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Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2013 Article ID 868252 13 pageshttpdxdoiorg1011552013868252

Research ArticleThree-Dimensional Mathematical Investigation of Dynamic andHydrostatic Pressure Distributions on Planing Hulls

Parviz Ghadimi Sasan Tavakoli Abbas Dashtimanesh and Seyed Reza Djeddi

Department of Marine Technology Amirkabir University of Technology Hafez Avenue No 424 PO Box 15875-4413 Tehran Iran

Correspondence should be addressed to Parviz Ghadimi pghadimiautacir

Received 15 May 2013 Accepted 27 July 2013

Academic Editor Jia-Jang Wu

Copyright copy 2013 Parviz Ghadimi et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A mathematical model is utilized in order to calculate three-dimensional pressure distributions on planing hulls This type ofmodeling is able to determine the hydrodynamic and hydrostatic pressures acting on the bottom of these hulls As a result the total3-dimensional pressure exerted on the planing hull as a sum of hydrostatic and hydrodynamic pressures can be evaluated Empiricalequations introduced in previous works have been used as the fundamentals for the present mathematical modeling method Theobtained results are compared against available experimental results and results of empirical equations in order to validate theproposed method The outcome of the R-squared tests conducted on these comparisons shows favorable accuracy of the resultsAfter evaluation of hydrodynamic pressure the effects of trim and deadrise angles and wetted length on the 3-dimensional pressuredistribution are analyzed Finally the total pressure on planing hull and the effect of velocity coefficients are studied

1 Introduction

Determining the exerted pressure on the planing hulls isessential to the study of their dynamics By computing thispressure it becomes possible to calculate the lift force centerof pressure vesselrsquos dynamic and water spray Complexitiesconcerned with 3-dimensional pressure distributions havemade their direct calculation improbable if not impossibleTherefore a common approach is to first measure thepressure distribution using 2-dimensional longitudinal andtransverse solutions which can be later used for calculationof 3-dimensional pressure distribution acting on the planinghull Additionally experimental methods can be used forcalculation of pressure distribution in this type of vesselAmong these experimental works results of Kapryan andBoyd [1] and Smiley [2 3] are of high importance On theother hand Wagner [4] used analytical methods to evaluatepressure distribution over a wedge in a water-entry problemwhich is identical to the pressure distribution acting on aninfinite planing plane

It was proved in Smileyrsquos experiments [3] that the longitu-dinal pressure distribution on the side sections of the centerline is less than that on the other sections Based on these

results empirical equations for evaluation of longitudinalpressure distribution were proposed Later Kapryan andBoyd [1] showed in their experimental studies that pressureon planing hulls will be reduced with an increase in thedeadrise angle

With the advances in computational and computerresources mathematical modeling methods for the analysesof planing vessels were introduced These methods takeadvantage of high speed and high efficiency of the computa-tional tools and bring about higher accuracy One of themostwell-known mathematical models is the method proposedby Savitsky [5] The basis for this modeling is the use ofexperimental results which lead to empirical equations thatcan be implemented for mathematical modeling in planinghulls in order to evaluate the lift force center of pressure andresistance

Numerical modeling of planing vessels has also gainedattention in the past decades Although this method is timeconsuming and expensive many works can be found inthe literature taking this approach A famous work in thisfield is the numerical study of Wellicome and Jahangeer [6]in which the 3-dimensional pressure distribution over theplaning hull was evaluated using computational techniques

2 Journal of Computational Engineering

Pressure distribution

Spray

(a)


(b)

Figure 1 Longitudinal (a) and transverse (b) pressure distribution on planing hulls [10]

Also in a recent attempt Ghadimi et al [7] used the methodof Smoothed-particle hydrodynamics (SPH) in order to studyflow around the planing plate and consequently find pressuredistribution over this surface

The initial modeling method of Savitsky [5] was latermodified and improved by Savitsky et al [8] by implementingwhisker spray and the resulting wetted area In anotherwork Morabito [9] studied the water spray and pressuredistribution over planing hulls As a result of his studiesa set of empirical equations was introduced for calculationof pressure distribution on planing hulls In the presentstudy empirical equations of Morabito [9] are used as thebasis of the mathematical model introduced for evaluation ofpressure distribution of a plning hull

2 Mathematical Formulation

One of the simplest techniques used for evaluation of planingvesselsrsquo dynamics is the planing plate model Sottorf [10]studied the pressure distribution on a planing plate in bothlongitudinal and transverse directions Therefore a plot ofthe longitudinal and transverse pressure distribution over aplaning plate was introduced by Sottorf [10] as shown inFigure 1 In the present paper the goal is to first calculatepressures on different longitudinal sectionswhich can later beused for 3-dimensional pressure distribution over the planinghull with consideration of transom stern effects and thencalculate the pressure alleviation which occur when gettingfarther from the center line

21 Stagnation Line andMaximum Pressure When a planingplane is moving along the water surface a stagnation lineappears on both sides of the center line It was shown that themaximum pressure on planing hulls occurs at the stagnationline [11] A spray area appears in front of the stagnation line onwhich the spray resistance force occurs while pressure areaappears in the aft direction in which drag and lift forces existThe stagnation line and spray and pressure areas for a planinghull are depicted in Figure 2

Based on fluid dynamics theories when a free flowcollides with an obstacle the maximum pressure occurs ata stagnation point In the case of a planing hull all thestagnation points lie on the stagnation line Experimentsby Smiley [3] proved that pressure distribution will fall onsections close to the stagnation line Therefore it can be

Pressure areaSpray area

Chine

Transom

Flow direction

Stagnation line

Keel

Spray edge

V

120572b

Figure 2 Bottom of a planing hull illustrating different sections [8]

concluded that at each section of the planing hull the max-imum pressure will occur at the stagnation line Moreoverthe pressure will also reduce along this line where at theintersection of the stagnation line and the center line themaximum possible pressure will appear Smiley [3] proposedthe following equation for the maximum pressure acting onthe planing hull

