method is described in (see also a short sketch in UsersGuide.ModelingOfFriction):

Otter M., Elmqvist H., and Mattsson S.E. (1999):

Hybrid Modeling in Modelica based on the Synchronous Data Flow Principle. CACSD'99, Aug. 22.-26, Hawaii.

More precise friction models take into account the elasticity of the material when the two elements are "stuck", as well as other effects, like hysteresis. This has the advantage that the friction element can be completely described by a differential equation without events. The drawback is that the system becomes stiff (about 10-20 times slower simulation) and that more material constants have to be supplied which requires more sophisticated identification. For more details, see the following references, especially (Armstrong and Canudas de Wit 1996):

Armstrong B. (1991):

Control of Machines with Friction. Kluwer Academic Press, Boston MA.

Armstrong B., and Canudas de Wit C. (1996):

Friction Modeling and Compensation. The Control Handbook, edited by W.S.Levine, CRC Press, pp. 1369-1382.

Canudas de Wit C., Olsson H., Åström K.J., and Lischinsky P. (1995):

A new model for control of systems with friction. IEEE Transactions on Automatic Control, Vol. 40, No. 3, pp. 419-425.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlangesAndSupport2 (Partial model for a component with two rotational 1-dim. shaft flanges and a support used for textual modeling, i.e., for elementary models), Rotational.Interfaces.PartialFriction (Partial model of Coulomb friction elements), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Clutch

Clutch based on Coulomb friction

Information

This component models a clutch, i.e., a component with two flanges where friction is present between the two flanges and these flanges are pressed together via a normal force. The normal force fn has to be provided as input signal f_normalized in a normalized form (0 ≤ f_normalized ≤ 1), fn = fn_max*f_normalized, where fn_max has to be provided as parameter. Friction in the clutch is modelled in the following way:

When the relative angular velocity is not zero, the friction torque is a function of the velocity dependent friction coefficient mu(w_rel), of the normal force "fn", and of a geometry constant "cgeo" which takes into account the geometry of the device and the assumptions on the friction distributions:

frictional_torque = cgeo * mu(w_rel) * fn

Typical values of coefficients of friction mu:

• 0.2 … 0.4 for dry operation,
• 0.05 … 0.1 when operating in oil.

When plates are pressed together, where ri is the inner radius, ro is the outer radius and N is the number of friction interfaces, the geometry constant is calculated in the following way under the assumption of a uniform rate of wear at the interfaces:

cgeo = N*(r0 + ri)/2

The positive part of the friction characteristic mu(w_rel), w_rel >= 0, is defined via table mu_pos (first column = w_rel, second column = mu). Currently, only linear interpolation in the table is supported.

When the relative angular velocity becomes zero, the elements connected by the friction element become stuck, i.e., the relative angle remains constant. In this phase the friction torque is calculated from a torque balance due to the requirement, that the relative acceleration shall be zero. The elements begin to slide when the friction torque exceeds a threshold value, called the maximum static friction torque, computed via:

frictional_torque = peak * cgeo * mu(w_rel=0) * fn   (peak >= 1)

This procedure is implemented in a "clean" way by state events and leads to continuous/discrete systems of equations if friction elements are dynamically coupled. The method is described in (see also a short sketch in UsersGuide.ModelingOfFriction):

Otter M., Elmqvist H., and Mattsson S.E. (1999):

Hybrid Modeling in Modelica based on the Synchronous Data Flow Principle. CACSD'99, Aug. 22.-26, Hawaii.

More precise friction models take into account the elasticity of the material when the two elements are "stuck", as well as other effects, like hysteresis. This has the advantage that the friction element can be completely described by a differential equation without events. The drawback is that the system becomes stiff (about 10-20 times slower simulation) and that more material constants have to be supplied which requires more sophisticated identification. For more details, see the following references, especially (Armstrong and Canudas de Wit 1996):

Armstrong B. (1991):

Control of Machines with Friction. Kluwer Academic Press, Boston MA.

Armstrong B., and Canudas de Wit C. (1996):

Friction Modeling and Compensation. The Control Handbook, edited by W.S.Levine, CRC Press, pp. 1369-1382.

