# Torsional Vibration Analysis

 Basic Elements Engines Machines

## Description

The Torsional Vibration Analysis model package (short TVA package) is made for steady state analysis of torsional powertrain models. The main focus of the package are marine and stationary drives systems containing combustion engines, transmission elements (as shafts, couplings, gears), and machines (as marine propeller, generator, pump). The package is divided into three sub-packages:

All model types can be developed and edited by the user for individual simulation tasks. The TVA package is based on several types from further packages as Mechanics,Signal Blocks, and Power Transmission.

The elements of this package are optimized for the following modeling principle: The motor / engine / drive component is on the left side of the screen or the model structure, respectively and the load component or machine (propeller, generator) on the right side. This principle is shown in figure 1:

Figure 1: Modeling principle with the TVA package

Figure 1 also defines the main rotation coordinate of the torsional system. The element connectors on the right side (driving side) are named ctr2. The connectors on their left sides (load sides) are named ctr1. The connectors define also the connector coordinate system or coordinate, respectively.

## Reduced / Equivalent Systems

Most elements of the package provide the possibility for the consideration of equivalent / reduced systems. 'Reduced' means, that all inertias, stiffness, dampings, as well as torque, and motion quantities will be transformed to a reference shaft (reference coordinate).

Usually, this reference shaft is the engine crankshaft. A system reduction is only reasonable, if there are any transmission ratios (gears) within the powertrain. If the engine shaft defines the reference coordinate, its speed can be named om1 (also gear input shaft). The transformation constant i21 is the component speed om2 divided by the reference speed om1:
`i21 = om2/om1`
where om2 is the speed of the gear output shaft.

Now, all inertias (J2), stiffnesses (k2), and damping constants (b2) of the gear output shaft can be transformed to the gear input shaft or the main shaft, respectively. The transformed values are also called 'reduced' or 'equivalent'. In the following formulas, they will be indicated by eq. The reduced system computes torques and motion quantities. These values can be transformed back into the real system (torques (T2eq), motion quantities (phi2eq, om2eq)).

• Reduced System:
• `J2eq = J2 * i21^2`
• `k2eq = k2 * i21^2`
• `b2eq = b2 * i21^2`

In this case, the input parameters of the elements are the equivalent/reduced values. The corresponding internal used value is always the real value.
Example: Input is J (element parameter): Then `Jint = J/i21^2` (Jint is the internally used real value).
• Real System:
• `J2eq = J2`
• `k2eq = k2`
• `b2eq = b2`

In this case, the input parameters of the elements are the real values. The corresponding internal used value is always equal to the input value.
Example: Input is J (element parameter): Then `Jint = J` (Jint is the internally used real value).

Following from these definitions, i21 must not be equal to zero. Signs of i21 will be taken into account. Hint: The gear transmission ratio i12 is defined vice versa: `i12 = om1/om2 = 1/i21`
The elements provide always result quantities of the real system, whereas the parameters are entered in reduced form.

## Stress Computation

Some elements are able to compute the torsional stress based on the inner torque and the effective cross section geometry.

The inner torque Ti is in elastic elements the sum of the spring torque Ts and the damping torque Td:
`Ti = Ts + Td`

In rigid elements, Ti is the constraint torque.

The torsional stress tau is
`tau = Ti/Wt`
with Wt, the section modulus of torsion: `Wt = 0.0625*pi*(do^4 - di^4)/do`
taking the effective inner diameter di and the effective outer diameter do into account.

Furthermore, these elements also compute the stress limits tau1 and tau2 depending on the kind of the corresponding shaft and the tensile strength Rm (parameter). For the internal approaches the enumeration kindTL is switched to "Internal Computation, Preset of Parameters" (default setting).
The shaft type can be selected in the basic elements by the enumeration kindShaft.

• Propulsion Shaft
`tau1 = max(((Rm+160)/18*cK*(0.35+0.93*do^(-0.2))*(3-2*speedratio^2)),((Rm+160)/18*cK*(0.35+0.93*do^(-0.2))*1.38))`
`tau2 = 1.7*tau1/cK^0.5`
• Auxiliary Shaft
`tau1 = 0.9*(Rm+160)/18*(0.35+0.93*do^(-0.2))`
`tau2 = 5.4*tau1`
• Crankshafts (combustion engine cranks)
`tau1 = (Rm+160)/18*0.55*(0.35+0.93*do^(-0.2))*max((3-2*(om_mean/omn)^2),1.38)`
`tau2 = 2.3*tau1`

The speed ratio (speedratio) is defined by: `speedratio = abs(om)/abs(omn)`
Furthermore, cK is the form factor, omn is the nominal speed, and om_mean is the mean speed (all parameters of the corresponding elements).

For individual approaches or values for the stress limits, change the enumeration kindTL to "Free Definition".
Then, you can setup tau1_in and tau2_in. It is possible to enter mathematical expressions.

The mean values of tau1 and tau2 will be added automatically as further curves to the SteadyState result window of tau. tau1 and tau2 are protected.

### Mean Values

For the use of mean vales in mathematical expressions use the function LTIOrderFilter().
Example: The mean value of the speed om at ctr2: "LTIOrderFilter(ctr2.om)"

## Spectral Power

All powers, provided as result quantities are computed as spectral powers. More information about the computation of spectral powers can be found in the UserManual section "Analysis".

Hint: Spectral powers are computed within Energy Blocks (names: ebe, ebk, ...). They will be visualized as rectangle white boxes in the diagram view.

## Initialization (Initial Values)

In general, it is required to set (transient) initial values also for Steady State Simulations. In a typical powertrain model, the initial speeds of all inertia elements (parameter om0) could be set to the start frequency and the intial angles (parameter phi0) as zero. Please take transmission ratios into account, if there are gear elements in the powertrain.
To ease this modeling step for models based on this TVA library, all elastic elements (like Spring-Damper) provides the option to set initial constraints (default setting, enumeration kindInit):

• kindInit=="No Deformation and no Velocity Difference": These initial constraints are active:
• No deformation: `ctr1.phi-ctr2.phi=0` (or `dphi=0`, resp.)
• No velocity difference: `ctr1.om-ctr2.om=0` (or `dom=0`, resp.)
• kindInit=="No Preload or Preload from Environment": In this case, please set the initial values in the connected inertia elements or Connections.

"Initial constraint" means, these equations are only valid while initialization computation. The solver computes the initial speeds and angles of the inertia elements of one side (e.g. at connector ctr2) from the initial values of the other side (e.g. connector ctr1). If your powertrain is modeled only by TVA library elements, you must only set initial values in one inertia element (please check the sample models).
A more detailed description of initial values in Steady State Simulations can be found in chapter Influence of Start Values And Transient Simulation on the Periodic Steady State.

## Main Results and Additional Results

The result quantities are grouped into Main Results and Additional Results. The Main Results are always active. The Additional Results can be activated by switching the enumeration results (Show...) to "all Results".

Depending on the setting of the enumeration results, the following action can be proceeded: Select a model element in the diagram view, right click on it and choose "All Protocols (Selection) on" in the upcoming context menu. Now, either only Main Results or all results (Main and Additional Results) will be protocoled.