119875max(12) 120588119881

2=

1205872tan2120591

1205872tan2120591 + 4tan2120573 (1)

inwhich119875max is themaximumpressure120588 is thewater density119881 is the advance velocity of the vessel 120591 is the trim angleand 120573 is the deadrise angle Also Morabito [9] introducedthe following formula for determination of the maximumpressure

119875max119902

= sin2120572 (2)

119902 =1

21205881198812 (3)

where 120572 is the angle between the stagnation and center lineswhich was defined by Savitsky [5] using (4) as in

120572 = tanminus11205872

tan 120591tan120573

(4)

Figure 3 shows the maximum pressure at different trimand deadrise angles

As clearly seen in Figure 1 the pressure alleviates justafter reaching its maximum valueThe pressure will vanish at

Journal of Computational Engineering 3

0

02

04

06

08

1

12

0 5 10 15 20 25 30 35 40 45 50

120591 = 2

120591 = 4

120591 = 8

120591 = 12

120591 = 24

120591 = 30

120573

Pm

axq

Figure 3 Maximum pressure versus deadrise angle at different trimangles using empirical equations of [9]

0

0005

001

0015

002

0025

003Pq

0X

minus1minus2 minus05minus15minus25minus35 minus3minus4

Figure 4 Longitudinal pressure distribution over a planing hullusing (6) as introduced by Smiley [3]

the transom due to the atmospheric conditions The pressuredecrease is calculated using (5) as proposed by Smiley [3]

119875119871

119902= 0006

12059113

11988323 (5)

The previous equation can be used for the aft areasof the stagnation point Pressure 119875

119871is the pressure at any

point behind the stagnation line and 119883 is the dimensionlessdistance between the stagnation line and the desired sectionwhich is given by (6)

119883 =119909

119887(6)

in which 119887 is the hullrsquos breadth An example of the calculatedpressure over the stagnation line using (6) for a planing hullwith a trim angle of 4∘ and a nondimensional wetted lengthof 4 is depicted in Figure 4

Equation (6) at 119883 = 0 yields to infinity ThereforeMorabito [9] proposed (7) for the maximum pressure valuewhich vanishes at119883 = 0

119875119871

119902=11986211988313

(119883 + 119870) (7)

0

001

002

003

004

005

006

008

007

0

Pq

X

minus2minus4minus6

Figure 5 Longitudinal pressure distribution over the center line ofa planing hull using (7) as introduced by Morabito [9]

Peak pressure line

Center line

Transom

Chine

Figure 6 Pressure distribution over the bottom of a planing boat[3]

in which

119862 = 000612059113 (8)

119870 =11986215

2588(119875max119902)15 (9)

An example of the calculated pressure distribution using(7) for a vessel with 120591 = 4 120582 = 4 and 120573 = 20 is shown inFigure 5 As clearly seen the pressure does not fade at thetransom stern In other words the effect of the transom sternhas not been taken into consideration

Morabito [9] proposed a transom correction factor tolongitudinal pressure distribution 119875

119879 in order to take into

consideration the effect of the transom stern on this distribu-tionThis factor which can be evaluated using (10) causes thelongitudinal pressure distribution to start decreasing from aregion close to transom (half breadth from the stern) andvanish at the transom stern It also causes the pressure tonever reach a value of unity close to the stagnation line (seeFigure 6)


Consider

119875119879=

(120582119910minus 119883)14

(120582119910minus 119883)14

+ 005

(10)

Here 120582119910is the distance between the stagnation line and

the transom stern at the desired cross-section which can becalculated for each longitudinal section with a nondimen-sional transverse distance of 119884 = 119910119887 from the center lineusing the following equation

120582119910= 120582 minus

(119884 minus 025)

tan120572 (11)

The diagram presented in Figure 5 is changed to the oneshown in Figure 7 by implementation of the transom sterneffect It is clear from Figure 7 that the longitudinal pressuredistribution is significantly affected close to the transomstern

22 Transverse Pressure Distribution and Longitudinal Distri-bution on Other Sections As stated in the introduction thepressure will alleviate along the transverse sections and willvanish at the chine Based on this fact the solution of Korvin-Kroukovsky [12] is used which introduces a factor for thetransverse pressure distribution This factor gives the ratioof pressure at a transverse section with fixed length to thepressure at the center line Equation (11) is given based on thesolution of Korvin-Kroukovsky [12] as follows

119875119884= [102 minus 005 (120573 + 5) 119884

14]05 minus 119884

051 minus 119884 (12)

The previous equation gives the pressure reduction basedon the distance from the center line but does not accountfor the pressure decrease close to the stagnation line in othersections

Using the swept wing theory Morabito [9] calculated themaximumpressure on each longitudinal sectionHe assumedthe velocity vector on the bottom of the planing hull to beconsisting of two components one along the stagnation line119881119904and the other normal to the stagnation line 119881

119899 Figure 8

depicts the bottom of the planing hull with the velocityvectors based on the method of Morabito [9]

Components of the velocity vector acting on the bottomof a planing hull are calculated using (13) as follows

119881119899= 119881 sin120572

119881119904= 119881 cos120572

(13)

in which 119881 is the advance velocity of the vessel By con-sidering 119875

119873as the dynamic pressure resulting from the

velocity component normal to the stagnation line Morabito[9] introduced the ratio of stagnation pressure 119875

119884Stag to thedynamic pressure 119875

119873at each longitudinal section using the

semiempirical equation (14) By multiplying (2) by this ratiomaximum pressure at each longitudinal section (119875max119902)119884which is the pressure at the stagnation line over that sectionis obtained as given in (15) This ratio has a value lower than

0

001

002

003

004

005

006

007

008

0

With transom effectWithout transom effect

Pq

X

minus1minus2minus3minus4minus5minus6

Figure 7 Effect of the transom correction factor on the longitudinalpressure distribution along the center line

