m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Ship design: buoyancy, stability, resistance and power

Connect hull geometry and displacement to initial stability, hydrodynamic resistance, propulsion and structural response.

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1. Hydrostatics and displacement

Definitions & inputs. ρ water density,g gravity,∇ displaced volume,m vessel mass,L length,B beam,T draft,CB block coefficient.

  1. Integrate hydrostatic pressure to obtain Archimedes’ buoyancy.

    FB=ρg∇,FB=mg⇒m=ρ∇F_B=\rho g\nabla,\quad F_B=mg\Rightarrow m=\rho\nabla

  2. The block coefficient describes volume relative to the enclosing box.

    ∇=CBLBT\nabla=C_B LBT

  3. A small added mass sinks a vessel according to waterplane area.

    ΔT≃Δm/(ρAwp)\Delta T\simeq\Delta m/(\rho A_{wp})

Interpretation. Waterplane area changes with draft, so large loading changes require the actual hydrostatic curves.

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2. Initial transverse stability

Definitions & inputs. KB center of buoyancy height,KG center of gravity height,Iwp transverse waterplane second moment,GM metacentric height,φ heel.

  1. Small heel shifts the center of buoyancy and defines the metacenter.

    BM=Iwp/∇,GM=KB+BM−KGBM=I_{wp}/\nabla,\quad GM=KB+BM-KG

  2. The horizontal buoyancy-weight lever gives righting moment.

    GZ≃GMsin⁡ϕ,MR=mgGZGZ\simeq GM\sin\phi,\quad M_R=mgGZ

  3. Liquid free surfaces reduce initial stability.

    ΔGMfree=ρtankIfree/(ρwater∇)\Delta GM_{free}=\rho_{tank}I_{free}/(\rho_{water}\nabla)

Interpretation. Positive GM alone does not establish large-angle stability, downflooding margin or damage survivability.

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3. Resistance and propulsion

Definitions & inputs. R total resistance,V speed,ρ density,S wetted area,CT coefficient,ηD propulsive efficiency.

  1. A nondimensional resistance coefficient scales the force.

    R=12ρV2SCTR=\tfrac12\rho V^2SC_T

  2. Tow power becomes delivered shaft power after propulsive losses.

    PE=RV,PD=PE/ηDP_E=RV,\quad P_D=P_E/\eta_D

  3. Viscous and gravity-wave similarity cannot generally both match in a small-scale water model.

    Re=VL/ν,Fr=V/gLRe=VL/\nu,\quad Fr=V/\sqrt{gL}

Interpretation. Propeller open-water maps, wake, thrust deduction and cavitation must be evaluated rather than replacing every loss by one constant efficiency.

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4. Structure and seakeeping

Definitions & inputs. M hull bending moment,I section second moment,y distance from neutral axis,k wave number,ω wave frequency,h water depth.

  1. Global hull-girder bending produces longitudinal stress.

    σ=My/I\sigma=My/I

  2. Linear gravity-wave dispersion connects wavelength and frequency.

    ω2=gktanh⁡(kh)\omega^2=gk\tanh(kh)

  3. Encounter frequency depends on vessel motion relative to wave propagation directionβ.

    ωe=ω−kVcos⁡β\omega_e=\omega-kV\cos\beta

Interpretation. Slamming, whipping, corrosion, fatigue and local buckling need additional load cases and models.

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Graphical worked example

Constant CT=0.005, wetted area 150 m² and water density 1025 kg/m³. Actual CT is not generally constant. X axis: Ship speed (m/s). Y axis: Illustrative resistance (N).
Constant CT=0.005, wetted area 150 m² and water density 1025 kg/m³. Actual CT is not generally constant. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Displaced volume

Definitions & inputs. L20m,B5m,T2m,CB.6.

  1. Choose the governing model and isolate the requested quantity.

    ∇=CBLBT\nabla=C_B LBT

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .6(20)(5)(2).6(20)(5)(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=120 m3\mathrm{Result}=120\ \mathrm{m^3}

Interpretation. Simple block-coefficient model.