Canudas de Wit C., Olsson H., Åström K.J., and Lischinsky P. (1995):

A new model for control of systems with friction. IEEE Transactions on Automatic Control, Vol. 40, No. 3, pp. 419-425.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Icons.Clutch (Icon of a clutch), Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelativeStates (Partial model for the compliant connection of two rotational 1-dim. shaft flanges where the relative angle and speed are used as preferred states), Rotational.Interfaces.PartialFriction (Partial model of Coulomb friction elements), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.OneWayClutch

Parallel connection of freewheel and clutch

Information

This component models a one-way clutch, i.e., a component with two flanges where friction is present between the two flanges and these flanges are pressed together via a normal force. These flanges may be sliding with respect to each other.

A one-way-clutch is an element where a clutch is connected in parallel to a free wheel. This special element is provided, because such a parallel connection introduces an ambiguity into the model (the constraint torques are not uniquely defined when both elements are stuck) and this element resolves it by introducing one constraint torque only instead of two constraints.

Note, initial values have to be chosen for the model such that the relative speed of the one-way-clutch ≥ 0. Otherwise, the configuration is physically not possible and an error occurs.

The normal force fn has to be provided as input signal f_normalized in a normalized form (0 ≤ f_normalized ≤ 1), fn = fn_max * f_normalized, where fn_max has to be provided as parameter.

The friction in the clutch is modeled in the following way. When the relative angular velocity is positive, the friction torque is a function of the velocity dependent friction coefficient mu(w_rel), of the normal force fn, and of a geometry constant cgeo which takes into account the geometry of the device and the assumptions on the friction distributions:

frictional_torque = cgeo * mu(w_rel) * fn

Typical values of coefficients of friction mu:

• 0.2 … 0.4 for dry operation,
• 0.05 … 0.1 when operating in oil.

The geometry constant is calculated - under the assumption of a uniform rate of wear at the friction surfaces - in the following way:

cgeo = N*(r0 + ri)/2

where ri is the inner radius, ro is the outer radius and N is the number of friction interfaces,

The positive part of the friction characteristic mu(w_rel), w_rel >= 0, is defined via table mu_pos (first column = w_rel, second column = mu). Currently, only linear interpolation in the table is supported.

When the relative angular velocity w_rel becomes zero, the elements connected by the friction element become stuck, i.e., the relative angle remains constant. In this phase the friction torque is calculated from a torque balance due to the requirement that the relative acceleration shall be zero. The elements begin to slide when the friction torque exceeds a threshold value, called the maximum static friction torque, computed via:

frictional_torque = peak * cgeo * mu(w_rel=0) * fn,   (peak >= 1)

This procedure is implemented in a "clean" way by state events and leads to continuous/discrete systems of equations if friction elements are dynamically coupled. The method is described in (see also a short sketch in UsersGuide.ModelingOfFriction):

Otter M., Elmqvist H., and Mattsson S.E. (1999):

Hybrid Modeling in Modelica based on the Synchronous Data Flow Principle. CACSD'99, Aug. 22.-26, Hawaii.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Modelica.Mechanics.Rotational.Icons.Clutch (Icon of a clutch), Modelica.Mechanics.Rotational.Interfaces.PartialCompliantWithRelativeStates (Partial model for the compliant connection of two rotational 1-dim. shaft flanges where the relative angle and speed are used as preferred states), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.IdealGear

Ideal gear without inertia

Information

This element characterizes any type of gear box which is fixed in the ground and which has one driving shaft and one driven shaft. The gear is ideal, i.e., it does not have inertia, elasticity, damping or backlash. If these effects have to be considered, the gear has to be connected to other elements in an appropriate way.

Extends from Modelica.Mechanics.Rotational.Icons.Gear (Icon of a rotational gear), Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlangesAndSupport2 (Partial model for a component with two rotational 1-dim. shaft flanges and a support used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.LossyGear

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

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.