Center lineTr

anso

mStagnation line

Chine

V

V

s

120572

120572

Vn

Figure 8 Components of the velocity vector for a planing hull [9]

the unity for all longitudinal sections except for the centerline and will decrease with the distance from the keel lineAfter multiplication the pressure at the stagnation line willalso alleviate with the distance from the center line and finallyvanish at the chine At 119884 = 0 the ratio yields to unity andgives the maximum dynamic pressure acting on the bottomof the planing hull as in

119875119884Stag

119875119873

= [102 minus 02511988414]05 minus 119884

051 minus 119884 (14)

119875max119902119884

=

119875119884Stag

119875119873

sin2120572 (15)

When the maximum pressure at each longitudinal sectionis calculated using (14) and (15) the only remaining taskis to evaluate the longitudinal pressure distribution for thedesired section Therefore (7) is used for this task with theexception that coefficients119862 and119870will now bemodifiedwiththe ones that take into account the effect of distance fromthe stagnation line The modified coefficients are calculatedusing (16) and (17) instead of (8) and (9) Finally the effect ofthe transom stern on the pressure distribution at the desiredsection is taken into consideration

119862 = 000611987511988412059113 (16)

119870 =11986215

2588((119875max119902)119884)15 (17)


Inputs

Calculate ratio of pressure at this section to pressure at center line section by

Total pressure

Yes

Yes

No

No

3D total pressure and dynamic

distribution

Transom effect

120591 120573 120574 C

Y = 0

determination of Py

Calculation ofC and K

Pstag

PN

Pmaxq

Y gt 05

X gt 120582y

X = 0

Y + h

X + h998400PT

PLq

PBq

Figure 9 Flowchart of the present algorithm in order to evaluate the pressure distribution over planing hulls

Clearly when 119884 = 0 119875119884in (16) will be equal to 1 and

therefore (16) and (17) will be identical to (8) and (9)

23 Hydrostatic and Total Pressure Based on the theoriesof fluid mechanics and buoyancy the force that is exertedby the fluid on the floating body is equal to the volumethat the body displaces [13] On the contrary in the caseof planing hulls this value is less than that of the displacedvolume [5 14] Shuford [14] assumed the buoyancy force tobe equal to the half of the displaced volume and Savitsky[5] took into account the effect of the wetted length for thecalculation of this force The main reason for the decrease ofthe buoyancy force can be sought in the fact that hydrostaticpressure distribution 119875

119861on the planing hull is altered It is

possible to take into account the effect of transom stern andbreadth on the hydrostatic pressure distribution in a way thathydrostatic pressure at any given point on the planing hullbody is calculated by the multiplication of these two factorsEquation (18) shows this simplification

119875119861

119902=120588119892119867 (119883 119884)

119902119875119879119875119884 (18)

Here 119892 is the gravitational accelerations and 119867 is thedepth of the given point Morabito [9] proposed (19) based

on the previous formula where 120572119882

is the angle between thecenter line and the calm water line which can be calculatedusing (20)The velocity coefficient (119862

119881) in (19) is given by (21)

and clearly the velocity increase will cause the hydrostaticpressure to decrease

119875119861

119902=2119875119879119875119884sin 120591

1198622

119881

(119883 + 119884(1

tan120572minus

1

tan120572119882

)) (19)

120572119882= tanminus1 ( tan 120591

tan120573) (20)

119862119881=

119881

radic119892119887 (21)

The total pressure acting on the planing hull as a sum ofdynamic and hydrostatic pressures is given by (22) as follows

119875Total119902

=119875119871

119902+119875119861

119902 (22)

24 Modeling Method The equations introduced thus far areused for 3-dimensional modeling in a way that the bottomof the planing hull is divided into a set of gridlines At eachlongitudinal section with a fixed breadth the longitudinal


Table 1 R-squared values for demonstrating the accuracy of thepredicted dynamic pressure distribution by the current model

Case 120573 120591 120582 119884 1198772

(a) 0 4 5120025 0906487849025 09106314030475 0914969083

(b) 0 30 1070025 0931204774025 0950345090475 0931461486

(c) 20 6 2360025 087950731025 08840247390475 0951144181

(d) 20 9 0950025 0906989641025 09270938960475 0853750853

(e) 40 12 4880025 0871029025 09811090475 0938936

(f) 40 24 2460025 0813929407025 0975481860475 082164695

dynamic pressure distribution is calculated while the effectof transom stern and pressure alleviation when getting closerto the chine are taken into account Therefore (19) and (20)are used for calculation of hydrostatic pressure at any givenpoint and consequently the total pressure would be the sumof both pressures calculated so far This type of modelingcan only give the dynamic or the total pressure acting onthe planing hull which can then be used to evaluate the 3-dimensional pressure distribution The parameters used asinput for calculations are deadrise angle trim angle averagedwetted length and velocity coefficient (transverse Froudenumber) In the case of only modeling the dynamic pressurethe transverse Froude number is not required as an input

A computer code is developed which uses two compu-tational loops as illustrated in Figure 9 In the first loopcalculations are performed with the alteration of 119884 while thesecond loop changes the value of119883 in order to achieve a fullyexpanded calculation over the bottom of the planing hullThe value of119883 (nondimensional distance from the stagnationline) is set to be positive in all equations and is only set asnegative for representations in various figuresThe reason forthe value of 119883 extending from 0 to 120582

119884at each longitudinal

section can be sought in the fact that at each section thepressure changes are calculated from the stagnation pointup to the transom stern The flowchart for the proposedalgorithm is shown in Figure 9