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Example 02. Displacement mass

Definitions & inputs. Seawater1025kg/m³,volume120m³.

  1. Choose the governing model and isolate the requested quantity.

    m=ρ∇m=\rho\nabla

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1025(120)1025(120)

  3. Evaluate the expression; the result uses the units shown.

    Result=123000 kg\mathrm{Result}=123000\ \mathrm{kg}

Interpretation. Static flotation.

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Example 03. Buoyancy force

Definitions & inputs. Same volume,g9.81.

  1. Choose the governing model and isolate the requested quantity.

    FB=ρg∇F_B=\rho g\nabla

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1025(9.81)(120)1025(9.81)(120)

  3. Evaluate the expression; the result uses the units shown.

    Result=1206630 N\mathrm{Result}=1206630\ \mathrm N

Interpretation. Equal to weight at equilibrium.

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Example 04. Freshwater volume

Definitions & inputs. Mass123000kg,ρ1000.

  1. Choose the governing model and isolate the requested quantity.

    ∇=m/ρ\nabla=m/\rho

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    123000/1000123000/1000

  3. Evaluate the expression; the result uses the units shown.

    Result=123 m3\mathrm{Result}=123\ \mathrm{m^3}

Interpretation. More volume must be displaced than in seawater.

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Example 05. Draft increment

Definitions & inputs. Added mass1000kg,Awp80m²,ρ1025.

  1. Choose the governing model and isolate the requested quantity.

    ΔT=Δm/(ρAwp)\Delta T=\Delta m/(\rho A_{wp})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1000/(1025(80))1000/(1025(80))

  3. Evaluate the expression; the result uses the units shown.

    Result=0.01219512 m\mathrm{Result}=0.01219512\ \mathrm m

Interpretation. Small increment only.

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Example 06. Rectangular waterplane inertia

Definitions & inputs. L20m,B5m.

  1. Choose the governing model and isolate the requested quantity.

    Iwp=LB3/12I_{wp}=LB^3/12

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20(53)/1220(5^3)/12

  3. Evaluate the expression; the result uses the units shown.

    Result=208.3333 m4\mathrm{Result}=208.3333\ \mathrm{m^4}

Interpretation. Roll about longitudinal axis.

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Example 07. Metacentric radius

Definitions & inputs. Iwp208.333m⁴,volume120m³.

  1. Choose the governing model and isolate the requested quantity.

    BM=Iwp/∇BM=I_{wp}/\nabla

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (20(53)/12)/120(20(5^3)/12)/120

  3. Evaluate the expression; the result uses the units shown.

    Result=1.736111 m\mathrm{Result}=1.736111\ \mathrm m

Interpretation. Idealized waterplane.

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Example 08. Metacentric height

Definitions & inputs. KB1m,BM1.7m,KG2m.

  1. Choose the governing model and isolate the requested quantity.

    GM=KB+BM−KGGM=KB+BM-KG

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1+1.7−21+1.7-2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7 m\mathrm{Result}=0.7\ \mathrm m

Interpretation. Positive initial stability.

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Example 09. Small-angle righting arm

Definitions & inputs. GM.7m,heel5°.

  1. Choose the governing model and isolate the requested quantity.

    GZ=GMsin⁡ϕGZ=GM\sin\phi

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .7sin⁡5∘.7\sin5^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=0.06100902 m\mathrm{Result}=0.06100902\ \mathrm m

Interpretation. Small-angle approximation.

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Example 10. Righting moment

Definitions & inputs. m123000kg,GM.7m,heel5°.

  1. Choose the governing model and isolate the requested quantity.

    MR=mgGMsin⁡ϕM_R=mgGM\sin\phi

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    123000(9.81)(.7)sin⁡5∘123000(9.81)(.7)\sin5^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=73615.31 N m\mathrm{Result}=73615.31\ \mathrm{N\,m}

Interpretation. Intact illustrative loading.

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Example 11. Free-surface correction

Definitions & inputs. ρtank1000,I10m⁴,displaced mass123000kg.

