Modelica.Mechanics.Rotational.Components

Information

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

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)
BearingFriction	Coulomb friction in bearings
Brake	Brake based on Coulomb friction
Clutch	Clutch based on Coulomb friction
OneWayClutch	Series 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
TorqueToAngleAdaptor	Signal adaptor for a Rotational flange with angle, speed, and acceleration as outputs and torque as input (especially useful for FMUs)
AngleToTorqueAdaptor	Signal adaptor for a Rotational flange with torque as output and angle, speed, and optionally acceleration as inputs (especially useful for FMUs)

Modelica.Mechanics.Rotational.Components.Fixed

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

Name	Description
phi0	Fixed offset angle of housing [rad]

Connectors

Name	Description
flange	(right) flange fixed in housing

Modelica.Mechanics.Rotational.Components.Inertia

Information

Rotational component with inertia and two rigidly connected flanges.

Parameters

Name	Description
J	Moment of inertia [kg.m2]
Initialization
phi	Absolute rotation angle of component [rad]
w	Absolute angular velocity of component (= der(phi)) [rad/s]
a	Absolute angular acceleration of component (= der(w)) [rad/s2]
Advanced
stateSelect	Priority to use phi and w as states

Connectors

Name	Description
flange_a	Left flange of shaft
flange_b	Right flange of shaft

Modelica.Mechanics.Rotational.Components.Disc

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.

Parameters

Name	Description
deltaPhi	Fixed rotation of left flange with respect to right flange (= flange_b.phi - flange_a.phi) [rad]

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft

Modelica.Mechanics.Rotational.Components.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.

Parameters

Name	Description
c	Spring constant [N.m/rad]
phi_rel0	Unstretched spring angle [rad]
Initialization
phi_rel	Relative rotation angle (= flange_b.phi - flange_a.phi) [rad]

Connectors

Name	Description
flange_a	Left flange of compliant 1-dim. rotational component
flange_b	Right flange of compliant 1-dim. rotational component

Modelica.Mechanics.Rotational.Components.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.

Parameters

Name	Description
d	Damping constant [N.m.s/rad]
useHeatPort	=true, if heatPort is enabled
Initialization
phi_rel	Relative rotation angle (= flange_b.phi - flange_a.phi) [rad]
w_rel	Relative angular velocity (= der(phi_rel)) [rad/s]
a_rel	Relative angular acceleration (= der(w_rel)) [rad/s2]
Advanced
phi_nominal	Nominal value of phi_rel (used for scaling) [rad]
stateSelect	Priority to use phi_rel and w_rel as states

Connectors

Name	Description
flange_a	Left flange of compliant 1-dim. rotational component
flange_b	Right flange of compliant 1-dim. rotational component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Rotational.Components.SpringDamper

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.

Parameters

Name	Description
c	Spring constant [N.m/rad]
d	Damping constant [N.m.s/rad]
phi_rel0	Unstretched spring angle [rad]
useHeatPort	=true, if heatPort is enabled
Initialization
phi_rel	Relative rotation angle (= flange_b.phi - flange_a.phi) [rad]
w_rel	Relative angular velocity (= der(phi_rel)) [rad/s]
a_rel	Relative angular acceleration (= der(w_rel)) [rad/s2]
Advanced
phi_nominal	Nominal value of phi_rel (used for scaling) [rad]
stateSelect	Priority to use phi_rel and w_rel as states

Connectors

Name	Description
flange_a	Left flange of compliant 1-dim. rotational component
flange_b	Right flange of compliant 1-dim. rotational component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Rotational.Components.ElastoBacklash

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.

Parameters

Name	Description
c	Spring constant (c > 0 required) [N.m/rad]
d	Damping constant [N.m.s/rad]
b	Total backlash [rad]
phi_rel0	Unstretched spring angle [rad]
useHeatPort	=true, if heatPort is enabled
Initialization
phi_rel	Relative rotation angle (= flange_b.phi - flange_a.phi) [rad]
w_rel	Relative angular velocity (= der(phi_rel)) [rad/s]
a_rel	Relative angular acceleration (= der(w_rel)) [rad/s2]
Advanced
phi_nominal	Nominal value of phi_rel (used for scaling) [rad]
stateSelect	Priority to use phi_rel and w_rel as states

Connectors

Name	Description
flange_a	Left flange of compliant 1-dim. rotational component
flange_b	Right flange of compliant 1-dim. rotational component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Rotational.Components.BearingFriction

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:

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 Witt 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., Astroem 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.

Parameters

Name	Description
useSupport	= true, if support flange enabled, otherwise implicitly grounded
tau_pos[:, 2]	[w,tau] Positive sliding friction characteristic (w>=0)
peak	peak*tau_pos[1,2] = Maximum friction torque for w==0
useHeatPort	=true, if heatPort is enabled
Initialization
startForward	true, if w_rel=0 and start of forward sliding
startBackward	true, if w_rel=0 and start of backward sliding
locked	true, if w_rel=0 and not sliding
Advanced
w_small	Relative angular velocity near to zero if jumps due to a reinit(..) of the velocity can occur (set to low value only if such impulses can occur) [rad/s]

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft
support	Support/housing of component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Rotational.Components.Brake

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 mue(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 * mue(w) * fn

Typical values of coefficients of friction:

      dry operation   :  mue = 0.2 .. 0.4
      operating in oil:  mue = 0.05 .. 0.1

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 mue(w), w >= 0, is defined via table mue_pos (first column = w, second column = mue). 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 * mue(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:

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 Witt 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., Astroem 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.