3 Validation

Experimental results of Kapryan and Boyd [1] are usedin order to validate the obtained results for the longi-tudinal dynamic pressure distribution They [1] evaluated

the pressure at various longitudinal sections for three dif-ferent planing hulls with deadrise angles of 0 20 and 40at multiple trim angles and averaged wetted lengths Theirexperiments were executed at 26 different cases that due tothe high volume of results in their study only two cases foreach deadrise angle are chosen for validation purposes Intheir experiments [1] parameter119883 behind the stagnation linehas positive values while in the present study this parameterhas negative values behind the stagnation line Therefore inorder to achieve a good comparison the results of [1] aremade negative Figure 10 shows the comparison between thecurrent results and the experimental results of [1] in differentconditions which prove that the proposed mathematicalmodel has favorable accuracy

In order to further support the accuracy of the obtainedresults against the experimental data R-squared values of theplots in Figure 10 have been presented in Table 1 The valuesof R-squared have been calculated using equation

1198772= 1 minus

119878119878res119878119878tot

(23)

where values of 119878119878res and 119878119878tot are obtained using

119878119878res = sum(119875

119902 Expminus119875

119902Mean)

2

119878119878res = sum(119875

119902 Expminus119875

119902 Predicted)

2

(24)

Here 119875119902Exp is the measured pressure by Kapryan andBoyd [1] and 119875119902Mean is the mean of the measured pressuresin each plot On the other hand 119875119902Predicted is the pressureobtained from the presentmathematicalmodel As evidencedin Table 1 the R-squared values are fairly close to 10indicating a favorable accuracy of the obtained results

In order to validate the obtained results for the hydrostaticpressure the exerted lift force by this pressure calculated from(25) is compared against the hydrostatic lift force coefficientof planing hulls proposed by Savitsky [5] Accordingly thehydrostatic pressure acting on the bottom of the planing hullis integrated over a planing plate and can be calculated from(26) as follows

1198621198710= 00055

12058225

1198622

119881

12059111 (25)

1198621198710= int

05

minus05

int

120582119910

0

119875119861

119902119889119909 119889119910 cos 120591 (26)

In order to have a comparison for the obtained resultsfrom the previous integration the curves of 119862

119871012059111 for both

methods and at four different transverse Froude numbersare illustrated in Figure 11 As evidenced in this figure theintegration of (26) gives results that are in agreement withthe results of (25) that proves the efficiency and accuracy ofthe present method for calculation of hydrostatic pressuredistribution The results of R-squared values of the data inFigure 11 which are presented in Table 2 affirm this claim


P

X

q

03504

02502015010050

0minus2minus4minus6X


0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X

0minus1 minus05minus15

Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X

0minus3 minus2 minus1

X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05

Mathematical modelingExp Kapryan and Boyd 1955



035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X

0minus3 minus2 minus1X


Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)

Figure 10 Comparison between the obtained results and the experimental results of Kapryan and Boyd [1] for the validation of dynamicpressure distribution

Table 2 R-squared values for hydrostatic lift coefficients on planinghulls

Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968

4 Results and Discussion

Theobtained results from the presentmathematicalmodelingare studied as parts of two main categories In the firstcategory the 3-dimensional dynamic pressure distributionover planing hull and different parameters affecting it arestudied The total pressure distribution and the effects ofdifferent parameters on this distribution are studied in thesecond category

41 Dynamic Pressure Based on the present mathematicalmethod the 3-dimensional dynamic pressure distribution

on planing hulls is modeled Afterwards the effect of trimangle deadrise angle and averaged wetted length on thisdistribution is studied

411 The Effect of Trim and Deadrise Angles In order tostudy the effect of trim and deadrise angles on the 3-dimensional dynamic pressure distribution themodeling hasbeen accomplished at different angles with a fixed wettedlength The obtained results are shown in Figures 12 13 14and 15 As seen in these figures the increase of trim angle at afixed deadrise angle causes the maximum pressure pressuredistribution and pressure level to rise which leads to anincrease in hydrodynamic lift force Moreover increase ofdeadrise angle at a fixed trim angle causes the maximumpressure and pressure level to alleviate and subsequentlydecreases the hydrodynamic lift force The effect of deadriseangle on the maximum pressure subsides with an increase ofthe trim angle

412 The Effect of Wetted Length In order to study the effectof the averaged wetted length on 3-dimensional dynamic


025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6

Figure 11 Comparison between the hydrostatic lift force coefficients on planing plate calculated using the proposed integration method andthe equation of [5]

pressure distribution over a planing hull a bodywith constantdeadrise and trim angle is studied It has been seen that anincrease in wetted length causes the pressure level to rise buthas no effect on the pressure distribution and the obtainedmaximum pressure (see Figure 16)

42 Total Pressure Thesumof dynamic and hydrostatic pres-sures at any given point gives the total pressure Thereforethe present method is able to model the 3-dimensional totalpressure distribution on the planing hull As a result the effectof the velocity coefficient on the total pressure distribution isstudied As clearly seen in Figure 15 at a constant trim angledeadrise angle and averaged wetted length an increase invelocity coefficient leads to a decrease in hydrostatic pressure

and consequently a decrease in total pressurewhichwill resultin a reduction of total lift force (see Figure 17)

5 Conclusion

In this article three-dimensional mathematical modeling ofdynamic and total pressure distribution over a planing hullis presented and the obtained results are validated againstexperimental results The calculated R-squared values ofthe corresponding data which are relatively close to 10indicate that the proposed method has favorable accuracyand efficiency Moreover the accuracy of the present modelin determination of the hydrostatic pressure distribution isshown to be favorable in comparison with the hydrostaticterm in the lift force coefficient equation This claim is also


minus35 minus3 minus25 minus2 minus15 minus1 minus05 0

minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4

minus35 minus3 minus25 minus2 minus15 minus1 minus05

minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6

Figure 12 Three-dimensional pressure distribution over planing hull for 120573 = 5 and 120582 = 3

minus4 minus35 minus3 minus25 minus2 minus15 minus1 minus05 0

minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6



minus5 minus4 minus3 minus2 minus1 0

minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6


minus6 minus5 minus4 minus3 minus2 minus1 0

minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35

Figure 16 The effect of averaged-wetted length on 3-dimensional dynamic pressure distribution for a planing hull with 120591 = 2 and 120573 = 10


