Modelica.Mechanics.Translational.Components

Information

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

Package Content

Name	Description
Fixed	Fixed flange
Mass	Sliding mass with inertia
Rod	Rod without inertia
Spring	Linear 1D translational spring
Damper	Linear 1D translational damper
SpringDamper	Linear 1D translational spring and damper in parallel
ElastoGap	1D translational spring damper combination with gap
SupportFriction	Coulomb friction in support
Brake	Brake based on Coulomb friction
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 position, speed and acceleration (usually, this is reference data from a control bus)
MassWithStopAndFriction	Sliding mass with hard stop and Stribeck friction
RelativeStates	Definition of relative state variables

Modelica.Mechanics.Translational.Components.Fixed

Information

The flange of a 1D translational mechanical system fixed at an position s0 in the housing. May be used:

• to connect a compliant element, such as a spring or a damper, between a sliding mass and the housing.
• to fix a rigid element, such as a sliding mass, at a specific position.

Parameters

Name	Description
s0	Fixed offset position of housing [m]

Connectors

Name	Description
flange

Modelica.Mechanics.Translational.Components.Mass

Information

Sliding mass with inertia, without friction and two rigidly connected flanges.

The sliding mass has the length L, the position coordinate s is in the middle. Sign convention: A positive force at flange flange_a moves the sliding mass in the positive direction. A negative force at flange flange_a moves the sliding mass to the negative direction.

Parameters

Name	Description
m	Mass of the sliding mass [kg]
L	Length of component, from left flange to right flange (= flange_b.s - flange_a.s) [m]
Advanced
stateSelect	Priority to use s and v as states

Connectors

Name	Description
flange_a	Left flange of translational component
flange_b	Right flange of translational component

Modelica.Mechanics.Translational.Components.Rod

Information

Rod without inertia and two rigidly connected flanges.

Parameters

Name	Description
L	Length of component, from left flange to right flange (= flange_b.s - flange_a.s) [m]

Connectors

Name	Description
flange_a	Left flange of translational component
flange_b	Right flange of translational component

Modelica.Mechanics.Translational.Components.Spring

Information

A linear 1D translational spring. The component can be connected either between two sliding masses, or between a sliding mass and the housing (model Fixed), to describe a coupling of the sliding mass with the housing via a spring.

Parameters

Name	Description
c	Spring constant [N/m]
s_rel0	Unstretched spring length [m]
Initialization
s_rel	Relative distance (= flange_b.s - flange_a.s) [m]

Connectors

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

Modelica.Mechanics.Translational.Components.Damper

Information

Linear, velocity dependent damper element. It can be either connected between a sliding mass and the housing (model Fixed), or between two sliding masses.

Parameters

Name	Description
d	Damping constant [N.s/m]
useHeatPort	=true, if heatPort is enabled
Initialization
s_rel	Relative distance (= flange_b.s - flange_a.s) [m]
v_rel	Relative velocity (= der(s_rel)) [m/s]
Advanced
stateSelect	Priority to use phi_rel and w_rel as states
s_nominal	Nominal value of s_rel (used for scaling) [m]

Connectors

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

Modelica.Mechanics.Translational.Components.SpringDamper

Information

A spring and damper element connected in parallel. The component can be connected either between two sliding masses to describe the elasticity and damping, or between a sliding mass and the housing (model Fixed), to describe a coupling of the sliding mass with the housing via a spring/damper.

Parameters

Name	Description
c	Spring constant [N/m]
d	Damping constant [N.s/m]
s_rel0	Unstretched spring length [m]
useHeatPort	=true, if heatPort is enabled
Initialization
s_rel	Relative distance (= flange_b.s - flange_a.s) [m]
v_rel	Relative velocity (= der(s_rel)) [m/s]
Advanced
stateSelect	Priority to use phi_rel and w_rel as states
s_nominal	Nominal value of s_rel (used for scaling) [m]

Connectors

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

Modelica.Mechanics.Translational.Components.ElastoGap

Information

This component models a spring damper combination that can lift off. It can be connected between a sliding mass and the housing (model Fixed), to describe the contact of a sliding mass with the housing.