Parameters

Name	Description
useSupport	= true, if support flange enabled, otherwise implicitly grounded
mue_pos[:, 2]	[w,mue] positive sliding friction coefficient (w_rel>=0)
peak	peak*mue_pos[1,2] = maximum value of mue for w_rel==0
cgeo	Geometry constant containing friction distribution assumption
fn_max	Maximum normal force [N]
useHeatPort	=true, if heatPort is enabled
Initialization
startForward	true, if w_rel=0 and start of forward sliding
startBackward	true, if w_rel=0 and start of backward sliding
locked	true, if w_rel=0 and not sliding
Advanced
w_small	Relative angular velocity near to zero if jumps due to a reinit(..) of the velocity can occur (set to low value only if such impulses can occur) [rad/s]

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft
support	Support/housing of component
heatPort	Optional port to which dissipated losses are transported in form of heat
f_normalized	Normalized force signal 0..1 (normal force = fn_max*f_normalized; brake is active if > 0)

Modelica.Mechanics.Rotational.Components.Clutch

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 mue(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 * mue(w_rel) * fn

Typical values of coefficients of friction:

      dry operation   :  mue = 0.2 .. 0.4
      operating in oil:  mue = 0.05 .. 0.1

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 mue(w_rel), w_rel >= 0, is defined via table mue_pos (first column = w_rel, second column = mue). 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 * mue(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:

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 Witt 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., Astroem 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.

Parameters

Name	Description
mue_pos[:, 2]	[w,mue] positive sliding friction coefficient (w_rel>=0)
peak	peak*mue_pos[1,2] = maximum value of mue for w_rel==0
cgeo	Geometry constant containing friction distribution assumption
fn_max	Maximum normal force [N]
useHeatPort	=true, if heatPort is enabled
Initialization
phi_rel	Relative rotation angle (= flange_b.phi - flange_a.phi) [rad]
w_rel	Relative angular velocity (= der(phi_rel)) [rad/s]
a_rel	Relative angular acceleration (= der(w_rel)) [rad/s2]
startForward	true, if w_rel=0 and start of forward sliding
startBackward	true, if w_rel=0 and start of backward sliding
locked	true, if w_rel=0 and not sliding
Advanced
phi_nominal	Nominal value of phi_rel (used for scaling) [rad]
stateSelect	Priority to use phi_rel and w_rel as states
w_small	Relative angular velocity near to zero if jumps due to a reinit(..) of the velocity can occur (set to low value only if such impulses can occur) [rad/s]

Connectors

Name	Description
flange_a	Left flange of compliant 1-dim. rotational component
flange_b	Right flange of compliant 1-dim. rotational component
heatPort	Optional port to which dissipated losses are transported in form of heat
f_normalized	Normalized force signal 0..1 (normal force = fn_max*f_normalized; clutch is engaged if > 0)

Modelica.Mechanics.Rotational.Components.OneWayClutch

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 maybe sliding with respect to each other Parallel connection of ClutchCombi and of FreeWheel. The element is introduced to resolve the ambiguity of the constraint torques of the elements.

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 and not two.

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. Friction in the clutch is modelled in the following way:

When the relative angular velocity is positive, the friction torque is a function of the velocity dependent friction coefficient mue(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 * mue(w_rel) * fn

Typical values of coefficients of friction:

      dry operation   :  mue = 0.2 .. 0.4
      operating in oil:  mue = 0.05 .. 0.1

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 mue(w_rel), w_rel >= 0, is defined via table mue_pos (first column = w_rel, second column = mue). 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 * mue(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:

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.

Parameters

Name	Description
mue_pos[:, 2]	[w,mue] positive sliding friction coefficient (w_rel>=0)
peak	peak*mue_pos[1,2] = maximum value of mue for w_rel==0
cgeo	Geometry constant containing friction distribution assumption
fn_max	Maximum normal force [N]
useHeatPort	=true, if heatPort is enabled
Initialization
phi_rel	Relative rotation angle (= flange_b.phi - flange_a.phi) [rad]
w_rel	Relative angular velocity (= der(phi_rel)) [rad/s]
a_rel	Relative angular acceleration (= der(w_rel)) [rad/s2]
Advanced
phi_nominal	Nominal value of phi_rel (used for scaling) [rad]
stateSelect	Priority to use phi_rel and w_rel as states
w_small	Relative angular velocity near to zero if jumps due to a reinit(..) of the velocity can occur (set to low value only if such impulses can occur) [rad/s]

Connectors

Name	Description
flange_a	Left flange of compliant 1-dim. rotational component
flange_b	Right flange of compliant 1-dim. rotational component
heatPort	Optional port to which dissipated losses are transported in form of heat
f_normalized	Normalized force signal 0..1 (normal force = fn_max*f_normalized; clutch is engaged if > 0)

Modelica.Mechanics.Rotational.Components.IdealGear

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.

