- The document discusses bridging the gap between steady-state and transient simulation for torsional vibrations under ice impact. - It introduces modeling methods that allow both transient and steady-state analysis to operate on the same model base using a unified framework based on ordinary differential equations. - It also discusses propeller modeling that incorporates established steady-state and transient methods and is being certified by classification societies for compliance with ice class simulation requirements.

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Bridging the Gap between Steady-State and Transient Simulation for
Torsional Vibrations under Ice Impact
Andreas Abel, Uwe Schreiber, ITI GmbH Dresden/Germany, {abel, schreiber}@itisim.com
Erik Werner, STRABAG Offshore Wind GmbH, Hamburg/Germany, erik.werner@strabag.com
Abstract
The transient simulation of ice impact scenarios by now became an integral part of classification
requirements. This development is forcing OEM and suppliers into simulation technologies which are
significantly different from the classical steady-state analysis commonly applied in the past. This
paper introduces modeling and simulation methods, which permit transient and steady-state analysis
to operate on the same model base. We also present recent developments in propeller modeling which
incorporate the established methods for both worlds and are in the process to be certified by
Germanischer Lloyd for compliance with the classification rules.
1. Introduction
The modeling and simulation of torsional vibration systems has been treating steady-state analysis and
transient simulation independently of each other very often or has been focusing on just one of the
aspects. Nowadays, both sides form essential parts of certification requirements, see e.g. GL (2012),
FSA (2010), IACS (2011). However, their mutual independence significantly increases modeling as
well as model management efforts.
Finding solutions, which allow combining the steady-state and transient simulation approaches into
one tool and permitting to execute both on the same model base, has the potential to streamline the
modeling and simulation process. Following such an approach introduces a number of challenges and
requirements:
• Nonlinear behavior: Generally, the classical steady-state approaches are based on linear
integral transformations such as Laplace or Fourier transform to relate time domain
representations of systems and signals into corresponding frequency domain representations.
Due to the linearity of the transformations only linear systems are easily transferable in such
an approach. In classical steady-state analysis for torsional vibrations the response of
nonlinear system components has been approximated in the frequency domain directly (e.g.
through frequency-dependent stiffness or damping characteristics), resulting in models, which
are not transferable back into the time domain.
• Modeling and result continuity: New methodologies for combining steady-state and transient
simulation have to take into account that established approaches have merge smoothly into
them. In particular it is necessary to preserve established ways of parameterization so that
existing data can be continued to be used for modeling. Also, new methods applied to steady-
state simulation in particular should be capable to reproduce results computed previously
using classical approaches (including nonlinearity modeling).
• Going beyond mechanical modeling: The restriction to the modeling of the mechanical sub-
system is usually acceptable for steady-state computations. But generally, the dynamic
behavior may be significantly affected by other system components too. Thinking e.g. of
transient ice impact analysis it becomes apparent that the dynamic response of the engine
control may considerably alter the overall driveline behavior and vibration response. So it is
desirable to include further modeling domains in a simulation framework.
Within the software SimulationX ITI has implemented a broad range of component libraries dedicated
to torsional vibration analysis (TVA). This includes the propeller modeling which is also discussed in

