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Latin American Journal of Solids and Structures
versão On-line ISSN 1679-7825
Lat. Am. j. solids struct. vol.11 no.14 Rio de Janeiro 2014
http://dx.doi.org/10.1590/S1679-78252014001400003
Analysis of the flexural mode response of a novel trimaran by segmented model test
Karim Akbari Vakilabadi^{I}; Mohammad Reza Khedmati^{I, *}; Habiballah Sayyari^{II}
^{I}Department of Marine Technology, Amirkabir University of Technology, Tehran, Iran
^{II}Supreme National Defense University, Tehran, Iran
ABSTRACT
A novel ship concept design is significantly an "adhoc" process. In the preliminary design stage of novel vessels, it is very important to be able to develop an initial estimate of the effects of stiffness and mass distribution on the longitudinal flexural natural frequencies due to different general arrangements in still water at zero speed to satisfy design specifications. For new emerging designs, this estimate has to be made based on a model test. The experiments should also be planned so that scales effects and other features that are not present in full scale case, are minimized. A model with a length of 1.5 meter has been selected. The model was cut into four segments longitudinally and connected by a backbone beam with three elastic hinges joining the four segments. Wet vibration tests were conducted on the model, showed significant influences on the flexural natural frequencies through variations in stiffness and different mass distributions. The whipping frequency was calculated with four degrees of freedom theoretical model to compare with the experimental results. The theoretical model shows a good agreement with the experimental results.
Keywords: Novel Trimaran; segmented model; elastic hinge; natural frequency; model test
1 INTRODUCTION
Marine vehicles with speeds above 30-35 knots which have recently been used primarily for both passenger transport and military applications are limited to high-speed monohulls and Multihulls. High forward speeds, complex structural outlines and the use of novel materials make the analysis of these vessels more challenging than conventional ships. During the last few years, there has been an increasing interest in the application of high speed multihull ships for military applications. When the high speed vessel concepts are up scaled, some new problems arise. The global hydrodynamic loads depend both on the ship's rigid body motions and its elastic deformation. These hydroelastic effects can be of particular importance for large high speed vessels. Springing is one of the challenges in ship-shape vessels. High speed and low rigidity are two features of a ship which generally increase the springing problems. The springing phenomenon may not influence the extreme structural responses, but the fatigue damage could be significantly increased (Friis-Hansen et al., 1995).
Hydroelasticity can be significant not only for steady state responces but also for those of the transient type. The flexibility of the structure in the area of slamming impact can influence the total slamming load (Kvalsvold, 1994), and the overall stiffness of the vessel's structure is important for global (whipping) responses. Ship classification societies have traditionally predicted the structural loads of ships using an empirical approach based on experience with previous vessels in contrast to an experimental analysis of the wave induced loads, Thomas et al. (2003a). Segmented monohull models have been traditionally developed to measure motions and loads at low Froude numbers as the work reported by Gerritsma et al. (1964) and Lloyd et al. (1979). This was followed by the evolution of low Froude number hydroelastic segmented models designed to replicate the whipping response of a full-scale vessel in addition to measuring motions and loads for the validation of numerical simulations, McTaggart et al. (1997). Dessi et al. (2005) have developed an elastically scaled high-speed monohull model used to measure dynamic loads and to replicate the modal response of the full-scale vessel with experimental results showing good correlation with finite element theory for dry tests in air. Kapsenberg et al. (2003) investigated a similar monohull model with an emphasis on slam loads and mathematically simulating whipping responses. Despite the extensive works have been conducted on the prediction and measurement of dry modal frequencies there is a lack of knowledge in validation of the theoretical wet modal frequency with the experimental wet modal frequency.
The model should be as simple as possible in order to provide easy and accurate measurements, and to keep production costs and time within reasonable limits. At the same time it should be sufficiently realistic for final calculations to be of practical values. Similarly, the model should be flexible enough to produce easily measurable hydroelastic responses, but not so flexible that qualitatively unrealistic phenomena occur. As discussed by Maeda (1991), flexible ship models can be constructed by different principles. The seakeeping Committee of 17th ITTC (ITTC 1984) also discusses various aspects of flexible model tests. A logical approach is to make the model continuously elastic (Watanabe et al., 1988). By this approach the stiffness distribution resembles a real ship, but it is difficult to adjust the stiffness once the model was produced. Alternatively, one can divide the ship hull into segments and connect the segments with a beam, or backbone, that has the proper stiffness. The backbone could be continuously elastic (Riska, 1994), or it could consist of stiff parts that are connected by elastic hinges (Troesch, 1984). Wereldsma (1974) has also proposed to use a completely stiff backbone. In the present experiments a novel wave piercer bow trimaran was utilized. The model was originially completely rigid, and the only practical way of making it flexible was to cut it into three segments. A rigid backbone frame with three elastic hinges was selected. According to variety of mass distribution due to different general arrangement, the vessels flexibility could be adjusted by changing the hinge geometry.
