Uncertainty analysis of ship model vertical center of gravity and transverse moment of inertia test

SHI Sheng-zhe， ZHENG Ya-xiong

(High Speed Hydrodynamic Laboratory, Special Vehicle Research Institute ofAviation Industry Corporation of China, Jingmen 448035, China)

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Uncertainty analysis of ship model vertical center of gravity and transverse moment of inertia test

SHI Sheng-zhe， ZHENG Ya-xiong

(HighSpeedHydrodynamicLaboratory,SpecialVehicleResearchInstituteofAviationIndustryCorporationofChina,Jingmen448035,China)

The usability of test results of ship model vertical center of gravity and transverse moment of inertia is generally depends on its uncertainty. Referring to the guidelines for uncertainty analysis in examination of liquid dynamic recommended by International Towing Tank Conference ( ITTC), the results were analyzed, bias limits and precision limits were calculated and total uncertainty was estimated. The total uncertainty of six tests on ship model vertical center of gravity is is 0.16% of the mean value, and the total uncertainty of six tests on ship model transverse moment of inertia is 5.66% of the mean value. The test results show that the total uncertainty of both the multiple tests and the single test is from the precision limits of ship model vertical center of gravity and transverse moment of inertia tests.Thus, the improved measurement system stability can enormously decrease the total uncertainty of multiple tests and the single test.

ship model test; vertical center of gravity; transverse moment of inertia; uncertainty analysis

Test uncertainty is quantitative analysis of the quanlity of test results. The usability of test results generally depends on uncertainty analysis[1]. In order to improve the test veracity of ship model towing tank, the international towing tank conference (ITTC) recommended tanks to offer test results and the usability of test results, too. China Ship Scientific Research Center (CSSRC) collected and translated the recommended procedures and guidelines of uncertainty analysis of resistance towing tank tests and computational fluid dynamics (CFD) of the ITTC[2]. ZHU et al.[3]calculated viscosity circumferential flow field of bare hull of submarine standard model SUBOFF and analyzed the uncertainty of CFD results using ITTC temporary procedure; ZHOU et al.[4-5]analyzed the uncertainty of form factor, Froude number and the breakwater resistance of a standard ship model; LIU et al.[6]mentioned a new method about uncertainty of wetted surface; SHI et al.[7]analyzed the uncertainty of form factor, wetted surface and Froude number, and presented the analysis results; SUN[8], YANG[9]and ZHOU et al.[10]made uncertainty analysis of CFD for ship and propeller using the procedures recommended by the ITTC.

Most uncertainty analysis of domestic ship model test was resistance test. No mention was made of ship model vertical center of gravity and transverse moment of inertia test. Referring to the guidelines for uncertainty analysis in examination of liquid dynamic of recommended by ITTC[11-14], this paper analyzes the uncertainty of ship model vertical center of gravity and transverse moment of inertia test and lays a foundation for the uncertainty analysis of rolling test in static water and transverse regular wave test at zero speed.

1 Test design

Preparative work of ship model test in static water and transverse regular wave at zero speed is the same as that in resistance test, except that the ballast of the ship model must to be moved to the required state (tonnage, fore draft and aft draft), and the vertical center of gravity and transverse moment of inertia of the ship model need to be adjusted. It is receivable to adjust the vertical center of gravity and transverse moment of inertia on simple edge shelves. The vertical center of gravity can be adjusted on the right-and-left edge shelf, and the transverse moment of inertia can be adjusted on the fore-and-aft edge shelf.

1.1 Principal dimensions of ship model

The resistance test of a 2.5 m fiberglass-reinforced plastics (FRP) ship model was conducted in a towing tank in China Special Vehicle Research Institute in 2012, and the ship model vertical center of gravity and transverse moment of inertia tests were conducted before the resistance test.

Table 1 Principal parameters of ship model

1.2 Measurement of vertical center of gravity

After the ballast of ship model moves to the required state (including test instrument in ship model) by right-and-left edge shelf, the support ship model moves weight for longitudinal balance of ship model horizontally, and then moves little weight along fore and aft to get vertical center of gravity by ship slope. Fig.1 shows the measurement setup of vertical center of gravity.

The vertical center of gravity is calculated by

wherezgis ertical center of gravity(m);Pis the weight of the small moving object(kg);Lis the distance of the small moving object(m);dOAis the distance between right-and-left edge and pointA(m), and its position in station 0 is the same height as right-and-left edge;Dmis the tonnage of ship model(kg); ΔHis the vertical distance of pointAwhen the small object moves before and after(m);HOKis the height of right-and-left edge(m)， and it is set at 0.33 m.