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detail in this paper, as well as solutions for engines, shafts, couplings and gears, which are tailored to
torsional vibration analysis requirements for ship propulsion systems. All models are implemented in
the Modelica modeling language, which can also be applied by the end user for customized modeling.
Within the software was realized a solution, where steady-state analysis and transient simulation can
be executed on one and the same models, making modeling and simulation significantly more
effective.
Coping with the limitations of the strictly linear relationship between time and frequency domain we
show how a combined application of time and frequency domain methods based on harmonic balance
can help to bring both worlds closer together.
A current driver for creating a closer relationship between transient and steady-state analysis is the
requirement for performing transient simulations for ice class certifications in addition to the classical
frequency domain TVA. The core aspect of considering ice impact in transient as well as potentially
in steady-state simulation is the modeling of the propeller and the propeller load generated in ice
impact situations. The modeling of propellers for transient and steady-state analysis in compliance
with the various standards thus forms the second major topic of this paper.
2. Joint Modeling for Time and Frequency Domain Simulation
2.1.Unified Framework
The simulation and analysis of models in time and frequency domain requires the selection of an
appropriate model description approach. Since torsional vibration analysis very often is based on
describing systems as intermeshed networks of lumped parameter elements, formulations as systems
of ordinary differential equations or differential algebraic equations (ODE or DAE) are an appropriate
way to describe the dynamics of a drive system as well as any other lumped-parameter system. In a
general form such equations look as follows:
)),(),((0 ptxtxf &= (1)
With x denoting the vector of states in the model, p the parameters and t the time. There are a large
number of tools available which are capable to solve such systems in transient analysis. If a model is
linear (such as in classical torsional analysis), Eq.(1) becomes a linear differential matrix equation in
)(tx and )(tx& .
The transition of the signals for a steady-state analysis into frequency domain takes place by using the
correspondences for harmonic signals
( ) )(ˆ)( ωxtx ↔ and ( ) )(ˆ)( ωω xjtx ↔& (2)
Where xˆ is a vector of complex numbers representing amplitude and phase of the respective signal at
frequency ω. For a linear system these correspondences transfer Eq.(1) into a system of algebraic
matrix equations, which can be solved independently for any frequency ω. The reduction to algebraic
equations is the strength of the classical frequency domain methods.
If the equation system is non-linear, there is no possibility to transfer the complete model into
frequency domain in such a straightforward way. But, there is still the possibility to transfer from a
system of differential equations into a system of algebraic equations, which also allow the com-
putation of steady state results. The starting point for this transformation is the assumption of an
existing harmonic steady-state solution, which allows expressing x as a Fourier series

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( )






⋅++= ∑=
N
k
p ktjkxx
T
t
xtx
1
0exp][ˆRe]0[ˆ)( ω (3)
In accordance with Eq.(2) the vectors ][ˆ kx represent amplitudes and phases in ( ) )(ˆ ωx for ω=ω0k.
px describes the periodicity of the signals, i.e. the advance over one fundamental vibration period of
the system (such as the advance of rotation angles by 4π per cycle in a four-stroke combustion
engine). Inserting Eq.(3) in Eq.(1) allows to create an algebraic equation system in ][ˆ kx . The solution
approach for such a system is known as harmonic balance (balancing amplitudes and phases of the
different orders k up to N in order to solve the equation system for ][ˆ kx ).
Without going further into details and referring to Abel and Nähring (2008), we would like to point
out that using harmonic balance it is possible to compute spectral results for steady-state operation
also for nonlinear systems and without a complete transformation of the system into the frequency
domain. This solution approach can use an equation system which is closely related to the original
transient differential equation system as seen in Eq.(1). Such approach has been realized in the
simulation software SimulationX by ITI, originating from a transient simulation tool and growing into
a combined time- and frequency-domain simulation environment in recent years.
2.2.Network Modeling Approach for Torsional Vibration Analysis
Network modeling methods are well established in modeling of physical systems, since they are
suitable for describing lumped-parameter physical systems in different domains (mechanical, fluid,
thermal, electrical, etc.). The modeling approach is based on fundamental balancing laws for across
quantities (such as angle or speed difference) and through quantities (such as torques), which exist in
a similar fashion in the different physical domains.
Fig. 1: Network model of a vessel driveline
In a network model the elements interact in a non-causal way, i.e. there is no prescribed direction of
propagation for particular physical quantities in the overall system model. As a consequence, model