Extends from Modelica.Mechanics.Rotational.Icons.Gear (Icon of a rotational gear), Modelica.Mechanics.Rotational.Interfaces.PartialElementaryTwoFlangesAndSupport2 (Partial model for a component with two rotational 1-dim. shaft flanges and a support used for textual modeling, i.e., for elementary models), Modelica.Thermal.HeatTransfer.Interfaces.PartialElementaryConditionalHeatPortWithoutT (Partial model to include a conditional HeatPort in order to dissipate losses, used for textual modeling, i.e., for elementary models).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.IdealPlanetary

Ideal planetary gear box

Information

The IdealPlanetary gear box is an ideal gear without inertia, elasticity, damping or backlash consisting of an inner sun wheel, an outer ring wheel and a planet wheel located between sun and ring wheel. The bearing of the planet wheel shaft is fixed in the planet carrier. The component can be connected to other elements at the sun, ring and/or carrier flanges. It is not possible to connect to the planet wheel. If inertia shall not be neglected, the sun, ring and carrier inertias can be easily added by attaching inertias (= model Inertia) to the corresponding connectors. The inertias of the planet wheels are always neglected.

The icon of the planetary gear signals that the sun and carrier flanges are on the left side and the ring flange is on the right side of the gear box. However, this component is generic and is valid independently how the flanges are actually placed (e.g., sun wheel may be placed on the right side instead on the left side in reality).

The ideal planetary gearbox is uniquely defined by the ratio of the number of ring teeth zr with respect to the number of sun teeth zs. For example, if there are 100 ring teeth and 50 sun teeth then ratio = zr/zs = 2. The number of planet teeth zp has to fulfill the following relationship:

zp := (zr - zs) / 2

Therefore, in the above example zp = 25 is required.

According to the overall convention, the positive direction of all vectors, especially the absolute angular velocities and cut-torques in the flanges, are along the axis vector displayed in the icon.

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.Gearbox

Realistic model of a gearbox (based on LossyGear)

Information

This component models the essential effects of a gearbox, in particular

• in component lossyGear
  • gear efficiency due to friction between the teeth
  • bearing friction
• in component elastoBacklash
  • gear elasticity
  • damping
  • backlash

The inertia of the gear wheels is not modeled. If necessary, inertia has to be taken into account by connecting components of model Inertia to the left and/or the right flange of component Gearbox.

Extends from Modelica.Mechanics.Rotational.Icons.Gearbox (Icon of a gear box), Modelica.Mechanics.Rotational.Interfaces.PartialTwoFlangesAndSupport (Partial model for a component with two rotational 1-dim. shaft flanges and a support used for graphical modeling, i.e., the model is build up by drag-and-drop from elementary components), Modelica.Thermal.HeatTransfer.Interfaces.PartialConditionalHeatPort (Partial model to include a conditional HeatPort in order to dissipate losses, used for graphical modeling, i.e., for building models by drag-and-drop).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.IdealGearR2T

Gearbox transforming rotational into translational motion

Information

This is an ideal mass- and inertialess gearbox which transforms a 1D-rotational into a 1D-translational motion. If elasticity, damping or backlash has to be considered, this ideal gearbox has to be connected with corresponding elements. This component defines the kinematic constraint:

(flangeR.phi - internalSupportR.phi) = ratio*(flangeT.s - internalSupportT.s);

Extends from Rotational.Interfaces.PartialElementaryRotationalToTranslational (Partial model to transform rotational into translational motion).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.IdealRollingWheel

Simple 1-dim. model of an ideal rolling wheel without inertia

Information

A simple kinematic model of a rolling wheel which has no inertia and no rolling resistance. This component defines the kinematic constraint:

(flangeR.phi - internalSupportR.phi) * radius = (flangeT.s - internalSupportT.s);

Extends from Rotational.Interfaces.PartialElementaryRotationalToTranslational (Partial model to transform rotational into translational motion).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.InitializeFlange

Initializes a flange with pre-defined angle, speed and angular acceleration (usually, this is reference data from a control bus)

Information

This component is used to optionally initialize the angle, speed, and/or angular acceleration of the flange to which this component is connected. Via parameters use_phi_start, use_w_start, use_a_start the corresponding input signals phi_start, w_start, a_start are conditionally activated. If an input is activated, the corresponding flange property is initialized with the input value at start time.

