The results of such analyses are primarily required during the transport preparation phase, from quotation to the design and engineering of the stowage plan, cribbing and sea fastenings. However, the same results and models can be of benefit during the transport phase and operation in the field as well.
Commonly used guidelines to carry out motion analyses as part of transport engineering are those by Noble Denton [1] or DNV GL [2]. In addition, the vessel and cargo will have their own specific limitations and restrictions on maximum allowable accelerations and motions.
Traditionally, marine transports are engineered to satisfy design criteria in terms of allowable wave heights. The “allowable wave height” can be calculated as the “allowable response level” divided by the “response level per unit wave height.” It follows that different responses may result in different allowable wave heights, depending on the allowable response level.
It is obvious that the allowable wave height also depends on the other wave parameters, like the wave period, spectrum shape and spreading.
Operational parameters, like the vessel heading and speed, may have a major effect on the response level in a certain sea state, and thus on the allowable wave height. The same applies for the vessel’s voyage plan. This is why weather routing is commonly applied. In general favorable wave headings for roll are unfavorable headings for pitch and the related accelerations. Detailed knowledge about the vessel’s seakeeping behavior makes it possible to do more advanced weather routing, namely by evaluating and optimizing for ship responses in the forecasted weather. Weather can refer to precipitation, fog, etc., but for the purposes of this paper it refers to waves in particular. Wind and current are also considered.
In fact, we are not referring to one allowable wave height, but to many allowable response levels. Each allowable response level implies a related allowable wave height, which again may depend on wave heading, etc. This results in a “minimum allowable wave height.” A balanced design of sea fastenings should therefore be based on a calculation of the expected levels of the relevant responses in the most likely wave environments.
The first part of this paper describes the calculation of typical design values such as the linear and angular motions and accelerations, which are used as input for the cribbing and sea fastening design. It will also be demonstrated how leg bending moments can be calculated in the same way. A general procedure for the calculation of the design values for motions, accelerations or leg bending moments includes:
– A vessel stability analysis to derive the proper mass and stability parameters
– Assessment of the environmental conditions which may be encountered
– A motion response analysis resulting in design motions, accelerations or other responses in critical locations on the vessel and cargo
The analysis sequence, as applied for the purposes of this paper, is shown in Figure 1. The obtained design values may serve as the criteria that should not be exceeded during the transport or operation. The second part of this paper describes how to give onboard operational support using the same methods and results as used in the design value calculation procedure.
Transfer functions
A hydrodynamic analysis in OCTOPUS starts with the calculation of a hydrodynamic database. This database does not depend on the loading condition. The procedure is as follows.
At first, a detailed hydrodynamic database (bhdb-file) is calculated. This extensive hydrodynamic database contains all the relevant hydrodynamic properties of the vessel for a range of drafts, speeds, headings and frequencies. The database contains:
– A definition of the geometry (3D)
– Radiation pressure distributions for the six modes of motion
– Diffraction pressure distributions for all wave headings
The hydrodynamic database can be calculated using any third-party 3D radiation/diffraction program. For ships with forward speed, DNV GL’s 3D-radiation/diffraction program WASIM [3] is used. Since WASIM is a time domain program, it would be necessary to model an autopilot to simulate a course-stable ship in waves and to solve the motion RAOs directly. For our purposes, it is not necessary to model an autopilot because WASIM is only used to solve the radiation and diffraction problem in the time domain. Snapshots of such simulations are shown in Figure 2 and Figure 3. The WASIM results of these particular simulations are transformed to the frequency domain by Fourier techniques. After that the pressure RAOs are converted to the OCTOPUS bhdb-format.
Figure 2 Snapshot of a simulation of a forced heave oscillation to calculate pressure distributions for added mass and damping calculation 
Figure 3 Snapshot of a simulation of a restrained vessel to calculate pressure distributions wave force calculation
The final step is a reduction of the database to a compiled hydrodynamic database (chdb-file) by section-wise integration of the pressures stored on the bhdb-file. This results in longitudinal distributions of added mass, damping and excitation forces, which can successively be used to rapidly evaluate any intermediate draft and trim without losing accuracy. This has been demonstrated by Rathje et al [4].
The actual loading condition
Potential flow models need to be extended with viscous damping effects, otherwise roll motion will be overestimated. A popular method is Ikeda’s roll damping method, which includes the following non-potential damping contributions:
– Frictional roll damping
– Eddy-making roll damping
– Lift roll damping coefficient
– Bilge keel roll damping
Since the viscous roll damping coefficient itself is a function of the roll amplitude and frequency, it results in a roll transfer function, which is nonlinear in the wave height. This implies that the linearized roll transfer function varies per sea state.