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components based on this approach are freely inter-connectable, reaching a high flexibility in the
modeling process. Also, this allows the creation of universal component libraries for assembling any
type of vibratory system with almost no modeling constraints. For classical TVA applications such
libraries may consist of engine components (crank mechanics, excitation models), driveline parts
(gears, couplings, dampers) as well as specific load models, in particular those with vibratory
characteristics (pumps, propellers – this modeling is introduced in more detail within this paper).
Within SimulationX, the modeling is based on the Modelica modeling language, www.modelica.org,
which provides a flexible and user-expandable modeling environment and permits to place a graphical
modeling frontend on top. Fig.1 shows a modeling example of a ship driveline, which can be used for
transient and steady-state analysis.
2.3.Simulation in Time and Frequency Domain
Models as seen in Fig.1 and formulated in the Modelica language can be simulated in both, time and
frequency domain. The simulation is moved from one mode to the other by toggling a switch and
setting appropriate simulation parameters. In transient simulations these are naturally start and stop
time. In a frequency-domain simulation the analysis range is defined through a start and stop value for
a selected reference quantity, such as rotational speed. Since a network model allows combining
components in arbitrary ways, the solver will not be able to identify automatically a reference point
for which the speed reference should be valid. In SimulationX it is therefore possible select any point
in the system as reference point. This has the additional benefit that results can be generated with
respect to various locations in a system, such as engine, propeller or other elements like pumps or
generators.
Since the harmonic balance methods generally address nonlinear models, they also have to take into
account that different orders are not superimposed independently from each other as it would appear
in a linear system. Instead, different frequency components modulate each other. With increasing
degree of nonlinearity the modulation effects increase. For this reason the number of considered
orders can be specified and an internal amount of additional orders is considered for improved
accuracy.
The modeling in such an approach can use techniques which are common and well established for
transient modeling of torsional vibrations in combustion engines drivelines. Namely the excitation
forces are feedback coupled to the dynamics of the system, such as for example:
• Mass forces: The piston mass excitation will respond to the crankshaft dynamics at any time
instant. The effect of the piston mass will vary depending on the instantaneous crank position.
Consequently, also the effective mass on the torsional system will vary over crank rotation.
When a (nonlinear) network model is created, this relationship is naturally incorporated when
relating rotary model parts (crank) and translatory parts (piston, pressure excitation) through
the crank equation.
• Pressure/torque excitation: In SimulationX, excitation models are given as functions of crank
angle and speed by sensing the respective quantities and computing the excitation from the
respective instantaneous values. This has the consequence that the excitation is responding to
variations of these quantities too, which arise e.g. from the torsional vibrations in the
driveline. In the real system such an effect is present also – it is the spring behavior of the gas
inside the piston.
Both effects are nonlinear. When computing steady-state behavior using harmonic balance, this type
of nonlinear relationships for mass and pressure/torque excitations is preserved in the analysis and is
visible in the results. This is a fundamental difference to classical steady-state analysis, where all
excitations are treated as if they would be externally generated excitation signals. Although the
harmonic balance results can be expected to be closer to the behavior of the real physical system,
these results may be significantly different from the results computed through a classical TVA
approach.

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Fig. 2: Transient and steady-state results from the same simulation model
We addressed this problem by providing dedicated model elements which allow to modify the model
such that it becomes equivalent to a classical steady-state analysis model by filtering signals in the
frequency domain. The linear time-invariant (LTI) filters allow altering spectral properties of their
input signals.
• LTI Order Filter: These filters are capable of filtering particular orders (including mean value)
from their input at the currently analyzed steady-state speed. By default they pass the mean
value and the signal portion growing linearly over one cycle (which are the first two
summands in the Fourier representation in Eq.(3)). If such a filter is applied to a speed or
angle signal derived from the drive train, it only passes the mean-value parts and thus the
excitation signal derived from it will not contain any oscillatory components. This is
equivalent to using an external excitation signal and thus allows matching the excitation to
classical TVA.
There are further useful applications of the LTI Order Filters. One is the handling of absolute
damping. In classical TVA the mean values (operating point) of the system is often not taken
into account in the analysis. Parameters such as absolute damping are examined only
according to their contribution to individual vibration orders. Quite often this leads to
parameter sets, where the absolute damping would create non-realistic load torque if applied
to the mean-value speed. In order to reproduce such results of classical TVA the order filter
can remove mean value components from a speed signal so that only vibration orders are
considered in damping torque computation.
• General LTI Filter: In classical TVA the specification of frequency-dependent parameters
such as stiffness or damping is a common approach and is achieved through prescribing the
parameter as a function of frequency and selectively applying appropriate values to the
different orders of an angle or speed. This methodology has been developed from frequency
domain consideration only and usually has no equivalent model in the time domain. In order
steady-state results
transient results
top: torque over time
bottom: torque(time)
over speed(time)
• simulation with propeller
blade excitation (4 blades)
• analysis of the inner torque
of propellerShaft is
showing a resonance
between 80 and 100 rpm
with 4
th
order