This paper presents the results on investigating the hydroelastic whipping response of a segmented wave piercer bow trimaran model with an emphasis on developing a theoretical model to predict the wet modal frequency which is validated with experiments. The effect of model stiffness and an weight distributionon the measured flexural response has been studied. A theoretical model of a four degree of freedom system using an theoretical hydrodynamic added mass was developed to predict the wet modal frequency of the model as a function of stiffness and mass distributions.
2 SELECTION OF MODEL WEIGHT DISTRUIBUTIONS ON A PROTOTYPE VESSEL
Light warships are typical of volume-limited ships and it is these types which are most suited to the wider and more convenient deck spaces above the waterline that the multihull offers. A more complex investigation into the choice of hullform must be based on the required attributes of the ship, such as arrangement, speed and seakeeping. Multihulls provide a good and useful deck layout with a greater deck area per ton of displacement compared to monohulls. Arrangement is a crucial parameter that affectes the weight distribution in the model test.
Estimation and prediction of weight distribution is an essential task in the design phase of a vessel, and the accuracy of this work will be crucial for the success of the project. Weight distribution is based on required general arrangement according to vessel missions. Prediction-based design uses a set of user specifications for a certain class of ships to predict the rest of the weight distribution parameters. The prediction of ship weight distribution is performed with an statistical method and international classification rules.
In the early stages of designing new ships it is necessary to estimate the weight per unit length of the vessel to predict the initial weight and cost. One of the most important issues for design of military vessels is a number of equipment, structural integrity, navigation facilities, armament and etc. Novel vessels have to be designed considering the limitations caused by these design requirements. For select the several logical general arrangement, shall be provide some class specific data for vessels.Â Table 1 give detailed The Weight Distribution for some selected Navy ship classes are presented in Table 1.
Table 2 presents the technical specification of the model which is based on the weight distribution presented in table 1. The ship to model scale ratio was 80 which is suitable for the size of the towing tank facility. Three weight distributions based on various general arrangements were considered in this experiment. Ship Model is divided in 20 stations longitudinally and each permanent segment weight is calculated.
3 THEORETICAL BACKGROUND
Wave induced slamming that occurs on a vessel, subjects the structure to an impulse load that responds with modal excitation that is characteristic to the design of high-speed craft, Thomas et al. (2003b). An impulse loading applied to the bow in head seas most often excites the first longitudinal mode of vibration in the vertical plane. The flexural frequency that is experienced by a prototype vessel as a result of wave excitation and slamming may be scaled to a model scale by applying a constant dimensionless wave encounter frequency between prototype and model scale and deducing the model scale flexural frequency as a function of the full-scale frequency and the square root of the full-scale to model scale length ratio as given by:
where
w_{m} = model scale modal frequency, w_{f} = full-scale modal frequency, l_{f} = full-scale length of vessel, l_{m} = model scale length.Â It is therefore possible to achieve the whipping frequency at model scale by scaling the full-scale frequency and designing the model with the capability of achieving a modal response that replicates the flexural dynamics of the full-scale vessel. Although the segmented model concept is an approximation of the full-scale vessel structural response, it facilitates a practical simulation of the whipping phenomena observed on a full-scale vessel.
A four degree of freedom theoretical model was developed in order to predict the longitudinal bending frequency of the model as a function of the effective stiffness and mass distribution of the model. Figure 1 shows the four degree of freedom model consists of a four rigid hull segments which are connected by three elastic hinges. The aft segment consists of aft main hull part and side hulls.