Solution of motion equations
To account for the nonlinear viscous damping behavior, the sea state dependent roll RAOs are solved in an iterative manner by applying the principle of stochastic linearization, as shown in Figure 4. The viscous damping is estimated using a start-value for the roll motion. The result is a roll RAO. This RAO is used to calculate the roll angle in a particular sea state. If the roll angle is equal to the assumed roll (which is calculated by solving the equations of motions, including non linear effects of roll damping), convergence has been achieved. As long as convergence has not been reached, a new roll damping is computed using a larger or smaller roll angle, the roll RAO is recalculated and a new roll response in the particular sea state is calculated. This loop is repeated until convergence has been obtained.Since the design sea state is not known prior to the calculation of the roll RAOs, the roll RAOs are initially computed for a range of sea states with different wave periods and wave heights. At a later stage, the short-term response statistics can be calculated by combining the design sea states with the RAOs for the best matching sea states.
Tuning of roll motions
Even when using the advanced approach outlined above, it may be difficult to achieve the required degree of correlation between measured and calculated roll motion or the related transverse accelerations. If model tests or full-scale results are available, a further tuning of roll motion or transverse acceleration can be applied.
This tuning of the roll angles or transverse accelerations can be done by modification of the viscous damping contribution. Since accelerations are the essential input for cribbing and seafastening design, as well as for the calculation of, for example, leg bending moments, tuning to accelerations is often the objective.
A practical way to increase the viscous damping is to increase the height of the bilge keel. This has an immediate effect on the magnitude of the total roll damping in a given sea state. Figure 5 and Figure 6 show the calculated roll motion and transverse acceleration as a function of the bilge keel height. The physical explanation for increasing the bilge keel height to a larger artificial bilge keel is to account for effects, which have not been modeled in a common seakeeping model, but which have a similar effect as a bilge keel. By varying the bilge keel height, roll motion or transverse accelerations can be tuned. Of course, this procedure can only be applied if reference material in the form of measurements is available.
Figure 7 Illustration of the calculation of the leg bending moment in longitudinal direction 
Figure 8 A semisubmersible heavy lift vessel with rig on deck and OCTOPUS monitoring and routing system on the bridge 
Figure 9 Procedure to short-term response statistics
Response combinations
By using the RAOs of the global ship motions, transfer functions of local accelerations or even leg-bending moments can be constructed. Combination of responses at the level of transfer functions guarantees the proper phase relations between the different response components, resulting in accurate and realistic response combinations.
An example of the definition of a leg bending moment in longitudinal direction is given by the following linear combination of responses (see also Figure 7).
Where mi = the mass of a leg element i, hi = the lever of the section under consideration, which is the distance between the section’s CoG and the jack house, and ¨xi= the local acceleration in x-direction, which itself is a linear combination of basic ship responses (rigid body accelerations plus a gravity component due to pitch).
Combination of response on the level of transfer functions takes away the discussion of combination or correlation factors between response components. In general, the response maxima will be slightly lower than those obtained by simply adding up the maxima of the individual responses as is sometimes still done. A lower and thus less conservative result implies fewer lashings or sea fastenings (i.e., cost saving), or, when using the same sea fastening, a higher degree of workability.
Design conditions and values
Short-term response statistics
Short-term response statistics can be calculated by combining the design sea states with the best matching set of RAOs with respect to sea states (see Figure 9). By assuming a Rayleigh probability distribution, the Most Probable Extreme (MPE) is given by:
Where m0 is the spectral moment or variance of the response. The zero-upcrossing period
and t is the reference period of a sea state, in seconds, typically three hours or 10,400 seconds. The significant amplitude of a response is given by:
which implies that
The number of cycles in a three-hour storm depends on the average response period of the response cycle. Depending on the response and seastate, this typically varies between six and 30 seconds (for roll, for example). This would result in:
The indicative optimum voyage
A metocean study starts with making an indicative route plan. The master will decide on the final route plan later. Based on the route plan and the expected date of departure, the wind, current and wave statistics can be obtained. In this example, a route has been calculated using the weather routing package SPOS [8]. SPOS is normally used for daily weather routing advice. Route optimization based on ship motion behavior can also be carried out. A maximum of 10 days of weather forecast is available. For periods further ahead, wave climatology is used. The SPOS wave climatology is based on more than 50 years of visual observations collected by the World Meteorological Organization. The climatology database in SPOS has been organized per month. For each location, a monthly average condition is used. This means a monthly average current, sea and swell (significant wave height, zero-upcrossing period and mean direction).