  1. Choose the governing model and isolate the requested quantity.

    ΔGM=ρtI/m\Delta GM=\rho_tI/m

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1000(10)/1230001000(10)/123000

  3. Evaluate the expression; the result uses the units shown.

    Result=0.08130081 m\mathrm{Result}=0.08130081\ \mathrm m

Interpretation. Subtract from uncorrected GM.

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Example 12. Froude number

Definitions & inputs. V5m/s,L20m.

  1. Choose the governing model and isolate the requested quantity.

    Fr=V/gLFr=V/\sqrt{gL}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5/9.81(20)5/\sqrt{9.81(20)}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3569608 \mathrm{Result}=0.3569608\ {}

Interpretation. Gravity-wave similarity.

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Example 13. Reynolds number

Definitions & inputs. V5m/s,L20m,ν10⁻⁶m²/s.

  1. Choose the governing model and isolate the requested quantity.

    Re=VL/νRe=VL/\nu

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5(20)/10−65(20)/10^{-6}

  3. Evaluate the expression; the result uses the units shown.

    Result=1×108 \mathrm{Result}=1\times10^{8}\ {}

Interpretation. Viscous similarity.

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Example 14. Resistance

Definitions & inputs. ρ1025,V5,S150m²,CT.005.

  1. Choose the governing model and isolate the requested quantity.

    R=ρV2SCT/2R=\rho V^2SC_T/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(1025)(25)(150)(.005).5(1025)(25)(150)(.005)

  3. Evaluate the expression; the result uses the units shown.

    Result=9609.375 N\mathrm{Result}=9609.375\ \mathrm N

Interpretation. Assumed fitted CT.

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Example 15. Effective power

Definitions & inputs. R10000N,V5m/s.

  1. Choose the governing model and isolate the requested quantity.

    PE=RVP_E=RV

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10000(5)10000(5)

  3. Evaluate the expression; the result uses the units shown.

    Result=50000 W\mathrm{Result}=50000\ \mathrm W

Interpretation. Tow power.

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Example 16. Delivered shaft power

Definitions & inputs. PE50kW,ηD.6.

  1. Choose the governing model and isolate the requested quantity.

    PD=PE/ηDP_D=P_E/\eta_D

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    50/.650/.6

  3. Evaluate the expression; the result uses the units shown.

    Result=83.33333 kW\mathrm{Result}=83.33333\ \mathrm{kW}

Interpretation. Simplified efficiency budget.

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Example 17. Propeller advance ratio

Definitions & inputs. V advance5m/s,n5rev/s,D2m.

  1. Choose the governing model and isolate the requested quantity.

    J=VA/(nD)J=V_A/(nD)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5/(5(2))5/(5(2))

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. Use advance velocity not blindly ship speed.

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Example 18. Hull beam stress

Definitions & inputs. M10⁶Nm,y1m,I.1m⁴.

  1. Choose the governing model and isolate the requested quantity.

    σ=My/I\sigma=My/I

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    106(1)/.110^6(1)/.1

  3. Evaluate the expression; the result uses the units shown.

    Result=1×107 Pa\mathrm{Result}=1\times10^{7}\ \mathrm{Pa}

Interpretation. Linear global bending.

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Example 19. Deep-water wavelength

Definitions & inputs. Period8s.

  1. Choose the governing model and isolate the requested quantity.

    λ=gT2/(2π)\lambda=gT^2/(2\pi)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    9.81(82)/(2π)9.81(8^2)/(2\pi)

  3. Evaluate the expression; the result uses the units shown.

    Result=99.92384 m\mathrm{Result}=99.92384\ \mathrm m

Interpretation. Deep-water linear wave approximation.

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Example 20. Added-load margin

Definitions & inputs. Mass limit150t,current123t.

  1. Choose the governing model and isolate the requested quantity.

    Δm=150−123\Delta m=150-123

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    2727

  3. Evaluate the expression; the result uses the units shown.

    Result=27 tonnes\mathrm{Result}=27\ \mathrm{tonnes}

Interpretation. Mass budget only; not stability approval.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.