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6

Figure 17 The effect of velocity coefficient on 3-dimensional total pressure distribution on a planing hull with 120591 = 2 and 120573 = 15

corroborated by its suitable R-squared values Mathematicalequations used in the present study take into account theeffect of transom stern and chine on dynamic and hydrostaticpressure distributions Using the proposed algorithm 3-dimensional pressure distribution over the bottom of aplaning plate is calculated

The effect of trim angle on the increase of pressurevalues and pressure level and the effect of deadrise angleon the reduction of these values are presented As seen inFigure 12 through Figure 15 at a constant trim angle andaveraged wetted length the effect of deadrise angle is studiedwhile the effect of trim angle is similarly analyzed with


a constant deadrise and averagedwetted length Furthermorethe independence of pressure distribution from the averagedwetted length is studied through various computational testswhich show that the pressure distribution and consequentlylift force are increased Finally the hydrostatic pressurealleviation and subsequently the decrease in total pressureacting on a planing plate are studied for an increase of thevelocity coefficient

Study of lift force center of pressure and influentialparameters for V-bottom hull forms and the pressure dis-tribution on asymmetric planing catamaran demihulls andtrimarans using empirical equations can be the subject offuture studies

References

[1] W J Kapryan and G M Boyd ldquoHydrodynamic pressuredistribution obtained during a planing investigation of fiverelated prismatic surfacesrdquo NACA Technical Note 1955

[2] R F Smiley ldquoA study of water pressure distribution duringlanding with special reference to a prismatic model havinga heavy loading and a 30-degree angle of deadriserdquo NACATranslation 1950

[3] R F Smiley ldquoAn experimental study of the water-pressuredistributions during landing and planing of a heavily loadedrectangular flat-plate modelrdquo NACA Technical Note 2453 1951

[4] HWagner ldquoPhenomena associatedwith impacts and sliding onliquid surfacesrdquo NACA Translation 1932

[5] D Savitsky Hydrodynamic Design of Planing Hulls vol 1Marine Technology 1964

[6] J F Wellicome and Y M Jahangeer ldquoThe prediction of pressureloads on planing hulls in calm waterrdquo Royal Institution of NavalArchitects no 2 pp 53ndash70 1978

[7] P Ghadimi A Dashtimanesh M Farsi and S Najafi ldquoInves-tigation of free surface flow generated by a planing flat plateusing smoothed particle hydrodynamicsmethod and FLOW3Dsimulationsrdquo Journal of Engineering Maritime Environment2012

[8] D Savitsky M F DeLorme and R Datla ldquoInclusion of whiskerspray drag in performance prediction method for high-speedplaning hullsrdquoMarine Technology vol 44 no 1 pp 35ndash56 2007

[9] M G Morabito On the spray and bottom pressures of planingsurfaces [PhD thesis] Stevens Institute of Technology 2010

[10] W Sottorf ldquoExperiments with planing surfacesrdquo NACA Trans-lation 1934

[11] J D Pierson and S Leshnover ldquoA study of the flow pressure andloads pertaining to prismatic vee-planing surfacesrdquo Report SITDL 50382 Davison Laboratory 1950

[12] B V Korvin-Kroukovsky and F R Cahbrow ldquoThe discontinu-ous fluid flow past an immersed wedgerdquo SMF Fund Paper 167Institute of Aeronautical Science 1948

[13] M White Frank Fluid Mechanics chapter 2 McGrowHill 4thedition 1998

[14] C L Shuford ldquoA theoretical and experimental study of planingsurfaces including effects of cross section and plan formrdquoNACA Technical Note 3939 1957

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks




Spray

(a)


(b)

Figure 1 Longitudinal (a) and transverse (b) pressure distribution on planing hulls [10]

Also in a recent attempt Ghadimi et al [7] used the methodof Smoothed-particle hydrodynamics (SPH) in order to studyflow around the planing plate and consequently find pressuredistribution over this surface

The initial modeling method of Savitsky [5] was latermodified and improved by Savitsky et al [8] by implementingwhisker spray and the resulting wetted area In anotherwork Morabito [9] studied the water spray and pressuredistribution over planing hulls As a result of his studiesa set of empirical equations was introduced for calculationof pressure distribution on planing hulls In the presentstudy empirical equations of Morabito [9] are used as thebasis of the mathematical model introduced for evaluation ofpressure distribution of a plning hull

2 Mathematical Formulation

One of the simplest techniques used for evaluation of planingvesselsrsquo dynamics is the planing plate model Sottorf [10]studied the pressure distribution on a planing plate in bothlongitudinal and transverse directions Therefore a plot ofthe longitudinal and transverse pressure distribution over aplaning plate was introduced by Sottorf [10] as shown inFigure 1 In the present paper the goal is to first calculatepressures on different longitudinal sectionswhich can later beused for 3-dimensional pressure distribution over the planinghull with consideration of transom stern effects and thencalculate the pressure alleviation which occur when gettingfarther from the center line

21 Stagnation Line andMaximum Pressure When a planingplane is moving along the water surface a stagnation lineappears on both sides of the center line It was shown that themaximum pressure on planing hulls occurs at the stagnationline [11] A spray area appears in front of the stagnation line onwhich the spray resistance force occurs while pressure areaappears in the aft direction in which drag and lift forces existThe stagnation line and spray and pressure areas for a planinghull are depicted in Figure 2

Based on fluid dynamics theories when a free flowcollides with an obstacle the maximum pressure occurs ata stagnation point In the case of a planing hull all thestagnation points lie on the stagnation line Experimentsby Smiley [3] proved that pressure distribution will fall onsections close to the stagnation line Therefore it can be