Fig.1 Measurement setup of vertical center of gravity

1.3 Measurement of transverse moment of inertia

The support ship model moves a small object along fore and aft at the same height, and adjusts the ship model to transverse balance. There is an initial angle by holding ship model. After clocking ten periods by stopwatch, the ship model transverse moment of inertia is got.

Fig.2 Measurement setup of transverse moment of inertia

Based on pendulum principle, when ship model gets instantaneous transverse force, the transverse time which circles fore-and-aft edge is

whereHOK′is the height of fore-and-aft edge(m)， and it is set at 0.587 m.

The transverse moment of inertia is

There are total bias limit and total precision limit described as

where the total bias limits of vertical center of gravity and transverse moment of inertia of the ship model can be calculated by

Whatever the precision limits of the single test or multiple tests are, the standard bias must come from multiple tests. If multiple tests can not be conducted, it is necessary to estimate a value for the precision limit by using reliable information.The precision limit of multiple tests is

whereMis test time which determines precision limit;Sdevis standard bias of multiple tests;Krefers to the procedures by ITTC,K=2.

The precision limit of single test is

P(S)=KSdev.

2 Uncertainty analysis

2.1 Bias limit

Bias limit is uncertainty part due to system effect, which can be got by the most allowable error of measurement equipment through combined standard uncertainty.

Distance is measured by altitudinal ruler and meter ruler. The most allowable error of altitudinal ruler, which takes for half width of distribution interval and has rectangular distribution, is 0.000 02 m. When the small object moves before and after, the bias limit of right-and-left edge,BHOK, and the bias limit of vertical distance of point A,BΔH, are given by

Themostallowableerrorofmeterruleris0.001m.Thebiaslimitofdistanceofthesmallmovingobject,BL, and bias limit of distance between right-and-left edge and point A,BdOA, are given by

The resolution of digital dynamometer to measure the tonnage of ship model is 0.02 kg, with rectangular distribution. The bias limit of tonnage of the ship model is

Theresolutionofdigitaldynamometertomeasuretheweightofthesmallmovingobjectis0.001kg,withrectangulardistribution.Thebiaslimitofweightofthesmallmovingobjectis

AccordingtoEq.(1),therelevantparametersareset:HOK=0.330 m,Dm=113.92 kg,P=0.168 kg,L=2.30 m,dOA=0.659 m and ΔH=0.049 3 m. Thus, the sensitivity coefficients of the vertical center of gravity are given by

=1,

According to Eq.(6)， the bias limit of vertical center of gravity,Bzg, is 1.638 1×10-4m. This value is 0.06% ofzg.

Table 2 Composition of total bias limit of ship model vertical center of gravity

DefinitionsValuePercentof(Bzg)2(Bzg)22．832×10-8?zg?HOKBHOK?è???÷23．364×10-911．88%?zg?DmBDm()22．153×10-110．08%?zg?PBP()22．480×10-887．58%?zg?LBL()25．291×10-140．00%?zg?dOABdOA()21．612×10-110．06%?zg?ΔHBΔH()21．166×10-100．41%

The resolution of digital stopwatch to measures time is 0.1 s. The stopwatch measures the time of ten transverse periods, thus the most allowable error of transverse period is 0.01 s, with rectangular distribution. The bias limit of transverse period is

Theright-and-leftedgeismeasuredbyaltitudinalruler,andthebiaslimitofright-and-leftedge,BHOK′, is 0.000 011 6 m.

LetT=1.59 s andHOK′=0.587 0. According to Eq.(3), the sensitivity coefficients of transverse moment of inertia are calculated by

高速列車的車頭外形影響著整輛列車的穩定性與整輛列車的阻力。那么什么樣的車頭外形才能減少阻力，提高穩定性呢？眾所周知，列車的頭，尾形狀影響著空氣壓差阻力、升力和橫風穩定性等因素。并且動車分為動力集中性與與動力分散型列車，而動力分散性列車則是將動力裝置分別安裝在車頭和車尾，過車頭車尾動力機車帶動中間的無動力機車。所以說像類似CRH380A這種動車組的車頭采用為可降低空氣阻力的流線型的動車，時速可以高達350公里每小時，但是在中國科學院氣體力學風洞研究所中，我們國家目前的理想高速列車模型中，時速可高達500公里每小時，但是為什么但目前為止，我沒還沒有一個500公里的高速列車下線運行呢？

According to Eq.(7)， the bias limit of transverse moment of inertia,BI, is 0.158 36 kg·m2, and this value is 1.41% ofI.