Figure 10 shows a calculated trans-North-Atlantic route for June based on wave and current climatology: the statistical average wave and current conditions at the time of passage. The route optimization has been carried out without considering motion criteria.
For each grid point and each point in time, the most likely significant height, period and mean direction of sea and swell are known. This information can be completed on the assumption of a JONSWAP spectrum shape with a gamma factor as a function of wave period and height, and a wave spreading function. For sea, a cos2 spreading function has been assumed. In case of swell, a cos8-spreading function was used.
With this information, short-term response statistics can be calculated for all headings and speeds. This provides the basis for calculation of an optimum route, taking into account maximum allowable response levels during the passage. The alternative route is shown in Figure 11. It is shown that in the same weather, a vessel specific optimum route is found much further south than the original route.
The result of this analysis is a route that statistically results in the fastest estimated time of arrival (ETA) and along which all the response criteria are satisfied. Relaxation of the criteria will result in a route that will converge to the Northern route, i.e., the route with fastest ETA if no other criteria are to be satisfied.
Design sea states
The design sea states can be further assessed using the indicative route in combination with a statistical wave database (Global Wave Statistics, ECMWF, those of classification societies, others). When the route plan is combined with wave scatter diagrams, a route-specific equivalent wave scatter diagram can be derived (Figures 12 and 13). From this, different methods are used to derive the extremes likely to be reached or exceeded once every 10 years, on average. A typical wave period range depends on the significant height of the design wave. Noble Denton [1] prescribes:
To calculate response statistics in waves, it is necessary to assume a spectrum shape. Often a Pierson Moskowitz or JONSWAP wave spectrum is used. Wave spreading may be applied.
Depending on the duration of exposure and the availability of a wave forecast, some reduction of the design wave height may be allowed. Some further reduction of wave heights may be considered when directionality and heading control is possible. This is only the case for self-propelled vessels with redundant propulsion systems.
Figure 12 Matching of an indicative route plan with a wave scatter diagram database. The passage time per wave area is accounted for. 
Figure 13 The route-specific wave scatter diagram
Voyage simulations
More elaborate analyses can be carried out when time series of waves are available. Using spectral data instead of derived wave parameters such as significant wave height, mean direction or zero-upcrossing period, ensures more accurate results, especially in multidirectional seas. Moreover, persistency effects are automatically included.
Voyage simulations can be carried out, for example, by using the Argoss w3c-database [6]. This historical worldwide wave database is the product of satellite observations and a third generation wave model. It covers a period of 15 years with a time resolution of three hours. The wave condition at a particular date, time and location is described by a distribution of the energy, the direction and the directionality, as a function of the frequency. The position list of the indicative voyage can be used as input for voyage simulations. These simulations are carried out for different dates and times of departure, and repeated until convergence is obtained after ‘N’ simulations. The design values can be derived by defining a required success rate after simulating ‘N’ voyages. Risk mitigating measures are not accounted for.
An example of a tool in which risk-mitigating measures can be modeled is SafeTrans [10]. The SafeTrans software is an engineering tool to calculate design values for marine transports and installations. For operations design, SafeTrans includes a decision mimic, which allows the user to take into account ship master decisions about postponing tasks or going for shelter given the weather forecast.
Onboard decision support
Onboard evaluation of responses
At some stage, the design values have been established, either by using calculations or simply from following rules or regulations. The seafastening design is finished and the transport is ready for departure. From that moment, it is up to the master to make the final decision regarding the route, speed and heading, taking into account the transport-specific operational limits, the weather forecasts, other operational input and his seamanship.
In this section there is an explanation of how the outlined methods and information can be used by the master in immediate and mid-term decision support for heading control in bad weather and response-based route planning.
Often the design values may be directly taken as the allowable values during the passage. However, for safety or comfort reasons, it may be advantageous to try to avoid the severest allowable conditions. This can be done by using a safety margin, which is achieved by reducing the allowable response level, to, for example, 75 percent of the design value. Note that the safety margins may be different for each response.
The procedure for operational support onboard is very similar to the design procedure and includes:
– Automatic processing of the actual loading condition, obtained from the loading computer or specified manually
– Calculation of hydrodynamic coefficients and wave forces for the actual draft and trim, using the pre-calculated hydrodynamic database
– Specification of the responses of interest (absolute or relative motions, accelerations, leg bending moments, etc)
– Calculation of the RAOs
– Specification of the statistical quantity and the corresponding allowable value for each response of interest
– Calculation of short-term response statistics by using the available wave information (wave radar measurement, observation by the master or weather/wave forecast)
– Evaluation of the response levels or probability of exceed with respect to the allowable values or criteria
– Presentation of the results in a non-academic style that is easily accessible to the mariner
A flow diagram of the procedure is shown in Figure 15.