Pressure areaSpray area

Chine

Transom

Flow direction

Stagnation line

Keel

Spray edge

V

120572b

Figure 2 Bottom of a planing hull illustrating different sections [8]

concluded that at each section of the planing hull the max-imum pressure will occur at the stagnation line Moreoverthe pressure will also reduce along this line where at theintersection of the stagnation line and the center line themaximum possible pressure will appear Smiley [3] proposedthe following equation for the maximum pressure acting onthe planing hull

119875max(12) 120588119881

2=

1205872tan2120591

1205872tan2120591 + 4tan2120573 (1)

inwhich119875max is themaximumpressure120588 is thewater density119881 is the advance velocity of the vessel 120591 is the trim angleand 120573 is the deadrise angle Also Morabito [9] introducedthe following formula for determination of the maximumpressure

119875max119902

= sin2120572 (2)

119902 =1

21205881198812 (3)

where 120572 is the angle between the stagnation and center lineswhich was defined by Savitsky [5] using (4) as in

120572 = tanminus11205872

tan 120591tan120573

(4)

Figure 3 shows the maximum pressure at different trimand deadrise angles

As clearly seen in Figure 1 the pressure alleviates justafter reaching its maximum valueThe pressure will vanish at


0

02

04

06

08

1

12

0 5 10 15 20 25 30 35 40 45 50

120591 = 2

120591 = 4

120591 = 8

120591 = 12

120591 = 24

120591 = 30

120573

Pm

axq


0

0005

001

0015

002

0025

003Pq

0X




119875119871

119902= 0006

12059113

11988323 (5)




119883 =119909

119887(6)



119875119871

119902=11986211988313

(119883 + 119870) (7)

0

001

002

003

004

005

006

008

007

0

Pq

X

minus2minus4minus6


Peak pressure line

Center line

Transom

Chine


in which

119862 = 000612059113 (8)

119870 =11986215

2588(119875max119902)15 (9)






Consider

119875119879=

(120582119910minus 119883)14

(120582119910minus 119883)14

+ 005

(10)



120582119910= 120582 minus

(119884 minus 025)

tan120572 (11)



119875119884= [102 minus 005 (120573 + 5) 119884

14]05 minus 119884

051 minus 119884 (12)



119899 Figure 8



119881119899= 119881 sin120572

119881119904= 119881 cos120572

(13)







0

001

002

003

004

005

006

007

008

0


Pq

X



Center lineTr

anso

mStagnation line

Chine

V

V

s

120572

120572

Vn



119875119884Stag

119875119873

= [102 minus 02511988414]05 minus 119884

051 minus 119884 (14)

119875max119902119884

=

119875119884Stag

119875119873

sin2120572 (15)


119862 = 000611987511988412059113 (16)

119870 =11986215

2588((119875max119902)119884)15 (17)


Inputs


Total pressure

Yes

Yes

No

No


distribution

Transom effect

120591 120573 120574 C

Y = 0

determination of Py


Pstag

PN

Pmaxq

Y gt 05

X gt 120582y

X = 0

Y + h

X + h998400PT

PLq

PBq







119875119861

119902=120588119892119867 (119883 119884)

119902119875119879119875119884 (18)




119881) in (19) is given by (21)


119875119861

119902=2119875119879119875119884sin 120591

1198622

119881

(119883 + 119884(1

tan120572minus

1

tan120572119882

)) (19)

120572119882= tanminus1 ( tan 120591

tan120573) (20)

119862119881=

119881

radic119892119887 (21)


119875Total119902

=119875119871

119902+119875119861

119902 (22)




Case 120573 120591 120582 119884 1198772

(a) 0 4 5120025 0906487849025 09106314030475 0914969083

(b) 0 30 1070025 0931204774025 0950345090475 0931461486

(c) 20 6 2360025 087950731025 08840247390475 0951144181

(d) 20 9 0950025 0906989641025 09270938960475 0853750853

(e) 40 12 4880025 0871029025 09811090475 0938936

(f) 40 24 2460025 0813929407025 0975481860475 082164695





3 Validation




1198772= 1 minus

119878119878res119878119878tot

(23)


119878119878res = sum(119875

119902 Expminus119875

119902Mean)

2

119878119878res = sum(119875

119902 Expminus119875

119902 Predicted)

2

(24)



1198621198710= 00055

12058225

1198622

119881

12059111 (25)

1198621198710= int

05

minus05

int

120582119910

0

119875119861

119902119889119909 119889119910 cos 120591 (26)


119871012059111 for both



P

X

q

03504

02502015010050



0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X


Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X


X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05




035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

02

04

06

08

1

12

0 5 10 15 20 25 30 35 40 45 50

120591 = 2

120591 = 4

120591 = 8

120591 = 12

120591 = 24

120591 = 30

120573

Pm

axq


0

0005

001

0015

002

0025

003Pq

0X




119875119871

119902= 0006

12059113

11988323 (5)




119883 =119909

119887(6)



119875119871

119902=11986211988313

(119883 + 119870) (7)

0

001

002

003

004

005

006

008

007

0

Pq

X

minus2minus4minus6


Peak pressure line

Center line

Transom

Chine


in which

119862 = 000612059113 (8)

119870 =11986215

2588(119875max119902)15 (9)






Consider

119875119879=

(120582119910minus 119883)14

(120582119910minus 119883)14

+ 005

(10)



120582119910= 120582 minus

(119884 minus 025)

tan120572 (11)



119875119884= [102 minus 005 (120573 + 5) 119884

14]05 minus 119884

051 minus 119884 (12)



119899 Figure 8



119881119899= 119881 sin120572

119881119904= 119881 cos120572

(13)







0

001

002

003

004

005

006

007

008

0


Pq

X



Center lineTr

anso

mStagnation line

Chine

V

V

s

120572

120572

Vn



119875119884Stag

119875119873

= [102 minus 02511988414]05 minus 119884

051 minus 119884 (14)