6. 407
to match such modeling in the harmonic balance approach, the general LTI filter can be
applied to a signal, acting directly in frequency domain on the spectrum of the input signals
and providing an easy way to implement frequency-dependent parameters in steady-state
analysis.
Due to usual non-transformability of frequency-dependent parameters to time domain alternative
models have to found for the time domain modeling. This is a particular challenge for consistent
modeling in both worlds and often a task not easy to solve. Orienting the analysis more strongly onto
time domain and network modeling allows exploring methods such as harmonic balance, which will
produce consistent results for transient and steady-state analysis.
3. Propeller Modeling
Having created a simulation framework for combining transient and steady-state analysis the
incorporation of propeller modeling into the software was a natural next step. Ice class requirements
for propeller excitations are described primarily in time domain due to the fact that ice impact is a
very non-stationary process and the resulting critical load scenarios are transient. At the same time the
regular propeller blade excitation and propeller loads are equivalently describable in time and
frequency domain. On the other hand, some propeller damping models do only exist in frequency
domain and respective modeling capabilities have to be created in time domain. The inclusion of
frequency-domain specific models allows keeping results in agreement with the still widespread
classical frequency domain simulation tools.
The presented propeller model computes the driveline loads due to ice impact according to various
classification rules and covers the ice classes:
• E1, E2, E3, E4, GL (2012),
• IC, IB, IA, IA super, FSA (2010) and
• PC1, PC2, PC3, PC4, PC5, PC6, PC7, IACS (2011).
It also permits a free customization of the ice class definitions within the framework of the used
computational background. The dependency on nominal and geometric parameters, propeller and
water inertia and damping is considered.
3.1.Ice Impact
Ice impact creates a pulsing load on the driveline with pulses whenever a propeller blade hits ice. In
order to define a unified framework for simulating such a process, major classification societies
defined a model assumption to be obligatory used in transient ice impact simulation.
This model requires the impact sequence to be modeled as a succession of half sine pulses. The
duration (in terms of angle) of the pulses depends on the amount of ice (small block, large block, two
blocks – termed Case 1 to 3 in the rules), whereas the amplitude of the pulses is defined through a set
of coefficients for the amount of ice (case 1 to 3 above), propeller type (ducted or open), propeller and
hub diameter, ice thickness, propeller pitch and pitch variability, drive type (engine, turbine, electric
motor), as well as propeller speed. For a complete ice impact sequence the pulses for the single blades
are to be superimposed according to the number of blades on the propeller, whereby single pulses may
mutually overlap. Fig.3 shows sample scenarios as they are listed in the various classification
requirements.
The implementation for transient simulation in SimulationX specifies the properties as stated above
through the parameter dialog of the element. This permits to handle the various possible configu-
rations in a straightforward and comprehensible way, Fig.4.

