# Types of Parameters

## Global Parameters

These parameters are defined globally in the model and provided with standard values.

Name

Comment

Default value

gravity

Gravitation

9.80665 m/s²

gravity3D

Gravity vector (3D)

{0.0,0.0,gravity} m/s²

pAtm

Atmosphere pressure

1.01325 bar

TAtm

Atmosphere temperature

20 °C

iSim

Counter for simulation run

• increments with each new simulation run
• in frozen, independent result curves the value of this counter is shown in the legend

0

nSim

Name of the simulation as text string: "text"

• if the value is not empty, it is shown in the legend of frozen, independent result curves instead of iSim
• used by the Variant Wizard
• Note: Use the function `String()`, to print current values in the legend of frozen curves:

empty

## Variable Element Parameters

### Parameter Input of the Elements Using Variable Parameters – Parameters are Effective Quantities

In general, when the elements are not specially tailored to variable parameters, the solution algorithm in SimulationX assumes that the parameters used are constant.

Spring elements are one example of this. The force F is calculated as follows in the spring element:

or the torque T for a rotary spring:

.

If we transpose the equation with respect to the stiffness k, we get

,

which is precisely true for constant stiffnesses.

Therefore, if you want to work with variable stiffnesses in the spring element, you have to use the so-called effective stiffness , since it is likewise defined as the quotient of torque (or force) and deformation (j or x):

The effective stiffness thus corresponds with the slope of a straight line passing through the working point and the zero point of the T-j-characteristic curve (cf. ). By all means, keff can be a function here. It does not have to be constant. For example, the following effective stiffness would be obtained for the torque in the coupling with the relation given in

.

If the expression is given as the parameter input for the stiffness of a spring element, the nonlinear relation will be precisely  described. No integration over the parameter is required. Initial values for the simulation start (integration constants do not have to be known).

The following section discusses how to proceed if the stiffness is given as a derivative.

### Integration over Variable Parameters – Parameters are Given as Derivatives

If variable parameters are given as derivatives, an integration over these parameters is required. We will illustrate this again using a nonlinear spring as an example.

In general the spring stiffness k is defined with the following relation:

where k is the derivative of the torque T with respect to the angle j. Thus, the stiffness corresponds with the slope of the tangent at the working point of the T-j-characteristic curve (Stiffness k and effective stiffness keff for nonlinear torque-deformation characteristic curves T(j)).

Figure 1: Stiffness k and effective stiffness keff for nonlinear torque-deformation characteristic curves T(j)

Transposing from the equation above yields the relation for the torque:

.

here is the integration constant.

Thus, the torque is calculated in the model by integration. Corresponding signal blocks are available for this.

If you want to directly transpose the above integration formula, you have to integrate over the differential angle j. section already tells you how this angle is obtained. This is also shown in Model structure of a coupling with nonlinear stiffness – integration of the stiffness over the angle . The integration takes place in the signal block Integral y over x of the Special Signal Blocks library. The torque applied at the output is once more put between the inertias of the coupling halves (J1 and J2) by means of the External Torque (coupling).

Figure 2: Model structure of a coupling with nonlinear stiffness – integration of the stiffness over the angle

In general, however, the integration during the simulation is made over time. You can also put down the integral for the torque for this. The torque is obtained as the integral over time by inserting .

, with

,

where w is the velocity. The simulation model then looks like the one shown in Model structure of a coupling with nonlinear stiffness – integration of the stiffness over time .

Figure 3: Model structure of a coupling with nonlinear stiffness – integration of the
stiffness over time

The Integrator of the Linear Signal Blocks library is required for the integration over time.

Note:

Three modeling possibilities for the given nonlinear relation are compiled in the samples model Mechanics/NonlinearCoupling.ism.

Known:

Solution 1: The torque depends on the difference of the turning angles
=> use of an external torque element (source) with formula.

Solution 2: The effective coupling stiffness is known. It can be entered in a
spring element.

Solution 3: Is the internal torque on the angle phi derived, we get the actual stiffness. In this case, it must be integrated and provide solutions according to Model structure of a coupling with nonlinear stiffness – integration of the stiffness over the angle " or Model structure of a coupling with nonlinear stiffness – integration of the stiffness over time ".

Both the result curves (cf. Identical results of different modeling variants of nonlinear relations in the element ") and the dynamic behavior of all three modeling possibilities are identical. Thus, all of these types of modeling are equivalent in terms of their results. You can choose any model, depending entirely on what you prefer or need to use.

Figure 4: Identical results of different modeling variants of nonlinear relations in the
element

The solutions presented here are exemplary. Of course, other solutions arise using the TypeDesigner or use of the functionality of Modelica.

## Initial Values

If a parameter is an initial value, the parameter control appears slightly different.

It contains a pin. If the pin is loose (like above) then the value is used as a hint for the solver to start the search for consistent initial values with that value. The finally used value might be different.

If the pin is stuck the value is obligatory. The solver will not modify it.

If there is no value given (the input field is empty) the value is interpreted as free and the solver will choose it appropriately. The initial value input field accepts only constant expressions.

## Result variables

Each model element calculates a number of result variables depending on its functionality and complexity. At every output time step all active result variables of all elements of the model are provided.

Figure 5: Control for Result Variables

The protocol attribute of a result variable can be switched on or off by clicking on the corresponding button. The protocol of a result variable can be shown in result windows (cf. section "Result Window") or can be saved to a file. The chosen unit of measurement is used as preset for the display. It has no effect on the simulation.