Let:
C1 centre of mass of segment-1 | θ rotation of hull segment |
C2 centre of mass of segment-2 | m total mass |
C3 centre of mass of segment-3 | m_{1} mass of segment 1 |
C4 centre of mass of segment-4 | m_{2} mass of segment 2 |
X_{1} distance from K_{1} to centre of mass of segment-1 | m_{3} mass of segment 3 |
X_{2} distance from K_{1} to centre of mass of segment-2 | m_{4} mass of segment 4 |
X_{3} distance from K_{2} to centre of mass of segment-2 | K_{1} stiffness of torsion spring-1 |
X_{4} distance from K_{2} to centre of mass of segment-3 | K_{2} stiffness of torsion spring-2 |
X_{5} distance from K_{3} to centre of mass of segment-3 | K_{3} stiffness of torsion spring-3 |
X_{6} distance from K_{3} to centre of mass of segment-4 | V shear force |
I_{1} moment of inertia of segment-1 | I_{2} moment of inertia of segment-2 |
I_{3} moment of inertia of segment-3 | I_{4} moment of inertia of segment-4 |
y vertical displacement |
The equations of motion representing this four degree of freedom system are
By substituting equation (1) in equation (2) we get V_{2},
Then:
By substituting equation (4) and equation (9) in equation (3) is obtained and by Eliminating from (6), (7) and (8) and
Considering results with the formulation of the following matrix
where
w_{n} : natural frequency
Evaluating the determinant of the matrix defined in equation 14 and excluding the solution at zero frequency for rigid body motion leads to the theoretical values of the natural frequencies of the four degree of freedom mass-spring system for which the determinant is zero. Table 3 shows a summary of the parameters that were input to the theoretical model and Figure 2 shows the results of the matrix determinant as a function of frequency for the evaluation of the wet modal frequency response of the hydroelastic segmented ship model.
It is observed from the results shown in Figure 2 that the wet modal frequency of the four degree of freedom system is 0 Hz for rigid body motion, 4.61Hz for the first longitudinal mode.
Figure 3 shows the model first longitudinal mode shape calculated for natural frequency.
4 MODEL DESCRIPTION
4.1 Hull Shape
The body plan of the hull and its outline are shown in figure 4 and 5 respectively. The length between perpendiculars is almost 1.5 meter. The vessel has a longitudinal symmetry body plan under the water line. A summary of the main particulars is given in table 2.
4.2 Stiffness distribution
The backbone frame is very stiff and the elastic hinges are located close to the gaps between the hull segments. There are three cut along the model length; one at the 0.25 L_{pp} from FORE perpendicular, one at the midship and another at position of outriggers connect to main hull.
4.3 Mass Distribution
The model has a total mass of 4.35 kilograms, and the Pitch radius of gyration is 0.36. The origin of the coordinate system located at the center of gravity. The xy-plane is parallel to the calm water surface when the ship is at rest at zero speed. Main contributions to the model's mass come from the hulls, which are made of fiber glass resin and backbone frame which is made of aluminum. The hull was cut longitudinally into three sections. Each rigid segment was attached to an aluminium square hollow section backbone beam. The segments of the model were joined together with three flexible elastic hinges that are interchangeable in order to vary the stiffness of the links. Separating the model into segregated compartments allowed the position and magnitude of ballast mass to be systematically located within the model to vary the configuration of ballast mass distribution according to three different general arrangements.
A schematic diagram of the elastic hinge is shown in Figure 6. The elastic hinge consists of a square hollow plug on either end that was connected at the center by a ï¬exible link of square cross section. Either end of the elastic hinge was inserted into the aluminum SHS (32x32x1.5) beam and was joined together with bolts to produce a highly rigid connection. The cross section dimensions and length of the ï¬exible link were varied in order to adjust the stiffness and ï¬exibility of the connection. Aluminum alloy was selected as the most suitable material for the manufacture of the elastic hinges as it was suï¬ciently light and was found to provide adequate strength and stiffness. The elastic hinges were manufactured on a CNC programmable machine at the Amirkabir University of Technology.
Experimental bending tests were performed on a sample of elastic hinges with different wall thickness t, in order to investigate the effect of wall thickness and weight on the ultimate strength and stiffness. Four elastic hinge samples were manufactured with a design stiffness of 3000 Nm/rad and wall thickness t, of 4 mm, 8 mm, 12 mm and 15 mm. The end plug of each elastic hinge was fixed to a suitable mechanism in a cantilever arrangement. Experiment performed with compression testing machine and the loads were incremented until such stage the failure load and failure mechanism was physically identified (Figure 7).
With reference to Figure 8 it is obvious that increases in the wall thickness responded with increases in the ultimate strength of the elastic hinge. Physical experiments were therefore undertaken in order to identify the effects of wall thickness on the stiffness of the elastic hinge.
