linear motions: surge, sway, and heave rotations: roll, pitch, and yaw

Transcript

1 heave yaw when the ship is treated as a rigid body, it has six degrees of freedom: three linear motions and three rotations as indicated in the figure at the left: body-fixed axes pitch, v roll, u sway linear motions: surge, sway, and heave rotations: roll, pitch, and yaw when the equations of motion are linearized, heave, pitch and surge are coupled in one set of three equations, and sway, roll and yaw are coupled in the other surge there is no linear coupling between pitch and roll in spite of this decoupling of the set of six linear equations of motion into two sets of three linearly coupled equations of motion, Froude in 863 commented that ships with a frequency frequency in roll that is half their frequency in pitch have very undersirable seakeeping characteristics -- a very astute observation for the time because in the linearized equations of motion roll and pitch are not coupled aparently the coupling i s nonlinear approximately a century later, Pauling and Rosenberg, in an effort to understand the coupling, conducted an experiment in which a model was attached to a sting that allowed only roll and pitch and placed it in a towing tank with a wave maker

2 heave yaw the essence of the two-degree-of freedom motion of the model in the wave tank is captured by the following equations: pitch, v roll, u sway surge u + μu + ω u = α uv + kcos Ω t + γ v + μ v + ω v = α u + f cos Ωt where u represents the roll motion and v represents the pitch motion introduce a small bookkeeping parameter ε and redefine: ) the damping coefficients: μ εμ and μ εμ ) the amplitudes of the excitation: k εk and f = εk 3) the coefficients of the nonlinear terms: α εα and α εα and u + εμ u + ω u = εα uv + ε kcos Ω t + γ v + εμ v + ω v = εα u + ε f cos Ωt

3 and u + εμ u + ω u = εα uv + ε kcos Ω t + γ v + εμ v + ω v = εα u + ε f cos Ωt MMS : assume approximations in the form ( ; ε) (, ) + ε (, ) and ( ; ε) (, ) + ε (, ) u t u T T u T T v t v T T v T T Du + ω u = D 0 iωt0 u + ω u = D Du + μd u αu v + ke Dv + ω v = iωt0 Dv + ω v = DDv + μdv αu + fe we obtain e ω and i T0 iωt0 u = A T v = A T e 0 0

4 iωt0 Du + ω u = DDu + μdu αuv + ke ω ω ω ω ω = + e + e + e e + e + e + i T0 i T0 i T0 i T0 i T0 iωt0 iω DA μ A cc α A A A A k cc = + ( ω ω ) ( ω + ω ) iωt0 i T0 i iω DA μ A e α AAe α AAe Ω + ke T i T 0 0 iωt0 Dv + v = DDv + Dv u + ke ω μ α ω ω ω = ω + μ e + α e + e + e + i T0 i T0 i T0 iωt0 i DA A cc A A f cc = ω + μ e α e α + e + iωt0 iωt0 iωt0 i DA A A AA k cc we digress to obtain what, if any, restrictions need to be placed on the equations for them to be consistent with physical requirements to answer the question 'what restrictions are necessary?', we consider two cases:

5 free-vibration cases to be considered: ) no internal resonance, no excitation ) internal resonance, no excitation ) in this case there are only secular terms and we so we require iβ μt DA + μa = 0 A = ae aβ = 0 & a + μa = 0 a = ce 0 as t iβ similarly, DA + μa = 0 A = ae A 0 as t NO problem here : free vibrations in a damped system decay, α and α are arbitrary

6 ) internal resonance: in this case there are both secular and small-divisor terms: ( ) iω( DA + μa) + αaa = ω = ω+ εσ Ai = ai i = i ω ω T0 iβi e 0 e, αaa i( σt+ β β) iω( a + iaβ + μa) + e = 0 4 αaa αaa a + μa+ sin γ = 0 & aβ + cosγ = 0 where γ = σt+ β β 4ω 4ω ( ω ω ) i T0 e 0 iω DA + μ A + α A = αa i( β β σt) iω( a + iaβ + μa) + e = 0 4 α a α a a + μ a sin γ = 0 & a β + cosγ = 0 4ω 4ω

7 ) cont'd: α aa α a a + μ a + sinγ = 0 a + μ a sinγ = 0 4ω 4ω α aa α a a β + cosγ = 0 β + cosγ = 0 a 4ω 4ω where α a αa γ = σ T + β β γ = σ + β β = σ + cosγ 4ωa ω in the steady state, a = a = γ = 0 4ωμa 4ωμa = sin = sin + = 0 aa γ aa γ add a a α α if α and α have the same sign, then both terms are either zero or positive; so, because the sum is zero, both terms must be zero, which is consistent with physical requirements: the free vibrations of a system with damping decay if α α and have different signs, then the first term is positive and the second is negative; though their absolute values must be the same, they could be non-zero, which would be inconsistent with physical requirements interesting result : if there is no internal resonance, α and α are arbitrary, but if there is internal resonance, they must have the same sign

8 αaa αa αa αa ) a = μa sin γ ) a = μa + sin γ 3) γ = σ + cosγ 4ω 4ω 4ωa ω in the steady state, a = a = γ = 0 a + a = 0 a comparison of the numerical solutions of equations ), ), and 3) for different values of the parameters: a -> a -> σ = 0

9 αaa αa αa αa ) a = μa sin γ ) a = μa + sin γ 3) γ = σ + cosγ 4ω 4ω 4ωa ω V = a + a 0 in the steady state? a i when α and α have the same sign, the motion decays, which is consistent with the physics for the free vibrations of a damped system a a

10 αaa αa αa αa ) a = μa sinγ ) a = μa + sinγ 3) γ = σ + cosγ 4ω 4ω 4ωa ω V = a + a 0 in the steady state? a i when α and α have different signs, the motion does not decay, which is inconsistent with the physics for the free vibrations of a damped system a we conclude that in our model equations for the ship with internal resonance α and α must have the same sign a

11 forced vibrations ( k = 0), external resonance Ω ω, no internal resonance recall ( μ ) iωt0 Du + ω u = DDu + Du αuv + ke ( DA μ ) ( ω ω ) ( ω + ω ) = iω + A α A A α A A + k iωt0 i T0 i T0 e e e e iωt 0 ω μ α e iωt0 Dv + v = DDv + Dv u + k = ω + μ e α e α + e + iωt0 iωt0 iωt0 i DA A A AA f cc μt+ iφ μt 0 e e cos eliminate secular terms from u : DA + μ A = A = c u c ωt + φ introduce a detuning factor: Ω = ω + εσ ( + i ) ( μ ) eliminate trouble some terms from u : iω DA + A = f e iσt μ e T+ iφ f μt f A = c e + u c e cos ω t + φ + cos Ω t + τ ( + σ ) iω μ σ ω μ iσ T the same as the solution of the linear problem

12 the equations of motion can be simplified : () φ() t let u = cθ t and v = c k u + μu + ωu = αuv + kcos Ω t + τ + μ ω c cos t c θ θ + θ = α θφ+ ( Ω + τ ) let cα = c = α c α v + μ v + v = u + f cos Ωt + + = ω α φ μφ ωφ θ c f cos c + ( Ωt ) c α let which is always real because = cα α = c = α c αα and α must have the same sign henceforth, we shall work with θ + μ θ + ω θ = θφ+ f cos Ω t + τ where f = k α α and τ is a constant φ + μ φ + ω φ = θ + f cos Ω t where f = fα where θ represents the roll motion and φ represents the pitch motion