Display of results
Effective onboard decision support in heavy weather requires that the ship responses have been calculated for all headings and speeds. The results for one particular sea state can be presented as a polar diagram (Figure 16) in which the radius of the diagram indicates the vessel speed. The same polar display can also be used to indicate resonance areas as formulated by the International Maritime Organization (IMO) [11]. After normalization of each response by dividing the calculated response by the allowable level, the condition in a particular sea state can be judged quickly, taking into account all the relevant responses simultaneously (the response envelope). For each speed and heading, the “maximum normalized response” (in terms of percentage of criterion) is evaluated. When below 75 percent, the condition is green; if it is over 100 percent, it will be red. Complex wave conditions like multidirectional confused seas as measured by wave-radar can be evaluated in the same objective manner.
In case of a weather forecast, weather windows can be calculated. An example is shown in Figure 17. The same normalization procedure as described above has been applied.
The successful implementation of these kinds of tools requires the acceptance of the system by the master and that he receives support from the office. Then a system that automatically calculates and updates diagrams like the examples in Figures 16 and 17 can and will be used to identify possible hazards and their consequences. The system will only assist the master to make the best decision for safe and effective ship operation in a particular condition, but ultimately the master takes the final decision himself.
Figure 18 shows a comparison between the measured accelerations (blue line) and the acceleration forecast (green line).
Figure 16 A polar diagram showing combinations of speed and heading that result in high (red) or low (green) responses, plus an indication of resonance areas according to the IMO guidelines [11] 
Figure 17 Display of weather windows after calculation of ship responses using the waves expected during a voyage plan 
Figure 18 Comparison between measured accelerations and the calculated accelerations based on wave forecast (Dockwise heavy transport vessel
Conclusion
A method for the robust and accurate calculation of ship responses in waves has been described. The ways of addressing the design values and how these values can serve as input for an onboard advisory system have been explained. The following conclusions and recommendations can be made:
– Consistency between the ship response calculation methods used during engineering and operation is of importance and has been ensured in the presented approach.
– The concept of “allowable wave height” is difficult to apply since each response has its own “allowable wave height”, which finally results in a “minimum allowable wave height.” Application of the “minimum allowable wave height” as sole operational criterion could result in too conservative sailing behavior.
– Knowledge about the impact of waves on the ship for all headings and speeds, however, allows effective and objective operational decision support with respect to speed, heading and route. This is not the case in the “allowable wave height” concept.
– The tools are available to calculate responses on board with the same accuracy as when using state-of-the-art engineering tools in the office. Implementation of the presented method has resulted in an effective proven operational support tool for heavy transports over sea.
Acknowledgements
The following companies are acknowledged for allowing the presentation of information in this paper: NMA Maritime & Offshore Contractors, Dockwise Shipping B.V. and Offshore Heavy Transport AS.
Notes
- Noble Denton (2005). General Guidelines for Marine Transportations, 0030/NDI REV 2.
- DNV GL (2008). Rules for planning and execution of Marine Operations.
- DNV GL (2005), WASIM User Manual.
- Adegeest, L. J. M., Drakogiannopoulos, J., Rathje, H. (2003). Concept and Implementation of an Innovative Shipboard Routing Assistance System’, RINA Conference Design & Operation of Container Ships.
- Amarcon (2007). Manual: Calculation of an OCTOPUS hydrodynamic database using WASIM. Amarcon, document M20070622.
- Melger, F., Adegeest, L. J. M. (2006). Connecting satellite data to decision support systems. Agency for Aerospace Programmes, Report nr. A430/NIVR 53502AR.
- Amarcon (2008). Manual: OCTOPUS-ONBOARD User Manual. AMARCON document.
- Meteo Consult (2007). SPOS Manual, version 6.
- DNV GL Class Note 30.5. Environmental conditions and Environmental loads.
- Aalbers A. B., Nataraja R., Anink S. (2005,). Design criteria for weather routed transport. RINA Heavy transport and lift conference, London.
- IMO (2007). Revised guidance to the Master for avoiding dangerous situations in adverse weather and sea conditions. MSC.1/Circ.1228.
- Journee, J. M. J. (2001). Verification and Validation of Ship Motions Program SEAWAY. TU Delft Report 1213.
- Adegeest, L. J. M. (2008). Response based weather routing and operation planning of heavy transport vessels. RINA Heavy transport and lift conference, London.