4.4 Data Acquisition
To measure the vertical acceleration a 3DM-GX1 instrument is used. Measured data were recorded digitally at sampling rate of 350 Hz.
5 TEST PROGRAM
5.1 Tests in air
In order to determine the eigen-frequencies, the corresponding mode shapes and the damping properties of the model, vibration tests were carried out. The model was suspended from four springs, one at the midpoint of each segment, and vibrations were excited by giving it manual impulse force. Impulse force in different longitudinal locations will excite different mode shapes. Estimates of the damping levels and the eigen-frequencies, as well as an overview of mode shapes, were obtained by analyzing the signals from the microstrain. Useful information was obtained for the First mode by this method. The two springs from which the model was suspended, have eigen-frequencies that much smaller than those of the model therefore their influence on the measured frequencies could be neglected.
5.2 Test in water
Experiments were performed at towing tank of Marine Engineering Research Center, Sharif University of Technology. Towing Tank Particulars is presented in Table 4. Tests were undertaken in still water and at zero speed with the model unrestrained and free to respond with dominant motions in heave and pitch. An impulse load was applied by hand at the stern of the model in order to excite the longitudinal mode of vibration. The stern was selected as the most appropriate position as this produces the most reliable and reproducible signal response. Model accelerations were recorded using an accelerometer that was located at the different position of the model (Figure 9).
The stiffness of the elastic hinge connecting the rigid hull segments was varied in order to investigate its effects on the flexural natural frequency response of the first longitudinal mode of vibration. Four elastic hinges of different stiffness were used to form the connection between the four rigid parts of the model. Bending tests were performed on the model by simply supporting it on either ends and applying a load at the connection point. It was found that the measured stiffness of both the hull and the backbone beam was considerably less than the theoretical stiffness of the elastic hinge. It was observed that although the stiffness of the elastic hinge was increased the overall measured stiffness approached the stiffness of the hull. This was due to the inherent flexibility of the fibre glass hull and aluminium backbone beam causing significant bending between the mounting points across each segment of hull and is discussed in further detail by Jason LavroffÂ et al. (2006).
The modal frequency response of the model was also evaluated as a function of the ballast mass distribution. According to three different general arrangements, the ballast masses were located within the designated compartments of the model to produce three distinct mass distributions while maintaining a constant trim angle for each elastic hinge stiffness configuration. Essentially this enabled the variation of both the ballast mass distribution and the effective stiffness of the model in order to observe the effects on the whipping frequency response. Table 5 presents three general arrangement configurations implemented during trials conducted on the trimaran hydroelastic model.
A series of tests in calm water at zero speed are conducted on a novel Trimaran vessel in order to predict its Flexural Mode Response. There were altogether 42 test conditions. A brief summary of the tests is presented in Table 6 and Figure 9 shows a sample test.
6 MODEL TEST RESULTS
The whipping response was clearly observed during all tests undertaken on the segmented model. Figure 10 displays a typical accelerometer response recorded on the trimaran model following an impulse load at the stern.
According to the power spectral analysis for a test case with an effective stiffness of 3000 Nm/rad, a ballast mass distribution corresponding to GA 1, the first longitudinal natural frequency is 3.1 Hz. The Difference between the measured frequency and the analytical solution is due to the configuration of AFT segment. In theoretical AFT segment was mono Hull and in experiment was three hulls.
7 CONCLUSIONS
A design methodology was developed to investigate the modal frequency of a trimaran ship model through the adjustment of stiffness with an elastic hinge and variety of general arrangements. These findings formulate an effective mechanism for attaining the natural frequency response of a segmented model for the purpose of replicating the whipping response of a full-scale vessel subject to slamming. It was shown that the full-scale whipping frequency could be scaled down to a model scale and through the adjustment of the appropriate parameters the target whipping frequency may be achieved.
A theoretical model was developed based on a four degree of freedom mass-spring system representing four rigid hull segments connected by a set of springs longitudinally. It was shown that the theoretical model gave accurate predictions of modal frequency when compared to experimental results based on the effective stiffness of the model. The validation of the theoretical model through experiments aids the on-going development of a future hydroelastic model. Because of the aft segment configuration the measured experimental natural frequency is different from the theoretical one.Â
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Received 25.12.2013
Accepted 14.10.2014
Available online 17.08.2014