As long as s_rel > s_rel0, no force is exerted (s_rel = flange_b.s - flange_a.s). If s_rel ≤ s_rel0, the contact force is basically computed with a linear spring/damper characteristic. With parameter n≥1 (exponent of spring force), a nonlinear spring force can be modeled:

   desiredContactForce = c*|s_rel - s_rel0|^n + d*der(s_rel)

Note, Hertzian contact is described by:

• Contact between two metallic spheres: n=1.5
• Contact between two metallic plates: n=1

The above force law leads to the following difficulties:

1. If the damper force becomes larger as the spring force and with opposite sign, the contact force would be "pulling/sticking" which is unphysical, since during contact only pushing forces can occur.
2. When contact occurs with a non-zero relative speed (which is the usual situation), the damping force has a non-zero value and therefore the contact force changes discontinuously at s_rel = s_rel0. Again, this is not physical because the force 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). Especially, often the following model is used (see, e.g., Lankarani, Nikravesh: Continuous Contact Force Models for Impact Analysis in Multibody Systems, Nonlinear Dynamics 5, pp. 193-207, 1994, pdf-download):

   f = c*s_rel^n + (d*s_rel^n)*der(s_rel)

However, this and other models proposed in literature violate issue (1), i.e., unphysical pulling forces can occur (if d*der(s_rel) becomes large enough). Note, if the force law is of the form "f = f_c + f_d", then a necessary condition is that |f_d| ≤ |f_c|, otherwise (1) and (2) are violated. For this reason, the most simplest approach is used in the ElastoGap model to fix both problems by using this necessary condition in the force law directly. If s_rel0 = 0, the equations are:

    if s_rel ≥ 0 then
       f = 0;    // contact force
    else
       f_c  = -c*|s_rel|^n;          // contact spring force (Hertzian contact force)
       f_d2 = d*der(s_rel);         // linear contact damper force
       f_d  = if f_d2 <  f_c then  f_c else
              if f_d2 > -f_c then -f_c else f_d2;  // bounded damper force
       f    = f_c + f_d;            // contact force
    end if;

Note, since |f_d| ≤ |f_c|, pulling forces cannot occur and the contact force is always continuous, especially around the start of the penetration at s_rel = s_rel0.

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

1. Curve 1 (elastoGap1.f) is the unmodified contact force, i.e., the linear spring/damper characteristic. A pulling/sticking force is present at the end of the contact.
2. Curve 2 (elastoGap2.f) is the contact force, where the force is explicitly set to zero when pulling/sticking occurs. The contact force is discontinuous when contact starts.
3. Curve 3 (elastoGap3.f) is the ElastoGap model of this library. No discontinuity and no pulling/sticking occurs.

Parameters

Name	Description
c	Spring constant [N/m]
d	Damping constant [N.s/m]
s_rel0	Unstretched spring length [m]
n	Exponent of spring force ( f_c = -c*|s_rel-s_rel0|^n )
useHeatPort	=true, if heatPort is enabled
Initialization
s_rel	Relative distance (= flange_b.s - flange_a.s) [m]
v_rel	Relative velocity (= der(s_rel)) [m/s]
Advanced
stateSelect	Priority to use phi_rel and w_rel as states
s_nominal	Nominal value of s_rel (used for scaling) [m]

Connectors

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

Modelica.Mechanics.Translational.Components.SupportFriction

Information

This element describes Coulomb friction in support, i.e., a frictional force acting between a flange and the housing. The positive sliding friction force "f" has to be defined by table "f_pos" as function of the absolute velocity "v". E.g.

       v |   f
      ---+-----
       0 |   0
       1 |   2
       2 |   5
       3 |   8

gives the following table:

   f_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 velocity "v" is not zero, the friction force is a function of v and of a constant normal force. This dependency is defined via table f_pos and can be determined by measurements, e.g., by driving the gear with constant velocity and measuring the needed driving force (= friction force).

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

   maximum_static_friction = peak * sliding_friction(v=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
useHeatPort	=true, if heatPort is enabled
f_pos[:, 2]	[v, f] Positive sliding friction characteristic (v>=0)
peak	peak*f_pos[1,2] = Maximum friction force for v==0
Initialization
startForward	true, if v_rel=0 and start of forward sliding
startBackward	true, if v_rel=0 and start of backward sliding
locked	true, if v_rel=0 and not sliding
Advanced
v_small	Relative velocity near to zero (see model info text) [m/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.Translational.Components.Brake

Information

This component models a brake, i.e., a component where a frictional force 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 velocity "v" is not zero, the friction force is a function of the velocity dependent friction coefficient mue(v) , 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_force = cgeo * mue(v) * fn

Typical values of coefficients of friction:

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

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

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

       frictional_force = 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.

Parameters

Name	Description
useSupport	= true, if support flange enabled, otherwise implicitly grounded
useHeatPort	=true, if heatPort is enabled
mue_pos[:, 2]	[v, f] Positive sliding friction characteristic (v>=0)
peak	peak*mue_pos[1,2] = Maximum friction force for v==0
cgeo	Geometry constant containing friction distribution assumption
fn_max	Maximum normal force [N]
Initialization
startForward	true, if v_rel=0 and start of forward sliding
startBackward	true, if v_rel=0 and start of backward sliding
locked	true, if v_rel=0 and not sliding
Advanced
v_small	Relative velocity near to zero (see model info text) [m/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.Translational.Components.IdealGearR2T

Information

Couples rotational and translational motion, like a toothed wheel with a toothed rack, specifying the ratio of rotational / translational motion.