For example, if "use_phi_start = true", then flange.phi is initialized with the value of the input signal "phi_start" at the start time.

Additionally, it is optionally possible to define the "StateSelect" attribute of the flange angle and the flange speed via parameter "stateSelection".

This component is especially useful when the initial values of a flange shall be set according to reference signals of a controller that are provided via a signal bus.

Extends from Modelica.Blocks.Icons.Block (Basic graphical layout of input/output block).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.RelativeStates

Definition of relative state variables

Information

Usually, the absolute angle and the absolute angular velocity of Modelica.Mechanics.Rotational.Components.Inertia models are used as state variables. In some circumstances, relative quantities are better suited, e.g., because it may be easier to supply initial values. In such cases, model RelativeStates allows the definition of state variables in the following way:

• Connect an instance of this model between two flange connectors.
• The relative rotation angle and the relative angular velocity between the two connectors are used as state variables.

An example is given in the next figure

Here, the relative angle and the relative angular velocity between the two inertias are used as state variables. Additionally, the simulator selects either the absolute angle and absolute angular velocity of model inertia1 or of model inertia2 as state variables.

See also the discussion State Selection in the User's Guide of the Rotational library.

Extends from Rotational.Interfaces.PartialTwoFlanges (Partial model for a component with two rotational 1-dim. shaft flanges).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.AngleToTorqueAdaptor

Signal adaptor for a Rotational flange with torque as output and angle, speed, and optionally acceleration as inputs (especially useful for FMUs)

Information

Adaptor between a flange connector and a signal representation of the flange. This component is used to provide a pure signal interface around a Rotational model and export this model in form of an input/output block, especially as FMU (Functional Mock-up Unit). Examples of the usage of this adaptor are provided in Rotational.Examples.GenerationOfFMUs. This adaptor has angle, angular velocity and angular acceleration as input signals and torque as output signal. Note, the input signals must be consistent to each other (w=der(phi), a=der(w)).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.TorqueToAngleAdaptor

Signal adaptor for a Rotational flange with angle, speed, and acceleration as outputs and torque as input (especially useful for FMUs)

Information

Adaptor between a flange connector and a signal representation of the flange. This component is used to provide a pure signal interface around a Rotational model and export this model in form of an input/output block, especially as FMU (Functional Mock-up Unit). Examples of the usage of this adaptor are provided in Rotational.Examples.GenerationOfFMUs. This adaptor has torque as input and angle, angular velocity and angular acceleration as output signals.

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.GeneralAngleToTorqueAdaptor

Signal adaptor for a rotational flange with torque as output and angle, speed and acceleration as input (especially useful for FMUs)

Information

Adaptor between a flange connector and a signal representation of the flange. This component is used to provide a pure signal interface around a Rotational model and export this model in form of an input/output block, especially as FMU (Functional Mock-up Unit). This adaptor has angle, angular velocity and angular acceleration as input signals and torque as output signal.

Note, the input signals must be consistent to each other (w=der(phi), a=der(w)).

Extends from Modelica.Blocks.Interfaces.Adaptors.PotentialToFlowAdaptor (Signal adaptor for a connector with potential, 1st derivative of potential, and 2nd derivative of potential as inputs and flow, 1st derivative of flow, and 2nd derivative of flow as outputs (especially useful for FMUs)).

Parameters

Connectors

Modelica.Mechanics.Rotational.Components.GeneralTorqueToAngleAdaptor

Signal adaptor for a rotational flange with angle, speed, and acceleration as outputs and torque as input (especially useful for FMUs)

Information

Adaptor between a flange connector and a signal representation of the flange. This component is used to provide a pure signal interface around a Rotational model and export this model in form of an input/output block, especially as FMU (Functional Mock-up Unit). This adaptor has torque as input and angle, angular velocity and angular acceleration as output signals.

Extends from Modelica.Blocks.Interfaces.Adaptors.FlowToPotentialAdaptor (Signal adaptor for a connector with flow, 1st derivative of flow, and 2nd derivative of flow as inputs and potential, 1st derivative of potential, and 2nd derivative of potential as outputs (especially useful for FMUs)).

Parameters

Connectors