Parameters

Name	Description
useSupport	= true, if support flange enabled, otherwise implicitly grounded
ratio	Transmission ratio (flange_a.phi/flange_b.phi)

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft
support	Support/housing of component

Modelica.Mechanics.Rotational.Components.LossyGear

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.

Parameters

Name	Description
useSupport	= true, if support flange enabled, otherwise implicitly grounded
ratio	Transmission ratio (flange_a.phi/flange_b.phi)
lossTable[:, 5]	Array for mesh efficiencies and bearing friction depending on speed
useHeatPort	=true, if heatPort is enabled

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft
support	Support/housing of component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Rotational.Components.IdealPlanetary

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

Name	Description
ratio	Number of ring_teeth/sun_teeth (e.g., ratio=100/50)

Connectors

Name	Description
sun	Flange of sun shaft
carrier	Flange of carrier shaft
ring	Flange of ring shaft

Modelica.Mechanics.Rotational.Components.Gearbox

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.

Parameters

Name	Description
useSupport	= true, if support flange enabled, otherwise implicitly grounded
ratio	Transmission ratio (flange_a.phi/flange_b.phi)
lossTable[:, 5]	Array for mesh efficiencies and bearing friction depending on speed (see docu of LossyGear)
c	Gear elasticity (spring constant) [N.m/rad]
d	(relative) gear damping [N.m.s/rad]
b	Total backlash [rad]
useHeatPort	=true, if HeatPort is enabled
T	Fixed device temperature if useHeatPort = false [K]
Advanced
stateSelect	Priority to use phi_rel and w_rel as states

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft
support	Support/housing of component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Rotational.Components.IdealGearR2T

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);

Parameters

Name	Description
useSupportR	= true, if rotational support flange enabled, otherwise implicitly grounded
useSupportT	= true, if translational support flange enabled, otherwise implicitly grounded
ratio	Transmission ratio (flange_a.phi/flange_b.s) [rad/m]

Connectors

Name	Description
flangeR	Flange of rotational shaft
flangeT	Flange of translational rod
supportR	Rotational support/housing of component
supportT	Translational support/housing of component

Modelica.Mechanics.Rotational.Components.IdealRollingWheel

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)*wheelRadius = (flangeT.s - internalSupportT.s);

Parameters

Name	Description
useSupportR	= true, if rotational support flange enabled, otherwise implicitly grounded
useSupportT	= true, if translational support flange enabled, otherwise implicitly grounded
radius	Wheel radius [m]

Connectors

Name	Description
flangeR	Flange of rotational shaft
flangeT	Flange of translational rod
supportR	Rotational support/housing of component
supportT	Translational support/housing of component

Modelica.Mechanics.Rotational.Components.InitializeFlange

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.

Parameters

Name	Description
use_phi_start	= true, if initial angle is defined by input phi_start, otherwise not initialized
use_w_start	= true, if initial speed is defined by input w_start, otherwise not initialized
use_a_start	= true, if initial angular acceleration is defined by input a_start, otherwise not initialized
stateSelect	Priority to use flange angle and speed as states

Connectors

Name	Description
phi_start	Initial angle of flange [rad]
w_start	Initial speed of flange [rad/s]
a_start	Initial angular acceleration of flange [rad/s2]
flange	Flange that is initialized

Modelica.Mechanics.Rotational.Components.RelativeStates

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.

Parameters

Name	Description
stateSelect	Priority to use the relative angle and relative speed as states
phi_nominal	Nominal value of the relative angle (used for scaling) [rad]

Connectors

Name	Description
flange_a	Flange of left shaft
flange_b	Flange of right shaft

Modelica.Mechanics.Rotational.Components.TorqueToAngleAdaptor

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 Mockup 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

Name	Description
use_w	= true, enable the output connector w (angular velocity)
use_a	= true, enable the output connector a (angular acceleration)

Connectors

Name	Description
flange
phi	Flange moves with angle phi due to torque tau [rad]
w	Flange moves with speed w due to torque tau [rad/s]
a	Flange moves with angular acceleration a due to torque tau [rad/s2]
tau	Torque to drive the flange [N.m]

Modelica.Mechanics.Rotational.Components.AngleToTorqueAdaptor

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 Mockup 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

Name	Description
use_w	= true, enable the output connector w (angular velocity)
use_a	= true, enable the input connector a (angular acceleration)

Connectors

Name	Description
flange
phi	Angle to drive the flange [rad]
w	Speed to drive the flange (w=der(phi) required) [rad/s]
a	Angular acceleration to drive the flange (a = der(w) required) [rad/s2]
tau	Torque needed to drive the flange according to phi, w, a [N.m]