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.Translational.Components.IdealRollingWheel

Information

Couples rotational and translational motion, like an ideal rolling wheel, specifying the wheel radius.

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.Translational.Components.InitializeFlange

Information

This component is used to optionally initialize the position, speed, and/or acceleration of the flange to which this component is connected. Via parameters use_s_start, use_v_start, use_a_start the corresponding input signals s_start, v_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_s_start = true", then flange.s is initialized with the value of the input signal "s_start" at the start time.

Additionally, it is optionally possible to define the "StateSelect" attribute of the flange position 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_s_start	= true, if initial position is defined by input s_start, otherwise not initialized
use_v_start	= true, if initial speed is defined by input v_start, otherwise not initialized
use_a_start	= true, if initial acceleration is defined by input a_start, otherwise not initialized
stateSelect	Priority to use flange angle and speed as states

Connectors

Name	Description
s_start	Initial position of flange [m]
v_start	Initial speed of flange [m/s]
a_start	Initial angular acceleration of flange [m/s2]
flange	Flange that is initialized

Modelica.Mechanics.Translational.Components.MassWithStopAndFriction

Information

This element describes the Stribeck friction characteristics of a sliding mass, i. e. the frictional force acting between the sliding mass and the support. Included is a hard stop for the position.

The surface is fixed and there is friction between sliding mass and surface. The frictional force f is given for positive velocity v by:

f = F_Coulomb + F_prop * v + F_Stribeck * exp (-fexp * v)

The distance between the left and the right connector is given by parameter L. The position of the center of gravity, coordinate s, is in the middle between the two flanges.

There are hard stops at smax and smin, i. e. if flange_a.s >= smin and flange_b.s <= xmax the sliding mass can move freely.

When the absolute velocity becomes zero, the sliding mass becomes stuck, i.e., the absolute position remains constant. In this phase the friction force is calculated from a force balance due to the requirement that the absolute acceleration shall be zero. The elements begin to slide when the friction force exceeds a threshold value, called the maximum static friction force, computed via:

   maximum_static_friction =  F_Coulomb + F_Stribeck

This requires the states Stop.s and Stop.v . If these states are eliminated during the index reduction the model will not work. To avoid this any inertias should be connected via springs to the Stop element, other sliding masses, dampers or hydraulic chambers must be avoided.

For more details of the used friction model see the following reference:

Beater P. (1999):

Entwurf hydraulischer Maschinen. Springer Verlag Berlin Heidelberg New York.

The friction model is implemented in a "clean" way by state events and leads to continuous/discrete systems of equations 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.

Optional heatPort

The dissipated energy is transported in form of heat to the optional heatPort connector that can be enabled via parameter "useHeatPort". Independently whether the heatPort is or is not enabled, the dissipated power is defined with variable "lossPower". If contact occurs at the hard stops, the lossPower is not correctly modelled at this time instant, because the hard stop would introduce a Dirac impulse in the lossPower due to the discontinuously changing kinetic energy of the mass (lossPower is the derivative of the kinetic energy at the time instant of the impact).

Parameters

Name	Description
smax	Right stop for (right end of) sliding mass [m]
smin	Left stop for (left end of) sliding mass [m]
L	Length of component, from left flange to right flange (= flange_b.s - flange_a.s) [m]
m	Mass [kg]
F_prop	Velocity dependent friction [N.s/m]
F_Coulomb	Constant friction: Coulomb force [N]
F_Stribeck	Stribeck effect [N]
fexp	Exponential decay [s/m]
useHeatPort	=true, if heatPort is enabled
Initialization
startForward	= true, if v_rel=0 and start of forward sliding or v_rel > v_small
startBackward	= true, if v_rel=0 and start of backward sliding or v_rel < -v_small
locked	true, if v_rel=0 and not sliding
s	Absolute position of center of component (s = flange_a.s + L/2 = flange_b.s - L/2) [m]
Advanced
v_small	Relative velocity near to zero (see model info text) [m/s]

Connectors

Name	Description
flange_a	Left flange of translational component
flange_b	Right flange of translational component
heatPort	Optional port to which dissipated losses are transported in form of heat

Modelica.Mechanics.Translational.Components.RelativeStates

Information

Usually, the absolute position and the absolute velocity of Modelica.Mechanics.Translational.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 position and the relative velocity between the two connectors are used as state variables.

An example is given in the next figure

Here, the relative position and the relative velocity between the two masses are used as state variables. Additionally, the simulator selects either the absolute position and absolute velocity of model mass1 or of model mass2 as state variables.

Parameters

Name	Description
stateSelect	Priority to use the relative angle and relative speed as states

Connectors

Name	Description
flange_a	(left) driving flange (flange axis directed in to cut plane, e. g. from left to right)
flange_b	(right) driven flange (flange axis directed out of cut plane)