119875max119902119884

=

119875119884Stag

119875119873

sin2120572 (15)


119862 = 000611987511988412059113 (16)

119870 =11986215

2588((119875max119902)119884)15 (17)


Inputs


Total pressure

Yes

Yes

No

No


distribution

Transom effect

120591 120573 120574 C

Y = 0

determination of Py


Pstag

PN

Pmaxq

Y gt 05

X gt 120582y

X = 0

Y + h

X + h998400PT

PLq

PBq







119875119861

119902=120588119892119867 (119883 119884)

119902119875119879119875119884 (18)




119881) in (19) is given by (21)


119875119861

119902=2119875119879119875119884sin 120591

1198622

119881

(119883 + 119884(1

tan120572minus

1

tan120572119882

)) (19)

120572119882= tanminus1 ( tan 120591

tan120573) (20)

119862119881=

119881

radic119892119887 (21)


119875Total119902

=119875119871

119902+119875119861

119902 (22)




Case 120573 120591 120582 119884 1198772

(a) 0 4 5120025 0906487849025 09106314030475 0914969083

(b) 0 30 1070025 0931204774025 0950345090475 0931461486

(c) 20 6 2360025 087950731025 08840247390475 0951144181

(d) 20 9 0950025 0906989641025 09270938960475 0853750853

(e) 40 12 4880025 0871029025 09811090475 0938936

(f) 40 24 2460025 0813929407025 0975481860475 082164695





3 Validation




1198772= 1 minus

119878119878res119878119878tot

(23)


119878119878res = sum(119875

119902 Expminus119875

119902Mean)

2

119878119878res = sum(119875

119902 Expminus119875

119902 Predicted)

2

(24)



1198621198710= 00055

12058225

1198622

119881

12059111 (25)

1198621198710= int

05

minus05

int

120582119910

0

119875119861

119902119889119909 119889119910 cos 120591 (26)


119871012059111 for both



P

X

q

03504

02502015010050



0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X


Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X


X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05




035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Consider

119875119879=

(120582119910minus 119883)14

(120582119910minus 119883)14

+ 005

(10)



120582119910= 120582 minus

(119884 minus 025)

tan120572 (11)



119875119884= [102 minus 005 (120573 + 5) 119884

14]05 minus 119884

051 minus 119884 (12)



119899 Figure 8



119881119899= 119881 sin120572

119881119904= 119881 cos120572

(13)







0

001

002

003

004

005

006

007

008

0


Pq

X



Center lineTr

anso

mStagnation line

Chine

V

V

s

120572

120572

Vn



119875119884Stag

119875119873

= [102 minus 02511988414]05 minus 119884

051 minus 119884 (14)

119875max119902119884

=

119875119884Stag

119875119873

sin2120572 (15)


119862 = 000611987511988412059113 (16)

119870 =11986215

2588((119875max119902)119884)15 (17)


Inputs


Total pressure

Yes

Yes

No

No


distribution

Transom effect

120591 120573 120574 C

Y = 0

determination of Py


Pstag

PN

Pmaxq

Y gt 05

X gt 120582y

X = 0

Y + h

X + h998400PT

PLq

PBq







119875119861

119902=120588119892119867 (119883 119884)

119902119875119879119875119884 (18)




119881) in (19) is given by (21)


119875119861

119902=2119875119879119875119884sin 120591

1198622

119881

(119883 + 119884(1

tan120572minus

1

tan120572119882

)) (19)

120572119882= tanminus1 ( tan 120591

tan120573) (20)

119862119881=

119881

radic119892119887 (21)


119875Total119902

=119875119871

119902+119875119861

119902 (22)




Case 120573 120591 120582 119884 1198772

(a) 0 4 5120025 0906487849025 09106314030475 0914969083

(b) 0 30 1070025 0931204774025 0950345090475 0931461486

(c) 20 6 2360025 087950731025 08840247390475 0951144181

(d) 20 9 0950025 0906989641025 09270938960475 0853750853

(e) 40 12 4880025 0871029025 09811090475 0938936

(f) 40 24 2460025 0813929407025 0975481860475 082164695





3 Validation




1198772= 1 minus

119878119878res119878119878tot

(23)


119878119878res = sum(119875

119902 Expminus119875

119902Mean)

2

119878119878res = sum(119875

119902 Expminus119875

119902 Predicted)

2

(24)



1198621198710= 00055

12058225

1198622

119881

12059111 (25)

1198621198710= int

05

minus05

int

120582119910

0

119875119861

119902119889119909 119889119910 cos 120591 (26)


119871012059111 for both



P

X

q

03504

02502015010050



0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X


Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X


X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05




035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Inputs


Total pressure

Yes

Yes

No

No


distribution

Transom effect

120591 120573 120574 C

Y = 0

determination of Py


Pstag

PN

Pmaxq

Y gt 05

X gt 120582y

X = 0

Y + h

X + h998400PT

PLq

PBq







119875119861

119902=120588119892119867 (119883 119884)

119902119875119879119875119884 (18)




119881) in (19) is given by (21)


119875119861

119902=2119875119879119875119884sin 120591

1198622

119881

(119883 + 119884(1

tan120572minus

1

tan120572119882

)) (19)

120572119882= tanminus1 ( tan 120591

tan120573) (20)

119862119881=

119881

radic119892119887 (21)


119875Total119902

=119875119871

119902+119875119861

119902 (22)




Case 120573 120591 120582 119884 1198772

(a) 0 4 5120025 0906487849025 09106314030475 0914969083

(b) 0 30 1070025 0931204774025 0950345090475 0931461486

(c) 20 6 2360025 087950731025 08840247390475 0951144181

(d) 20 9 0950025 0906989641025 09270938960475 0853750853

(e) 40 12 4880025 0871029025 09811090475 0938936

(f) 40 24 2460025 0813929407025 0975481860475 082164695





3 Validation




1198772= 1 minus

119878119878res119878119878tot

(23)