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Fig. 3: Ice impact torque profile according to GL (2012), FSA (2010), IACS (2011)
Fig. 4: Propeller parameterization for transient ice impact simulation
In addition the parameterization provides the possibility the override the standard configuration
options by user-defined torque profile parameters. In the simulation the propeller response will not be
replayed like an external signal defined for a particular reference speed, but the generated torque
profile will dynamically respond to the condition of the simulated driveline by adjusting amplitudes
and angle growth rate to the current rotation speed of the propeller and thus reflect the effect of drive
speed reduction due to the load increase caused by the ice impact.
Fig. 5: Model from the certification test set and tests results
In order to allow a certification of the models a test set has been generated, where each model in the
set reproduces a particular behavioral aspect of the propeller as well as specific parameter
combinations, excluding any dynamic interaction with a driveline model, which potentially modifies
the results such, that a clear verification becomes impossible. This test set and the documented

8. 409
reference results allow the quick verifying of the correct behavior of the models after model
modifications or the appearance of new software releases. Fig.5 shows a test set example model,
generating and displaying torque load results for a particular ice class and varying ice amounts.
When connected to a driveline model the propeller excitation will vary with the dynamic state of the
driveline and the transient response of the overall system will depend on various system parameters
such as the mass-elastic properties, but also for instance the reaction of the engine speed control. Fig.6
shows such a model with a simple mean-value engine model, so that the observed driveline
oscillations are exclusively attributed to the propeller excitation. The propeller excitation itself is
composed of a propeller load model, regular propeller blade excitations (visible through slight torque
and speed fluctuations in stationary operation before the ice impact) and the shown ice impact torque.
As response to the ice impact the engine speed drops and is later re-adjusted by the speed controller.
Fig. 6: Transient response model for a four-bladed propeller
For simulation in steady state the ice impact specification is kept with the only exception that the
torque load is considered as an infinite sequence of ice hits.
3.2.Propeller Load Modeling
For modeling of propeller loads in frequency domain there exists a number of approaches, see e.g.
Ker Wilson (1956). These are composed of descriptions for the mean value load (not affected by
oscillatory components) such as propeller or combinator curves and models for the damping of oscil-
latory components in the propeller vibrations. Typical damping assumptions are classical damping
models of Archer, Schwanecke or Frahm, but also standard damping assumptions such as Lehr’s
damping. The damping models usually depend on the mean values of torque and speed, as well as the
vibration orders.
In frequency-domain modeling and stationary operation the separation between mean value and
vibratory behavior is straightforwardly described and used in computations. In contrary, in time-
domain transient simulation mean values are not clearly defined for non-stationary signals and also
the estimation of mean values from stationary signals requires the observation of the signal over at
least one cycle of an oscillation. For low-frequency portions in non-stationary signals this can mean
that the “mean” value may change transiently in shorter time intervals than the low frequency portions
themselves. In this case it becomes impossible to distinguish between the two aspects.
2-stroke Diesel engine, 7000kW @ 116rpm
FP open propeller, Ice Class E1/IC
engine
crankshaft
flywheel
intermediateShaft
flange
propellerShaft
E1/IC
propeller
engineTorque
setSpeed
controller

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Considering this it becomes questionable whether the classical steady state damping models are
transferable at all into a non-stationary time-domain analysis. This question is not yet clearly
answered. For the modeling of propellers applicable to transient and steady-state analysis in
SimulationX we eventually made the decision to not apply the steady-state damping models to the
time domain. So, only the propeller load curves are commonly used for both analyses and in transient
simulation use short-time filters for mean value estimation. Damping for the propeller models in
transient simulation is described by a viscous damping coefficient, applied to the deviation between
mean value speed and current speed of the propeller. How well such an approach correlates with the
results computed in a steady-state analysis and with the classical propeller damping models is subject
to further research. The same applies to the establishment of guidelines for a consistent parameteri-
zation of transient and steady-state modeling in order to achieve at least similar results.
Fig.7 shows the parameterization of the propeller model for different propeller load and damping
scenarios. In Fig.8 the certification test setup for applying and measuring the propeller damping
according to Schwanecke is displayed. In this analysis the propeller is set to a mean value speed and a
specific first-order oscillation. The chart shows the resulting damping torque.
4. Model Certification
The analysis of non-stationary torsional vibrations in particular under ice impact is a fairly recent
extension of the various class rules. The computational implementation of these rules for software
vendors is a step into new territory and the respective solutions have to be proven to be compliant
with the class rules. At the same time transient simulation is characterized by a multitude of dynamic
interactions between the different elements in a complete model, which might obscure the actual
behavioral aspects of the model properties to be verified.
Fig. 7: Propeller load parameterization

