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Special Properties of Model Elements

The properties dialog windows of the following element types contain special pages for the Steady-State Simulation.

*Spring-Damper-Backslash*of the libraries*Linear Mechanics*and*Rotational Mechanics**Elastic Friction*of the library*Rotational Mechanics**Elastic Coupling*of the library*Power Transmission/Couplings**Disc Clutch*of the library*Power Transmission/Couplings**Gear*of the library*Power Transmission/Transmission Elements**Belt Drive*of the library*Power Transmission/Transmission Element*

Furthermore, the library *Special Signal Blocks* provides element types with behavioral description in the frequency domain which are only fully supported with the Periodic Steady-State Simulation. Some remarks on these types are given in the subsection *LTI Element Types* below.

Figure 1: Special Page for the steady-state simulation in the properties dialog of the element type Spring-Damper-Backslash. On the right-hand side the choices for the friction approach are shown.

## Available Damping Formulations

The Steady State Analysis provides three damping models:

- Viscous Damping (linear, frequency-independent)
- Relative Damping (nonlinear, frequency-dependent)
- Lehr's Damping (nonlinear, frequency-dependent)

The combobox for the Damping offers the following choices:

*"Steady State: Viscous Damping, Transient: Parameter Page"**"Steady State: Relative Damping, Transient: Parameter Page"**"Steady State: Lehr's Damping Factor, Transient: Parameter Page"**"Steady State & Transient: Parameter Page"**"Steady State & Transient: Relative Damping"**"Steady State & Transient: Lehr's Damping Factor"*

For the first three choices the backslash is switched off during the Steady-State Simulation. Exactly the same damping description is used for Steady-State Simulation and the Time Domain Simulation if one of the last three items is selected. Especially for the item *4* all parameters from the Parameter Page of the element can be used. If you select item *5* or *6* all parameters up to the Contact Stiffness are switched off on the Parameter Page.

## Viscous Damping (linear, frequency-independent)

The damping torque of the frequency independent damping is linearly dependent on the speed difference between the connectors of the element:

*b* is the corresponding damping constant. Viscous damping is often modeled in this form.

In the Steady-State Simulation the damping coefficient *b* is employed in the same form as in the Time Domain Simulation. Therefore, in the Frequency Domain all orders are damped equally:

## Relative Damping (nonlinear, frequency-dependent)

The relative damping *ψ* is the ratio of the damping losses per period (area enclosed by the angle-torque-curve) divided by the maximal elastic deformation energy (reversible deformation of the spring)

It is therefore a unitless quantity.

For certain damping phenomena, as e.g. Coulombs damping, the damping torque keeps constant over the excitation frequency if the deviation amplitude does not change. In the case of constant relative damping an equivalent damping coefficient may be calculated for the dominating excitation frequency

*ω* with the help of the spring stiffness *k* for which the damping losses are the same:

This variability of the damping constant with the dominating excitation frequency *ω* is taken into account in the Steady-State Simulation if relative damping is selected. Since this damping ansatz usually reflects some nonlinear physical phenomenon the computed results unavoidably are approximations.

## Lehr's Damping (nonlinear, frequency-dependent)

This quantity is the damping parameter of the normalized differential equation of the damped harmonic oscillator. It is directly related to the relative damping:

For damping phenomena, as e.g. Coulomb's damping, the damping factor keeps constant over excitation frequency. Analogous to the relative damping an equivalent speed dependent damping constant can be defined:

The variability of the damping constant over the dominating excitation frequency is taken into account for the Steady-State Simulation. Since this damping behavior usually models some nonlinear physical phenomenon the calculated results necessarily are approximations.

## Calculation of the Damping Torque and the Damping Force for the Nonlinear Damping

If one chooses Lehr's damping factor *D* for the rotational element I the damping torque is calculated using a smoothened version of Reid's approach

with the Limit Speed Difference . The smaller the better the -term corresponds to the signum function from the original form of Lehr's ansatz but the more difficult becomes the numerical solution of the system of equations. Thus, you should choose small enough to get the desired damping effect but large enough to avoid convergence problems. For angular differences of the Spring-Damper which are close to unbiased harmonic oscillations the spectral component of the damping torque corresponding to the dominating spectral component results to

Lehr's Damping factor *D* and the relative damping *ψ* are related by the equation

If one replaces Lehr's Damping factor *D* in the first two equations by the right-hand side of the third equation, one obtains the damping torque in dependence of *ψ*.

The *Linear Spring-Damper-Backslash* element is designed in an analogous way as the rotatory one. If one selects Lehr's damping factor the damping force is calculated according to the equation

with the limit speed difference (see comments on above). The damping force spectral component corresponding to the dominating spectral component of the displacement difference of the Spring-Damper approximately results to

## Additional Result Quantities

On the page Steady-State Simulation of the Properties Dialog of the *Spring-Damper-Backslash* element there is the Power of the Spectral Components available as a result quantity.

The orders of for the rotatory and the linear *Spring-Damper-Backslash* elements are computed corresponding to the following equations

or

(the superscript star marks the complex conjugated number). The real part of the complex power is the power caused by the damping of the element, the imaginary part is the reactive power oscillating between the Spring and the connected masses. As the sum signal of the spectral power the overall damping power of the element is collected.

During the Time Domain Simulation the instantaneous power

flowing into the *Spring-Damper* (considered as a two-port) is assigned to the result quantity .

## LTI Element Types

The library *Special Signal Blocks* contains element types modeling Linear Time Invariant subsystems (short LTI systems, a well known notion from system theory). This includes:

- LTIOrderFilter multiplies the spectral components by fixed user-defined complex numbers. One important application is to filter out all oscillations in an angle signal and let only pass the signal components corresponding to the uniform circular motion
- LTIMultJ multiplies the spectral components (at positive frequencies) with the imaginary unit . This LTI-system plays a crucial part in the modeling of material damping (see above).
- LTIFilter filters the input signal with a user defined frequency-domain transfer function. The real- and imaginary part of the transfer function are input as curves by the user.

One of the most remarkable properties of the LTI systems is that they can easily be treated in the frequency domain whereas they often do not fit into the scheme of delay-differential-algebraic equations. For that reason the full operation of those LTI systems is only available in the Periodic Steady-State Simulation. During a time domain simulation the input signal is just multiplied by the value of the transfer function for zero frequency (i.e. the amplification of the mean value component).

Important Properties of LTI systems are:

- The complex amplitudes and for the oscillation orders of the input signal
*x*and*y*the output signal , respectively, are related by the equation where*ω*denotes the phase velocity of the fundamental oscillation component with order 1.

The LTIOrderFilter is an exception to this rule. Its transmission factor only depends on the oscillation order*k*and not on the phase velocity. (It generalizes the concept of the Linear Time Invariant system.) - The mean value components and of the input- and output-signal, resp., are related to each other by the equation

whereas stand for real and imaginary part and is the derivative of the transfer function with respect to the phase velocity. - The period component of the output signal results to

from the period component of the input signal. - LTI systems with non-vanishing period component in the input signal may not have integrating behavior since in this case the output signal would not be representable as a linear combination of ansatz functions (time-quadratic signal component).
- The real part of the frequency domain transfer function of the LTI in dependence of the phase velocity
*ω*is symmetric with respect to the origin. The imaginary part is anti-symmetric with respect to the origin. I.e., there yields

where is the conjugated complex of . (This follows from the assumption of real time-domain signals.)

Since therewith the values of the transfer function at negative phase velocities are completely determined by the values of the transfer function at positive phase velocities the user is only required to preset the transfer function for positive phase velocities.