119878119878res = sum(119875

119902 Expminus119875

119902Mean)

2

119878119878res = sum(119875

119902 Expminus119875

119902 Predicted)

2

(24)



1198621198710= 00055

12058225

1198622

119881

12059111 (25)

1198621198710= int

05

minus05

int

120582119910

0

119875119861

119902119889119909 119889119910 cos 120591 (26)


119871012059111 for both



P

X

q

03504

02502015010050



0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X


Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X


X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05




035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Case 120573 120591 120582 119884 1198772

(a) 0 4 5120025 0906487849025 09106314030475 0914969083

(b) 0 30 1070025 0931204774025 0950345090475 0931461486

(c) 20 6 2360025 087950731025 08840247390475 0951144181

(d) 20 9 0950025 0906989641025 09270938960475 0853750853

(e) 40 12 4880025 0871029025 09811090475 0938936

(f) 40 24 2460025 0813929407025 0975481860475 082164695





3 Validation




1198772= 1 minus

119878119878res119878119878tot

(23)


119878119878res = sum(119875

119902 Expminus119875

119902Mean)

2

119878119878res = sum(119875

119902 Expminus119875

119902 Predicted)

2

(24)



1198621198710= 00055

12058225

1198622

119881

12059111 (25)

1198621198710= int

05

minus05

int

120582119910

0

119875119861

119902119889119909 119889119910 cos 120591 (26)


119871012059111 for both



P

X

q

03504

02502015010050



0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X


Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X


X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05




035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










P

X

q

03504

02502015010050



0minus2minus4minus6

03

Pq

Pq

03504

02502015010050

03025

02

015

01

005

0

03

Y = 0025 120582 = 512 Y = 025 120591 = 4 120573 = 0 Y = 0475

(a)

X

Pq

04

06

08

1

02

0

Pq Pq

04

06

08

1

02

0

04

0607

02

001

03

05

0minus2 minus1

X

0minus2 minus1

X


Y = 0025 120582 = 107 Y = 025 120591 = 30 120573 = 0 Y = 0475

(b)

Pq

02

015

01

005

0

Pq

Pq

02

015

01

005

0

X


X


X

00

002

004

006

008

01

012

minus2 minus1

Y = 0025 120582 = 236 Y = 025 120591 = 6 120573 = 20 Y = 0475

(c)

Pq

X


X

0minus1 minus05

X

0minus1 minus05




035

02502015010050

03

Pq

Pq

025

02

015

01

005

0

025

02

015

01

005

0

03

Y = 0025 120582 = 095 Y = 025 120591 = 9 120573 = 20 Y = 0475

(d)

Figure 10 Continued


X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










X


X


X


Pq

000200400600801

014012

Pq

0

002

004

006

008

01

014

012

0

002

004

006

008

01

012016

Y = 0025 120582 = 488 Y = 025 120591 = 12 120573 = 40 Y = 0475

(e)

025

02

02

03

03

04

05

015

0101005

0 0

X


X



Pq Pq 02

03

04

05

01

0

Pq




Y = 0025 120582 = 246 Y = 025 120591 = 24 120573 = 40 Y = 0475

(f)



Case 119862119881

1198772

(a) 1 092417071(b) 2 0901327451(c) 4 0924163963(d) 6 0924163968








025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










025

02

015

01

005

00 2 4 6

120582

CL0120591

11

(a) 119862119881 = 1

0 2 4 6

400E minus 02

300E minus 02

200E minus 02

100E minus 02

500E minus 02

600E minus 02

000E + 00

120582

CL0120591

11

(b) 119862119881 = 2

IntegratedSavitsky

0 21 43 5

140E minus 02

120E minus 02

100E minus 02

800E minus 03

600E minus 03

400E minus 03

200E minus 03

000E + 00

120582

CL0120591

11

(c) 119862119881 = 4

IntegratedSavitsky

0 21 43 5

400E minus 03

300E minus 03

200E minus 03

100E minus 03

500E minus 03

600E minus 03

000E + 00

120582

CL0120591

11

(d) 119862119881 = 6





5 Conclusion




minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus05

0

050

00501

01502

xb

yb

Pq

0

002

004

006

008

01

012

(a) 120591 = 2


minus05

0

050

01020304

xb

yb

Pq

005

01

015

02

025

03

(b) 120591 = 4


minus05

0

050

0102030405

xb

yb

Pq

0

000501015020250303504045

(c) 120591 = 6



minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

025

Pq

0

005

01

015

02

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

0

005

01

015

02

025

03

035

xb

yb

(c) 120591 = 6




minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus05

0

050

001002003004

Pq

00005001001500200250030035

xb

yb

(a) 120591 = 2


minus05

0

050

00501

01502

Pq

0

002

004

006

008

01

012

014

xb

yb

(b) 120591 = 4


minus05

0

050

01020304

Pq

005

01

015

02

025

xb

yb

(c) 120591 = 6



minus05

0

050

0002000400060008

001

Pq

12345678times103

xb

yb

(a) 120591 = 2


minus05

0

050

001002003004

Pq

0

0005

001

0015

002

0025

003

xb

yb

(b) 120591 = 4

minus4minus3

minus2minus1

0

minus05

0

050

005

01

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120591 = 6




minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus05

0

050

002004006008

01

xb

yb

Pq

0001002003004005006007008

(a) 120582 = 15


minus05

0

050

002004006008

Pq

0001002003004005006007

xb

yb

(b) 120582 = 25


minus05

0

050

002004006008

Pq

0

001

002

003

004

005

006

007

xb

yb

(c) 120582 = 35



minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(a) 119862119881 = 2


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(b) 119862119881 = 4


minus05

0

05minus002

0002004006008

01

0001002003004005006007008

Pq

xb

yb

(c) 119862119881 = 6







References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












References

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article three-dimensional mathematical ...downloads.hindawi.com/archive/2013/868252.pdf ·...

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