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Fig. 8: Test setup and test result for steady-state propeller damping according to Schwanecke
For this reason Germanischer Lloyd as one of the drivers and certifying agent in the development of
the new ice rules and ITI as provider of a simulation tool for transient and steady-state vibration
analysis have decided to establish a well-defined procedure for:
• Measuring and evaluating individual behavioral aspects of simulation model objects (namely
propeller models) in transient and steady-state simulation
• Defining how the behavior is validated against the class rules
• Establishing a procedure how the model compliance can be checked continuously and in
particular after release changes in models and/or simulation environment
Whereas ITI as software developer is executing the verification sequence and result generation,
Germanischer Lloyd verifies and testifies the compliance with the class rules. Eventually the
compliance will be confirmed by issuing a certification by Germanischer Lloyd that the modeling
approach and simulation results obtained in SimulationX are in accordance with the class rules.
4.1.Certification test report
The main task for the model certification was to find appropriate test scenarios, whose simulation
results can be recomputed manually or by other computation software. By this, the test scenarios are
for testing only one feature (e.g. only mean load or only ice impact load). Every test scenario has been
described in an separate chapter of the certification test report. Fig.9 shows a sample page of this
report for testing the propeller blade excitation with 1st
and 2nd
harmonic:
Fig. 9: Sample page of the certification test report
scenario
parameters
simulation results
(usually reaction torques from
the test environment)
expected results incl. equations
and description for re-
computations
result: test is passed or not passed

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4.2.Automatically testing the certified model for new software releases
The certified simulation results from the certification test report are frozen to the test models. ITI’s in-
house test engine runs all models and compares the current simulation results with the stored
reference results. All newly computed results must accord to the reference results within the limits of
numerical accuracy. Only after this the test has been passed. This procedure becomes part of the
standard SimulationX software tests and only after full compliance a new release will be published. In
addition the permanent testing approach allows an easy re-initiation of the certification process and a
renewal the compliance certificate issued by Germanischer Lloyd if this should become necessary.
5. Conclusions
Bridging the gap between steady state simulation in the frequency domain and transient simulation in
the time domain for non-linear models poses considerable challenges to simulation engineers and tool
providers. This is primarily caused by the linear nature of the model transformations between time
and frequency domain. As a consequence both worlds have been quite strongly separated in the past
when it came to the description of the behavior of non-linear phenomena, which has led to non-
transferable solutions on both sides.
In this paper we have demonstrated, that it is generally possible to implement modeling methods,
which allow executing transient as well as steady-state simulations on the very same model and are
consistently applicable to linear as well as non-linear models. This opens new possibilities in torsional
vibration analysis as well as other fields.
A dedicated propeller model was created in collaboration with the Germanischer Lloyd, which works
in time and frequency domain and computes the driveline loads due to ice impact according to various
classification rules.
It has to be noted nevertheless, that this process is still under way and some of the established
methodology especially in steady-state analysis does not (yet?) fit very well into the presented
framework. Such topics remain subject to further research and maybe open a perspective into
rethinking the way how such kind of analyses should be performed in the future.
References
ABEL, A., NÄHRING, T.(2008), Frequency-domain analysis methods for Modelica models,
6th
Int. Modelica Conf. 2, Bielefeld, pp.383-391
FSA (2010), Finnish-Swedish Administration / Transport Safety Agency, TraFi/31298/03.04.01.00/
2010
GL (2012), Guidelines I – Part1 – Chapter 2 – Section 13 – Machinery for Ships with Ice Classes,
Germanischer Lloyd, Hamburg
IACS (2011), IACS Unified Requirements – Polar Class, UR I3 Req.2011
KER WILSON, W. (1956), Practical Solution of Torsional Vibration Problems, Chapman & Hall