Table 3 shows the composition of total bias limit of ship model transverse moment of inertia. It can be seen that the transverse period almost accounts for 100% of the total bias limit of ship model transverse moment of inertia. Improving time measurement precision is the most important, which can reduce the bias limit of transverse periodT.

Table 3 Composition of total bias limit of ship model transverse moment of inertia

DefinitionsValuePercentof(BI)2(BI)22．508×10-2?I?TBT?è???÷22．508×10-299．99%?I?DmBDm()21．918×10-70．01%?I?zgBzg()21．918×10-70．00%?I?HOK′BHOK′()29．114E×10-100．00%

2.2 Precision limit

Precision limit is uncertainty part due to random effect, which results from lack of reiteration, such as random error, instability, impossible accurately re-installation test state, and so on.

For more exact precision limit, the ship model needs to be re-installed before every test, and then the standard bias of a series of tests are evaluated . There are six times in the test. This is the best method including random errors, such as installation error, longitudinal slope and transverse slope.

Table 4 Standard bias of ship model vertical center of gravity and transverse moment of inertia test

TesttimeΔH(m)z(m)T(s)I(kg·m2)10．04830．2841．5911．25120．04850．2841．6011．52730．05030．2851．6011．52340．05010．2851．6312．35850．04910．2841．5710．70460．04920．2841．5510．165Average0．04930．2841．5911．255Standardbias0．000520．75553

Table 4 shows the average and standard bias of ship model vertical center of gravity and transverse moment of inertia test.

Accodting to Eq.(8), the precision limit of multiple tests of ship model vertical center of gravity is

4.216 4×10-4m.

Thisvalueis0.15%ofzg.

According to Eq.(9), the precision limit of single test of ship model vertical center of gravity is

Pzg=K×Sdevzg=2×0.000 52=

1.032 8×10-3m.

Thisvalueis0.36%ofzg.

According to Eq.(8), the precision limit of multiple tests of the transverse moment of inertia of the ship model is

This value is 5.48% ofI.

According to Eq.(9), the precision limit of single test of ship model transverse moment of inertia is

PI=K×SdevI=2×0.755 53=1.511 1 kg·m2.

Thisvalueis13.43%ofI.

2.3 Total uncertainty

According to Eqs.(4) and (5), combining bias limits and precision limits of single test and multiple tests, total uncertainty are evaluated.

The total uncertainty of the multiple tests of ship model vertical center of gravity is

This value is 0.16% ofzg.

The total uncertainty of the single test of ship model vertical center of gravity is

This value is 0.37% ofzg.

The total uncertainty of the multiple tests of ship model transverse moment of inertia is

This value is 5.66% ofI.

The total uncertainty of the single test of ship model transverse moment of inertia is

This value is 13.50% ofI.

Table 5 shows the composition of total uncertainty. It is helpful to systematical analysis of the total uncertainty of ship model vertical center of gravity and transverse moment of inertia test.

Table 5 Composition of total uncertainty

3 Conclusion

1) Improving measurement system precision for the weight of small moving object and right-and-left edge can reduce the total bias limit of ship model vertical center of gravity. Improving time measurement precision is the most important, which can reduce the bias limit of transverse period, and furthermore, can reduce the total bias limit of ship model transverse moment of inertia.

2) Whatever total uncertainty of multiple tests or single test is, primary part of ship model vertical center of gravity and transverse moment of inertia comes from precision limit. Impoving the stability of measurement system can reduce precision limit, and furthermore, can reduce total uncertainty of multiple tests and single test.

3) This test is conducted six times repeatedly. The total uncertainty of multiple tests is half of single test. Therefore, it can be proven that the method of multiple tests is more effective to reduce total uncertainty than single test.

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船模重心高度和橫向轉動慣量測量試驗的不確定度分析

史圣哲， 鄭亞雄

(中國特種飛行器研究所 高速水動力實驗室， 湖北 荊門 448035)

船模重心高度和橫向轉動慣量測量試驗測量結果的可用性很大程度上取決于其不確定度的大小。 參照ITTC推薦規程中試驗流體動力學不確定度分析規范， 對船模重心高度和橫向轉動慣量測量試驗進行了不確定度分析， 給出了船模重心高度和橫向轉動慣量的偏差極限、 精密度極限和總不確定度。 重心高度的六次試驗平均值的總不確定度占平均值的0.16%， 橫向轉動慣量的六次試驗平均值的總不確定度占平均值的5.66%。 船模重心高度和橫向轉動慣量測量試驗多次試驗平均值的總不確定度和單次試驗的總不確定度主要來自精密度極限， 提高測量系統的穩定性可以極大的降低多次試驗平均值的總不確定度和單次試驗的總不確定度。……